Instructor’s Manual
Microeconomics
Third edition
Hugh Gravelle
Ray Rees
For further instructor material
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ISBN-13: 978-0-273-65892-4 / ISBN-10: 0-273-65892-1
Pearson Education Limited 2007
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----------------------------------Third edition 2007
© Pearson Education Limited 2007
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ISBN-13: 978-0-273-65892-4
ISBN-10: 0-273-65892-1
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Gravelle and Rees: Microeconomics Instructor’s Manual, 3rd edn
Contents
Chapter 2
The Theory of the Consumer
1
Chapter 3
Consumer Theory: Duality
21
Chapter 4
Further Models of Consumer Behaviour
39
Chapter 5
Production
60
Chapter 6
Cost
69
Chapter 7
Supply and Firm Objectives
83
Chapter 8
The Theory of a Competitive Market
100
Chapter 9
Monopoly
112
Chapter 10
Input Markets
130
Chapter 11
Capital Markets
145
Chapter 12
General Equilibrium
154
Chapter 13
Welfare Economics
160
Chapter 14
Market Failure and Government Failure
167
Chapter 15
Game Theory
178
Chapter 16
Oligopoly
196
Chapter 17
Choice under Uncertainty
207
Chapter 18
Production under Uncertainty
221
Chapter 19
Insurance, Risk Spreading and Pooling
226
Chapter 20
Agency, Contract Theory and the Firm
242
Chapter 21
General Equilibrium under Uncertainty and
Incomplete Markets
255
Appendices
277
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Gravelle and Rees: Microeconomics Instructor’s Manual, 3rd edn
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Chapter 2
The Theory of the Consumer
Exercise 2A
1. Suppose two indifference curves intersect at a point x0, as shown in Fig. 2A.1. Then
there exist bundles x′, x″ such that x′ ~ x0 and x0 ~ x″ but x′ > x″ which violates
transitivity. Note that we do not require the non-satiation axiom for this. The same
result would follow if x″ > x′.
2. The answer is given by diagrams in Fig. 2A.2 (examples are used to label the axes).
3. The assumption of non-satiation rules out the possibility of bliss points (refer to the
answer to question 4 of Exercise F). It ensures that consumers will always want to
consume more goods than are available given the resources the economy possesses. As
a result, we have relative scarcity and the need for the allocation of scarce resources
among competing uses. Thus the usefulness of microeconomics rests upon some kind of
non-satiation assumption.
4. See Fig. 2A.3.
4. (a) The utility function is quasi-concave but not strictly quasi-concave. Blue and
red matches are perfect substitutes (we assume the consumer cares only about
incendiary properties and not colour). Therefore utility depends only on the total
number of matches: reducing the number of blue matches by one is fully compensated
by increasing the number of red matches by one. MRS21 is therefore constant and equal
to one.
4. (b) Assuming the consumer has the usual configuration of legs, right and left shoes of
the same style, colour, quality, size, etc. are perfect complements. Utility depends on the
number of pairs of shoes the consumer possesses. Thus the consumer would be
indifferent between two pairs of shoes and two pairs of shoes plus one left shoe. Thus
the indifference curves take the extreme kinked shape shown here. We would say that
right and left shoes have to be consumed in fixed proportions, here one-to-one. The
utility function is concave but not strictly concave. MRS21 = −dx2/dx1 is zero along a
horizontal segment of an indifference curve and undefined along a vertical segment
(and conversely for MRS12).
5. The correct answer here is (c), as may perhaps have been guessed from the fact
that it is the most complicated answer. (a) is wrong because it refers to total rather than
marginal valuations. (b) is wrong in a more subtle way. Strict convexity does
not refer to absolute amounts of the marginal variations in the two goods required
to stay on an indifference curve, but only on how the ratio of these amounts varies
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Fig. 2A.1
Fig. 2A.2
as we move along an indifference curve. It is quite consistent with strict convexity that,
at a bundle consisting of a lot of water and a few diamonds, the consumer would give up
very little water for an extra diamond. In water-diamond space the indifference curves
could be everywhere very steep. The point is that at a bundle with more water and
fewer diamonds, he would give up more water for an extra diamond.
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Fig. 2A.3
Exercise 2B
1. Suppose that p1 = p1(x1), with p1′ > 0, but that p2 is constant. Then, we can write the
consumer’s budget constraint as x2 = (M − p1(x1)x1)/p2 ≡ B(x1) and we have
B′(x1) = −
1
(p + x1 p1′ ) < 0.
p2 1
Thus again we have a negatively-sloped budget constraint. We could interpret
p1 + x1 p1′ as the marginal price or marginal cost of x1 to the consumer, since it is
the derivative d[p1(x1)x1]/dx1, where p1(x1)x1 is the total cost of buying x1. Thus the slope
of the budget line, or curve, at a point is the ratio of the marginal cost of x1
to the price (= marginal cost) of x2.
If we want the budget set still to be a convex set, we require B(x1) to be a concave
function, i.e. we require
B″(x1) = −
1
( 2 p1′ + x1 p1′′) < 0
p2
or
2 p1′ + x1 p1′′ > 0.
This cannot be guaranteed without further restrictions on the function p(x1). If this
function is convex, so that p1′′ ≥ 0, then we have immediately that B″(x1) < 0 and the
budget set is a convex set. If, on the other hand, p1(x1) is non-convex, then we require
2 p1′( x1 ) > − x1 p1′′( x1 ) at all x1 in the interval [0, I1], where I1 satisfies
p1(I1)I1 = M.
If this condition does not hold then we may have non-convexity of the budget set which,
as we saw in Appendix D, can cause local optima to be non-unique or, worse, nonglobal. Accordingly, a point satisfying the Lagrange multiplier conditions may not then
be a true solution to the constrained problem.
In the case where the condition is satisfied, Fig. 2B.1 illustrates the budget set.
Choose the unit of measurement for x2 such that its price is p2 = 1. Then, in the figure,
the slope at B(x1) at a point such as x0 measures the marginal cost or price
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Fig. 2B.1
Fig. 2B.2
B′ ( x10 ), while the slope of the line from x′ to M gives the average cost or price p( x10 ).
To see the latter, note that if p2 = 1 expenditure on good 2 is x20 and so expenditure on
good 1 is M − x20 = p1 ( x10 ) x10 , while the slope of the line is (M − x20 ) / x10 . Thus marginal
exceeds average cost or price.
2. Denote the connection charge by C, let p0 be the price of the first n units of electricity
and let p1 be the price of the remainder, with p0 > p1. Let the price of the consumption
good be normalized at 1. Then Fig. 2B.2 shows the consumer’s budget constraint on the
assumption that her income M is large enough that she is able to buy more than n units
of electricity if she wants to.
If she buys no electricity then she spends M on the consumption good. If she wants
to buy any positive quantity of electricity she pays C. Denoting the consumption good by
c and electricity demand by e the function defining the budget constraint is
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Fig. 2B.3
c =
M
for e = 0,
c =
M − C − p0e
c =
M − C − p0n − p1(e − n) = M − C − (p0 − p1)n − p1e
for 0 < e ≤ n,
for n < e.
Since p1 < p0 the second segment of the budget constraint is flatter than the first. The
intercept on the e-axis is
B=
M − C − ( p0 − p1 ) n
p1
The marginal price of electricity is p0 for 0 < e ≤ n and p1 for e > n. The average price of
electricity is
F0
=
C
+ p0
e
F1
=
C + ( p0 − p1 ) n
+ p1
e
for 0 < e ≤ n,
for n < e.
In each case therefore the average price falls with e but is always above the marginal
price.
The effects of changes in the connection charge and in prices are shown in
Fig. 2B.3.
If, say, C falls to C′ then the budget constraint shifts to B′. If with the fixed charge at
C, p0 increases to p0′ then the budget constraint becomes B″. Finally, if with C and p0
fixed, p1 falls to p1′ then the constraint becomes B′″.
Supplementary questions
(i) On the budget constraint of Fig. 2B.2, superimpose indifference curves appropriate
to each of the following types of consumer:
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Fig. 2B.4
(a) one who buys no electricity;
(b) one who buys between zero and n units of electricity;
(c) one who buys more than n units
(ii) Is the consumer’s feasible set convex? Ilustrate some problems that may arise.
3. Let g be the amount of garbage removed, so that the amount retained by the
consumer is x = G − g, with G the total amount produced. Let p be the cost per unit of
garbage removed. Finally, let y be the consumption good, with price set at 1, and M the
consumer’s income. Then the budget constraint is
y + pg = M
or
y − px = M − pG.
The budget constraint in (x, y)-space is graphed in Fig. 2B.4, on the assumption that
pG < M. The slope of the line is p.
Supplementary questions
(i) Show the effects on the budget constraint of a rise in M; a fall in c; and increase in *.
(ii) Superimpose the indifference curves for each of the following types of consumer:
(a) one who has all her garbage removed;
(b) one who has some but not all garbage removed;
(c) one who has no garbage removed.
What happens to the first type of consumer if c* > M?
Exercise 2C
1. Given a budget constraint as in Fig. 2B.2, the conditions of the Existence Theorem are
satisfied (given that the utility function is continuous) since the feasible set is nonempty, closed and bounded. It is not, however, convex, and so we may have multiple
global optima (an indifference curve could be tangent to both linear segments) and local
optima that are not global (two indifference curves each tangent to one linear segment).
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Fig. 2C.1
2. We can use the Kuhn-Tucker conditions as a general framework within which to
organize the answer to this question. Thus suppose we want to solve
max u(x1, x2)
x1 ,x2
s.t. p1x1 + p2 x2 ≤ M,
x1, x2 ≥ 0,
where u is quasi-concave. The Kuhn-Tucker conditions, which are both necessary and
sufficient for an optimum are,
ui − λ*pi
≤
0
xi* ≥ 0
xi* [ui − λ*pi] = 0
p1 x1* + p2 x2* − M
≤
0
λ* ≥ 0 λ*[ p1 x1* + p2 x2* − M ] = 0
i = 1, 2,
Now take two cases:
(a) One of the goods, say x1, is a bad, i.e. u1 < 0. From the Kuhn-Tucker conditions we
see that if x1* > 0, then u1 = λ*p1 at the optimum. This tells us that a necessary condition
for positive consumption of a bad at the optimum is that its price is negative – one must
be paid for consuming it. If p1 > 0 then the condition can only be satisfied with u1 < λp1
implying x1* = 0. If you have to pay to consume a bad then you choose a zero quantity.
(b) Suppose that the consumer is satiated with x1 when she reaches a consumption
I1; i.e. u1 > 0 for x1 < I1 and u1 = 0 for x1 ≥ I1, for any consumption x2. We now show that,
if u2 > 0 everywhere in the feasible set, and both prices are positive, the consumer is
always at an equilibrium at which x1* < I1, so that u1 ( x1* , x2* ) > 0. Thus for i = 2,
the Kuhn-Tucker condition gives u2 = λ*p2 > 0 and so λ* > 0, and the budget constraint
must be binding. For i = 1, if x1* ≥ I1 > 0 we must have u1 = 0 < λ*p1, which then implies
x1* = 0, which is a contradiction. x1* > 0 implies u1 = λ*p1 > 0 implying in turn x1* < I1.
Fig. 2C.1 illustrates the solution.
(c) Now let (I1, I2) be the bliss point, so that xi < Ii ⇒ ui > 0 and xi ≥ Ii ⇒ ui = 0.
If the feasible set does not contain the bliss point, then we can show, as in ( b), that
x i* < Ii, or neither good is consumed to satiation. On the other hand, if (I1, I2) is
feasible then it satisfies the Kuhn-Tucker conditions. Thus, if (I1, I2) is not feasible,
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Fig. 2C.2
setting, say x1* = I1 must imply x2* < I2. But then, just as before, u2 = λ*p2 > 0 ⇒ λ* > 0
and so x1* = I1 > 0 is inconsistent with the Kuhn-Tucker conditions. On the other hand,
if (I1, I2) is feasible and the consumer chooses it then u1 = u2 = 0 and so λ* = 0 and the
budget constraint is not binding (though it may pass through (I1, I2)). Fig. 2C.2
illustrates.
(d) If Ii is defined as the value of xi at which i changes from being a good to a bad,
then in effect (I1, I2) is a bliss point and given that both prices are positive the analysis
is just as in case (c).
Turning now to question 4 of Exercise 2A, first recall that in the case of red and blue
matches we have MRS21 = u1/u2 = 1. We can then distinguish three cases:
(i)
x1* > 0, x2* > 0. From the Kuhn-Tucker conditions this implies ui = λ*pi or
u1/u2 = p1/p2 = 1. Thus a necessary condition for this solution possibility is that
p1 = p2: both types of matches will be bought only if they have the same price.
(ii)
x1* > 0, x2* = 0, implying u1 = λ*p1, u2 ≤ λ*p2 or (u1/u2) ≥ p1/p2. Thus a necessary
condition for this case is p1 ≤ p2: we have a solution in which, say only red matches
are bought only if their price is no higher than that of blue matches.
(iii) Similarly x1* = 0, x2* > 0 implies p1 ≥ p2.
Thus we see that the solution depends essentially on the relative prices of the perfect
substitutes as Fig. 2C.3 shows.
In the case of left and right shoes the Kuhn-Tucker conditions cannot be used
because the utility function is not everywhere differentiable. Fig. 2C.4 however
illustrates the solution, which must always occur at a kink of an indifference curve if
both prices are positive, whatever their price ratio. It will never be optimal to buy
superfluous shoes.
3. (a) If ui > 0, a consumer always increases utility by moving from a point below the
budget line to a point on it. So a utility maximizing consumer will always choose a point
on his budget line.
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Fig. 2C.3
Fig. 2C.4
3. (b) At point x′ in text Fig. 2.8 the consumer can achieve higher indifference curves by
moving rightward along the budget line, thus feasible consumption bundles exist which
yield higher utility and x′ would not be chosen.
4. Let t be the amount of the transactions cost.
(a) Lump sum transactions cost. The consumer can at most buy xi0 = (M − t)/pi. The
budget constraint simply shifts inward in a parallel fashion. The transactions cost is
equivalent to a reduction in income.
(b) Proportional to price. If ti = ki pi, then the budget constraint becomes simply
∑(1 + ki)pi xi ≤ M, and we have a perfectly standard problem with effective prices
(1 + ki)pi.
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(c) Decreasing with quantity bought. If ti = ti(xi) with ti′ < 0, the budget constraint is
∑(pi + ti(xi))xi ≤ M. In the 2-good case we then have that the slope of the budget
constraint is given by
dx2
( p + t + x1t1′)
=− 1 1
0
dx1
( p2 + t2 + x2t2′ )
In this case we cannot say whether the budget constraint has a positive or negative
slope at any point, without further specification of the functions ti(xi). (Note the
contrast with the case in which ti′ > 0, i = 1, 2.)
5. The marginal utility of income is not uniquely defined, but rather changes with any
positive monotonic transformation of the utility function. Thus if v = T[u], then
∂v
∂u
= T′
∂M
∂M
where T′ > 0. This also implies that in ordinal utility theory we cannot talk of the
“diminishing marginal utility of income” since
∂2 v
∂ 2 u ∂u
=
+
T ′′
T
′
∂M 2
∂M 2 ∂M
and so even if ∂2u/∂M 2 < 0, we can always choose a transformation T[.] with T″ > 0
sufficiently to make ∂2v/∂M 2 > 0.
Exercise 2D
1. We keep the notation used in answering question 3 of Exercise 2B. Fig. 2D.1 carries
out the analysis in terms of g, the amount of garbage removed. Note that ∂u/∂x = −∂u/∂g
and so if x is a bad, with negative marginal utility, g is a good. In the analysis we assume
that g is not a Giffen good, and we take a rise in the price of garbage removal.
2. We keep the notation used in the answer to question 2 of Exercise 2B. Fig. 2D.2 gives
the answer, for a change in C and p0. Note that the effect of a change in C is zero if the
consumer does not buy e, and is equivalent to an income effect if she does. A change in
p0 affects demand both where e* < n and where e* > n.
Supplementary question
(i) Analyze the effect of a change in p1. Is this always zero if e* < n?
3. (a) In the case of red and blue matches, Fig. 2D.3 illustrates the Marshallian and
Hicksian demands. With p2 given, the Marshallian demand curve for x1 is the vertical
axis for p1 > p2, any value in [0, x10 ] for p1 = p2, and then the rectangular hyperbola given
by x1 = M/p1, for p1 < p2. The Hicksian ‘demand curve’ on the other hand is a stepped
function given by the vertical axis for p1 > p2, any value in [0, x10 ] for p1 = p2, and the
vertical line corresponding to demand at x10 for p1 < p2. To see this, note that any budget
line flatter than some indifference curve I0 which makes I0 the highest attainable
indifference curve must touch it at x10 . Thus the substitution effect is zero for p1 > p2,
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Gravelle and Rees: Microeconomics Instructor’s Manual, 3rd edn
Fig. 2D.1
Fig. 2D.2
Fig. 2D.3
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Fig. 2D.4
undefined at p1 = p2 and zero at p1 < p2. This implies that for p1 < p2, any increase in
demand for x1 resulting from a fall in p1 is due solely to an income effect (the horizontal
distance between the Marshallian and Hicksian demand curves).
3. (b) In the case of left and right shoes, Fig. 2D.4 illustrates the Marshallian and
Hicksian demands. At all prices p1 with given p2 the Hicksian demand curve is vertical,
indicating a zero substitution effect. Changing relative prices with the requirement that
the consumer remains on the same indifference curve always results in the solution at
the kink of the indifference curve. On the other hand the Marshallian demand curve has
a negative slope, and so the entire demand increase for x1 following a fall in p1 is due to
the income effect.
4. According to Hicks, real income is based on the achievable level of utility while for
Slutsky real income is based on the bundle of goods that can be bought. An advantage
of the Slutsky definition is that it is based on something observable (see also the theory
of revealed preference in section 4A). In text Fig. 2.15, because of the convexity of the
indifference curve, the reduction in money income required to cancel out the increase in
real income in the Hicksian sense must be larger than that required to cancel out the
increase in real income in Slutsky’s sense. Hence, if the good x1 is a normal good, the
income effect is greater in the Hicksian case than in Slutsky’s case, and the Hicksian
demand curve will lie further inside the Marshallian demand curve than does the
Slutsky demand curve.
Supplementary question
(i) Carry out the analysis for text Fig. 2.15 in the case in which x1 is an inferior (though
not Giffen) good.
5. We decompose the price effect into income and substitution effects because we want
to be more precise about the circumstances under which the Marshallian demand curve
will have a negative slope. Before this analysis, all we can say is that this slope may be
positive, negative or zero. It appears that the theory has no predictive or explanatory
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content and cannot be refuted because it is consistent with anything we might observe.
After the analysis we can say that the theory predicts that normal goods will certainly
have negatively sloped Marshallian demands, but that inferior goods may or may not
have. The analysis also classifies the determinants of the steepness of the demand curve
in terms of the closeness of substitutes and the strength of income effects. Finally it
gives an explanation of the existence of Giffen goods, whose Marshallian demand
curves have positive slopes. In the next chapter we consider a more precise statement
of income and substitution effects in the form of the Slutsky equation, which is one of
the most important equations in economic theory.
6. (a) Applying Lagrange’s method to the problem max u(x) s.t. px = M, where u(x) is
the Cobb-Douglas function and p is a price vector, gives the n + 1 first-order conditions
α i x1α . . . xiα −1 . . . x nα − λpi
=
0
∑pi xi − M
=
0
1
i
n
i = 1, . . . , n
Take any of the first n conditions, say the first, and divide through the remaining
n − 1 to obtain
α i x1α . . . xiα −1 . . . xnα
p
= i
α 1 x1α −1 . . . xiα . . . xnα
p1
1
i
1
n
i
i = 2, . . . , n,
n
and cancelling terms gives simply
α i x1 pi
=
α 1 xi p1
i = 2, . . . , n
α i p1
x1
α 1 pi
i = 2, . . . , n.
or
xi =
Then substituting into the budget constraint gives
n
α p
αi
pi i 1 x1 = p1 x1 1 +
= M.
α 1 pi
i= 2
i= 2 α 1
n
p1x1 +
∑
∑
But
n
αi
αi
1
=
=
∑
∑
α1
i= 2 α 1
i=1 α 1
n
1+
and so we have the Marshallian demand function
x1 =
α1M
p1
.
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But since the choice of x1 was completely arbitrary, we can write any demand function
as
xi = α i
M
,
pi
i = 1, . . . , n.
The main properties of these demand functions are:
(i)
expenditure on each good, pixi, is a constant proportion of income, αiM, at all
prices;
(ii) the Marshallian demand curve is therefore a rectangular hyperbola with slope
− a i M / pi2 and elasticity −1;
(iii) the Engel curves are rays through the origin with slopes ∂xi /∂M = αi /pi and income
elasticities (∂xi /∂M)(M/xi) = 1. There are no inferior goods;
(iv) ∂xi /∂pj = 0, i ≠ j, and demands depend only on their own prices. Therefore there are
no (Marshallian) substitutes and complements.
Supplementary questions
(i) Sketch a typical Marshallian demand curve and Engel curve for this case.
(ii) On an indifference curve diagram, show what property (iv) must imply about the
way in which an optimal point changes when the price of a good falls.
6. (b) Define Vi = xi − ki, i = 1, . . . , n. If we then define ? ≡ M − ∑ ni =1 piki (> 0 by
assumption, otherwise no solution exists), we can then formulate the problem as
max V1α V2α . . . Vnα s. t.
1
2
n
Vi
n
∑pV = ?
i
i
i=1
which is identical in form to the problem for the Cobb-Douglas case. Thus, we can write
the solution as Vi = αi?/pi, implying the Marshallian demand functions for the xi,
xi
=
=
n
M −
p j k j + ki
pi
j =1
αi
∑
(1 − αi)ki +
α iM
pi
−
αi
∑p k .
pi j ≠ i
j
j
The properties of these functions are then:
(i)
they are downward-sloping, since ∂xi /∂pi = −αi(M − ∑ j ≠ i p j k j )/ pi2 < 0.
However, they are no longer rectangular hyperbolas with unit elasticity since we
have
pi xi = pi (1 − α i ) ki + α i M −
pj kj ,
j≠i
∑
and so expenditure on xi is a linear increasing function of pi, and elasticity varies
with pi;
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(ii) Engel curves are again linear, since ∂xi /∂M = αi /pi, but they now have non-zero
intercepts [(1 − αi)piki − αi∑j≠i pjkj]/pi. Moreover, in general these intercepts vary
with i. There are still, however, no inferior goods, αi /pi > 0.
(iii) We now have ∂xi /∂pj = −αikj /pi < 0, i, j = 1, . . . , n, i ≠ j. Thus all goods are
(Marshallian) complements. Though this may be thought an improvement on the
Cobb-Douglas case, where cross-effects are zero, restricting all goods to be
complements is still not very reasonable.
The constants ki are typically interpreted as subsistence level requirements for the
goods. Thus Vi could be thought of as demand net of the subsistence requirement,
∑ ni =1 piki the expenditure required to buy the “subsistence bundle”, and ? is a kind of
“discretionary income”, i.e. income over and above that required to buy the subsistence
bundle. We can then interpret the cross-price elasticities ∂xi /∂pj as follows. An increase
in pj causes an increase in the expenditure required to provide the subsistence level of
good j, which in turn reduces discretionary income ? and so, since every good is
normal, we have a reduction in demand for good i. If we totally differentiate the above
expression for pixi we obtain
d(pixi) = −αikjdpj
i, j = 1, . . . , n, i ≠ j
Thus expenditure on good i falls by the proportion αi of the amount by which ? falls,
−kjdpj. With pi fixed, this implies the partial derivative given above.
Supplementary question
(i) Obtain an expression for the price elasticity of demand for this demand function and
comment on its properties.
6. (c) Setting up the utility maximization problem as a Lagrangean yields the first-order
conditions
fi′( xi* ) − λpi = 0 i = 1, . . . , n
n
−
∑ p x* + M
i
i
=
0.
i=1
The precise forms of the Marshallian demand functions will of course depend on the
functions fi(.), which we have not specified. However, we can carry out a general
qualitative analysis of the properties of these functions quite easily. To do so, we use the
comparative statics methods of Appendix I rather than the simpler duality methods of
chapter 3. Also to simplify notation we assume just two goods, n = 2. Then, totally
differentiating through the above first-order conditions and using Cramer’s Rule to solve
for the demand derivatives gives
∂x1 − λp22 + x1 p1 f2′′
=
∂p1
D
λp1 p2 + x2 p1 f2′′
∂x1
=
∂p2
D
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Gravelle and Rees: Microeconomics Instructor’s Manual, 3rd edn
− p1 f2′′
∂x1
=
∂M
D
− λp12 + x2 p2 f1′′
∂x2
=
D
∂p2
λp1 p2 + x1 p2 f1′′
∂x2
=
D
∂p1
∂x2
− p2 f1′′
=
∂M
D
where D = −( p12 f2′′+ p22 f1′′) > 0 from the second-order conditions. From these three
equations we can write the Slutsky equations
λp2j
∂xi
∂x
= −
− xi i ,
∂M
D
∂pi
∂xi
λp1 p2
∂x
=
− xj i ,
∂p j
D
∂M
i, j = 1, 2, i ≠ j.
The first term in each of these equations is the substitution effect of a change in the
corresponding price, the second term is the income effect. One interesting result of this
functional form is that since λp1p2/D > 0, the goods must be what we call later Hicksian
substitutes – the possibility of their being Hicksian complements is ruled out.
Next note that if the fi′′ < 0, then each good is a normal good and must have a
negatively sloped Marshallian demand. This assumption is that there is “diminishing
marginal utility” for each good. Thus this assumption together with the additively
separable form, implies that there are no inferior goods and so no Giffen goods. This
was in fact the structure that underlay Marshall’s theory of demand, which ordinal
utility theory supplanted. We cannot, however, make the assumption fi′′ < 0 in an
ordinal theory, since it cannot survive a positive monotonic transformation of the utility
function. What we can note is that the assumption that the utility function be strictly
quasi-concave implies (see question 7 of Appendix B) that
( f1′)2 f2′′+ ( f2′)2 f1′′< 0
(since the cross-partial derivatives fij = 0 in the additively separable case). Indeed, this
can be seen in precisely the condition that D > 0, if we substitute fi′ = λpi from the firstorder conditions. Now, the above condition does not imply fi′′ < 0, i = 1, 2. It would be
possible to construct examples in which, say f1′′ < 0 and f2′′ > 0 were consistent with
the condition. For this we would have to restrict the relative variations in f1′ and f2′,
since the above condition implies
2
f ′
f1′′< − 1 f2′′
f2′
and this would have to hold over the whole domain of the function. Ruling out such
cases and taking fi′′ < 0, i = 1, 2, we have that both goods are normal and so their
Marshallian demands have negative slopes.
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17
Supplementary question
(i) A consumer is said to have strongly or additively separable preferences if her utility
function can be written
u(x) = F fi ( xi )
i
∑
where F is any increasing function. Show that the Cobb-Douglas, Stone-Geary, and
additively separable utility functions are all examples of strongly separable preferences.
Exercise 2E
1. In (a) of Fig. 2E.1, the horizontal axis is at the level of the consumer’s initial
endowment of x2, and she is a net seller of x1 and buyer of x2. In (b) of the figure the
vertical axis is at the level of the consumer’s initial endowment of x1, and she is a net
buyer of x1 and seller of x2. In each case we take an increase in p1 relative to p2, so that
the budget constraint rotates clockwise through the initial endowment point. We also
assume that each good is a normal good. Then we see that in case (b), the consumer’s
net demand for x1 certainly falls, and the figure decomposes the overall change into an
income and substitution effect as before. The analysis here is exactly similar to the case
in which the consumer has money income. In (a) however, we see that the consumer’s
net supply of x1 could increase or decrease, depending on the relative strengths of the
income and substitution effects. Thus we have an ambiguous response to the price
change in this case even though x1 is a normal good.
The reason for the difference in the two cases is that the income effect of a price
change depends on whether the consumer is a buyer or seller of the good. If she is a
buyer, an increase in price reduces her real income, and so if the good is normal the
income effect reinforces the substitution effect. On the other hand, if the consumer is a
seller of the good, an increase in its price increases her real income, and so if the good
is normal this will lead her to increase her consumption of it (reduce net supply). This
income effect therefore goes in the opposite direction to the substitution effect, which
is to reduce consumption of x1 (increase net supply) following the price increase.
Supplementary questions
(i) Work through the analysis of Fig. 2E.1 for the case of a price fall.
(ii) Analyze the effects of a rise in price p1 when x1 is an inferior good.
2. The slope of the consumer’s offer curve at a point shows the consumer’s subjective
marginal rate of exchange between the two goods. Thus a point on the curve, such as
x′, shows that the consumer is prepared to exchange I1 − x1′ of x1 for x2′ − I2 of x2, to
give an average rate of exchange of ( x2′ − I2)/(I1 − x1′ ). The slope of the offer curve at,
say, x′, shows the relation between the extra bit of x1 she is prepared to trade for an
extra bit of x2 given that she holds ( x1′, x2′ ) of the goods.
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Fig. 2E.1
3. The offer curve is the locus of tangency points between a budget line, reflecting a
particular price ratio, and an indifference curve. That is, it shows the (x1, x2)-pairs at
each of these tangency points, since it shows the trades the consumer wishes to make.
4. (a) In a world of complete certainty, the model would be appropriate for a market
in stocks and shares. Each participant in the market has given initial holdings of
shares. Given the prices of shares, investors will adjust their portfolios, selling some
stocks and buying others, until equilibrium is reached: a set of prices is achieved
at which each investor is content to hold her particular portfolio and no further
trades take place. However, the assumption of complete certainty is a very restrictive
one and rules out many of the aspects of stock markets, for example, speculation and
portfolio diversification, that we would really want to study. For analysis of this, see
chapters 19, 21.
4. (b) A market in secondhand cars is again one which it is apt to treat as an “initial
endowments” model. Consumers are initially endowed with a stock of used cars of
varying vintages, qualities, types, etc., there is a flow of old cars out of the market and a
flow of new cars onto the market. At various prices consumers could be buyers of used
cars, sellers of used cars and buyers of new cars. In equilibrium, a set of prices prevails
at which the total stock of new and used cars is held by consumers with no-one wishing
to make further trades. However, such a static equilibrium is likely to last for only a
short time, because of depreciation (change in quality) of used cars and the continual
flow of new cars with changes in style, design, performance etc. Thus trading in the car
market is likely to be a virtually continuous activity.
4. (c) This is a special case of the general model in which the endowment of one good,
bread, is zero while that of the other is positive. Thus the consumer will almost certainly
be a buyer in the market for bread and a “seller of leisure time”, or to put it more
conventionally, a supplier of labour. The price of leisure time is the wage rate, since
taking an hour of leisure means giving up the hourly wage that could have been earned
by working instead. If (contrary to general usage) we measure x2 on the horizontal axis
and x1 on the vertical axis, and define the price ratio as w/p1, where w is the wage rate
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Gravelle and Rees: Microeconomics Instructor’s Manual, 3rd edn
19
and p1 is the price of bread, then Fig. 2E.1 analyzes the supply of labour (I2 − x2) in
exchange for bread, and shows the ambiguous effect of an increase in the wage rate. A
much fuller analysis of this model is carried out in section 4C below.
Supplementary question
(i) Explain why the analysis of Fig. 2E.1 implies that the supply of labour may increase
or decrease as the real wage increases.
Exercise A1
1. Let I2 be the subsistence level for x2. Then the consumer’s demand functions are now:
x1 = (M − p2I2)/p1; x2 = I2. That is, the consumer with lexicographic preferences will buy
only the minimum amount of x2 he needs for subsistence and will spend the rest of his
income on x1.
2. Suppose now there are no minimum subsistence levels but there is a satiation level
for x1 denoted V1. Then, assuming M/p1 > V1, his demand functions are: x1 = V1; x2 =
(M − p1V1)/p2, a vertical line and a rectangular hyperbola respectively. (Incidentally,
we should not really have referred to the “marginal utility of x2” in this question since
the utility function does not exist for this ordering.)
3. For n goods, the statement of the lexicographic ordering is for any two unequal
bundles x′ = ( x1′, . . . , xn′ ), x″ = ( x1′′, . . . , xn′′):
x1′ > x1′′
⇒ x′ Ɑ x″
x1′ = x1′′ and x2′ > x2′′
⇒ x′ Ɑ x″
... ... ...
x1′ = x1′′, x2′ = x2′′, . . . , xn′ −1 = xn′′−1 and xn′ > xn′′
... ...
⇒ x′ Ɑ x″.
Without subsistence and satiation levels, the demand functions would be: x1 = M/p1;
x2 = . . . = xn = 0. The consumer spends all his income on the first good. With subsistence
but without satiation levels, the demand functions would be: x1 =
(M − ∑ in=1 piIi)/p1; xi = Ii, i = 2, . . . , n, where Ii is the subsistence level for good i.
Without subsistence but with satiation levels the demand functions would be:
xi = Vi, i = 1, . . . , k − 1; xk = (M − ∑ ik=−11 piVi)/pk, k = 1, . . . , n. Here, Vi is the subsistence
level for good i and k is the first value of i such that ∑ ik=1 piVi ≥ M.
4. We have to confess that we find lexicographic preferences implausible as a general
formulation of preferences both for individual goods and for groups of goods. The
reasonable observation that one must have minimum levels of water, food, sleep
(“leisure”) and shelter to survive can be handled by introducing subsistence levels into
the standard framework set out in this chapter. That is, we define a consumer’s
consumption set C as the set of consumption bundles it is physically (as opposed to
economically) feasible for the consumer to choose, and we could then define
C = {x ∈ R+n | x ≥ I}
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Gravelle and Rees: Microeconomics Instructor’s Manual, 3rd edn
where I = (I1, . . . , In) is a subsistence vector. Consumer theory can only then apply to
those people whose consumption sets have a non-empty intersection with their budget
sets, i.e. who can literally afford to live.
On the other hand lexicographic preferences together with subsistence levels are
appropriate for those individuals with strong addictions – say to alcohol or hard drugs.
Otherwise evidence suggests that consumers have preferences which allow them to
trade off among goods.
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21
Chapter 3
Consumer Theory: Duality
Exercise 3A
1. Let x1* and x2* be the Hicksian demand functions, then xi* , i = 1, 2, are the solutions
to the maximization problem
min p1x1 + p2x2
x1 , x 2
s.t.
E = x1a x2b ,
a + b = 1.
With λ as the Lagrange multiplier, the Lagrange function is
L = p1x1 + p2x2 + λ(E − x1a x2b )
and so the first-order conditions are
Lx = p1 − λax1a −1 x2b = 0,
(3.1)
Lx = p2 − λbx1a x2b−1 = 0,
(3.2)
Lλ = E − x1a x2b = 0.
(3.3)
1
2
(3.1) and (3.2) give
p1
ax2
=
p2
bx1
⇒ x2 =
p1 b
x1
p2 a
(3.4)
substitute (3.4) into (3.3)
p b
E = x 1 x1
p2 a
b
a
1
b
p b
= 1 x1
p2 a
−b
p b
⇒ x1* = E 1
p2 a
−b
and by symmetry,
−a
−a
p a
x2* = E 2 .
p1 b
(3.5)
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Gravelle and Rees: Microeconomics Instructor’s Manual, 3rd edn
Let m(p1, p2, u) be the expenditure function;
m(p1, p2, u) = p1 x1* + p2 x2*
p −a a −a
p −b a b
1
= p1 E + p2 E 2
p1 b
p2 b
b
a
a
= Ep11−b p2b + Ep21−a p1a
b
b
b
a
a
= Ep1a p2b + Ep2b p1a
b
b
−a
−a
a b a − a a b
= + p1 p2 E
b b
Now
v = u2
= ( x1a x2b )2
= x12a x22b .
min ∑pixi
H = x12a x22b ,
s.t.
a+b=1
L = p1x1 + p2x2 + λ ( H − x12a x22b )
Lx = p1 − 2aλ x12a −1 x22b = 0
(3.6)
Lx = p2 − 2bλ x12a x22b−1 = 0
(3.7)
Lλ = H − x12a x22b = 0
(3.8)
1
2
Dividing (3.6) by (3.7) gives
p1
ax2
=
p2
bx1
⇒ x2 =
p1 b
x1
p2 a
just as in (3.4) above. Substitute (3.4) in (3.8)
2b
p b
H = x 1 x12b
p2 a
2a
1
2b
p b
= 1 x12
p2 a
since a + b = 1
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23
−2 b
p
x12 = H 1
b
p2 a
,
−b
p b
x1* = H 1 .
p2 a
1
2
By symmetry,
p a
x2* = H 2
p1 b
−a
1
2
Substituting in the expenditure function yields
p b
m(p1, p2, v) = p1 H 1
p2 a
1
2
−b
p a
+ p2 H 2
p1 b
−a
1
2
Simplifying yields
a b a − a
m(p1, p2, v)* = + H p1a p2b
b b
1
2
Now substituting for H = u gives
1
2
a b a − a
m(p1, p2, v)* = + up1a p2b
b b
as before. Then we have the derivatives:
a b a − a a b
∂m( p1 , p2 , u)
= + p1 p2
∂u
b b
−a
b
1 a a
∂m( p1 , p2 , u)
−
= + p1a p2b v
2 b b
∂v
1
2
Thus although the values of the expenditure function and the Hicksian demands are
unaffected by the transformation of the utility function, the measure of the ‘marginal
cost of utility’, ∂m/∂u, which is the reciprocal of the marginal utility of income, ∂u/∂m,
does depend on the specific utility function used. You should confirm for the
transformation used here that:
∂m ∂m dv
.
=
∂u
∂v du
2. For perfect complements, the indifference curves take the form shown in Fig. 3A.1.
As the figure shows x1* = x2* . The equilibrium must satisfy the budget constraint
p1 x1* + p2 x2* = M
and so
M
= x1* = x2* = u.
p1 + p2
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Fig. 3A.1
Hence,
u* =
M
p1 + p2
is the indirect utility function. Inverting the indirect utility function gives the
expenditure function
m(p, u) = (p1 + p2)u.
In the perfect substitute case, u = ax1 + bx2. As the figure shows, in maximizing utility
we have two main cases determined by the relative slopes of the budget line and the
linear indifference curves (where these slopes are equal the solution is at any point on
the budget constraint).
(a)
a
b
a p1
or equivalently
.
>
>
b p2
p1 p2
Then we have a corner solution with x1 = M/p1, x2 = 0. Then u = aM/p1 > bM/p2.
(b)
b
a
p1 a
.
> or equivalently
>
p2 b
p2 p1
Then we have a corner solution with x2 = M/p2, x1 = 0. Then u = bM/p2 > aM/p1. These
results allow us to write the indirect utility function as
aM bM
a b
,
u = max
= M. max , .
p1 p2
p1 p2
Inverting this gives the expenditure function
p p
M = u min 1 , 2
a b
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Gravelle and Rees: Microeconomics Instructor’s Manual, 3rd edn
25
Supplementary question
(i) Obtain the above result on the expenditure function in the case of perfect substitutes
by solving the expenditure minimization problem.
[Hint: use Fig. 3A.1 and obtain the values for M in cases (a) and (b).]
3. (a) If u(x) is strictly quasi-concave, there is a unique solution x* to the problem
max u(x) s.t. px = m, and we assume at this solution every xi* > 0. Moreover, there is a
unique solution V to the problem min px s.t. u(x) = u*, and we also have that u(x*) = u*
= u(V). Suppose that x* ≠ V. Then px* ≠ pV.
Either px* > pV. But given the strict quasi-concavity of u, this implies there exists
I = kx* + (1 − k)V, k ∈ (0, 1) such that u(I) > u(x*) and pI < px* = M, thus contradicting
the optimality of x*.
Or px* < pV. But this immediately contradicts the optimality of V since u(x*) = u* and so
x* is feasible for the expenditure minimization problem. Thus the assumption x* ≠ V
leads to a contradiction and we have x* = V.
3. (b) The first-order conditions for the utility maximization and expenditure
minimization problems are, respectively
ui(x*) − λ*pi = 0
px* = M
i = 1, . . . , n
pi − µ*ui(V) = 0
u(V) = u
i = 1, . . . , n.
But we just saw that if u(V) = u(x*), then V = x*, and so:
λ* = ui(x*)/pi,
µ* = pi/ui(x*),
⇒ λ* = 1/µ*.
4. First define: Vi ≡ xi − ci,
i = 1, 2. Then note that
M = p1x1 + p2x2 = p1V1 + p2V2 + p1c1 + p2c2
We can then write the problem as
min p1V1 + p2V2 + p1c1 + p2c2
V1, V 2
s.t. V1a V 2b = u
This gives the first-order conditions
p1 − aµ V1a −1 V2b = 0
p2 − bµ V1a V 2b−1 = 0
V1a V 2b = u
and Hicksian demand functions:
V1 = (a/b)b(p1/p2)−bu
V2 = (a/b)−a(p2/p1)−au
or
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Gravelle and Rees: Microeconomics Instructor’s Manual, 3rd edn
x1 = (a/b)b(p1/p2)−bu + c1
x2 = (a/b)−a(p2/p1)−au + c2
Substituting into the expression for M gives the expenditure function:
M = [(a/b)b + (a/b)−a] p1a p2b u + p1c1 + p2c2.
The difference to the expenditure function in question one is the term p1c1 + p2c2, which
can be interpreted as the cost of the required subsistence bundle, a kind of ‘fixed cost of
survival’.
5. Suppose that the optimal bundle has positive consumption of both goods. The
Lagrangean is
f(x1) + x2 + µ[u − p1x1 − p2x2]
and the first order conditions on x1 and x2 are
f′(x1) − µp1 = 0
1 − µp2 = 0
From the condition on x2 we can substitute 1/p2 for µ to write the condition on x2
as f′(x1) = p1/p2. Thus the Hicksian demand for good 1 depends only on relative prices
and not on the required utility level: x1 = h1(p1/p2). Any increase in the required utility
level is met entirely by an increase in x2. Using the constraint, the Hicksian demand for
good 2 is x2 = u − f(h1(p1/p2)) = h2(u, p1/p2). The expenditure function is
m(u, p) = p1h1(p1/p2) + p2[u − f(h1(p1/p2))]
The slope of an indifference curve is
dx2
= −f′(x1)
dx1
so that the indifference curves have the same slope along vertical lines in (x1, x2) space.
Supplementary question
(i) After reading section 3B, derive the indirect utility function and the Marshallian
demands and show that good 1 has a zero income elasticity of demand so that all
additional income is spent on good 2.
Exercise 3B
1. The Hicksian demand function is derived from the problem
min
∑px
i
i
s.t. u(x1, x2, . . . , xn) = E
i
yielding the first-order conditions:
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Gravelle and Rees: Microeconomics Instructor’s Manual, 3rd edn
pi ui ( x*)
=
pn un ( x*)
27
i = 1, 2, . . . , n − 1,
u(x*) = E.
If the price vector is multiplied by k > 0, the conditions become
kpi
u ( x*)
= i
kpn un ( x*)
which clearly will leave the solution unchanged. Thus,
xi* = Hi(p1, . . . , pn, u) = Hi(kp1, . . . , kpn, u)
and the Hicksian demand function is homogeneous of degree zero in prices. Identifying
Hi with the function f in Euler’s Theorem and pj with xi we have
n
∂H i
j =1
j
∑ ∂p p = 0.
j
2. From question 6(a) of Exercise 2D we have that the consumer’s Marshallian demands
are:
x1 = aM/p1;
x2 = (1 − a)M/p2.
Substituting into the utility function gives
a
aM (1 − a ) M
u=
p2
p1
1− a
= a a (1 − a )1−a p1− a p2− (1−a ) M
Roy’s Identity says that in general:
∂u
= −λxi.
∂pi
Then,
(a)
∂
[a a (1 − a )1−a p1− a p2− (1−a ) M ] = − a 1+a (1 − a )1−a p1− (1+a ) p2− (1−a ) M
∂p1
(b) From the first-order conditions we find
λ = ax1a −1 x21 − a / p1
So, −λx1 = − ax1a x21−a / p1 .
(c) Given the indirect utility function, we can use (a) to write:
∂u
au
=−
∂p1
p1
and given the direct utility function we can use (b) to write
−λx1 = −
au
p1
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3. u = a a (1 − a )1−a p1− a p2− (1−a ) M . Inverting this gives the expenditure function
Alternatively, solving the expenditure-minimization problem gives: M = m(p, u) =
a−a(1 − a)−(1−a) p1a p2(1−a ) v
p
x1 = 1
p2
− ( 1− a )
a
1− a
1− a
u
−a
a
p a
x2 = 1
u
p2 1 − a
and so we have the expenditure function:
m(p, u) = p1x1 + p2x2
p
= p1 1
p2
− ( 1− a )
a
1− a
1− a
a
−a
p a
u + p2 1
u
p2 1 − a
a 1−a a − a
= p p u
+
.
1 − a
1 − a
a
1
1− a
2
We therefore need to show that:
a
1 − a
1− a
a
+
1 − a
−a
= a−a(1 − a)a−1
Now:
−a
a a
+ 1 = a−a(1 − a)a(1 − a)−1
1 − a 1 − a
= a−a(1 − a)a−1
as required.
4. (a) Given the Marshallian demand functions
xi = Di(p, M)
i = 1, . . . , n.
the adding up property implies
∑ p D (p, M) = M.
i
i
(3.9)
i
Allowing pj to change with M constant, differentiating through (3.9) gives
Dj(p, M) +
∂Di
∑ p ∂p = 0 j = 1, . . . , n.
i
i
(3.10)
j
as required. Differentiating through (3.9) with respect to M holding prices constant gives
∂Di
∑ p ∂M = 1
i
i
as required.
© Pearson Education Ltd 2007
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29
4. (b) Multiply through the jth equation in (3.10) by pj /M, and each term pi∂Di /∂pj by xi /xi
(= 1) to obtain
p ∂Di
p j D j ( p, M )
si e ij = 0
pi xi j
+
= sj +
xi ∂p j
M
i
i
∑
∑
as required.
Multiply through (3.11) by xiM/xiM to obtain
M ∂Di
∑ p x x ∂M = ∑ s η = 1
i
i
i
i
i
i
i
as required.
4. (c) Euler’s Theorem states that for a function f(x1, . . . , xn), if f is homogeneous of
degree zero then
∑ f x = 0.
i
i
i
We know that the Marshallian demand functions are homogeneous of degree zero in
prices and income, so
p1
∂Di
∂Di
∂Di ∂Di
+ p2
+ . . . + pn
+
M =0
∂p1
∂p2
∂pn ∂M
or
∂Di
n
∂Di
∑ p ∂p + ∂M M = 0
i = 1, . . . , n.
j
j =1
j
Dividing through by xi gives
∑e + η = 0
ij
i
j
as required.
4. (d) From the symmetry property we derive immediately:
∂D j
∂pi
∂D j
∂Di
∂Di
+ xj
− xi
.
∂p j
∂M
∂M
=
Multiplying through by pi and summing over i gives
∂D j
∂Di
∂Di
∂D j
∑ p ∂p = ∑ p ∂p + x ∑ p ∂M − ∂M ∑ p x .
i
i
i
j
i
i
i
i
i
j
i
i
Using the Engel aggregation result, homogeneity and the fact that ∑i pi xi = M gives
∂D j
∂D j
∂Di
∂D j
∑ p ∂p = ∑ p ∂p + x − M ∂M = − M ∂M
i
i
i
i
i
j
j
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and so we have the Cournot aggregation result
∂Di
∑ p ∂p + x = 0.
i
j
i
j
5. The Slutsky equations for goods i and j are
∂Di ∂H i
∂Di
=
− xj
∂p j
∂p j
∂M
∂D j
∂pi
=
∂H j
− xi
∂pi
∂D j
∂M
where ∂Hi/∂pj = ∂Hj/∂pi by symmetry. Call this term S. Then if ∂Di/∂pj < 0 (complements)
and ∂Dj/∂pi > 0 (substitutes), we must have
xi
∂D j
∂M
< S < xj
∂Di
∂M
as the necessary and sufficient condition for this to happen. If S < 0 (Hicksian
complements) then good j must be inferior, good i could be normal or, if inferior, have a
sufficiently smaller (in absolute value) income effect than j. If S > 0 (Hicksian
substitutes) then i must be normal and j could be inferior or normal with a sufficiently
smaller income effect.
Exercise 3C
1. In Fig. 3.6 of the text, simply interpret B as the initial equilibrium and A as the postprice change equilibrium. Then EV in the figure becomes CV and CV becomes EV.
We make the same reinterpretation of the areas under the Hicksian demand curves
in Fig. 3.6(b). If u1 is now the initial level of utility, and u0 the final level, with
p1 = ( p11 , p2 , . . . , pn ) the initial price vector and p0 = ( p10 , p2 , . . . , pn ) the final price vector,
we now have
p
∂m
dp1 =
H1 ( p, u1 )dp1
p ∂p
p
1
p10
∫
EV = m(p0, u0) − m(p1, u0) =
p
∂m
dp1 =
H1 ( p, u0 )dp1
p ∂p
p
1
∫
1
1
p10
1
1
∫
0
1
CV = m(p0, u1) − m(p1, u1) =
∫
1
1
0
1
1
1
In terms of the expression of CV and EV in terms of indirect utilities, we now have that
they must respectively satisfy
u*(p1, M0) = u*(p0, M0 + CV) = u1
u*(p1, M0 − EV) = u*(p0, M0) = u0
(compare these to [C.1] and [C.2] in the text). Note that here, because we wish to stress
the relation between CV or EV and the area under the Hicksian demand curve, we have
defined them so that they are always positive, regardless of the direction of the price
change.
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31
2. (a) A contour of the utility function is defined by
E = f(x1) + x2
Differentiating totally gives
f′(x1)dx1 + dx2 = 0 ⇒
dx2
= −f′(x1).
dx1
Thus the slope of the indifference curve depends only on x1. For given x1, every
indifference curve has the same slope for all values of x2.
2. (b) Solving
max f(x1) + x2
x1 , x 2
s.t. p1x1 + p2x2 = M
gives the first-order conditions
f′(x1) − λp1 = 0
1 − λp2 = 0
p1x1 + p2x2 − M = 0
For x1 we have
f′(x1) = p1/p2 ⇒ x1 = f′−1(p1/p2) ≡ φ(p1/p2).
Then substituting into the budget constraint we have
x2 = M/p2 − φ(p1/p2)/p2
This gives the indirect utility function
u = f[φ(p1/p2)] + M/p2 − φ(p1/p2)/p2
and the expenditure function
M = φ(p1/p2) + p2[u − f[φ(p1/p2)]].
Differentiating gives the Hicksian demand function for x1
∂M
= φ′/p2 − f′φ′ = H1(p1, p2).
∂p1
Note that this Hicksian demand function is independent of the utility level. For given p2
(which could always be set equal to 1) there is only one Hicksian demand curve,
whatever the utility level.
2. (c) Since the demand function x1 = φ(p1/p2) does not contain M as an argument, we
must have ∂x1/∂M = 0 and so x1 has zero income elasticity. To show that CV and EV are
identical for any price change from p10 to p11 we can use the expenditure function just
derived to write:
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Gravelle and Rees: Microeconomics Instructor’s Manual, 3rd edn
CV = φ ( p10 / p2 ) + p2 [u 0 − f [φ ( p10 / p2 )]] − φ ( p11 / p2 ) − p2 [u0 − f [φ(p1/p2)]]
EV = φ ( p10 / p2 ) + p2 [u1 − f [φ ( p10 / p2 )]] − φ ( p11 / p2 ) − p2 [u1 − f [φ(p1/p2)]]
which are identical when the terms in u have been cancelled. In other words since there
is only one Hicksian demand curve whatever the level of utility the area under it
between two prices will be the same whatever the level of utility.
2. (d) From the first order conditions we have the marginal utility of income λ = 1/p2,
and so ∂λ/∂p1 = 0. Thus condition [C.8] of the text is satisfied and the Marshallian
consumer surplus is an appropriate money measure of a utility change. In fact, as we
may expect from the result in 2(c), the Hicksian and Marshallian demand functions for
x1 are identical in this case. To show this we can use Roy’s Identity, which gives
∂u
φ ( p1 / p2 )
.
= λx1 = −
p2
∂p1
Differentiating the indirect utility function gives
∂u
= f′φ ′/p2 − φ ′/ p22 = −φ/p2
∂p1
Then, multiplying this through by −p2 gives
φ′/p2 − f′φ ′ = φ
which, recalling the expression for H1 in (c) above, gives the result.
3. From our earlier results on the Cobb-Douglas case (question 6(a) of exercise 2D) we
have the demand functions
x1 =
M
2 p1
and x2 =
M
2 p2
Hence the indirect utility function is
1/ 2
M M
v* =
2 p1 2 p2
=
1/ 2
1 −1/ 2 −1/ 2
p1 p2 M
2
So,
v0 =
1
(1)−1/2(1)−1/2100
2
= 50
−1 / 2
1 1
v = (1)−1/2100
2 4
1
= (2)(100) = 100
2
1
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33
The expenditure function is
M = 2vp11 / 2 p21 / 2
and
CV = m(p0, v0) − m(p1, v0)
= 2v0 [( p10 )1 / 2 p21 / 2 − ( p11 )1 / 2 p21 / 2 ]
= 50
EV = m(p0, v1) − m(p1, v1)
= 2v1
[( p10 )1 / 2 p21 / 2 − ( p11 )1 / 2 p21 / 2 ]
= 100.
The first-order conditions in this example yield
λ=
x2 x1
=
.
p1 p2
So, ∂λ/∂p1 = − x2 / p12 ≠ 0; ∂λ/∂p2 = − x1 / p22 ≠ 0 and so condition [C.8] is not satisfied. Thus
the Marshallian consumer surplus is not a valid (exact) measure of the change in utility
for these preferences.
4. (a) Using the Slutsky equation, the requirement that the Marshallian cross-price
effects are equal can be written as
∂D j
∂Di
= Hij − Dj DiM = Hji − Di DjM =
∂p j
∂pi
where Hij, Hji are the cross-substitution effects and DiM, DjM the effect of income on
demand. Since cross-substitution effects are equal, equality of Marshallian cross-price
effects implies
Dj DiM = Di DjM ⇒
DiM M D jM M
=
Di
Dj
so that all goods must have the same income elasticity of demand: η1 = . . . = ηn = η.
But from the Engel aggregation property of Marshallian demands (question 4(b) of
Exercise 3B)
1=
∑ s η = ∑ s η = η∑ s = η.
i
i
i
i
i
i
i
4. (b) The expression for CV and EV in terms of the expenditure function in [C.3] and
[C.5] are well-defined since the expenditure function is well-defined. Alternatively, the
condition for path-independence of the integrals in [C.3] and [C.5] when more than one
price changes is that the cross-effects of prices on the Hicksian demands are equal and
this is ensured by the equality of cross-substitution effects.
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Gravelle and Rees: Microeconomics Instructor’s Manual, 3rd edn
Exercise 3D
1. To simplify notation assume only 3 commodities, x1, x2, x3 and that p2 = kp20 ,
p3 = kp30 , so that p2/p3 = p20 / p30 , a constant. k itself may vary. We choose as our
composite commodity
xc = p20 x2 + p30 x3
and the price of this commodity is k. The idea is that we can analyze demands x1, xc in
terms of p1 and k. Consider the expenditure function m(p1, p2, p3, u) derived in the usual
way. Since pi = kpi0 , i = 2, 3, we can re-write this as
m(p1, p2, p3, u) ≡ m(p1, kp20 , kp30 , u) ≡ µ(p1, k, u)
and we can work with µ as the expenditure function for the composite commodity case.
Thus we have
∂µ ∂m ∂p2 ∂m ∂p3
= p20 x2 + p30 x3 = xc
=
+
∂k ∂p2 ∂k ∂p3 ∂k
using Shephard’s Lemma. Thus xc is the composite commodity corresponding to price k.
The expenditure function µ can be taken as fully describing preferences in the case
where we treat x2 and x3 as a composite commodity.
To show that µ has the properties of an expenditure function (we just showed that it
satisfies Shephard’s lemma) first note that it is concave in p1, k. To see this, let
L = λk′ + (1 − λ)k″
0≤λ≤1
for values k′, k″ of k, and let
pi′ = k′pi0 ,
pi′′ = k′′pi0
i = 1, 2.
Now concavity of the expenditure function m implies
m(F1, F2, F3, u) ≥ λm( p1′, p2′ , p3′ , u) + (1 − λ ) m( p1′′, p2′′, p3′′, u)
where
Fi = λpi′ + (1 − λ ) p1′′,
i = 1, 2, 3.
Then for i = 2, 3,
Fi = λki′ pi0 + (1 − λ) k′′pi0 = Lpi0 .
Thus
m(F1, Lp20 , Lp30 , u) ≥ λm( p1′, k ′p20 , k ′p30 , u) + (1 − λ ) m( p1′′, k′′p20 , k′′p30 , u)
implying, given the definition of µ:
µ(F1, L, u) ≥ λµ( p1′, k ′, u) + (1 − λ ) µ( p1′′, k′′, u)
and so µ is concave.
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35
To show that ∂µ/∂k ≥ 0, we note simply
∂µ ∂m 0 ∂m 0
=
p2 +
p3 ≥ 0
∂k ∂p2
∂p3
by the properties of the expenditure function m.
Finally, µ is clearly homogeneous of degree one in p1, k, since for α > 0
u(α p1, α k, u) ≡ m(α p1, α kp20 , α kp30 , u) ≡ m(α p1, α p2, α p3, u)
and so the result follows from the linear homogeneity in prices of the expenditure
function m.
2. This function is the constant elasticity of substitution (CES) function, which is also
widely used as a production function. In that context it is analyzed quite fully in
questions 6, 7 and 8 of Exercise 5A. Here we note that it is a homothetic function: in
fact, it is the basic form for the class of homothetic functions which have constant
elasticity of substitution (and which contains the Cobb-Douglas function as a special
case). Recall from [D.10] of the text the definition of a homothetic function: any positive
monotonic transformation of a linear homogeneous function is homothetic. Then note
that:
− β1
f(x1, x2) = [α 1 x1− β + α 2 x2− β ]
is linear homogeneous, since for k > 0,
− β1
f(kx1, kx2) = [α1(kx1)−β + α2(kx2)−β ]
− β1
= [k−β (α 1 x1− β + α 2 x2− β )]
= k (α 1 x1− β + αx2− β )
− β1
= kf(x1, x2)
Since the function is linear homogeneous then trivially it is homothetic, but then by
defining any differentable transformation
− β1
T [(α 1 x1− β + α 2 x2− β ) ]
with
T′ > 0
we obtain the entire family of homothetic utility functions with constant elasticity of
substitution.
3. From the solution to question 4 of Exercise 3A we have the expenditure function for
the Stone-Geary utility function as
m(p1, p2, u) = Ap1α p21 −α u − p1c1 − p2c2
where A ≡ ((α /(1 − α))1−α + (α /(1 − α))−α). Then simply define
b(p) ≡ Ap1α p21−α
and a(p) ≡ −(p1c1 + p2c2).
4. We shall answer this question for the case in which the utility function is written
u = u1(x1) + u2(x2) + . . . + un(xn)
ui′ > 0, ui′′ < 0, all i
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Gravelle and Rees: Microeconomics Instructor’s Manual, 3rd edn
(for the more general but complicated case of any positive monotonic transformation of
this function see A. Deaton and J. Muellbauer, Economics and Consumer Behaviour,
Cambridge, 1980, Chapter 5). Maximizing u subject to the constraint ∑pixi = m gives the
first-order conditions
ui′( xi* ) = λpi
i = 1, . . . , n
m − ∑ pi xi* = 0.
Write the Slutsky equation for this model as:
∂x
∂xi
= sij − x j i
∂p j
∂m
i, j = 1, . . . , n
where sij is the Hicksian or compensated demand derivative. The first step in the proof
is to show that
sij = µ
∂xi ∂x j
∂m ∂m
i, j = 1, . . . , n,
i≠j
for some given µ > 0 independent of i and j. If we then show that ∂xi/∂m > 0 for all i, this
immediately establishes that sij > 0 for all i, j, implying that all pairs of goods are
Hicksian substitutes. First, differentiate through the first-order conditions for the ith
good with respect to m and to pj, to obtain:
ui′′
∂xi
∂λ
= pi
∂m
∂m
ui′′
∂xi
∂λ
= pi
∂p j
∂p j
i = 1, . . . , n.
Use the second equation to solve for ui′′/ pi and substitute into the first to obtain
∂λ ∂λ /∂p j ∂xi
=
∂m ∂xi /∂p j ∂m
i = 1, . . . , n.
Since an exactly similar equation holds for good j, we have
∂λ /∂p j ∂xi
∂λ ∂λ /∂pi ∂x j
=
=
∂m ∂x j /∂pi ∂m ∂xi /∂p j ∂m
i, j = 1, . . . , n,
Thus, we can write
∂x j ∂λ ∂xi
∂xi ∂λ ∂x j
=
∂p j ∂pi ∂m
∂pi ∂p j ∂m
i, j = 1, . . . , n,
i ≠ j.
We now make use of the following facts:
∂2 u
∂2 u
∂
∂λ
=
=
=
( − λxi )
∂pi ∂m ∂m∂pi ∂m
∂pi
∂λ
∂x
= − λ i + xi
∂m
∂m
i = 1, . . . , n,
© Pearson Education Ltd 2007
i ≠ j.
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37
and similarly for ∂λ/∂pj. Then, using the Slutsky equations to substitute for ∂xi /∂pj and
∂xj /∂pi, and the above fact to substitute for ∂λ/∂pi and ∂λ/∂pj, we obtain
∂x ∂λ
∂x ∂x
∂xi ∂λ
∂x ∂x
+ λ i j = s ji − xi j x j
+λ j i .
sij − x j
xi
∂m ∂m
∂m ∂m
∂m ∂m
∂m ∂m
We leave it to the reader to check the tedious details of expanding the brackets,
cancelling terms and regrouping, but if you do this correctly you will obtain:
sij
∂x
∂λ x j
∂x
∂x ∂x ∂x j
− x j i = −λ xi j − x j i i
xi
∂m ∂m ∂m
∂m ∂m
∂m
∂m
(remember sij = sji) and so, defining µ ≡ −λ/(∂λ/∂m) we have
∂xi ∂x j
∂m ∂m
sij = µ
i, j = 1, . . . , n,
i≠j
as required. Note that we require that ∂λ/∂m(= ∂2u/∂m2) < 0, which cannot hold for
ordinal utility functions in general, but does follow from the special form of the utility
function taken here, since from the first-order conditions we have ui′′ = pi∂λ/∂m and ui′′
< 0 by the assumed form of the utility function.
Next, we need to make use of Euler’s Theorem. This states that if the function
f(x1, . . . , xn) is homogeneous of degree zero, then
∑ fi xi = 0.
Now the Hicksian demand functions Hi(p, u) are homogeneous of degree zero in prices,
and so we have the n equations
∑ (∂H /∂p )p = ∑ s p = 0
i
j
j
ij
j
j
i, j = 1, . . . , n.
j
Using the expression derived earlier for sij (i ≠ j) and writing out this equation for any
one i gives
sii pi +
∂xi ∂x j
∂x j
∂xi
∑ µ ∂m ∂m = s p + µ ∂m ∑ ∂m
ii
i
j≠i
j≠i
= sii pi + µ
∂xi
∂xi
1 − pi
∂m
∂m
=0
where we have used
pi
∂x
∂xi
pj j = 1
+
∂m j ≠ i
∂m
∑
from the budget constraint. Then we can solve for
sii = −
µ ∂xi
∂xi
1 − pi
pi ∂m
∂m
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Gravelle and Rees: Microeconomics Instructor’s Manual, 3rd edn
Now we know that sii < 0 and µ > 0. But this then rules out the possibility that
∂xi/∂m < 0, since if that were true the above would give sii > 0. Thus for this special
utility function only normal goods are possible. In that case of course every sij > 0
(i ≠ j), and only Hicksian substitutes are possible. Note another implication of this
special functional form: the Hicksian demand derivatives sii, sij (i ≠ j), as well as the
Marshallian demand derivatives, can all be estimated simply from knowledge of the
income derivatives ∂xi /∂m.
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Gravelle and Rees: Microeconomics Instructor’s Manual, 3rd edn
39
Chapter 4
Further Models of Consumer Behaviour
Exercise 4A
1. The preference assumptions of section 2A are:
Completeness
Transitivity
Reflexivity
Non-satiation
Continuity
Strict convexity
Non-satiation will imply the first behavioural assumption that the consumer spends
all her income.
The second behavioural assumption is that only one commodity bundle x is chosen
by the consumer for each price and income situation. This implies there cannot be
straight stretches of indifference curves, which is ensured by the assumption of strict
convexity.
The third behavioural assumption is that there exists one and only one price and
income combination at which each bundle is chosen. This rules out kinked indifference
curves or curves that are not continuous.
The fourth behavioural assumption, that of consistency, is implied by transitivity of
preferences.
2. If
MI =
p1 x1 p1 x 0
<
= LP
p0 x 0 p0 x 0
this means that at prices in the current period the bundle x1 is cheaper than the bundle
x0. We know the consumer spends all her income. The fact that she chose x1 in the
current period merely reveals that she could not afford x0, it does not say anything
about her preferences. Similarly,
MI =
p1 x1 p1 x1
>
= PP
p 0 x 0 p 0 x1
implies that 1/(p0x0) > 1/(p0x1) or p0x0 < p0x1. The fact that an individual chose x0 at prices
p0 is an indication that she cannot afford x1 at the base period prices. Again, we
© Pearson Education Ltd 2007
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Gravelle and Rees: Microeconomics Instructor’s Manual, 3rd edn
Fig. 4A.1
Fig. 4A.2
learn nothing about her preferences over x0 and x1. In Fig. 4A.1 the consumer is
observed to choose x1 when the budget line is NN which represents income M1 and
prices p1. This tells us nothing about whether the bundle x1 is preferred to x0 as x0 is not
affordable at prices p1 and income M1. In this case p1x1 < p1x0 so that
MI =
p1 x 1
p1 x 0
<
= LP.
p0 x 0
p0 x 0
In Fig. 4A.2 the individual chooses x0 on budget line MM which represents prices p0
and income M0. Since x1 is not affordable we do not know what her preferences are for
x1 versus x0. Since p0x0 < p0x1 then
MI =
p1 x1 p1 x1
>
= PP
p 0 x 0 p 0 x1
and we know nothing about which bundle is preferred.
© Pearson Education Ltd 2007
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41
3. The MI and LP indices are calculated using average consumption bundles for the
group of consumers,
p1
n
MI = 0
p
n
∑x
s1
∑x
s0
1
∑x
s0
∑x
s0
p
n
LP = 0
p
n
s
s
s
s
Here, n is the number of consumers in the group. The n’s can be cancelled so that this
weighting scheme is no different than using the sum of consumption bundles.
Conditions (a) or (b) given in the text must hold for MI > LP to imply that all consumers
are better off.
4. The Paasche price index for one consumer is
PP =
p1 x1
.
p 0 x1
For a group of individuals we can use the sum of consumption bundles,
PP =
p1 ∑ s x s1
p 0 ∑ s x s1
MI =
p1 ∑ s x s1
.
p0 ∑ s x s 0
Assume MI < PP and divide both sides by p1 ∑s xs1 to give 1/(p0 ∑sxs0) < 1/(p0 ∑s xs1) or
p0 ∑s xs0 > p0 ∑s xs1. Taking the case of two consumers a and b, we can write
p0xa0 + p0xb0 ≥ p0xa1 + p0xb1.
(4.1)
This condition implies that at least one of p0xa0 > p0xa1 or p0xb0 > p0xb1 holds, but both
inequalities do not necessarily hold. We cannot infer that both consumers are worse off
in the current period, only that at least one of them is.
Assume that the bundles bought by the consumers at given prices are proportional,
i.e. xa1 = kxb1 and xa0 = kxb0. We can rewrite (4.1) as
p0kxb0 + p0xb0 > p0kxb1 + p0xb1
or
(1 + k)p0xb0 > (1 + k)p0xb1.
This implies p0xb0 > p0xb1 so that individual b is worse off in the current period. Similarly,
we can show that p0xa0 > p0xa1; both individuals are worse off. As discussed in the text,
for consumers to have equiproportionate expenditure patterns for all price vectors we
must have either identical consumers with identical income or consumers with identical
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homothetic preferences. These remarks hold whether we use average or total
consumption bundles in the price indices.
5.
PQ =
p1 x1
p1 x 0
If PQ ≤ 1 then p1x1 ≤ p1x0 implying that the bundle chosen in the current period, x1 is
cheaper at current prices than x0. Thus we do not know whether x1 is actually preferred
to x0 because x0 cannot be purchased at p1 given income M1(= p1x1).
LQ =
p0 x1
p0 x 0
If LQ ≥ 1, then p0x1 ≥ p0x0 and again we do not know if x0 is preferred to x1 because x1
cannot be purchased at income M0 (= p0x0) and prices p0.
If we cannot tell if one individual is better or worse off we certainly cannot say
whether a group of consumers have a better or worse standard of living.
6. For an individual pensioner to be better off we must have
MI =
p1 x1 p1 x 0
≥
= LP
p0 x 0 p0 x 0
or
p1 x 0
p1 x1 ≥ p0 x 0 0 0 .
px
The government increases individual pensioner’s income in proportion to the rise in LP.
Thus,
p1 ∑ x s1
p1 x1 = p0 x 0 0 s s 0
p ∑s x
where the Laspeyres index is assumed to be calculated by the sum of consumption
bundles.
However, we cannot tell whether this government scheme will leave an individual
pensioner better off. If the individual’s LP measure rose by more than the total index,
i.e.,
p1 x 0
p1 ∑ s x s1
>
p0 x 0 p0 ∑ s x s 0
then for the individual MI might be less than LP and we cannot say whether the
individual has been made better off in the current period compared to the base period.
We may note that total pensioners’ money income will increase by the rise in the
total LP index, so that
p1
∑
s
x s1 = p 0
p1 ∑ x s 0
x s0 0 s s0 .
p ∑s x
s
∑
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From this we can conclude that at least some pensioners are made better off.
If the government raised pensioners’ income by the increase in the Paasche index
then total money income in the current period would be as follows
p1
∑
x s1 = p 0
s
p1 ∑ x1
x s0 0 s 1
p ∑s x
s
∑
or
p1 ∑ s x s1 p1 ∑ s x1
=
p0 ∑ s x s 0 p0 ∑ s x1
or MI = PP.
As for the preceeding example, we cannot infer whether individual pensioners are
better or worse off. But we do know that for some individuals at least MI = PP, so that
some pensioners are definitely worse off.
If prices fell, the reverse scheme would be to reduce pensioner’s income
proportionally to the fall in prices. Our previous conclusions hold for a price increase or
decrease.
Exercise 4B
1. (a) In Fig. 4B.1, the upper boundary of the feasible set is defined by the wage line R
M1 which plots income as a function of labour supplied M1 = R + w1z, given the wage
rate w1.
1. (a) (i) A fixed proportional income tax (t) would change the wage line:
M2 = R + w1(1 − t)z
The slope of the line is reduced to w1(1 − t) lowering the boundary of the feasible set
(wage line R M2).
1. (a) (ii) Overtime payments imply that a higher wage rate w′ is paid for hours in
excess of a certain value z*. The constraint is then given by the relation
M = R + wz
z ≤ z*
M = R + wz* + w′(z − z*)
z > z*.
This is graphed in Fig. 4B.2. Note that the feasible set is no longer convex and two local
optima (one of which may not be global) are possible.
1. (a) (iii) Unemployment benefit is paid when z = 0, but is not paid if z > 0. This
introduces a discontinuity in the relation between M and z, and a nonconvexity in the
budget constraint. Thus in Fig. 4B.3, M = R + u for z = 0 and M = M + wz for z > 0,
where u is unemployment benefit. This implies that a local optimum may not be global
(for example, a tangency point in the interior of the diagram may involve a lower
indifference curve than that passing through R + u).
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Fig. 4B.1
Fig. 4B.2
Fig. 4B.3
1. (a) (iv) Fixed hours of work reduce the feasible set to two points: z = K, the fixed
number of hours, or z = 0.
1. (b) (i) If the labour supply curve is expressed as a relation between hours worked
and the gross wage, then imposition of a proportional tax shifts the labour supply curve
upward by the amount of the tax (compare the standard analysis of a tax on a good
produced in a competitive market). However, it is more usual to regard the gross wage
as an exogenous constant and to take hours worked as a function of the net of
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Fig. 4B.4
tax wage U = (1 − t)w. Then variations in the rate of tax t cause movements along the
labour supply curve rather than a shift in it.
1. (b) (ii) Labour supply now becomes a function of two wage rates, the standard wage
w and the overtime wage w′. Some care must then be taken in discussing the labour
supply curve. Fig. 4B.4 illustrates some possibilities. In (a) of the figure the individual
is initially in equilibrium earning the standard wage and increases in this cause
straightforward reductions in labour supply. In (b), on the other hand, an increase in the
standard wage, with the overtime wage unchanged (αβ is parallel to α ′β ′), causes a
jump in labour supply to an equilibrium at the overtime wage. In (c), increases in the
standard wage cause reductions in labour supply, with the overtime wage unchanged,
because of an income effect. Finally, in (d), increases in the overtime wage reduce
labour supply and a jump to an equilibrium at the standard wage cannot take place (for
an increase in the overtime wage: a jump could occur for a decrease).
1. (b) (iii) The effect of unemployment benefit is to introduce a discontinuity into the
labour supply function. Fig. 4B.5 illustrates. The indifference curve I0 passes through the
point R + u, and at a wage rate of w0 the individual would be just indifferent between
setting z = 0 and receiving u and setting z = z0 and receiving R + w0z0. At higher wage
rates she will work, at lower wage rates she chooses unemployment.
1. (b) (iv) With fixed hours of work the individual will in general be at a wage-hours pair
that is off her labour supply curve – only by chance will the wage and fixed hours of
work correspond to a point of tangency in the indifference curve in the diagram.
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Fig. 4B.5
2. Two interpretations of “the same tax revenue” are possible: (a) the same tax revenue
if the same amount of labour is supplied or (b) the same tax revenue at the new amount
of labour supplied. Interpretation (a) gives an unambiguous answer if the individual has
no non-labour income, as Fig. 4B.6 shows.
In the figure, 0M1 is the pre-tax budget constraint and 0M2 the budget constraint with
a proportional income tax. With the proportional tax the initial point chosen is A and
the amount of tax paid is equal to the vertical distance between 0M2 and 0M1 at ZA. All
points along the line TT yield the same tax revenue for the government.
With a progressive tax the marginal rate of tax increases with income and therefore
with the amount of labour supplied. Hence the slope of the after-tax budget constraint
must decrease with z. If the progressive tax schedule is to yield the same tax revenue at
a constant labour supply it must be of the form 0M3 which cuts 0M2 from above at A.
Hence the new optimum choice must be to the left of A, at B say.
With the second interpretation of “unchanged” tax revenue the progressive tax
schedules which yield the same tax revenue at the new amount of labour could be like
0M4 or 0M5 depending on the individual’s preferences. With 0M4 labour supply is reduced
at the new optimum where I4 is tangent to 0M4 at C. With different preferences a
constant tax revenue might require a tax schedule like 0M5 in which case labour supply
is reduced at D where the indifference curve Î5 is tangent to 0M5.
Note that the second interpretation of an unchanged tax revenue is less sensible
since it requires that tax authorities have detailed knowledge of preferences.
3. In Fig. 4B.7, MT represents the target income. The individual is indifferent between
points that achieve the target income, thus the indifference ‘curve’ is a horizontal line.
The ‘target income’ hypothesis gives insufficient information to allow any other
indifference curves to be drawn. Since the individual supplies just enough labour to
meet the target income the higher the wage rate the less will be the labour supplied. The
labour supply curve is negatively sloped, and, since wz = MT = constant, it is in fact a
rectangular hyperbola.
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47
Fig. 4B.6
Fig. 4B.7
4. (a) If all prices increase by a factor k the constraint on the labour supply decision
in problem [B.5] in the text is kpy = wz + R. This is equivalent to the constraint
py = (w/k)z + R/k. Thus the effect of the price change is equivalent to a reduction
in the real wage rate and of real unearned income by a factor 1/k. In terms of text
Fig. 4.2(a) the intercept of the wage line is shifted down to R/k and its slope is reduced.
If leisure is a normal good the effect of the increase in consumer prices on labour
supply is ambiguous: the substitution effect of the reduction in the real wage rate will
reduce labour supply but the income effect of the reduction in the real wage and the
reduction in real unearned income will tend to increase it.
4. (b) If all prices and the money wage rate increase by a factor k the constraint on the
labour supply decision is kpy = kwz + R which is equivalent to py = wz + R/k. In
Fig. 4.2(a) the effect is to shift the intercept of the wage line downwards but there
is no change in the real wage and hence no change in the slope of the wage line. Labour
supply is increased or reduced depending on whether leisure is a normal good or an
inferior good.
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5. When preferences are weakly separable between all goods and leisure the utility
function [C.1] can be written
u = u(x, L) = g(r(x), L)
where r(x) is the sub-utility function for the vector of consumption goods. For a given
L utility is maximized subject to the budget constraint px + wL ≤ R + wT ≡ F only if r(x)
is maximized subject to px ≤ F − wL. The Lagrangean for this first-stage problem is
r(x) + λ(F − wL − px)
and the first-order conditions on the n consumption goods, assuming a non-corner
solution for simplicity are
ri(x) − λpi,
i = 1, . . . , n
where ri = ∂r/∂xi.
The optimal Marshallian demands are xi* ( p, F − wL) and the indirect sub-utility
function is
R(p, F − wL) = r(x*(p, F − wL)).
Note that the amount of leisure L affects the optimal demand for the consumption
goods only via its effect on the amount of income available to be spent on them. The
indirect sub-utility function R(p, F − wL) has all the properties of the indirect utility
functions of section 3B. In particular Roy’s Identity holds:
∂R( p, F − wL )
= − R F ( p, F − wL) xi* ( p, F − wL )
∂pi
which yields a result we will use below:
*
R p F = R Fp = − R FF xi* − R F xiF
i
i
(4.2)
The second stage of the utility maximization problem is to choose L so as to maximize
g(R(p, F − wL), L) = u(x*(p, F − wL), L) = G(L; p, F, w)
and the first-order condition is
GL(L; p, F, w) = −gr(R, L)wRF + gL(R, L) = 0
(4.3)
and the optimal demand for leisure is L*(p, F, w).
Consider first the effect on the demand for leisure of an increase in F. Using the
simple comparative-static procedure on (4.3) we get
∂L* − GLF g rr R F wR F + g r wR FF − g Lr R F
=
=
∂F
GLL
GLL
(4.4)
Now using the same procedure for the effect of an increase in the price of the i’th
consumption good, we have, using (4.3),
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* − g Lr R F xi*
GLp = g rr R F wR F xi* + g r wR FF xi* + g r wR F xiF
(4.5)
i
*
= xi*[w( grr RF RF + gr RFF ) − g Lr RF ] + gr wRF xiF
*
= − xi*GLF + g r wRF xiF
Using the first-order condition (4.3) to substitute in the last term, the Slutsky equation is
∂L* − GLp
∂L* g L
=
= − xi*
+
xiF
∂pi
GLL
∂F GLL
i
(4.6)
The first part of (4.6) is the income effect of a change in the price of the i’th
consumption good, showing the effect of the reduction in real income or purchasing
power on the demand for leisure. The second term is the substitution effect. The
assumption of weak separability means that the substitution effect of a change in the
price of the consumption good arises via the change in the total expenditure on
consumption goods. If the i’th good is normal then the cross-substitution effect leads to
a reduction in the demand for leisure (remember GLL is negative from the second-order
condition). Thus whether leisure and the i’th consumption good are Hicksian
complements or substitutes depends on whether the i’th consumption good is normal or
inferior.
Exercise 4C
1. (a) For the two good case the absolute value of the slope of the full budget constraint
(F) is
s≡−
dx2 ( p1 + wt1 )
=
dx1 ( p2 + wt2 )
We want to know what happens to s when w increases. Taking its derivative with
respect to w:
∂s t1 ( p2 + wt2 ) − ( p1 + wt1 )t2
=
∂w
( p2 + wt2 )2
=
t1 p2 − t2 p1
( p2 + wt2 )2
This derivative will be less than zero if t1p2 < t2p1 or,
t1 p1
<
t2 p2
as in [C.5] of the text. Hence if good 1 is less time intensive than good 2, an increase in
the wage rate will reduce the absolute value of the slope of the full budget line, i.e. it
will become flatter.
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1. (b) Since w influences all the full prices (ρi = pi + wti) and the full income (F = R +
wT) its effects are more complicated than a change in income or a single price.
Consider first the effect of w on the consumer’s maximized utility. Allowing for its
effects on the full prices and full income the partial derivative of the indirect utility
function [C.9] with respect to w is
∂v( ρ, F )
=
∂w
∂v ∂ρ i
∂v ∂F
∑ ∂ρ ∂w + ∂F ∂w = −∑v x t + v T = v [T − ∑t x ] = v z
F
i i
F
F
i
i
F
i
where we have used Roy’s identity for the effects of the full prices on v : vi =
∂v(ρ, F)/∂ρi = −vFxi(ρ, F). This is just another version of Roy’s identity: the effect
on utility of an increase in the price of a commodity (labour) that the individual sells
is the quantity sold (z = T − ∑tixi), which is the rate at which income increases with
price, times the marginal utility from additional income.
The effect of w on the consumer’s demand for goods is also complicated because w
alters all full prices and full incomes. Recall from section 3B that the Marshallian and
Hicksian demands are equal if the required utility level in the full cost minimization
problem is set at the maximized utility achieved in the utility maximization problem:
xi(ρ, F) = hi(ρ, v(ρ, F))
Hence the effect of full income on the Marshallian demand for good i is
xiF(ρ, F) = hiuvF
and the effect of w on xi is
∂xi ( ρ, F )
=
∂w
∂hi ∂ρ j
∂v
∑ ∂ρ ∂w + h ∂w = ∑ h t + h v z = ∑ h t + zx
iv
j
j
ij j
iu
F
j
ij j
iF
(4.7)
j
where hij = ∂hi/∂ρj is the cross substitution effect of the full price ρj on the constant
utility (Hicksian) demand for good i. The effect of w on the demand for good i has been
decomposed into an income effect (zxiF) and the sum of n substitution effects. In
general, the effect of an increase in the wage on the demand for goods is ambiguous for
two reasons. First, the good may be normal or inferior so that the increase in real
income or utility caused by the increase in the wage rate may increase or reduce
demand. Second, the change in the wage rate alters all the relative full prices and so
leads to n substitution effects, rather than just one as in the case of change in a single
price.
If we assume that there are just two goods we can get some insight into the
substitution effects of the wage increase. Condsider the sum of the substitution effect
terms in (4.7) in the case of good 1:
h11t1 + h12t2
(4.8)
The Hicksian demand functions are homogeneous of degree zero (see section 3A) so
that we can use Euler’s Theorem (see section 5C) to establish
h11ρ1 + h12 ρ2 = 0
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This enables us to substitute −h11ρ1/ρ2 for h12 in (4.8) to get
h11t1 − h11
t
t
ρ1
t2 = h11 1 − 2 ρ 1
ρ2
ρ1 ρ 2
(4.9)
Since the own full price substitution effect h11 is always negative, the wage substitution
effect is also negative if good 1 has a greater time intensity than good 2 (recall our
discussion of [C.6]). (4.9) indicates that in the two-good case the wage substitution
effect on good 1 is proportional to the own full price substitution effect on good 1. The
full price of good 2 falls relative to the full price of good 1 when w increases if good 1 is
more time intensive than good 2 and so the wage substitution effect is in the same
direction as the own full price substitution effect, leading to a decrease in demand for
good 1. If good 1 is less time intensive than good 2 an increase in w is equivalent to
reduction in the relative full price of good 1 and so the wage substitution effect would
increase the demand for good 1.
For good 2 the substitution effect would be
h21t1 + h22t2
From Euler’s Theorem we have
h21ρ1 + h22 ρ2 = 0
or
ρ2
ρ1
h21 = h22
Substituting this into the substitution effect:
h22t2 − h22
t
t
ρ2
t1 = h22 2 − 1 ρ 2
ρ1
ρ 2 ρ1
We know that h22 < 0 and with good 1 more time intensive than good 2
t2
ρ2
−
t1
ρ1
<0
The substitution effect is positive. An increase in the wage rate causes more of good 2 to
be consumed at a constant utility.
The effect of an increase in the wage rate on labour supply is given by the Slutsky
equation
∂z( p, F )
=−
∂w
i
∑ ∑ t h t + z( ρ, F )z ( ρ, F )
i
ij j
F
j
The last term is the income effect and its sign is ambiguous. The first term is the ownsubstitution effect which for the two-good case can be written
t1h11t1 + t1h12t2 + t2h21t1+ t2h22t2 = t12 h11 + 2h12t1t2 + t22 h22
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Fig. 4C.1
Fig. 4C.2
since we know h12 = h21. For a strictly concave utility function h(p, u) is strictly concave
in prices and the above quadratic form is negative definite. This means the own wage
substitution effect is positive.
2. (a) The equation of the full budget line F is
R − wT
p + wt1
x2 =
− 1
x1
p2 + wt2
p2 + wt2
If unearned income increases F will shift out in a parallel fashion. In Fig. 4C.1 the
optimal bundle changes from A to B, but we cannot in general determine whether
consumption of x1 or x2 will increase or decrease. This depends on whether they are
normal or inferior goods.
2. (b) If p1 increases the slope of F becomes steeper (F1) in Fig. 4C.2 (a). If p2 increases
the slope of F becomes less steep and its intercept on the x2 axis falls, as in Fig. 4C.2 (b).
In both cases the change in the optimal bundle is ambiguous and depends on
preferences.
2. (c) An increase in t1 will have the same impact on the feasible set as an increase in p1.
Likewise a change in t2 is comparable to a change in p2. Again, the impact on the optimal
bundle depends on the structure of the indifference curves.
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3. This question can be answered simply by noting that
∂ρ i
= 1;
∂pi
∂ρ i
= w,
∂ti
where ρi is the full price of good i. Putting the problem in the standard form max
u(x) s.t. ρx = F yields the indirect utility function v(ρ, F), the expenditure function
F = f(ρ, u), the Marshallian demands xi = Di(ρ, F) and the Hicksian demands
xi = Hi(ρ, u). Then we have
∂v
∂v ∂ρ i
=
= − λxi
∂pi ∂ρ i ∂pi
where λ is the marginal utility of full income;
∂v
∂v ∂ρ i
=
= − λwxi .
∂ti ∂ρ i ∂ti
From the Slutsky equation
∂Di ∂H i
∂Di
=
− xi
∂ρ i ∂ρ i
∂F
we have
∂Di ∂Di ∂ρ i ∂Di
=
=
∂pi ∂ρ i ∂pi ∂ρ i
and the Slutsky equation is unchanged. We then have
∂Di ∂Di ∂ρ i
∂H i
∂Di
=
=w
− wxi
.
∂ti
∂ρ i ∂ti
∂ρ i
∂F
4. Since ρi = pi + wti the effect of an increase in the money price on the demand for good
i is
∂xi ( ρ, F ) ∂xi ( ρ, F ) ∂ρ i ∂xi ( ρ, F )
=
=
∂pi
∂ρ i
∂pi
∂ρ i
and so the elasticity of demand for good i with respect to its money price is
ep =
i
∂xi pi ∂xi ρ i pi
p
=
= eρ i .
∂pi xi ∂ρ i xi ρ i
ρi
i
Thus if two individuals have the same full price demand elasticity the individual with
the greater wage rate will have the smaller money price demand elasticity since ρi/pi
will be smaller for her.
5. The model in section 4C of the text does not consider the time spent travelling to
work explicitly. One approach might be to regard ‘travelling to work’ as one of the
consumption goods xi, and so a reduction in its ti causes a fall in its full price ρi. We
could then consider compensating and equivalent variations associated with this fall.
However, a more realistic approach is to take time spent travelling to work as some
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fixed value tz > 0 (given that z > 0) and so the time constraint in the problem could be
written
∑tI xi + z = T − tz = }.
The model could be analyzed as before, with } taking the place of T. Thus full income
now takes the value | = R + w}. We then have the indirect utility function v( ρ, | ),
where ∂v/∂| = λ is the marginal utility of full income. It follows that the effect of a
change in tz on utility is given by:
∂v ∂v ∂ |
=
= − λw.
∂tz ∂ | ∂tz
If then tz falls from some initial level tz0 to some new level tz1 , we can define CV and EV
measures by
v(ρ, | 0) = v(ρ, | 1 − CV)
v(ρ, | 1) = v(ρ, | 0 + EV).
But then obviously
CV = EV = | 1 − | 0 = w(tz0 − tz1 )
and so we simply have to value the fall in commuting time at the individual’s wage rate
(as long as this does not change).
Exercise 4D
1. (a) Let xo denote the market good used as an input in domestic production and write
the household production function as
y = h(t1, t2, xo)
The budget constraint must be rewritten as
2
∑
i=1
2
xi + px o =
∑ w (T − t )
i
i
i=1
where p is the price of xo. The rest of the model is as before. The optimality condition
with respect to xo can be written as
ρh3(t1, t2, xo) = p
so xo is used to the point at which its marginal value product in household production
equals its market price. The rest of the model is as before.
1. (b) Let Li denote i’s leisure, the direct consumption of own time. The utility functions
are rewritten as
ui = ui(xi, yi, Li),
i = 1, 2
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It is useful now to introduce zi as i’s market labour supply and to write the individual
time constraints explictly as
ti + zi + Li ≤ T,
i = 1, 2
In the Lagrange function in [D.4], replace the previous utility functions with the new,
write income as wizi so that the budget constraint is ∑iwizi − ∑ixi, and add the term
∑ τ (T − t − z − L )
i
i
i
i
i
where τi are Lagrange multipliers. We consider only the possibility that the secondary
earner may supply no market labour, so we assume all variables other than possibly z2
are stictly positive at the optimum. The first order conditions on the x, y, L, t and z are
now
u1x = λ = σ ux2
uy1 = µ = σ uy2
u1L = τ 1
σ uL2 = τ 2
µhi (t1, t2) = τi,
i = 1, 2
λw1 = τ1
λw2 ≤ τ2, z2 ≥ 0, z2(λw2 − τ2) = 0
If z2 > 0 at the optimum, nothing much changes, we just have some additional conditions
uLi
= wi
uxi
uLi wi
=
uyi
ρ
which are just standard equalities of marginal rates of substitution and price ratios
(remember the price of x is 1). The price of leisure in this case is the wage. However,
more interesting is the case of a corner solution where z2 = 0, since then we have
uLi
= ρh2 (t1 , t2 ) ≥ w2
uxi
uLi
w
= h2 (t1 , t2 ) ≥ 2
uyi
ρ
In this case, the opportunity cost of leisure may no longer be the wage, if the secondary
earner is not supplying market labour, but rather is in general given by the marginal
value of time spent in household production. The second condition says that the
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marginal rate of substitution between leisure and the domestic good is equated to the
secondary earner’s marginal product of time spent in producing the domestic good.
1. (c) This is largely a matter of notation. Write x = [x1, . . . xn] as a vector of market
goods, y = [y1, . . . , ym] as a vector of household goods, and H(y, t1, t2) as the household’s
transformation function, giving the technological possibilities of producing the
household goods with time. There will now be a vector of implicit prices of household
goods ρ = (ρ1, . . . , ρm) and we also need the vector of prices of market goods p =
(p1, . . . , pm). If we assume that the transformation function can be represented by the m
production functions
yj = hj(t1j, t2j)
i.e. there is no joint production, then the model is essentially as before. However, there
is in general the possibility of joint production, where a given time input produces more
than one good (e.g. looking after the baby while cooking dinner).
2. Note that, given positive market labour supplies of both household members, an
alternative way of solving the model is to derive the cost function for the household
good by solving
min C =
ti
∑w t
i i
s.t. y ≤ h(t1, t2)
i
for given y. This results in a cost function C(w1, w2, y) and it is a standard result (see
section 6B) that if h(., .) is homogeneous of degree 1, this can always be written as
C(w1, w2, y) = c(w1, w2)y
where c(w1, w2) is the cost of producing 1 unit of y. Thus we set ρ = c(w1, w2).
3. If the market and domestic goods are perfect substitutes, we can rewrite the utility
functions as ui(xi + yi) and solve just as before. We obtain from the first order
conditions the condition
ρ=
wi
=1
hi (t1 , t2 )
i = 1, 2
Recall that the price of the market good is 1. Thus the implicit price of the domestic
good at the optimum is always equal to that of the market good. The household
produces the domestic good up to the point at which its marginal cost is equal to the
market price. If this is less than it wants to consume at that price, it buys the rest in,
if more, it sells the surplus on the market. This illustrates nicely that the household
in this model can be thought of as a small economy. What we have just described
is precisely the optimum of a small country in international trade producing and
consuming a good which is available on the world market at an exogenously given
price.
Suppose that the bought in domestic labour d is a perfect substitute for
the secondary earner’s domestic labour. Then the production function becomes
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h(t1, t2 + d). The price of d is w < w2. Intuitively we would expect that the secondary
earner will supply no domestic labour if a perfect substitute is available at a lower wage,
and this is confirmed by the relevant first order conditions, which are
µh2(t1, t2 + d) ≤ λw2, t2 ≥ 0, t2[µh2(t1, t2 + d) − λw2] = 0
µh2(t1, t2 + d) − λw = 0
Thus if w < w2 we must have t2 = 0. However if the two forms of labour are less than
perfect substitutes then we would write the production function as h(t1, t2, d) and it is
quite possible to have all three types of labour input positive at the optimum.
4. The value of household production in this model is ρy. Unfortunately, the implicit
price of the household good is usually unobservable, since we do not have data on
domestic output (time use studies give detailed data on domestic inputs). However,
consider the budget constaint in [D12]. This implies that
ρy = T ∑ wi + π − ∑ xi
i
i
If we are able to observe the household’s wages and value of consumption of market
goods, which seems reasonable, we could take the difference between the value of the
household’s time endowment and its consumption as a measure of the value of
household production. This could involve an error. There are three possible cases:
(a) The household production function has constant returns to scale. In that case it is
straightforward to show that π = 0. Thus there is no error.
(b) Household production has diminishing returns to scale. In that case π > 0 and the
measure would understate the true value of household production.
(c) Household production has increasing returns to scale. In that case π < 0 and the
measure would overstate the true value of household production.
If we are prepared to accept that in practice returns to scale in household production
are roughly constant, then we have quite a simple practical measure of the value of
household production, at least in this simple model.
5. If the households face the same wage rates and have identical preferences then there
are only 2 possible reasons for the difference in labour supply of the secondary earner:
differences in distributional preferences; and differences in the household production
function. For example, if the secondary earner’s leisure is a normal good and she
obtains a higher income share in one household than another, then that household
could show a lower secondary earner labour supply. However, we focus here on the
difference in production functions. For simplicity, assume the primary earner in each
household supplies labour only to the market, and that the household production
functions are simply
yh = khth
h = 1, 2
with k1 > k2 and th the secondary earner labour supply to domestic production in
household h. Here we can think of the kh as productivity parameters which reflect the
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physical and human capital available to the household. It follows that the implicit price
of the household good is
ρh =
w2
kh
which is the opportunity cost of the time needed to produce 1 unit of the domestic good.
The household budget constraint can be written as
∑x + ρ y = T∑w
ih
i
h
h
i
h = 1, 2
i
This tells us immediately that household 1, with the higher domestic productivity, must
have the higher utility possibilities, since its budget constraint lies everywhere above
that of household 2 except at the point yh = 0. The slopes of the budget constraints in
(x, y)-space are respectively 1/ρ1 > 1/ρ2. An income measure of the difference in their
welfares would be the compensating or equivalent variations defined by the achieved
utility levels of the two households: how much income could we take away from
household 1 to put its members on the same utility levels as those reached at the
optimum of household 2; or how much income must we give household 2 to put its
members on the same utility levels as those in household 1? Note that we could not use
the difference in their money incomes from market labour supply as a measure of the
difference in welfare, for the simple reason that the secondary earner’s labour supply,
and therefore market income, in household 1 could be greater or less than the
secondary earner’s labour supply and income in household 2. To see why, note the we
can write market labour supply of the secondary earner as
lh = T − th = T − yh(ρh)/kh
That is, market labour supply is total time minus the amount of time required to meet
the household’s demand for the domestic good, which is a function of the relative price
of that good. Differentiating with respect to kh gives
∂lh yh
ρ ∂y
y
= (1 + h h ) = h (1 − eh ) ⱀ 0
∂kh kh
yh ∂ρ h
kh
where eh is the price elasticity of demand for the domestic good. This shows the two
opposing effects of, say, an increase in the household’s domestic productivity. The time
input required to produce a given amount of the domestic good would fall, and this
would increase the time allocated to the market. But the fall in the opportunity cost of
the domestic good would (normally) increase the demand for it, and this would tend to
increase time spent in domestic production and reduce market labour supply. Thus if
demand for the domestic good is inelastic (eh < 1), the former effect dominates, while if
this demand is elastic (eh > 1) the latter effect dominates. The important consequence
of this simple demonstration is that a household’s total market income may be a poor
indicator of its welfare relative to another household when household production is
taken into account and secondary earner labour supply varies across households.
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6. If the domestic good is a household public good, we can write the utility functions as
ui = ui(xi, y)
i = 1, 2
where y is the output of the public good. The production function is as before. The
relevant optimality condition will now become, with the same notation as before
uy1 + σ uy2 = λρ
This corresponds to the usual condition of equality between the sum of marginal
utilities of the public good to its marginal cost (see Section 13B). Note that the overall
demand and total output of the good, and hence the allocation of time, will depend, if
tastes differ, also on the utility distribution with in the household, as expressed by the
value of the shadow price of the utility constraint, σ.
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Chapter 5
Production
Exercise 5A
1. The casual system was clearly ‘co-ordination by the market’ in that the employment
contract extended over the period of just one job – the unloading of a ship – and new
contracts were entered into each time a ‘job’ appeared. It existed in the ports industry
because of the day to day variability in the flow of jobs which depended on the rate at
which ships arrived to be unloaded and reloaded, and on the non-storability of output.
2. If each individual contracts with every other individual the required number of
contracts is
n!
n( n − 1)( n − 2)! n( n − 1)
=
=
( n − 2)!( 2 !)
( n − 2)!( 2 !)
2
If there is coordination by a central coordinator only n contracts are required.
3. (a) The small size of the group and the artistic nature of the work might well make
the producers’ cooperative more effective than the conventional firm. Shirking would be
discouraged through peer pressure and the presence of shirking would be easy to
detect. Individuals could be rewarded by receiving the profits on their own sales. Each
member of the cooperative would benefit from access to a central studio and potterymaking supplies. A disadvantage might be difficulties in agreeing on how to divide up
the costs of operation and generally deciding on the level of investment to make in a
studio and equipment.
The conventional firm might be less effective because artists on a fixed wage would
have less incentive to be productive. However, the firm would have no difficulty
planning the scale of operation and the level of investment to be made.
3. (b) A producers’ cooperative would be difficult to organize. Workers could be paid
according to the number of motor cycle parts produced, but the quality control might be
a major problem. Unlike the pottery workers in (a) workers could not easily identify
their own output so that ‘pride of work’ would be diminished. Workers would have
an incentive to shirk if there were little chance of being detected. A penalty scheme
could presumably be devised whereby a person’s pay would be docked if she were
discovered to be making defective products. Major problems would be likely to exist in
getting any agreement on production plans.
The conventional firm has the advantage of being able to make decisions quickly, but
also suffers from the problem of quality control. A centrally controlled firm is perhaps in
a better position to design penalty schemes to discourage shirking.
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3. (c) A producers’ cooperative would have the same problems described in (b). The
coordination of production decisions would be very difficult without the central
authority of a firm. Typically, a manager would be appointed to run the cooperative
subject to some sort of democratic oversight by the workers. The problem here is to
trade off the manager’s accountability to the workforce against the need for effective
decision-taking.
Exercise 5B
1. Production is output-efficient when the output produced is the maximum possible
from the input combination. Production is technically efficient when it is impossible to
reduce the use of any input without reducing output. Output-efficiency is necessary for
technical efficiency since, if the firm is not producing the maximum possible output
from an input combination, it can reduce the use of one or both inputs without reducing
output. If the firm is output-efficient it is on an isoquant. If the firm is operating on the
upward sloping segment of an isoquant (see Fig. 5.2 in the text), then it is output
efficient but not technically efficient since it could move ‘south-west’ down the
isoquant, reducing the use of both inputs and producing the same output.
2. The contours of a function f(z1, z2) have the slope dz2/dz1 = −f1/f2 which is positive only
if the partial derivatives f1, f2 are of opposite sign. We did not rule out the possibility that
marginal product could be negative so that isoquants could be positively sloped. In
chapters 2 to 4 we assumed that marginal utilities were always positive so that
indifference curves were always negatively sloped.
3. If a firm’s technology is represented by a production function y = f(z1, z2), then the
only permissible transformation of the production function is proportional: a change of
units in which the output or the inputs are measured. Such transformations cannot alter
the sign of marginal product, only its magnitude. The utility function used in chapters 2
to 4 is an ordinal function, unique only up to a positive monotonic transformation. Thus
any statement about diminishing marginal utility is meaningless since it is always
possible to find another numerical representation of preferences which contradicts it.
4. (a) Fixed Proportions Technology (Leontief). The isoquant map for the single process
fixed proportions production function:
y = min(z1/β11, z2/β12)
is shown in Fig. 5B.1. The economic region is the ray from the origin through the
corners of the isoquants: all points off this ray are technically inefficient since it is
possible produce the same output after reducing the use of one input and holding the
use of the other constant.
4. (b) Process 2 is technically inefficient if β11 < β21 and β12 < β22 so that process 2
requires more of both inputs to produce any specified output. Fig. 5B.2 shows the
isoquants for both production processes for one unit of output. The input vector zi
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Fig. 5B.1
Fig. 5B.2
produces one unit of output when used in process i (i = 1, 2). The vector za = kz1
(0 ≤ k ≤ 1) will therefore produce k units of output and similarly zb = (1 − k)z2 will
produce (1 − k) units of output. Hence the input combination zc which is just the sum of
za and zb can produce one unit of output and is on the unit isoquant. Varying k traces the
negatively sloped portion of the unit isoquant.
4. (c) Fig. 5B.3 illustrates the unit isoquant and input requirement set with
four technically efficient production processes. Processes whose input vectors for
producing one unit of output lie inside the unit output input requirement set generated
by combinations of other production processes are not technically efficient.
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Fig. 5B.3
5. Partially differentiating the Cobb-Douglas production function
y = z1α z2β
with respect to z1 and z2 yields
MP1 = α z1α −1 z2β = α z1α z2β / z1 = αy/z1
MP2 = β z1α z2β −1 = β z1α z2β / z2 = βy/z2
Thus
MRTS 21 =
MP1 α z2
=
MP2 β z1
so that the marginal rate of technical substitution is independent of the output level and
varies only with the input proportions. The isoquants have the same slope along rays
from the origin and the Cobb-Douglas production function is homothetic.
6. Write the Constant Elasticity of Substitution production function as y = f(g(z1, z2))
where g = δ 1 z1α + δ 2 z2α and f = Ag1/α. Partially differentiating with the respect to input i
and using the function of a function rule gives
∂y
∂g A −1
= f′
= g αδ i ziα −1
∂zi
∂zi α
1
α
= Aα δ i A1−α ( g 1/α )1−α ziα −1 = Aα δ i y1−α ziα −1
y
= Aα δ i
zi
1−α
Hence
MRTS 21 =
MP1 δ 1 z2
=
MP2 δ 2 z1
1−α
and so MRTS is independent of output and depends only on the input proportions.
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7. (a) First note that using the implicit function rule on y − f(z1, z1r) = 0 yields
dz1
fz
= g ′( r ) = − 2 1 < 0
dr
f1 + f2 r
dz2 dz1 r
f1 z1
=
= z1 + g ′( r ) z1 =
>0
dr
dr
f1 + f2 r
Now write MRTS as m(z1, z2) to get
dm
dz
dz
( m2 f1 − m1 f2 ) z1
= m1 1 + m2 2 =
dr
dr
dr
f1 + f2 r
Hence
σ=
dr m
dm r
=
( f1 z1 + f2 z2 ) f1
( m2 f1 − m1 f2 ) z1 z2 f2
=
( f1 z1 + f2 z2 ) f1 f2
( m2 f1 − m1 f2 ) z1 z2 f 22
Finally, substituting in the denominator for the partial derivatives of m with respect
to z1 and z2:
m1 =
( f2 f11 − f1 f21 )
,
f 22
m2 =
( f2 f12 − f1 f22 )
f 22
yields the required expression for the elasticity of substitution.
7. (b)(i) The marginal rate of substitution of the Leontief fixed proportions production
function of question 4(a) is zero for r < : ≡ β12/β11, undefined at r = : and infinite for
r > :. Hence the elasticity of substitution is not defined for the Leontief production
function. However, looking ahead to chapter 6, we can use another definition of the
elasticity of substitution in terms of the change in the cost minimizing input ratio to
changes in the input price ratio:
σ=
% change in z2 / z1
d( z2 / z1 ) ( p1 / p2 )
=
% change in p1 / p2 d( p1 / p2 ) ( z2 / z1 )
(5.1)
where p1/p2 is the input price ratio. (Compare [B.8].) As we show in chapter 6, if
the production function is differentiable, a non-corner solution to the problem of
minimizing the cost of producing a given output will have p1/p2 = f1/f2. In such cases the
two definitions of the elasticity of substitution are equivalent. In the Leontief case the
cost minimizing input ratio is always :, provided both input prices are positive. Hence
the elasticity of substitution in (5.1) is zero.
7. (b)(ii) The log of the MRTS for the Cobb-Douglas production function is
ln m = ln α z2 /β z1 = ln α /β + ln r
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so that
σ=
dr m dr m
d ln r
=
=
=1
dm r
r dm d ln m
7. (b)(iii) The log of the MRTS for the CES production function is (see question 6)
ln m = ln δ1/δ2 + (1 − α) ln r and so σ = 1/(1 − α)
7. (c) The marginal products of the linear production function are constant: MPi = ai,
(i = 1, 2). Hence the isoquants are straight lines with slope −a1/a2 and MRTS is constant
at all input ratios: dm/dr = 0, implying that σ = (dr/dm)(r/m) is infinite.
8. (a) With α = 1 the CES production function in question 6 reduces to
y = A(δ1z1 + δ2z2)
(5.2)
which is the same form as the linear production function in question 7(c) with ai = Aδi.
The alert reader will notice that in the definition of the CES production function in
question 6 we required that δ1 + δ2 = 1, so that the coefficients have the same dimension.
However, in (5.2) the coefficients Aδi appear to have the dimension of [output]/[input i].
The answer to this difficulty is that the CES production function could be written as
y = A[δ1(k1z1)α + δ2(k2z2)α]1/α = Ag1/α
(5.3)
where the coefficients ki have the dimension [output]/[input i]. The production
function is now explicitly dimensionally homogeneous. In the definition in question 6
we tacitly chose the units in which the inputs are measured so that the coefficients
ki = 1 and do not appear explicitly in the production function.
8. (b) Choose units so that ki = 1, (i = 1, 2) in (5.3). Take the log of the CES form, with
g = δ 1 z1α + δ 2 z2α , to get
ln y = ln A + (1/α) ln g
which is undefined at α = 0. However, we can use L’Hôpital’s Rule:
lim
α →0
ln g
α
=
limα →0 d ln g /α
limα →0 dα /dα
= lim
α →0
1 dg
g dα
Since
lim dg /dα = lim(δ 1 z1α ln z1 + δ 2 z2α ln z2 ) = δ 1 ln z1 + δ 2 ln z2
α →0
α →0
and limα→0 g = δ1 + δ2 = 1 we have
lim ln y = ln A + δ1 ln z1 + δ2 ln z2
α →0
which is the log of a Cobb-Douglas production function.
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8. (c) Let us use the definition (5.3). Consider an input bundle where
k1z1 ≤ k2z2
Since limα→−∞ δ 1/α = 1, there exists an < < 0 such that
α<<⇒
δ 11/α Ak1 z1 ≤ δ 12/α Ak2 z2
⇒ δ1(Ak1z1)α ≥ δ2(Ak2z2)α
⇒ 2δ1(Ak1z1)α ≥ δ1(Ak1z1)α + δ2(Ak2z2)α = Aαg
⇒ (2δ1)1/αAk1z1 ≤ Ag1/α = y
Hence
lim ( 2δ 1 )1/α Ak1 z1 = Ak1 z1 ≤ lim y
α → −∞
α → −∞
(5.4)
Next, notice that when α < 0
δ1(Ak1z1)α ≤ δ1(Ak1z1)α + δ2(Ak2z2)α = Aαg
which implies
δ 1/1 α Ak1z1 ≥ Ag1/α = y
and so
lim δ 11/α Ak1 z1 = Ak1 z1 ≥ lim y
α → −∞
α → −∞
(5.5)
Hence (5.4) and (5.5) together imply limα→−∞ y = Ak1z1. Repeating the analysis for the
case in which k2z2 ≤ k1z1 we conclude that
lim A[δ1(k1z1)α + δ2(k2z2)α] = A min(k1z1, k2z2)
α → −∞
where the right hand term is the Leontief production function of question 4(a) with
βi1 = 1/Aki.
Notice that since the elasticity of substitution of the CES form is 1/(1 − α) and
limα→−∞1/(1 − α) = 0 we have confirmed the results in question 7 that σ = 0 for the
Leontief production function.
Exercise 5C
1. (a) If y = f(z) is homogeneous of degree t, we have
f(sz) = stf(z)
and differentiating both sides partially with respect to zi gives fi(sz)s = stfi(z) so that
fi(sz) = st−1fi(z)
(5.6)
so that the marginal products are homogeneous of degree t − 1 and vary with the scale
of production unless f(z) is linear homogeneous (t = 1).
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1. (b) Since MRTS is just the ratio of the marginal products (5.6) implies that, for
homogeneous functions, MRTS is independent of the scale of production. The isoquants
have constant slopes along rays from the origin. Compare the discussion of homothetic
preferences in chapter 3.
2. (a) Multiplying all inputs by s > 0 gives
a1sz1 + a2sz2 = s(a1z1 + a2z2) = sf(z)
so that the linear production function has constant returns.
2. (b) The Leontief production function also has constant returns
min(sz1/β1, sz2/β2) = s min(z1/β1, z2/β2) = sf(z)
2. (c) But with the Cobb-Douglas production function
(sz1)α(sz2)β = sα + β z1α z2β = sα+βf(z)
so that there are increasing, constant or decreasing returns as α + β is greater than,
equal to or less than 1.
2. (d) The CES production function has constant returns:
A[δ1(sz1)α + δ2(sz2)α]1/α = A[(δ 1 z1α + δ 2 z2α ) sα ]1/α
= A[δ 1 z1α + δ 2 z2α ]1/α s
= sf(z)
3. Suppose that f(z) is homogeneous of degree t. We must show that there is a linear
homogeneous function ᐉ(z) and an increasing transformation G(ᐉ) such that G(ᐉ(z)) =
f(z). Suppose that we define ᐉ(z) ≡ [f(z)]1/t. Then ᐉ(z) is linear homogeneous:
ᐉ(sz) = [f(sz)]1/t = [stf(z)]1/t
= s[f(z)]1/t = sᐉ(z)
Hence if we also define G(ᐉ) ≡ ᐉt we have
Gᐉ(z) = ([f(z)]1/t)t = f(z)
and f(z) is indeed a homothetic function.
An example of a homothetic but non-homogeneous function is
y = A(δ 0 + δ 1 z1α + δ 2 z 2α ) 1 /α
(5.7)
which is the general form of the CES with an additional constant term δ0.
Supplementary Questions
(i) Show that (5.7) is not homogeneous.
(ii) Show that (5.7) is an increasing transformation of a linear homogeneous function
of z.
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4. Since f(z) is linear homogeneous Euler’s Theorem [C.4] implies that
f1z1 + f2z2 = y
so the numerator in
σ=
f1 f2 ( f1 z1 + f2 z2 )
z1 z2 [2 f12 f1 f2 − f11 ( f2 )2 − f22 ( f1 )2 ]
(5.8)
is f1 f2y. The partial derivatives of a linear homogeneous function are homogeneous of
degree zero (see [C.3]) and so, applying Euler’s Theorem again,
f11z1 + f12z2 = 0 ⇒ f11 = −f12z2/z1
f21z1 + f22z2 = 0 ⇒ f22 = −f12z1/z2
(remember that f12 = f21). Hence the denominator in (5.8) can be written
z1z2[2f12 f1 f2 + z2 f12( f2)2/z1 + z1 f12( f1)2/z2]
which can be rearranged to give
f12[ f1z1( f1z1 + f2z2) + f2z2( f1z1 + f2z2)] = f12( f1z1 + f2z2)2 = f12y2
and so
σ=
f1 f2
f12y
Exercise 5D
1. For z2 = z20 output is maximized when the variable input is at the level z1* . Any further
increase in z1 will place the firm on a lower isoquant and so reduce output. Hence z1* is
on the ridge line R1 which is the boundary of the economic region. Since output is
maximized at z1* the marginal product of z1 must be zero.
2. Average product decreases with z1 up to z10 and then increases, so that AP1 has
a local minimum at z10 for given z2. Hence its derivative [D.3] is zero at this point and
AP2 = MP1.
Supplementary Question
(i) Why must the MP1 curve cut the AP2 curve from below at z10 and from above at z1′ ?
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Chapter 6
Cost
Exercise 6A
1. Assume first that there are no transactions costs and suppose that the market price of
the asset is p0 now at date 0 and p1 and p1 one year later at date 1. To use the asset for
one year the firm must buy it at date 0 and resell it at date 1. By purchasing the asset at
date 0 and reselling it at date 1 the firm reduces its income stream at date 0 by −p0 at
date 0 and increases it by p1 at date 1. The cost to the firm is
p0 −
p1
1+ r
since £1 of income at date 1 has a present value of 1/(1 + r). If the price of the asset is
constant, p0 = p1, the cost of using the asset for a year is p0r/(1 + r).
The asset’s infinite durability is not sufficient to ensure that its price is constant over
time. We would need to assume that there is no technical progress or change in the
value of the output produced by the asset. The implication of the asset having a finite
life is that its price will decline over time so that p1 < p0.
If the firm already owns the asset at date 0 it could sell it for p0 and lend out the
proceeds for one year at the interest rate r, giving an income stream of 0, p0(1 + r). If it
uses the asset for a year and then sells it at date 1 its income stream from this
transaction is 0, p1. Hence the difference in its income stream from using the asset for a
year is 0, p0(1 + r) − p1 which has the same present value as the case in which the firm
does not own the asset at date 0.
If there is no second-hand market the cost of using the asset from date 0 to date 1 is
just p0. If the firm already owns the asset at date 0 its opportunity cost is zero.
If there are transactions costs of t per transaction and the firm buys the asset at date
0 and resells at date 1 the opportunity cost is (assuming that the transactions costs are
incurred by the purchaser, so that the market prices are net of transactions costs)
p0 + t −
p1
.
1+r
Exercise 6B
1. The isoquants for the single process Leontief technology are rectangular, so that the
cost minimizing input bundles are always at the ‘corner’ of the relevant isoquant, where
the minimum amounts of the two inputs are used to produce the required output level.
Hence the cost minimizing conditional input demands are zi = βiy, (i = 1, 2) and the cost
function is C(p, y) = (p1β1 + p2β2)y The cost function is linear in output, yielding
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Fig. 6B.1
Fig. 6B.2
the total cost curve in part (b) of Fig. 6B.1 and the horizontal marginal and average cost
curves in part (c). As part (a) shows, changes in relative input prices (changes in the
slope of the isocost lines) have no effect on the optimal bundle. The conditional input
demands do not vary with the input prices, only with output.
2. In Fig. 6B.2 the isoquant I is a straight line with slope −α1/α2 since marginal products
are constant: MPi = αi. If the isocost lines are steeper than I (p1/p2 < α1/α2), as in the
figure, cost is minimized by using only input 1 to produce output. Hence when p1/α1 <
p2/α2 the conditional input demands are z1 = y/α1, z2 = 0 and total cost is C = p1y/α1
Conversely when the isocost lines are flatter than the isoquants cost is minimized when
only z2 is used. Thus p1/α1 > p2/α2 implies conditional input demands z1 = y/α1, z2 = 0 total
cost is C = p1y/α1. The cost function is therefore C(p, y) = y min(p1/α1, p2/α2) and the cost
curves have the same form as those of parts (b) and (c) of Fig. 6B.1.
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Supplementary Questions
(i) What is the interpretation of pi /αi?
(ii) What is the cost function and the conditional input demands when the isocost lines
have the same slope as the isoquants?
3. Note that since there is strict essentiality no output is produced if either input
is zero. Hence the solution to the cost minimization problem must have positive
amounts of all inputs. Forming the Lagrangean
L( z, λ ) = ∑ pi zi + λ ( y − Az1α z21−α )
the first order conditions are
p1 − λαAz1α −1 z21−α = 0
(6.1)
p2 − λ (1 − α ) Az1α z2−α = 0
(6.2)
y − Az1α z21 −α = 0
(6.3)
From (6.1) and (6.2) we have the usual condition for cost minimization that the input
price ratio equals the ratio of marginal products:
p1
α z2
=
p2 1 − α z1
which implies
z2 =
p1 1 − α
z1
p2 α
(6.4)
Substituting (6.4) in the constraint (6.3) gives
p 1−α
y = Az1 1
z1
p2 α
1 −α
α
p 1−α
= Az1 1
p2 α
1 −α
and so the conditional demand for input 1 is
y p 1−α
z1 = 1
A p2 α
α −1
(6.5)
Substituting (6.5) into (6.4) gives the conditional demand for input 2 as
p 1 − α y p1 1 − α
z2 = 1
p2 α A p2 α
α −1
y p 1−α
= 1
A p2 α
α
(6.6)
Note that, since 0 < α < 1 the conditional input demand for zi is decreasing in pi and
increasing in pj.
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z2
slope −p1s /p2
EP
I2
I1
I0
slope −p1b /p2
z1
z01
Fig. 6B.3
Using (6.5) and (6.6), the cost function is
y p 1−α
C ( p, y) = p1 1
A p2 α
=
α −1
y p 1−α
+ p2 1
A p2 α
y
α −1 1−α 1 − α
p1 p1 p2
α
A
α −1
α
1−α
+ p2 p2−α p1α
α
α
1 − α α −1 1 − α α 1
= yp1α p21−α
+
α A
α
Since the production function has constant returns, the cost function is linear in
the output and the cost curves are of the same form as those in parts (b) and (c) of
Fig. 6B.1.
4. The isocost lines will be kinked at z1 = z10 , as in Fig. 6B.3. When z1 < z10 its marginal
opportunity cost is the selling price p1s and when z1 > z10 it is the price at which more of
the input can be bought: p1b > p1s. Hence the isocost lines are steeper to the right of z10
than to the left. The cost minimizing solutions for different output levels are shown in
Fig. 6B.3 and EP is the expansion path. Note that at the output y1 the optimal input
combination is at the kink in the isocost line C1 and so changes in p1s, p1b or p2 may have
no effect on the cost minimizing input combination at this output.
The cost function is kinked at the output (y1 in Fig. 6B.3) at which the cost
minimizing demand for input 1 is z10 because the marginal opportunity cost of z1 jumps
from p1s to p1b. Fig. 6B.4 shows the total, average and marginal cost curves for a simple
case in which the technology has constant returns to scale.
Recalling the expression for long run marginal cost [B.8], long run marginal cost
jumps from p1s/f1 to p1b/f1 at output y1.
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Fig. 6B.4
Supplementary questions
(i) Show the effect on EP of small increases in p1s, p1b and p2.
(ii) Explain the shape of the average cost curve in part (b) of Fig. 6B.4.
5. (a) Homogeneity implies that the slope of isoquants is constant along rays from the
origin. Hence the expansion path defined by the tangency of isoquants and isocost lines
is also a ray from the origin and the cost minimizing input ratio is independent of the
required output. Let X = (z1(p, 1), z2(p, 1)) be the cost minimizing input combination for
producing an arbitrary reference output level of W. Hence C(p, W) = pX.
Because f(z) is homogeneous of degree n, the input vector s1/nX will produce sW units
of output:
f(s1/nX) = (s1/n)nf(X) = sf(X) = sW
since f(X) = W. Further, s1/nX will also be the cost minimizing vector for producing y = sW
units since it is on the same ray from the origin as X and therefore on the expansion
path. Hence the minimum cost of producing sW is
C(p, sW) = s1/npX = s1/nC(p, W)
and the cost of producing y = sW is
y
C(p, y) = C(p, sW) =
W
1/ n
C(p, W) = y1/nC(p, W)W−1/n = y1/nb(p)
as required since the arbitrary reference output W is constant.
5. (b) Since X is cost minimizing for W, the cost minimizing bundle for y is s(y)X and
C(p, y) = s(y)C(p, W)
(6.7)
The factor of proportionality s(y) is defined by
F(f(s(y)X) = F(s(y)f(X)) = y
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(where we have written the homothetic production function as an increasing
transformation F of a linear homogeneous production function f). We can invert (6.8)
to get
s(y)f(X) = F −1(y) ⇒ s(y) =
F −1
f ( X)
(6.9)
Using (6.7) and (6.9), marginal cost is
Cy(p, y) = s′(y)C(p, W) = C(p, W)/F ′f(X)
and so the elasticity of cost with respect to output is
Eyc =
=
Cyy
s′( y)C ( p, W) F
=
C ( p, y)
s( y)C ( p, W)
F
F ( f ( s( y) X ))
=
F ′f ( X ) s( y)
F ′f ( s( y) X )
where the last expression is the reciprocal of the scale elasticity of a homothetic
function (see text, page 104).
6. Denote the input price ratio p1 /p2 by ρ and the cost minimizing input ratio z2 /z1 by
r = r(ρ), so that relative exenditure is ρ/r and
dρ / r 1
dr 1
ρ dr 1
= 2 r − ρ = 1 −
= (1 − σ )
dρ
r
dρ r
r dρ r
(where the last follows from the definition of the elasticity of substitution and the fact
that cost-minimization implies that ρ = f1/f2). Thus if the relative price of input 1
increases, relative expenditure on it increases only if the elasiticity of substitution is
less than 1.
Supplementary question
(i) How do relative expenditures vary with ρ in the case of a Cobb-Douglas production
function?
7. Since it is not cost minimizing to use more of the input than necessary, total cost is a
step function of output, as in part (a) of Fig. 6B.5. Over the range y ∈ (0, J] average cost
is p/y, so the average cost curve is a rectangular hyperbola over this range, with
lim y→0 C /y = ∞ and lim y→ J C /y = p/J. For y ∈ (J, 2J] average cost is 2p/y with lim y→ J C /y
= 2p/J and lim y→2 J C /y = p/J. Generally, for y ∈ (nJ, (n + 1)J], (n = 1, 2 . . .) we have
lim C/y = np/(n + 1)J
y→ nJ
and
lim C/y = p/J
y→( n +1 ) J
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Fig. 6B.5
Hence the average cost curve is a set of discontinuous ‘scalloped’ segments as part (b)
of the figure. As output becomes large relative to the capacity J of the indivisible input,
the average cost curve tends to a horizontal line with height p/J. The derivative of total
cost with respect to output is zero for y ∈ (nJ, (n + 1)J) and is undefined for y = nJ.
Hence the marginal cost curve is the horizontal axis in part (b) for the open intervals
(nJ, (n + 1)J) and undefined at y = nJ. Although marginal cost is not defined at nJ, the
additional cost incurred in increasing output by some discrete amount is well defined.
The increase in cost from producing ∆ ∈ (0, y < J] is p.
Exercise 6C
1. The isoquants for a firm with four fixed proportions processes is shown in Fig. 6C.1.
The Leontief production function for process i is
z z
y = min 1 , 2
β 1i β 2i
and we have assumed that β11 > β12 > β13 > β14. With input 2 fixed at z20 , and assuming that
input 2 has a marginal opportunity cost of zero, the short run expansion path is the
horizontal line at z20 . The firm will minimize cost by using as little as possible of the
costly input 1 to produce the required output. Thus its expansion path is the horizontal
line abcd for output of y1 (corresponding to isoquant I1) or more. For output of less than
y1, such as y0 corresponding to isoquant I0, the firm’s cost is the same at all points on the
vertical segment ef of the isoquant I0. Thus for output y ∈ (0, y1) all non-negatively
sloped curves from the vertical axis to point a are expansion paths, including the
horizontal line z20 fa.
The firm’s cost curves are shown in Fig. 6C.2. For y ∈ [0, y1) additional output is
produced by using process 1 only and requires only β11 additional units of z1 to produce
an extra unit of output. Hence marginal cost over this range of output is p1β11.
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Fig. 6C.1
Fig. 6C.2
For y ∈ [y1, y2) a mixture of process 1 and process 2 is required to produce more
output given that the maximum amount of input 2 is fixed. Extra output is produced by
increasing the use of process 2 and reducing the use of process 1. Similarly to produce
extra output when y ∈ [y2, y3), the firm substitutes process 3 for process 2. The firm
substitutes processes which require larger amounts of input 1 to produce an additional
unit of output for those which require smaller amounts as its output increases.
As it moves along abcd from segment ab to segment bc to segment cd it requires
larger amounts of z1 to produce more output. Its variable cost curve is shown in part (a)
of Fig. 6C.2 and its marginal cost curve as an increasing step function in part (b). At y4
no more output can be produced given the constraint on z2: further increases in z1 do
not increase output and the shout run marginal cost curve is vertical.
Average variable cost for any output is the slope of a line from the origin to the
variable cost curve in part (a). Hence for output up to y1 the SMC and AVC curves in
part (b) coincide, but for y > y1 the AVC curve lies below SMC.
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Fig. 6C.3
2. The short run total cost curves are shown in part (a) of Fig. 6C.3 and the marginal and
average curves in part (b). Since marginal product is always positive with a CobbDouglas production function y = z1α z2β the firm will always wish to use all the fixed input
available to minimize the amount of the costly variable input required. Thus we can
invert the production function at the level of the fixed input X2 to get the cost minimizing
level of the variable input as
z1 = ( yX2− β ) = y k
1
1
α
α
−α
where k = X . Hence variable cost is
β
1
α
p1y k
and short run marginal cost
1−α
SMC = p1y kα −1
α
which is increasing in y if α ∈ (0, 1). Since the marginal product of input 1 is never zero,
no matter what the required output, the SMC curve never becomes vertical.
3. See the ridge line R in Fig. 5.2. of the text. The firm would never choose to operate
above such a ridge line since it could always reduce the amount of the variable input
required by moving back down the positively sloped section of the isoquant. Hence the
expansion path would be the ridge line up to z2 = z20 and the horizontal line at z20
thereafter.
4. The isocost lines in Fig. 6C.4 are negatively sloped for z ≥ z20 , since the marginal
opportunity cost of input 2 is p2 for z ≥ z20 . For z ≤ z20 input 2 has zero marginal
opportunity cost and the isocost lines are vertical. Assuming that MP2 is positive, the
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Fig. 6C.4
firm will always use all its contracted-for workers (who have zero opportunity cost to
it). It will hire more workers if their marginal product is sufficiently large that the
isoquants are flatter at z20 than the isocost line. (Recall the interpretation of the shadow
value (µj) of the fixed input on page 132 of the text.)
The expansion path in Fig. 6C.4 is the horizontal line at z20 up to the isoquant I0 and
the positively sloped curve EP thereafter. The STC curve lies above LTC up to y0 and
coincides with LTC thereafter.
5. Refer to page 117 of the text for the properties of the long run cost function.
Property (a): Short run cost is increasing in y if more of the variable inputs are required
to produce more output and the prices of variable inputs are positive. Increases in any
input price must increase total cost if a positive amount of that input is used.
Property (b): The solution (zv(p, y, zk0 ), zk(p, y, zk0 )) to the short run cost minimization
problem at prices p is also the solution at prices tp since
S(p, y, zk0 ) = pvzv(p, y, zk0 ) + pkzk(p, y, zk0 )
≤ pvzv + pkzk,
all feasible (zv, zk)
implies that
tpvzv(p, y, zk0 ) + tpkzk(p, y, zk0 ) ≤ tpvzv + tpkzk,
all feasible (zv, zk)
Hence S(tp, y, zk0 ) = tS(p, y, zk0 ).
Property (c): For continuity see Takayama (1985, pp253–254). For concavity, let ( zvi , zki )
be cost minimizing at prices pi. Then
tS ( p1 , y, zk0 ) = tp1v z1v + tp1k z1k ≤ tpv3 zv3 + tpk3 zk3
(1 − t) S ( p2 , y, zk0 ) = (1 − t ) pv2 zv2 + (1 − t) pk2 zk2 ≤ (1 − t) pv3 zv3 + (1 − t) pk3 zk3
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and adding the left hand sides of these expressions and the right hand sides gives
tS ( p1 , y, zk0 ) + (1 − t) S ( p2 , y, zk0 ) ≤ [tp1v + (1 − t) pv2 ]zv3 + [tp1k + (1 − t ) pk2 ]zk3
Defining
p3 = ( pv3 , pk3 ) = (tp1v + (1 − t) pv2 , tp1k + (1 − t) pk2 )
we have
tS ( p1 , y, zk0 ) + (1 − t) S ( p2 , y, zk0 ) ≤ S ( p3 , y, zk0 )
which establishes that the short run cost function is concave in prices.
Property (d): Define the function
g( p, p0 , y, zk0 ) = S ( p, y, zk0 ) − pv zv ( p, y, zk0 ) − pk zk ( p, y, zk0 )
Now compare g with G on page 117 of the text and apply exactly the same argument to
establish
∂g
∂S
=
− zl ( p0 , y, zk0 ) = 0,
∂pl p = p ∂pl p = p
l
0
l
l
ᐉ = v, k
0
l
which is Shephard’s Lemma for the short run cost function.
Exercise 6D
1. Suppose that there are two inputs (physical plant and fuel), with each plant having a
fixed proportions technology
z z
y = max 1 , 2
β 1i β 2i
and the fixed input level z20i . The maximum output from plant i is yi0 = z20i /β 2 i and
its constant short run marginal cost up to this capacity is p1β1i. Labelling the plants so
that β11 < β12 . . . < β1n, the firm’s short run marginal cost curve is shown in Fig. 6D.1.
The firm uses plant 1 to produce output of y10 or less, plants 1 and 2 to produce
y ∈ ( y10 , y10 + y20 ) and so on.
2. (a) The Kuhn-Tucker conditions are necessary and sufficient in this cost minimization
problem if the objective function in [D.1] is convex. (See Appendix H.) This requires
that C(y) is convex in y. But
C(ty + (1 − t)0) = C(ty) = F + vtαyα
and
tC(y) + (1 − t)C(0) = tC(y) = tF + tvyα
where t ∈ (0, 1). Hence, when α ∈ (0, 1] the cost function is concave for all output levels
since C(ty + (1 − t)0) > tC(y) + (1 − t)C(0). When α > 1, the cost function is concave for
small enough t (small enough output) since
limt→0 C(ty) = F > limt→0 tC(y) = 0
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Fig. 6D.1
Hence the Kuhn-Tucker conditions cannot be used to identify the optimal least cost
plant allocation: when the plant cost functions are concave it may be less costly to
allocate all output to one plant so as to incur only one fixed cost.
2. (b) When α = 1, C(y) = F + vy and average cost is v + F/y which declines with y.
Hence there are economies of scale for all output levels. When α > 1 the cost function is
of the form shown in Fig. 6.14 in the text. There are economies of scale (declining
average cost) up to W, defined by the equality of marginal and average cost:
F
+ vWα−1 = αvWα−1
W
and we can solve for
1
F
F
W=
=
2v
v(α − 1)
α
1
3
when α = 3.
2. (c) When α = 1, so that C = F + vy, marginal cost is constant and equal in the two
plants. It is never cost minimizing to use both plants and incur two fixed costs.
When α > 1 it is cost minimizing to use two plants for sufficiently large total output
because the saving in variable costs will more than offset the additional fixed cost. The
critical total output J above which it is cheaper to use two plants is defined by
J
F + vJα = 2F + 2v
2
α
and solving we have
1
F
4F
=
J=
1−α
3v
v(1 − 2 )
α
1
3
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Thus, since W < J, there is a range of output over which there is subadditivity and
diseconomies of scale.
Exercise 6E
1. (a) Recall from Appendix I of the text that a function is concave if the principal
minors of its Hessian alternate in sign, with the first prinipal minor being negative. The
second derivatives of C are
1 − 1 −
1 −
C ii = − yi − yi y j = − yi (1 + y j )
4
4
4
3
2
3
2
1
2
3
2
1
2
1 −
yi y j
4
1
2
C ij =
1
2
Since Cii < 0 the first principal minor is negative. The second principal minor is
C11C22 − C122 =
[
1
−
( y1y2 ) (1 + y1 )(1 + y2 ) − ( y1y2 ) −1
16
3
2
1
2
1
2
]
which is positive since the first term in square brackets is
− 32
− 32
1
2
1
2
( y1y2 ) + ( y1y2 ) −1 + ( y1y2 ) ( y1 + y2 )
which is greater than the second term. Hence C is concave in (y1, y2).
1. (b) Refer to the text definition [E.5] of multi-product economies of scale. We have, at
t = 1,
Etc =
∑ C C = ∑ 2 y (1 + y ) C
1
yit
i
=
− 12
i
i
1
2
yi
j
i
[
1
y1 + 2( y1y2 ) + y2
2C
1
2
1
2
1
2
1
2
1
2
1
( y + y2 ) + ( y1y2 )
= 2 1
( y1 + y2 ) + ( y1y2 )
1
2
1
2
]
1
2
1
2
< 1
This holds for all output vectors, so there are global economies of scale.
1. (c) There are economies of scope over y ∈ [0, y0] if [E.7] holds over this interval.
Since
1
2
C(y1, 0, p) = y1
1
2
C(0, y2, p) = y2
and
we have
1
2
1
2
C(y1, 0, p) + C(0, y2, p) = y1 + y2
1
2
1
2
< y1 + y2 + ( y1y2 )
= C(y1, y2, p)
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Thus C does not exhibit economies of scope: separate production of the two goods is
less costly.
1. (d) Since Cij = 14 ( y1y2 )
− 12
> 0, the cost function does not have cost complementarity.
1. (e) To see that C is not globally subadditive it suffices to show that there is some pair
of output vectors y′ = ( y1′ , y2′ ), y ′′( y1′′, y2′′) such that
C(y′ + y″, p) < C(y′, p) + C(y″, p)
Setting y′ = (y1, 0) and y″ = (0, y2), we see that the inequality in part (c) above
establishes that the cost function is not globally subadditive. Hence economies of scope
are necessary for subadditivity. The example in the text ([E.9]) shows that they are not
sufficient.
Supplementary question
(i) Sketch the contours of C in output space. Is it quasi-convex or quasi-concave? Use
the contours to illustrate the various concepts in parts (a) to (e).
2. (a) In the definition of cost complementarity [E.10], choose output vectors y1 = 0,
y2 = y, y3 = y, so that [E.10] implies
C(2y, p) − C(y, p) < C(y, p) − C(0, p) = C(y, p)
or
C(2y, p) < 2C(y, p)
Hence costs increase less than proportionately with output: there are economies of
scale.
2. (b) Now let y1 = (0, 0), y2 = (y1, 0), y3 = (0, y2) and apply [E.10] to get
C(y1 + y2, p) − C(y1, 0, p) < C(0, y2, p)
which implies [E.7] so that there are economies of scope.
Supplementary question
(i) Over what, if any, ranges of outputs do the following functions exhibit economies of
scale, economies of scope; subadditivity?
1
4
1
4
C = y1 + y2 − ( y1y2 )
1
4
C = 1 + (y1 + y2)2
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Chapter 7
Supply and Firm Objectives
Exercise 7A
1. Assuming that both inputs are used at the optimum, the first order conditions are
pfi − pi = 0,
i = 1, 2
These conditions can be rearranged to yield
f1 p1
=
f2 p2
p=
p1 p2
=
f1
f2
the first of which is the usual cost minimization condition that the slope of the isoquant
equals the slope of the isocost line (see [B.4] in chapter 6). Recalling the discussion of
LMC in section 6B, we see that the second of these conditions is the requirement that
the firm adjust its inputs (and hence its output) until price equals marginal cost.
2. (a),(b) The firm will plan to produce where p = LMC and p ≥ LAC. The LAC and LMC
curves for firms with diminishing returns to scale (diseconomies of scale) and constant
returns to scale are shown in parts (a) and (b) of Fig. 7A.1.
2. (c) A firm with increasing returns to scale will have LMC < LAC and planning to
produce where p = LMC would be planning to produce at a loss. It could do better by
closing down and avoiding the loss.
3. (a) C(y, p) increases with pi if input i is used by the firm and so average cost
increases at all output levels, including that for which LAC is minimized.
3. (b) The output W at which LAC is minimized satisfies LAC = LMC or
C(W, p) − Cy(W, p)W = 0
Using the implicit function rule gives
−( C p − C yp W) C p − C yp W
∂W
=
=
∂pi C y − C yy W − C y
C yy W
i
i
i
i
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Fig. 7A.1
Fig. 7A.2
Since LMC cuts LAC from below at W, Cyy(W, p) > 0. The numerator can be rearranged,
using Shephard’s lemma, to get
Cp − Cyp W = zi −
i
i
∂zi
W = zi(1 − ez y)
∂y
i
where ez y is the elasticity of demand for input i with respect to output. Thus W increases
or decreases with pi as the demand for input i decreases or increases more than in
proportion with output.
i
2. (c) Since Cyp = ∂zi/∂y, LMC increases or decreases with pi as zi is normal or
regressive. In Fig. 7A.2(a) we have assumed ez y = 1. In part (b) note that the fact the
input is regressive implies both that the LMC curve shifts down and that the output at
which LAC is minimized increases.
i
i
Supplementary question
(i) What is the effect of input prices on W if the production function is Cobb-Douglas,
CES or Leontief?
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Fig. 7B.1
Exercise 7B
1. In part (a) of Fig. 7B.1 the firm wrongly forecast a price of pwf and installed plant
which gave it the SMCw curve through w. The actual price is pa > pwf. Given its mistaken
forecast and actual SMCw curve the firm maximises profit with output ya. The firm’s
total cost at ya is the area under LMC up to output yw plus the area under SMCw between
yw and ya. (Remember that at output yw short and long run cost are equal since the
installed plant was designed to produce yw.) Hence its profit given its wrong forecast is
the area gawh. If its forecast had been right it would have installed plant which gave it
the SMCr and produced and output of yr. Its profit would then have been the area grwh.
Thus its wrong forecast reduces its profit by the area war. The firm loses from a wrong
forecast because it finds itself with the inappropriate plant size. This reduces its profit
in two ways: it has a higher marginal cost and therefore earns less profit over the output
range yw to ya and it produces a different (smaller) output.
Part (b) of the figure illustrates the case in which the wrongly forecast price exceeds
the actual price: pwf > pa. The firm’s cost given its actual wrongly chosen plant size is the
area under its LMC up to yw less the area under its SMCw curve between yw and ya.
Hence its profit is the area grwh less the area war. If it had correctly forecast that price
would pa it would have installed plant such that it faced the SMCr curve and produced
output yr. Its profit with a correct forecast would have been the area grwh. Hence its
reduction in profit from the wrong forecast is the area war.
2. Comparing the size of the areas war in parts (a) and (b) of Fig. 7B.1 it is apparent that
the answer depends on the precise curvature of the long and short run marginal cost
curves. The firm could be better or worse off being overly optimistic compared with
being overly pessimistic.
3. At each date the firm will make two decisions. Given its fixed input, the price of
output and the actual price of the variable input it chooses its output level. It also
makes a forecast of the prices of the variable input, the fixed input and its output and
chooses a plant size such that its forecast LMC is equal to the forecast output price.
The firm’s actual output decision is made by comparing current output price with
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Fig. 7B.2
actual short run marginal cost (determined by the actual variable input price and the
amount of the fixed plant chosen last period). The firm’s forecast prices are relevant for
its plant size decision but have no direct influence on its actual output in any period,
which is determined by equating actual short run marginal cost and actual output price.
Supplementary question
(i) Illustrate the firm’s decisions over output and plant size in a diagram analogous to
Fig. 7.3 in the text.
4. See Fig. 7B.2. The discussion of the relationship between short and long run cost
functions in section 6B of the text explained why the short run total cost curve is
tangent to the long run total cost curve from above and hence why SMC cuts LMC from
leads it to install
below at the tangency output. In Fig. 7B.2 the firm’s forecast of pt+1
f
1
a plant which gives rise to the SMC curve in period t + 1. When the actual price turns
out to be pat+1 it produces the output yat+1 . It correctly forecasts that the price in period
t + 2 will be the same as the actual period t + 1 price: p tf+ 2 = pat+ 2 = pat+1 and chooses
its plant for period t + 2 so that LMC = ptf+2 . In period t + 2 it faces the SMC 2 curve
and produces yat+2 . Hence its long run supply response is greater than its short run
response. This must always be the case for small output price variations since for
small output variations around the planned output the SMC curve is always steeper than
the LMC curve.
The cost curves in Fig. 6.11 in the text are an example where the relationship
between long and short run responses to a large price change would be the same as to a
small price change. Fig. 7B.3 gives an example where the long run response would be
smaller than the short run response for a large price change like that from p′ to p″.
Exercise 7C
1. (a) Refer to the definition of separability in production (text page 110). Since
production is separable the firm’s cost function for the output vector (y1, . . . , yn) is
C(y1, . . . , yn) =
∑ C (y )
i
i
i
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Fig. 7B.3
where C i(yi) is the cost function resulting from minimizing wzi to produce yi subject to
the production function yi = f i(zi) for product i. The profit function if production is
separable can be written
π = ∑ ( piyi − C i ( yi ))
i
and the first order conditions on output i for maximization of π are
pi − C yi (yi) = 0
i = 1, . . . , n
If each product division was told to maximize its own profits π i = piyi − C i(yi) it would
also choose an output satisfying pi = C yi (yi). Hence separate maximization of division
profits implies maximization of total firm profits. The actions of a division (input and
output choices) do not affect the profit of any other division. Note that, in addition
to separability in production (which implies that the cost function of the firm is
separable), this result requires that there is no interdependence in the firm’s markets, so
that changes in input or output decisions have no effect on the prices faced by any other
division.
1. (b) Suppose that the firm’s cost function is
C(y1, y2) = C 0 + C 1(y1) + C 2(y2)
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where C0 is a fixed administrative cost which must be incurred to produce any output
but which does not vary with the output of either good. The rule for allocating the fixed
cost is that division i is debited with kiC0, ∑ki = 1, ki ≥ 0. Hence the profit of division i is
piyi − C i(yi) − kiC 0
i = 1, 2
and separate profit maxization leads to
pi − C yi ( yi ) − C 0
dk i
=0
dyi
(7.1)
This is only compatible with firm profit maximization if dki/dyi = 0 and for the three
allocation rules we have
ki =
p1a1
p1a1 + p2a2
⇒
dk i
=0
dyi
ki =
p1y1
p1y1 + p2y2
⇒
dk i
>0
dyi
⇒
dk i
>0
dyi
C1
k = 1
C + C2
i
Thus only the first of these allocation rules leads to profit maximization for the firm as a
whole. Since the firm operates in competitive markets the product prices, and thus the
shares ki, are not affected by output changes. If any of the firm’s goods were sold in
monopolized markets then even this first allocation rule would not lead to profit
maximization for the firm as a whole. Note that in the specification of the first
allocation rule we need some dimensionality constants so that the prices of different
goods can sensibly be added and the share ki is not affected by arbitrary changes in the
units in which the goods are measured.
1. (c) No. Consider the following case, in which for simplicity the cost allocation shares
are independent of the division outputs and the output levels chosen maximize the
firm’s profit. Suppose that at these output levels
p1y1 − C1 − k1C 0 < 0
p2y2 − C 2 − k2C 0 > 0
p2y2 − C 2 − C 0 < 0
∑( pi y i − C i − k i C 0 ) = ∑( pi y i − C i ) − C 0 > 0
The firm would first be led to close down the “unprofitable” division 1. It would then
find that after attributing all the central administrative costs to the remaining division 2
that division 2 now makes a loss and should also be shut down. Thus the firm would
cease production despite the fact that it makes a positive profit by operating both
divisions.
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1. (d) A sufficient condition for shutting down a division is that its separate contribution
to any central costs is negative:
piyi − Ci < 0
(7.2)
Clearly, shutting down a single division which makes a non-negative contribution
cannot increase overall profit. However, it is possible that after closing all divisions
which satisfy (7.2) the firm will find that even though all its remaining divisions have
piyi − C i ≥ 0 it may be better off closing down entirely because ∑(piyi − C i) − C 0 < 0
where the sum is over the divisions which make positive contributions.
Supplementary question
(i) Would separate maximization of division profits imply maximization of firm profits if
(a) the firm was a monopolist for some of its products? (b) was a monopsonist for some
its inputs? (c) if some of its products were substitutes?
Exercise 7D
1. Part (a) of Fig. 7D.1 shows a production possibility set with the property that for
some relative prices the firm’s optimal netput vector is not unique and over some ranges
of relative prices the optimal netput does not vary with relative prices. Part (b) plots the
resulting supply function for the output (positive netput) y2 against the price ratio r =
p1/p2. The firm’s optimal netput is not unique at price ratio r′ since at this price ratio the
slope of the isoprofit lines is equal to the slope of the upper boundary of the production
possibility set between a and b. The kink at a means that for all r ∈ [r″, r′] the firm
chooses the same netput a. Similarly for all r ∈ [r′, r′″] the firm chooses b. Thus we get
the vertical segments of the supply function in part (b). Note that at the price ratio r′ the
supply function is drawn as a horizontal line: strictly speaking there is a supply
correspondence at this price ratio since r′ does not map into a unique netput.
Supplementary questions
(i) Is the supply function continous if PS is convex, or if strictly convex, or if concave?
(ii) What are the implications of the PS in Fig. 7D.1 for the firm’s demand for its input
(netput 1)?
2. Let prices of the output and the inputs increase from (p, w) at date 0, when the firm
makes its production plan and purchases its inputs, to (kp, kw), k ≥ 1 at date 1, when
production takes place, the output is sold and the firm is taxed. Its recorded profit at
date 1 for tax purposes is kpy − wz so that it pays tax of t(kpy − wz). At date 0 the firm
correctly anticipates the price changes and makes its input decision to maximize its
after tax profit
k(py − wz) − t(kpy − wz) = (1 − t)kpy − (k − t)wz
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Fig. 7D.1
subject to its production function. Equivalently, the firm chooses its output to maximize
)(y, k, t) = (1 − t)kpy − (k − t)C(y, w)
(The firm would choose the same input mix to produce output y at prices w and at
prices (k − t)w and so its cost function, allowing for the tax and the change in prices, is
C(y, (k − t)w) = (k − t)C(y, w).)
The first order condition for maximization of ) is
)y = (1 − t)kp − (k − t)Cy(y, w) = 0
(7.3)
k−t
Cy(y, w) ≥ Cy(y, w)
k(1 − t)
(7.4)
which implies
p=
Hence the combined effect of the tax on recorded profit and inflation is to make the
firm act as if it faced a marginal cost curve which has been shifted upward. It will
therefore produce a smaller output than it would if either there was no inflation (k = 1)
or no tax (t = 0).
More formally, we can apply the simple comparative static method (text, Appendix I,
page 699) to examine the effect of an increase in the tax. Partially differentiating (7.3)
with respect to t gives
)yt = −(kp − Cy) ≤ 0
which is negative when k > 1 (see (7.4)).
Supplementary question
(i) Show that increases in the rate of inflation reduce the firm’s output when t > 0. What
happens if there is a correctly anticipated deflation?
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Exercise 7E
1. If the entrepreneur has an initial endowment of income J the intercepts of the P(E)
and I0 curves in text Fig. 7.8 shift up by this amount, but otherwise the analysis is
unchanged. The point of this question is to say that the absence of endowed income in
the analysis in the text does not really matter.
2. We know for the quasi-linear utility function that indifference curves in (E, y)-space
are vertical displacements of each other (see question 2, Exercise 3C), and so this utility
function gives the type of preferences for which utility maximization and profit
maximization (in the absence of a market in entrepreneurial input) are equivalent. Quite
simply, for any contour
G(E) + y = E
we have that
dy
= − G′( E )
dE
and so since this slope depends only on E and not on y indifference curves have the
same slopes along any vertical line. We leave to the reader the details of working
through the model, as in [E.2] to [E.7], with this utility function.
3. This question is exploring a little more deeply the factors underlying the P(E)
function in the model, and is concerned more with comparative-statics than whether
the firm is profit-maximizing. A general outline of the analysis is given in Fig. 7E.1.
Essentially, an increase in output price p or a fall in input price pz can be expected to
change the P(E) curve in the way shown. The location of the new equilibrium is
ambiguous, for the familiar reason that both income and substitution effects are
involved. The substitution effect of the kind of change shown would always increase
the entrepreneur’s effort supply, since the marginal return to effort has increased.
However, at any level of E the entrepreneur is now better off, and if leisure is normal
this will tend to reduce effort supply. Of course, we know that in the quasi-linear case
of question 2, income effects are absent and effort supply will increase. Given that
the effects of the change on E are ambiguous in the general case, we should not
be surprised if the effects on q and z are also ambiguous, as can be confirmed by
carrying out the comparative-statics. Of more interest is the question: suppose the
entrepreneur’s preferences correspond to the case in which she maximizes profits,
so that the introduction of these preferences makes no difference to the equilibrium
choice; then could it be the case that the comparative-statics responses might still
differ? To explore this question we adopt the quasi-linear preferences of question 2 and
set up the following model.
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Fig. 7E.1
The entrepreneur seeks to solve
max g(E) + y
s.t. y
=
pq − pzz
q
=
f(E, z)
g(E) + y
≥
u0
where u0 is her reservation utility. We assume, as illustrated in Fig. 7.8 of the text, that at
the equilibrium the constraint is non-binding. We can use the constraints to simplify the
problem to
max g(E) + pf(E, z) − pzz
E ,z
with first-order conditions
g′(E*) + pfE(E*, z*) = 0
pfz(E*, z*) − pz = 0.
Clearly, the condition on employment of z is exactly that which would be satisfied for
a profit-maximizing firm (and this is true also for a more general utility function),
while the first condition essentially expresses the tangency solution of Fig. 7.8. For the
comparative-statics, differentiating through totally gives the system
g ′′ + pf EE
pf
zE
pf Ez dE − f E dp
=
pfzz dz dpz − fz dp
From the second-order conditions the matrix, denoted D, has determinant D > 0.
We obtain the comparative-statics effects
∂E p( fz f Ez − f E fzz )
=
>0
∂p
D
∂E − pf Ez
=
<0
∂pz
D
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∂z pf E fzE − fz ( g ′′ + pf EE )
=
>0
∂p
D
∂z g ′′ + pf EE
=
<0
∂pz
D
The signs are unambiguous if we assume, reasonably, that g″, fzz and fEE are all negative
and that fzE > 0 (the inputs are “cooperant”). For these assumptions a straightforward
profit-maximizing model would give the same results, and so we can confirm that if the
special assumptions on preferences hold under which an entrepreneur chooses a profitmaximizing equilibrium, her comparative-statics responses will also be those predicted
by a profit-maximizing model, at least qualitatively.
If there is free entry into the industry, we expect it to take place until the “marginal
entrepreneur”, for whom entry is just worthwhile, is earning just about her reservation
utility. Any entrepreneurs with lower opportunity costs or who are more productive
than this will therefore be earning rents in long-run equilibrium. If all entrepreneurs are
identical then long-run equilibrium occurs at the point of tangency between P(E) and I0
in Fig. 7.8 of the text.
4. If the entrepreneur prefers to work for herself than for someone else, even though
work of any kind creates a disutility, we can model this by distinguishing two kinds of
effort: Eo is the work for herself; Em is work supplied to the market. The utility function
is then u(Eo, Em, y), with uo < 0, um < 0, uy > 0, in an obvious notation. The idea that work
for herself is preferable could be expressed by restricting
−
dE m uo
=
>1
dEo um
implying that an hour’s work on the market trades off for less than an hour’s work for
oneself, at any given income level. We also specify of course that u is strictly quasiconcave. We can then formulate the entrepreneur’s problem as
max u(Eo, Em, y)
s.t. y = P(Eo + EB) + wEm − wEB,
Eo + Em = T,
Eo , Em , EB ≥ 0
where EB is the entrepreneurial input bought in from the market at wage rate w. Note
that we assume that bought-in input is just as productive as the entrepreneur’s own
input. It is also also necessary to specify non-negativity conditions explicitly, since
corner solutions are quite possible.
Substituting for y in the utility function, we obtain first-order conditions
u0 + uyP′ − λ
≤ 0,
Eo ≥ 0,
Eo[u0 + uyP′ − λ] = 0
uy(P′ − w)
≤ 0,
EB ≥ 0,
EB[uyP′ − w] = 0
um + uyw − λ
≤ 0,
Em ≥ 0, Em[um + uyw − λ] = 0.
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The first result we can establish is that if working for oneself is always preferable to
working in the market, we cannot have simultaneously Eo > 0, Em > 0 and EB > 0. For
suppose this is true. Then the first-order conditions are strict equalities. Then w = P ′,
and eliminating λ gives
uo + uyw = um + uyw ⇒ uo = um or
uo
=1
um
which we have assumed cannot be true. In fact there are two possible cases:
1. Eo > 0, EB > 0, Em = 0. In that case it is straightforward to show that we have an
equilibrium given by conditions [E.11]–[E.13] of the text.
2. Eo > 0, Em > 0, EB = 0. From the second condition we have (non-trivially) P ′ < w,
and so Eo is extended beyond the point at which a wage line is tangent to the P(E )
curve. The extent of this difference is determined by the condition
uo um
=
+ ( w − P ′)
uy uy
so that the lower return for work within the firm than for work outside simply
reflects the stronger preference for the former.
5. Arranging private patients in order of fee per minute gives, we assume, a function
P(E) (ignore the discreteness of the real world and assume this is differentiable). The
fee per unit time offered by the health service is equivalent to the wage rate w in the
model of the entrepreneurial input market. Thus we can represent the physician’s
choice of time allocation by this model, and his possible equilibria are described by Fig.
7.9 of the text (assuming he can also hire other physicians to work for him at the health
service fee, and that he has no preference for working for himself). Thus he will work
both for the health service and for himself if he is at the equilibrium γ shown in Fig. 7.9.
If he is now forced to choose either to work for himself or for the health service, then
we have to find (a) the point of tangency of an indifference curve with P(E); (b) the
point of tangency of an indifference curve with the wage line 0w; and compare the
utility levels achieved in each case. As Fig. 7.9 is drawn, the physician will choose to
work for himself, since the point in (a) will certainly be above the point in (b). However,
it is quite possible to construct a figure which gives the result that he would choose the
health service (Hint: 0w must intersect P(E)). If he was previously at a point such as γ,
then the introduction of the forced choice will certainly reduce his income and utility,
though the effects on his effort are ambiguous, this could rise or fall. In the case drawn
in Fig. 7.9 effort must fall, but again cases can be constructed in which effort would
increase.
Finally, return to the case where he can supply effort both to the health service and
private practice. A tax on private earnings shifts the P(E) curve in Fig. 7.9 downward
while leaving 0w unchanged. If, as we might expect, the new curve is flatter at every E
(the tax reduces the marginal return to effort supplied to private practice) then effort
supplied to private practice will certainly fall, while that supplied to the health service
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could rise, fall or remain unchanged depending on the size of the income effect of the
tax (suppose the new overall equilibrium is somewhere along UU in Fig. 7.9 of the text).
Exercise 7F
1. With F = 0, [F.3] of the text becomes
pfN = pf(N*)/N*
The comparative-statics now become
f
p
f
pfNNdN + f N − dp + pf N* − p = 0
N
N
N
But the second and third terms are zero by the first-order condition, and so we must
have dN = 0. Diagrammatically, both fN and f/N increase by the same amount at N*, and
so the employment level is unchanged. Thus a fixed cost is required to achieve the
strong form of Ward’s result. At the same time, the result that an increase in output
price leads to no increase in output and employment is still very striking.
2. Let NO denote outside workers so that the total labour force is NI + NO. The production
function is f(NI + NO), and the market wage is w. Then the inside workers seek to
maximize
yI = [pf(NI + NO) − wNO]/NI.
We assume that at the equilibrium, NO > 0. Then the first-order condition is:
∂y I
= (pf′ − w)/NI = 0 ⇒ pf′ = w
∂N O
Note there is no condition with respect to NI because we assume there is a fixed
number of inside workers. Thus the firm acts just like a profit-maximizer in choosing
its aggregate employment level and hence the number of outside workers. The
comparative-statics will also be identical.
3. The income of the representative worker is again
y = [pf(Nl, N) − F]/N
where F is the fixed cost of capital. First, proceed by solving
max u = v[(pf(Nl, N) − F)/N] − l
l, N
giving first-order conditions:
v′pf1 − 1 = 0 ⇒ pf1 = 1/v′
v′
(pf1l* − (pf − F)/N*) = 0 ⇒ pf1l* = (pf − F)/N*
N
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where f1 is the marginal product of labour. The first condition says that given the
optimal number of workers N* (note that N is treated as a real number rather than an
integer), the amount of time each works, l*, is set so that the marginal revenue product
of labour is just equal to the worker’s marginal rate of substitution between income and
effort, 1/v′. This can be thought of as the worker’s supply price. The second condition
says that the number of workers (each working the optimal time l*) is adjusted to the
point at which an additional worker adds just as much to revenue ( pf ′l*) as she is paid
(( pf − F)/N*), just as in the Ward-Vanek model.
The notation in the comparative-statics will be simplified if we denote ( pf − F )/N
by y and its partial derivatives by yj, j = l, N, p. Moreover, note that yN = 0 from the
first-order condition. Then differentiating through the first-order conditions gives the
system:
v′′yl pf1 + v′pf11 N
pf + pf lN − y
11
l
1
v′pf11l dl − v′f1
=
dp
pf11l 2 dN y p − f1l
Then solving for dl gives
∂l − y p v′pf11l
=
>0
|D |
∂p
The sign on this derivative follows from the facts that f11 < 0 because of the strict
concavity of the production function, and |D| > 0 from the second-order condition (D is
the square matrix in the above system). Thus an increase in price, since it increases the
marginal value product of work, increases each worker’s labour supply. The lack of
ambiguity in this case arises because income effects are excluded by the special form of
utility function.
Solving for dN gives
∂N
f
= v′′( pf1 )2 − f1l + f11 Nl /| D | < 0
N
∂p
To explain this result, note that we again have f11 < 0, |D| > 0. The first term in the
numerator is also negative if f/N > f1l (since v″ < 0 and recall yl = pf1). This is assured
because f is strictly concave in Nl.
Thus, although this model is in many respects less special and more ‘reasonable’ than
the simple Ward-Vanek model, we still have the result that the number of workers falls
when output price increases. This is because of the sharing rule, y = ( pf − F )/N: an
increase in price increases the cost of the marginal worker by more than she adds to
revenue, so she is fired.
4. (a) When choosing her consumption level consumer-owner i will take the price set by
her firm as given. She will however realise that she will receive a proportion θi of any
change in profit generated by her purchase and will take this into account in choosing
her consumption of the firm’s product. Thus she realises that the firm’s profit π is
defined implicitly by
p ∑ xj(p, Jj + θjπ) − c(x) − π = 0
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where xj(p, Jj + θjπ) is the demand by j. In general an increase in purchases by i changes
profit directly by altering revenue and costs and indirectly because the change in firm
profit alters the demand by all owners. Hence, using the implicit function rule
dπ
p − c ′( x)
=
dxi 1 − [ p − c ′( x)] ∑ i≠ j x jJθ j
(7.2)
where x jJ is the change in demand induced by an increase in j ’s income.
Fortunately we have assumed that changes in income have no effect on demand by
any owner ( x jJ = 0, ∀j) and so (7.2) simplifies to
dπ
= p − c′(x)
dxi
The Lagrangean for the consumer-owner’s choice of xi is
L = ui(xi, yi) + λ[Ji + θiπ − yi − pxi]
(7.3)
and the first order conditions are
dπ
Lx = uxi + λ θ i
− p = uxi − λ[p − θi(p − c′(x))] = 0
dxi
(7.4)
Ly = uyi − λ = 0
(7.5)
i
i
plus the budget constraint. Thus i acts as if she faced a price of Yi = p − θi[p − c′(x)].
4. (b) An increase in the price p has two effects on i: as a consumer she is worse off but
as a part-owner she may be better or worse off depending on whether the firm’s profit is
increased. Using the implicit function rule on (7.1) and remembering our assumption
that the income elasticity of demand is zero for all owners, we have
x + [ p − c ′( x)]x p
dπ
=
= x + [p − c′(x)]xp
dp 1 − [ p − c ′( x)] ∑ x jJθ j
where xp = ∑∂xj /∂p.
From the envelope theorem (Appendix J) the marginal effect of an increases in p on
i, given that she has chosen xi optimally, is the partial derivative of her Lagrangean (7.3)
with respect to p
dπ
∂L
= λ θ i
− xi
∂p
dp
= λ{θi[x + (p − c′)xP] − xi}
( p − c ′ ) x p p xi
= λx θ i 1 +
−
p
x x
( p − c ′)
= λx θ i 1 +
e − δ i
p
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Setting this expression equal to zero and rearranging gives the optimal price – marginal
cost margin in the question.
Note that the weights attached to the conflicting interests of i as shareholder and
consumer are shown by her share of the profit θi and of total consumption δi. If she does
not consume any of the product she will wish the firm to maximize profit. If she owns a
smaller proportion of the firm than her share in its output θi < δi she will wish the firm to
price below marginal cost.
Since the price which is optimal for i varies with θi and δi shareholders will disagree
about the price the firm should set unless θi/δi is the same for all shareholders.
Supplementary question
(i) Many companies offer their shareholders the opportunity to buy their products
at prices below those offered to non-shareholders. For example British Home Stores
gives shareholders vouchers entitling them to 10% off the marked price of goods sold
in their stores. Why do firms use this form of price discrimination? What are the optimal
discriminatory prices at which goods would be sold to shareholders and to nonshareholders?
5. The proportional rate of income tax satisfies the public sector budget constraint
G = π + tJ and so
t=
G−π
J
Hence we can write the Lagrangean for the i’th consumer-taxpayer’s consumption
decision as
L = ui(xi, yi) + λ[(1 − t)Ji − yi − pxi]
( J + π − G)
= ui(xi, yi) + λ
Ji − yi − pxi
J
J
J
= ui(xi, yi) + λ i ( J − G) + i π − yi − pxi
J
J
Thus apart from the first term, this is identical in form to the Lagrangean (7.3) in
question 1 with Ji /J replacing θi. For the purposes of this problem we can therefore
think of i as the “owner” of a proportion of the public sector firm. Hence the price
which each “owner” prefers the public firm to set will depend on their income relative
to total income and their share in total consumption. Thus we would expect individuals
with higher incomes to favour a price closer to the profit maximizing level.
The analogy between ownership of shares in a private firm quoted on the stock
exchange and “ownership” of a public sector firm via the taxation system can be helpful
but it should not be taken too far. In particular the two situations are certainly not
equivalent in respect of the ability of the individual to cease being an “owner”. The
taxpayer “owner” of a public firm can only vary her share of the firm by changing her
income relative to total income and can only divest herself completely by emigrating.
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She does not have one crucial ownership right: the ability to sell her share in the firm’s
profits or losses.
6. The i’th consumer’s budget constraint is
Ji +
px
x
π − yi − pxi = Ji + i (px − c(x)) − yi − pxi
pxi
x
= Ji −
c( x )
xi − yi
x
Denoting average cost by A(x) ≡ c(x)/x the Lagrangean is
L = ui(xi, yi) + λ[Ji − A(x)xi − yi]
and since x = ∑xj the first order condition on the choice of xi is
uxi − λ[A(x) + A′(x)xi] = 0
Thus the individual acts as if faced with a price, equal to average cost, which may vary
with the amount bought. If average cost is constant then A = c′ and she acts as if facing a
perfectly competitive firm which sells at a price equal to marginal cost.
The price paid has no effect on the individual’s budget constraint since any payment
made is returned to her via the cooperative dividend and she bears only a proportion
xi /x of the total cost of the firm. Hence the nominal price is irrelevant.
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Chapter 8
The Theory of a Competitive Market
Exercise 8A
1. For the firm the effective supply function is (from [A.4] of the text):
yj = sj(p, w(z(y(p)))) = sj(p)
dy j
dp
= s jp + s jw w′( z) z ′( y)
dy
= s ′j ( p) > 0
dp
because sjp > 0, w′(z) < 0 (pecuniary economies), sjw < 0, z′(y) > 0, dy/dp > 0.
For the market
y=
∑ y = ∑ s ( p) = s( p)
j
j
j
j
∑ j s jp
dy
=
0
dp 1 − w′( z) z ′( y) ∑ j s jw
With pecuniary external economies w′(z) < 0. This ensures that the supply function for
the individual firm slopes upward, but we now have ambiguity regarding the sign of the
slope of the market supply function.
External pecuniary economies might result if input suppliers had production
processes with increasing returns to scale.
If labour is the variable input one could imagine circumstances in which the
expansion of output in a given industry in a particular city would encourage skilled
workers to locate in that region, thereby reducing an individual firm’s cost of hiring in
terms of training expenses or actual wages paid.
2. yj = sj(p, a)
diseconomies.
sjp > 0
sja < 0. We are not including any possible pecuniary external
a = a(y(p))
ay > 0
yj = sj(p, a(y(p)))
dy j
dp
= s jp + s ja a ′( y)
dy
0
dp
where sjp > 0, sja < 0, a′(y) > 0 and dy/dp > 0. Thus the effective supply of a firm could
have a positive, negative, or zero slope.
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Fig. 8A.1
The change in industry supply as a result of an increase in p is
dy
=
dp
dy j
∑ dp = ∑ s + a ′(y) dp ∑ s
dy
jp
j
ja
j
∑ j s jp
dy
=
>0
dp 1 − a ′( y) ∑ j s ja
because a′(y) > 0 and ∑jsja < 0. The effective industry supply is positively sloped.
3. If the supply function is continuous and non-decreasing only for p > p0, then we may
have the case shown in Fig. 8.2 (b) of the text, in which an equilibrium does not exist. If
the supply function must be continuous and non-decreasing for all p ≥ 0, then three
cases are possible:
1. D(p0) = s(p0), in which case p0 is obviously an equilibrium.
2. D(p0) > s(p0). Then under the given assumptions an equilibrium must exist, since
for a sufficiently high price p1, we have D(p1) < s(p1) and the conditions for
existence are met.
3. D(p0) < s(p0). In that case it is possible that s(p0) > D(p0) at all p ≥ 0, in which case
we have an equilibrium at p = 0 as long as the excess supply of the good can be
disposed of costlessly (see Fig. 12.1 (d) of the text).
4. In Fig. 8A.1 the labour demand curve is continuous and strictly decreasing, the labour
supply curve is backward-bending. There is no equilibrium in this market.
5. (a) A per unit tax of amount t is applied. The pre-tax equilibrium is at p0 and q0 in
Fig. 8A.2. If the producer pays the tax we can consider the tax as an increase in
production costs which shifts the supply curve up vertically by t. The intersection of S′
and D shows what consumers will pay (pc). Producers remit t to the government and
thus receive ps (= pc − t).
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Fig. 8A.2
If the consumer pays the tax we may create a net-of-tax demand curve D′ which is
vertically below D by the amount t. The intersection of S with D′ shows what producers
will receive in the marketplace. This is ps as before. The consumer remits t to the
government and receives pc(= ps + t). Q′ is traded as before. Hence pc, ps, and the
quantity traded are independent of who pays the tax to the government formally.
The incidence of the tax refers to the way in which the actual prices paid by buyers
and received by sellers change. As compared to the initial equilibrium price p0, the price
to buyers rises by pc − p0, and the price received by sellers falls by p0 − ps, and these two
amounts (which sum to t) give the incidence of the tax. Then, we note that since the
post-tax equilibrium is independent of who formally pays the tax, so is the incidence of
the tax.
5. (b) Assume that the maximum price pmax is below the initial equilibrium (otherwise it
is not binding). If sellers pay the tax, the supply curve in (a) of Fig. 8A.3 shifts to S′. The
price consumers pay remains at pmax while the price sellers receive falls to pmax − t, where
t is the amount of the tax. Thus the entire incidence of the tax falls on sellers. If buyers
pay the tax, then the demand curve shifts down to D′ in (b) of Fig. 8A.3. However, this
simply reduces the amount of excess demand at pmax. The price sellers receive remains
at pmax, and buyers pay the price pmax + t. Thus in this case the entire incidence of the tax
falls on buyers.
Supplementary question. What happens if, in (b) of Fig. 8A.3, the tax paid by buyers
is so large that D′ intersects S at a price below pmax?
5. (c) If there is a minimum price above the initial (no-tax) equilibrium, and sellers pay
the tax, then the supply curve may shift as in (a) of Fig. 8A.4. Buyers continue to pay
pmax, sellers now receive pmax − t, and so they bear the entire incidence of the tax.
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Fig. 8A.3
Fig. 8A.4
Supplementary question: What happens if, in (a) of Fig. 8A.4, the tax paid by sellers is
so large that S′ intersects D at a price above pmax?
If buyers pay the tax, then the demand curve shifts to D′ in (b) of Fig. 8A.4, the
price sellers receive remains at pmin, excess supply increases, and buyers pay the price
pmin + t, and so they bear the entire incidence of the tax.
Exercise 8B
1. In (a) of Fig. 8B.1 the supply curve is steeper than the demand curve. Marshall’s
process is stable. Below y* the demand price is above the supply price so that sellers
will expand production. Likewise when y is greater than y* the supply price exceeds
the demand price and sellers contract output. The Walrasian TP is unstable as excess
demand increases with price.
In (b) of Fig. 8B.1 the demand curve is steeper than the supply curve. As the arrows
indicate Marshall’s process will now be unstable. The Walrasian TP is stable as excess
demand decreases as price increases.
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Fig. 8B.1
2. Cobweb cycles are more likely in the market for coffee because of the delay between
planting and harvesting.
Lettuce grows very rapidly so that farmers’ planting plans would be based on
expectations only a few weeks into the future. There is less chance of errors in
forecasting which could cause cobweb cycles.
In the case of the market for lawyers, if it were competitive there is every reason to
expect a cobweb phenomenon. There is quite a long supply lag, since it takes around
five years to qualify as a lawyer. Moreover, young people embarking on a law degeree
tend to observe current earnings of lawyers rather than to form rational expectations.
However, the market for lawyers is not competitive, in that entry is controlled by the
legal profession, and so we do not observe fluctuations in earnings which characterize
the cobweb.
3. (a) In equilibrium we have
z( p, a) = 0 = D( p, a) − s( p)
Totally differentiate to get
Da( p, a)da + Dp( p, a)dp − s′( p)dp = 0
Rearrange to get
Da ( p, a )
dp
=−
da
D p ( p, a ) − s ′( p)
We are interested in the conditions that make dp/da > 0. We have assumed Da( p, a) is
positive; for the overall expression to be positive we require
Dp( p, a) − s′( p) < 0
or
Dp( p, a) < s′( p).
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Fig. 8B.2
Fig. 8B.3
3. (c) Consider Fig. 8B.2 where the initial excess demand function is Z( p, a0) and there
are three possible initial equilibria at α0, β0 and γ0. α0 and γ0 are stable under Walrasian
TP, whereas β0 is unstable. After a increases to a1 the excess demand function shifts to
the right and there are again three equilibria at α1, β1 and γ1.
Since under Walrasian TP it is impossible for an unstable equilibrium to be reached
the new equilibrium after the shift in Z must be at one of the new stable equilibria: α1 or
γ1. If the initial equilibrium was the unstable one at β0 then the new equilibrium will be at
α1. After the demand shift there is positive excess demand at the original equilibrium
price and under Walrasian TP the price will be driven up.
3. (d) The previous result does not hold under the Marshallian adjustment process
as Fig. 8B.3 shows. The ( peculiar) demand and supply functions generate an excess
demand function which has the same form as in Fig. 8B.2 but now the stable equilibrium
is at β0 and the other two equilibria are unstable. If the initial equilibrium is at α0 a
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rightward shift in the demand curve leads to price and quantity decreasing to a new
equilibrium at β1. After the demand shift the demand price is less than the supply price
at the initial quantity and so price and quantity must fall under the Marshallian process.
If the initial equilibrium was at γ1 the same argument shows that the price
and quantity will fall and no equilibrium will be established. Finally, if the initial
equilibrium was at β0 the Marshallian process will drive price and quantity up to a new
equilibrium at β1.
4. Adaptive expectations:
We set up the linear model
yt = a + bpte
(the supply function)
xt = α − βpt
(the demand function)
pte − pte−1 = k( pt −1 − pte−1 )
(0 < k < 1)
Setting yt = xt for equilibrium at each t gives:
a + bpte = α − βpt.
Solving for pte :
pte =
α − βpt − a
b
and
pte−1 =
α − βpt −1 − a
b
Substituting gives
α − βpt − a
b
+
α − βpt −1 − a
b
( k − 1) − kpt −1 = 0
Rearranging, we have the first order linear difference equation
pt =
k(α − a )
β
bk
+ 1 − k − pt −1 .
β
Given the initial condition p0 = F, the solution to this equation is
bk
pt = p* + (F − p*) 1 − k −
β
t
where p* = (α − a)/(b + β) is the equilibrium price. Then we see that as t → ∞, pt → p* if
−1 < 1 − k −
bk
β
<1
that is if
2
b
>1+ > 0
k
β
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Fig. 8C.1
Fig. 8C.2
If b and β are both positive (demand and supply curves have the usual slopes) the right
hand inequality is satisified. Then the actual price will not diverge monotonically from
equilibrium. However, it could diverge with oscillations of increasing amplitude if k and
b/β are sufficiently large so as to violate the left hand inequality.
Exercise 8C
1. (a) As in Fig. 8C.1, each firm’s long run average cost and long run marginal cost
curves will be horizontal as firms can expand output with no change in average cost.
The long run market supply curve will also be horizontal at the constant long run
average cost. The equilibrium output of each firm and the number of firms in the
industry are indeterminate.
1. (b) As in Fig. 8C.2, each firm’s long run average and marginal cost curves are
everywhere upward sloping. p* is the only long run equilibrium price where profits are
zero. In the long run we would have a very large number of firms producing an
infinitesimally small amount at price p*. The long run supply curve is horizontal at p*.
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Fig. 8C.3
Fig. 8C.4
1. (c) As in (a) but with input prices increasing as market output increases. Although a
firm could individually expand output at constant long run average cost, when all firms
expand output average cost will increase. The long run market supply curve will
therefore be upward sloping.
2. In Fig. 8C.3 we have an equilibrium price of p0 with a very large number of firms each
producing an infinitesimal amount. If demand increases to D′(p) market output will
expand via entry of new firms. Input prices will drop causing the LAC and LMC curves
of the typical firm to shift down to LAC′ and LMC′. The equilibrium price drops to p1.
The intersection of p1 and p0 and D′(p) are two points on the long run market supply
curve S(p).
In the case in which each firm has an increasing returns production function there
is no determinate equilibrium. Long-run average and marginal costs are falling (see
Fig. 8C.4), and so if price is taken as given – the firm’s perceived demand curve is
horizontal – there is no profit-maximizing output for the firm.
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Fig. 8C.5
The output at which p = MC is a profit minimum, not a maximum (what is the sign
of the second derivative of the profit function at that point?). In other words, the firm
would seek to expand output indefinitely beyond this point. If increasing returns to
scale exist over a range of outputs which is large relative to market output, we would
expect the structure of the market to become imperfectly competitive.
3. (a) Naive expectations: The initial market equilibrium is at (y0, p0) in Fig. 8C.5.
Firms will be making zero profits. The actual number of firms and their output are
indeterminate. In year 1 demand increases to D′. Output can only expand along the
short run supply function which reflects a fixed number of firms. Price in year 1 rises to
p1 which must imply the firms are making positive profits. If firms expect p1 to prevail in
year 2 new firms will enter the market and existing firms will expand production.
However, the model as it stands tells us nothing about this process – truly naive firms
would seek to expand by an infinite amount since if they really believe price will remain
at p1 then every scale of output yields positive profits.
Rational expectations: Under rational expectations the year 1 short run equilibrium is
(p1, y1) as before. However, firms know that in year 2 p0 is the only price at which
planned outputs which maximize profits at that price can actually be sold. Existing
firms will expand capacity and new firms will enter to expand market output to y2. The
market moves to its full long run equilibrium in year 2.
Note, however, there is still something of an indeterminacy in the theory. How
does each firm know by how much to expand so that aggregate output from existing
and new sellers just equals the new equilibrium y2? What makes all these expansion
decisions consistent? In fact there is nothing in the theory which explains this, and so
the best conclusion is that even in the rational expectations case the perfectly
competitive adjustment process under constant returns to scale is indeterminate.
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Fig. 8C.6
3. (b) Naive expectations: The initial equilibrium is at (y0, p*) in Fig. 8C.6. Demand
increases to D′(p) in year 1 and output expands along the short run supply curve s′(p) to
y1 for the market (W for the firm). Firms plan for year 2 as though p1 would be the price
forever. Existing firms will expand output to W2 where long run marginal cost (LMC)
intersects p1. But firms are making positive profits at p1 which entices new firms to enter
the market. Each new firm, given that it has identical costs to existing firms, would
want to enter at a scale W2. But again the theory does not tell us anything about the
number of new firms that will enter the market and so again the adjustment process is
left unspecified.
Rational expectations: Under rational expectations we would expect price to jump to p1
initially in year 1 (assuming the increase in demand was unanticipated). In year 2 firms
would expect p* to prevail and would produce an ‘infinitesimally small’ amount with
total market supply at y2.
3. (c) Naive expectations: The initial equilibrium in the market is at (y0, p0) in Fig. 8C.7.
Individual firm production is indeterminate, but we assume it is at W0. Demand increases
to D′(p) in year 1. Output expands along s(p); price in year 1 will be p1. Firms plan for
year 2 assuming p1 will prevail. Output will expand along long run supply S(p) to y2.
Individual firms’ long run average cost curves will shift up to p1 as input prices rise.
Market output y2 can only be sold for price p2. At price p2 planned market output for
year 3 will be determined by the long run supply curve S(p). This process will continue
until the final new market equilibrium of (y*, p*) is attained. Individual firms’ long run
average costs will shift as market output changes, coming to rest at p*.
Rational expectations: We move directly from (y1, p1) in year 1 to (y*, p*) in year 2.
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Fig. 8C.7
Fig. 8C.8
4. The initial equilibrium is (y0, p0) in Fig. 8C.8 with each firm producing W0. Demand
increases to D′(p). Market output expands along the short run supply curve and price in
year 1 rises to p1. Existing firms expand output to W1 and make positive profits. New
firms will enter the market and drive profits to zero. Since input prices are constant and
all firms have identical costs, the long run supply curve must be horizontal at p0, the
minimum long run average cost for all firms.
Assuming rational expectations firms will be aware that p0 is the only profit
maximizing price where their sales expectations will be met. Thus in year 2 all firms will
again produce W0. Total market supply will have increased to y2 with the increase over y0
met entirely by an increase in the number of firms serving the market.
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Chapter 9
Monopoly
Exercise 9B
1. Let q = f(z1, z2) be the monopolist’s production function (with the standard
properties), wi the price of input zi, i = 1, 2, p = D0(q) the inverse demand function in the
current period (the short run) and p = D1(q) that in the next period (the long run). Note
that the next period demand function is assumed to be known with certainty. In the
current period the firm’s choices are subject to the constraint z2 ≤ z20 , while next period
both inputs are unconstrained. The firm’s problems are therefore:
this period (the short run):
max
q 0 , z1
D0(q0)q0 − w1z1 − w2 z20
s.t. q0 = f(z1, z20 )
planning for next period (the long run):
max
q1 , z1 , z2
D1(q1)q1 − ∑wizi
s.t. q1 = f(z1, z2)
Note in the short run problem we use the previous idea that the firm will find it optimal
to set z2 = z20 , given that associated with the fixed input is a fixed cost. Then, from the
first-order conditions for the first problem we obtain
p0 + q0 D0′ = w1 / f1 ( z1 , z20 )
which is the marginal revenue = marginal cost condition (recall that marginal cost =
input price / marginal product).
From the first-order condition for the second problem we obtain:
p1 + q1 D1′ = w1 / f1 ( z1* , z2* ) = w2 / f2 ( z1* , z2* ).
Since z1* and z2* are chosen without constraint, the wi/fi ratios in this case are long-run
marginal cost. Note that if we set z2 = z2* , then short-run marginal cost in the next
period is w1 / f1 ( z1 , z2* ). It is straightforward to show that the z1* obtained as a solution
to the long-run problem also solves the short-run problem next period, of
max
q1 , z1
D1(q1)q1 − w1z1 − w2z2*
s.t. q1 = f ( z1 , z2* )
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Fig. 9B.1
For suppose there exists some X1 ≠ z1* such that
D1(f(X1, z2* )) f ( X1 , z2* ) − w1X1 − w2 z2*
>
D1 ( f ( z1* , z2* )) f ( z1* , z2* )) − w1 z1* − w2 z2*
Then this contradicts the fact that ( z1* , z2* ) solves the long-run maximization problem.
2. In Fig. 9B.1 we show the monopoly’s profit-maximizing equilibrium for the case in
which there are increasing returns to scale. Note that the second-order condition is
satisfied.
Supplementary question: Draw figures illustrating the case of increasing returns to
scale in which the monopoly’s second-order conditions are not satisfied.
3. (a) Let the monopolist’s revenue function be R(q, a), with Ra > 0, Rqa > 0, so that
increases in a increase both total and marginal revenue at each output. Then the
monopoly’s first-order condition on optimal output q* is
Rq(q*, a) − C′(q*) = 0.
Differentiating totally and rearranging gives
Rqa
dq *
=−
da
Rqq − C ′′
where we require that Rqq ≠ C″. Since we normally assume R is strictly concave in q, Rqq <
0, and if C″ ≥ 0 (non-decreasing marginal cost) then we immediately have dq*/da > 0.
However, if C″ < 0 and is sufficiently large relative to Rqq that Rqq − C″ > 0, then we may
have dq*/da < 0. Note, however, that the second-order condition for a maximum is that
Rqq − C″ < 0, and so if q* is a true maximum we have that increasing a increases output.
For the effect on price, write the demand function as q = D(p, a), Dp < 0, Da > 0, and
write the profit function as
π(p, a) = pD(p, a) − C(D(p, a)).
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Then the first-order condition is
πp = D + pDp − C′Dp = 0
and the second-order condition is
πpp = 2Dp + pDpp − C ′′D p2 − C′Dpp < 0.
From now on we assume the second-order condition (which is essentially equivalent to
the condition Rqq − C″ < 0) is satisfied. Now totally differentiating through the first-order
condition gives
πppdp + [Da − C″DpDa +(p − C′)Dpa] da = 0
and so rearranging gives
[
D a − C ′′D p D a + ( p − C ′) D pa
dp
=−
da
π pp
]
Since πpp < 0 the sign of dp/da will be that of the bracketed expression in the numerator,
which of course is πpa. Since Dp < 0, and since we know that p > C′ at the equilibrium, it
is sufficient for this expression to be positive that Da > 0 and Dpa, C″ ≥ 0. However,
perverse results are possible, even with πpp < 0, if Dpa < 0 and C″ < 0.
3. (b) In this case the monopolist’s profit function becomes C(q) + tq where t is the tax
per unit of output. In effect the monopolist’s marginal cost curve shifts up by the
amount of the tax. The first-order condition is now
R′(q) − C′(q) − t = 0
and the second-order condition is unchanged. The comparative-statics effect on output
is
dq
1
=
<0
dt R ′′ − C ′′
where the sign follows from the second-order condition. Since the demand function is
unchanged, we have the effect on price as simply
dp /dq
dp dp dq
=
=
>0
dt dq dt R ′′ − C ′′
since the demand function has a negative slope.
3. (c) After tax profits are defined as
) = (1 − τ)(R(q) − C(q))
where τ is the profit tax rate. From the first-order condition
(1 − τ)[R′ − C′] = 0
we see that the profit tax leaves the equilibrium unaffected. Overall, we see that the
comparative-statics effects are broadly similar to those for the competitive firm, except
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that in the case of a demand shift there is greater scope for ambiguity of results under
monopoly.
4. We assume that you, the regulator, have full knowledge of the cost and demand
functions of the monopoly and are free to choose any instrument. Then four possible
methods are:
1. Set a maximum price F = D(Z) = C′(Z). The monopolist will then produce output Z
because this maximizes its profit at this maximum price: in effect it is confronted
with a horizontal demand curve at F for outputs q < Z, and so none of these
outputs yields higher profit than Z.
2. Set a minimum output Z. The monopolist will then maximize its profit by
producing Z, since any larger output makes less profit and lower outputs are not
feasible.
3. Pay a subsidy to the monopolist of å, where this satisfies
R′(Z) = C′(Z) − å.
Diagrammatically, the subsidy shifts the monopolist’s marginal cost curve
downward until it intersects the marginal revenue curve at Z (note that for this we
may need å > C′(Z)). Then the monopolist’s profit maximizing output given this
subsidy is Z.
4. Pay consumers a subsidy per unit of s*, where s* satisfies
R′(Z) + s* = C′(Z).
Diagrammatically, the subsidy shifts the demand and marginal revenue curves of
the monopolist upward until the marginal revenue curve cuts the marginal cost
curve at Z.
Other possible policies would be to tax substitutes or subsidise complements for the
monopoly output (thus shifting its demand and marginal revenue curves upward) or
subsidising its inputs (shifting its marginal cost curve downward). However, such
indirect methods are ‘second best’ (see section 14C of the text) because they create
resource allocation distortions in the market concerned. For the rest of this answer we
consider only the four measures given above.
A problem with policies 3 and 4 is that they greatly increase the profit of the
monopoly and are a cost to the public purse. Both these problems can be met by
simultaneously imposing a lump-sum profit tax on the monopoly. This leaves profitmaximizing output unchanged at Z, reduces the monopoly profit and finances the
subsidy. With this addition, there is nothing on resource allocation grounds to differentiate between the four measures. Choice among them would require information
on the administrative costs and political feasibility of each. In reality the main problem
is that the regulator is unlikely to possess exact information on the cost function, and
possibly the demand function, of the monopoly, and furthermore policies must be
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formulated in light of the fact that costs and demands change over time in a way which
may well be influenced by the type of policy adopted.
5. (a) A supply curve is a relationship between price and quantity supplied. For a firm in
a competitive market we derive it by taking the first-order condition on output
p = C′(q)
and inverting it to obtain output as a function of price
q = C′−1(p) = S(p).
In the case of a monopoly, the first-order condition
R′(q) = C′(q)
cannot be solved to give output as a function of price: in choosing output (price) the
monopoly simultaneously chooses price (output).
5. (b) If the monopolist buys an input in a competitive market then it can be said to have
a demand function for that input. Thus we can represent the monopoly’s input choice
problem as:
max p[ f(z1, z2)] f(z1, z2) − w1z1 − w2z2
where p[.] is its inverse demand function and f(. , .) its production function. Then its
first-order conditions are
(p + fp′)fi = wi
i = 1, 2
where the left hand side is its marginal revenue product (MRPi) (marginal revenue
× marginal physical product) of input i. These equations can be solved to give
zi = zi(w1, w2), the input demand functions. Note that the left hand side can be written
pfi + fp′fi. Since p′ < 0, while pfi is the marginal value product (MVPi), this shows
that MRPi < MVPi, and so a monopoly uses less of the inputs than would be the case if
the competitive market condition MVPi = wi prevailed. This is, of course, another way of
expressing the point that the monopoly distorts the allocation of resources by
restricting output of the monopoly good. Note, finally, that from the above condition we
obtain f1/f2 = w1/w2, indicating that the monopoly produces its (restricted) output in a
cost-minimizing way.
6. We assume throughout that the bidding for the monopoly franchise is competitive and
that all bidders would be equally efficient in production. Then
(a) Price and output will be set at the profit-maximizing (monopoly) levels, and the
winning bid will equal the monopoly profit.
(b) Price and output will be at the point of intersection of the demand and long-run
average cost curves, since this gives the lowest break-even price.
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(c) Let å < 1 be the share of revenue paid to the government. Then, given å the
monopolist will seek to maximize
(1 − å)R(q) − C(q)
implying the output that satisfies the condition
R′(Z) = C′(Z)/(1 − å).
Since 1 − å < 1, this implies that output is lower and price higher than in either of cases
(a) and (b). The value of å that will win the franchise is that which satisfies:
(1 − å)R(Z) − C(Z) = 0
since that just exhausts monopoly profit. Then, these two equations jointly determine Z
and å.
Supplementary question: Show that in the case of linear demand p = a − bq and costs
C = cq, a > c > 0, we have å = 1 − (c/a).
7. The firm’s demand functions are pi = Di(q1, q2), i = 1, 2, and so its revenue function is
R( q1 , q2 ) =
∑ q D (q , q ).
i
i
1
2
i
Its profit function is
π(q1, q2) = R(q1, q2) − C1(q1) − C2(q2),
where we assume separability of costs. Then the first-order conditions for maximum
profit are
Ri − Ci′ = 0
i = 1, 2,
where
Ri = pi + qi
∂D j
∂Di
,
+ qj
∂qi
∂qi
i, j = 1, 2, i ≠ j
is ‘total marginal reveue’ (TMR) of good i. The TMR takes account of the demand
interdependence between the outputs, and exceeds the partial MR pi + qi(∂Di/∂qi) if the
goods are complements, and is less if they are substitutes. As the second-order
condition for this problem (see p. 700 of the text), we require R11 − C1′ < 0 and
R11 − C 1′′ R12
> 0.
R 21 R 22 − C 2′′
The second condition,
( R11 − C1′′)( R22 − C 2′′) > R12 R21
can be thought of as placing restrictions on the demand functions for the two goods to
ensure a well-behaved problem.
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8. The monopoly’s profit function is
q1−α − cq
and its first-order condition is
1
(1 − α )q
−α
(1 − α )
=c⇒q=
.
c
α
For this to be well-behaved we clearly require 0 < α < 1. Since α is the inverse of the
(constant) elasticity of demand in this case we require demand elasticity to exceed
unity. The intuition for this is as follows. Suppose demand elasticity equals (is less than)
unity. Then reductions in output and increases in price leave total revenue unchanged
(increase it) while reducing total costs and so profit must increase. Thus the firm will
want to set output as close to zero as possible, selling this at an infinitely high price. But
there is no solution to this problem since profit can always be increased by reducing
output. The non-existence is created by a discontinuity in the firm’s profit function at q
= 0.
Supplementary question: Graph the firm’s profit function for the cases in which
α ≥ 1.
9. A monopolist with zero marginal costs maximizes profit by maximizing revenue,
implying that marginal revenue is zero. But
1
MR = p 1 + = 0 ⇒ e = −1.
e
The value of the Lerner index at this point is obviously unity.
Exercise 9C
1. We would expect that institutions have a lower demand elasticity for journals than
individual academics, who in turn have a lower demand elasticity than students. Thus
we predict subscription rates to be highest for institutions and lowest for students,
which is in fact the case.
To compare the pricing policies of profit maximizing publishers and learned societies
we need to specify an objective function for the letter. Let us assume they want to
maximize the total number of subscriptions. For simplicity, we assume just two groups
of subscribers, with subscriptions x1 and x2 respectively. The inverse demand functions
are p1(x1), p2(x2), and the cost function is C(x1 + x2). Thus the profit maximizing
publisher wishes to
max π = p1(x1)x1 + p2(x2)x2 − C(x1 + x2)
x1 , x 2
while the learned society wishes to maximize
x1 + x2
s.t. p1(x1)x1 + p2(x2)x2 − C(x1 + x2) ≥ B
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where we impose a break-even constraint on the learned society’s activities. The firstorder conditions for the latter are
1 + λ[MRi − MC] = 0
∑pixi − C(x1 + x2) − B ≥ 0
i = 1, 2
λ≥0
λ[∑pixi − C(x1 + x2) − B] = 0
We must have λ ≠ 0 and so the break-even constraint is binding. From the first two
conditions we have
MR1 − MC = MR2 − MC = −1/λ < 0
which in turn implies MR1 = MR2. Thus, just as the profit-maximizing publisher, the
learned society equalizes marginal revenues and so will set a higher subscription rate in
the market with the less elastic demand. The main difference is that its subscription
rates overall will be lower (MRi < MC) and subscription volume higher, because it is
simply concerned with meeting its profit constraint (this is assumed to be for less than
maximum profit).
2. We would expect the car manufacturers’ demand for sparking plugs to be far
more elastic than that of individual consumers. The manufacturers can shop around
for alternatives to any one supplier, on a worldwide scale, or indeed could set up
manufacture of the plugs themselves. Individual consumers, on the other hand, have
sparking plugs in their cars replaced as part of an overall service, and typically accept
what the garage gives them.
3. In general, we could take house price as an indicator of the income and wealth of the
indiviual purchaser. The higher the purchaser’s income the greater her willingness to
pay and the smaller, it would be assumed, her price elasticity of demand. Thus relating
fees to house price is a way of approximating price discrimination according to demand
elasticity.
4. This is a clear example of second-degree price discrimination. The low demand types
will choose 1 unit at 50p, the high demand types 2 units at 90p. In terms of the analysis
of this section we have: x1* = 1, F1* = 50p; x2c = 2, F2* = 90p.
5. The buyer faced with a full-line force has a choice of not buying the monopolized
good and buying the complementary good on a competitive market, or buying the
monopolized good and the complementary good at a price above the competitive level.
She will do this if the excess paid on the complementary good is less than her consumer
surplus on the monopolized good. Thus the full-line force can be seen as a way of
scooping out some consumer surplus on the monopoly good, in conditions where nonlinear pricing may not be feasible. It may also have the longer-run effect of eliminating
competition in the market for the complementary good and extending its monopoly to
that market.
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6. (i) Let R1(q1) be the revenue function for the good in the US and R2(q2) that in Japan,
expressed in dollars and yen respectively. Also let:
q11 be the amount produced and consumed in the US
q12 the amount produced in the US and shipped to Japan
q21 the amount produced in Japan and shipped to the US
q22 the amount produced and consumed in Japan.
Intuitively we would not expect both q12 and q21 to be positive at the optimum, but it is
useful to see how that emerges out of the mathematics. The production cost functions
are C1(q11 + q12) in the US, in dollars, and C2(q21 + q22) in Japan, in yen, with Ci″ > 0, i = 1, 2.
Suppose that transport cost per unit is $t, regardless of whether the good is shipped
from Japan to the US or conversely. Finally, the exchange rate, in dollars per yen, is y.
Then the firm’s profit in dollars is
π = R1(q1) − C1(q11 + q12) + y[R2(q2) − C2(q21 + q22)] − t(q12 + q21)
and the firm seeks to maximize this subject to the constraints:
q1 ≤ q11 + q21
q2 ≤ q12 + q22
qij ≥ 0, i, j, = 1, 2
(we assume qi > 0, i = 1, 2, at the optimum, (i.e. some output is sold in both markets).
The first-order conditions are:
R1′ − λ 1 = 0
yR′2 − λ 2 = 0
− C1′ + λ 1 ≤ 0
q11* ≥ 0
q11* ( λ 1 − C1′) = 0
− C1′ − t + λ2 ≤ 0
q12* ≥ 0
q12* (λ2 − C1′ − t) = 0
− yC 2′ + λ2 ≤ 0
* ≥0
q22
* (λ2 − yC 2′ ) = 0
q22
− yC 2′ − t + λ2 ≤ 0
* ≥0
q21
* (λ1 − yC 2′ − t) = 0
q21
together with the functional constraints, which, given Ri′ > 0 at the optimum, will be
binding, i.e. λi > 0. We now consider the main solution possibilities.
* > 0, λ 2 < C 1′ + t, λ 1 < yC 2′ + t
1. q11* > 0, q 22
In this case q12 = q21 = 0, production and sales take place in each country
separately. From the first four conditions we see that this implies R i′ = C i′ , i = 1, 2,
the straightforward monopoly solution. At the optimum, we see that it would not
be worth producing additional output in the country with the lower marginal cost
(there is nothing to say that these marginal costs – or the marginal revenues –
must be equal across countries at the optimum) because adding transport cost
brings the “marginal delivered cost” in the other country above the marginal
revenue in that country. Fig. 9C.1 illustrates this solution, with the US as the
country with the lower marginal revenue = marginal cost solution.
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Fig. 9C.1
Fig. 9C.2
* > 0, q12* > 0, λ 1 < yC 2′ + t.
2. q11* > 0, q 22
In this case some US production is exported to Japan. We have as conditions:
R1′ = C 1′ ; yR2′ = yC 2′ = C 1′ + t
Fig. 9C.2 illustrates this solution. The intuition is as follows. If marginal revenue
in the US differed form marginal costs there, it would be possible to increase
profit by varying sales and production in the US alone, regardless of exports. If
marginal revenue in Japan (in dollar terms) differed from the marginal cost of
producing and transporting a unit of US output, it would be possible to increase
profit by varying exports. Finally, if the marginal cost of producing a unit of
output in Japan differed from that of producing and shipping a unit from the US it
would be possible to increase profit by substituting between US and Japanese
output.
* > 0, q12* > 0, q 21
* > 0.
3. q11* > 0, q 22
We shall show that this case is impossible: we would not simultaneously observe
exports from the US to Japan and conversely. The first-order conditions in this
case yield:
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Fig 9C.3
R1′ = C 1′ = yC 2′ + t
yR 2′ = yC 2′ = C 1′ + t
The first line gives C 1′ − yC 2′ = t, the second gives C 1′ − yC 2′ = −t, which is a
contradiction given that t > 0.
4. q11* > 0, q12* > 0, λ 2 < yC 2′, λ 1 < yC 2′ + t.
In this case sales in both the US and Japan are supplied by US production, with
* = q21
* = 0. From the first-order conditions we obtain:
q22
R1′ = C1′,
yR′2 = C1′ + t
which are self-explanatory. Essentially, it is cheaper to supply Japan entirely from
the US, implying the Japanese marginal cost function is everywhere above the US
marginal cost plus transport cost function over the relevant range of outputs.
Fig. 9C.3 illustrates.
Clearly nothing much would be added by considering cases in which Japan is the
exporting country.
6. (ii) The comparative-statics effects depend on which is the initial equilibrium. Let us
take case (1), in which there is production in both countries and the US exports to
Japan. Substituting to eliminate the constraints gives the first-order conditions:
R1′( q11* ) − C1′( q11* + q12* ) = 0
* + q12* ) − C1′( q11* + q12* ) − t = 0
yR2′( q22
* + q12* ) − C 2′( q22
*) = 0
R2′( q22
* . Note that the first-order conditions are the
in the three non-zero unknowns q11* , q12* , q22
same whether the firm is US- or Japanese-owned. A Japanese firm would presumably
want to maximize profit expressed in yen, but multiplying through the firm’s profit
function by 1/y would leave the solution unchanged.
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We use the standard comparative-statics methods of Chapter 2 to find the effects of a
change in the exchange rate y:
C ′′( R ′′ − C 2′′) R 2′
dq11*
= − 1 2
<0
dy
D
R ′( R ′′− C1′′)( R2′′− C 2′′)
dq12*
= − 2 1
>0
dy
D
*
dq22
R ′ R ′′( R ′′− C1′′)
<0
= 2 2 1
dy
D
where D is the 3 × 3 Hessian determinant and from the second-order conditions D < 0.
The signs on these comparative-statics effects reflect the assumptions that C1′′ > 0,
Ri′′ < 0 and Ri′ > 0. The intuitive explanation is as follows. If the dollar devalues against
the yen, y increases. This increases the marginal revenue of exports from the US to
Japan in dollar terms, which is why q12* increases. However, in yen terms the marginal
revenue curve for Japanese sales does not shift, and so the increased exports reduce
marginal revenue in yen, causing a corresponding fall in Japanese production. However,
since
* R2′C 2′′( R1′′− C1′′)
dq12* dq22
+
=
>0
dy
dy
D
there is still an overall expansion in total Japanese sales. There is a fall in US sales
because the increased exports raise marginal cost in the US, thus requiring an increase
in marginal revenue of domestic sales. Again, however, there is a net increase in US
production since
dq11* dq12*
R ′( R ′′− C 2′′) R1′′
+
=− 2 2
>0
dy
dy
D
In a similar way one can carry out the comparative-statics analysis for the other
cases.
7. If we impose the constraint that, in the model of first-degree price discrimination the
fixed charges must be zero, we have third-degree price discrimination: the monopolist
can identify each buyer’s type and prevent arbitrage between them. Then, if we take
conditions [C.14]–[C.16] in the text, Fi = 0 implies that we can delete [C.15], since the Fi
are no longer choice variables. [C.16] becomes
vi(pi, 0) ≥ Ei
βi ≥ 0 βi[vi − Ei] = 0 i = 1, 2.
Now if this constraint is non-binding βi = 0 and so [C.14] becomes
pi + xi /xi′ = c
i = 1, 2,
which is precisely the condition that MR1 = MR2 = c, as derived earlier in the section. But
we know that, since Ei = vi ( pi0 , 0), we must have βi = 0 for any pi < pi0 . Thus provided
[C.14] has a solution with xi > 0, i = 1, 2, we must have pi < pi0 and the rest goes
through.
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8. In the model with linear demands and constant marginal cost, as long as ai > c, i = 1, 2
we can without loss of generality set c = 0 (given each ai > c, this is simply a change of
origin for measuring the constant terms in the demand functions). Then under thirddegree price discrimination with linear demands we maximize the total profit (revenue)
function
π ∑( a iqi − biqi2 )
The first-order conditions yield
qi* = ai /2bi
i = 1, 2,
implying prices and profits
pi* = ai /2, i = 1, 2, π * = ∑ a i2 /4bi .
Note also that the sum of outputs is
a b + a2b1
.
q1* + q2* = 1 2
2b1b2
Where a uniform price must be charged, we can add to the above maximization problem
the constraint p1 = p2, i.e.
a1 − b1q1 − a2 + b2q2 = 0.
With λ as the Lagrange multiplier on this constraint the first-order conditions become:
a1 − 2b1q1 − b1λ = 0,
a2 − 2b2q2 + b2λ = 0,
a1 − a2 − b1q1 + b2q2 = 0.
This is a linear system in the three unknowns q1, q2 and λ. Solving this system gives
a 2 b1 b2 − 2a 1 b1 b2 − a 1 b22
Z1 =
,
−( 2b1 b22 + 2b2 b12 )
Z2 =
a 1 b1 b2 − 2a 2 b1 b2 − a 2 b12
.
−( 2b1 b22 + 2b2 b12 )
The interesting thing now is that if we take the sum Z1 + Z2 we obtain:
Z1 + Z2 =
( a 1 b22 + a 1 b1 b2 ) + ( a 2 b1 b2 + a 2 b12 )
2( b1 b22 + b12 b2 )
=
( a 1 b2 + a 2 b1 )( b1 + b2 )
2b1 b2 ( b1 + b2 )
=
a 1 b2 + a 2 b1
= q1* + q 2* .
2b1 b2
Thus, total market output is the same with or without price discrimination. What differs
is the division of this total output between the two markets. Fig. 9C.4 illustrates.
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Fig. 9C.4
In both price discrimination and non-discrimination cases total output is set where
its marginal revenue is zero (= marginal cost). Under price discrimination, the price of
the less elastic good is higher and that of the more elastic good lower than under price
uniformity. Profits are lower in the latter case because marginal revenues in the two
sub-markets are not equalized.
Supplementary question: Now allow c > 0 and suppose that a1 > c > a2. What do you
think happens to the profit maximizing solution under both price discrimination and
uniformity? Illustrate diagrammatically.
9. To show that under second degree price discrimination both types will be offered
separate contracts, suppose to the contrary that they are offered the same contract (x0,
F 0). The argument in the text establishes that (x0, F 0) must lie on E1, and we can never
have x0 > x1 (refer to the text for notation) and so x0 < x1* . Consider the type 2
indifference curve passing through (x0, F 0). Since at x* on this curve dF/dx = −1, x0 < x*
and the strict convexity of the indifference curve implies that at (x0, F 0) we must have
that the indifference curve is steeper at x0 than at x*. At x* the slope of the indifference
curve dF/dx = −c and so if at x0 dF/dx < −c, we have that dF > cdx, and so a small
movement along the curve will increase the firm’s profit on a type 2 contract. Thus,
since such a move would not violate self-selection, (x0, F 0) cannot be optimal.
10. Self-selection by quality difference
This is answered simply by redefining the variable x. Let each consumer now demand
one unit of the good, and x now measures the quality of the good. The cost parameter c
now measures the production cost of a ‘unit of quality’. Then, assuming two types
of buyers with utility functions Ui(xi), defined on quality, the analysis goes through
just as before. The fact that the seller makes a profit on sales to type 2 buyers can
be interpreted as the result, stated in the question, that the differential between the
charges for the two types of quality exceeds the extra cost of producing that quality.
This can be easily shown by rearranging condition [C.28] in the text:
F *2 − F 1* = U 2 ( x2* ) − U 2 ( x1* )
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Fig. 9C.5
Then, as Fig. 9C.5 shows, at the second-best equilibrium the right hand side of the above
equation must exceed c( x2* − x1* ), the cost of the additional quality of the unit of the
good sold to type 2 buyers.
Exercise 9D
1. (a) Let the inverse demand function be p(x) = a − bx so that revenue is R(x) = ax −
2bx2. With constant marginal cost of c and zero fixed cost, profit is R(x) − cx and the
profit maximising quantity satisfies
R′(x) − c = a − 2bx − c = 0
and is
x* =
a−c
2b
Profit maximising price is
p* = a − bx* =
a+c
2
and maximised profit is
π* = (p* − c)x*
Welfare is measured as the area under the demand curve (willingness to pay) minus
the area under the marginal cost curve (total cost) and hence is maximised at the output
where marginal cost curve (height c) cuts the demand curve (height p(x)). Thus the
welfare maximising price po and quantity xo are defined by
po = p(xo) = a − bxo = c
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so that welfare maximising quantity is twice the monopoly profit maximising quantity:
xo =
a−c
= 2x*
b
When the demand curve is linear and marginal cost is constant the expression in the
text for the monopoly welfare loss (MWL) on the right hand side of [D.1] is exact:
MWL =
1
( p* − p o )( x o − x*)
2
and so subsitituting for po and xo
MWL =
1
1
1
( p* − c )( 2x* − x*) = ( p* − c ) x* = π *
2
2
2
Supplementary question
(i) How would the answer change if the firm had a fixed cost?
1. (b) With the constant elasticity demand function x(p) = kp−α, the inverse demand
function is p(x) = (k/x)−1/α, and revenue is R(x) = k1/αx(α−1)/α. The elasticity of demand is
−α and we assume that α > 1 so that demand is inelastic. Profit is maximised when
α − 1 1/α −1/α
R′(x) − c =
k x −c=0
α
and the profit maximising quantity and price are
α − 1
x* = k
αc
p* =
α
αc
α −1
Welfare maximising price is po = c and welfare maximising quantity is
xo = kc−α
Monopoly welfare loss is the area between the demand curve and marginal cost curve
between x* and xo:
MWL =
∫
xo
x*
= k1/α
[ p( x) − c]dx =
α
(α − 1)
xo
∫ [k x
x*
1 /α
−1 / α
− c]dx
[(xo)(α−1)/α − (x*)(α−1)/α] − c(xo − x*)
= ∆WTP − ∆C
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The first term can be written as
∆WTP = k
1/α
( α −1 )/ α
a
( kc −α )(α −1)/α − k α − 1
α
(α − 1)
c
α
1/α
= k k
= kc
1−α
( α −1 )/ α
1−α α − 1 a −1 1−α
c −
c
α
(α − 1)
α
α − 1 a −1
1 −
(α − 1) α
α
and the second as
a
1−α α − 1 a 1−α
α − 1
−α
∆C = c ( kc ) − k
= k c −
c
α
α c
= kc
1−α
α − 1 a
1 −
α
Total expenditure at the monopoly price is
α
α − 1
α c α − 1
k
R(x*) = p*x* =
= k
α −1 αc
αc
α − 1
= kc1−α
α
and thus
α −1
[
][
− 1−( α )
MWL ∆WTP − ∆C (α −1 ) 1 − ( α )
=
=
α −1
R
R
( αα−1 )
α
α − 1
=
α
−α
α − 1
−
α
α −1
−1
α −1
a −1
α − 1
−
α
1−α
α −1
a
]
α − 1
+
α
which depends only on the demand elasticity. You should plot MWL/R against α > 1 and
show that it declines monotonically to zero as α increases (demand becomes more
elastic).
2. (a) Assume that marginal costs do not vary with output whether or not there is Xinefficiency and that the demand curve is linear. In Fig. 9D.1 the marginal cost curve
with no X-inefficiency is at c0 and with X-inefficiency is at c1. Given the marginal cost
curve at c1 the monopolist sets a price of p* and output is x*. If costs were minimised
the welfare maximising output would be at xo where the marginal cost curve c0 cuts the
demand curve and consumers pay a price equal to the marginal cost of production. If
output was increased to xo consumers would be willing to pay the area under the
demand curve between x* and xo for the additional output. The additional cost of this
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Fig. 9D.1
output, with no X-inefficiency would be the area under the c0 marginal cost curve. The
difference between the consumers’ willingness to pay and the additional costs is the
area acd. With production with no X-inefficiency there would also be a cost saving on
the initial x* output equal to the area c1edc0. Thus the total welfare loss from an Xinefficient monopolist is the sum of the areas acd and c1edc0.
2. (b) If the X-efficient marginal cost curve was at c1 the welfare loss would be the sum
of the usual triangle abe plus the increase in the fixed costs.
2. (c) This part of the question is designed to remind you that the calculations of welfare
loss made so far rest on the assumption that £1 has the same marginal social value
whether it accrues to the owners of the firm, their employees, or consumers.
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Chapter 10
Input Markets
Exercise 10A
1. In Fig. 10A.1 the curves D(p1, p2, p, z20 ) plot the market demand for input 1 against its
price, holding the level of the fixed input 2 (and thus the number of firms), the price of
input 2 and output price constant. The curves are just the horizontal sums of the MRP
curves of the individual firms, as shown in text Fig. 10.1 with z2 held constant at its
initial level. When the price of input 1 falls from p10 to p11 firms find that their SMC has
fallen and they increase their output. If the increased output has no effect on p all firms
world move down their MRP curve generated by the initial output price p0. But if the
increase in the output of firms leads to a reduction in the price of output, say to p1, then
their MRP curves are shifted down. Aggregating these new lower MRP curves across all
firms gives D(p1, p2, p1, z20 ). Thus when the reduction in the price of input 1 induces a
reduction in the price of output the increase in demand (z10 to z11 ) is less than would be
indicated by the constant output price demand curve D(p1, p2, p0, z20 ). The short run
industry demand curve, allowing for the induced change in output price is shown by
D(p1, p2, z20 ).
To compare the long and short run industry responses to a change in the price of
input 1 we consider a simple case in which all firms have identical linear homogeneous
quasi-concave production functions f(z1, z2). Linear homogeneity implies that we can
treat the industry as if it were composed of a single firm which takes the output price as
given when it chooses its inputs. Since we are interested in industry demand for inputs
rather individual firm demands the fact that the size of individual firms is indeterminate
under the assumption of linear homogeneity does not matter. The long run equilibrium
will be characterised by profit maximizing choices of the two inputs
pfi(z1, z2) − pi = 0,
i = 1, 2
(10.1)
and the market clearing condition
p = p(f(z))
where p(f) is the inverse demand function for the industry’s output. Note that we do not
impose a zero profit condition to determine the number of firms since linear
homogeneity implies that LMC = LAC and LMC = pi/fi = p from the profit maximization
conditions. Substituting the inverse output demand function into (10.1) and totally
differentiating with respect to the input price p1 gives
p ′ f12 + pf11 p ′ f1 f 2 + pf12 ∂z1 /∂p1 1
=
2
p ′ f 2 f1 + pf 21 p ′ f 2 + pf 22 ∂z 2 /∂p1 0
130
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131
Fig. 10A.1
Using Cramer’s rule gives the long run effect on industry demand for input 1 of an
increase in its price:
∂z1 ( p1 , p2 ) p ′ f22 + pf22
=
<0
∂p1
∆
(10.3)
where
∆ = ( p ′ f12 + pf11 )( p ′ f 22 + pf22) − (p′f1f2 + pf12)2 > 0
by the second order conditions on the profit maximization problem and the downward
sloping demand curve for the industry’s output: p′ < 0.
In the short run, input z2 cannot be varied and so profit maximization with the output
price taken as given requires that the choice of the variable input z1 satisfies
pf1(z1, z20 ) − p1 = 0
(10.4)
(so that p = SMC). The other requirement for short run equilibrium is that the product
market clears and p − p(f(z1, z20 )) = 0. Substituting the market clearing condition into
(10.4) and differentiating with respect to p1 gives
∂z1 ( p1 , p2 , z 20 )
1
=
<0
2
∂p1
p ′ f1 + pf11
(10.5)
To compare the long and short run responses set the fixed input in (10.5) at its long
run level. Then multiply (10.3) and (10.5) by ∆ ( p′ f12 + pf11) < 0 and note that since
∆ < ( p ′ f12 + pf11 )( p ′ f 22 + pf22)
we have established that
∂z1 ( p1 , p2 ) ∂z1 ( p1 , p2 , z 20 )
<
∂p1
∂p1
Thus the long run reduction in demand for an input as a result of an increase in its price
is greater than the short run reduction.
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2. Let us suppose that the market for input 2 is stable in the Marshallian and Walrasian
sense (recall section 8B) and has an upward sloping supply function for input 2. Let
z2 = z2(p2), z2′ > 0 be supply function for input 2 and Di(p1, p2), (i = 1, 2) be the demand
functions for the two inputs. We are interested in the effect of an increase in the price of
input 1 on the demand for input 1 given that changes in p1 shift the demand curve in the
market for input 2 and hence cause a change in p2, which then shifts the demand curve
for input 1. Equilibrium in the market for input 2 requires
z2(p2) − D2(p1, p2) = 0
which implicitly defines the equilibrium price of input 2 as a function of the price of
input 1: p2 = p2e ( p1 ) with
dp2e
D 21
=
dp1 z 2′ − D 22
(10.6)
where D2i denotes the partial derivative of D2 with respect to pi. Since we know that
ceteris paribus increases in the price of an input reduce the demand for it (Dii < 0) the
equilibrium price of input 2 increases or decreases with p1 if the inputs are substitutes
(D21 > 0) or complements (D21 < 0). (If the inputs are substitutes an increase in p1 shifts
the demand curve for input 2 to the right and hence increases its equilibrium price.)
Now consider the effect of an increase in p1 on the demand for input 1, allowing for
the induced change in the price of input 2:
dD1 ( p1 , p2e ( p1 ))
dpe
= D11 + D12 2 > D11
dp1
dp1
Since we can usually assume (see question 10A.4) that the cross-price effects on input
demands are equal the last term in this expression must be positive (see (10.6)). Hence
the implication of the interaction of the two markets is that the demand curve for input
1 allowing for induced changes in p2 is less steep than the ceteris paribus demand
curve. The reason is that the induced changes in p2 counteract the direct effects of p1.
3. In choosing her input levels a monopolist takes account of the fact that she faces a
downward sloping demand curve for her product. Thus monopolist m’s demand
function for an input zim ( p1 , p2) does not depend on the price of her output and the
total demand for input i by all the monopolists is just ∑m zim ( p1 , p2). The market demand
curve for input i in the rather unusual case where all the firms are monopolists is the
horizontal sum of their individual demand curves.
4. Recall the discussion of the firm’s profit maximizing input choices zk(p1, p2) and its
maximum profit function π *(p1, p2) (text pages 211–213). The profit maximizing input
demands are, using Hotelling’s Lemma,
zi(p1, p2) = −
∂π * ( p1 , p2 )
∂pi
so that
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Fig. 10B.1
∂z i
∂ 2π*
=
∂p j ∂p i ∂p j
But, by Young’s Theorem, if the partial derivatives of a function are differentiable the
cross partial derivatives are equal. The profit function π* has continuous partials with
respect to input prices if the production possibility set is strictly convex so that there
are no jumps in the profit maximizing net output as result of small changes in relative
prices. Hence if the firm has a strictly convex production possibility set the second
order cross partials will be equal and input k is a substitute for input i if and only if
input i is a substitute for input k.
Exercise 10B
1. Fig. 10B.1 shows how shifts in the supply curve for the input can lead to the same
quantity being demanded at two different prices (part (a)) and different quantities at the
same price (part (b)). In a competitive input market such supply shifts would trace out
the market demand curve with higher prices being associated with a smaller quantity.
2. Suppose that the firm is a monopsonist in the market for input 1 but buys input 2 in a
competitive market. The Lagrangean for the problem of minimizing the expenditure
necessary to produce a required output level is
p1(z1)z1 + p2z2 + λ[y − f(z1, z2)]
where p1(z1) is the inverse supply function for input 1. The first order conditions are
p1′ (z1)z1 + p1 − λf1 = 0
p2 − λf2 = 0
plus the constraint. Rearranging we get
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Fig. 10B.2
f1 p1′ z1 + p1 p1
=
>
f2
p2
p2
(10.7)
so that cost minimization implies that the marginal rate of technical substitution is
greater than the input price ratio. Since the marginal cost of input 1 to the firm exceeds
its price, the firm minimizes its cost by using less of z1 and more of z2 to produce a given
output than it would if it treated the price of input 1 as unaffected by its action.
Fig. 10B.2 illustrates for a simple case in which the production function is
homothetic and p2′′ ≥ 0. The slope of the firm’s isoquants is constant along rays from
the origin. The firm’s isocost curves reflect the fact that the price of input 1 increases as
the firm buys more of it:
dz2
p ′ z + p1
=− 1 1
dz1
p2
and
d 2 z2
p′′z + 2 p1′
=− 1 1
<0
2
dz1
p2
As the reader should confirm the assumption p2′′ ≥ 0 implies that ∑pizi is strictly convex
and so has quasi-convex contours, as shown in the figure. The isoquants become
more steeply sloped as the amount of z1 increases. Hence the expansion path
is not a ray from the origin (as would be the case if the firm treated p1 as a parameter)
but the curve EP. The firm’s cost function can be derived in the usual way from
its expansion path.
Supplementary question
(i) Show that the assumption p1′′ ≥ 0 implies that expenditure is a strictly convex
function of (z1, z2). Is the assumption plausible? What would be the consequences if
p1′ > 0 but p1′′ < 0?
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Fig. 10B.3
3. In Fig. 10B.3 we assume that the market demand curve under competition is identical
to the monopsonist’s MRP curve under monopoly. Without a minimum wage the
competitive equilibrium is a wage of wc and employment of zc. Imposing a minimum
wage of w1 > wc raises the market wage to w1, creating an excess supply of labour since
supply at this wage exceeds demand.
Under monopsony the initial equilibrium without a minimum wage is a wage wm and
employment of zm. With a minimum wage of w0 > wm the firm faces an effective
marginal buyer cost of w0 for z < X0. However for z ≥ X0 the firm must increase the wage
above w0 along the supply curve to generate an increase in supply. Hence for z ≥ X0 its
marginal buyer cost curve is the segment of MBC to the right of X0. Faced with a
marginal buyer cost curve which is discontinous at X0 the firm will choose to employ
X0 > zm workers at a wage of w0 > wm. Thus a minimum wage which does not exceed
the competitive wage will increase employment in a monopsonized labour market. A
higher minimum wage, such as w1 will generate a marginal buyer cost curve which is the
minimum wage line up to X1 and the MBC curve thereafter. The firm’s optimum
employment level will then be at z1 at the minimum wage of w1. At this wage there is an
excess supply of labour but the level of employment is greater than without a minimum
wage. If the minimum wage is w2 there will be an excess supply and the level of
employment is reduced. Thus, under monopsony, imposing a binding minimum wage
increases employment if it is less than U.
4. In Fig. 10B.4 we assume that the supply functions of the two groups are linear. Recall
the discussion of third degree monopoly price discrimination (text pages 195–196). The
firm will buy small amounts of the input (up to z0) from group 1 since over this range the
marginal buyer cost of input 1 MBC1(z1) is less than MBC2(0). For larger amounts the
firm will allocate its purchases between the two groups so as to equate the marginal
buyer costs. Hence the firm’s overall marginal buyer cost curve MBC in part (c) of the
figure is the MBC1(z1) curve up to z0 and the horizontal sum of the MBCi(zi) curves
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Fig. 10B.4
thereafter. The profit maximizing input level is z* where the MBC curve cuts the firm’s
marginal revenue product curve MRP. The firm buys zi* of input i at price pi* , equating
the marginal buyer costs but paying a higher price to group 1 which has a more elastic
supply. (Recall from text page 218 that MBCi = pi(1 + 1/e is ).)
It is best to model the effect of legislation which forbids paying different wages to
the two groups by examining the solution to the firm’s problem of maximizing R(z1 + z2)
− ∑pi(zi)zi subject to the constraint p1 − p2 ≤ δ. The two groups supply an identical input
and so the firm’s output and thus its revenue depends only total employment. We
assume that revenue is an increasing but concave function of total employment: R′ > 0,
R″ < 0. We assume that group 1 has the more elastic supply function and would be
offered a higher price if discrimination was legal. When discrimination is illegal we set δ
= 0. If discrimination is legal the firm’s prices are not constrained and we can capture
this by making δ sufficiently large that the constraint p1 − p2 ≤ δ does not bind. To ensure
a well behaved problem we assume that expenditure on the inputs is a convex function
of (z1, z2). ( pi′′ ≤ 0, i = 1, 2 is sufficient but not necessary for convexity – see question 2
above.)
The Lagrangean for the legally restrained monopsonist is
R(z1 + z2) − ∑pi(zi)zi + λ[p1 − p2 − δ ]
and the first order conditions in the case in which the constraint is binding and both
groups are employed are
R′ − p1′ z1 − p1 + λp1′ = 0
R′ − p2′ z2 − p2 − λp2′ = 0
p1 − p2 − δ = 0
Now totally differentiate the first order conditions with respect to δ to get
R ′′ − p1′′( z12 − λ ) − p1′
R ′′
p1′
R ′′
R ′′ − p2′′( z + λ ) − p2′
2
2
− p2′
p1′ ∂z1 /∂δ 0
− p2′ ∂z 2 /∂δ = 0
0 ∂λ /∂δ 1
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Cramer’s rule yields
∂z1
= [a + ( z22 + λ ) p1′ p2′′]∆−1
∂δ
(10.8)
∂z2
= −[a + ( z12 − λ ) p2′ p1′′]∆−1
∂δ
(10.9)
where a = p1′ p2′ − ( p1′ + p2′ ) R ′′ > 0 and ∆ is the bordered Hessian of the system and is
positive by virtue of the second order conditions (see Appendix I).
In general the effects of forbidding monopsonistic discrimination depend on rather
fine details of the groups’ supply functions and the firm’s revenue function. If we
assume (as in Fig. 10B.4) that the supply functions are linear ( pi′′ = 0) then we see from
(10.8) and (10.9) that relaxing the legal constraint (increasing δ) will increase z1 and
reduce z2. This accords with intuition: the legal constraint forces the firm to reduce w1
and increase w2 compared with the situation in which it is unconstrained. In the linear
case the changes in employment of two groups are exactly offsetting and so total
employment is unchanged. When the firm is not allowed to discriminate it pays both
groups the same wage and therefore faces a supply function S which is the horizontal
sum of S1 and S2. This curve plots the average cost of labour to the firm and gives rise to
the discontinous marginal buyer cost curve abcde. Since the discrimination and no
discrimination marginal buyer cost curves coincide for total employment greater than X
there is no difference in the profit maximizing total employment. The firm is obviously
worse off if it is not allowed to discriminate since it is subject to an additional binding
constraint. The difference in profit is shown in diagram by the area bcd which is the
difference in the firm’s input cost (the difference in the areas under the discrimination
and no discrimination marginal buyer cost curves.)
Supplementary question
(i) Under what circumstances will forbidding discrimination lead to the paradoxical
result that the wage employment of group 1 falls and the employment of group 2
increases? What happens to prices in this case?
5. For given (z1, p1) the firm chooses (z2, . . . , zn) to maximize
) = R(f(z1, z2, . . . , zn)) − p1z1 −
n
∑ p z = )(z , z , . . . , z , p , p , . . . , p )
i
i
1
2
n
1
2
n
i=2
The optimal (z2, . . . , zn) depend on the input prices (p2, . . . , pn) and on (p1, z1):
Xi = Xi(p1, z1, p2, . . . , pn), i = 2, . . . , n. The maximum profit function for given (p1, z1) is
π = π(p1, z1, p2, . . . , pn) = )(z1, X2, . . . , Xn, p1, p2, . . . , pn)
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Fig. 10B.5
The marginal effects of increases in p1 and z1 are
n
∂π
∂)
∂ ) ∂X i
∂)
=
+
=
= − z1
∂p1 ∂p1 i = 2 ∂z i ∂p1 ∂p1
∑
n
∂π
∂)
∂ ) ∂Xi ∂ )
=
+
=
= R ′f1 − p1
∂z1 ∂z1 i=2 ∂zi ∂z1 ∂z1
∑
(Remember ∂)/∂zi = 0, i = 2, . . . , n.) Hence the slope of indifference curves in (p1, z1)
space, after allowing for the fact that changes in (p1, z1) induce changes in (z2, . . . , zn), is
dp1
π
R ′f1 − p1
= − z1 =
dz1 dπ = 0
π p1
z1
Since π is decreasing in p, lower indifference curves in Fig. 10B.5 correspond to higher
profit: the firm is better off on I1 than on I0. The firm will maximize profit by choosing
the (p1, z1) combination which gets it onto the lowest feasible indifference curve. What
(p1, z1) combinations are feasible depend on the input market conditions which the firm
faces.
If the firm was a competitive buyer of the input it would treat p1 as a parameter: it
would be constrained to choose a (p1, z1) combination on the horizontal line with height
equal to the given p1. For example, if p1 = p10 it would maximize profit by choosing an
input level of z10 where its indifference curve I0 is tangent to the horizontal line p10 p10 .
Hence the slope of its indifference curve at the profit maximizing (p1, z1) combination
would be zero and so R′f1 ( z10 ) = p10 . Similarly, if the firm acted as if the input price was a
parameter equal to p11 , it would choose the profit maximizing input level z11 , where I1 is
tangent to the horizontal line p11 p11 and its marginal revenue product is equal to the
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input price. The firm’s demand curve for the input is the locus of such points of
tangency, i.e. its MRP1 curve. Notice that to the left of the MRP1 curve the firm’s
indifference curves are positively sloped and to the right of it they are negatively sloped.
This follows from the fact that to the left of MRP1 increases in z1 at given p1 increase
profit and to the right of it they reduce profit.
When the firm acts as a monopsonist it is constrained by the supply curve of the
competitive input suppliers. It therefore maximizes profit by choosing the (p1, z1)
combination on S1 which yields the highest profit. This is at ( p1* , z1* ) where the
indifference curve I* is tangent to S1. Since the supply curve is positively sloped the
point of tangency between I* and S1 must also occur where I* is positively sloped and
R′f1 ( z1* ) > p1*
The monopsonist will maximize profit by demanding less of the input than would a firm
which treated the input price as a parameter.
6. To focus on the welfare loss from monopsony consider an isolated labour market
where the single employer of z uses it to produce an output x = f(z) which is sold on a
competitive product market at a price of p which is unaffected by the monopsonist’s
employment and output level. There are many workers in the isolated labour market
and each supplies one unit of labour. The inverse labour supply function is w(z) which
shows the wage that workers in this isolated labour market must be paid to induce them
to supply z. Thus w(z) is the height of the supply curve S in Fig. 10B.6. The monopsonist
chooses to employ z* workers at a wage of w* where its marginal buyer cost MBC
equals the value of the marginal product pf′(z).
Because the labour supply curve slopes upward infra marginal workers receive an
economic rent since the wage they get exceeds the wage they require to supply labour
(see text page 220). The total economic rent is the area between the wage line and the
supply curve: area w*ce. (Compare the consumer surplus in section 4C when there is no
income effect.) The monopsonist’s revenue is pf(z) which is the area under the value of
the marginal product curve VMP. Thus monopsonist profit is the area under VMP minus
the cost of labour w(z)z: area dacw*.
Refresh your memory of the discussion of the monopoly welfare loss (section 9D).
Assuming that £1 has the same marginal social value whether it accrues to workers or
to the monopsonist, welfare is the sum of the economic rent of workers and the profit
of the monopsonist. (Because the product market is competitive the price of the
product is unaffected by the monopsonist’s output and so we can ignore the welfare
of consumers.) Welfare is thus the area under the VMP curve minus wz (monopsonist
profit) plus wz minus the area under the labour supply curve (economic rent). Welfare
is maximised at z0 where the area under the VMP curve minus the area under the labour
supply curve is maximised. Alternatively, welfare is the value of output minus the cost
of labour:
W(z) = pf(z) −
∫ w( z)dz
z
0
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Fig. 10B.6
which is maximised when
W′(z) = pf′(z) − w(z) = 0
The welfare loss from monopsony is
W(z0) − W(z*) = p[f(z0) − f(z*)] −
zo
∫ w( z) dz
z*
or the difference between the area under VMP curve and the labour supply curve
between z* and z0.
In Fig. 10B.6 in moving from the welfare maximising (w0, z0) to the monopsony profit
maximising (w*, z*) the monopsonist gains by an amount equal to the area w0fcw*
minus the area afb. The workers lose economic rent equal to the area w0fcw* plus the
area cfb. Thus the welfare loss is the area abc, which is the sum of areas afb and cfb.
Exercise 10C
1. The rent maximizing union chooses (w, z) to maximize [C.1] subject to the inverse
industry demand function w = wd(z) which gives the maxmimum per unit wage wd that
can be charged for a supply of z ie the height of D in text Fig. 10.5. Substituting wd(z) for
w in [C.1] the union maximand is
wd(z)z −
∫ ω ( 0) d 0
z
0
(remember ω (z) is the reservation wage – the lowest per unit wage at which z will be
supplied). The first order condition when the optimal z is positive is
wd′ ( z) z + wd(z) − ω (z) = 0
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which can be rearranged to give
wd ( z) − ω ( z)
= wd′ ( z)
z
the left hand side of which is the slope of the union’s indifference curve (see [C.2]) and
the right hand side is the slope of the industry demand curve. Hence the union chooses
(w*, z*) where the union indifference curve is tangent to the industry demand curve.
2. Since the union will not force members to work at wage w when u(w) < û(U), U is
defined only for u(w) ≥ û(U). The slope of the indifference curve U(w, z) in [C.5] is
dw
[u( w) − û( U)]
=−
≤0
dz U
zu ′( w)
(10.10)
The curvature of the union indifference curves is established by differentiating (10.10)
with respect to z:
dw
d2w
−1
dw
− ( u − û) u ′′z
+ u ′
u ′zu ′
=
2
2
dz
dz
dz U ( zu ′)
Hence the indifference curves have the usual quasi-concave shape if the marginal utility
of income is non-increasing: u″ ≤ 0.
Partially differentiating (10.10) with respect to w and z gives
∂( dw/dz)
−1
=
{u′zu′ − ( u − û) u′′z}
∂w
( zu′)2
∂( dw/dz) u − û
=
∂z
( zu′)2
which shows that the indifference curves get flatter as z increases for given w and, if
u″ ≤ 0, steeper as w increases for given z.
The preferences represented by [C.6] are a special case of those represented by [C.5]
with u(w) = û(U) for w = U and u′ = û′ = 1, so that
dw
( w − U)
=−
dz U
z
Hence, as text page 223 suggested, the indifference curves are rectangular hyperbolas
with horizontal axis at w = U.
The assumption (a) that u(w) > û(U) at w = U means that union members prefer to
be in work rather than unemployed at the same wage. Assumption (b) that du/dw >
dû/dU at w = U means that union members get greater utility from an extra £1 when
employed than when they are unemployed at the same wage. Assumption (a) implies
that unemployment per se has a cost to union members (loss of self esteem, say). This
would mean that U is defined for some w < U and that the indifference curves extend
below the horizontal line at U.
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Fig. 10D.1
Exercise 10D
1. The contract curve in (w, z) space is defined by (a) the tangency of the indifference
curves of the firm and the union (so that the agreement is efficient) and (b) by the
individual rationality constraints that both parties will not accept an agreement which
makes them worse off than without an agreement. With U given by [C.1] and (hence
indifference curves by [C.2]) the contract curve must satisfy the tangency condition
R ′f ′ − w
w−ω
=−
z
z
which is equivalent to
R′(f(z))f′(z) − ω (z) = 0
Since w does not appear in this equation (ω (z) is the reservation wage, whereas w
is the actual wage paid) the contract curve is a vertical line in (w, z) space at z*
determined by the intersection of the supply curve (plotting ω (z)) and the MRP
(plotting R′f′). The individual rationality requirements imply that in Fig. 10D.1 the
union will not accept a wage less than ω (z*) and the firm will not pay a wage greater
than pAP(z*). Thus the contract curve is the vertical line c1c1.
With U given by [C.3] the tangency condition is
R ′f ′ − w
w
=−
z
z
which implies
R′(f(z))f′(z) = 0
The contract curve in Fig. 10D.1 is now the vertical line c2c2 at z** where MRP is zero.
The firm will not pay more than pAP(z**) and the union will accept only non-negative
wages.
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2. (a) The tangency condition for the contract curve is
R ′f ′ − w
u( w) − û( U)
=−
<0
z
z
(10.11)
(See the answer to question 10C.2 for the union’s indifference curve.) Note that since
the union gets ûX in the absence of any agreement it will only make an agreement which
gives u(w) > û(U) and so the indifference curves must have negative slope at the
contract curve. Rearranging the tangency condition gives the implicit function of the
contract curve:
F(w, z) = (R′f′ − w)u′(w) + u(w) − û = 0
with partial derivatives
Fw = −u′ + (R′f′ − w)u″ + u′ = (R′f′ − w)u″ > 0
Fz = (R′f″ + R″f′f′)u′ < 0
The sign of Fw follows from the assumption that u″ < 0 and the fact that the firm’s
indifference curves are tangent to the negatively sloped union indifference curves,
which implies that R′f′ − w < 0. The sign of Fz arises from the fact that MRP declines
with z. Thus the slope of the contract curve is
F
dw
( R ′f ′′ + R ′′f ′f ′) u ′
=− z =−
>0
dz
Fw
( R ′f ′ − w) u ′′
2. (b) (i) Neither the firm nor the union will wish to increase effective labour n beyond
n0 for any given w: union members get the same utility when employed whether working
or playing cards and the revenue to be shared between union and firm would be
reduced by increasing n beyond n0. Conversely if n < n0 making idle employed union
members work effectively would increase revenue and leave their utility unchanged.
Hence for n ≤ n0 all union members who are employed will be used effectively: z = n and
ᐉ = 0. The union and firm indifference curves will be as in (10.11) and the contract curve
will be positively sloped in (w, z) space up to z = n0.
Beyond z = n0 any additional union members who are employed play cards and the
firm’s profit is
R(f(n0)) − (n0 + ᐉ)w = R(f(n0)) − wz
Its indifference curves therefore have slope dw/dz = −w/z. Since union members are
indifferent when employed between working effectively and being idle, the form of the
union objective function is the same for all levels of z. Hence the tangency condition for
contract curve for the region where z ≥ n0 is
−
w
u−û
=−
z
zu ′
which yields the implicit equation
F(w, z) = u(w) − û − u′(w)w = 0
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Fig. 10D.2
Since this equation does not depend on z the contract curve is horizontal for z ≥ n0.
In Fig. 10D.2 we assume that members have the same utility function whether
employed or not. Hence the union will never make an agreement with a wage less than
U. The contract curve is abc. If the actual bargain is along the horizontal segment bc, say
at d, the agreement has zd workers employed at a wage of w0 of whom n0 are
productively employed and zd − n0 are idle in the firm.
(ii) Let the revenue function of the firm be R(f, α) where increases in the shift
parameter α increase revenue for a given output and increase the marginal revenue
product of labour: Rα > 0, Rfα > 0. The positively sloped segment of the contract curve
now satisfies
F(w, z, α) = (Rf f′ − w)u′(w) + u(w) − û = 0
Since Fα = Rf α f′u′ > 0, we see that
∂w
F
= − α < 0,
∂α
Fw
F
∂z
=− α >0
∂α
Fz
Increases in the demand shift parameter α shift the positively sloped segment of the
contract curve down and to the right. An increase in α also increases the output at
which marginal revenue Rf is zero. Hence n0 also increases with α. However, the
equation for the horizontal segment of the contract curve is unaffected by α and so w0
does not change. If the actual bargain is shifted along the horizontal segment as a result
of the change in α the demand shift would have no effect on the wage rate.
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Chapter 11
Capital Markets
Exercise 11B
1. (a) Expenditure on consumption cannot be negative. In the case where A > 0
(borrowing), consumption in period 0 cannot exceed endowed income plus borrowing
and consumption in period 1 cannot exceed endowed income less the repayment of the
loan.
1. (b) Redraw Fig. 11.2 with the point of tangency between the indifference curve and
the wealth line (i) above and to the left of the endowment point, (ii) at the endowment
point.
1. (c) The consumer would have no consumption in one period.
1. (d) The consumer would have linear indifference curves and would choose to spend
all her wealth on consumption in only one of the periods.
1. (e) ρ is the subjective rate of interest: the consumer is willing to accept 1 + ρ
additional consumption in period 1 in exchange for one unit less consumption in period
0. Thus with the stated values of ρ she is willing give up one unit in period 0 in exchange
for 0.8, 1 and 1.2 units of consumption in period 1 respectively.
2. The Lagrangean for the intertemporal utility maximization problem is
L = M 0α M11−α + λ[V0 − M0 − M1/(1 + r)]
The problem is formally identical to the standard utility maximization problem of
chapter 2, except for the notational changes: p0 = 1, p1 = 1/(1 + r), M = V0. We can draw
on the results from question 2D.6 to write the demand functions for current and future
consumption as
M 0* = V0α
M1* = V0(1 − α)(1 + r)
3. No. δ = u1/u0 > 1 implies that the consumer values £1 in the future more highly
than £1 now (corresponding to ρ = 1 − (u1/u0) < 1). The market determined discount
factor µ = 1/(1 + r) is the present value of £1 received in period 1. The consumer is in
equilibrium when her indifference curve is tangent to her wealth line:
−u0/u1 = −1/δ = −(1 + r) = −1/µ ⇒ δ = µ
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Fig. 11B.1
4. Let the interest rates for borrowing and lending be rb > rl. The slope of the budget or
wealth line to the left of the endowment point, where the consumer lends, is −(1 + rᐉ).
To the right of the endowment point R the consumer borrows, so the slope of the
wealth line is −(1 + rb). Thus the budget line is kinked at R in Fig. 11B.1, where three
possible solutions are shown. When the consumer does not borrow or lend ρ(R0, R1) ∈
[rb, rᐉ].
Exercise 11C
1. (a) From [C.3], [C.4] and the envelope theorem, the slope of PP is
dD1 dD1 /dK 1
∂D1 /∂K 1
p f ( L* , K 1 )
=
=− 1 K 1
=
− pK
pK
dD0 dD0 /dK 1
(11.1)
where L*1 = L*1 ( p1 , w, K 1 ). It is apparent that an increase in pK flattens PP.
Applying the usual comparative static procedure to the first order condition p1 fL − w
= 0 on L1 shows that
−1
f f
∂( dD1 /dD0 ) −1
∂L*
=
f K + p1 f KL 1 =
f K − KL L
∂p1
dp1
pK
f LL
pK
As the reader should check by partially differentiating fK/fL with respect to L, this
expression is negative (PP becomes steeper as p1 increases) if L is a normal input. (See
text Fig. 6.4.)
Similarly
∂dD1 /dD0
p
∂L*
p f
= − 1 f KL 1 = − 1 KL
pK
pK f LL
∂w
∂w
and so an increase in w flattens or steepens PP as fLK is negative or positive.
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1. (b) An increase in the depreciation rate means that less capital is available next
period for a given expenditure this period. Hence D1 is smaller for given D0 and PP
pivots down about its intercept on the D0 axis.
1. (c) If disinvestment is impossible and there is no depreciation then K1 ≥ K0 and the
maximum D0 is p0 f ( L*0 , K 0 ) − wL*0 . PP is discontinuous, dropping to the horizontal axis
at this level of D0.
2. Total differentiation of the first order conditions [C.11] to [C.13] gives
p0 f L L
0
0
∂L*0 /∂α −VL α
p1 f L K (1 + r ) −1 ∂L*1 /∂α = −VL α
p1 f K K (1 + r ) −1 ∂K 1* /∂α −VK α
0
0 0
0
p1 f L L
1 1
p1 f L K (1 + r ) −1
1
1
0
1
1
1
1
1
1
where α = p0, p1, pK, w. Using Cramer’s rule we get
∂K 1*
=
∂α
p0 f L L
0
−VL α
0
p1 f L L
−VL α
0
p1 f L K (1 + r ) −1
−VK α
0
0
0
1 1
1
1
1
1
∆
2
Making the appropriate substitutions and using the second order conditions which
imply ∆ < 0; VL L < 0, (t = 0, 1), yields
t t
∂K 1*
= p0 f L L p1 f L L (1 + r ) −1 ∆−1 < 0
∂pK
0
0
1 1
∂K 1*
= p0 f L L p1 (1 + r ) −2 [ f L f L K − f L L f K ]∆−1 > 0 if K1 normal
∂p1
0
0
1
1
1 1
1
∂K 1*
= p0 f L L p1 (1 + r ) −2 f L K ∆−1
∂w1
0
0
1
1
Referring back to question 1 we see that if the parameter change makes the PP curve
flatter for given K1 then the optimal period 1 capital stock is smaller.
3. (a) Draw the wealth line tangent to PP at Ç, so that
p1 f K ( L*1 , K0)/pK = (1 + r)
1
3. (b) Draw a very flat wealth line so that PP is everywhere steeper than the wealth line
and the optimum has D0 = 0 and
p1 f K ( L*1 , Ñ1)/pK > (1 + r)
1
at Ñ1 = [p0 f(L0, K0) − wL0 + pKK0]/pK.
4. Just use [B.9] and [C.8].
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5. Let the amount which the firm borrows be B. Then the present value of the cash flows
available to the shareholders is
V = V(L0, L1, K1, B) = D0 +
D1
1+ r
= p0 f(L0, K0) − wL0 − pK(K1 − K0) + B +
1
[p1 f(L1, K1) − wL1 − B(1 + r)]
(1 + r )
(11.2)
Then clearly variations in B have no effect on the value of the firm to its owners: ∂V/∂B
= 1 − (1 + r)/(1 + r) = 0. The shareholders will not care about the way in which the
investment plan is financed because they have access to the capital market on exactly
the same terms as the firm: they can therefore nullify the effects of B on their period 0
and period 1 incomes by suitable lending or borrowing on their own account. They care
only about the present value of the cash flows, not their timing.
6. (a) When interest payments are tax deductible the present value of the firm’s cash
flows after payment of a tax at the rate t on its dividends is
V = D0 +
D1
1+ r
= [p0 f(L0, K0) − wL0 − pK(K1 − K0) + B](1 − t)
1
[p1 f(L1, K1) − wL1 − B(1 + r)](1 − t)
+
(1 + r )
(11.3)
We see that (11.3) is just (11.2) multiplied by 1 − t. The tax shifts PP radially inwards,
without altering its slope at given K1. It will therefore have no effect on the firm’s
optimal decisions:
V ( L*0 , L*1 , K 1* , B * ) ≥ V(L0, L1, K1, B)
⇒ (1 − t)V ( L*0 , L*1 , K 1* , B * ) ≥ (1 − t)V(L0, L1, K1, B)
Note again that since
∂(1 − t)V/∂B = (1 − t)[1 − (1 + r)/(1 + r)] = 0
the level of borrowing by the firm has no effect on its present value.
6. (b) With interest payments not tax deductible the present value of the firm’s cash
flows is
V = D0 +
D1
1+ r
= [p0 f(L0, K0) − wL0 − pK(K1 − K0) + B](1 − t)
1
{[p1 f(L1, K1) − wL1 − B](1 − t) − rB}
+
(1 + r )
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The effect of additional borrowing (holding all other decision variables constant) on the
wealth of the shareholders is
∂V
1−t
− rt
= (1 − t) −
−1+ r =
∂B
1+ r
1+ r
and so the firm will reduce its borrowing to zero if possible and finance its investment
by reducing D0. If the firm is forced for some reason to finance additional investment by
borrowing, so that B = pKK1 then the first order condition on K1 becomes
p1 f K
rt
dV
=0
− pK −
= (1 − t)
dK 1
1+ r
1+ r
1
instead of [C.13]. The other first order conditions on L0, L1 are essentially unchanged by
the tax. Totally differentiating the first order conditions with respect to t and applying
Cramer’s rule gives
p1 f K
r 1
∂K 1*
+ pK −
= (1 − t)2 p0 f L L p1 f L L −
<0
∂t
1+ r ∆
1+ r
1
0
0
1 1
since the first order condition on K1 implies that the penultimate term must be negative.
7. The monopolist’s first order conditions are similar to [C.11]–[C.15] except that
marginal revenue MRt replaces price pt. For example, [C.15] becomes
r=
MR1 f K
pK
1
−1
The marginal rate of return on investment is still equated to the interest rate. However
the average rate of return is
D1 − pK K 1
>r
pK K 1
since there is no competition to drive the average return down to equal the market rate
of interest.
8. Remembering that variations in K1 alter the optimal labour input in period 1, the rate
of change of the slope of PP is
d[− p1 f K ( L*1 , K 1 )/ pK ]
1
dK 1
= −
p1
dL*
f KL 1 + f KK
pK
dK 1
Using the earlier comparative static result (question 1) that
dL*1
f
= − LK
dK 1
f LL
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Fig. 11C.1
the bracketed term in (11.5) can be rearranged as
(fLL fKK − fKL fKL)
1
f LL
which is negative if the production function is concave. (See Appendix I.) Hence the
slope of PP becomes flatter as K1 increases if the production function is concave.
9. Assume that the borrowing rate is greater than the lending rate: rb > rl. With a single
owner there are three possible optimal production and consumption decisions,
illustrated in Fig. 11C.1.
In case (i) the owner will prefer the production plan a and will then lend to achieve
her optimal consumption. Since she wishes to lend, the relevant rate of interest for
evaluating changes in the firm’s cash flows is the lending rate. Hence a is at the
tangency between PP and a wealth line with slope −(1 + rᐉ). If she wishes to lend this
is the optimal production plan. In case (ii) the owner’s optimal consumption plan
involves borrowing and she prefers the production plan b. At b the value of the firm at
the interest rate rb is maximized where the wealth line with slope −(1 + rb) is tangent to
PP. If she wishes to borrow this is the optimal production plan. In case (iii) the owner
does not borrow or lend on the capital market and consumes the firm’s cash flows. Thus
her optimal consumption and production plans coincide.
The owner’s efficient feasible consumption plans available by production and
transactions in the capital market when borrowing and lending rates differ are those
along the budget curve cabd. The optimal production plan a, b or an intermediate point,
cannot be determined without knowledge of the owner’s preferences. Separation of
production and consumption decisions is not possible if the interest rate the owner
faces depends on his consumption decision. Even worse problems arise if there is more
than one owner. It is possible that some owners wish to borrow and would therefore
prefer production plan b, whilst others wish to lend and prefer production plan a. Under
these circumstances it is not clear what is the firm’s objective function. A similar
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difficulty arises under uncertainty when the incompleteness or imperfection of markets
means that owners could evaluate production decisions differently. See section 21F.
10. The dividend payable would be increased by the proceeds from selling the assets
after production is complete. If the assets had not depreciated:
D1 = p1 f(L1, K1) − wL1 + pKK1
The PP curve would be shifted upward and would have a steeper slope
p1 f K + pK
dD0
=−
dD1
pK
1
The level of investment would be increased.
11. (a) The present value of the firm’s cash flows (shareholders’ wealth) without the
project is V 0 = ∑Dt(1 + r)−t and with it is
V 1 = ∑[Dt + Rt](1 + r)−t = ∑Dt(1 + r)−t + ∑Rt(1 + r)−t = V 0 + NPV
Thus accepting projects with a positive net present value will increase shareholder
wealth.
11. (b) In many types of projects NPV(r) is monotone decreasing with r. (One example
is a project in which there is a sequence of negative cash flows followed by a sequence
of positive cash flows.) If NPV(r) is monotone decreasing with r then NPV(r) > 0 ⇔
i > r and the two criteria are equivalent.
11. (c) (i) NPV(i) = 0 is a polynomial equation in i which has T – 1 roots for a project
with T cash flows. Only if the cash flows satisfy certain conditions will it be the case
that there is a single positive real root with NPV decreasing at this root. In general there
may be multiple roots, in which case it is not clear which of the roots should be
compared with the market rate of interest. (ii) If the market rate of interest differs in
different periods NPV can still be easily calculated but there is no obvious market rate
with which the internal rate of return can be compared. (iii) The internal rate of return
criterion can also be misleading for mutually exclusive projects. Consider project A
with cash flows −10, 15 which has an internal rate of return of 50% and project B with
cash flows −100, 120 and internal rate of return 20%. Project A has a smaller NPV at a
rate of interest of 10%. Thus choosing the project with the larger internal rate of return
could lead to wealth not being maximized.
Supplementary questions
(i) Show that increases in r always reduce NPV if the project consists of a sequence of
negative cash flows followed by a sequence of positive cash flows.
(ii) What is the formula for NPV when the market rate of interest differs in different
time periods?
(iii) Construct a project with two positive internal rates of return.
(iv) How could you adapt the internal rate of return criterion so that it could be used to
decide between mutually exclusive projects?
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Fig. 11D.1
Exercise 11D
1. (a) A simple example has the consumer with utility function
u = min(M0/α0, M1/α1)
and endowment (R0, R1), with R1/R0 < α1/α0. The indifference curves are rectangular and
have corners on the ray with slope α1/α0. The consumer will never borrow. (Draw the
diagram.)
1. (b) Same preferences as in (a) but with R1/R0 > α1/α0.
1. (c) In Fig. 11D.1 a reduction in r shifts the consumer from a to b so that c0 is reduced
and borrowing is reduced. The wealth effect bc outweighs the substitution effect ac.
Since increases in r can make the consumer better or worse off, depending on whether
he is a lender or borrower, to guarantee that Ac is negatively sloped we must assume
that the substitution effect is always larger than the absolute value of the wealth effect.
2. The precise effects depend on the consumer preferences and the changes in the
feasible set. The latter are the same for all consumers.
2. (a) The consumer’s wealth line is shifted out and his endowment point shifted
vertically upward. If M0 is a normal good he will increase M0 and since R0 is unchanged
his borrowing is increased (or lending is reduced).
2. (b) A tax on endowed income shifts the endowment point inward along the ray with
slope R1/R0.
2. (c) The budget constraint is kinked at the endowment point with a slope −(1 + r) to
the right for borrowing and −(1 + r(1 − t)) to the left for lending.
3. and 4. The implications of the changes depend on whether the horizontal sum of the
individual Ac curves is shifted up or down. This depends on the particular assumptions
about preferences and endowments of the different consumers.
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Fig. 11D.2
5. In Fig. 11D.2 aM* is the substitution effect, ab is the wealth effect and bM** the
production effect.
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Chapter 12
General Equilibrium
Exercise 12B
1. With the βhi (individual shareholdings) taken as exogenously given, substituting from
[B.5] in the text into [B.2] allows us to write the consumer’s wealth as
Wh =
∑β ∑ p S (p , p ,..., p )
hi
i
j
ij
1
2
n
j
Since the Sij are homogeneous of degree 0 in prices, W h is homogeneous of degree 1 in
prices, i.e.
Wh ( kp10 , kp20 , . . . , kpn0 ) = kW h ( p10 , p20 , . . . , pn0 )
(Note we have supressed the βhi in the function W h because they are assumed not to
change). When we solve the problem
max uh
s.t.
∑p V
j
hj
≤ Wh = W h(p1, p2, . . . , pn)
j
the parameters of the problem (excluding initial endowments of goods and shares) are
simply the prices pj and so the consumer’s net demands can be written as functions of
these alone. Since both sides of the budget constraint are homogeneous of degree 1 in
prices, demands will be unaffected by an equiproportional increase in prices and so the
net demand functions are homogeneous of degree zero.
2. As the price of a good falls to zero, the demand for it could go to infinity, i.e. its
demand curve has no quantity intercept (for example a Cobb-Douglas utility function
has this feature). The simple form of the non-satiation axiom implies this: more of any
one good is always preferred to less. The solution is to modify this axiom to one of local
non-satiation: for any one good taken individually there is always a satiation point,
beyond which more is indifferent to less, but we assume that every consumer is always
in the region of the consumption space at which she is non-satiated in at least one good.
She will always be on her budget constraint. The reason for being concerned with this
problem is that if, at some points in the set of price-vectors, the consumer’s demand for
a good goes to infinity, we cannot assume the continuity of the mapping from prices to
excess demands, which plays such an important role in the existence proof.
It does seem reasonable to restrict attention to economies in which resources overall
are relatively scarce, even if consumers can be satiated in particular commodities.
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3. It is clear enough that if supply cannot meet demand some consumption plans cannot
be realized. If supply exceeds demand then suppliers cannot sell all they planned but if
unsold output is costless to dispose of then there are no resource implications for
having excess supply and so no modification needs to be made to supply functions
themselves in situations of excess supply. Firms will however prefer to sell output at a
positive price if they can, rather than throw it away, and this of course underlies the
process by which excess supply results in bidding down prices.
Exercise 12C
1. Zero degree homogeneity allows the normalization of the price vectors to make the
price set closed and bounded as well as convex. This, and the continuity of the excess
demand functions then allows Brouwer’s Theorem to be used to establish existence of a
fixed point. Walras’ Law is important in defining the mapping back from the set of
excess demands to the set of price vectors. First, it is used (see question 6 of this
Exercise) to establish that the denominator in [C.20] is non-zero and so the renormalization can be applied. It is then used repeatedly in establishing that the fixed
point of the composite mapping really is an equilibrium.
2. In Fig. 12.3 of the text, point d is to the right of point c because point c generates the
point γ in excess demand space, where z1 > 0 and z2 < 0. Thus, under the rule defining
the reverse mapping k, p1 must be raised and p2 lowered, implying a rightward move
along the line ab. The reason f is to the left of e follows similarly.
3. (a) The mapping is essentially based on the linear functions p′j + kj zj, where p′j ≥ 0
and kj > 0 are constants. Thus any sequence of zj-values tending to a limit, say z 0j will
cause p′j + kj zj to tend to the limit p ′j + k j z 0j . The max(.,.) introduces a kink in
this function, at that z 0j = − p′j / k j , but no discontinuity there, since lim z →z p′j + kj zj = 0 =
p′j + k j z 0j in this case also. Then the denominator is the sum of continuous functions
and is itself therefore continuous, while, as we show below, its value is also bounded
away from zero, so the domain of the mapping is in effect restricted to z-vectors which
do not create difficulties due to the unboundedness of pj.
j
0
j
3. (b) If zj is a linear function of p′j we have
zj = aj + bj p′j
aj ≥ 0
and so
pj = max(0, p′j + kj(aj + b j p′j ))
= max(0, ajkj + (1 + b j k j ) p′j )
If bj < 0 (which we tend to assume intuitively even though Giffen goods may exist) and
bj kj < −1, then the graph in question will, for aj > 0, be a negatively-sloped line until it hits
the p′j axis, at which it then coincides with the axis; or, if aj = 0, it is simply the axis.
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4. If p′ and p″ are such that ∑ p ′j = ∑ p ′′j = 1 and F = kp′ + (1 − k)p″, 0 ≤ k ≤ 1, then
Fj = kp′j + (1 − k) p′′j
and ∑Fj = k ∑ p′j + (1 − k) ∑ p′′j = 1 as required.
5. This is simply a plane with vertices at
(1, 0, 0), (0, 1, 0), (0, 0, 1).
6. Suppose to the contrary that
max[0, p′j + kjzj(p′)] = 0
for every j. Since kj > 0 for all j while p′j > 0 for at least one j, this implies that zj(p′) ≤ 0
for all j and zj(p′) < 0 for at least that one j. But then we cannot have
∑ p′j z j (p′) = 0
as required by Walras’ Law, since at least one term of the sum is strictly negative and
there are no positive terms to cancel it out. So, Walras’ Law ensures the mapping in
[C.16] is well-defined.
Exercise 12D
1. The essential answer to the question can be given algebraically, the sketch of the
figure is left to the reader. We have two excess demand functions z1(p1, p2), z2(p1, p2).
Normalizing prices by setting p2 = 1 and letting p1 now denote the relative price p1/p2, we
note that gross substitutability implies ∂z2/∂p1 > 0. Also, Walras’ Law states
p1z1(p1) + z2(p1) = 0
If p1* is the equilibrium relative price, Walras’ Law holds at this point too, with, also
z1 ( p1* ) = z2 ( p2* ) = 0
Now let p1* fall. By gross substitutability, z2 must fall also – becoming negative – while
therefore, from Walras’ Law, z1 must increase – becoming positive. The converse
follows from increasing p1 from p1* . Thus, graphed against p1, z1(p) is negatively-sloped
and z2(p) is positively-sloped. It follows that if we begin with p10 > p1* and reduce p1 we
move toward equilibrium and if p10 < p1* and we increase p1 we again move toward
equilibrium. Thus the tatonnement process would be stable in this case.
If Y1 > p1* and Y2 < p2* then gross-substitutability implies
z1(Y1, Y2) < 0, z2(Y1, Y2) > 0
Then we must have
( p1* − Y1)z1(Y1, Y2) + ( p2* − Y1)z2(Y1, Y2) > 0
which again gives
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p1* z1(Y1, Y2) + p2* z2(Y1, Y2) > Y1z1(Y1, Y2) + Y2z2(Y1, Y2) = 0
as required.
3. Gross-substitutability refers to the effects of price changes on excess demands
zj(p) = Dj(p) − Sj(p)
Since
∂z j
∂pk
=
∂d j
∂pk
−
∂S j
∂p k
>0j≠k
goods could be (Marshallian) complements in demand as long as they were sufficiently
strong substitutes in supply, or conversely and still satisfy this assumption. Using the
Slutsky equation we have
∂z j
∂pk
= s ik − D k
∂d j
∂pk
−
∂S j
∂pk
>0j≠k
where sik > 0 if the goods are Hicksian substitutes and sik < 0 if they are Hicksian
complements. Both cases are clearly consistent with gross substitutability given that the
other two terms in the expression can take either sign.
Exercise 12E
1. (a) If two consumers are at different points in an Edgeworth box then the sum of
their consumptions of at least one good is not equal to the total amount available.
1. (b) If 1 is at a point southwest of 2, the sum of the consumptions of both goods is less
than their total supply. If 1 is northeast of 2 then the sum of consumptions of each good
exceeds the available supply.
1. (c) At a single point on a vertical side of the box, the consumers are both consuming
x2 but one of them is consuming all of x1. At a single point on the horizontal side we
have the reverse.
1. (d) Rotate the figure for consumer 2 until its origin is diagonally opposite 1’s origin,
and superimpose 2’s figure on 1’s, ensuring that the lengths of the sides reflect available
quantities of goods.
1. (e) The bottom right hand corner of the box 2’s origin.
1. (f) This is somewhat ambiguous, since scales could be chosen for measuring
quantities of the goods such that the box is always square. However, assuming the same
distance on each axis is chosen for ‘one unit’ of each good, the box is square if there are
equal numbers of units of the goods.
A sufficient condition for no trade to take place would be that the consumers’
marginal rates of substitution are equal at the initial endowment point (indifference
curves are tangent there). If this is not the case (indifference curves intersect) some
mutually beneficial trade can always be found.
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I2
02
I1
α
C
I2M
M
l
R2
I1
01
Fig. 12F.1
2. The contract curve would lie along a side of the box if there is a corner solution to the
exchange problem, with one consumer receiving the entire amount of one of the goods
in equilibrium (see (c) of the previous question). At such a point it need no longer be the
case that the consumers’ marginal rates of substitution are equal. We have the following
cases, where MRS12i is consumer i’s marginal rate of substitution between the goods:
(i) if the equilibrium is on the lower horizontal side (2 consumes all the y) then
MRS121 ≥ MRS122 : 2 would not be prepared to compensate 1 at the rate required to move
into the interior of the box (1’s indifference curve is at least as steep as 2’s at the
equilibrium point;
(ii) if on the left hand vertical side then MRS121 ≤ MRS122 at the equilibrium;
(iii) if on the upper horizontal side then MRS121 ≤ MRS122 ;
(iv) if on the right hand vertical side then MRS121 ≥ MRS122 .
The reader should now take Fig. 12.5 of the text and sketch each of these cases.
3. Here, if each consumer’s indifference curves had linear segments then a ‘tangency’
could take the form of the coincidence of these segments for the consumers.
Exercise 12F
1. Essentially 1 should find the allocation which maximizes her utility subject to the
constraint that 2 be on his offer curve. In Fig. 12F.1 (which is based on Fig. 12.7) if
1 offers 2 the price ratio implied by the line α l, 2 will choose point M which makes 1
better off than any other point on 2’s offer curve, including the competitive equilibrium
E. However, this point is not in the core (and cannot be, because it can never be an
intersection point of offer curves) and so can be improved upon. For example, 1 could
be made still better off, and 2 no worse off if they traded along I 2M to point C. Note,
however that point C could not be reached if there were a constraint on the nature
of the trading process, which ruled out any form of exchange except that in which 1
must offer 2 a given price at which all units are to be traded. Diagrammatically, 1 can
only offer 2 a given line from the initial endowment point. This constraint then makes M
the best point for 1. This example relates closely to the discussion of the welfare effects
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of monopoly and price discrimination in Chapter 14. As a link between the
two, note that if the initial endowment point were such that 1 owned all of one of the
goods, then the above analysis would go through essentially unchanged but would
explain why, when a monopolist is contrained to sell all units of his good at the same
price, there is an economic inefficiency in the sense that the resulting allocation is not
in the core of the economy. To reach a point in the core other than a competitive
equilibrium, ‘price discrimination’ or ‘non-linear pricing’ must be feasible, under which
different units of the good are sold at different prices.
2. If x* is chosen at price p* and x′ is strictly preferred to x* then it must cost more, i.e.
p*x′ > p*x*, otherwise it would have been chosen. If x′ and x* are indifferent, but x* is
chosen at point p*, then strict convexity of preferences again implies that px′ > p*x*
(review the discussion of the expenditure function in Chapter 3).
3. Just go through equations [F.1] to [F.11] for this 4-consumer case.
4. Since consumers of the same type have identical preferences and initial endowments
they solve exactly the same optimization problem. If preferences are strictly convex, the
solution to the problem is unique and so they all consume the same bundle in
equilibrium.
5. Simply set xB′ 1 = xB′ 2 and y B′ 1 = y B′ 2 in [F.15] and [F.16] of the text and note that the
proof goes through just as before, noting that B1 would be indifferent to the trade
(therefore why not make it?) while A1 will be strictly better off.
6. All we need to do is let A2 drop his deal with B2 and return to the initial endowment
point. He can then trade with A1 to reach c0 in text Fig. 12.9.
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Chapter 13
Welfare Economics
Exercise 13B
1. Rewrite the Lagrangean [B.5] interchanging the two individuals so that the first two
terms are u2(x21, x22, z2) + λ[E1 − u1(x11, x12, z1)]. The first order conditions are identical in
form to [B.6] to [B.11] except that u1i , u1z are interchanged with ui2 , u z2 . The tangency
conditions are also identical in form since the Lagrange multiplier λ on the utility
constraint cancels out. In terms of text Fig. 13.1, it does not matter whether we
determine FF by maximizing u1 for given u2 or vice versa.
2. (a) It is sufficient to assume strictly quasi-concave utility functions and strictly
concave production functions.
2. (b) The endowment constraint zh ≤ Kh binds if individual h’s marginal rate of
substitution between the input and one of the goods it produces (the marginal cost of
his input in terms of that good) is less than the marginal product of his input in
producing that good. If we interpret zh as labour then the marginal cost of labour can be
plausibly assumed to become arbitrarily large as zh tends to Kh, so that the constraint
will not bind when the input is labour.
2. (c) The tangency conditions are replaced by suitable inequalities (as for example in
the consumer optimization example in section 2C).
2. (d) The multipliers measure the rate at which the maximized value of the objective
function changes as there is a marginal change in the relevant constraint parameter.
Thus λ is the rate at which u1 falls as E2 increases (it is the reciprocal of the slope of FF
in Fig. 13.1). ρi is the rate at which u1 increases if the material balance constraint xi ≥
∑xhi is relaxed to xi + ∆i ≥ ∑xhi where ∆i is a small endowment of good i. ωi is the value
of relaxing [B.3] by increasing the endowment of input h. µi is the rate at which u1
increases if the output produced by (zi1, zi2) is increased from f i(zi1, zi2) to f i(zil, zi2) + ∆i.
Note that µi = ρi: relaxations in [B.1] and [B.2] are equally valuable.
Exercise 13C
1. (a) The Lagrangean is
L = W(u1, u2, . . . , uH) +
∑ ρ [x − ∑ x ]
i
i
ih
i
+
∑ ω [z − ∑ z ] + ∑ µ [f (z , . . . , z ) − x ]
i
h
h
160
h
h
ih
i
i
i1
i
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Fig. 13C.1
with first order conditions for a non-corner solution
∂L
∂x ih
= Wh uih − ρi = 0,
h = 1, . . . , H;
∂L
∂zh
= Wh uzh + ωi = 0,
h = 1, . . . , H
∂L
∂zih
= −ωh − µ i f hi = 0,
h = 1, . . . , H;
∂L
∂xi
= ρi − µi = 0,
i = 1, . . . , n
i = 1, . . . , n
i = 1, . . . , n
plus the constraints.
1. (b) The Wh terms can be cancelled out by suitable rearrangement to give a set of
tangency conditions which are identical in form to the efficiency conditions of section
B. For example write the conditions on xih and xjh as Wh uih = ρi, Wh u hj = ρj and divide one
equation by the other to give
u ih ρi
=
,
u hj ρ j
h = 1, . . . , H;
i, j = 1, . . . , n;
i≠j
2. See Fig. 13C.1. The welfare contours are right angled about the 45° line in (u1, u2)
space. Only if the FF curve was upward sloping as it crossed the 45° line would the
Rawlsian optimum not be at the 45° line where the individuals have equal utility. For
example, it may be necessary to give one of the individuals a larger income than the
other if that individual is very productive and cannot be induced to provide large output
in any other way: there is a trade off between equity and total output. Note that the
Rawlsian optimum is always Pareto efficient.
3. The swf contours of a utilitarian are straight lines with slope −1, as in Fig. 13C.1. The
optimal allocation will not in general have equal utilities. If FF is symmetric about the
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Fig. 13C.2
45° line the utilitarian and Rawlsian optima coincide. This requires that the individuals
have identical preferences and productivities.
4. The slope of the contours of W in (x1h, x2h) space if W = W(u1, u2) are
dx 2 h Wh u1h u1h
=
=
dx1 h Wh u2h u2h
which depends only on the bundle consumed by h. A non-paternal swf is weakly
separable in the individual consumption bundles.
5. In Fig. 13C.2 a is measured horizontally and, without any loss in generality, we have
set a1 = 0 (no restriction) and a2 = 1 (say a complete ban). OS is the origin for the
smoker, who prefers a larger income yS and smaller a (less restrictive law). ON is the
origin for the non-smoker who prefers more income yN and a more restrictive law.
The vertical side of the box measures the total income of the two parties. The initial
position is at b.
After the ban the new position is at c. N is better off at c than b and could pay up to
ce and still be better off as a result of the change from a1 to a2. Thus CV12N = ce. S is
worse off and would have to be compensated with cd additional income: CV12S = −cd.
The ban passes the Hicks-Kaldor compensation test since ∑i CV12i = de > 0
A move from a2 to a1 (c to b) makes N worse off ( CV21N = −bf) and S better off
( CV21S = bg). Since ∑i CV21i = gf > 0 removing the ban also passes the Hicks-Kaldor
compensation test.
A paradox requires
CV12S + CV12N > 0
and
CV21S + CV21N > 0
which implies
( CV12N + CV21N ) + ( CV12S + CV21S ) > 0
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Fig. 13C.3
If a is a normal good for N, then, since he is better off at c than b, the amount he is
willing to pay for the increase in a is less than the amount she would have to be paid in
compensation for the decrease: CV12N < − CV21N or CV12N + CV21N < 0. If a is a normal good
for S then the amount she must be paid in compensation for the increase in a is greater
than the amount which she would be willing to pay for the reduction in a: − CV12S > CV21S
or CV12S + CV21S < 0. Hence if a is normal for both parties (13.1) cannot be satisfied and
so the paradox cannot arise. Equivalently: a necessary condition for a paradox is that a
is an inferior good for at least one of the parties.
6. In Fig. 13C.3 a2 passes the Scitovsky test relative to a1 and a3 passes the Scitovsky test
relative to a2. But a1 passes the Scitovsky test relative to a3.
7. From the definition of the compensating variation
vh(yh − CV h, p1) = vh(yh, p0)
and so the change in W as p changes from p0 to p1 is
W(v1(y1, p1), v2(y2, p1)) − W(v1(y1 − CV 1, p1), v2(y2 − CV 2, p1))
=
∑∫
0
h
=σ
CV h
∑∫
h
Wh vyh ( yh − CV h − Sh, p1)dSh
CV h
0
dSh = σ
∑ CV
h
h
if Wh vyh = σ, h = 1, 2, so that the social marginal utility of income is constant and equal
across all consumers.
Exercise 13D
1. If there is no market in one of the goods we must specify how its level is determined
at the equilibrium of the economy. An obvious assumption is that the individuals have
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an endowment of this good (say good 0) and since they cannot trade it or firms produce
it, they are forced to consume their endowment of the good 0. For goods for which
there are markets consumers will adjust their consumption until uih = λhpi where λh is
consumer h’s marginal utility of income and pi the market price of good i. The marginal
rate of substitution between good i and the unmarketed good 0 is therefore
u0h
u0h
=
u ih λ h p i
and since there is no market in which consumers can adjust their consumption of good
0 until u0h = λhp0 their marginal rates of substitution between the unmarketed good and
the marketed goods will not be equalised.
2. Let xih be the net demand for good i by individual h (the difference between
consumption and endowment). Then from the budget constraints x1h + px2h = 0, we can
write
uh(x1h, x2h) = uh(−px2h, x2h)
where p is the price of good 2 in terms of good 1. Individual 2 treats p as a parameter
and announces her true utility maximizing demand x22(p) at p where x22(p) satisfies
du2/dx22 = − u12 p + u22 = 0
and we assume for simplicity that x22
′ ( p) < 0. The Walrasian equilibrium p is determined
by the equilibrium condition for market 2:
x21 + x22(p) = 0
(13.2)
(remember Walras Law implies that with both individuals’ budget constraints binding
equilibrium in n − 1 markets implies that there is equilibrium in the remaining market).
Individual 1 realises that (13.2) implies that p depends on his announced demand:
p = p(x21) with p′(x21) = −1/ x22
′ ( p) > 0. He therefore chooses x21 to maximize
u1 = u1(−p(x21)x21, x21)
so that the first order condition is
−[p + x21p′(x21)] u11 + u21 = 0
Hence the equilibrium satisfies
u21
1
1 u22
=
≠
=
u11 p + x 21 p ′ p u12
and is not Pareto efficient.
See Fig. 13D.1 (and compare Fig. 12.7). Individual 1 realises that individual 2 will
always announce a demand on her offer curve αR2 through her endowment α. By
announcing a demand of V21 individual 1 can ensure an equilibrium price of Y and get to
β where his utility is maximized along αR2. Compared with the Walrasian equilibrium
price p* at E, individual 1 has restricted his supply of the good he is selling and driven
up its relative price, making himself better off and individual 2 worse off.
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Fig. 13D.1
Exercise 13E
1. Let ti be the per unit tax on good i, so that firms get the net price pi and consumers pay pi + ti. Since all consumers face the same relative gross of tax price ratios (pi
+ ti)/(pj + tj) for goods consumed, their MRS for these goods are equal. The budget
constraint for consumer h is
∑ ( p + t )x − wz − ∑ β π = 0
k
i
i
hi
hk
h
k
Consumers’ MRSiz between consumption of good i and supply of the input is set equal to
the relative price ratio w/(pi + ti). But firms adjust their use of the input in production of
good i until the net of tax value of the increase in the output of good i equals the price
of the input : piMPi = w. Hence MPi = w/pi > MRSiz. Consumers and firms place different
marginal values on the input in terms of good i, thus violating [B.7]. An equal rate
proportional purchase tax at the rate t would also violate [B.7] and would be equivalent
to an income tax at the rate θ = t/(1 + t) since
∑ p (1 + t)x − wz − ∑ β π = 0
k
i
hi
hk
h
k
and
∑ p x − w(1 − θ)z − (1 − θ) ∑ β π = 0
k
i
hi
hk
h
k
are equivalent constraints.
2. Since potential tax payers can always emigrate we cannot think of any non-distorting
tax or subsidy.
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Exercise 13F
1. One example is (a, b, c)1, (a, c, b)2, (b, c, a)3 which yields [a, b], [b, c], [a, c].
3. See text page 310. To demonstrate that acyclicity is less demanding consider the
preference orderings (a, b, c)1, (a, b, c)2, (c, a, b)3, (b, c, a)4 with a majority voting rule
which ranks two alternatives as indifferent if they have the same number of votes. Then
[a, b], [b, c] but a and c are indifferent. Hence transitivity is violated but acyclicity is not.
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Chapter 14
Market Failure and Government Failure
Exercise 14B
1. (a) A lump sum or proportional profits tax has no effect on monopoly output but
transfers income from the owners of the monopoly to the beneficiaries of the
government expenditure or reductions in other taxation financed with the proceeds.
1. (b) The inefficiency is worsened: a tax on sales reduces output because marginal
revenue is reduced.
1. (c) Since marginal revenue with a per unit subsidy of s is d(p + s)x/dx = p + s + p′x,
setting s = −p′x leads the monopolist to produce where p = MC.
1. (d) Setting pmax = p* where p* is defined by the intersection of the demand and
marginal cost curves induces an efficient output.
1. (e) Competition for the right to be a monopolist will lead the highest bidder to make a
bid equal to the maximum profit that the monopoly could earn. The successful bidder
will then choose a profit maximizing output. The bidding regime merely appropriates
the monopoly profit for the government.
1. (f) Competition amongst bidders will drive the profit from the monopoly to zero and
the bid price down to average cost, not to marginal cost.
Supplementary question
(i) Draw diagrams to show that a price equal to average cost could be better or worse
than the profit maximizing price.
2. (a) First degree price discrimination. Type i consumers have quasi-linear utility
Ui(xi) + yi and their marginal rates of substitution between the monopolised good and
the composite commodity are
MRS i =
dyi
U ′( x )
= − i i = −U i′( xi )
dxi
1
The marginal rate of transformation between the composite commodity and the
monopolised good is the marginal cost of the monopolised good: the amount of the
composite commodity that must be given up to produce one additional unit of the
monopolised good. The monopoly sets a price to type i equal to marginal cost: pi = c and
each type chooses xi so that
U 1′ (x1) = p1 = c = p2 = U 2′ (x2)
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Hence marginal rates of substitution are equal for the two types and equal to the
marginal rate of transformation. Thus the necessary conditions for Pareto Efficiency
are satisfied. (There are no changes in the allocation which will make either type of
consumer or the monopolist better off without making someone else worse off.)
2. (b) Second degree price discrimination. Recall from page 202 that type 2 consumers
who place a higher marginal value on the good (have a higher marginal rate of
substitution) face a price equal to marginal cost, so that MRS2 = p2 = c. But type 1
consumers face a price above marginal cost because the monopolist does not observe
consumer types and has to offer a menu of prices and fixed charges which induces the
consumers to reveal their types honestly. Thus MRS1 = p1 > c.
If type 2 consumers could trade with type 1 consumers they could offer to buy some
of the type 1’s consumption at a price which is above marginal cost and below the price
that the type 2 consumers are paying. Both types would be better off. The monopolist
would be no worse off, provided that both types continued to buy the same amount
from the monopolist, so that the arbitrage merely redistributed consumption between
purchasers.
2. (c) Third degree price discrimination. The consumers in the separated markets face
different prices (and both prices are above marginal cost) so that marginal rates of
substitution are not equalised and are greater than the marginal rate of transformation.
Consumers buying in the low price market could resell the good to consumers in the
high price market at a price between the low and high prices set by the monopolist and
both types of consumers would be better off. The monopolist would be no worse off
provided that the amounts sold by the monopolist to the two markets did not alter.
3. Let p = p(x, q) be the inverse demand function relating the price (consumers’ marginal
willingness to pay) to output x and quality q, with px < 0, pq > 0. Let c(x, q),
cx > 0, cq > 0 be the cost function. Using willingness to pay (the area under the
demand curve) less cost as the measure of social benefit from the good, the marginal
benefit from quality improvement is
∫ p ( 8, q)d8 − c (x, q)
x
0
q
q
whereas the marginal effect on profit is
pq(x, q)x − cq(x, q)
In general the marginal social benefit and the marginal private benefit to the monopolist from quality changes will not be equal. In Fig. 14B.1 for example, increasing q
from q0 to q1 increases revenue by the rectangle (p1 − p0)x, whereas the benefit to
consumers increases by the shaded area abcd. Since the firm’s reward for increasing
quality differs from the social benefit it will choose the wrong quality. In the example in
the figure it will choose too low a quality.
The fact that quality is a public good for consumers increases the difficulty of
forming a coalition to share the costs of bribing the firm to vary its quality level.
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Fig. 14B.1
Fig. 14B.2
Supplementary questions
(i) Draw a diagram in which the monopolist has too great an incentive to improve
quality.
(ii) What restrictions on p(x, q) ensure that the monopolist’s quality choice is welfare
maximizing at any given quantity?
4. (a) Assume that the welfare function is
W = −x − π (x)L
so that P and D have equal weight. The socially optimal care level x* minimizes the sum
of the cost of care and expected accident costs and satisfies
Wx = −1 − π ′(x*)L = 0
See Fig. 14B.2.
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4. (b) With the three legal regimes D’s objective function u(x) is
No liability
Strict liability
Negligence
u = −x
u = −x − π(x)L
u = −x − π(x)L
u = −x
if x < x0
if x ≥ x0
where x0 is the due care standard set by the courts: the defendant is held to be negligent
if care is less than the due care standard.
No liability leads to no care and strict liability to efficient care. See Fig. 14B.2 for the
negligence regime. Under such a regime D’s objective function is discontinuous at the
due care level x0: her expected costs are shown by the upper curve plotting x + π(x)L up
to x0 and by the 450 line beyond x0. Consider the level of care V defined by x* + π(x*)L =
V. When x0 < V the optimal care level for D is x = x0 and when x0 > V her optimal care is
x*. Hence if the due care standard is x0 = x* or x0 > V D will take an efficient amount of
care. Note that a high enough due care standard is equivalent to strict liability and
induces efficiency.
4. (c) The welfare function is now
W = −x − y − π(x, y)L
and the efficient care levels satisfy
Wx = −1 − πx(x*, y*)L = 0
Wy = −1 − πy(x*, y*)L = 0
If D is not liable she sets x = 0. However P’s care is efficient (for given x) since he bears
the full cost of the accident and so chooses y to minimize y + π(x, y)L.
If D is strictly liable she takes an efficient level of care (for given y) but P takes no
care. Since he never bears any of the accident costs he has no incentive to take care.
Under a negligence regime if D takes due care (x = x0), she will not be liable and P will
bear the full accident cost and will be motivated to take efficient care (for given x).
Hence if the due care standard is set at x0 = x* both parties will take efficient care.
5. (a) and (b) The equilibrium input level is determined by the breakeven condition
(p − t)f(L) = (w + θ)L
where t, θ are taxes on output and the input respectively. Hence the equilibrium input is
Ö = Ö((p − t)/(w + θ)). Since Ö′ > 0 increases in either tax will reduce the equilibrium
input and output, the efficient input level L* can be achieved by setting suitable taxes so
that Ö((p − t)/(w + θ)) = L*. Since the equilibrium depends on the relative price ratio the
optimal taxes are not unique.
5. (c) A firm with monopoly power over output would choose L to maximize p(f(L))f(L)
− wL. There would be no over exploitation since the resource is no longer free access
but the firm would choose an inefficiently small input level.
6. Fig. 14B.3 plots the value of the average and marginal product curves.
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Fig. 14B.3
Invert the production function to get the input requirement function L(q). Average
cost is wL(q) so that the breakeven condition is pq = wL(q) or
pq/w − L(q) = rq − L(q) = 0
where r = p/w. Hence the supply function is
dq
−q
−q
=
=
dr r − L ′( q) ( p/ w) − (1 f ′)
Since pf′ < w at the equilibrium, dq/dr > 0 when f′ > 0 and dq/dr < 0 when f′ < 0. Hence
the backward bending supply curve in Fig. 14B.4. With r = r0 the equilibrium is at q0.
Let a(q), a′ > 0 be the cost to a driver of making a trip along a road where q is the
number of drivers making the trip. (The cost includes the value of the time taken over
the trip. As the number of drivers increases the time each takes increases and so the
cost of the trip for each driver increases.) Each driver decides to make the trip if the
benefit from it exceeds its cost. If we rank the drivers in decreasing order of the size
of their benefits we can derive a function b(q), b′ < 0 which gives the benefit to
the marginal driver when there are q drivers. The equilibrium number of drivers is
determined by b(q) = a(q). However the social cost of q trips is qa(q) so that the
marginal social cost of an additional trip is
a(q) + qa′(q) > a(q)
Each driver takes account only of the cost of a trip to them (a) and ignores the fact that
their journey increases the costs of all other drivers making the trip. Thus the private
marginal cost is the average social cost. The difference between private and social
marginal cost is the congestions cost qa′(q).
We can illustrate this in Fig. 14B.4 if we let the horizontal axis measure the number
of trips and the vertical axis measure b and a. The equilibrium number of trips is q0
where the demand curve for trips b(q) cuts the marginal private cost curve a(q). The
efficient number of trips is q** where marginal social benefit and marginal social cost of
trips are equal. This could be achieved by levying a toll of t = q**a′(q**) per trip.
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Fig. 14B.4
7. The material balance constraints are yi ≥ ∑hxhi, i = 1, . . . , n; q ≥ q1 = . . . = qH, where
yi is the net output of commodity i (yi < 0 if commodity i is an input), xhi is the
net consumption (consumption less endowment) of commodity i by individual h,
q is the output of the public good, qh is individual h’s consumption of the public
good. The production function constraint is written implicitly (see section 5E) as
g(y1, . . . , yn, q) ≤ 0.
The Lagrangean for the welfare maximization problem is
L = W(u1, . . . , uH) +
∑ ρ (y − ∑ x ) − µg(y , . . . , y , q)
i
i
i
hi
1
n
(14.1)
h
where uh(xh1, . . . , xhn, qh) is h’s utility function defined in terms of net consumption (see
section 2E). We assume that all the constraints bind so that we can substitute q for qh.
With a non-corner solution the first order conditions are
∂L
= Whuih − ρ i = 0,
∂xhi
∂L
= ρi − µgi = 0,
∂yi
∂L
∂q
=
h = 1, . . . , H;
i = 1, . . . , n
i = 1, . . . , n
∑ u − µg = 0
h
q
q
h
plus the constraints.
Use the conditions on the xhi to substitute pi / uih for Wh in the condition on q,
substitute µgi for pi and rearrange to get
− uqh
∑u
h
h
i
=−
gq
gi
The left hand side is the sum of consumers’ marginal rates of substitution between the
public good and commodity i and the right hand side is the marginal rate of
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substitution between q and i. Thus efficiency requires that consumers’ summed
marginal valuations of the public good in terms of commodity i should be equal to its
marginal cost in terms of commodity i.
If the public good was optional (e.g. a broadcast) the material balance constraint on
the public good becomes q ≥ qh, h = 1, . . . , H. Adding ∑hλh(q − qh) to (14.1) yields first
order conditions which are very similar to those previously derived except for
∂L
= − µg q +
λh = 0
∂q
h
∑
∂L
= Wh uqh − λ h = 0,
h
∂q
h = 1, . . . , H
Substituting for λh gives the same efficiency conditions as before.
9. Suppose ∑j≠i Tj > 0. Then i gets
Ti +
if
vi
∑T > 0
j
j≠i
and
−Ti = −
∑T
if
j
Ti +
j≠i
∑T ≤ 0
j
j≠i
Hence if vi ≥ − ∑j≠i Tj then i should announce Ti ≥ vi and get a payoff of vi. On the other
hand if vi < − ∑j≠i Tj then i should announce Ti ≤ vi and get a payoff of −Ti = −∑j≠i Tj
Conversely suppose ∑j≠i Tj ≤ 0. Then i gets
vi − Ti = vi +
∑T
j
if
Ti +
j≠i
∑T > 0
j
j≠i
and
0
if
Ti +
∑T ≤ 0
j
j≠i
Hence if vi + ∑j≠i Tj ≥ 0 then i should announce Ti ≥ vi ≥ −∑j≠i Tj and get a payoff of vi − Ti
= vi + ∑j≠i Tj. On the other hand if vi + ∑j≠i Tj < 0 then i should announce Ti ≤ vi ≤ − ∑j≠i Tj
and get a payoff of 0.
Thus if i knows all the other announcements he cannot do better than announce
Ti = vi ie tell the truth whatever the strategies of the other players.
9. (b) Individuals 3 and 4 are better off without the project but will not individually
misrepresent their preferences to prevent it given that all other individuals tell the truth
because each would have to pay the tax ∑ j ≠ i v j . Suppose that they both agree to report
Ti < −25, i = 3, 4. Then since T4 + v1 + v2 < 0 individual 3 can report T3 < −25 and not have
to bear the tax because her report does not alter the decision. Similarly, since T3 + v1 + v2
< 0 individual 4 can report T4 < −25 and also not have to bear the tax because his report
does not alter the decision either. Since such reports stop the project and do not lead to
either of them bearing the tax, individuals 3 and 4 have an incentive to collude.
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Exercise 14C
1. (a) Monopoly i has the profit function
πi(p1, p2) = piDi(p1, p2) − ci(Di(p1, p2))
The private sector monopoly 2 chooses p2 to maximize its profit, so that
π22(p1, p2) = D2 + (p2 − c2′ ) D22 = 0
(14.2)
holds. This first order condition implicitly defines the profit maximizing price as a
function of the price of the public monopoly: p2 = f(p1). The welfare criterion is the sum
of consumer surplus and firm profit:
W=
∑ ∫ D dp + ∑ π = W ( p , p )
∞
i
pi
i
i
i
1
2
i
and
Wi = −Di + πii = (pi − ci′) Dii
(We assume that income effects are zero so that Dij = Dji and the sum of consumer
surpluses does not depend on the path over which prices vary.)
The welfare maximizing p1 given that p2 = f(p1) satisfies
dp
dW
= W1 + W2 2 = ( p1 − c1′ ) D11 + ( p2 − c 2′ ) f ′( p1 ) = 0
dp1
dp1
and suitable manipulation and division by p1, p2, D1, D2 gives the expression in the text.
1. (b) Partial differentiation of π22 with respect to p1 gives
π221 = D21 + (p2 − c2′ ) D221 − c 2′′D22 D12
Now if the demand function for good 2 is of the form D2 = a(p1) − bp2, a > 0, b > 0 and
firm 2 has constant marginal cost ( c 2′′ = 0) then the usual comparative static analysis
establishes that
sign f ′(p1) = sign π221 = sign D21
Thus if the goods are substitutes (Dij < 0) the optimal second best price of good 1
exceeds its marginal cost.
Exercise 14D
1. (a) There are always majorities for increasing q from qi to qi+1 if i < M = (n + 1)/2.
Conversely for reducing q from qi to qi−1 if i > M. At qM a majority will vote against an
increase or a decrease.
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Fig. 14D.1
1. (b) At qM we have B M′ (qM) = c′(qM)/n, whereas the condition for efficiency in public
good supply is ∑ Bi′ (q*) = c′(q*). Thus the median voter’s preferred supply is efficient
only if BM
′ (q) = ∑ Bi′ (q)/n ie the median voter’s marginal benefit equals the mean
marginal benefit. If tastes ( Bi′) are symmetrically distributed then q* = qM.
2. (a) Suppose we ignore the preferences of bureaucrats and define an efficient
allocation as one which maximizes the benefit to consumers less expenditure: B(q) −
C(q) − w. Then the efficient allocation has B′(q*) = C′(q*) and w = 0.
2. (b) The bureau wishes to maximize its total allocation which is equal to its actual
expenditure A = E(q) = C(q) + w. Politicians know q and B(q) but cannot tell how much
of actual expenditure is wasted. They are willing to allocate up to the value of the total
benefits: A = kB(q), k ∈ [0, 1]. The bureaucrats will always set k = 1 for given q ie they
will extract the maximum amount that politicians are willing to pay for q. Thus the
bureaucrats’ problem is to choose q and w to maximize C(q) + w subject to C(q) + w =
B(q). Depending on the benefit and cost functions there are two types of solution,
illustrated in Fig. 14D.1.
(i) If C = C1(q), the bureau will maximize A at q1, where C1(q1) = B(q1). There is no
waste since w = 0. Note that the efficient output is q* < q1.
(ii) If C = C2(q) the bureau will set q = q2 where B′(q2) = 0 since any larger q reduces B
and therefore A. Since B(q2) > C2(q2) and the bureaucrats wish to maximize total
expenditure they set w = B(q2) − C2(q2), again giving a total expenditure equal to the
benefit.
3. (a) The welfare function is
W=
∞
∫ D( 6)d 6 + pD(p) − C(D(p))
p
which is maximized when
dW/dp = −D(p) + D(p) + [p − C(p)]D′(p) = 0
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so that the optimal price is marginal cost. In what follows it is easier to have output x
rather than price as the choice variable. Invert the demand function to get p = p(x)
which is the height of the demand curve or consumers’ marginal willingness to pay for
the good. We can write the welfare function equivalently as the difference between
consumers’ total willingness to pay for the good less cost of production:
W=
∫ p( 8 )d 8 − C(x)
x
0
which is maximized when
dW/dx = p(x*) − C′(x*) = 0
3. (b) Assume that the mark up pricing constraint is binding in that at the unregulated
profit maximizing output level x0 we have p(x0) > (1 + k)C(x0)/x0. Since the firm is on its
demand curve and faces a binding markup constraint its output is determined by
p(x)x − (1 + k)[C(x) + w] = R(x) − (1 + k)[C(x) + w] = 0
(14.3)
Thus its only degree of freedom is the amount of waste w and (14.3) implicitly defines
output as a function of waste: x = x(w). By increasing w the firm increases its recorded
average cost [C(x) + w]/x and so can move up its demand curve, increasing its price and
reducing its output. From (14.3)
dx
1+k
=
<0
dw R ′( x) − (1 + k)C ′( x)
(remember R′ < C′ since the pricing constraint forces the firm to increase output beyond
the profit maximizing level where R(x0) = C′(x0)).
The regulated firm’s profit function is
Π(w) = R(x(w)) − C(x(w)) − w
and
Π′(w) = (R′ − C′)
dx
kR ′
−1=
dw
R ′ − (1 + k)C ′
(14.4)
There are two types of profit maximizing solution.
(i) If Π′(0) > 0 the firm sets w > 0. Since the numerator in (14.4) is negative, Π′(0) > 0
requires R′(x(0)) < 0. In this case the price constraint increases the firm’s output beyond
the point at which marginal revenue is zero. The firm will therefore choose to produce
wastefully in order to increase recorded average cost to push up price and reduce
output until revenue is maximized: R′(x(w)) = 0.
(ii) If Π′(0) ≤ 0 the firm sets w = 0 and output and price are x(0) and p(x(0)). The
price constraint means that profit is proportional to revenue:
Π=R−c−w=R−
px
k
= R( x( w))
1+k
1+k
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Thus it will choose w (and hence x) to maximize revenue. Since x′(w) < 0 it sets w = 0 if
R′(x(0)) > 0.
3. (c) Yes. If the firm has increasing average cost the output at which average cost
equals price could be worse than the output determined by marginal revenue equals
marginal cost. This is true whether waste is positive or zero. (Draw a diagram.)
Supplementary questions
(i) Draw diagrams to illustrate these two solution types.
(ii) Can markup regulation reduce welfare if the firm has decreasing average cost?
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Chapter 15
Game Theory
Exercise 15B
1. The payoff matrix for this game is
C
NC
C
2, 2
5, 0
NC
0, 5
1, 1
where C denotes confess and NC not confess. Since the numbers in the table represent
prison sentences, the smaller the number the better. Thus regardless of what the other
does, it is a dominant strategy to confess, but then each receives a longer sentence than
if both did not confess. The idea that the logical pursuit of self interest can make
everyone worse off than if they did not behave in this way, contrary to the results of
neoclassical welfare economics, in which self seeking behaviour leads to a Pareto
efficient outcome (see Chapter 13), is considered to be a very important lesson of this
game. It suggests the need for rules, norms or social conditioning which could lead
people to choose the socially preferable outcome rather than the individually rational
one. Note of course that “society” in this case means the players of the game.
2. Firm B’s profit function is
vB = 40qB − qAqB − qB2
and so maximising gives
qB =
40 − q A
2
which is B’s reaction function. Firm A’s problem is then
max v A = 40q A − q AqB − q 2A
qA
s. t.
qB =
40 − q A
2
The simplest way to solve is just to substitute for qB in A’s profit function, to give
1
max v A = 20q A − q 2A
q
2
A
yielding qA = 20. This yields a Nash equilibrium because A is maximising its profit given
B’s optimal choice, and B is doing the same thing. The outputs (20, 10) are mutually best
replies.
178
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3. Here we have to find the Cournot or Cournot-Nash equilibrium. We just saw that B’s
reaction function is
qB =
40 − qA
2
Proceeding in exactly the same way with A we find that its reaction function is
qA =
40 − qB
2
A Nash equilibrium is a point ( q AN , qBN ) which satisfies both reaction functions, since
then these outputs are mutually best responses. This gives the pair of linear equations
1
q AN + qBN = 20
2
1 N 1 N
q A + qB = 20
2
2
Solving these then gives q AN = qBN = 13 13 .
4. In a game with no proper subgames, the only subgame is the entire game itself. Thus a
Nash equilibrium of the entire game is subgame perfect.
5. (Outline) The game tree begins with a node at which B can choose either to enter or
not. If not, the payoffs are zero for B and the monopoly profit for A. If it moves along the
branch corresponding to entry, we then have a subgame corresponding to the Cournot
game analysed in question 3. A, let us say, has the choice of any output level. All these
therefore lie in B’s information set. Corresponding to each possible output of A, B can
choose any possible output level. We have just seen that the Nash equilibrium of this
subgame is (13 13 , 13 13 ). Replacing this subgame by its payoffs (177, 177), clearly A will
prefer to enter than not enter. Thus we have the same solution as before.
6. Consider the fifth city. Since there are no cities left in which to deter entry, the
incumbent will certainly accept entry rather than incur a loss to deter it. But then in the
fourth city, the entrant knows that entry is going to take place in the fifth city, and so
there is no gain to the incumbent in fighting to deter entry in city four, and so entry will
take place and the incumbent will not fight it. But then the same is true in the third,
second and finally the first city. Thus it never pays to fight entry. The only subgame
perfect strategy is to accept entry in each city. This is thought of as a paradox, because
one might have thought that for a cost of only −1 in the first city, the incumbent could
fight entry, thus deterring entry in all remaining cities and preserving its monopoly
profit. The backward induction argument shows that this cannot be an equilibrium.
7. Recall that each player at the beginning of the second period is assumed to know
what action the other chose in the first period. It follows that a strategy in the repeated
game must specifiy an action for the first period, and an action for the second period
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contingent on what the other player chose in the first. The game is symmetrical, and we
can list player i = A, B’s strategies as follows:
s1i = (10, 10, 10)
s2i = (10, 10, 12)
s3i = (10, 12, 10)
s4i = (10, 12, 12)
s5i = (12, 10, 10)
s6i = (12, 10, 12)
s7i = (12, 12, 10)
s8i = (12, 12, 12)
where the first number is the output chosen in the first period, the second is output in
the second period if the other player chose 10 in the first period, and the third is output
in the second period if the other chose 12 in the first period. Using Fig. 15.9 in the text
(and rounding where necessary) we have the payoff matrix
s1A
s2A
s3A
s4A
s5A
s6A
s7A
s8A
s1B
s2B
s3B
s4B
s5B
s6B
s7B
s8B
380, 380
380, 380
394, 362
394, 362
396, 360
396, 360
410, 342
410, 342
380, 380
380, 380
394, 362
394, 362
378, 374
378, 374
389, 353
389, 353
362, 394
362, 394
373, 373
373, 373
396, 360
396, 360
410, 342
410, 342
362, 394
362, 394
373, 373
373, 373
378, 374
378, 374
389, 353
389, 353
360, 396
374, 378
360, 396
374, 378
372, 372
386, 354
372, 372
386, 354
360, 396
374, 378
360, 396
374, 378
354, 386
365, 365
354, 386
365, 365
342, 410
353, 389
342, 410
353, 389
372, 372
386, 354
372, 372
386, 354
342, 410
353, 389
342, 410
353, 389
354, 386
365, 365
354, 386
365, 365
For example consider the strategy pair s3A = (10, 12, 10) and s7B = (12, 12, 10).
A chooses 10 in the first period and B chooses 12. A responds in the second period
by choosing 10 and B responds to A’s choice of 10 in the first period by choosing 12.
From Fig. 15.9, this leads to the payoffs (342, 410).
The payoffs for A that are highest in their columns, and the payoffs for B that are
highest in their rows, are shown in bold. We see that there is a unique Nash equilibrium
at the strategy pair ( s8A , s8B ), and this is again worse for both players than the
cooperative strategies such as ( s1A , s1B ). In this game however there are no strictly
dominant strategies.
8. We take the Stackelberg game shown in Fig. 15.3 and suppose it is now infinitely
repeated. The choice of 13 13 by A and 10 by B gives payoffs of 222 for A and 167 for B,
which are therefore better than the one-period Nash equilibrium payoffs for both firms.
Consider the following trigger strategies:
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A will produce 13 13 in periods t, t + 2, t + 4, . . . . as long as B produces 10 in periods
t + 1, t + 3, . . . . If however B deviates in any period t + j by producing 13 13 , A will
produce 20 in every period t + j + 1, t + j + 3, . . . thereafter.
Consider now the payoff to B from deviating. He makes an immediate profit gain of
10, but then loses 67 in every period thereafter. Thus it does not pay to deviate if
10 ≤ 67/r
or
r ≤ 6.7
where r is the one period interest rate. If this condition is satisfied he will never deviate.
Obviously also it never pays A to deviate, given that B will not deviate. Thus the
cooperative solution can be supported by the trigger strategy as long as the interest rate
is less than 670%.
Exercise 15C
1. Since B knows its own type, it could use the information in Fig. 15.11 to calculate the
probabilities of A being of each type. However, it does not need to do this since it can
observe A’s output decision before it has to choose its own output. It knows that if A is
type A1, it will optimally produce 20.4, knowing that B2’s reaction function then leads it
to produce 9.3, and B0’s reaction function will lead it to produce 10.3. It will in that case
set the probability that A is of type A1 to 1. Likewise, if A produces 20.9, B knows that A
is type A0 and sets the probability of this to 1. It then makes the optimal response, given
its own type, as correctly predicted by A.
2. If the incumbent is uncertain about the entrant’s costs, we have a Bayesian game. For
concreteness, suppose that B’s costs are £1 per unit with probability π and £2 per unit
with probability 1 − π. In the post-entry duopoly game, the incumbent will have a
reaction function derived from maximising its expected profit (recall its cost per unit is
£1 with certainty)
HA = (41 − qA − π qB1 − (1 − π)qB2)qA − qA
yielding the reaction function
qA =
40 − π qB 1 − (1 − π )qB 2
2
The type B1 entrant will maximise
HB1 = (41 − qA − qB1)qB1 − qB1
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giving the reaction function
qB 1 =
40 − qA
2
while type B2 maximises
HB2 = (41 − qA − qB2)qB2 − 2qB2
to give the reaction function
qB 2 =
39 − q A
2
Substituting for qB1 and qB2 in A’s reaction function and solving for qA gives
q ABN =
4
(10.25 − 0.25π)
3
Thus for example if we assume π = 0.5, the PBE of the post entry duopoly game is
q ABN = 13.5
q BBN1 = 13.25
q BBN2 = 12.75
At this post entry equilibrium each type of firm B finds it profitable to enter and will do
so. We could, as in the text, suppose that the incumbent could make the threat of
producing an output of 27 32 which, if believed, would deter entry by each type of firm B,
but this cannot be a PBE because it is not optimal for A in the continuation game
beginning at the node following B’s entry, given its probability beliefs about B’s type.
3. (a) This really just repeats the discussion of the Finitely Repeated Prisoner’s Dilemma
in the text, with a different payoff matrix. Thus the PBE paths are
A
BR
BN
t=1
C
C
C
t=2
C
C
C
t=3
C
R
C
t=4
R
R
C
and we have to find the critical value of π at which, for the payoffs in the question, A
would not wish to deviate from this equilibrium. Using the notation in the text, the value
to A of the equilibrium path is
7
VA1 ( C , C , C , R ) = 1 + π
2
found by assuming that both types of B play these equilibrium strategies. The possible
deviating paths for A, and their associated payoffs, found just as shown in the text, are
as follows
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3
VA1 ( C , C , R, R ) = 2 + π
2
VA1 ( R, C , R, R ) =
1 3
+ π
2 2
VA1 ( R, R, R, R ) =
3
2
VA1 ( C , R, R, R ) =
5
2
VA1 ( C , R, C , R ) =
3 3
+ π
2 2
VA1 ( R, R, C , R ) =
1 3
+ π
2 2
VA1 ( R, C , C , R ) =
3 3
+ π
2 2
There are 23 = 8 possible paths because all candidate paths must end in R. The condition
under which the first deviation is not preferable to the equilibrium is
7
3
1
1+ π ≥ 2+ π ⇔π ≥
2
2
2
It is easy to see that if this is satisfied so will be all the other non-deviation conditions,
and so π = 12 is the critical probability.
3. (b) There is a typo in the text. The question should read:
“Let the number of repetitions be n > k, and show that, as n increases, the minimum
value of π such that the rational B will play tit-for-tat for the first n − k periods, with k
depending on π but not on n, will decrease.”
The answer is drawn from the paper by Kreps et al (1982). There they show that in
a game with n periods, in any Pareto-undominated PBE (they use the sequential
equilibrium concept, which is equivalent in this context) both players choose C in each
period for sure as long as there are at least k periods left in the game, where, in our
example,
k = 2+
8
π
This does not rule out that both will both play C later than this, but the complexity of
the analysis is such that only this rather loose upper bound on the number of periods at
the end of the game in which R will be played can be placed. Thus for example we just
saw that for π ≥ 0.5 in our four period game R would be played by some player only in
the last two stages, while for π = 0.5 we have k = 18, which is greater than the number of
stages in that example. Bearing this in mind, it is still true that, since k varies inversely
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with π, the larger is n, the smaller is the minimum value of π consistent with
n − k > 0.
Exercise 15D
1. The strategic form game matrix is
Player A
H
T
Player B
H
T
2,0
0,2
0,2
2,0
There is no Nash Equilibrium in pure strategies: for each cell (pure strategy
combination) one of the players can do better by changing their strategy.
Since each player only has two pure strategies their mixed strategy is uniquely
defined by the probability that they play a particular strategy. Denote player A’s mixed
strategy as the probability pA that she plays H and B’s mixed strategy by the probability
pB that he plays H. The expected payoffs are
VA(pA) = 2pApB + 2(1 − pA)(1 − pB)
VB(pB) = 2pA(1 − pB) + 2(1 − pA)pB
The marginal payoffs are
VA′ (pA) = 4pB − 2
VB′ (pB) = 2 − 4pA
We know there is no pure strategy Nash equilibrium so that these marginal payoffs must
be zero and the mixed strategy equilibrium is pA = 1/2, pB = 1/2.
A is willing to randomise (choose a probability of H which is strictly greater than
zero and less than 1) if pB = 1/2 and similarly B is willing to randomise if pA = 1/2. Thus pA
= 1/2, pB = 1/2 are a Nash Equilibrium in mixed strategies in that neither can do better by
deviating if other does not.
If we now change the game so that A gets £3 when the coins match (draw the
strategic form matrix of payoffs) her expected payoff changes to
VA(pA) = 3pApB + 3(1 − pB)(1 − pB)
and her marginal payoff to
VA′ (pA) = 6pB − 3
Hence if pB = 1/2 then pA = 1/2 is still a best reply by A and the unique Nash equilibrium
in mixed strategies is the same as before. The change in payoffs does not alter relative
to the payoffs from the pure strategy combinations and so she has no incentive to alter
her mixed strategy.
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2. The payoff matrix for scissors, paper, stone is
Player A
Scissors
Paper
Stone
Scissors
0,0
0,1
1,0
Player B
Paper
1,0
0,0
0,1
Stone
0,1
1,0
0,0
There are no pure strategy Nash equilibria: whatever the pair of strategies (cells) at
least of the players does better by deviating to another strategy given the strategy of the
other player.
Denote the probabilities that player A chooses scissors by pA1 and the probability that
he chooses paper by pA2 and analogously for player B. The expected payoffs (since the
probability of A choosing stone is 1 − pA1 − pA2) are
VA = pA1pB2 + pA2(1 − pB1 − pB2) + (1 − pA1 − pA2)pB1
VB = pB1pA2 + pB2(1 − pA1 − pA2) + (1 − pB1 − pB2)pA1
The marginal payoffs for A are
∂VA/∂pA1 = pB2 − pB1
∂VA/∂pA2 = (1 − pB2 − pB1) − pB1
and analogously for player B. We know that there is no pure strategy Nash equilibrium
so that 0 < pik < 1 for all strategies k = 1, 2, 3 for both players i = A, B. Hence the
marginal payoffs from pA1 and pA2 (and the analogous payoffs for player B) must be zero.
Thus pB1 = pB2 and pB2 − 2pB1 = 1 implying pB1 = pB2 = pB3 = 31 Similarly for player A’s
probabilities. Thus the unique mixed strategy has both players choosing scissors, paper,
and stone with equal probabilities and getting expected payoffs of zero.
3. The payoff matrix is
Firm A
1
2
Firm B
1
2
10,5 0,0
0,0 5,10
There are two pure strategy Nash equilibria in which the firms choose the same location.
There is also a mixed strategy Nash equilibrium. Letting pA, pB be the probabilities
that players A,B choose location 1, the expected payoffs are
VA(pA) = 10pApB + 5(1 − pA)(1 − pB)
VB(pB) = 5pApB + 10(1 − pA)(1 − pB)
and the marginal payoffs are
VA′ (pA) = 10pB − 5(1 − pB) = 15pB − 5
VB′ (pB) = 5pA − 10(1 − pA) = 15pA − 10
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Hence the mixed strategies pA = 2/3, pB = 1/3 constitute a Nash equilibrium in that
neither player would gain by deviating if the other did not.
Notice that at the pure strategy equilibrium where both choose location 1 which can
be characterised as pA = 1, pB = 1, the constant marginal payoffs are both positive
(VA′ (pH) = 10, VB′ (pH) = 5) so that neither player wishes to randomise (reduce their
probability of choosing location 1). Similarly at the pure strategy equilibrium where
both choose location 2 the marginal payoffs are negative (VA′ (pH) = −5, VB′ (pH) = −10),
so neither player wishes to increase their probability of choosing location 1 from zero.
The expected payoffs for the players at the three equilbria are
VA
10
5
3 13
(pA = 1, pB = 1)
pA = 0, pB = 0)
pA = 2/3, pB = 1/3
VB
5
10
3 13
VA + VB
15
15
6 32
The mixed strategy equilibrium leads to a smaller total payoff because the probability of
the firms choosing the same location and getting the total payoff of 15 is pApB +
(1 − pA)(1 − pB) = 4/9, so that the expected total payoff from the mixed strategy
equilibrium is 15 × ( 49 ) = 6 32 . The mixed strategy equilibrium is also Pareto inefficient in
terms of expected payoffs: both players would be better off moving to either of the pure
strategy equilibria. But since this is a non-cooperative game they have no means of
doing so.
Supplementary question
(i) If the game is cooperative so that binding agreements are possible what bargain
would the two firms make if no side payments are possible and the game is played once
only?
Exercise 15E
1. Consider any two agreements a′ and a″ in P. If the parties agree to use a randomizing
device which selects a′ with probability t and a″ with probability (1 − t), individual i will
get expected utility of tui(a′) + (1 − t)ui(a″). Thus even if there is no agreement â in P
such that
u(â) = tui(a′) + (1 − t)ui(a″),
i = 1, 2
they can always choose a randomizing device which is equivalent in expected utility
terms to this agreement. Hence the set of feasible utility combinations is convex
whatever the characteristics of the set A of physical agreements.
In Fig. 15E.1 the set of physical agreements without any randomization maps into the
set U′. Randomization over the set of physical agreements convexifies the set of utility
outcomes. For example, an agreement to choose a′ and a″ each with probability 12 ,
enables the parties to achieve the point û, even though the physical agreement â which
would yield this outcome is not feasible. Thus all outcomes in the area R are
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u2
â
u′
R
u
u″
a″
U′
a′
P
u1
Fig. 15E.1
Fig. 15E.2
achievable by a suitable randomization and the utility payoff set U is the union of U′ and
R. (U is the convex hull of U′.)
2. (a) In part (a) of Fig. 15E.2 the individuals have initial wealth yi. An agreement gives
xi − yi to i. The set P of feasible agreements is the triangle satisfying ∑xi − ∑yi ≤ 1, xi − yi
≥ 0. In part (b) U is the set of utility outcomes derived from the agreements in P. Along
the upper boundary of U the agreements satisfy ∑xi − ∑yi = 1. This boundary is strictly
convex: changing the agreement to give more to individual 1 increases her utility but
reduces the utility of individual 2. Since marginal utility declines with wealth ( ui′′ =
−1/xi < 0) the rate of increase of individual 1’s utility declines and the rate of decrease of
individual 2’s increases as the agreement shifts them down ab from left to right.
The set U′ is also derived from P but with u1 = (1/2)logx1 − logy2, u2 = 2logx2.
2. (b) Let s = x1 − y1 be the amount of the $1 given to individual 1. Since both individuals
have positive marginal utility from wealth and the Nash solution satisfies E the
agreement generated by the Nash solution must have ∑xi − ∑yi = 1 so that x2 − y2 =
1 − s. Hence the Nash product to be maximized by s is
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N(s; y1, y2) = [u1(y1 + s) − u1(y1)][u2(y2 + 1 − s) − u2(y2)]
Now
Ns = u1′ ( y1 + s)[u2(y2 + 1 − s) − u2(y2)] − u2′ ( y2 + 1 − s)[u1(y1 + s) − u1(y1)]
so that
Ns(0, y1, y2) = u1′ ( y1 )[u2 ( y2 + 1) − u2(y2)] > 0
Ns(1, y1, y2) = − u2′ ( y2 )[u1 ( y1 + 1) − u1(y1)] < 0
Hence the Nash bargain must have s ∈ (0, 1) and both parties share in the gain from the
agreement. With the particular preferences assumed the Nash product is maximized
when
N s ( s, y1 , y2 ) =
y + 1 − s
y + s
1
1
log 2
log 1
−
=0
y1 + s
y2
y2 + 1 − s
y1
2. (c) Consider the sign of
N s (1/2, y1 , y2 ) =
y + s
y + s
1
1
log 2
log 1
−
y1 + s
y2 y2 + s
y1
(15.1)
If the sign depends only whether y1 is larger or smaller than y2 then, since Ns is
decreasing in s, we can make a definite conclusion about whether the richer or poorer
individual gets a larger or smaller share. Multiplying (15.1) by (y1 + s)(y2 + s) we see that
the sign of (15.1) is the sign of
y + s
y + s
( y2 + s)log 2
− ( y1 + s)log 1
y2
y1
(15.2)
Obviously at y1 = y2 this expression is zero and the Nash bargain has s = 1/2. Now
consider the derivative
y + s
y + s
d
y+s
+ log
( y + s)log
=1−
y
y
dy
y
(15.3)
Readers should sketch the graphs of the terms in (15.3) to convince themselves that the
derivative is negative for all finite positive y. Thus if y1 > y2 the second term in (15.2) is
smaller than the first term and so (15.2) is positive, which implies that (15.1) is also
positive. Hence we have established that a larger share is given to the richer of the two
individuals.
The result does not hold for all utility functions. It is apparent from the first order
condition for maximizing the Nash product that the solution depends on the ratios
ui′ /[( ui ( xi ) − ui(yi)]. When the utility function is concave an increase in endowed
income yi will reduce marginal utility but is will also reduce the denominator since
ui′( xi ) − ui′( yi ) < 0. Thus the effect of a change in endowed income depends on the
magnitude of the first and second derivatives of the utility function. You may like to
think about this again after reading the discussion of risk aversion in section 17E.
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Supplementary question
(i) Compare two “divide a dollar” bargaining problems which are identical except that
in one of them individual 1 has utility u1(x1) and in the other she has preferences
represented by
g(x1) = G(u1(x1)),
G′(u1) > 0, G″(u1) < 0, G(u1(y1)) = 0
Thus in the second problem her utility function is a concave transformation of her
utility function in the first problem. (She is more risk averse in the second problem – see
section 17E.) Show that individual 1 will get a smaller share at the Nash solution in the
second problem compared with the first.
3. If there is no agreement no union members are employed by the firm. When there is
no employment the union and the firm are unaffected by w, so that we can, with no loss
in generality, assume that the disagreement wage is U. The disagreement event in (z, w)
space is (0, U) and yields utilities π(0, U) = 0, U(0, U) = Uz0. In Fig. 10.7 the set of
agreements P is the triangle bounded by the pAP and UU curves and the vertical axis.
Since P is closed and bounded and contains the disagreement event, and the utility
functions are continuous, the set of utility payoffs U is closed, bounded and contains
π(0, U), U(0, U). It is not necessary to assume that randomization is possible to show
that U is convex. Remember that efficient bargains have z = z* (where pAP(z*) = U), so
that movements along the upper right boundary of U can occur only through changes in
w. Hence the slope of the upper boundary of U in (π, U) space is dπ/dU = πw/Uw = −1.
Since all points along pAP give π = 0 and all points along UU give U = Uz0, the set U in
(π, U) space is the triangle with vertices (0, Uz0), (0, (pAP − U)z* + Uz0), ((pAP − U)z*,
Uz0).
The Nash product is
N = [R(f(z)) − wz][(w − U)z]
which is maximized when
Nz = (R′f′ − w)(w − U)z + (w − U)(R − wz) = 0
(15.4)
Nw = −z(w − U)z + (R − wz)z = 0
(15.5)
Using (15.5) to substitute for R − wz in (15.4) gives
0 = (R′f′ − w)z + (w − U)z = (R′f′ − U)z
implying (R′f′ = U, which is the text equation [10D.6]. Employment is z* which
maximizes the total surplus R − Uz. The wage w* determines how this maximized
total surplus is shared. From (15.5) we get w*z* = [R(f(z*)) − Uz*]/2, so that the wage
w* = (pAP − U) is halfway along the contract curve CC in Fig. 10.7. The reader
should subsitute this wage into the firm and union utility functions and note that
π(z*, w*) − π(0, U) = (R − Uz*)/2
U(z*, w*) − U(0, U) = (R − Uz*)/2
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so that the Nash solution splits the gain from trade equally between the firm and the
union.
4. The first order conditions for maximization of the asymmetric Nash product N =
π α[(w − U)z]1−α are
Nz = α (R′f ′ − w)π α−1[(w − U)z]1−α + (1 − α)π α[(w − U)z]−α(w − U) = 0
(15.6)
Nw = −α zπ α−1[(w − U)z]1−α + (1 − α)π α[(w − U)z]−αz = 0
(15.7)
Rearranging (15.7) to get
π α[(w − U)z]−α =
α
1−α
π α−1[(w − U)z]1−α
(15.8)
substituting in (15.6) and cancelling terms gives
R′f ′ − w + (w − U) = R′f ′ − U = 0
and so the level of employment is not affected by the degree of asymmetry in the
bargaining solution. The agreement will maximize the total surplus.
We can manipulate (15.8) to get
( w* − U) z* 1 − α
=
π ( z*, w*)
α
The greater is α the smaller is the right hand side of this equation and so the left hand
side must be reduced by reducing w* which reduces the numerator and increases the
denominator. Thus the greater is the weight on the firm’s utility in the Nash product the
greater its share of the gain from trade.
Exercise 15F
1. The dates at which offers can be made are 0, 1, 2, . . . , T − 1, T. Suppose that the
number of dates (T + 1) is odd, so that player A makes the last offer at date T. The
subgame beginning at that date consists of her making an offer and player B deciding to
reject or accept it. The Nash equilibrium is an offer xT = 1 by A which is accepted by B:
rTb = 0. At date T – 1 player B makes an offer. Since A will get 1 (her payoff from the
period T game) if there is no agreement at T − 1, she will reject all period T − 1 offers
which give her less than the discounted value of her period T payoff: δaxT = δa. Given his
rational anticipation of what will happen in the period T game, B’s optimal period T − 1
offer is yT−1 = 1 − δa. At period T − 2 A knows that B will get a payoff of 1 − δa if there is
no agreement in period T − 2 and so her optimal offer is
xT−2 = 1 − δbyT−1 = 1 − δb(1 − δa) = 1 − δb + δaδb
which B would accept since he is indifferent between δbyT−1 at date T − 2 and yT−1 at date
T − 1.
At date T − 3 B’s optimal offer is yT−3 = 1 − δaxT−2. Similarly at date T − 4 A will make
the just acceptable offer
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xT−4 = 1 − δbyT−3 = 1 − δb + δaδb − δbδaδb + (δaδb)2
Repeated application of the same arguments shows that at date 0 her offer (and payoff)
is
T /2
x0 = (1 − δb)
∑ (δ δ ) + (δ δ )
i
a
b
(T/2)+1
a b
i= 0
Taking the limit as T → ∞ player A gets the payoff from the game of
x0 =
1−δb
1 − δ aδ b
which is identical to her payoff from the infinite horizon game of the text.
Now consider a version of the finite horizon game in which the number of periods is
even, so that player B makes the last offer, although player A makes the first offer. The
optimal offers are now yT = 1, xT−1 = 1 − δb, yT−2 = 1 − δa(1 − δb) and so on. The reader
should be able to show that
T /2
x0 = (1 − δb)
∑ (δ δ ) + (δ δ )
i
a
b
a b
i= 0
(T/2)+1
xT−1
which as T → ∞ has the same limit as in the game in which A has the last move, since
the differences in the last terms in the relevant equations for x0 vanish as T → ∞. Thus in
the limit of the finite horizon game what matters is which player makes the first offer,
not who makes the last offer.
2. (a) and (b) A’s strategy in the T + 1 period game is a sequence of offers (x0, . . . , xT)
and B’s is a sequence of rejection rules (r0, . . . , rT). Let βt be the minimum payoff that B
can get from the SPE of the game starting at date t. Then at date t − 1 A’s optimal offer,
which is accepted by B, is δbβt. Hence B’s minimum payoff from the SPE of the game
starting at period t − 1 is βt−1 = δβt. Thus the date 0 optimal offer by A is δ TβT . But at date
T B will accept any non-negative offer, since he gets 0 if there is no agreement. Hence
A’s optimal offer at date T is xT = 1 and βT = 0. Thus the SPE strategies are xt = 1, rt = 0, t
= 0, . . . , T.
2. (c) B’s rejection rules are credible only if rt ≤ rt+1, which implies that
rt+k ≥ rtδ b− k , k > 1
(15.9)
But, since the equilibrium rejection rules must also satisfy rt ≤ 1 and δb ∈ (0, 1), (15.9)
can be satisfied for large k only if rt = 0, t = 0, 1, . . . . Hence A’s SPE strategy is xt = 0,
t = 1, . . . .
Supplementary question
(i) Suppose there is only one period and A and B make simultaneous offers x ≥ 0, y ≥ 0,
which they get only if x + y ≤ 1. What are the Nash equilibria?
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3. A’s SPE offer and payoff when the day is split into n periods is
x*( n ) =
1 − δ bn
1 − δ 1b/n
=
1 − δ anδ bn 1 − (δ a δ b )1/n
Now lim n →∞ δ 1i / n = δ 0i = 1, so we must use L’Hôpital’s rule to evaluate the limit of the
above ratio:
d(1 − δ 1b/n )/dn
n →∞ d (1 − (δ δ )1/ n )/dn
a b
lim x*( n ) = lim
n →∞
Now
dδ 1i/n
δ 1/n log δ
=− i 2 i
dn
n
and so
δ b n −2 log δ b
log δ b
=
/
−
1
2
n
n→∞ (δ δ )
n log δ aδ b log(δ aδ b )
a b
lim x*( n ) = lim
n→∞
Differentiation of this expression for x* shows that A’s payoff is increasing in δa and
decreasing in δb. (Remember logδi < 0).
Exercise 15G
1. Let F(b) be the distribution function for b. B will buy if b ≥ p so that the probability of
a sale is 1 − F(p). S gets zero benefit from the asset if it is not sold and will therefore set
the price p to maximize her expected revenue R(p) = [1 − F(p)]p. As we can see from
Table 15G.1 it is never optimal to have p < ᐉ or p ≥ h.
Since marginal revenue is discontinuous at ᐉ there are two types of solution. (i) If
h < 2ᐉ marginal revenue is negative for p > ᐉ and positive for p < ᐉ, hence p* = ᐉ.
(ii) When h > 2ᐉ marginal revenue is positive at ᐉ and so p* = h/2.
In case (i) the expected gain from trade is Eb = (h + ᐉ)/2. Since p* = ᐉ trade is certain
to take place. Thus the seller’s monopoly power creates no inefficiency. In case (ii) the
expected gain from trade is
h
1
3h 2
bdb =
< Eb
4( h − l)
h−l p
∫
Some potential gains from trade are lost because with probability F(h/2) > 0 the realised
benefit is less than the price and B does not buy.
p ∈ (0, ᐉ)
p=ᐉ
p ∈ (ᐉ, h)
p=h
p>h
F(p)
R(p)
1
p
1
p
(h − p)/(h − ᐉ)
p(h − p)/(h − ᐉ)
0
0
0
0
Table 15G.1
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R′(p)
1
(h − 2p)/(h − ᐉ)
0
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Supplementary questions
(i) Illustrate the two solution types in a diagram. Show that in case (ii) variations in ᐉ
have no effect on the solution.
(ii) How is the answer affected if S places the value c ∈ (ᐉ, h) on the asset or if she must
incur c to produce the asset after promising to sell it to B?
2. If q < G = h/ᐉ, S sets p = ᐉ. Trade takes place whatever B’s actual type and the
expected gain from trade is maximized. The outcome is efficient.
If q > G, S sets p = h. There is no trade with probability (1 − q). A fully informed
regulator R could effect a Pareto improvement by ordering S to set p ∈ [0, ᐉ] if R
observes b = ᐉ. This outcome is not feasible if b is not observed by R. Whether R can
make a Pareto improvement then depends on whether R can intervene before or after B
learns his type.
(i) Suppose R can intervene before B observes b. R could order B to pay p0 to S
before B learns b and p1 = ᐉ when the asset is exchanged. Since B would otherwise get
an expected benefit of zero he is better off if
q(h − p1) + (1 − q)(ᐉ − p1 − p0) = q(h − ᐉ) − p0 > 0
S would otherwise get q(h − ᐉ) and is better off if
p0 + p1 = p0 + ᐉ > qh
Both parties can be made better off if p0 ∈ (qh − ᐉ, q(h − ᐉ)). Note that if the parties can
sign a binding contract fixing (p0, p1) before b is revealed the regulator is unnecessary.
(ii) If R can intervene only after b is revealed to B no Pareto improvement is possible.
In order to increase the total gains from trade the price p1 to a buyer with a value of ᐉ
must be reduced to ᐉ. To make the seller no worse off she must be compensated with at
least p0 = q(h − ᐉ) > 0. But then the type ᐉ buyer has an expected value of −p0 and is
worse off. There is no set of feasible payments which will ensure that trade always
takes place and that all individuals (the seller, the high value buyer and the low value
buyer) are no worse off. In case (i) there are only two individuals: the seller and the
buyer who does not yet know his type. In case (i) all individuals would be willing to
accept the regulator’s intervention. In case (ii) at least one of the parties would always
veto it.
Supplementary question
(i) Show that when p0 is chosen before b is revealed the suggested regulatory
mechanism also works if b is distributed uniformly over (ᐉ, h).
3. We can show that if q < G there is no equilibrium with p0 ∈ (60, h). Such a p0 is rejected
by type ᐉ. We know that a0(p0, h) = 1 for p0 ∈ (60, h] cannot be part of a PBE (see text
pages 396–397). Nor is a strictly mixed strategy for type h part of a PBE since this would
require a0 = A0 [G.11], which is impossible since q < G < 1. Thus only a0 = 0 can be part of
a PBE for p0 ∈ (60, h]. But then, since both types reject such p0, q1(p0) = q and the optimal
p1 = ᐉ. Since setting p0 ∈ (60, h) and p1 = ᐉ gives an expected payoff to S
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Fig. 15G.1
of δᐉ, she would be better off setting p0 = 60, p1 = ᐉ and getting an expected payoff of
qh(1 − δ) + δᐉ.
The two candidates for a PBE therefore have S playing p0 = 60, p1 = ᐉ or p0 = ᐉ, p1 = ᐉ.
But since q < G, S gets a higher payoff from the low price strategy.
4. If the union knew the firm’s type i it would make the take or leave it offer (zi, wi)
which maximizes U (z, w) subject to the firm breaking even: iR(d(zi)) − wizi ≥ 0.
Since the pAP curve for a type h firm lies outside that for a type ᐉ firm it must be true
that hR(f(zᐉ)) − wᐉzᐉ > 0. Thus the union will either set (zᐉ, wᐉ) and get U(zᐉ, wᐉ) for sure,
or it will set (zh, wh) and get an expected payoff of qU(zh, wh). The critical value
G = U(zᐉ, wᐉ)/U(zh, wh) determines the type of solution.
5. When S can commit herself to her offers we do not have to impose the requirement
that whenever it is her turn to move her strategy should be optimal from that point on
given the information she has acquired. This requirement need only be imposed at the
start of the game when S commits herself to a strategy for the whole game. Her strategy
is defined by (p0, λ). Fig. 15G.1 shows the possible strategies. We consider how the
equilibrium is derived for the case of a strong seller: q > G = ᐉ/h.
At
p0 = 60(λ) = h(1 − δλ) + δλᐉ
(15.10)
the type h buyer is indifferent between accepting p0 and rejecting it and facing the
probability λ of p1 = ᐉ next period. The line 60 in the figure plots (15.10). There are four
types of strategies which the seller can adopt. (i) She can set p0 ∈ [0, ᐉ] which yields a
certain payoff of p0 whatever the value of λ. Clearly the best such strategy is p0 = ᐉ
which yields v1 = ᐉ. (ii) Strategies in the region ii with p0 ∈ (ᐉ, 60) result in the type h
accepting the first period offer and the type ᐉ accepting the second period offer if it is ᐉ.
These strategies yield qp0 + (1 − q)λδᐉ. This function is increasing in p0 and λ and,
because q > ᐉ/h, its contours are flatter than the 60 line. The supremum of the payoff
function in this region is v2 = qh which is the limit of qp0 + (1 − q)λδ ᐉ as p0 → h, λ → 0.
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(iii) The payoff along the curve 60 is q60 + (1 − q)λδᐉ = qh(1 − λδ) + λδᐉ and the optimal
strategy in this region is λ = 0, 60(0) = h with payoff v3 = qh. (iv) The region iv has
p0 ∈ (60, h]. Neither buyer type buys in the first period so the payoff is δ [λᐉ + (1 − λ)qh].
Since q > ᐉ/h the supremum of the payoff function is v4 = δ h which is achieved as
p0 → h, λ → 0.
Comparison of the payoffs from the different types of strategies shows that the
optimum strategy when the seller is strong is to set p0 = h, λ = 0 which yields exactly the
same payoff to the seller as in the single period model. You should now be able to use
similar reasoning in the weak seller case to establish that the optimum strategy has
p0 = ᐉ.
6. (a) Denote the offer in the second period by the type b buyer as pb1. At date 1 S’s
optimal strategy is accept any offer pb1 ≥ 0 so that the optimal offer by the buyers at that
date are ph1 = pᐉ1 = 0. At date 0 S realises that a type b buyer will refuse any offer p0 by
her such that b − p0 < δ b since the buyer will get a discounted payoff of δ b by refusing p0
and making his optimal offer in period 1. Thus an offer p0 = (1 − δ )ᐉ is certain to be
accepted and an offer p0 = (1 − δ )h will be accepted with probability q. Thus S will set
p0 = (1 − δ )h if q > ᐉ/h and p0 = (1 − δ ) ᐉ if q < ᐉ/h.
•
6. (b) At date 1 S has observed B’s offer p0 and has updated her belief from q to q1(p0).
She will then choose λ by reference to [G.5]. Consider the case in which q < ᐉ/h. The
equilibrium has both types of buyer making the same offer p0(ᐉ) = p0(h) = δ ᐉ and being
willing to accept any period 1 offer from S which satisfies p1(p0) ≤ b. The seller will
accept any p0 ≥ δ ᐉ and will set p1(p0) = ᐉ. This is an equilibrium because at date 1 S has
received no information from the period 0 offer which is the same for both types of
buyer. Since q < ᐉ/h her optimal period 1 offer is ᐉ, which would yield her ᐉ at date 1.
Thus she would accept any offer p0 ≥ δ ᐉ from B at date 0 and reject all other offers. If a
type b buyer makes the just acceptable offer at date 0 he gets b − δ ᐉ. The type b buyer
will not set a lower price since this would not be accepted and the would face p1 = ᐉ
next period. This would yield 0 < ᐉ(1 − δ ) to a type ᐉ and δ (h − ᐉ) < h − δ ᐉ to a type h.
Note that the seller sets p1 = ᐉ even if she receives an out of equilibrium offer p0 ≠ δ ᐉ.
We assume that such offers convey no information about the buyer’s type.
The case where q > ᐉ/h is much more complicated and the reader should consult the
discussion of a similar model in David Canning, “Bargaining theory”, in F. Hahn (ed.),
The Economics of Missing Markets, Information and Games, Oxford University Press,
1990.
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Chapter 16
Oligopoly
Exercise 16B
1. Write the inverse demand functions in [B.2] of the text as the linear system:
β1q1 + γ q2 = α1 − p1
γ q1 + β2q2 = α2 − p2
Then using Cramer’s Rule, we have,
q1 =
α 1 − p1 γ
α 2 − p2 β 2
β1
γ
γ
β2
=
β 2 (α 1 − p1 ) − γ (α 2 − p2 )
β 1β 2 − γ 2
=
α 1 β 2 − α 2γ
β 2 p1
γ p2
−
+
2
2
β 1β 2 − γ
β 1β 2 − γ
β 1β 2 − γ 2
and the solution of q2 follows similarly.
We require βjαi − γ αj > 0 because this is the value of qi when p1 = p2 = 0. For the
above solution to be possible we require β1β2 ≠ γ 2, but for qi to vary negatively with pi
we require the stronger condition that β1β2 > γ 2.
2. Inspection of the relevant rows of Table 16.1 shows that qiM = 1/2Zi.
3. The individual outputs qiM and Zi are indeterminate in the homogeneous goods case
because the firms’ marginal costs are identical and constant, so that the firms’ outputs
are perfect substitutes in supply. There is nothing in the models that allows us to solve
for individual outputs. For example, the joint profit function in the monopoly case is
Π(q1 + q2) = α (q1 + q2) − γ (q1 + q2)2 − c(q1 + q2).
Maximizing this first with respect to q1 then q2 gives
α −c
= q1 + q2
2γ
α −c
= q1 + q2
2γ
which, of course, are the same condition, and so we can only solve for total output.
196
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4. In general terms the joint-profit-maximizing problem is
max Π1(q1, q2) + Π2(q1, q2)
yielding the first-order conditions
Π ii ( q1M , q2M ) + Π ij ( q1M , q2M ) = 0
i, j = 1, 2, i ≠ j.
Now if, as is the case in this model, Π ij ( q1M , q2M ) < 0, then Π ii ( q1M , q2M ) > 0, i.e.
an increase in qi, with qj held fixed at q Mj , will increase profit. Since qiR maximizes Πi(qi,
q Mj ), we must therefore have qiR > qiM as required (recall that Πi is strictly concave in
qi).
5. We have that
Âi ≡ (ai + bici)/2bi;
Åi ≡ φ / 2bi.
Then, simply using the definitions of ai, bi and φ from question 1 gives:
Âi =
( β jα i − γα j ) + β j ci
2β j
and this must be positive by virtue of the assumption that βjαi > γαj. We then have
Åi Åj =
φ2
4bi b j
=
γ2
4β i β j
and since γ 2 < βi βj we certainly have 0 < 1 − Åi Åj < 1. It follows that the pair of
simultaneous linear equations (derived from [B.24] of the text)
p1 − Å1 p2 = Â1
−Å2 p1 + p2 = Â2
has a solution (since 1 − Å1Å2 ≠ 0) and that this solution, given in [B.25] of the text,
implies positive prices.
6. It will simplify the algebra without losing real generality if we set ci = cj = 0 in the
model. Then the Bertrand equilibrium prices can be written (using the definitions given
in question 1):
piB =
2β i ( β jα i − γα j ) + γ ( β iα j − γα i )
4β i β j − γ 2
.
If we can show that piB > 0 then the Bertrand equilibrium is profitable. The term 4βiβj −
γ 2 > 0. We can re-write the numerator as
αi(βiβj − γ 2) + βi(βjαi − γ αj)
and each of these terms is positive given the assumptions explained in question 1.
The Cournot-Nash equilibrium will be more profitable than the Bertrand if i’s price
piC > piB , i = 1, 2. To obtain piC substitute the expressions for qiC , q Cj into i’s inverse
demand function to obtain
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α i ( 4 β 1 β 2 − γ 2 ) − β i ( 2β j α i − γα i ) − γ ( 2β iα j − γα i )
p =
4β 1 β 2 − γ 2
C
i
Then
piC − piB = γ 2αi > 0
where the details of the algebra are left to the reader.
7. In the discussion of the Stackelberg model (see p. 407 of the text) we sketched the
profit contours for firm 1 as strictly concave relative to the q1-axis and, since lower q2
implies higher profits for firm 1, this is equivalent to asserting that the firm’s profit
function is strictly quasi-concave. To prove this, write, say firm 1’s profit contour as
(α1 − c1 − γ q2)q1 − β 1 q12 = Π 10
where Π10 > 0 is some given (feasible) profit level. We can solve this explicitly for q2 as a
function of q1:
q2 =
α 1 − c1 β 1
Π0
−
q1 − 1
γ
γ
γ q1
Differentiating gives
dq2
β
Π0
= − 1 + 12
dq1
γ γ q1
Note that by setting this derivative to zero we obtain the value of q1 at which the contour
is at its maximum, as
Π0
q1 = 1
β1
1/ 2
implying that the higher is Π1, the larger is the output at the peak of the profit contour,
as Fig. 16.2 of the text illustrates. To confirm the concavity of the contour, differentiate
again to get
d 2 q2
2Π 10
=
−
< 0.
γ q13
dq12
8. In case (a) of the Edgeworth model, each firm’s capacity is at least sufficient to
supply the entire market at a zero price, i.e. G ≥ α /γ. This means that if firm i sets pi > 0,
firm j captures the entire market by setting pj = pi − ε, and it certainly has enough
capacity to supply this. Then repeat the reasoning underlying the Bertrand result for the
homogeneous output case to rationalize the Nash equilibrium at p1 = p2 = 0.
Exercise 16C
1. The joint-profit maximum is found by solving
max[100 − (q1 + q2)](q1 + q2) − q12 − 2q22
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First-order conditions are
100 − 4q1 − 2q2 = 0
100 − 2q1 − 6q2 = 0
yielding outputs, prices and profits:
q1M = 20; q2M = 10; pM = 70; Π1M = 1000; Π 2M = 5000.
To derive the Cournot-Nash equilibrium we obtain the reaction functions by solving:
max q1 (100 − (q1 + q2))q1 − q12
max q2 (100 − (q1 + q2))q2 − 2q22
giving
q1C =
100 − q2
;
4
q2C =
100 − q1
.
6
This results in the Cournot-Nash equilibrium
q1C = 21.74; q2C = 13.04; pC = 65.22; Π 1C = 945.25; Π C2 = 510.39.
Then we see that firm 2’s profit at the Cournot-Nash equilibrium, at 510.39, is higher
than its profit at the joint-profit maximum, 500. One way of explaining this is to note
that at the joint profit maximum the firms’ marginal costs must be equalized, while at
the Cournot-Nash equilibrium there is nothing to bring about this result. It follows that
the firm with the higher marginal cost function (here firm 2) may produce lower output
at the joint-profit-maximizing position and, in the absence of side-payments, this gives it
lower profit.
2. Though in principle the profit frontier is derived by solving
max Πi (q1, q2)
s.t.
Πj (q1, q2) ≥ Π j ,
in practice it is easier to solve by introducing the parameter λ ∈ (0, 1) and formulating
the problem as
max λΠ1(q1, q2) + (1 − λ)Π2(q1, q2)
the profit functions in this case being
Π1 = 100q1 − (q1 + q2)q1 − q12
Π2 = 100q2 − (q1 + q2)q2 − 2q22 .
From the first-order conditions we obtain
100λ − λ4q1 − q2 = 0
100(1 − λ) − q1 − (1 − λ)6q2 = 0.
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Solving for q1 and q2 by Cramer’s Rule gives
q1 =
600λ (1 − λ ) − 100(1 − λ )
24λ (1 − λ ) − 1
q2 =
400λ (1 − λ ) − 100λ
.
24λ (1 − λ ) − 1
Since we must have q1 ≥ 0, q2 ≥ 0, this implies λ ∈ [1/6, 3/4] (just set qi = 0 and solve for λ
in each case). Then the following table gives the values for q1, q2, Π1, Π2 as λ varies over
this interval.
λ
q1
0.167
0.300
0.400
0.500
0.600
0.700
0.75
0
13.86
17.65
20.00
21.85
23.76
25.00
q2
16.65
13.37
11.76
10.00
7.56
3.47
0
Π1
0
816.49
934.39
1000.00
1064.97
1164.48
1250.00
Π2
833.33
615.42
553.54
500.00
419.35
228.43
0
Note that joint profits are maximized at λ = 0.5. At the outputs q1 = 20, q2 = 10, the firms’
marginal costs are equalized, since,
C1′ = 2q1 = 40
C 2′ = 4q2 = 40
As the firms move around the profit frontier, they redistribute profit by redistributing
outputs, and this leads to a violation of the cost-minimization condition of equality of
marginal costs. For example, at λ = 0.7 we have
C1′ = 2q1 = 47.52;
C 2′ = 4q2 = 13.88
and so overall profit could be increased by expanding q2 and contracting q1. Note that,
because costs are increased, the sum of firms’ outputs falls as they move away from the
joint-profit-maximizing position.
If we replace the cost functions in the above model by C1 = q1, C2 = 2q2, the analysis
becomes considerably simpler. Note that now firm 1’s marginal cost, 1, is always below
that of firm 2, which is 2. So, joint-profit-maximization involves firm 1 producing the
entire output, so we solve
max 100q12 − q1
q1
giving the solution q1* = 49.5, Π1* = 2450.25.
3. We can treat this as a straightforward problem in comparative-statics. Given the
problem in [C.6] of the text, we have the first-order conditions
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Π21 + λΠ11 = 0
Π22 + λΠ12 = 0
Π1 − Π10 = 0
We are interested in the sign of ∂λ /∂Π10 , since (as in [C.9]) we have that ∂Π 2 /∂Π 10 = −λ.
Differentiating the first-order conditions totally gives the system
Π 211 + λΠ 111
Π + λΠ
121
221
Π 11
Π 212 + λΠ 112
Π 222 + λΠ 122
Π 12
Π 11 dq1 0
Π 12 dq2 = 0
0 dλ dΠ 10
Call the determinant of the 3 × 3 matrix D, and note that the second-order conditions
imply D > 0 (see Appendix I, and note here we have a case where n = 2, m = 1). Then
applying the standard comparative-statics procedure gives
∂λ
= {(Π211 + λΠ111)(Π222 + λΠ122) − (Π221 + λΠ121)(Π212 + λΠ112)}/D
∂Π 10
and so it remains to evaluate the bracketed terms. Recall the profit functions
Πi(q1, q2) = (αi − ci − γ qj)qi − β i qi2
i, j = 1, 2, i ≠ j
It follows that we have
Π211 = Π122 = 0
Π111 = −2β1;
Π222 = −2β2
Π112 = Π121 = Π212 = Π221 = −γ
We then have
4λβ 1 β 2 − (1 + λ ) 2 γ 2
∂λ
=
D
∂Π 10
Note that at the joint-profit-maximizing point λ = 1, and so we have
∂λ
4
2
= (β1β2 − γ ) > 0
0
∂Π1 D
Since λ is the absolute value of the slope of the profit frontier, this tells us that the
frontier is certainly strictly concave in a neighbourhood of that point.
To construct the profit frontier for the homogeneous output example, note that in
this example the firms have identical marginal costs. It follows that the profit frontier is
a line with slope −1 through the joint-profit-maximizing point (recall the discussion
surrounding [C.3] of the text). From Table 16.2 we have that maximum joint profit is
20.25, and so the equation for the profit frontier is
Π 2M = 20.25 − Π1M
Π1M , Π 2M ≥ 0.
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This is graphed in Fig. 16.11 of the text. Since it is linear, it is concave.
4. In the case of homogeneous products the (inverse) market demand function is p = 100
− (q1 + q2) and the firms have identical constant marginal costs of 1. It is immediate that
firm j minimaximizes firm i by setting q jX to satisfy
100 − q jX = 1
since in that case market price equals marginal cost, and i cannot make a positive profit
at any output: minimax profit is zero.
5. The reason a price-setting duopoly can sustain a wider set of collusive allocations
than a quantity-setting duopoly is evident from Fig. 16.9. The one-shot Nash equilibrium
involves a lower profit in the price-setting than in the quantity-setting case, and so larger
punishments can be threatened in the former case.
6. The example used in the chapter has the demand function p = 10 − (q1 + q2) and the
firms’ cost functions are Ci = qi, i = 1, 2. From question 3 we have that the profit frontier
in this case is given by
Π2 = 20.25 − Π1
(16.1)
To find the set of weakly renegotiation-proof (WRP) profit pairs, we proceed by finding
the equations of the curves defining the boundaries of the set (refer to Fig. 16.11 of the
text). Suppose that the given profit pair is ( Π1* , Π *2 ), which is to be sustained by a pair
of punishment outputs ( q1P , q2P ). If ( Π1* , Π *2 ) is WRP, there must exist a punishment
output q2P (start by taking firm 1 as the cheat) that satisfies
max{Π 1 ( q1 , q2P )} ≤ Π 1* .
q1
(16.2)
That is, q2P must be such as to make firm 1’s punishment profit no greater than the
collusive profit, even if it makes its best response to 2’s punishment output (in general
q1P will give firm 1 less profit than this). To translate (16.2) into a condition involving
only q2P and Π1* , solve
max Π1 ( q1 , q2P ) = max(10 − ( q1 + q2P ))q1 − q1
q1
q1
to obtain
Z1 = (9 − q2P )/2
and then rewrite (16.1) as
1
Π1(Z1, q2P ) = ( 9 − q2P ) 2 ≤ Π 1*
4
(16.3)
(we leave the substitution for Z1 and rearrangement of terms to the reader).
Thus if q2P satisfies (16.3), firm 1 can never do better when it is punished than when
it colludes.
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To induce firm 2 to carry out the punishment rather than renegotiate back to the
original agreement, we require
Π2 ( q1P , q2P ) = (10 − ( q1P + q2P ))q2P − q2P ≥ Π *2 .
(16.4)
Since we want to determine the upper boundary of the WRP set, we can choose q1P to
be the most favourable to firm 2, which means setting q1P = 0. This gives us the
condition
Π2 (0, q2P ) = (9 − q2P )q2P ≥ Π *2 .
(16.5)
Thus (16.3) and (16.5) give a pair of conditions with which to solve for the upper
boundary of the WRP set. We can derive an equation for this by first solving (16.3) as an
equality to obtain
q2P = 9 − 2 Π 1*
and then substituting into (16.5) and rearranging to get
18 Π 1* − 4Π 1* ≥ Π *2 .
(16.6)
Taking the equality in (16.6) gives the function which is graphed as the upper boundary
of the WRP set in Fig. 16.11 of the text. Given the symmetry of the model, it is easy to
see that the lower boundary is obtained by exchanging subscripts in (16.6).
We can locate the points at which the boundary curves meet the profit frontier by, of
course, solving (16.1) and (16.5) simultaneously. We obtain from them the quadratic
3Π 1* − 18 Π 1* + 20.25 = 0
and so solving and discarding the infeasible root gives Π1* = 2.25. This then gives
Π *2 = 18.
Note that in all this nothing has been said about discount rates, we have simply
found the set of profit pairs which can be sustained by punishments involving less profit
for one firm and more for the other. Implicit in all this is the idea that if the firm being
punished reneges in the punishment phase the punishment will be reimposed from the
beginning (as with Abreu’s simple penal code). Fudenberg and Maskin (1986) showed
that there always exists some range of discount rates for which cheating can be
deterred, and so implicit in the above discussion is also the assumption that the
discount rate falls in that range.
Moving on to strong renegotiation proofness (SRP), this criterion restricts attention
to WRP profit pairs that are Pareto-undominated, i.e. that lie along the profit frontier.
This means that as well as having to satisfy conditions (16.3) and (16.4), the punishment
outputs q1P and q2P must generate profit pairs along the profit frontier. We know that
these profit pairs satisfy the condition
q2P = 4.5 − q1P
(refer to Table 16.2 of the text). Again, to define the boundary points of the SRP set take
the worst possible punishment for, say, firm 1, as q1P = 0, which then gives q2P = 4.5. But
clearly then, since the left hand side takes the value 20.25, this condition is satisfied for
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any Π *2 on the profit frontier, and so places no constraint on the solution. The only
constraints is (16.3). With q2P = 4.5, this becomes Π 1* ≥ 5.06. Thus the upper boundary
point of the SRP set is (5.06, 15.19). By symmetry, the lower boundary point is (15.19,
5.06).
It is easy to see why (5.06, 15.19) cannot be in the SRP set. If this were chosen as an
allocation, there is no value of q1P (since it cannot be negative) that could be used to
punish a deviation by firm 1 from the punishment pair (0, 4.5). On the other hand, for
any point to the right of (5.06, 15.19) (and to the left of (15.19, 5.06)) on the profit
frontier, it is always possible to find a point to its left which both is an effective
punishment point for a deviation, and which is itself capable of being enforced by threat
of a further move to the left. This is why the SRP set is an open set (a similar argument
can be used to explain why the WRP set is also open).
Exercise 16D
1. If there is a constant marginal cost of output in each firm, then all that happens is that
in Fig. 16.14 of the text, the reaction functions shift leftward by amounts that depend on
the value of the marginal cost, their slopes unchanged. This can be seen by introducing
a term, say cx1, into the expression for firm 1’s profit in [D.3] of the text, where c is the
constant marginal cost, and nothing that in [D.4] the constant term in the reaction
function then becomes (35 − c/2). The same can be shown for [D.6]. Of course, we
require c not to be “too large”, otherwise no output would be profitable.
2. This question requires us to work through the model taking c, rather than F, as the
parameter that will determine the various possible cases. Since we know that capacity
never exceeds output we shall let xi denote both output and capacity for firm i. The
counterparts of [D.3] and [D.5] are
max[100 − c − (x1 + x2)]x1
x1
and
max[100 − c − (x1 + x2)]x2
x2
yielding reaction functions
x1 = (100 − c − x2)/2;
x2 = (100 − c − x1)/2
and the Cournot-Nash outputs become functions of c:
x1c = (100 − c)/3; x2c = (100 − c)/3.
Thus the lower is c the higher the Cournot-Nash outputs. In Fig. 16.14 of the text,
reducing c shifts R1 and R2 out parallel to themselves, so that point c moves in a northeasterly direction along a 45° line. At the same time O1 is unchanged because it does not
depend on c (capacity is a sunk cost). Thus the range of outputs within which the
incumbent will choose to precommit capacity certainly shrinks as c increases. The
intersection of O1 and R2 is now given by the solution of
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x1 = 50 − 0.5x2
x2 = (100 − c − x1)/2
yielding x1 = (100 + c)/3, x2 = 2(50 − c)/3. Thus the incumbent will choose capacity in the
interval
[(100 − c)/3, (100 + c)/3]
which is clearly wider than a point if and only if c > 0, and the width of the interval
increases with c.
At the Cournot-Nash point ( x1c , x2c ) the firms’ profits are
π 1c = (100 − 2c)2/9;
π c2 = (100 − c)2/9 − F
With F = 300, entry will certainly not take place if π c2 ≤ 0, or equivalently if c ≥ 48. We
therefore assume in what follows that c < 48.
As in the text example, there are two cases in general.
1. The point of intersection of O1 and R2 is profitable for the entrant. Since at this
point x1 = (100 + c)/3, x2 = 2(50 − c)/3, this corresponds to
π2 = [2(50 − c)]2/9 − 300 > 0
or c < 24. It follows that for 0 < c < 24, any capacity choice the incumbent makes
in the interval [(100 − c)/3, (100 + c)/3] will be profitable for the entrant, and the
best the incumbent can do is maximize his profit given that entry will take place.
The Stackelberg leadership problem in this case is
max[100 − c − (x1 + (100 − c − x1)/2)]x1
x1
with solution
x1s = (100 − c)/2.
In the text, we had c = 30 and so xs = 35. This solution shows that the output of the
incumbent, and therefore the resulting profit, falls as c increases. Note that as in
the text example, the Stackelberg solution always happens to coincide with the
monopoly solution. The entrant’s corresponding output is
x2s = (100 − c)/4
and so the entrant’s post-entry output and profits also fall as c increases. Note
that the ratio of x1s to x2s is constant whatever the value of c.
2. If c lies in the interval 24 ≤ c < 48, then the entrant’s profit becomes zero
somewhere in the interval [(100 − c)/3, (100 + c)/3]. The significance of the output
level V1 (notation as in the text) is that if the incumbent sets capacity at or above
this level he can deter entry. For given F it is of interest to see how V1 varies with
c. With F = 300, V1 must satisfy
π2 = [100 − c − (x1 + x2)]x2 − 300 = 0
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where x2 is given by the reaction function
x2 = (100 − c − x1)/2.
Substituting into the expression for π2 and solving for x1 gives
V1 = 100 − c1 ± 34.64.
Clearly then the higher is the marginal capacity cost the lower will be the output
at which entry is just profitable and so the lower the capacity level to which the
incumbent must precommit if he wants to deter entry. The discussion of the
possible sub-cases in the text then applies directly.
3. Let v < 30 be the price at which the incumbent could sell off capacity post-entry, while
30 is the cost of expanding capacity pre- and post-entry. The effect of introducing v is to
change the form of the incumbent’s post-entry reaction function, O1. Given a capacity
commitment L1, the incumbent’s post-entry reaction function is found by solving
max π 1 = [100 − v − (x1 + x2)]x1
x1
s.t. x1 ≤ L1
yielding the reaction function
(100 − v − x2 )/2 for x1 < L1
x1 =
otherwise
L1
This reaction function has the same slope as R1 and O1 in Fig. 16.14 of the text, but lies
between them (for 0 < v < 30). The analysis then goes through exactly as before, but
with respect to this reaction function. Nothing in the results changes qualitatively, but
quantitatively the profitability and the effectiveness of capacity precommitment will be
reduced. Intuitively, capacity is less of a sunk cost and this reduces its precommitment
value.
4. If in this model F = 0, the entrant certainly makes a profit at the intersection of O1 and
R2 in Fig. 16.14 of the text. Then the only case possible is that where the incumbent acts
as a Stackelberg leader, choosing a capacity L*1 = 35, and allowing entry.
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Chapter 17
Choice under Uncertainty
Exercise 17C
1. (a) and (b) The largest amount p that an individual with initial income y and the
utility function u = √y will pay for the prospect which yields either 900 or 400 with equal
probabilities is defined by
0.5√(y + 900 − p) + 0.5√(y + 400 − p) = √y
(17.1)
After some tedious manipulation it is possible to solve this equation for
p = 650 −
15625
y
(Start by multiplying through by 2, square both sides, rearrange, square both sides
again . . . ) The individual will never be willing to pay the expected value of the prospect
but the amount he would pay increases with his initial income.
2. (a) Instead of (17.1) we have
0.5a(y + 900 − p) + 0.5a(y + 400 − p) = ay
which solves for p = 650.
2. (b) Instead of (17.1) we have
0.5(y + 900 − p)2 + 0.5(y + 400 − p)2 = y2
(17.2)
which solves for
p = 650 ±
1
√(4y2 − 5002)
2
and we are interested only in the positive root of the second term. (If we take the
negative root p will be zero for y ≈ 696.42 and (17.2) will be violated.) The prospect
becomes more valuable to the individual the greater his initial income.
3. Daniel Bernoulli suggested that the gamble is not infinitely valuable because
individuals are concerned with expected utility rather than expected income.
Specifically he proposed the utility function u(y) = log y so that
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Eu =
∞
∞
∑ π s u(ys ) = ∑ 2− s log 2s
1
1
= s2− s log 2
1
∞
∑
= 2 log 2
Diminishing marginal utility is not sufficient to ensure that prospects with arbitrarily
small probabilities of arbitrarily large payoffs have a finite expected utility. Consider a
utility function u(y) and a prospect with payoffs (y1, y2, . . . ) with probabilities πs = 2−s
such that u(ys) ≥ 2s. For the individual with the utility function u this prospect has an
infinite expected utility:
Eu =
∞
∞
∑ π u(y ) ≥ ∑ 2 2 = ∞
s
1
−s
s
s
1
To avoid this kind of difficulty it is necessary to assume that the utility function is
bounded. See K J Arrow, Essays in Risk Bearing, North Holland, 1970.
4. Let πi be the probability that the punter’s horse wins in race i, bi be the bet placed on
the race i and wi = kibi be the amount won if the horse wins race i. If the punter wins on
race 1, b2 = w2 = k1b1; if the punter then wins on race 2, b3 = w2 = k2b2 = k2k1b1 and, if the
last race is also won, w3 = k3b3 = k3k2k1b1.
The prospect of the bet on the third race is P 3 = (π3, 1 − π3; k3k2k1b1, −b1). The second
race is a prospect whose prizes are the prospect of the third race with probability π2 and
the loss of the original stake with complementary probability: P 2 = (π2, 1 − π2; P 3, −b1).
The first race is a prospect with prizes of P 2 with probability π1 and −b1 with
complementary probability: P 1 = (π1, 1 − π1; P 2, −b1). The rational equivalent of the
compound prospect P 1 is the prospect P = (π1π2π3, 1 − π1π2π3; k3k2k1b1, −b1). Betting in each
race separately gives three simple prospects P si = (πi, 1 − πi; kibi, −bi).
Exercise 17D
1. (a) This is inconsistent with the version of expected utility theory set out in section C:
utility depends on whether she has gambled as well as on her income. It is possible to
describe her preferences over income by two utility functions ui(y), i = g, n where the
subscript indicates whether she has gambled or not. If we assume that ug(y) = −∞ she
would never gamble because her expected utility from any gamble would always be less
than her expected utility without the gamble.
1. (b) If she believes (wrongly) that the probability of winning is zero then her reasons
for rejecting the gamble are consistent with expected utility theory.
1. (c) This is consistent: she has declining marginal utility.
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Fig. 17D.1
1. (d) This is consistent if we interpret “cannot afford” as placing a very high value on
income lost relative to income gained. It is a statement about the shape of her utility
function.
1. (e) This is consistent: she is better off with some other gamble.
1. (f) This is consistent: the rationale for not accepting the gamble relates to feasibility,
not her preferences.
2. One example is when income varies with health and health directly affects
preferences. This is ruled out by axiom 3 (equivalent standard prospects). The value of
v1 at which y1 is equivalent to (v1; yu, yL) will not be unique if the individual also cares
about health.
3. The apparent contradiction arises because the purchase of unfair insurance implies
risk aversion and acceptance of an unfair bet implies risk preference. One obvious,
though not necessarily satisfactory, explanation for such behaviour is that preferences
are state dependent. (One buys unfair health insurance because health state affects
the marginal utility of income. See section 19B.) Another possibility, suggested by
Friedman and Savage (Journal of Political Economy, 56, 1948, 297–304) is that the
utility function has both concave and convex segments, as in Fig. 17D.1. Suppose that
the individual initially faces the prospect P a = (πa; y1, y2) with expected income J a and
buys an insurance policy with a premium of ra leaving him with a certain income of yca =
J a − ra. If he is now offered the unfair gamble of the risky prospect P b = (πb; y3, y4) with
expected income J b < yca he would accept it. Yet a third explanation is that one obtains
pleasure from gambling of certain kinds, e.g. on horse racing, but not of others, e.g. on
falling sick.
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4. From the definition of a concave function [D.6]
tg(y1) + (1 − t)g(y2) = t[a + bv(y1)] + (1 − t)[a + bv(y2)]
= a + b[tv(y1) + (1 − t)v2)]
≤ a + bv(ty1 + (1 − t)y2)
= g(ty1 + (1 − t)y2),
t ∈ [0, 1]
(where the concavity of v is used at the third step in the argument). Hence g(y) is
concave.
Define ys = tys1 + (1 − t)ys2, t ∈ [0, 1] and use the definition of concavity
vs(ys) ≥ tvs(ys1) + (1 − t)vs(ys2)
which implies
πsvs(tys1 + (1 − t)ys2) ≥ tπsvs(ys1) + (1 − t)πsvs(ys2)
Hence
H(y1, . . . , yS) =
∑ π v (y )
s
s
s
s
≥t
∑ π v (y ) + (1 − t)∑ π v (y )
s
s
s1
s
s
s
s2
s
= tH(y11, . . . , yS1) + (1 − t)H(y12, . . . , yS2)
which establishes the concavity of the expected utility function in (y1, . . . , yS) space.
Hence the contours of the expected utility function are quasi-concave if the utility of
income function v(y) is concave.
5. Reducing yL to yL′ means that the probability of getting the better outcome yu must be
increased if the individual is to be indifferent between y and the standard prospect.
Conversely in Fig. 17.2(b).
6. Herrman has diminishing marginal utility.
7. Let P 1 = (π ; y, y) be a certain prospect and P 2 = (π ; y + g, y − ᐉ) be a risky prospect
with π g − (1 − π)ᐉ = 0. Let ua = √y and ub = [ua(y)]4 = y2, so that ub is an increasing
monotone transformation of ua. Since ua is a concave function P 1 is preferred to P 2 if
u = ua(y), but ub is a convex function and P 2 is preferred to P 1 if u = ub(y). To be more
specific try y = 100, g = 30, ᐉ = 15, π = 1/3.
8. Let ycu , ycg be the certainty equivalents under the two utility functions. Then
u( ycu ) = Eu(y) ⇒ a + bu( ycu ) = a + bEu(y)
and since g(y) = a + bu(y) we have
g( ycu ) = Eg(y)
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and so ycu = ycg . Since the risk premium is defined as expected income less certainty
equivalent income we have also established that the risk premium is unaffected by a
positive linear transformation of the utility function: ru = Ey − ycu = Ey − ycg = r g.
9. Let Ag, Av be the coefficients of absolute risk aversion for the two utility functions.
Since g′(y) = f ′(v)v′(y) and g″(y) = f ′(v)v″(y) + f ″(v)v′v′ we have
Ag =
− g ′′ − f ′v′′ − f ′′v′v′
=
g′
f ′v′
=
− v ′′ f ′′
−
v′
v′
f′
= Av −
f ′′
v′ > A v
f′
since f ″ < 0. Inserting this result in the approximation for the risk premium shows that
for ‘small’ gambles the risk premium is larger for the utility function g.
More generally, we can use the definition of certainty equivalent income [D.2]. Apply
Jensen’s inequality (the expected value of a concave function of a variable is less than
the value of the function evaluated at the expected value of the variable) to note that
Eg = Ef(v) < f(Ev)
(17.3)
From the definition of certainty equivalent income
g( ycg ) = Eg(y)
and
v( ycv ) = Ev(y)
we see that (17.3) implies
g( ycg ) < f ( v( ycv )) = g( ycv )
and so ycg < ycv since g is increasing in y. Hence
rg = Ey − ycg > Ey − ycv = rv
10. Differentiation gives Table 17D.1.
Constant absolute risk aversion. Integrating
− u′′
d log u′
=−
=A
u′
dy
v
a − be−Ay
a + b 1y− R
a + b log y
1− R
v′
Abe−Ay
by−R
by−1
v″
A = −v″/v′
2
−Ay
−A be
A
−R−1
−bRy
Ry−1
−by−2
y−1
Table 17D.1
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gives
−log u′ = Ay + k0
when A is a constant. Hence
log u′ = −k0 − Ay
and so
u′ = K0e−Ay
where log K0 = −k0. Integrating again gives
u = K1 − K0e−Ay
as required.
Constant relative risk aversion. Assume that R ≠ 1. Integrating
− u′′y − d log u′
=
=R
u′
d log y
gives
log u′ = −R log y − k0
and so u′ = K0y−R where log K0 = −k0. Integrating again gives
u = K1 +
K 0 y 1− R
1− R
Supplementary question
(i) What is the form of the utility function when the constant relative risk aversion is
unity: R = 1?
11. (a) For risk aversion we require v″ = 2c < 0 which implies c < 0. Marginal utility will
be positive if v′ = b + 2cy > 0 which requires b > 0 and y < −b/2c.
11. (b) Expected utility is
Ev = a + bEy + cEy2 = a + bEy + c[σ 2 + (Ey)2]
where σ 2 = Ey2 − (Ey)2 is the variance. The slope of indifference curves in (σ 2, Ey) space
is
dσ 2
b + 2cEy
=−
dEy
c
which is negative for Ey < −b/2c so that the indifference map is as shown in Fig. 17D.2.
For Ey < −b/2c the slope of the indifference curve is positive, declining with Ey and
unaffected by σ 2. In Fig. 17D.2 we have drawn the indifference curves with Ey on the
vertical axis and σ 2 on the horizontal axis, as is conventional in finance theory. Thus
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σ
Fig. 17D.2
each indifference curve now become steeper as Ey increases. The amount of additional
expected income required to compensate for an increase in risk is increasing: risk
aversion is increasing with mean income.
Exercise 17E
1. Differentiating the slope of the indifference curve [E.3] with respect to y1,
remembering that around the indifference curve y2 is a function of y1, gives
dy
−π 1
d( dy 2 / dy1 )
v ′( y 2 ) v ′( y1 ) − v ′( y1 ) v ′′( y 2 ) 2
=
2
dy1
dy1
π 2 v ′( y 2 )
=
−π 1
π v ′( y1 )
v ′( y 2 ) v ′( y1 ) + v ′( y1 ) v ′′( y 2 ) 1
>0
2
π 2 v ′( y 2 )
π 2 v ′( y 2 )
So indifference curves get flatter as y1 increases if the individual is risk averse (v″ < 0).
More formally, expected utility is a weighted sum of concave functions and is therefore
also concave and so has quasi-concave contours. If the individual is risk preferring
(v″ > 0) indifference curves are concave to the origin.
2. A risk neutral individual has v(y) = a + by, (b > 0) and so using [E.3]
dy2
π v ′( y1 )
π
=− 1
=− 1
dy1
π 2 v ′( y2 )
π2
so that indifference curves are negatively sloped straight lines which show
combinations of income (prospects) with the same mean.
3. From [E.3], an increase in π1 makes the indifference curves steeper. Intuitively: a
larger increase in y2 is required to compensate for a unit reduction in y1 if the probability
of state 1 has increased.
4. Use of [E.14] shows that decreasing absolute risk aversion implies that the slope of
the indifference curves will decrease along lines parallel to and below (y1 > y2) the 45°
line and increase along lines parallel to and above (y2 > y1) the 45° line.
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Fig. 17E.1
5. (a) From question 17C.10 we know that constant absolute risk aversion implies that v′
= bAe−Ay and so the slope of the indifference curve [E.3] is
dy 2
π bAe − Ay
π
=− 1
= − 1 e − A( y − y )
− Ay
dy1
π2
π 2 bAe
1
1
2
2
so that the slope is constant along lines parallel with the 45° line where y1 − y2 is
constant. Attitude to risk depends on the absolute difference between the two payoffs.
5. (b) Since constant relative risk aversion (not equal to 1) implies v′(y) = by−R (when
R ≠ 1) the slope of the indifference curve is
dy2
π by − R
π y
= − 1 1− R = − 1 1
dy1
π 2 by2
π 2 y2
−R
which is constant if along rays from the origin where y1/y2 is constant. Attitude to risk
depends on the relative size of the two payoffs.
6. Since g′(y) = f ′(v(y))v′(y) the slope of the indifference curves when preferences are
represented by g is
dy2
π f ′( v( y1 )) v ′( y1 )
=− 1
dy1
π 2 f ′( v( y2 )) v ′( y2 )
Now f ″ < 0 implies f ′(v(y1)) < (>) f ′(v(y2)) if y1 > (<) y2, so that the indifference curves
are flatter below the 45° line and steeper above it. Fig. 17E.1 shows the effect of the
transformation on the indifference curves and the certainty equivalent income and risk
premium. (Refer to question 17D.9 for a proof that the transformation reduces certainty
equivalent income and increases the risk premium.)
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The endowment point is E. The indifference curve I Eg is more “bowed in” than the
indifference curve I v and cuts the certainty line lower down than the I Ev indifference
curve through the same endowment point. Certainty equivalent income for the risky
income at E with preferences g(y) is ycg and with preferences v(y) is ycv . The respective
risk premia are r g, rv.
7. (a) Indifference curves in (y1, y2) space are rectangular about the 45° line. (Compare
the isoquants for the Leontief production function.)
7. (b) Differentiation gives f ′(u) = α u−α−1 > 0 and f ″(u) = −α 2u−α−2 < 0.
7. (c) The argument is very similar to that in question 8(c) in section 5B where the
relationship between the Leontief and CES production functions was examined. An
individual who ranks prospects by reference to the expected utility V = ∑πsus would also
be willing to rank them by any increasing monotone transformation H = H(V),
(H′ > 0) of expected utility. Thus H is an equivalent representation of his preferences
towards risky prospects. In particular prospects can be ranked by V = − ∑ π s us−α and by
H = −V −1/α which is an increasing transformation of V for α > 0.
Let us = u(ys). Suppose for a particular prospect that u1 = min(u1, . . . , uS), so that
−u1 = max(−u1, . . . , −uS) Now limα→∞ π −s 1/α = 1, so that there exists an < > 0 such that for
α > < ⇒ −π 1−1/α u1 = max( −π 1−1/α u1 , . . . ,−π −S1/α uS )
⇒ −π 1 u1−α = min( −π 1 u1−α , . . . ,−π S uS−α )
⇒ − Sπ 1 u1−α ≤
∑ −π u = V
−α
s
s
⇒ −(Sπ1)−1/αu1 ≥ V−1/α = −H
⇒ lim( Sπ 1 ) −1/α u1 = u1 ≤ lim H
α →∞
α →∞
(17.4)
Next note that, since we can always choose the original utility function so that us > 0,
−π 1 u1−α ≥
∑ −π u = V
s
−α
s
implying
π 1 u1−α ≤ −V ⇒ π 1−1/α u1 ≥ −V −1/α = H
⇒ lim π1−1/αu1 = u1 ≥ lim H
α →∞
α →∞
(17.5)
Hence from (17.4) and (17.5)
lim H = u1
α →∞
Repeating the argument for the instances in which us, s = 2, . . . , S is the minimum utility
level we see that
lim H = min(u1, . . . , uS)
α →∞
Hence projects are ranked by their minimum utility level or, equivalently, by their
minimum income level.
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Exercise 17F
1. Differentiate g(y) = (y − J)2 twice with respect to y to get g″ = 2 > 0. Hence g(y) is a
convex function of y and E(g(y)) (the variance of y) is increased by a mean preserving
spread in the distribution.
2. Write the two prospects as
P a = (.75, 0, .25, 0; 10, 22.727, 100, 1000)
P b = (0, .99, 0, .01; 10, 22.727, 100, 1000)
Then
γ1 = 0.75, γ2 = −.99, γ3 = .25, γ4 = −.01
so that, although ∑γsys = 0 (requirement (a)), requirement (b) is not satisfied.
It is possible to write P a equivalently as
P a = (0, .75, .25, 0; 3 13 , 10, 100, 120)
and to construct a mean preserving spread of P a
P c = (0.7, 0, 0, 0.25; 3 13 , 10, 100, 120)
Calculation shows that the variance of P c is 2552.08 and with u = √y, Eu = 4.1079. With u
= log y, the expected utilities of P a, P c are 2.8782 and 2.0999 respectively.
3. The change from F1 to F2 is a mean preserving spread if and only if it reduces the
expected value of a concave function of y. (See text pages 481–482). But since y1 and y2
differ only in location and scale and in fact have the same mean, we know from text
page 476 that all risk averters are made worse off if and only if σ2 > σ1. Hence F2 is a
mean preserving spread of F1.
Supplementary question
(i) Prove that sign Cov(h(y), y) = sign dh/dy to put the plausible assertion on text
page 476 that Cov(v′, y) < 0 if v is concave ([F.9]) on a more rigorous footing.
4. Since y(γ) is a linear function of y, the random variables y(γ) are all members of a
class differing only in location and scale (text page 475). The distribution function of
y(γ) is
G = Prob[9(γ) ≤ y] = Prob[γ (9 − J) + J ≤ y]
y − (1 − γ ) J
= Prob[9 ≤ (y − (1 − γ))J/γ) = F
γ
= G(y, γ)
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Since
y − (1 − γ ) J y − J
Gγ (y, γ) = − f
2
γ
γ
we have
−1
y′
9 − (1 − γ ) J
( y − J )d 9
γ
y′
∫ G ( 9, γ )d 9 = γ ∫ f
0
γ
2
0
(17.6)
Now Ey(γ) = Eγ (y − J) + J = J so that variations in γ do not alter the mean. Hence
(recall [F.19])
∫G = 0
1
0
γ
Next note that the derivative of (17.6) with respect to y′ is −f(⋅)(y′ − J)γ−2 which is nondecreasing for y′ ∈ (0, J) and non-increasing for y′ ∈ (J, 1). Since (17.6) is zero at y′ = 0
and at y′ = 1 it must therefore be non-negative for all y′ ∈ (0, 1). Because both
conditions [F.18] and [F.19] are satisfied, increases in γ constitute a mean preserving
spread.
To show that an increase in γ makes all risk averters worse off differentiate Ev(y(γ))
with respect to γ :
d
Ev(γ (y − J) + J) = Ev′(y(γ))(y − J)
dγ
= Ev′E(y − J) + Cov(v′, y)
= Cov(v′, y)
and Cov(v′, y) < 0 if v″ < 0.
Exercise 17G
1. The rate of change of prudence is
dP ( y) d − v ′′′( y)
−1
[v″″v″ − (v′″)2]
=
=
dy
dy v ′′( y) ( v ′′)2
which has the same sign as −[v″″v″ − (v′″)2] = [(v′″)2 − v″″v″]. Hence, since the first term
is positive and risk aversion is equivalent to v″ < 0, an individual has decreasing
prudence only if the fourth derivative of the utility function is negative: v″″ < 0.
2. The table has the derivatives of the three functions from Question 10, Exercise 17D.
v
v′
v″
v′″
P = −v′″/v″
P′
a − be−Ay
Abe−Ay
−A2be−Ay
A3be−Ay
A
0
a + by1−R/(1 − R)
by−R
−Rby−(1+R)
(1 + R)Rby−(2+R)
(1 + R)y−1
–
a + bln y
by−1
−by−2
by−3
y−1
–
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Supplementary questions
(i) From the answer to Question 8, Exercise 17E, show that the rate of change of
absolute risk aversion has the same sign as A(y) − P(y) and then compare A and P for
the three utility functions in the table to confirm your answers to Question 10, Exercise
17E about the sign of A′(y).
(ii) We can define relative prudence as Py by analogy with relative risk aversion (Ay). Is
relative prudence increasing, decreasing or constant for the three utility functions in the
table?
3. Expected utility is
E = v(y0 − a) + Ev(J1 + z + (1 + i)a)
where random period 1 income y1 = J1 + z is written as the sum of its mean J1 and a
random component z with zero mean (Ez = 0). The first order condition on the choice of
saving is
Ea = −v′(y0 − a) + (1 + i)Ev′(J1 + z + (1 + i)a) = 0
so that the marginal utility of consumption in the first period equals discounted second
period expected utility of consumption. Or: the marginal rate of substitution between
first and second period consumption is equal to rate at which consumption can be
transferred between periods by saving: Ev′(J1 + z + (1 + i)a)/v′(y0 − a) = 1/(1 + i) (recall
the savings decision under certainty from chapter 11B).
3. (a) Recalling the method of simple comparative statics for problem with a single
choice variable (Appendix I)
Eay
∂a*
[− v′′( y0 − a )]
=−
=−
>0
∂y0
Eaa
Eaa
0
(remember that Eaa < 0 for a maximum and the individual is risk averse). Intuitively,
since first period income is higher, marginal utility in the first period is smaller, and the
individual responds by reducing consumption in the first period by saving more, thereby
increasing future consumption and reducing expected marginal utility in the second
period.
3. (b) The marginal effect of an increase in expected future income is
EaJ
∂a*
[(1 + i) Ev ′′( J1 + z + (1 + i)a )]
=−
=−
<0
Eaa
Eaa
∂J1
1
An increase in expected future income reduces the expected marginal utility of
consumption in the second period and so first period consumption must be increased by
a reduction in saving.
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3. (c) The effect of an increase in the rate of interest is
∂a*
E
[Ev ′( J1 + z + (1 + i)a )] + a[Ev ′′( J1 + z + (1 + i)a )]
= − ai = −
Eaa
Eaa
∂i
=−
[Ev ′( J1 + z + (1 + i)a )] a[Ev ′′( J1 + z + (1 + i)a )]
−
Eaa
Eaa
=−
[Ev′( J1 + z + (1 + i)a )]
∂a*
+a
∂J1
Eaa
The first term is the substitution effect and is positive: an increase in the rate of
interest increases the opportunity cost of current consumption and so, holding expected
utility constant, saving will increase. However, an increase in i increases future
consumption at any given level of savings and increases in future consumption make
saving less valuable. The “income” or more precisely, wealth effect has a definite sign
(unlike the wealth effects examined in chapter 10B) because we have had to make more
restrictive assumptions about preferences under uncertainty in order to represent them
by an expected utility function. But the income and substitution effects are of opposite
sign so that the overall effect of an increase in the price of current consumption on
saving is ambiguous without further restrictions on preferences.
3. (d) We can write
∂a *
E
E[v ′( y1 ) + a(1 + i) v ′′( y1 )]
= − ai = −
∂i
Eaa
Eaa
=−
v′′
v′′
1
1
E 1 + a(1 + i) v′ = −
E 1 + [y1 + a(1 + i) − y1 ] v′
v′
v′
Eaa
Eaa
=−
v′′
v′′
1
E 1 + y1 v′ − [y1 − a(1 + i)] v′
v′
v′
Eaa
=−
1
E[[1 − R(y1)]v′ + [(y1 − a(1 + i))]A(y1)v′]
Eaa
Since y1 > a, A > 0, and v′ > 0, then if relative risk aversion is less than 1 (R < 1)
increases in the rate of interest increases saving.
4. The first order condition is now written
Ea = −v′(y0 − a) + (1 + i)Ev′(J1 + βz + (1 + i)a) = 0
and so
Eaβ = (1 + i)E[zv″(J1 + βz + (1 + i)a)] = 0
Since by construction Ez = 0
E[zv″(J1 + βz + (1 + i)a)] = EzEv″ + Cov(z, v″) = Cov(z, v″(J1 + βz + (1 + i)a))
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Now increases in z will lead to increases in consumption and so to increases or
decreases in v″ depending on whether v′″ is positive or negative. Thus a Sandmo
increase in risk increases saving if prudence (−v′″/v″) is positive. Moreover, recall
Question 8, Exercise 17E, we see that a sufficient condition for a Sandmo increase in
risk to increase saving is that absolute risk aversion is increasing which implies that
v′″ > 0.
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Chapter 18
Production under Uncertainty
Section 18B
1. Suppose that there is only output price uncertainty. The effect of an increase in B on
the marginal value of output is, from [B.3],
∂Vx
= Ev″(psx* − c(x*) + B)[ps − c′(x*)]
∂B
Define p0 ≡ c′(x*) and y0 ≡ p0x* − c(x*). Since ys is increasing in ps
ps ≥ p0 ⇒ A(ys) ≤ A(y0)
⇒ −v″(ys) ≤ A(y0)v′(ys)
⇒ −v″(ys)[ps − c′(x*)] ≤ A(y0)v′(ys)[ps − c′(x*)]
(18.1)
Similarly,
ps < p0 ⇒ A(ys) > A(y0)
⇒ −v″(ys) > A(y0)v′(ys)
⇒ −v″(ys)[ps − c′(x*)] < A(y0)v′(ys)[ps − c′(x*)]
(18.2)
(the last inequality follows because ps ≤ p0 = c′(x*)). Since (18.1) and (18.2) cover all
possible values of ps we can multiply each inequality through by −πs < 0 and sum over s
to get
Ev″(ys)[ps − c′(x*)] > −A(y0)Ev′(ys)[ps − c′(x*)] = 0
where the last equality follows from the first order condition [B.3] on x. Hence when
there is diminishing absolute risk aversion an increase in lump sum income reduces the
marginal value of output and so reduces the output level chosen.
Supplementary questions
(i) Repeat the above procedure to establish that an increase in lump sum income
increases the input level if there technological uncertainty and diminishing absolute risk
aversion.
(ii) Show that when there is price and technological uncertainty the assumption of
diminishing absolute risk aversion is not sufficient to determine the effect of lump sum
income on the input level.
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2. The variance of income [B.1] is
Var(ys) = Var(psx) = x2Var(ps)
which is increasing in output.
A risk averse firm with quadratic utility v = a + bys + cys2 has expected utility
V = a + bEys + cEys2 = a + bEys + c(Eys)2 + cVar(ys)
with c < 0, Eys < −b/2c. (See question 11, exercise 17D.) The marginal value of output is
dys
dVar( y s )
+c
Vx = (b + 2cEys)
dx
dx
At the output at which expected income is maximized, we have dEys/dx = 0 and so
Vx = cdVar(ys)/dx < 0. Hence the firm with a quadratic utility function chooses a
smaller output than a firm which is risk neutral and maximizes expected income.
3. (a) In state s the producer chooses output xs* to maximize v(ys). Since utility is
increasing in income (v′(ys) > 0) this implies maximization of ys and the first order
condition on output in state s is
dys
= ps − c ′( xs* ) = 0
dxs
(18.3)
and the optimal state s output is xs* = g(ps). Note that, since the cost function is not
state dependent, the supply function determining output in each state s is state
independent.
3. (b) The maximized utility in state s is v( ys* ) = v(psg(ps) − c(g(ps)) + B) and expected
utility is
Ev(ys) = Ev(psg(ps) − c(g(ps)) + B)
3. (c) Intuitively: the firm cannot be worse off if it chooses output after observing
ps rather than before, since it has the option of setting xs = x* where x* maximizes
Ev(psx − c(x) + B). More formally, since xs* = g(ps) maximizes ys,
v(psg(ps) − c(g(ps)) + B) ≥ v(psx* − c(x*) + B)
which implies
Ev(psg(ps) − c(g(ps)) + B) ≥ Ev(psx* − c(x*) + B)
3. (d) The firm is better (worse) off a result of a mean preserving contraction in the
distribution of ps if its maximized state s utility v(psg(ps) − c(g(ps)) + B) is concave
(convex) in ps. (See the discussion of mean preserving spreads in section 17F.) The
derivative of v( ys* ) with respect to ps is, using the envelope theorem,
dv( ys* )
dy *
∂y *
= v ′( ys* ) s = v ′( ys* ) s = v ′( ys* ) g( ps )
dps
dps
∂ps
Differentiating again with respect to ps we have
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dg( p s ) v ′′
dg ps g( ps ) v ′
v ′′( y s* ) g( ps ) 2 + v ′( y s* )
=
p s g( p s ) +
dp s
dps g( ps ) ps
v′
p g( p s ) g( p s ) v ′
= ε ps − R( y s* ) s
(18.4)
ps
*
y
s
where εps is the elasticity of supply with respect to ps and R = −v″ys/v′ is relative risk
aversion.
It is obvious from the first order condition (18.3) on state s output that the supply
elasticity is positive. Hence if risk aversion is “small” or B is “large” relative to profit
psg(ps) − c(g(ps)), (18.4) is positive and v( ys* ) is convex in ps. In these circumstances the
owner of the firm is worse off as a result of a price stabilization scheme which is a mean
preserving contraction of the price distribution. Thus the timing of decisions is crucial
in evaluating the gains and losses from price stabilization.
4. There is diminishing absolute risk aversion if
d( − v′′( y)/ v′( y))
1
=−
(v′″v′ − v″v″) < 0
dy
( v′)2
Hence v′″ < 0 is sufficient, but not necessary, for diminishing absolute risk aversion.
Supplementary question
(i) Use a Taylor’s series expansion on v(ys) about Eys to get an expression for Ev(ys)
which shows that v′″ < 0 can be interpreted as a dislike of positive skewness in the
distribution of ys.
5. Partially differentiate the first order condition [B.34] on z with respect to w to get
Vzw = − Ev ′′( ys ) z[ pfs′( z) − w] − Ev′(ys)
(18.5)
Hence if the firm is risk neutral (v″ = 0), the firm reduces the demand for the input when
its price increases because Vzw = −Ev′ < 0
Differentiate [B.34] partially with respect to B to get
VzB = Ev″(ys) ( pfs′ − w)
and substitute this into (18.5), yielding
∂z
V
V
Ev ′
= − zw = zB z +
∂w
Vzz Vzz
Vzz
=−
∂z
Ev ′
z+
∂B
Vzz
The second term is the substitution effect which is always negative since Vzz < 0. The
first term is the ambiguous income effect. The increase in w makes the firm worse off
and if there is declining absolute risk aversion the income effect reinforces the
substitution effect.
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6. When the market demand has constant elasticity of ε the demand function is q = pε
and so log q = ε log p and
Var(log q) = ε 2 Var(log p)
Hence
Var(log pq) = Var((1 + ε) log p) = (1 + ε)2Var(log p)
and the variance of log pq is less than the variance of log q if and only if ε > − 12 .
Exercise 18C
1. (a) If Eps = F < pf the expected profit from selling one unit forward and then buying a
unit on the spot market (to deliver the promised unit) is positive. Hence arbitrarily large
expected profits can be made by such selling forward and the market cannot be in
equilibrium if there are any traders who care only about expected income.
1. (b) We can write the first order condition on the sale of futures contracts [C.3]
(letting xf < 0 indicate purchases) as
(pf − F)Ev′(ys) − Cov(v′(ys), ps) = 0
When pf > F the first term is positive so that Cov(v′, ps) > 0 is required under contango. If
there is risk aversion contango implies that ys and ps are negatively correlated. But from
[C.1] this requires that the ps(fs − xf) are negatively correlated. If production is not
stochastic the ps(fs − xf) are negatively correlated only if the firm sells more forward
than it produces (xf > f(z)), entering the spot market to make up the difference between
its promised delivery and its output.
2. Let state 1 be fine weather and state 2 be wet weather. Denote the premium for cover
q against wet weather by ρ. Incomes in the two states are
y1 = p1 f1(z) − wz − ρ q + B
y2 = p2 f2(z) − wz + (1 − ρ)q + B
Assume that fine weather is preferable to wet weather (p1 f1(z) > p2 f2(z)). The firm’s
choice of input and cover satisfy
Vz = π1v′(y1) [ f1′( z) − w] + π2v′(y2) [ f2′( z) − w] = 0
(18.6)
Vq = −π1v′(y1)ρ + π2v′(y2)(1 − ρ) = 0
(18.7)
Since insurance is actuarially fair (ρ = π2), (18.7) implies v′(y1) = v′(y2) and cover is
set at the level which equates the state contingent incomes:
q* = p1 f1(z) − p2 f2(z)
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Fig. 18C.1
The equality of marginal utilities of income across states means that we can also divide
(18.6) through by v′ to get
π 1 [ f1′( z) − w] + π 2 [ f2′( z) − w] = 0
so that the availability of actuarially fair insurance leads the insured producer to act as
if risk neutral.
Fig. 18C.1 illustrates (compare text Fig. 18.3). By increasing z the firm can move
along the state contingent production possibility frontier FF. Since dys/dz = ps fs′( z) − w
the frontier has the slope
dy2 p1 f1′( z) − w
=
dy1 p2 f2′( z) − w
Note that p1 f1(z) > p2 f2(z) implies that FF lies below the 45° line. If there is no insurance
the risk averse producer chooses b0 where her indifference curve is tangent to FF.
(Rearrange (18.6) to derive the tangency condition.) The lines a0b0, a*b* have slope
−π1/π2 and are iso-expected income lines. A risk neutral producer would choose point b*
where expected income is maximized.
When insurance is available the risk averse producer could still choose a production
plan at b0 and then insure to get to a0. However, she does even better by choosing to
produce the state contingent incomes at b* and insuring to a*.
Without insurance the owner of the firm can alter her state contingent incomes only
by changing her input choice. The insurance market permits the exchange of state
contingent incomes and allows the owner to separate production and consumption
decisions. The production plan is made to maximize expected income and the resulting
risk traded away via the insurance market to leave the owner with a risky income but
certain consumption. Compare the separation of intertemporal consumption and
investment in chapter 11. Note that since the introduction of insurance alters the
production decisions it will have consequences for the input and output markets.
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Chapter 19
Insurance, Risk Spreading and Pooling
Exercise 19B
1. (a) The slope of the budget constraint [B.5] is −(1 − kπ2)/kπ2 > −(1 − π)/π. Increases in
π flatten the budget constraint and reduce the slope of the indifference curves. (See
question 4 for the effect on the demand for cover.)
1. (b) If the insurer incurs administrative costs of k0 + kq whether the accident occurs or
not, the breakeven premium (in £) is P = π q + k0 + kq = k0 + (π + k)q for cover of q. Hence
dy1/dq = −(π + k) and dy2/dq = 1 − (π + k). The budget line has slope dy2/dy1 =
−[1 − (π + k)]/(π + k) and starts from a point (y − k0, y − L − k0) instead of (y, y − L).
In Fig. 19B.1 contracts along bc, except b, break even. The feasible set is point a plus bc
except for point c.
2. The marginal value of cover is (see [B.8])
Vq = −π 1 v1′ (y − pq)p + π 2 bv′1 (y − L + (1 − p)q)(1 − p)
and so increases in a have no effect on the demand for insurance. Increases in b
increase the marginal utility of income if there has been an accident and so increase the
demand for cover. The state dependence of utility functions affects decisions which
transfer income between states only if marginal utility depends on the state.
Full cover would be bought if b ≥ π1p/π2(1 − p) since this implies that Vq(q, ⋅) > 0 at
q < L.
3. With full cover, actuarially fair insurance the insured has expected utility of
v(y − π2L) > π1v(y) + π2v(y − L)
and would be willing to pay up to β, defined implicitly by
v(y − π2L − β) = π1v(y) + π2v(y − L)
over and above the fair premium π2L for such a policy. Thus the certainty equivalent
income for no cover is yc = y − π2L − β. Since expected income is J = y − π2L we see that
β = J − yc is the risk premium.
4. (a) Partially differentiate Vq with respect to π2, remembering that π1 = 1 − π2:
Vqπ = v′(y − pq)p + v′(y − L + (1 − p)q)(1 − p) > 0
2
which implies that demand for cover increases with the accident probability.
226
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Fig. 19B.1
4. (b) Since
VqL = −π2v″(y − L + (1 − p)q)(1 − p) > 0
the demand for cover increases with the loss.
4. (c) Because p varies with π2, the effect of an increase in the accident probability on
the marginal value of cover is, from (19.1) and [B.18],
Vqπ + Vqp
2
dp
= [v1′ p + v2′ (1 − p)] − [π 1 v1′ + π 2 v2′ ]k − kVqyq
dπ 2
(19.2)
where vs′ = v′(ys). Make use of the first order condition [B.8] to substitute π 1 v1′ p/π 2 for
v2′ (1 − p) and π 1 v1′ p /(1 − p) for v2′π 2 and collect terms to write the first two terms of
(19.2) as
π
π 2π 1
π p
p
v1′ p 1 + 1 − kπ 1 v1′ 1 +
−π1 − 1
= v1′ k π 2 +
1 − p
1 − p
π2
π2
=
v1′ k
[(1 − p) − π1(1 − p) − π1p]
(1 − p)
=
v1′ k
(1 − p − π1)
1− p
=
v1′ kπ 2
(1 − k) < 0
1− p
Even though the substitution effect (the second term in (19.2)) outweighs the effect of
the change in the slope of the indifference curves (the first term in (19.2)) the overall
effect is ambiguous because of the income effect. However, if insurance is a normal
good (so that the income effect of a rise in π2 and therefore in p reduces the demand
for insurance) we see that the demand for cover will decrease when the accident
probability increases.
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Supplementary question
(i) Illustrate these comparative static results in a series of diagrams.
5. The first order condition is
Vq = [v(y − P) − v(y − q − P + q)]dF − P ′
+ (1 − P ′)
= −P′
∫ v′(y − P (q))dF
q
0
∫ v′(y − L − P + q)dF
L1
q
∫ v′(y − P )dF + (1 − P ′)∫ v′(y − L − P + q)dF = 0
q
L1
0
q
Partially differentiate with respect to y to get
Vqy = − P ′
∫ v′′(y − P )dF + (1 − P ′)∫ v′′(y − L − P + q)dF
q
L1
0
q
and, using the definition of absolute risk aversion to replace v″ with −Av′,
Vqy = P′
∫ A(y − P)v′(y − P)dF
q
0
− (1 − P′)
∫ A(y − L − P + q)v′(y − L − P + q)dF
L1
q
Now y − L − P + q < y − P when q < L. Hence, if A declines with income, A(y − L − P + q)
> A(y − P) and so
Vqy < P′
∫ A(y − P)v′(y − P)dF − (1 − P′) ∫ A(y − P)v′(y − L − P + q)dF
q
L1
0
q
= A(y − P)[P′
∫ v′(y − P)dF − (1 − P′) ∫ v′(y − L − P + q)dF]
q
L1
0
q
= A(y − P)[−Vq(q*, ⋅)] = 0
where q* is the optimal cover, so that Vq(q*, ⋅) = 0. Hence an increase in income reduces
the demand for cover if and only if the insured has decreasing absolute risk aversion.
Exercise 19C
1. If the small uninsurable risk is characterised (as in [C.1]) by a reduction in income in
states 2 and 4 then clearly expected utility is reduced because an increase in D reduces
income in these states with no offsetting increase in income in states 1 and 3:
dEu dq
dEu
dEu
= −f2u′(y2) − f4u′(y4) < 0
+
=
dq dD D = 0 dD D = 0
dD D = 0
(remember that insurance against the insurable risk is chosen optimally so that
dEu/dq = 0).
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More interesting is the effect of small uninsurable risk which is a fair bet ie has
a zero expected value. To ensure this we can write the state contingent incomes as
y1 = x + kD − pq, y2 = x − D − pq, y3 = x + kD − L + (1 − p)q, y4 = x − D − L + (1 − p)q with
k = (f2 + f4)/(f1 + f3) = τ /(1 − τ). Now the effect of small uninsurable fair risk is
dEu
= f1 u1′k − f2 u2′ + f3 u3′ k − f4 u4′
dD D = 0
= ( f1 k − f 2 ) u2′ + ( f 3 k − f4 ) u4′
= [ f1( f2 + f4) − f2( f1 + f3)]
= ( f1 f4 − f2 f3)
u2′
u4′
+ [ f3( f2 + f4) − f4( f1 + f3)]
1−τ
1−τ
u2′ − u4′
1−τ
(19.3)
(19.4)
(where ui′ = u′(yi) and remember that with D = 0, y1 = y2, y3 = y4.)
Consider a number of cases. (a) The insured faces an actuarially fair premium on the
insurable risk, in which case she buys insurance until marginal utility is equal in all
states. Hence (19.4) is zero: with fair insurance a small uninsurable fair bet makes the
individual no worse off. Intuitively, since marginal utility is equalised across all states
the insured is risk neutral towards small bets. (b) Insurance against the loss L is
actuarially unfair so that u′(y2) < u′(y4). Now, as some simple manipulation shows,
( f1 f4 − f2 f3) is positive or negative if and only if f4/( f3 + f4) is greater or less than f2/( f2 + f4).
But f4/( f3 + f4) is the conditional probability of the uninsurable loss ocurring given that
the insured loss has ocurred and f2/( f2 + f4) is the conditional probability of the uninsurable loss occuring given that the insured loss has not ocurred. If the former exceeds
the latter the losses are positively correlated. Hence with positively correlated losses a
small uninsurable risk reduces expected utility. Conversely if the losses are negatively
correlated expected utility is increased.
Exercise 19D
1. (a) The project makes the individual worse off because utility without the project is
V(0) = log 10000 = 9.210 and expected utility with the project is
V(1) = 0.5 ln(20000) + 0.5 ln(1000) = 8.406
1. (b) With n individuals in a syndicate sharing the project proceeds each gets an
expected utility of
10000
9000
V(n) = 0.5 ln 10000 +
+ 0.5 ln 10000 −
n
n
The syndicate size which would leave the members no worse off with the project is
defined by V(0) = V(n) or
10000
9000
ln(10000) = 0.5 ln 10000 +
+ 0.5 ln 10000 −
n
n
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Multiplying through by 2 and anti-logging gives
100002 = (10000 + 10000/n)(10000 − 9000/n)
which solves for the unique breakeven { = 9.
1. (c) Assume for the moment that n is a continuous variable. Then a necessary
condition for the expected utility of syndicate members to be maximized with respect to
n is V′(n) = 0. Equivalently, since multiplying the maximand by a positive constant (2)
does not affect the solution, the optimal size for syndicate members satisfies
0 = 2V′(n) =
=
−10000 n −2
90000 n −2
−
((10000 n + 10000)/ n ) ((10000 n − 9000)/ n )
−10 n −1
9 n −1
−
10 n + 10 10 n − 9
Multiplying through by n and rearranging gives the unique solution n* = 18.
1. (d) The compensating variation p(n) for a syndicate member is defined by
10000
9000
0.5 ln 10000 +
− p( n) + 0.5 ln 10000 −
− p( n ) = ln 10000
n
n
It is the maximum amount that would be paid for membership of the syndicate. With
N identical individuals, all counting equally in social welfare, we can use the sum of
compensating variations as the welfare measure. (The N − n non-members have
a compensating variation of zero.) Using [D.8], the welfare measure is W = np(n) =
500 − nᐉ(1/n), where ᐉ is the risk loading. We know that limn→∞ nᐉ(1/n) = 0, so that
provided N is large the socially optimal syndicate size is n** = N.
1. (e) If the owner gives away the fraction λ of the project his expected utility is
V = 0.5 ln(10000 + (1 − λ)10000) + 0.5 ln(10000 − (1 − λ)9000)
Using the answer to part (c), this is maximized when λ = (n* − 1)/n* = 17/18.
1. (f ) Assume that there are no competing projects. Since the owner is a monopolist and
knows all potential buyers’ utility functions, it is plausible that he will price shares in
the project so that buyers are just indifferent between buying and not buying. He sells a
proportion λ of the project to n individuals at a price p. Adopting a more general
notation, let the utility function of the owner and the buyers be u, ub respectively, the
income without the project of the owner and the buyers be y, yb respectively, their
incomes with the project in state s be ys, ybs respectively and the project payoff in state s
be zs. Then p satisfies
Eub(ybs) = Eub(yb + λzs/n − p) = Eub(yb + ᐉ(n, λ) + λes/n) = u(y)
(19.6)
where es = zs − Ezs = zs − K. (See [D.6].) Again using the definition of the risk loading
(ᐉ = λX/n − p), the total revenue from share sales is np = λK − nᐉ(n, λ) which is maximized for given λ by making n as large as possible: n = N − 1.
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With given λ, the owner gets
V = Eu(ys) = Eu(y + pn + (1 − λ)zs) = Eu(y + λK − (N − 1)ᐉ + (1 − λ)zs)
= Eu(y + K − (N − 1)ᐉ(n, λ) + (1 − λ)es)
The effect of an increase in the proportion of the project sold off is
dV
dy
dl
= Eu′( ys ) s = Eu′( ys ) − es − ( N − 1)
dλ
dλ
dλ
= −Eu′(ys)Ees − Cov(u′(ys), es) − Eu′(ys) (N − 1)
= −Cov(u′(ys), es) +
Eu′( ys )Cov( ub′ ( ybs ), es )
Eub′ ( ybs )
dl
dλ
(19.7)
where we have used the definition of the risk loading (19.6) to get
− Eub′ ( ybs )e s
− Cov( ub′ , e s )
dl
=
=
dλ ( N − 1) Eub′ ( ybs ) ( N − 1) Eub′ ( ybs )
(Remember Ees = 0.) At λ = 0 the second term in (19.6) is zero because Cov( ub′ ( ybs , e s ))
= 0 since ybs = yb if the owner does not sell any of the project. At λ = 1 the first term is
zero. (Cov(u′(ys), es) = 0 because ys is non-stochastic if the owner sells all the project.
Hence, remembering that the Cov terms are negative if there is risk aversion, the
optimal λ is positive but less than 1. The owner will not sell all of the project. As λ
increases more risk is borne by buyers and less by the seller. This makes the owner
more willing to bear risk and eventually reduces the proceeds from sales to risk averse
buyers.
Exercise 19E
1. (a) Denote the variance of the rates of return by σ 2, the random rate of return on
security i by hi and the amount invested in security i by Di. The realised value of the
portfolio is w = ∑1n Di(1 + h)i and the variance of the portfolio value is
Var(w) = Var(∑ Di(1 + hi)) = ∑ Di2 Var(1 + hi) = σ 2 ∑ Di2
(The covariances are zero since the rates of return are independently distributed.) Since
Var(w) is convex and symmetric in (D1, . . . , Dn), it is minimized when D1 = . . . = Dn = y/n.
1. (b) The minimized portfolio variance is
Var(w) = σ 2∑y2/n2 = σ 2y2/n
which vanishes as n → ∞.
Exercise 19F
1. A proof is constructed along the following lines: we know that the self-selection
constraint for high-risk types, [F.9], must be imposed, and so the discussion of [F.9]
goes through as before and the equilibrium contract for high-risk types is (πhL, L). If
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constraint [F.9] is binding for low-risk types then they are on the low-risk type
indifference curve passing through this point – Ah in Fig. 19.8 of the text. But since the
low-risk indifference curve is steeper than the high-risk indifference curve at this point
(draw in a low-risk indifference curve in Fig. 19.8), it is obviously always possible to
increase the low-risk types’ expected utility while still breaking even and continuing to
satisfy [F.9], i.e. by moving along the Ih indifference curve through Ah. But this then
means that constraint [F.9] is no longer binding for low-risk types.
2. We leave the reader to construct the figures from the more formal analysis given in
the answer for question 3.
3. Differentiating totally through [F.11] of the text and rearranging gives
[π l (1 − π h ) vl′1 − π h (1 − π l ) vl′2 ]dZl = π h vh′ [vh′ − vl′2 ]dL + [vh′ L + ( v( yl 2 ) − v( yl 1 )]dπ h
− Zl [(1 − π h ) vl′1 + π h vl′2 ]dπ l
where vl1′ and vl′2 are low-risk types’ marginal utilities at their state 1 and 2 incomes yl1
and yl2 respectively, and vh′ is a high type’s marginal utility (the same across states). It
can quickly be established that the coefficient of dZl is negative, because πh > πl , (1 − πl )
> (1 − πh ), and vl′2 > vl1′ (since yl1 > yl2). We denote this term by δ < 0. Then comparativestatics effects are:
(i)
∂Zl π h [vh′ − vl′2 ]
=
>0
δ
∂L
An increase in the size of loss increases the coverage low-risk types can obtain. Note
that if [F.11] is to hold we must have (since utility functions are identical):
y − πl Zl > y − πh L > y − L + (1 − πl )Zl
Then, since v″ < 0, this implies vl′2 > vh′ and the sign of ∂Zl /∂L follows from δ < 0.
(ii)
∂Zl
v ′ L + ( v( yl 2 ) − v( yl 1 ))
0
= h
∂π h
δ
The sign of this effect is ambiguous, since the term vh′ L > 0 while v(yl2) − v(yl1) < 0. A
change in πh changes both sides of [F.11] in the same direction, and whether Zl will have
to be increased or decreased (or left unchanged) to maintain the equality depends on
the parameters of the problem.
(iii)
− Z [(1 − π h ) vl′1 + π h vl′2 ]
∂Zl
= l
>0
∂π l
δ
An increase in the low risk type’s probability of loss increases the degree of coverage
because it reduces the attractiveness of the low-risk contract to the high-risk types.
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4. If insurers could not observe how much insurance an individual buys in total, then the
analysis falls apart, because a high-risk type could simply take out several low-risk type
contracts, effectively moving upwards along the line Bly0 in Fig. 19.8 until he achieves
the best point he can. Thus the separating equilibrium can only be sustained if buyers
can be rationed to the total level of coverage Zl.
5. This is essentially explained by the answer to question 4. It may be costly to monitor
every buyer of insurance to ensure that she is not taking out more total coverage than Zl
at the low-risk premium. However, if every buyer knows that, in the event she makes a
claim, the total coverage she will in fact receive will be restricted to Zl, then it is a waste
of money to buy more coverage than this ex ante. Note, however, that sellers of
insurance must still exchange information on the identity of claimants, but this involves
lower costs than monitoring the identity of all buyers of insurance.
6. In Fig. 19.9, an increase in risk-aversion of low-risk types will increase the curvature
of the indifference curve I d′ about its intersection point with the 45° line. Thus, to
obtain a tangency point the line S* must rotate to the right. This widens the set of values
of γ for which the separating equilibrium will exist. If risk aversion of high-risk types
increases, this increases the curvature of the indifference curve I h in Fig. 19.9, thus
shifting point d upwards along the line Bly0, implying that the I d′ indifference curve
must also be higher. This again therefore widens the set of γ -values for which the
separating equilibrium exists.
7. The tax paid on each high risk contract is T so that the low risk breakeven line Sᐉ
satisfies
Pᐉ = πᐉqᐉ + T
The proceeds of the tax per head of the total population of high and low risks are λT
since the proportion of low risks is λ. The amount of subsidy per high risk contract is
therefore λT/(1 − λ) and the high risk breakeven line Sh satisfies
Ph = πhqh −
λT
(1 − λ )
At the intersection of Sᐉ and Sh the low and high risk contracts offer the same amount of
cover qᐉ = qh = q. By offering this pair of contracts the insurer would have expected
costs, after taxes and subsidies of
λ[πᐉq + T] + (1 − λ) π h q −
λT
= λπᐉq + (1 − λ)πhq = èq
(1 − λ )
The pooling contract breakeven line S satisfies
Ü = [λπᐉ + (1 − λ)πh]q = èq
Thus the intersection of the breakeven lines Sᐉ, Sh is also on the pooling contract
breakeven line.
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Fig. 19F.1
8. Fig. 19F.1 retains the assumption of Fig. 19.10 that there are equal numbers of high
and low risks and illustrates the set of possible cross-subsidy contracts generated as the
level of cross-subsidy T varies. Increasing T shifts the post-subsidy break-even line Äᐉ
for the low risk contract down and the post-subsidy break-even line Äh for the high risk
contract up. Hence the high risks’ contract Ãh moves up the 45° line. The low risks’
contract ã is defined by the intersection of the high risk indifference curve Éh through Ãh
with the post-subsidy low risk break-even line Äl. Hence as T increases the low risk
contract shifts to the left, tracing out the locus dã.
In the text the cross-subsidy arose from government intervention. However there
can also be a Wilson anticipatory equilibrium in which competing firms offer crosssubsidising contracts without the need for government imposed taxes and subsidies. In
Fig. 19F.1 competition amongst firms for low risk customers ensures that if they offer
cross-subsidising separating contracts the only possible candidate for an equilibrium is
the pair Ãh, ã where the expected utility of low risks is maximized given the selfselection and break-even constraints. This pair of contracts is clearly not a RothschildStiglitz Nash equilibrium since some firm could make a profit by offering only the
contract ã which attracts only low risks if all other firms continue to offer the pair Ãh, ã.
But if a firm offers only the contract ã and attracts only low risks the other firms will
have a higher ratio of bad to low risks and will make a loss on their cross-subsidising
pair of contracts. These will be withdrawn and all risks will buy ã which will make a
loss. Hence the cross-subsidy pair Ãh, ã is an anticipatory equilibrium. Note that since
the locus dã lies above the fair odds pooling line except at d the pooling contract a
cannot be an anticipatory equilibrium if cross-subsidy is feasible.
If the indifference curves of the low risks are sufficiently steeply sloped the best
contract for them will be at d where there is no cross-subsidy. Hence with the
anticipatory equilibrium concept and feasible cross-subsidy an equilibrium will always
exist and it will be separating.
The anticipatory cross-subsidy equilibrium is also constrained Pareto efficient in that
any intervention by a planner who is subject to the same informational constraints as
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the firms can only make one type better off by making the other type worse off. The
anticipatory cross-subsidy equilibrium maximizes the expected utility of the low risk
types subject to the constraints that firms break even over the pair of contracts offered
and that insureds self-select. A planner would be subject to exactly the same constraints
and could not make any Pareto improvement using tax and subsidy instruments. For a
fuller discussion see K. Crocker and A. Snow, “The efficiency of competitive equilibrium
in insurance markets with asymmetric information”, Journal of Public Economics, 26,
1985, 207–219.
Exercise 19G
1. (a) With the insured paying a lump sum tax T and receiving a proportionate subsidy s
on her care expenditure, her expected utility is
V(a, P, T, y, q, L, s) = [1 − π(a)]v1(y1) + π(a)v2(y2)
(19.8)
where y1 = y − P − T − (1 − s)a and y2 = y − L − P − T − (1 − s)a + q. Since the insurer
cannot observe a the insurance contract terms P, q cannot be made conditional on a
and so in choosing her care level the insured takes the insurance contract as given.
Assuming that the optimal amount of care is positive, her care choice satisfies the first
order condition
Va = π ′(a)[v2(y2) − v1(y2)] − (1 − s)[(1 − π ) v1′( y1 ) + π v2′ ( y2 )] = 0
(19.9)
and the second derivative with respect to a is
Vaa = π ″(a)[v2 − v1] − 2π ′( a )[v2′ − v1′](1 − s) + Evi′′
(19.10)
where vi′ is marginal utility in state i.
The second term in (19.9) is the expected marginal utility of income times the “price”
(1 − s) of care or the utility cost of increased care expenditure. The first term is the gain
from increased expenditure on care which reduces the probability of state 2. Note that
the insured will only take care if this first term is positive, which requires that her utility
in state 2 is smaller than in state 1. Thus if she is sufficiently well compensated for the
accident that v2 ≥ v1 she will take no care at all.
Note also that the plausible assumptions of risk aversion ( vi′′ < 0) and diminishing
returns to care (π ″ > 0) are not sufficient to ensure that V is concave in a. We must
place stronger restrictions on preferences and the technology for Vaa < 0. If the insured
has less than full compensation for accident v2 < v1 the first term is negative. The last
term is negative by risk aversion. However the middle term could be positive or
negative. For example if marginal utility of income does not depend directly on the state
and y2 < y1 then v2′ > v1′ and it is possible that Vaa > 0.
1. (b) The optimal care level is a*(P, T, y, q, L, s). Inspection of Va shows that second
order cross partial derivatives satisfy VaP = VaT = −Vay and Vaq = −VaL, so that
a *P = aT* = − ay* and aq* = − a L* . We will therefore just derive the effects of P and q on the
amount of care. (Note that an increase in P reduces income in both states, an increase
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in q increases income in state 2 only.) For future use we will also derive the effect of a
change in the subsidy of care s.
Partially differentiating Va with respect to P, q and s gives
VaP = VaT = −Vay = −π ′[v2′ − v1′] + (1 − s) Evi′′
(19.11)
Vaq = −VaL = π ′v2′ − (1 − s)π v2′′
(19.12)
Vas = π ′[v2′ − v1′]a + Evi′ − (1 − s) Evi′′a = Evi′ + aVay
(19.13)
The comparative static responses are in general ambiguous without further
assumptions. Thus consider the effect of an increase in cover q. Intuitively one would
expect that if income in the accident state is increased, thus increasing utility in that
state, this would lead to a reduction in care because the marginal benefit of care
π ′[v2 − v1] is reduced. As (19.12) shows this neglects the effect of an increase in q on the
marginal cost of care. Because income in state 2 is greater, marginal utility is smaller so
that the marginal cost of care is reduced. Hence the optimal amount of care could
increase or fall with q. Rearranging Vaq we see that care is reduced only if
−π ′
− v ′′
> 2
(1 − s)π
v2′
(19.14)
where the right hand side is the coefficient of absolute risk aversion in state 2. Hence if
the insured is not too risk averse an increase in cover will reduce care.
Using (19.13) we can also derive a Slutsky equation for an increase in the subsidy
rate:
a s* =
−Vas − Evi′ Vay − Evi′
=
−
=
+ a *a y*
Vaa
Vaa
Vaa
Vaa
(19.15)
The first term is the positive substitution effect (remember an increase in s corresponds
to a reduction in the price of care) and the second is an ambiguous income effect.
Referring to −Vay = VaP in (19.11) shows that risk aversion (which implies Evi′′ < 0) is
insufficient to ensure that care is normal good.
Supplementary question
Suppose that the insured has state dependent constant absolute risk aversion
preferences with vi = Ki − ki exp(−α yi). What is the interpretation of the coefficients Ki,
ki i = 1, 2? What restrictions must be placed on them to yield a sensible specification
with intuitively plausible comparative static properties?
2. Competition ensures that the insurer has zero expected profit M = P − qπ. The insurer
is also constrained by the fact that care cannot be made an explicit term of an
enforceable contract. Hence insurers realise that care is chosen to maximize the
insured’s expected utility given the premium and the cover: a = a*(P, T, y, q, L, s) and
that their expected profit constraint must be written
M = P − qπ(a*(P, T, y, q, L, s)) = 0
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which defines the premium implicitly as a function of the cover (and y, L, T, s):
P = P(q, T, y, L, s)
(19.17)
Using the implicit function rule on (19.16) gives
Pq = −
Mq
MP
=
π + qπ ′( a*)aq*
1 − qπ ′( a*)a *P
(19.18)
The assumption that care is an inferior good ay* = − a *P < 0 is sufficient, though not
necessary, to ensure that an increase in the premium with cover held constant increases
expected profit: M P = 1 − qπ ′a *P > 0. This, coupled with the assumption that care is
reduced by an increase in cover, implies that Pq > π, since
Pq > π ⇔ qπ ′a q* + π > π (1 − qπ ′a *P ) ⇔ [π a *P + a q* ]qπ ′ > 0
Hence, under these assumptions, insurers will set a premium which increases more than
in proportion to the cover since they know that an increase in cover increases the
accident risk.
Competition amongst insurers means that the insurance contract will maximize
the insured’s expected utility subject to the insurer breaking even (the insurer’s
participation constraint) and to the insured’s choice of care taking no account of its
effect on the insurer’s expected profit (the insured’s incentive compatibility constraint).
Both constraints are embodied in (19.16). Using (19.16) to substitute for P in (19.17), the
marginal value of cover is
dV
= Va [a q* + a *P Pq ] + Vq + VPPq = Vq + VPPq
dq
= π v2′ − Evi′ Pq
(19.19)
(Remember that a is chosen so that Va = 0.) From (19.19) we see that
dV
= π v2′ − Evi′π = π (1 − π )( v2′ − v1′ )
dq q=0
which is positive if marginal utility is state independent since then y1 > y2 implies
v2′ > v1′.
3. The equilibrium in the insurance market has the insured choosing an optimal care
level given the insurer’s break even constraint and the information asymmetry which
prevents care being directly controlled by the contract. Assuming that the optimal cover
is positive, it is defined by dV/dq = 0 and depends on the parameters which the insurer
and insured take as exogenous: q = q**(y, T, L, s).
Substituting for a and q, the per capita public sector budget constraint can be written
as
T − sa*(P(T, y, q**, L, s), T, y, q**, L, s) = 0
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defining the lump sum tax implicitly as function of the subsidy rate and the other
exogenous variables:
T = }(y, L, s)
(19.20)
Using the implicit function rule we see that
} s s=0 =
a* + s dads*
= a*
*
1 − s da
dT
(19.21)
With identical individuals treated identically the social planner’s problem is to
choose s, T, a, q, P to maximize V subject to (a) the per capita public sector budget
constraint, (b) the insured’s choice of care, (c) the break even constraint on the
insurance contract and (d) the insured’s choice of cover. Using (19.20) and (19.21) we
can use the four constraints to substitute for T, a, q, P to reduce the planner’s problem
to choosing the single policy instrument s to maximize
V(a*, P(}, y, q**, L, s), }, y, q**, L, s)
(19.22)
The derivative of V with respect to s is
dV
da* da* dV dq** dq**
= Va
+
}s +
+
}s
ds
dT
ds dT dq ds
+ VP [Ps + PT}s] + VT}s + Vs
= VP [Ps + PT}s] + VT}s + Vs
(19.23)
where we have used the fact that Va = 0 and dV/dq = 0.
Using (19.23) and the implicit function rule on (19.22) we get
qπ ′a s* + qπ ′aT* } s
dV
= VP
− Evi′a* + Evi′a*
ds s=0
1 − qπ ′a P*
= VP qπ ′
a s* + aT*a*
1 − qπ ′a *P
a * − a y*a*
= − Evi′qπ ′ s
>0
1 − qπ ′a *P
(19.24)
From (19.24) we see that the numerator in the fraction in (19.24) is the substitution
effect on care of an increase in the subsidy and is positive. Hence, since π ′ < 0, we have
established that introducing a small subsidy for care improves on the competitive
insurance market equilibrium even though the insurance contract mitigates the moral
hazard produced by the insurer’s inability to control the level of care directly via the
insurance contract and the subsidy must be financed by a tax on the insured. The
market is not constrained efficient because the planner has an instrument (the subsidy)
which is not available to the insured and insurer.
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4. To derive the full information first best allocation suppose that the government can
observe a and control it directly. Hence it has no need of tax and subsidy instruments
and its budget constraint is P − π(a)q = 0. The Lagrangean for its problem is
L = [(1 − π(a)]v1(y − P − a) + π(a)v2(y − L − P − a + q) + λ[P − π(a)q]
and the first best premium, cover and care satisfy
LP = − Evi′( yi ) + λ = 0
Lq = π v2′ ( y2 ) − λπ = 0
La = π ′(a)[v2 − v1] − Evi′( yi ) − λπ ′(a)q = 0
(19.25)
The first two conditions imply that λ = v2′ = Evi′ = v1′ so that there is perfect insurance in
that the marginal utility of income is equalised. Rearranging the condition on the
amount of care gives
v − v1
− q = 1
Evi′
π ′( a ) 2
where the left hand side is the social marginal benefit of care taking account of its effect
on the expected utility of the insured and the expected costs of the insurer which fall at
the rate q as the accident probability is reduced.
When the government cannot directly control a it must rely on the care subsidy to
influence it indirectly. The government budget constraint is now P + T + sa − π q = 0.
Since the effect of P and T on the budget constraint and the insured’s utility are
identical we can set T = 0 and let the premium be set both to finance the care subsidy
and the cost of cover. The government is subject to its budget constraint and to the
constraint that the insured will choose her care solely for its effect on her expected
utility, ignoring its effect on the government budget constraint. Despite this it is possible
to induce a first best solution. The insured’s choice of care satisfies (if it is positive)
π ′(a)(v2 − v1) − (1 − s) Evi′ = 0
(19.26)
However the first best choice of care satisfies (19.25). If the government chooses the
subsidy so that
s=−
π ′( a )q
Evi′
then (19.26) will imply that (19.25) holds. Hence by also setting P and q, subject to the
government budget constraint, to ensure that there is perfect insurance, the first best
can be achieved.
The only difficultly with this argument arises if utility is state independent. Then
perfect insurance ( v1′ = v2′ ) implies v1 = v2 in which case the insured has no incentive to
take any care and will set a = 0. However if insurance is only slightly less than perfect so
that v2 − v1 = −ε < 0 with ε “small” then the allocation can be made as close as desired to
the first best.
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Exercise 19H
1. (a) Separating equilibrium. Refer to Figure 19.14 of the text, and let s2 and s3
respectively denote the s-values at which the indifference curves I1 and I2 cut a
horizontal line drawn from ”. Then if s* ∈ [s2, s3] we have a separating ICE. If type 1
chooses s = 0, it receives θ1, and if type 2 chooses s ≥ s* ∈ [s2, s3] it will receive ”. Type 1
is in fact indifferent between (0, θ1) and (s2, ”), and if it chooses the latter this is still
consistent with P’s beliefs.
(b) Pooling Equilibrium. There is also a pooling equilibrium with s* = 0. It is
impossible to choose s < 0, so no type receives θ1, everyone chooses s = 0 and receives
”. This is also consistent with P’s beliefs.
2. Take P’s beliefs to be as given on page 548 of the text. Applying the arguments
underlying [H.1] and [H.2] in the text results in the inequalities
s* ≥
θ 2 − θ1
c1
; s* ≤
θ 2 − θ1
c2
(19.27)
But if c2 > c1 these inequalities cannot both be satisfied and there is no separating ICE
with signalling.
3. (a) We obtain a result as in the lemons model if r2 > ”. Then no type 2 worker would
accept the pooling contract, and only type 1 workers are employed at wage θ1.
(b) If r2 < θ1, then we have essentially the same results as before. If θ2 > r2 > θ1 then
there are two possible cases. Refer to Figure 19.14 in the text. The point s0 continues to
have the interpretation that type 1 will never choose s to the right of this in order to
receive θ2. However s1 is no longer relevant as the upper bound of possible equilibrium
values of s, since if r2 > θ1 this violates the participation constraint for type 2 that
θ2 − c2s* ≥ r2
(19.28)
In fact this defines a new upper bound of the interval given by
s1′ =
θ 2 − r2
c2
< s1
(19.29)
Then as long as s1′ ≥ s0 , we have that separating ICE in which type 1 chooses s = 0 and
receives θ1 and type 2 chooses s* ∈ [s0, s1′ ] and receives θ2 exist. However, if r2 is
sufficiently high that s1′ < s0 , then no amount of s acceptable to type 2 can achieve
separation, since they would allow type 1 to signal s1′ , receive θ2 and be better off than
at (0, θ1). Then if r2 ≤ ” only a pooling equilibrium is possible, while if r2 > ” we have the
lemons case as in (a).
4. In a pooling equilibrium, either
(i) P would offer r2 so both types would work and her profit is
πθ2 + (1 − π)θ1 − r2
or
(ii) P would offer 0, so only type 1 would work and her profit is (1 − π)θ1.
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If πθ2 − r2 > 0 therefore she chooses (i), if πθ2 − r2 < 0 she chooses (ii).
In any separating equilibrium, P will be able just to pay the reservation
values r2 and 0, and so her profit is
π (θ2 − r2) + (1 − π)θ1.
(19.31)
Thus in the case of a pooling equilibrium (i) the difference in profit is
π (θ2 − r2) + (1 − π)θ1 − [πθ2 + (1 − π)θ1 − r2] = (1 − π)r2 > 0
(19.32)
while in the case of pooling equilibrium (ii) the difference is
πθ2 + (1 − π)θ1 − π r2 − (1 − π)θ1 = π (θ2 − r2) > 0
In each case the separating equilibrium yields a higher profit.
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Chapter 20
Agency, Contract Theory and the Firm
Exercise 20B
1. If instead of [B.7] of the text we set up the problem as:
max πhv(y1) + (1 − πh)v(y2) − ah
s.t.
Kh ≥ z0
where z0 is the reservation expected utility for P, then the first-order conditions [B.8],
[B.9] will be the same, except for being multiplied through by the Lagrange multiplier
µ = 1/λ. Condition [B.10] will be replaced by the condition Kh = z0. The interpretation of
the second-best solution and its comparison with the first-best remains unchanged. The
only difference is that P receives her reservation utility in both first- and second-best,
and it is actually A, the manager, who is made worse off by the asymmetry of
information.
2. It may not be possible to find a J that satisfies [B.4]. For example, if π is very large
then J will have to be small to offset the fact that the “punishment state” is unlikely to
happen. There may be a lower-bound on J, i.e. a limit to the extent to which P can
punish A, which then makes it impossible to find a sufficiently small J. In that case, P
would have to devise an incentive-compatible contract of the second-best kind we
analyze in section 20D of the text.
3. If P wants to enforce the low-effort contract, she would solve the problem:
max Kl = πl(x1 − y1) + (1 − πl)(x2 − y2)
s.t. πlv(y1) + (1 − πl)v(y2) − al ≥ H0.
As in the high-effort case, the risk-neutrality of P means that y1* = y2* and A is fully
insured, with
π l v( y1* ) + (1 − π l ) v( y2* ) − al = H 0 .
It is then obvious that there is no incentive-compatibility problem; A would clearly not
choose ah > al if offered this contract since it would make him worse-off. Thus P would
have no problem in achieving the low effort level if that was optimal for her.
4. This question reinforces the point that it is P’s risk-neutrality that leads her to offer A
a certain payment in the first-best case. If P is risk-neutral, the first-best contract (for
the high-effort case) is found by solving:
max áH = πhU(x1 − y1) + (1 − πh)U(x2 − y2)
s.t. πhv(y1) + (1 − πh)v(y2) − ah ≥ v0
242
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yielding the conditions
πhU′(x1 − y1* ) + λπhv′ ( y1* ) = 0
−(1 − πh)U′(x2 − y2* ) + λ(1 − πh)v′ ( y2* ) = 0
together with the constraint. These yield the conditions
U ′( x1 − y1* )
U ′( x2 − y2* )
=
v ′( y1* )
v ′( y2* )
which implies a tangency point of indifference curves in (y1, y2)-space; again we have a
Pareto-efficient risk-sharing solution. Now suppose y1* = y2* . They v ′( y1* ) = v ′( y2* ) and
the right-hand side of the equation is 1. But if x1 ≠ x2, we would then have x1 − y1* ≠
x2 − y2* , and the left-hand side is unequal to 1, a contradiction. Thus we must have
y1* ≠ y2* .
5. If µ = 0, from [B.13] and [B.14] we have
1
1
=λ=
v′( W1 )
v′( W2 )
implying W1 = W2 as in the first-best case. But then, in the constraint [B.12], with W1 = W2 = W
we have ah ≤ al, which is false, and so we must have µ > 0. This is another reflection of
the fact that a constant payment cannot be incentive-compatible.
If λ = 0, the in [B.14] we have
1
µ (π h − π l )
=−
(1 − π h )
v′( W2)
The right hand side is negative since πh > πl and we just saw that µ > 0. But the left hand
side cannot be negative, since v′ > 0. Thus we have a contradiction and we must have
λ > 0.
6. Comparative-statics of contracts.
We note that [B.15] and [B.16] “solve for” W1 and W2, since ah, al are given: we have two
equations in two unknowns. The parameters of the model are the effort levels ah, al and
probabilities πh, πl. Totally differentiating [B.15] and [B.16] gives the system:
(1 − π h ) v2′ dy1
da h − [v( W1 ) − v( W2 )]dπ h
π h v1′
(π − π ) v − (π − π ) v dy = da − da − [v( W ) − v( W )][dπ − dπ ].
l
1′
h
l
2′
2
l
1
2
h
l
h
h
We derive just one comparative-statics result here, leaving the remainder to the reader.
Solving for ∂y1/∂π l gives
∂y1 −(1 − π h ) v2′[v( W1 ) − v( W2 )]
=
> 0.
∂π l
−(π h − π l ) v1′v2′
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To explain this sign; vi′ = v ′( Wi ), and so v1′, v2′ > 0. Since W1 > W2 and πh > πl, the rest of the
argument follows immediately. The intuitive argument is that if the probability of the
high outcome in the low effort state increases, this increases the relative attractiveness
to A of choosing low effort, and so to maintain incentive-compatibiltiy it is necessary to
increase the payoff in the better state. It can also be shown that this change reduces W2.
But then, in Fig. 13.4, this rightward move from β along I h0 must put P on a lower
(straight line) indifference curve, that is, the agency cost has increased.
An even simpler version of the comparative-statics can be undertaken using
condition [B.21]. We could rewrite this as
∆v = ∆a/∆π
where ∆v is the difference in utility at incomes y1 and y2. Then, this utility difference
must clearly increase (y1 and y2 must increasingly diverge) the greater the difference in
effort levels and the smaller the difference in probabilities of the better state at each
effort level.
Exercise 20C
1. From [C.12] of the text, we have
dy*
rP
=
=1
dx rP + rA
if A is risk-neutral since then rA = 0. Thus the first best contract has y*(x) = x − k with k
the constant of integration chosen to satisfy the agent’s reservation utility constraint:
∫ ( x − k) f(x, a*)dx = V + a*
x1
0
x0
where a* is A’s first best optimal a. In effect, A is fully insuring P, or equivalently, A is
buying the production opportunity from P at a price k. If P cannot observe a, she
nevertheless still knows what k is, and so she simply gives the first best payment
function y*(x) = x − k. Now in fact, as long as she receives k, P does not really care what
a is chosen by A, but the interesting thing is that A will choose the first best optimal a*.
To see this, note that the first term in condition [C.8] of the text becomes
k
∫ f ( x, a)dx = 0
x1
x0
a
since the integral in this expression is always zero. Thus the condition becomes
∫ ( x − k) f (x, a)d − 1 = 0
x1
a
x0
x
But this is precisely what we would get if we solved A’s maximization problem:
max
a
∫ ( x − k) f(x, a)dx − a
x1
x0
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and we know of course that x − k and a* satisfy A’s reservation constraint. Thus, the
first-best solution is available under asymmetric information when A is risk-neutral.
2. We can again use [C.12] of the text to good effect. We have
dy*
rP
=
dx rP + rA
Where rP = −u″/u′ and rA = −v″/v′. Evaluating the relevant derivatives from the given
utility functions, we have
(1 − kp )( x − y − d p ) −1
dy*
=
>0
dx (1 − kp )( x − y − d p ) −1 + (1 − kA )( y + d A ) −1
Thus y* increases monotonically with x given that ki ∈ (0, 1), i = P, A. For further
restrictions, revert to the simpler notation of [C.12] and differentiate totally:
1
dy*
d
=
{(rP + rA)drP − rP(drP + drA)}
2
dx [rP + rA ]
=
1
rP
( drP + drA )
drP −
rP + rA
rP + rA
=
1
dy
dy
drA
1 − drP +
rP + rA
dx
dx
Thus
d 2y*
1
dy drP dy drA
+
=
1 −
2
dx
rP + rA
dx dx dx dx
Evaluating the total derivatives drP /dx, drA/dx, we have
dy
drP
= −(1 − kp)(x − y − dp)−2 + (1 − kp)(x − y − d p ) −2
dx
dx
dy
= − 1 − (1 − kp)(x − y − dp)−2 < 0
dx
dy
drA
= −(1 − kA)(y + d A ) −2
<0
dx
dx
Thus since dy/dx ∈ (0, 1) we have from the above expression that d2y*/dx2 < 0. Thus the
optimal payment function y*(x) is a strictly increasing strictly concave function.
3. Note from the first-order conditon [C.18] in the text
∂Eu
∂ 2 EV
= −é
>0
∂a
∂a 2
since on the given assumptions EV is strictly concave in a. This implies that at the
second-best solution P would always want more a than is in fact supplied – her
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expected utility at the second best solution is increasing in a. The answer to the
question however follows simply from A’s reservation constraint. In the first best this is
∫ v(y* ( x)) f ( x, a*)dx = a* = V
x1
0
x0
and in the second best
∫ vW( x)) f ( x, â )dx = â + V
x1
0
x0
Suppose â = a*. We know that in the second best, because of the departure from the
optimal risk-sharing, W(x) yields a distribution of utilities which, at probabilities f(x, a*)
gives less expected utility than y*(x), and so the second reservation constraint could
not be satisfied with â = a*. We must therefore have â < a* − to accommodate a lower
expected utility of income A has to be given lower effort.
4. The treatment of this example is set out quite fully in Holmstrom (1979), to which the
reader is referred.
5. Denote the density and distribution functions for the state of the world parameter θ
by h(θ) > 0 and H(θ). Hence
F(x, a) = Pr[g(a) + θ ≤ x] = Pr[θ ≤ x − g(a)] = H(x − g(a))
For first degree stochastic dominance we require
Fa(x, a) = −h(x − g(a))g′(a) ≤ 0
and so a positive marginal product of effort ensures stochastic dominance.
CDFC requires
Faa(x, a) = h′g′g′ − hg″ ≥ 0
Hence the simple assumption of a uniform distribution of states (h′ = 0) and a declining
marginal product of effort g″ < 0 guarantee CDFC. However if the marginal product of
effort is constant and the distribution of states is negative exponential we have Faa > 0.
Note that a triangular distribution would not satisfy CDFC without further assumptions
about the rate of decline of the marginal product of effort.
The MLRC is
∂( fa / f ) 1
= 2 (fax f − fa fx)
∂x
f
=
1
g ′h ′h h ′ h ′′
(−hh″g′ + h′h′g′) =
−
2
h2 h h′
h
= −g′
∂ 2 log h
≥0
∂θ 2
since f = Fx = h(x − g(a)), fx = h′(x − g(a)), fa = −h′(x − g(a))g′(a), fax = −h″g′.
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Clearly, even with the assumption that the marginal product of effort is positive but
decreasing and the simplest possible form of uncertainty (additive) strong additional
assumptions on the distribution of states is required to generate the plausible result that
the payment schedule is increasing in the outcome (see [C.21]). For example, the
assumption that the distribution is uniform implies that MLRC is satisfied but (20.1) is
zero so that the payment schedule is determined solely by the risk aversion of the
Principal and the Agent with incentives playing no part in influencing its slope. With a
uniform distribution higher output is no more or less likely to arise from greater effort
and so there is no need to include an incentive term in the reward schedule.
6. First we find an equivalent but more convenient way of representing the preferences
of A and P. Define z = y − 12 a 2 as the “net income” of A and note that
Ez = E(y − 12 a 2 ) = δ0 + δ1Ex + δ2Em − 12 a 2 = δ0 + δ1(ka + µ) − 12 a 2
(20.2)
σ 2z = Var(y − 12 a 2 ) = Var(y) = δ 12 Var ( x) + δ 22 Var ( m) + 2δ1δ2Cov(x, m)
= δ 12σ 2 + δ 22ψ 2 + 2δ1δ2σψρ
(20.3)
Effort affects mean net income but has no effect on the variance of net income. This
simplifies the results considerably.
Since z is normally distributed (it is a linear combination of normally distributed
variables θ and m) and A has an exponential utility function, his expected utility can be
written as a function of the mean and variance of net income z. (Those familiar with
probability theory will recognise the similarities with derivation of the moment
generating function for linear combinations of normally distributed random variables.)
We get
Ev(z) = − Ee −α z = − e E ( −α z)+ Var ( −αz ) = − e
1
2
− α Ez + 12 α 2σ 2z
= − e −αV
(20.4)
Since (20.4) is increasing in V = Ez − 12 ασ 2z (dv/dV = αe −αV > 0), maximizing Ev(z) is
equivalent to maximizing
V(a; δ0, δ1, δ2) = δ0 + δ1(ka + µ) − 12 a 2 − α2 [δ 12σ 2 + δ 22ψ 2 + 2δ 1δ 2σψρ]
(20.5)
Since x − y = x − δ0 − δ1x − δ2m = (1 − δ1)x − δ2m − δ0 is normally distributed with
E(x − y) = (1 − δ1)(ka + µ) − δ0
(20.6)
Var(x − y) = (1 − δ1)2σ 2 + δ 22ψ 2 − 2(1 − δ1)δ2σψρ
(20.7)
the same argument as above means that maximizing P’s expected utility Eu(x − y) is
equivalent to maximizing
U(a; δ0, δ1, δ2) = (1 − δ1)(ka + µ) − δ0 − β2 [(1 − δ 1 ) 2 σ 2 + δ 22ψ 2 − 2(1 − δ1)δ2σψρ] (20.8)
Given the incentive scheme A chooses a to maximize (20.5). The first order condition
is Va = δ1k − a = 0, so that
a = δ1k
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The principal designs the contract to maximize her objective function (20.8) subject
to the participation constraint that A will accept the contract and to the incentive
compatibility constraint that A’s effort satisfies (20.9). Hence the Lagrangean is
L = U(a; δ0, δ1, δ2) + λ[V(a; δ0, δ1, δ2) − V0] + γ [kδ1 − a]
(20.10)
where V0 is the minimum level of V which A will accept. (Alternatively we could
substitute kδ1 for a to get the equivalent problem of choosing δ0, δ1, δ2 to maximize U(kδ1,
δ0, δ1, δ2) subject only to the participation constraint V(kδ1, δ0, δ1, δ2) ≥ V 0.) The first order
conditions are (remembering that Va = 0 from the first order condition on A’s choice of
a)
Ua + λVa − γ = Ua − γ = (1 − δ1)k − γ = 0
(20.11)
Uδ 0 + λVδ 0 = −1 + λ = 0
(20.12)
Uδ 1 + λVδ 1 + γ k = −(ka + µ) + β (1 − δ1)σ 2 − βδ2σψρ
+ λ[(ka + µ) − αδ1σ 2 − αδ2σψρ] + γ k = 0
(20.13)
Uδ 2 + λVδ 2 = −βδ2ψ 2 + β (1 − δ1)σψρ + λ[−αδ2ψ 2 − αδ1σψρ] = 0
(20.14)
plus the constraints.
Use (20.11) and (20.12) to substitute (1 − δ1)k for γ and 1 for λ in (20.13) and (20.14).
Now solve (20.14) for
δ2 =
σρ[β (1 − δ 1 ) − αδ 1 ]
ψ (α + β )
(20.15)
and substitute for δ2 in (20.13) and solve for δ1. We get
δ1 =
k 2 + σ 2 β (1 − ρ 2 )
k 2 + σ 2 (α + β )(1 − ρ 2 )
(20.16)
δ2 =
−σρα k 2
ψ (α + β )[k 2 + σ 2 (α + β )(1 − ρ 2 )]
(20.17)
(δ0 is determined by substituting for δ1, δ2 in the participation constraint.)
The intuition behind these results can be brought out by examining the effects
of varying the parameters (differentiate (20.16) and (20.17) with respect to the
parameters). Alternatively consider some special cases.
First best: If P can observe a, (20.9) is no longer a constraint and γ ≡ 0. The first order
condition on a will be Ua + λVa = (1 − δ1)k + δ1k − a = 0. P will direct A to set a = k where
the marginal cost of effort to A is equal to the value of the marginal product of effort.
With γ = 0 (20.13) and (20.14) together imply that δ2 = 0 and δ1 = β /(α + β). Incentives and
risk sharing can be separated because P can control a directly and use the contract to
share risks efficiently.
Pure risk sharing: k = 0. Since A’s effort does not affect output, the contract is purely
for risk sharing and δ1 = β /(α + β), δ2 = 0. Note (i) that since linking A’s reward to the
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message increases the variances of P and A’s incomes, risk is minimised by setting
δ2 = 0; if P is risk neutral she cares only about her mean income (20.6) which in the
current setting is equivalent to assuming that her preferences are represented by (20.8)
with β = 0. She will then optimally bear all the risk. Compare the first best contract.
Perfectly informative message: ρ = 1. The message is perfectly correlated with the
state: m = t0 + t1θ. Hence Em = t0 + t1µ = 0, Var(m) = ψ 2 = t12σ 2 and t0 = −ψµ/σ, t1 = ψ/σ, so
m = ψ(θ − µ)/σ. Observing the message is equivalent to observing the state, and hence
since output is also observed, to observing the agent’s effort a = (xt1 − m − t0)/t1k. P can
therefore direct A to supply the first best effort a = k. The reward schedule is
y = δ0 + x −
σα
α
m = δ 0 + ka + θ −
(θ − µ )
ψ (α + β )
(α + β )
= δ0 + ka +
µα + θβ
α +β
Risk neutral agent: α = 0. Since A is risk neutral setting δ1 = 1, δ2 = 0 leads to optimal
incentives for effort and there is no need to worry about efficient risk sharing since the
risk neutral party (A) bears all the risk. In effect A buys the firm from P who receives a
fixed payment −δ0.
7. Suppose that the contract reduces the agent’s income by δ when there is low profit
and the principal receives the signal that the agent’s effort was low. The agent now has
three income levels: with probability πi profit is high and he gets y1; with probability
(1 − πi)(1 − ri) profit is low but the signal indicates high effort and he gets y2; with
probability (1 − πi)ri profit is low and the signal indicates low effort so that he is
punished and gets y2 − δ. We can show that the optimal contract will make the agent’s
reward depend on profit and the signal by showing that the owner’s expected income is
increased by increasing δ from zero and adjusting the other terms of the contract y1, y2
to keep the binding participation and incentive compatibility constraints satisfied. We
do this by using the Envelope Theorem (Appendix J).
For a given δ the optimised value of the Lagrangean for the principal’s problem is,
Ö = πh(P1 − W1) + (1 − πh)(P2 − W2 + rhδ )
+ λ{πhv(W1) + (1 − πh)[(1 − rh)v(W2) + rhv(W2 − δ )] − Eh − E0}
+ µ{πhv(W1) + (1 − πh)[(1 − rh)v(W2) + rhv(W2 − δ )]
− Eh − πᐉv(W1) + (1 − πᐉ)[(1 − rᐉ)v(W2) + rᐉv(W2) − δ )] − Eᐉ}
where W1 and W2 are the optimal payments to the agent. Using the Envelope Theorem the
marginal value of an increase in δ is
∂Ö
= (1 − πh)rh − λ(1 − πh)rhv′(W2 − δ)
∂δ δ =0
− µ(1 − πh)rhv′(W2 − δ ) + µ(1 − πᐉ)rᐉv′(W2 − δ )
= (1 − πh)rh − v′(W2)[(λ + µ)(1 − πh)rh − µ(1 − πᐉ)rᐉ]
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Using the first order condition on W2 (rearrange [20B.4]) we can substitute
1 − πh = v′(W2)[(λ + µ)(1 − πh)rh − µ(1 − πᐉ)]
for (1 − πh) in the first term in (20.18) to get
∂Ö
= v′(W2){[(λ + µ)(1− πh)rh − µ(1 − πᐉ)]rh
∂δ δ =0
− [(λ + µ)(1 − πh)rh − µ(1 − πᐉ)rᐉ]}
= v′(W2)(rᐉ − rh)(1 − πᐉ)µ > 0
(remember that µ > 0 because the incentive compatibility constraint binds and that
rᐉ > rh because the signal is more likely to correctly indicate low effort if low effort
was supplied).
We see that the ability to imperfectly monitor effort is more valuable to the principal
as: (a) the signal is better able to discriminate between low and high effort (rᐉ − rh
increases); (b) the likelihood of low profit given low effort (1 − πᐉ) increases; (c) the
value of relaxing the incentive compatiblility constraint increases and (d) the marginal
utility of income increases. The intuition behind (d) is that if marginal utility of income
is greater then the “cost” of punishment is greater for the agent and the punishment is
more effective in inducing high effort.
Exercise 20D
1. Given
dyi ψ x ( x,θ i )
=
dxi
v ′( yi )
partially differentiating with respect to θi gives
∂ dyi v ′ψ xθ ψ xθ
=
=
∂θ i dxi ( v ′)2
v′
From [D.7] we have ψxθ < 0. Thus we have condition [D.13] of the text. What it means is
that if we fix a point (xi, yi), and increase θi, then the slope of the indifference curve
decreases. Intuitively, a given increase in output requires a smaller income increase as
the agent’s productivity increases, because he can achieve the output increase with a
smaller effort increase.
2. If A is risk-neutral his utility function can be written as v = yi − ai. Then, when we
substitute ai = ψ(xi, θi) we obtain the quasi-linear form:
v = yi − ψ(xi, θi)
It follows that the slope of the indifference curve
dyi
1
=ψ x =
dxi
xa ( a , θ i )
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which is independent of yi. Thus in this case his indifference curves are vertical
displacements of each other. If he is risk-averse then from [D.12] of the text we have
dyi ψ x ( x,θ i )
=
dxi
v ′( yi )
and
∂ dyi
ψ x v ′′
>0
=−
∂yi dxi
v′
since ψx > 0, v′ > 0 and v″ < 0. Thus for fixed xi as yi increases the slopes of successive
indifference curves increase.
3. The model of adverse selection is essentially identical to the model of second-degree
price discrimination in section 9C and the answers to questions (i) and (ii) are given
therefore on pages 200–204 of the text. Then we have:
(iii) If the agent is risk-neutral the slopes of indifference curves for a given type are
equal along a vertical line in (xi, yi)-space. Consequently, the ‘no distortion at the top’
result, that x2* and V2 satisfy the same tangency condition, means that the slopes of the
indifference curves have to be the same, and so in the risk-neutral case we must have
x2* = V2.
(iv) To show that >2 > é2 use condition [D.19] of the text to obtain
>2 − é2 =
π
v ′( W1 )
>0
We can give these multipliers their ‘shadow price’ interpretation: the loss of expected
utility to the principal from a small tightening of the low productivity type’s reservation
utility constraint is greater than the loss of utility to P from a small increase in the lower
type’s utility at his optimal contract (V1, W1).
4. As the discussion surrounding Fig. 20.12 of the text makes clear, the higher is π the
smaller the distortion in the type 1 contract, and so in Fig. 20.12 the higher the Î2
indifference curve must be to maintain self-selection, and so the higher must be W2. If the
indifference curves rise very sharply then we could find that the W2 required to achieve
self-selection exceeds the x2 produced. Thus the constraints yi ≤ xi should be imposed.
If the constraint was binding for i = 2, that would mean that there is a larger distortion
away from first best in the type 1 contract as well as a distortion from the first best
condition in the solution for x2.
5. The analysis is essentially as in the two-agent case, with distortions in the types 1 and
2 contracts but with the ‘no distortion at the top’ result holding. However, only the
lowest-productivity type can be held to his reservation utility. We now also have the
possibility that types 1 and 2 may be offered the same contract.
6. The Lagrangean for the problem of maximizing the expected value of the welfare
function subject to the participation and incentive compatibility constraints is
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L = ∑πi[B(xi) − θixi − kTi] + ∑λi[(p(xi) −θi)xi + Ti]
+ µ1[(p(x1) − θ1)x1 + T1 − (p(x2) − θ1)x2 − T2]
+ µ2[(p(x2) − θ2)x2 + T2 − (p(x1) − θ2)x1 − T1]
where pi = p(xi) is the inverse demand function of consumers generated by p = B′(xi).
The first order conditions for a non-corner solution are
LT = −πik+ λi + µi − µj = 0
(20.19)
Lx = (πi + λi + µi − µj)(pi − θi) + (λi + µi − µj)p′(xi)xi − µ2(θi − θj) = 0
(20.20)
i
i
for i = 1, 2; i ≠ j, plus complementary slackness conditions on the participation and
incentive compatibility constraints.
Notice that the participation constraint on the low cost firm never binds:
0 ≤ (p1 − θ1)x1 + T1 < (p1 − θ2)x1 + T1 ≤ (p2 − θ2)x2 + T2
where the first inequality follows from the participation constraint on firm 1, the second
from θ2 < θ1 and the third from the incentive compatibility constraint on firm 2. Hence λ2
= 0.
Suppose first that the transfers have no social cost (k = 0). In this case the first best
allocation in which both types of firms price at marginal cost pi = θi is achievable. If the
regulator also sets T1 = 0 and T2 = (θ1 − θ2)x1 then firm 1 breaks even and firm 2 is just
indifferent between producing x1 at price p1 = θ1 and getting T1 = 0 and producing x2 at
price p2 = θ2 and getting T2. In fact any transfers which satisfy T2 − T1 ≥ (θ1 − θ2)x1 will
satisfy all the constraints. Because there are no costs to the transfers the regulator can
use them to satisfy the incentive compatibility constraints without the need to adjust
prices away from their first best levels.
When transfers have a social cost the regulator will wish to balance the cost of the
distortions introduced by prices diverging from marginal cost against the cost of the
transfers. Using the conditions on the Ti to substitute πik for λi + µi − µj, remembering
from the discussion in the text that the incentive compatibility constraint on the type 1
firm will not bind, and rearranging the conditions on the outputs gives
p1 − θ 1 =
µ 2 (θ 1 − θ 2 ) − π 1 kp′( x1 ) x1
>0
(1 + k)π 1
p2 − θ 2 =
−π 2 kp′( x2 ) x2
>0
(1 + k)π 2
Thus we see that because transfers are costly the price set for the low cost firm exceeds
marginal cost. The low cost firm is induced to choose its output and transfer pair by
being offered a more profitable output and a smaller transfer than in the first best
solution achievable when transfers are costless.
7. Consider condition [D.53] in the text and note that if we were to let πh get arbitrarily
small, there is nothing to stop the right hand side becoming negative. But this would
then present a problem, because the derivative on the left hand side is supposed always
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to be positive. The problem arises because we have implicitly assumed that we always
obtain an interior solution with sH > 0. It is possible however that we could have a
corner solution in which sH = 0. We should in fact formulate the problem with the
explicit constraint sH ≥ 0. We will now show that doing this resolves the mathematical
difficulty we just pointed out, and also has an interesting intuitive explanation.
Thus if we add to the problem defined in [D.46], [D.47] and [D.48] the condition
sH ≥ 0, we have to replace [D.50] with the Kuhn-Tucker condition
−λπH + γ − µ ≤ 0
sH ≥ 0
sH(−λπH + γ − µ) = 0
The case examined in the text assumes sH > 0, so γ = µ + λπH, and then substituting into
[D.52] and rearranging gives [D.53]. Suppose however we have sH = 0 at the optimum.
Then γ ≤ µ +λπH and using this in [D.52] gives instead of [D.53] the condition
ψ ′(θH – àH) ≥ 1 −
λ
πL
[ψ ′(θ H − àH ) − ψ ′(θ L − àH )]
(1 + λ ) π H
This fixes up the mathematics, since the right hand side can be negative without
violating the condition.
For the economic intuition, refer again to Fig. 20.17 in the text. As we slide rightward
along the H type’s indifference curve away from point H* we are trading off two effects.
On the one hand we are incurring a cost from distorting the equilibrium of the H type,
causing it to supply less effort and incur higher costs, but at the same time we are
receiving a benefit by reducing the rent that has to be paid to the L type. We stop at the
optimum when the marginal cost of the first is just equal to the marginal benefit of the
second. However, suppose πH is very small. Then the expected marginal cost of the
distortion is small relative to the expected marginal benefit of the rent reduction, and so
it could pay to go on increasing the distortion until we can go no further, i.e. until sH = 0.
This then implies from [D.47] that aH = ψ(aH) = 0, so the H type supplies no cost reducing
effort. This could imply that its costs are so high as to exceed u, so that it would not be
required to produce. This would imply offering the H type a contract that induces it not
to produce at all. The regulator knows then that the only firm in the market would be
the L type, and so she can offer it the first best contract, leaving it with a zero rent.
8. Let (åL, åH, àL, àH) be the optimal solution to the single period problem as found on
page 596 of the text. The proposition is that offering these in both periods is optimal,
given that the regulator can commit to doing this in the second period, which implies
ignoring the information on the firm’s type she has gained in the first period. The proof
is by contradiction. We suppose that there are alternative menus ( sL′ 1 , sH′ 1 , c L′ 1 , c H′ 1 ) in
period 1 and ( sL′ 2 , sH′ 2 , cL′ 2 , cH′ 2 ) in period 2 that yield a better solution to the two-period
problem. We then show that in the single period problem, a mixed strategy involving
these two menus with appropriately chosen probabilities must in that case yield a better
solution for the regulator than (åL, åH, àL, àH). This contradicts the optimality of the latter.
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Thus if ( sL′ 1 , sH′ 1 , c L′ 1 , c H′ 1 ) and ( sL′ 2 , sH′ 2 , c L′ 2 , c H′ 2 ) are optimal in the two-period problem
they must satisfy the constraints for that problem. Again it suffices to consider only the
participation constraint of the high cost type and the incentive compatibility constraint
of the low cost type. In the two period case these are respectively
sH′ 1 − ψ (θ H − c H′ 1 ) + δ [sH′ 2 − ψ (θ H − c H′ 2 )] ≥ 0
i.e. the present value of profit over the two periods must be non-negative, with δ the
discount factor, and
sL′ 1 − ψ (θ L − c L′ 1 ) + δ [sL′ 2 − ψ (θ L − c L′ 2 )] ≥ sH′ 1 − ψ (θ L − c H′ 1 ) + δ [sH′ 2 − ψ (θ L − c H′ 2 )]
i.e. it cannot pay the low cost firm to claim to be high cost. Next, note that these
inequalities continue to hold if we divide through each by 1 + δ. But this then implies
that in the single period problem, choosing ( sL′ 1 , sH′ 1 , c L′ 1 , c H′ 1 ) with probability 1/(1 + δ )
and ( sL′ 2 , sH′ 2 , c L′ 2 , c H′ 2 ) with probability δ /(1 + δ ) must be feasible since it satisfies these
constraints in that problem. Moreover, if in the two-period problem
(1 + δ ) u −
∑ π [λs′ − (1 + λ )c ′ − ψ (θ − c ′ )] +δ ∑ π [λs′ − (1 + λ )c ′ − ψ (θ − c ′ )]
i
i1
i1
i
i1
i2
i
i= L , H
i2
i
i2
i= L , H
> (1 + δ ){u −
∑ π [λå − (1 + λ )à − ψ (θ − à )]}
i
i
i
i
i
i= L , H
then offering ( sL′ 1 , sH′ 1 , c L′ 1 , c H′ 1 ) and ( sL′ 2 , sH′ 2 , c L′ 2 , c H′ 2 ) with probabilities 1/(1 + δ ) and
δ /(1 + δ ) respectively must yield higher expected utility to the regulator than offering
(åL, åH, àL, àH) for certain (just divide through by 1 + δ ). This gives the contradiction.
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Chapter 21
General Equilibrium under Uncertainty and
Incomplete Markets
Exercise 21B
1. Given that the individuals are risk averse and therefore have convex-to-the-origin
indifference curves, a contract curve outside the 45° line can only arise if (a) the
individuals have different probability beliefs, or (b) if at least one of them has state
dependent marginal utility. As Fig. 21B.1 shows, if the individuals have identical
probability beliefs and state independent marginal utility their indifference curves have
slope −π1/π2 at their 45° lines. Hence they can only be tangent between the 45° lines.
They both share in any social risk in that both have incomes which are positively
correlated with total income.
2. (a) and (b). When probability beliefs are identical the contract curve is defined by
va′ 1 ( ya 1 ) vb′1 ( y1 − ya 1 )
=
va′ 2 ( ya 2 ) vb′2 ( y2 − ya 2 )
(21.1)
Now consider the different types of LRT (text page 609).
(i) With the quadratic type (21.1) is
ρ a − ya 1 ρ b − y1 + ya 1
=
ρ a − ya 2 ρ b − y2 + ya 2
which solves for the equation of the contract curve as
ya 2 =
ρ a ( y2 − y1 ) ρ a + ρ b − y2
+
ya 1
ρ a + ρ b − y1 ρ a + ρ b − y1
Note that for economically sensible results we assume that neither individual would be
satiated if given the total income ys available in any state: ρi > ys, i = a, b; s = 1, 2. The
contract curve is an upward sloping straight line.
(ii) With the exponential form (21.1) is
exp( − ya 1 / ρ a ) exp[−( y1 − ya 1 )/ ρ b ]
=
exp( − ya 2 / ρ a ) exp[− y2 − ya 2 )/ ρ b ]
or
exp[(ya1 − ya2)/ρa] = exp[−(y1 − ya1 − y2 + ya2)/ρb]
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Fig. 21B.1
Taking logs and rearranging gives
ya 2 =
ρ a ( y2 − y1 )
+ ya 1
ρa + ρb
The contract curve is parallel to and between the 45° lines.
(iii) For the logarithmic type (21.1) is
ρ a + ya 2 ρ b + y2 − ya 2
=
ρ a + ya 1 ρ b + y1 − ya 1
so that
ya 2 =
ρ a ( y2 − y1 ) ρ a + ρ b + y2
+
y
ρ a − ρ b − y1 ρ a + ρ b + y1 a 1
When ρa = ρb = 0, so that u = log yis, the contract curve is the diagonal of the Edgeworth
Box.
(iv) Finally, with the power form (21.1) is
ρ a + τ a ya 1
ρ a + τ a ya 2
−1/τ a
ρ + τ b ( y1 − ya 1 )
= b
ρ b + τ b ( y2 − ya 2 )
−1/τ b
This yields ya2 as a linear function of ya1 only if τa = τb = τ, so that we can raise both sides
to the power −τ and then solve for
ya 2 =
ρ a ( y2 − y1 ) ρ a + ρ b + τ y2
+
y
ρ a + ρ b + τ y1 ρ a + ρ b + τ y1 a 1
1. (c) With one risk averse individual (say individual b) and identical probability beliefs,
the contract curve is defined by
va′ 1 ( ya 1 )
=1
ua′ 2 ( ya 2 )
(since vbs′ is a positive constant). Hence if the marginal utility of income is state
independent, the contract curve is the 45° line for individual a where ya1 = ya2.
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Supplementary questions
(i) Draw the contract curves for the various cases in parts (a) and (b) of the question.
What are the effects of increases in: the difference between the total state contingent
incomes; ρa?
(ii) What does the contract curve look like in part (c) if a has state independent
marginal utility or different probability beliefs to b?
3. (a) Write the first order condition [B.7] as an implicit equation to get
dyas
π bs vbs′
<0
=
π as vas′′ + λπ bs vbs′′
dλ
An increase in the distributional weight on b reduces a’s income in all states.
3. (b) Similarly
dyas
− vas
′
=
>0
dπ as π as vas
′′ + λπ bs vbs′′
4. Since
v ′′ d log v ′
=
= − ( ρ + τ y ) −1
v′
dy
integrating, when τ ≠ 0, gives
log v′ = K − τ −1log(ρ + τ y) = log[k1(ρ + τ y)−1/τ]
(21.2)
where K = log k1. Hence
v′ = k1(ρ + τ y)−1/τ
(21.3)
Now if τ = 1 we have v′ = k1(ρ + y)−1 and integration gives
v = k0 + k1 log(ρ + y)
Since utility functions representing preferences satisfying the axioms of expected utility
theory are unique up to linear transformations we can choose k0 = 0, k1 = 1 to yield the
logarithmic type of LRT.
If τ ≠ −1, τ ≠ 0 integrating (21.3) gives
v = k0 + k1
1
(ρ + τ y)(τ−1)/τ
τ −1
which is identical to the power form after setting k0 = 0, k1 = 1.
With τ = −1 the power form becomes
1
v = − (ρ − y)2
2
and multiplying by 2 gives the quadratic form on text page 609.
Finally, if τ = 0 we have
d log v ′
1
=−
dy
ρ
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so that log v′ = K − y/ρ and v′ = k1 exp(−y/ρ) where K = log k1. Hence v = k0 − k1
exp(−y/ρ), and k0 = 0, k1 = 1 gives the exponential form.
Since we have considered all possible values of τ and placed no restrictions on ρ, the
four types derived above are the only possible utility functions with LRT.
5. (a) With exponential utility functions the sharing rule [B.7] is given by
πas exp(−yas/ρa) = λπbs exp[−(ys − yas)/ρb]
which implies
log π as −
yas
ρa
= log λπ bs −
ys − yas
ρb
Collecting terms in yas and dividing through by ρ a−1 + ρ b−1 gives the result.
5. (b) With logarithmic utility functions, [B.7] is
π as
ρ a + yas
=
λπ bs
ρ b + ys − yas
which implies
ρb + ys − yas = >(ρa + yas)
which rearranges to give the result.
6. Differentiate the sharing rule in question 5(a) with respect to ωa
ω
dyas
ys
1
1
log a +
=−
−
2
(ω a + ω b )
dω a
λω b (ω a + ω b )ω a (ω a + ω b )2
=
1
1
− yas +
ωa + ωb
ωa
which is positive if ωayas < 1 where ωayas is the coefficient of relative risk aversion.
Note that an increase in ωa reduces a’s marginal share of the income in state s: the
variability of yas is reduced.
7. (a) Yes, numerous studies show that most individuals are very bad at assessing
probabilities.
7. (b) When the individuals have the same probability beliefs and marginal utility is state
independent, the sharing rules do not depend on the probability beliefs. Hence the
allocation in each state is the same whether beliefs are correct or not. Only if it is
possible to transfer total income between states will it matter that the individuals’
unanimous beliefs are incorrect.
7. (c) From question 3(b) we know that an increase in πas will increase yas. Thus
compared with the allocation in which individuals have the same correct beliefs, the
allocation with a having incorrect beliefs will have yas larger (and therefore ybs smaller)
when a incorrectly believes πas to be larger than it actually is.
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8. (a) and (b). The analysis here is very similar to that of sections 2E and 11B in which
the individual financed consumption by selling endowments. Fig. 11.1 shows the effect
(with suitable relabelling) of changing the relative price of state contingent incomes and
of increasing the endowment of state 1 income.
8. (c) We can write the consumer’s budget constraint as wi − ∑s ps yis ≥ 0 where wi =
∑s ps Wis is wealth or the market value of endowed state contingent incomes. The utility
maximization problem now has the same mathematical form as that of chapter 2,
although the interpretation of the choice variables and the budget constraint differs. The
utility maximizing demands depend on prices and wealth: Dis(p, wi). The demand for yis,
s = 1, . . . , S will increase with w because of the additive separability of preferences over
state contingent incomes. The first order conditions imply
π i1 vi′1 ( yi1 )
p1
= ... =
π iS viS′ ( yiS )
pS
Since marginal utility is positive an increase in wi must be used to buy more of at least
one of the state contingent incomes, say yi1. But since vis′′ < 0 the increase in yi1 reduces
the ratio π i1 vi′1 / p1 and, if the first order conditions are to continue to hold, all the other
ratios π is vis′ / ps must also be reduced, which requires that the other yis must increase.
Hence all state contingent incomes are normal goods and if an increase in ps makes
the individual worse off the wealth effect will reinforce the substitution effect. From
Roy’s Identity the effect of an increase in ps on expected utility is θi(Dis − Wis). Thus the
statement in the question is in fact correct only for states of the world for which the
individual has a positive net demand.
9. (a) If a has no information on what state of the world has occurred his income and
utility functions must be the same in all states, otherwise he can infer the state from yas
or vas. In the Edgeworth Box in Fig. 21B.2 the initial allocation must be on a’s 45° line
and the slope of his indifference curves at the 45° line is −πa1/πa2.
9. (b) Suppose that the individuals wrote a contract which purports to shift them from
the initial endowment at c0 to the allocation at c1. The contract requires a to give up
income if state 2 occurs in exchange for additional income if state 1 occurs. If b is
selfish and rational she will always tell a that state 2 has occurred, irrespective of the
true state. Hence the contract will actually generate the allocation c2. Similarly a
contract which purports to move them to c3 will actually move them to c4.
9. (c) The set of allocations which are feasible via contracts between the parties is
the rectangular line through c4c0c2. The only point on this curve which satisfies the
individual rationality constraints that it leaves both individuals no worse off than at the
no contract point is the original endowment point c0. Thus the core is the single
allocation c0.
10. See Fig. 21B.3. Without the information device the individual has a budget line
through his endowment point W with slope −p1/p2 and initial wealth w = ∑spsWs. His
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Fig. 21B.2
Fig. 21B.3
optimal state contingent consumption plan is y*. His maximized expected utility is
V(p, W) = ∑ vs ( ys* ).
When he acquires the information device he knows what the state will be before
trading starts on the markets in state contingent claims. On acquiring the correct
information that state s will occur he spends his entire wealth to buy claims to y s* * =
w/ps of income in state s and gets utility of vs(w/ps). Hence his vector of state contingent
consumption is (w/p1, . . . , w/pS) at y**. His expected utility is now ∑sπsvs(w/ps) and he is
on the indifference curve I** through y**.
The maximum he would be willing to pay for such a device (the value of the
information) is δ defined by ∑sπsvs(w/ps − δ ) = ∑ vs ( ys* ). In terms of the figure, δ is
measured by the vertical or horizontal distance from y** on I** to a on I*.
If there is public information before the markets for state contingent claims open
that state s will occur, the price of claims to state s income will be 1 and the price to
claims in income in all other states will be zero. No trade will take place on the markets
since no one will be willing to buy claims to states which will not occur.
All individuals will be forced to consume their endowments if there is public
information about the state before the markets open. Thus no individual is better off,
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and some will be worse off, compared with a situation in which there is no information
device. In Fig. 21B.3 the public information shifts the individual from I* to Î.
Exercise 21C
1. Refer to text page 103 and note that a function which is homogeneous of degree
k has partial derivatives which are homogeneous of degree k − 1. Hence, although
doubling all prices and income does not alter utility, it halves the gain in utility from an
extra £1. Since marginal utility of income is homogeneous of degree −1
∑ vyp pi + vyyy = −vy
i
(dropping the s subscript to save notation).
This equation shows that assumptions about the marginal utility of income can have
surprisingly strong implications about the pattern of consumption. For example, if we
assume that the marginal utility of income is constant with respect to prices (vyp = 0,
i = 1, . . . , n) we are also assuming that
i
−
vyy y
yy
=1
so that there is constant relative aversion to income risk, of unity. The indirect utility
function must have the form v(p, y) = α (p) + log y. But this implies that preferences are
homothetic and that the income elasticity of demand for all goods is unity. (See text
pages 68–70 and note that taking the log of [D.13] there gives the above form with
α (p) = −log a(p)).
2. If u(xs) is concave then v(ps, ys) is concave in ys (text page 622). Denoting ∑πsCVs by ä
we can write
v( ps2 , ys − CVs) = v( ps2 , ys − ä) + vy ( ps2 , y1 − ä)(ä − CVs) + vyy ( ps2 , 9s )( ä − CVs )2 /2
for some 9s ∈ (ys − ä, ys − CVs). Hence
Ev( ps2 , ys − CVs) − Ev( ps2 , ys − ä)
= Evy ( ps2 , ys − ä)(ä − CVs) + Evyy ( ps2 , 9s)(ä − CVs)2/2
= Cov(vy ( ps2 , ys − ä), ä − CVs) + Evyy ( ps2 , 9s)(ä − CVs)2/2
(21.4)
Although we have assumed that the marginal utility of income does not vary directly
with the state of the world this is not sufficient for the Cov term to be zero. The Cov
term will be zero if we also assume that marginal utility of income is not affected by the
prices which vary across states of the world. If we make these assumptions (21.4) will
be negative (since vyy < 0) and the expected ex post compensating variation is less than
the ex ante compensating variation (see [C.45]).
3. The demand for good i is Di(p, y) = − v p ( p, y)/vy(p, y) so that
o
∂Di
1
[v p y vy − v p vyy ]
=−
∂y
( vy )2
i
i
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Thus vyy = 0 is neither necessary nor sufficient for demand for good i to be unaffected by
income.
4. Since good 1 is chosen before the state of the world is known xs1 = x1, s = 1, . . . , S. In
state s the other goods are chosen to maximize u(x1, xs2, . . . , xsn) subject to ∑ 2n psi xsi ≤ ys
− ps1x1. The Lagrangean for the state s problem is
Ls = u(x1, xs2, . . . , xsn) + λs[ys − ps1x1 −
n
∑p x ]
si
si
(21.5)
2
Her demand for xsi, i = 2, . . . , n is Dsi(ps2, . . . , psn, ys − ps1 − x1) and her state s indirect
utility function is vs = v(x1, ps2, . . . , psn, ys − ps1 − x1). Before the state of the world is
known she chooses x1 to maximize the expected value of the indirect utility function, so
that her choice of x1 satisfies
Evs1(x1, ps2, . . . , psn, ys − ps1 − x1) − Evsy(x1, ps2, . . . , psn, ys − ps1 − x1)ps1
= Eu1(x1, xs2, . . . , xsn) − Eλs ps1 = 0
where we have used the envelope function on the Lagrangeans (21.5). Note that the
demand for good 1 (D1) depends on the anticipated incomes and prices of all the goods
in all states. The maximized expected indirect utility is V = Evs(D1, ps2, . . . , psn, y2 − ps1 −
x1).
The consumer is made better or worse off by a mean preserving contraction in the
price of good 1 if Evs is increased or decreased. The derivative of vs with respect to ps1 is,
using the envelope theorem on (21.5),
vsp =
s1
∂Ls
= −λsx1 = −vsyx1
∂ps1
Partially differentiating again with respect to ps1
vsp p =
s1 s1
− ∂( vsy x1 )
∂ps1
= vsyy x12
Thus if she is averse to income risks vsyy < 0 her state s indirect utility function is
concave in ps1 and hence its expected value is increased by a mean preserving
contraction in the distribution of ps1. (See section 17F.)
Exercise 21D
1. The Pareto efficiency conditions are derived by maximizing the expected utility of,
say, the producer subject to (i) the consumer getting at least a specified level of
expected utility (Ec ); (ii) the production function constraints: xsc ≤ fs(z), s = 1, . . . , S and
(iii) the constraints on the availability of the composite commodity: ys + ysc ≤
Ws + Wsc ≡ Ws, s = 1, . . . , S. Since both individuals have positive marginal utility and
the marginal product fs′ is also positive, all the constraints bind and we can substitute
fs(z) for xsc and Ws − ys for ysc to write the Lagrangean as
L = Evs(ys, z) + λ[Eusc ( Ws − ys, fs(z)) − Ec )]
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The first order conditions are
c
fs′( z) = 0
Lz = Evsz + λEusx
c
= 0,
Ly = πsvsy − λπ s usy
(21.6)
s = 1, . . . , S
s
(21.7)
plus the expected utility constraint on the consumer.
Rearranging (21.7) gives
π s vsy π s usyc
=
π l vly π l ulcy
(21.8)
which is the usual condition for efficient risk bearing that marginal rates of substitution
c
, we
across states are equalized for different individuals. Since, from (21.7), vsy = λusy
c
c
can divide each vsz term in (21.6) by vsy and each usx term by λusy to write the condition
for efficient production as
uc
v
E sx
fs′ = E sz
c
usy
vsy
The left hand side is the expected marginal value product of the input in terms of
income and the left hand side is its expected marginal cost in terms of income.
1. (a) With a market in state contingent income claims the producer can buy (qs > 0) or
sell (qs < 0) claims to income in state s at price psy, subject to the budget constraint that
the total value of sales and purchases must be non-positive:
∑p q ≤0
sy
(21.9)
s
s
Given her trades in the state contingent income markets and her correct expectations
about the spot price of her output, she correctly anticipates that her state s income will
be
ys = ps fs(z) + qs + Ws
Her Lagrangean for her choice of input and state contingent income trades is
L = Evs(ps fs(z) + qs + Ws) − µ
∑p q
sy
s
s
with first order conditions
Lz = Evsz + Evsy ps fs′( z) = 0
Lq = πsvsy − µpsy = 0,
s
s = 1, . . . , S
(21.10)
(21.11)
plus the budget constraint. Her optimal input choice is z*(W, p, π, py) and her net or
market demand for state s income is Qs(W, p, π, py).
The consumer also buys ( qsc > 0) or sells ( qsc < 0) claims to state contingent income
at prices psy, subject to a budget constraint: ∑ s psy qsc ≤ 0, yielding the state s budget
constraint
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y sc + ps x sc = Wsc + qsc
His demand for xsc = Ds ( ps , Wsc + qsc ) maximizes his state s direct utility usc ( ysc , xsc ) with
the first order condition
c
c
usx
− usy
ps = 0
(21.12)
to give state s indirect utility
vsc ( ps , Wsc + qsc ) = usc ( Wsc + qsc − ps Ds , Ds )
Before the state is revealed he chooses his state contingent income trades and the
Lagrangean for this problem is
Lc = Evsc ( ps , Wsc + qsc ) − µ c
∑p q
c
sy s
s
and the first order conditions are
c
c
Lcq = π s vsy
− µ c psy = π s usy
− µ c psy = 0,
s = 1, . . . , S
c
s
(21.13)
(Note that the assumption of rational expectations means that the correct spot prices
are used in the indirect utility functions when the state contingent income trades are
chosen before the state is revealed.) The consumer’s net demand function for state s
contingent income is Qsc (Wc, p, π, py).
The economy is in equilibrium when the markets for state contingent incomes clear:
Qs(W, p, π, py) + Qsc ( W c , p, π, py) = 0,
s = 1, . . . , S
and the spot markets for the producer’s output clear:
fs(z*) − Ds(ps, Wsc + Qsc ) = 0,
s = 1, . . . , S
Since all agents face the same prices for state contingent income trades, (21.11) and
(21.13) imply
πsvsy/µ = psy = π s vsyc / µ c = π s usyc / µ c
and dividing this through by the analogous condition for another state ᐉ shows that the
conditions for Pareto efficient risk bearing (21.8) are satisfied.
From (21.11) we can write (21.10) as
Evsz +
∑ π v p f ′ = Ev + µ ∑ ρ p f ′
s
sy
s
s
sz
sy
s
s
s
(21.14)
s
From (21.12) and (21.13) we have
ps =
c
c
c
usx
usx
usx
π
=
=
c
c
usy
vsy
µ c psy s
and substituting in (21.14) gives
Evsz + ( µ / µ c )
∑π u f ′ = 0
s
c
sx
s
s
Since for some choice of Ec in the Pareto efficiency problem we will have λ = µ/µ c, we
see that there is efficient production.
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1. (b) With markets for the state contingent delivery of the producer’s output both
parties can contract at prices psx for delivery of one unit of x if and only if state s occurs.
The producer can sell (buy) (ζs > (<)0 claims at prices psx to get a certain wealth from
production of ∑s psxζs. Her wealth or budget constraint for trading in state contingent
income claims is now
∑p ζ −∑p q ≥0
sx
s
s
sy
sy
s
and her state s income is
ys = Ws + qs + ps[fs(z) − ζs] + B
The following arbitrage condition must hold:
psy ps = psx,
s = 1, . . . , S
in equilibrium. Suppose it did not for some s and, in particular, that psy ps > psx. If the
producer reduced her sales of output on the state contingent commodity market by
one unit her wealth would fall by psx and her state s income increase by ps. If she
simultaneously sold claims to ps of state s income on the state contingent income
market her state s income would fall by ps and her wealth would increase by psy ps. The
net effect of these transactions would be to leave her state s income unchanged but to
increase her wealth by psy ps − psx > 0. Analogous arguments apply when psy ps < psx.
The arbitrage condition means that transactions in state contingent income claims
and state contingent commodities are equivalent means of transferring income across
states via market exchanges. Thus define the net transfer to state s via market
exchanges as
δs ≡ qs − psζs = qs − psxζs/psy
so that we could write state s income as ys = Ws + psfs + δs + B and the budget constraint
as
∑ p ζ − ∑ p q = −∑ δ ≥ 0
sx
s
sy
s
sy
s
s
s
Since what matters is the net transfer we could cast the analysis in terms of choice of
the δs. Equivalently we can set arbitrary values for the ζs (or the qs) and let the producer
choose the qs (or the ζs). It is simplest to assume that the producer always sets ζs = fs(z)
so that state s income is
ys = Ws + qs + B
and the budget constraint is
∑ p f ( z) − ∑ p q ≥ 0
sx
s
sy
s
sy
s
With this formulation the Lagrangean for the producer’s problem is
∑ p f ( z) − ∑ p q ]
L = Evs(Ws + qs + B, z) + µ[
sx
s
s
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The first order conditions are
Lz = Evsz + µ ∑ psx f s′ = 0
Lq = πsvsy − µpsy = 0,
(21.15)
s = 1, . . . , S
s
(21.16)
plus the budget constraint.
The consumer chooses the xsc and the qsc via contracts on the markets for the state
contingent commodity and state contingent income. We assume that the consumer buys
all her requirements of x on the state contingent commodity markets and does not enter
the spot market for the commodity when the state is revealed. (It is easy to
show that the arbitrage condition implies that the consumer is indifferent as to the
mix of spot and state contingent commodity market transactions which yield a given
consumption level xsc .) In this economy the spot markets do not operate because the
producer sells all her output on the contingent commodity markets and the consumer
buys all his requirements on the contingent commodity markets.
The consumer’s budget constraint is
∑p q +∑p x ≤0
sy
c
s
sx
s
c
s
s
and his Lagrangean is
∑p q +∑p x ]
Lc = Eusc ( Wsc + qsc , xsc ) − µ c [
sy
c
s
s
sx
c
s
s
The first order conditions are
c
Lcq = π s usy
− µ c psy = 0,
s = 1, . . . , S
(21.17)
c
Lcx = π s usx
− µ c psx = 0,
s = 1, . . . , S
(21.18)
c
s
c
s
plus the budget constraint.
It is apparent from (21.16) and (21.18) that the market in state contingent income
claims again leads to efficient risk bearing because the parties face the same relative
c
/µ c
prices of claims to state contingent incomes. Using (21.17) to substitute psx = π s usx
in (21.15) shows that the producer’s choice of input satisfies
Evsz + ( µ / µ c )
∑π u f ′ = 0
s
c
sx
s
s
and is therefore Pareto efficient.
2. Writing the first order condition in the question as
Evsy ps f s′ + Evsz − tEvsy f s′ = 0
Comparing this with [D.17] (after setting psa = ps since there are rational expectations),
we see that setting
t=−
c
) Ds psz
E ( vsy − λvsy
Evsy f s′
ensures that the producer’s choice implies ∂W/∂z = 0.
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Note that the partial derivative of the first order condition on z with respect to t
(holding z constant but remembering tfs = Hs) is
∂ dEvs
∂y
= ( Evsyy + Evszy ) s − Evsy fs′ = − Evsy fs′ < 0
∂t dz
∂t
since ∂ys/∂t = 0. Hence the producer’s input level can be reduced by increasing t or vice
versa.
3. Drop the s subscript to save notation and recall that unity elasticity of demand with
respect to price implies
−1 =
∂ log D
∂ log p
Integrating and using Roy’s Identity,
−vc
log D = log c p = −log p + K(y)
vy
c
= 0 implies that vc is additively separable in y and p and can be written vc(p, y) =
But vyp
a(p) + k1g(y). Hence
− a ′( p)
log D = log
= −log p + K(y)
k1 g ′( y)
so that
a ′( p) = −
k( y) k1 g ′( y)
p
where log k(y) = K(y). Hence
a(p) = −k(y)k1g′(y) log p + m
and since a is not a function of y we must have k(y)k1g′(y) = constant = k0. Since the
utility function is unique up to linear transformations we can always set m = 0 to give
[D.19].
4. The producer’s state s income is given by [D.20] and her forward sales satisfy [D.21].
However, because she does not choose z, her optimal forward sale is xf(W, p, pf , π, z).
The consumer’s optimal forward and spot demands are still given by Df(Wc, p, pf , π) and
Ds(ps, Wsc + (ps − pf)Df). Hence equilibrium of the forward and spot markets requires
xf(W, p, pf , π, z) − Df(Wc, p, pf , π) = 0
fs(z) − Ds(ps, Wsc + (ps − pf)Df) = 0,
s = 1, 2, 3
instead of [D.24] and [D.25]. The equilibrium forward and spot prices thus depend on
the level of z:
pf = pf(z, ⋅), ps = ps(z, ⋅),
s = 1, 2, 3
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Proceeding as on text pages 629–631, the marginal social value of z is
∂p f
∂p f
∂W
∂p
c c ∂ps
xf
= Evsy ps fs′ + Evsz + Evsy ( f s − x f ) s + x f
+ λEvsy
− Df
∂z
∂z
∂z
∂z
∂z
∂p
∂p
∂p
c
= Evsy ps fs′ + Evsz + E ( vsy − λvsy
) Ds s + D f f − s
∂z
∂z
∂z
(Use the market clearing conditions and Roy’s Identity.) At the level of z chosen by the
producer the first two terms sum to zero and so
∂p
∂p
∂W
∂p
c
) Ds s + D f f − s
= E ( vsy − λvsy
∂z
∂z
∂z
∂z
(21.19)
As in the case where there are no futures markets the input choice of the producer is
not in general constrained Pareto efficient because she neglects the effect of her
changes in z on the market clearing prices and the consequent transfers of income
across states. Since there are no markets in state contingent income claims there is
in general inefficient risk bearing so that changes in forward and spot prices which
alter state contingent incomes have efficiency implications. Unfortunately, as total
differentiation of the market clearing conditions with respect to z shows, the effect of z
on the equilibrium prices is extremely complicated.
Exercise 21E
1. (a) The slope of the budget line AB in text Fig. 21.3 is given by [E.4] and does not
depend on the initial holdings. A reduction in the initial holding in firm 1 shifts the point
â1Y 1 inward along the ray 0Y 1. The value of the initial holdings is reduced but since the
prices of shares is unchanged the budget line slope is unaltered.
1. (b) Write δ for the denominator of [E.4] and differentiate [E.4] with respect to M1:
Y
Y −2
d( dy2 /dy1 ) 1 Y21 Y11 −2
= 2
−
M1 Y12 − 22 − 12 M1 Y11
dM1
δ M 2 M1
M 2 M1
=−
=
=
Y M − Y12 M 2
1 Y21 M1 − Y11 M 2
Y12 − 22 1
Y11
2
δ M1
M1 M 2
M1 M 2
2
1
δ 2 M12 M 2
(Y12Y21 − Y11Y22 )
Y11Y21 Y12 Y22
−
<0
δ 2 M12 M 2 Y11 Y21
The inequality follows from inspection of Fig. 21.3 where the slope of 0Y 1 (Y12/Y11) is less
than the slope of 0Y 2 (Y22/Y21).
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Fig. 21E.1
An increase in the price of shares in firm 1 therefore makes the budget line steeper:
acquiring additional state 1 income by trading in the stock market becomes relatively
more expensive. Firm 1 yields relatively more income in state 1 than firm 2 and so an
increase in the price of its shares makes it more expensive to acquire claims to state 1
income: the budget line becomes steeper. Since the individual can always choose not to
trade in the markets for shares the endowment point W is always feasible. Hence the
budget line pivots through W.
1. (c) The denominator in [E.4] is negative since m1Y11 > m2Y21. Hence an increase in Y11
makes the absolute value of [E.4] smaller. The budget line AB becomes flatter because
the point A moves horizontally to the right. The budget line also becomes flatter when
Y12 increases because this shifts the point A vertically upwards.
2. The rate of return on firm j shares is hj2 = Yjs/Mj − 1 (see [E.5]). Consider Fig. 21E.1 in
which point a at (1 + h11, 1 + h12) is the state contingent income vector attainable by
spending £1 on buying shares in firm 1. Point b is similarly the state contingent income
vector obtainable by spending £1 on buying shares in firm 2. By suitable choice of
(λ1, λ2), λj ≥ 0, ∑λj = 1 it is possible to achieve any point on ab. Such a mixture of shares
is an asset with random rates of return hs = ∑λj hjs. In particular the point c where
income is the same in both states is attainable. The composite security or portfolio
corresponding to c has a certain rate of return h0. If the rates of return in the firms are
such that both points a and b are on the same side of the 45° line a point on the 45° line
is attainable if short sales are permitted ie if the restriction λj ≥ 0 is lifted. (See the
discussion in the text.)
There are two circumstances in which it will not be possible to achieve a certain rate
of return when there are as many firms as states. (i) The first is when two of the firms
have the same rates of return. In the two state, two firm case h1s = h2s, s = 1, 2 implies
that the line ab collapses to a single point and no change in the state distribution of
incomes is possible by exchanging shares in the two firms. (ii) The second case in
which a certain rate of return is not possible is when the line ab has a slope of 1 and is
parallel to the 45° line, as for example if firm 2’s rates of return generated point b′ in the
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figure. In this case the line ab′, no matter how extended, does not cut the 45° and there
is no composite asset with a certain rate of return.
The second case is not economically sensible since firm 1 yields a larger rate of
return in all states of the world. Clearly there would be no demand for shares in firm 2
and its market price M2 would fall. This would increase its rate of return (see [E.5]) and
the point b′ would be shifted out along the ray 0b′ until its rates of return are not
dominated by firm 1. Depending on investors’ preferences the market price would
adjust until the markets clear and a point like b″ is attainable by spending £1 on firm 2
shares. But now suitable purchases and (short) sales will make all points along ab″ and
its extension feasible, including a point on the 45° line.
More formally: it is possible to achieve a certain rate of return by a suitable mixture
of shares in the two firm, two state case if the following equations can be solved for λ1,
λ2, h0:
λ1h11 + λ2h21 = h0
λ1h12 + λ2h22 = h0
λ1 + λ2 = 1
or, in matrix form,
h11 h21
h
12 h22
−1 −1
−1 λ 1 0
−1 λ 2 = 0
0 h0 −1
These equations can be solved for λ1, λ2, h0 only if the determinant ∆ of the system is
non-zero. Since the determinant is ∆ = (h21 − h11) − (h22 − h12) the system has a solution
only if one firm’s rate of return does not exceed that of the other by the same amount in
both states. This rules out a configuration like a, b′ in Fig. 21E.1. If the determinant of
the system is non-zero we can use Cramer’s rule to solve for
h
h0 = − 11
h12
h21
h22
If the vectors of the firms’s rates of return are linearly independent we can solve for
a non-zero certain rate of return achievable by a suitable combination of the shares.
This is the spanning condition. Note that if the vectors are linearly dependent the
solution has h0 = 0. This is not economically sensible. Linear dependence means that ab
is ray from the origin, so that a certain rate of return of zero is obtainable at the origin.
But if ab is a ray from the origin one firm’s rates of return are larger than the other’s in
every state. The price of the dominated firm’s shares would adjust until the rates of
return were equal and ab would collapse to a single point. But then ∆ = 0 and the system
would have no solution: a certain rate of return is not obtainable. Hence, allowing for
the adjustment of share prices, we see that the spanning condition is a necessary and
sufficient condition for the construction of a composite security with a certain rate of
return.
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Fig. 21E.2
3. Fig. 21E.2 shows the effect of the various changes on the budget constraint. The
initial budget constraint is AB and is constructed as in text Fig. 21.5. and has slope
(h2 − h0)/(h1 − h0).
3. (a) An increase in wealth w shifts the budget line outward to A′B′ with no change in
slope. The demand for the risky asset is unaffected if there is constant relative risk
aversion which implies that the slope of indifference curves is constant along rays from
the origin.
3. (b) Let the return on the risky asset increase from hs to hs + k. The budget line is now
BA″: the income attainable from investing only in the risky asset is larger in both states
whilst the certain income attainable from investing only in the safe asset is unaffected.
More formally: the right hand side negative numerator in [E.11] is reduced and the
positive denominator increased, thus reducing the absolute value of [E.11]. The effect
on demand will depend on risk aversion: if the individual has decreasing absolute
(relative) risk aversion the wealth effect of the increase in the risky returns increases
the amount (proportion) of initial wealth invested in the risky asset, thus reinforcing the
substitution effect.
3. (c) An increase in the return on the safe asset pivots the budget line to B″A making it
steeper. If there is diminishing risk aversion the substitution and wealth effects will
work in opposite directions.
3. (d) An increase in the probability of the high return state 1 has no effect on the
budget constraint which is defined solely by initial wealth and the rates of return on safe
and risky assets. An increase in π1 makes the indifference curves steeper and hence
leads to an increase in the demand for the risky asset.
3. (e) With only two states a mean preserving spread requires a change in the state
contingent rates of return as well possibly as a change in probabilities. The simplest
case to analyse is a change in the rats of return with constant probabilities. Expected
income is constant along lines with a slope of −π1/π2 (the slope of the indifference
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Fig. 21E.3
curves at the 45° line). Thus a movement down the constant expected income line
through A away from the 45° line holds expected income constant but is clearly riskier:
the rate of return in state 1 has increased whilst that in state 2 has fallen. The effect
is to steepen the budget line to BA′″. The individual is worse off and, if there is
diminishing risk aversion, the wealth effect now reinforces the substitution effect and
the demand for the risky asset is reduced.
4. (a) With a proportional tax at the rate t on final wealth we have
ys = [w(1 + h0) + D(hs − h0)](1 − t)
instead of [E.6]. The budget constraint in Fig. 21E.3 is shifted inward but its slope is
unchanged. If the individual has constant relative risk aversion investment in the risky
asset is unchanged: B′C′/B′A′ is equal to BC/BA so that the proportion of initial wealth
invested in the risky asset is unchanged. Since D/w is unchanged and the tax does not
alter w the demand D for the risky asset is also unchanged if there is constant relative
risk aversion.
4. (b) With a tax at the rate θ on the income from investments we have
ys = (w − D)[1 + h0(1 − θ)] + D[1 + hs(1 − θ)] = w[1 + h0(1 − θ)] + D(hs − h0)(1 − θ)
instead of [E.6]. Using the equation for y1 to solve for D and substituting in the equation
for y2 gives the equation for the budget line as
y2 = w[1 + h0(1 − θ)] +
y1 − w[1 + h0 (1 − θ )]
( h2 − h0 )
( h1 − h0 )(1 − θ )
so that the slope of the budget line is unaffected by the tax. The effect of the tax is to
shrink the budget line from BA to B′A″, reducing the final wealth achievable by
investing entirely in the safe or the risky asset. When θ = 1 and all investment income is
taxed away the budget line collapses to the point (w, w) on the 45° line.
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If there is constant relative risk aversion the tax shifts the individual from C to C′,
increasing the proportion of initial wealth invested in the risky asset from BC/BA to
B′C′/B′A″. In this case taxation increases risk taking (measured by the demand for the
risky asset).
5. The effect on the slope of the indifference curve of movement along a straight line in
(y1, y2) space defined by y2 = a + by1 is given by
d( −π 1 v ′y1 ) /π 2 v ′( y2 )
−π 1
[v″(y1)v′(a + by1) − v′(y1)v″(a + by1)b]
=
dy1
π 2 ( v ′( y2 ))2
v′( a + by1 ) v′( y1 )
−
= −γ
b
v′′( a + by1 ) v′′( y1 )
(21.20)
= γ [T(a + by1) − bT(y1)]
(21.21)
where γ = π1v″(y1)v″(y2)/π2(v′(y2))2 and T(y) = −v′(y)/v″(y) is risk tolerance. Thus
indifference curves have constant slope along a straight line if and only if
T (a + by1) = bT(y1)
(21.22)
for all y1. Now (21.22) implies
dT (a + by1 )
= T ′(a + by1)b = bT ′(y1)
dy1
or T ′(a + by1) = T ′(y1). Hence (21.22) implies that T(y) is linear: T = α + βy, so that
T ′ = β is constant. Conversely if T is linear the “income consumption curve” is also
linear since it is always possible to find constants a, b such that
T(a + by1) = α + β (a + by1) = b(α + βy1) = bT(y1)
implying that (21.21) is zero.
6. Let the proportion of initial wealth invested in the certain asset be δ0 so that c = δ0w,
and the amount invested in risky asset j be δj(w − c) = δj(1 − δ0)w. Hence
ys = w[δ0(1 + h0) + (1 − δ0)
J
∑ δ (1 + h )]
j
js
1
= w[δ0(1 + h0) + (1 − δ0)(1 + hJs +
J −1
∑ δ (h − h ))]
j
js
Js
1
= wzs(δ )
(21.23)
remembering that ∑1J δ j = 1. Note that zs depends on the investment decision but not on
w.
Marginal utility in state s is v′(ys) = (ρ + τys)−1/τ (with τ = 1 for the logarithmic utility
function). In the case in which ρ = 0 marginal utility can be written
v′(ys) = (τ ys)−1/τ = w−1/τ(τ zs)−1/τ
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so that the first order conditions are
Ev′(wzs(δ )) w
∂zs
∂z
= w−1/τ E[τ zs(δ )]−1/τ w s = 0, i = 0, . . . , J − 1
∂δ i
∂δ i
(21.24)
Dividing each of the first order conditions by w1−1/τ leaves a set of conditions
determining the optimal δ0, . . . , δJ−1 which do not depend on w. Hence the proportion of
wealth invested in each asset, including the safe asset, is independent of wealth.
Now consider the case in which ρ ≠ 0 and denote the investment in the safe and risky
assets as
c = α + δ0(w − α)
Dj = δj(1 − δ0)(w − α)
(Note that c + ∑ Dj = w). Choice of the δj completely determines investment in the safe
and risky assets: by setting δ0 = (c − α)/(w − α) and δj = Dj/(w − α), j = 1, . . . , J and
investment plan (c, D1, . . . , DJ) with c + ∑ Dj = w can be achieved whatever the value
of α .
With this reformulation of the investment problem, write state s wealth as
ys = α (1 + h0) + (w − α)[δ0(1 + h0) + (1 − δ0)[1 + hJs +
J −1
∑ δ ( h − h )]]
j
js
Js
1
= α (1 + h0) + (w − α)zs(δ )
(21.25)
For given α , the optimal investment plan satisfies the first order conditions on δ0, . . . ,
δJ−1:
Ev′(α (1 + h0) + (w − α)zs(δ ))(w − α)
∂zs
∂δ i
= E{ρ + τ [α (1 + h0) + (w − α)zs(δ )]}−1/τ(w − α)
∂zs
∂δ i
= E{ρ + τα (1 + h0) + τ (w − α)zs(δ )}−1/τ(w − α)
∂zs
=0
∂δ i
(21.26)
With different α different δ0, . . . , δJ−1 would satisfy these conditions but the optimal
portfolio c, D1, . . . , DJ would be unchanged. Suppose we set
α = −ρ/τ(1 + h0)
Then the conditions (21.26) simplify to
E[τ(w − α)zs(δ )]−1/τ(w − α)
= (w − α)−1/τE[τ zs(δ )]−1/τ(w − α)
∂zs
∂δ i
∂zs
= 0, i = 0, . . . , J − 1
∂δ i
(21.27)
Dividing these conditions through by (w − α)1−1/τ gives a set of equations which
determine δ0, . . . , δJ−1 but do not depend on w − α. Hence the proportion of wealth in
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Fig. 21E.4
excess of α invested in the assets is independent of w − α. Thus setting α = −ρ/τ(1 + h0)
and then choosing the δ0, . . . , δJ−1 determined by (21.26) yields an optimal portfolio.
Moreover since δi, i = 1, . . . , J − 1 does not vary with w − α the share of the risky part of
the portfolio invested in each risky asset is independent of w:
Di
δ (1 − δ 0 )( w − α )
δ
= Ji
= Ji
J
∑1 D j ∑1 δ j (1 − δ 0 )( w − α ) ∑1 δ j
7. (a) Before the state is revealed the price p of a share in the firm entitling the holder to
income Ys/N in state s and the price of a call option entitling the holder to buy one share
at a price of p0 is c. An individual with initial wealth w can buy n shares and q call
options subject to the budget constraint w = np + qc. Given these transactions the state
contingent incomes are
y1 = n
Y1
Y
Y
q
+ q 1 − qp0 = n 1 + (Y1 − p0 N )
N
N
N N
(21.28)
y2 = n
Y2
N
(21.29)
Using the budget constraint to substitute for n = (w − qc)/p in (21.28) and then using the
resulting equation for y1 to substitute for q in (21.29) gives the equation for the budget
line in (y1, y2) space as
( Npy1 − wY1 ) Y2
y2 = w −
c
Y1 ( p − c ) − p0 Np pN
The slope of the budget line is
− cY2
dy2
=
dy1 Y1 ( p − c ) − p0 Np
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In Fig. 21E.4 purchases of the call option enable all points along the budget line from
B where q = 0 to A where q = w/c to be reached. Since the option is only worth
exercising if state 1 occurs, purchases of the option transfer are a means of transferring
income between states by reducing income in both states by c and increasing income in
state 1 by Y1/N − p0. By selling call options (q < 0) it is possible to reduce income in state
1 and increase state 2 income. Hence all points along BA and its extension are
achievable via the share and option markets. Adding a market in options which are only
worth exercising if state 1 occurs is equivalent to creating a market in an artificial firm
with a payoff vector (Y1 − Np0, 0), so that together the payoff vectors of the original and
the artificial firm satisfy the spanning condition.
7. (b) The total number of securities (firm shares and options) must be at least equal to
the number of states and there must be a subset of S of them with linearly independent
payoff vectors. One firm could be sufficient if there were S − 1 options each with
different exercise prices p0i such that Y1/N > p01 > Y2/N > p02 > . . . > p0S −1 > YS/N.
Exercise 21F
1. In the case in which each firm’s output in one state is a linear combination of its
outputs in the other two states, constrained Pareto efficiency is possible because the
number of states is now effectively the same as the number of firms, and firms will be
able to deduce (effective) shareholders’ discount rates from market data.
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Appendices
Exercise A
1. (a) We can represent the choice variables as x1, x2, x3, where x1 = 1 if you go home on
foot and zero otherwise; x2 = 1 if you go home by bus, and zero otherwise; and x3 = 1 if
you go home by train, and zero otherwise. The feasible set is the set of triples
S = {(x1, x2, x3)} = {(1, 0, 0), (0, 1, 0), (0, 0, 1)}.
If ti denotes how long it takes to get home by mode i = 1, 2, 3, and ci denotes the money
cost of mode i, then time taken is
3
T=
∑t x
i
i
i =1
and money cost is
3
C=
∑c x
i
i
i =1
We can postulate that you have an objective function V(T, C) which expresses your
preferences over time-cost combinations. For example, suppose that this takes the
linear form V = wT + C, w > 0. Note that we expect the ‘best’ way of getting home to
minimize V. Thus the problem is to minimize V by choosing one of the triplets in S.
A more direct approach to the problem would define the feasible set
S′ = {(t1, c1), (t2, c2), (t3, c3)}
i.e. in terms of the time-cost pairs corresponding to each mode. You must still have
some function such as V, however, which places a relative valuation on time and money.
Fig. A.1 then illustrates a solution. (We assume walking is costless, so we ignore
depreciation on shoes etc.) Clearly, each mode could be optimal for some value of w.
We take the case in which the bus is best: time is too valuable to you to go on foot, but
not so valuable that you take the train. Note that minimizing V means getting onto the
lowest possible line.
Supplementary questions
(i) What observable economic variable could you take as an estimate of w, if you
wanted to construct the function V?
(ii) Draw lines which yield first (0, t1), then (c3, t3) as solutions and interpret what they
imply about w.
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Fig. A.1
(iii) Assume now that your objective is to minimize time spent travelling, subject to not
spending more than a given amount of ". Use Fig. A.1 to reconsider the solution
possibilities.
(iv) Now do the same assuming the objective is to minimize cost subject to not taking
longer to get home than a specified time Q.
(v) Suppose you were indifferent between taking the bus or the train. What does that
tell us about your value of w?
1. (b) Let xi, i = 1, . . . , n denote the quantity of foodstuff i, and x0 the calorie-free,
expensive, all-vitamin tablet. Let pi, ci and vij be respectively the price, calorie count and
the content of vitamin j = 1, . . . , m, in foodstuff i. Let Pj denote the minimum amount of
vitamin j you require, " your minimum possible calorie consumption, and M your
maximum possible expenditure on food. Then your problem is
n
minimize C =
∑c x
i
i
i =1
subject to :
n
∑c x
i
≥
",
≤
M,
i
≥
Pj
j = 1, . . . , m,
xi
≥
0
i = 0, 1, . . . , n.
i
i =1
n
∑px
i
i
i=0
n
∑v x
ij
i=0
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Supplementary questions
(i) Suppose the calorie-free tablet does not exist, there are only two foodstuffs, one
vitamin, and the parameter values are
p1 = 4,
c1 = 40,
v1= 20,
" = 1200,
p2 = 2.5,
c2 = 40,
v2 = 10,
M = 100.
P = 400,
Graph the constraints and identify the feasible set. Then find a feasible, calorieminimizing pair of quantities ( x1* , x2* ).
(ii) Suppose now that the calorie-free tablet also exists, it contains 50 of the vitamin,
and costs 10 per unit. What is the optimal solution in this case?
1. (c) The price of each good sold at market B is less than the corresponding price at
market A. Then either the consumer will do all his shopping at market A, or all at
market B. The choice variables are xiA , xiB , i = 1, . . . , n, the quantities of each good
bought at supermarket A or B respectively. The corresponding prices are piA , piB ,
i = 1, . . . , n, with piA > piB , all i, by assumption. Let M be available income, and C
the extra cost incurred by shopping at market B rather than at market A. Then, if he
shops at market A his feasible set is determined by the constraint,
n
∑p x ≤M
A
i
A
i
xiA ≥ 0
i = 1, . . . , n,
i =1
and if at market B by
n
∑p x ≤ M −C
B
i
B
i
xiB ≥ 0
i = 1, . . . , n.
i =1
If he shops at market A he will choose the optimal consumption vector
VA = ( V1A , . . . , V nA ), and if he shops at B, he will choose the optimal consumption vector
VB = ( V1B , . . . , V nB ). Then, in general, all we can say is that his choice of market depends
on whether or not he prefers VA to VB.
Supplementary questions
(i) Suppose piA = kpiB , k > 1. Also assume that whichever market he shops in the
consumer spends all available income. Find a condition under which the consumer will
certainly shop at B. (The condition does not involve the optimal vectors VA, VB.)
(ii) You know the vectors VA, VB, all the prices, M and C. Can you frame conditions
under which: the consumer certainly buys at A; he certainly buys at B? Is this
information enough to allow you to predict his choice in all cases?
2. The answers are given by Fig. A.2.
Exercise B
1. (a) Let f(x*) ≥ f(x) all x ∈ S, so that x* is a global maximum (f is the objective
function, S the feasible set, and x* ∈ S). Take a small neighbourhood of points around
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Fig. A.2
x*, and take any point x′ both in this neighbourhood and in S (if there is no such point
then the proposition is trivially true). Then
f(x*) ≥ f(x) all x ∈ S and x′ ∈ S ⇒ f(x*) ≥ f(x′)
and so x* is also a local maximum.
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Fig. B.1
1. (b) Let x* and x′ be global maxima of f over the set S with x* ≠ x′. Then by definition
f(x*) ≥ f(x′), all x ∈ S, and f(x′) ≥ f(x), all x ∈ S. In particular, f(x*) ≥ f(x′) and f(x′) ≥
f(x*), which then implies f(x*) = f(x′).
1. (c) For a linear function, f(I) = kf(x′) + (1 − k) f(x″) for I = kx′ + (1 − k)x″, 0 ≤ k ≤ 1.
Comparison with the definitions of concavity and convexity then gives the result.
2. The answers are given by the Fig. B.1.
3. The function is quasi-concave if B′ is a convex set, and strictly concave if B′ is a
strictly convex set.
4. If f is concave then
f(I) ≥ kf(x′) + (1 − k)f(x″)
0≤k≤1
for all x′, x″ ∈ X, the domain of the function. It follows that if we choose x′, x″ such that
f(x′) = f(x″), then the above inequality must hold for these. This implies quasi-concavity
from the definition.
An example of a quasi-concave, non-concave function is
y = x12 x22 .
Supplementary question
(i) Use the answer to question 7 below to show that this function is (strictly) quasiconcave. Use the definition of a concave function, and the points (0, 0), (1, 1), with
k = 1/2, to show that this function is not concave. Sketch the function in three dimensions (or get a computer to do so).
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5.
(a) This feasible set is non-empty, closed, unbounded and convex.
(b) This feasible set is non-empty, not closed, bounded and convex.
(c) This feasible set is non-empty, closed, bounded and convex.
(d) This feasible set consists of the single point (3, −2) and so is non-empty, closed,
bounded and convex.
(e) This feasible set is empty.
(f) This feasible set is non-empty, closed, unbounded and non-convex.
(g) This feasible set is non-empty, closed, unbounded and non-convex.
(h) This feasible set is non-empty, closed, bounded and non-convex.
(i) This feasible set is non-empty, closed, bounded and convex.
(j) This feasible set consists of two points (1, 3), (2, 0) and is non-empty, closed,
bounded and non-convex.
Supplementary question
(i) Explain the above answers.
6. A set is closed if it contains whatever boundary points it has, but it can still be
unbounded if its boundaries do not fully enclose it, for example, the set of numbers
x ≥ 0. A set is not closed if it does not contain all its boundary points, for example the
set of numbers satisfying 0 ≤ x < 1.
7. We have
dx2
f (x , x )
=− 1 1 2 .
dx1
f 2 ( x1 , x 2 )
Hence:
d dx2
d f1 ( x1 , x2 )
=−
dx1 dx1
dx1 f 2 ( x1 , x2 )
=−
dx
dx
1
f f + f12 2 − f1 f 21 + f 22 2
2 2 11
f 2
dx1
dx1
where we use the fact that x2 is implicitly a function of x1. Then substituting dx2/dx1 =
−f1/f2 gives
d 2 x2
f
1
= − 2 f 2 f11 − f12 f1 − f1 f 21 + f 12 22
2
dx1
f2
f2
=−
1
{ f12 f 22 − 2 f1 f 2 f12 + f 22 f11 }
3
f2
Where we use Young’s Theorem, f12 = f21. Since, for strict quasi-concavity d 2 x2 /dx12 > 0,
multiplying through by −1 gives the required result.
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Fig. C.1
Supplementary question
(i) Explain why it is neither necessary nor sufficient for strict quasi-concavity that
fi > 0, fii < 0, i = 1, 2.
Exercise C
1. From Fig. C.1 we see that it is possible to have existence of a solution when the
conditions of the theorem are not met. Existence of a solution therefore does not imply
that the conditions are met, and so they are not necessary conditions for existence.
They do however guarantee that a solution does exist (they rule out cases in which a
solution does not exist) and so they are sufficient.
2. If a solution exists then there must exist a feasible point, i.e. the feasible set is nonempty. Therefore non-emptiness is a necessary condition for existence.
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Fig. D.1
Exercise D
1. (a) See Fig. D.1.
1. (b) It has been assumed that f1, f2 > 0, and so increasing x1 and x2 in text Fig. D.2 must
increase the value of the objective function. Thus points along the contour c yield the
same value of the objective function as points along ab, and points above the contour,
since they involve higher x1 and x2, yield higher values of the function.
2. If the feasible set S in text Fig. D.2 is not closed, in that it does not contain the points
on the segment ab, then no solution exists. Between any point in S and a point on ab we
can always find another point yielding a higher value of the function. Thus no maximum
exists.
3. A local minimum is a global minimum if: (a) the objective function is quasi-convex,
and (b) the feasible set is convex. The proof is precisely as given in the text, if we note
that minimizing f(x1, x2) is equivalent to maximizing −f(x1, x2), and that if f(x1, x2) is
quasi-convex, then −f(x1, x2) is quasi-concave.
Supplementary question
(i) Draw the counterpart of text Fig. D.2 for a minimization problem in which the origin
(a) is, and (b) is not a solution.
4. This question anticipates the discussion of the next appendix. If B is strictly convex
then the solution will be a unique point of tangency rather than the set of points along
ab (see text Fig. E.1 (c) for an illustration).
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Exercise E
1. See Fig. D.1.
2. Given a minimization problem in which the feasible set is convex and the objective
function is non-constant and quasi-convex, a solution is unique if (a) the feasible set is
strictly convex or (b) the objective function is strictly quasi-convex (or both). The proof
follows that in the text exactly, if we again note that minimizing f(x) is equivalent to
maximizing −f(x), and f(x) (strictly) quasi-convex implies −f(x) (strictly) quasi-concave.
3. In text Fig. E.1 (a), the feasible set is not strictly convex because its two lower
boundaries, the x1- and x2-axes are linear. Its upper boundary is defined as
U B = {x ∈ S|x is a boundary point of S and x′ Ⰷ x ⇒ x′ ∉ S}
It is easy to show that if we are maximizing f(x1, x2) over S, with f1, f2 > 0, then a solution
must be in U B. Suppose there are two such solutions, x* and x**, both in
U B. Then, if S is strictly upper convex then I = kx* + (1 − k)x**, 0 < k < 1 lies in the
interior of S and the rest of the proof goes through as before.
Supplementary question
(i) Show that if we are maximizing a function f(x) with fi > 0, i = 1, . . . , n, over a nonempty, closed, bounded convex set S, then any solution must lie on the upper boundary
of S.
Exercise F
1. See Fig. F.1.
2. Without further specification, a satisfactory point could be anywhere in the feasible
set, and may or may not be affected by a change in the boundaries of the set.
“Satisficing” is not a solution principle, while optimizing is.
3. We assume that the objective function is strictly quasi-concave. The feasible set is the
line ab in text Fig. F.1 (a). The diagrams in Fig. F.2 answer the question.
4. Recall that one of the fundamental concepts in microeconomics is relative scarcity.
People always want to consume more commodities than can be made available with the
resources an economy possesses. This creates a need for allocation, and
microeconomics is concerned with the way a market economy solves this allocation
problem. Now if everyone in the economy possesses a bliss point, and resources and
technology were such that everyone could attain their bliss point, then relative scarcity
would disappear, and so too would the microeconomic problem. Production and
distribution would simply have to be organized so as to give each person the
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Fig. F.1
Fig. F.2
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Fig. G.1
consumption vector corresponding to the bliss point. We would not need a price
mechanism to help allocate scarce resources among competing ends. We would have
reached Utopia. On the other hand, if people have bliss points but the total amounts of
commodities required to reach them all are beyond the capacity of the economy to
produce, then we again have relative scarcity and microeconomics is a relevant – indeed
essential – field of study. In other words people may have bliss points, but this is not
very important if they are locally non-satiated at levels of output feasible for the
economy.
Exercise G
1. See Fig. G.1.
In case (a) f ′(x*) > 0, and in case (b) fi(x*) > 0 at the optimal point.
2. If for this constrained minimization problem we form the Lagrange function
L(x, λ) = f(x) −
m
∑ λ [g (x) − b ]
j
j
j
j =1
and minimize this function with respect to x and λ, then we obtain precisely conditions
[G.15] and [G.16]. This is simply the familiar point that an unconstrained maximum or
minimum must occur at a point at which all partial derivatives are zero.
3. (a) The Lagrange multiplier associated with the constraint on the balance of
payments deficit is often called the “shadow price of foreign exchange”. It shows by
how much GNP in the economy could be increased if the constraint on the deficit were
relaxed slightly, for example by an exogenous increase in foreign exchange reserves.
The Lagrange multiplier on the skilled labour constraint would be called the shadow
wage of skilled labour, since it shows by how much GNP would increase for a marginal
increase in the amount of skilled labour. This may well differ from the actual wage rate
in the economy, hence the term “shadow”.
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Fig. G.2
3. (b) The Lagrange multiplier in this case is simply the marginal rate of return to
investment in the firm. The firm should compare it to the cost of borrowing, since if this
is lower, investment should be expanded; and also compared to the return of lending
outside the firm, since if this is higher, investment should be contracted.
4. (a) See Fig. G.2.
There are five solution possibilities:
(i) point α, where x1 = 0, x2 > 0. In this case the b1 constraint is non-binding,
(ii) a point along αγ, such as β, where x1, x2 > 0, while the b1 constraint is again nonbinding
(iii) point γ, where x1, x2 > 0 and both constraints bind,
(iv) a point along γ ε such as δ, where x1, x2 > 0, while the b2 constraint is non-binding,
(v) point ε, where x1 > 0, x2 = 0, and the b2 constraint is non-binding.
4. (b) λ*1 is the shadow price of the b1-constraint. It shows the rate at which the
optimized value of the objective function changes when b1 changes. If λ*1 = 0, this must
imply that a (very small) change in b1 leaves the solution unaffected. Thus b1 must be
non-binding at the optimum, and we have either case (i) or case (ii) above.
Exercise H
1. We may have both x2* = 0 and L2 = f2 − λ*a2 = 0, since this does not violate [H.20].
Therefore we cannot say that x2* = 0 ⇒ L2 < 0. In terms of text Fig. H.4, we can interpret
the case in which x2* = 0 and f2 = λ*a2 as being that in which the contour c1 and the
constraint line are just tangent at ( x1*, 0), since then we have:
a1
f ( x *, 0)
= 1 1
.
a2
f ( x *, 0)
2
1
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In terms of the original discussion of non-negativity conditions in text Fig. H.1, this is
like the case illustrated in Fig. H.1 (c). It so happens that the necessary condition for the
case without negativity conditions holds at the point x2* = 0. This is clearly something
of a special case, but, all the same, you should always resist the temptation to conclude
that Li < 0 when x i* = 0.
2. In text Fig. H.4, a1 and a2 would be the prices of goods 1 and 2 respectively, and the
contours c0, c1, c2 would be indifference curves. Then we have the case in which the
consumer spends all her income on good 1. Though this may seem rather special in
the 2-good case, in reality a consumer buys positive amounts of a small subset of
all the commodities that are in fact available – corner solutions are perfectly typical.
3. Given the problem max f(x) s.t. xi ≥ bi, i = 1, . . . , n (for simplicity ignore functional
constraints) we can proceed in either of two ways:
(a) Define Vi = xi − bi, and write the problem as
max f(V1 + b1, . . . , Vn + bn)
Vi ≥ 0
s.t.
i = 1, . . . , n.
Note that ∂f/∂xi ≡ ∂f/∂Vi since the bi are constants. Thus applying [H.4] we have
fi ≤ 0,
Vi* ≥ 0,
Vi* fi = 0.
But Vi* = x i* − bi, so this becomes
fi ≤ 0,
x i* ≥ bi ,
( xi* − bi ) fi = 0.
Thus, if we have an interior solution, with xi* > bi, we must have fi = 0, while at a corner
solution with xi* = bi we have fi ≤ 0. We can extend this to take account of functional
constraints just as before.
(b) Define the Lagrange function
L(x, λ) = f(x) +
∑ λ ( x − b ).
i
i
i
i
Then the Kuhn-Tucker conditions are
Li = fi + λ*i = 0
Lλ i = xi* − bi ≥ 0,
λ*i ≥ 0,
λ*i ( x i* − bi) = 0.
Then:
if
λ*i > 0, fi = − λ*i < 0 and xi* = bi
if
xi − bi > 0, λ*i = 0 and fi = 0
if
λ*i = ( x i* − bi) = 0, then fi = 0.
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Fig. H.1
Supplementary questions
(i) Develop necessary conditions for the problem max f(x) s.t. xi ≤ bi, i = 1, . . . , n.
(ii) Suppose f(x1, x2) is a strictly quasi-concave utility function, a consumer has a
standard budget constraint, and must consume at least minimum subsistence levels s1, s2
of the two goods. Apply one of the above procedures and interpret the solution
diagrammatically.
(iii) Interpret the Lagrange multipliers in approach (b).
4. Detailed answers depend on exactly how the new constraint relates to the two initial
constraints. Suppose it enters as illustrated in Fig. H.1.
Then, clearly the α and β solutions are still possible (as are corner solutions on the
original two constraints) but solutions at γ are ruled out. Instead, we have three new
solution possibilities, at δ, at ε, or at a point like φ along δε. The Lagrange function is
now
2
2
2
f ( x1 , x2 ) − λ 1 a i x i − b1 − λ 2 c i x i − b2 − λ 3 e i x i − b3 .
i =1
i =1
i =1
∑
∑
∑
The Kuhn-Tucker conditions are now
Li
= fi − λ*1 a i − λ*2 c i − λ*3 e i ≤ 0
xi* ≥ 0 xi* Li = 0 i = 1, 2,
Lλ 1 = ∑ a i xi* − b1 ≤ 0,
λ*1 ≥ 0
λ*1 Lλ = 0,
Lλ 2 = ∑ c i xi* − b2 ≤ 0,
λ*2 ≥ 0
λ*2 Lλ = 0,
Lλ 3 = ∑ e i xi* − b3 ≤ 0,
λ*3 ≥ 0
λ*3 Lλ = 0.
1
2
3
Then, at α, set λ*2 = λ*3 = 0; at β, set λ*1 = λ*3 = 0; at φ, set λ*1 = λ*2 = 0; at δ, set λ*2 = 0; at ε,
set λ*1 =0. Then, for each of these cases the Kuhn-Tucker conditions can be used to
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Fig. H.2
characterize the optimal pair ( x1* , x2* ), (which are both positive at all these points –
again you should also add the cases ( x1* , 0) and (0, x2* )). Essentially, then, the KuhnTucker conditions allow a systematic working through of solution possibilities.
5. In this problem, for one unit of good i the consumer must pay ai units of money and ci
ration points. If ration points cannot be exchanged for money, the two constraints must
be imposed separately. Then, in Fig. H.2, at an α-type solution the consumer is spending
all her endowment of money but is left with surplus ration coupons (since
α is below the b2 constraint). In other words her preferences are such that she wants
to buy relatively more of good x2, whose money price is relatively higher than its
points price ((a2/a1) > (c2/c1)). A β-type on the other hand has preferences such that his
optimal consumption bundle requires all his ration points but leaves him with money
left over. Only a solution at γ involves spending exactly all of both money and points
endowments.
As the figure shows, there is no reason in general why everyone would be at a point
such as γ – if preferences vary over individuals, the other solution types are also
perfectly possible. Thus, there will be people with “spare” ration points, and also people
with “spare” money. In microeconomics we regard it as a basic fact of human nature
that in such a situation people will want to trade (if, for some reason, the rationing
authority has declared this illegal, then there will be a “black market” in ration points).
The important point is that such trade makes everyone better off. To see this, let p be
the money price of a ration that is established in the market for points. Then the two
constraints can be collapsed into one, since multiplying through the ration constraint by
p and summing we have
(a1 + pc1)x1 + (a2 + pc2)x2 ≤ b1 + pb2.
This is a valid operation because p is in dimensions £/points, and so multiplying through
the points constraint puts it in the dimension of money. Intuitively, we could think of a
consumer selling her entire points endowment b2 for pb2. Then, to buy a unit of xi she
must pay ai as the price, and then pci as the money cost of ration points she
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needs for the purchase. Thus we could think of (ai + pci) as the “full price” of a unit of xi
and b1 + pb2 as her “full income”.
What can we say about the new single budget constraint? Diagrammatically, we can
say that it passes through the point γ in Fig. H.2, and has a slope intermediate between
those of the two separate constraints. But this then means that the feasible set expands,
as in the figure. Thus both α and β types are made strictly better off, as we would
expect. But in general γ -types will also be better off unless it so happened that their
marginal rate of substitution at ( x1γ , x2γ ) was exactly equal to (a1 + pc1)/
(a2 + pc2).
To establish the result illustrated in the above figure, note first that at ( x1γ , x2γ ) we
have, since it is an intersection point
a1 x1γ + a2 x2γ = b1
c1 x1γ + c2 x2γ = b2
and so it must follow that
(a1 + pc1 ) x1γ + (a2 + pc2 ) x2γ = b1 + pb2,
so the new constraint must pass through point γ. Next, we have
a1 c1
< ⇒ a1 c2 < a 2 c1
a 2 c2
⇒ a1a2 + a1pc2 < a1a2 + a2pc1
⇒ a1(a2 + pc2) < a2(a1 + pc1)
⇒
a1 a 1 + pc1
<
a 2 a 2 + pc 2
In a similar way we can show that (a1 + pc1)/(a2 + pc2) < (c1/c2). Thus the slope of the
single budget constraint lies between that of the initial constraints. (Note that a similar
result could be established if we initially assumed (a1/a2) > (c1/c2).) Thus trade in ration
points in general makes everyone better off.
We have not yet discussed how the price p is determined. Essentially, this will be by
the demand and supply of ration coupons. Intuitively, the more α-type people there are
relative to β-types, the greater will be the amount of “excess coupons” relative to
“excess money”, and so the lower is p, the money price of coupons. In turn, the lower is
p the flatter will be the single budget constraint in the figure and the larger is the gain of
those with “excess money” (the β-types) and the smaller that of those with “excess
coupons” (the α-types).
Supplementary question
(i) Suppose that instead of being given an endowment of generalized ration points, with
each good having a given points price, rationing is effected by giving the consumer
endowments of coupons which can only be spent on a particular good, i.e.
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Fig. H.3
x1-ration coupons and x2-ration coupons. There is still of course a money budget
constraint. Carry out an analysis, along the above lines, for this type of rationing. When
will markets in coupons exist? Should trade in coupons be permitted?
6. Assume for concreteness that (a1/a2) < (c1/c2). Then,
a1 c1
< ⇒ a1 c2 < a 2 c1
a 2 c2
⇒ λ*1 a1a2 + λ*2 a1c2 < λ*1 a1a2 + λ*2 a2c1
⇒ a1( λ*1 a2 + λ*1 c2) < a2( λ*1 a1 + λ*2 c1)
⇒
a1 λ*1 a1 + λ*2 c1
<
.
a 2 λ* a + λ* c
1 2
2 2
The other inequality is established in a similar way, as are the two corresponding
inequalities for the case (a1/a2) > (c1/c2).
7. If, at γ in text Fig. H.5, we have that the contour is tangent to the b1 constraint, then
we must have
f1 a 1
=
.
f2 a 2
From conditions [H.27], [H.28] we see that this must imply that λ*2 = 0. Thus in [H.30]
we have in this case both that λ*2 = 0 and c1 x1* + c2 x2* − b2 = 0, since the solution is on
the b2 constraint line at γ. This suggests that the b2-constraint is in some sense nonbinding. We illustrate this in Fig. H.3, where the contour is tangent to the b1 constraint.
Suppose the b2-constraint shifts outward to the dashed line. Then the solution is
unaffected: point γ is still both feasible and optimal. Thus for a relaxation of the
constraint or an increase in b2, the optimized value of the objective function will be
unchanged: dV/db2 = 0, where V = f ( x1* , x2* ) , at γ. On the other hand, if the b2-constraint
shifts inward to the dotted line, point γ is no longer feasible, the solution must change,
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and the optimized value of the objective function must fall. Since this resulted from a
decrease in b2 we have dV/db2 > 0. That is, at γ the derivative of V with respect to b2 is in
this case not unique: viewed as a function of b2, it has a discontinuity at that value of
b2 at which the b2-constraint passes through the point of tangency of the objective
function contour with the b1-constraint. In such cases, the Kuhn-Tucker conditions are
made to yield the correct answer by choosing λ*2 = 0. Note that in this kind of case, if
we were carrying out the comparative-statics analysis with respect to the b2-constraint,
we would have to specify the direction of change in the constraint, because of this
discontinuity.
Exercise I
1. From [I.7] we have
∂x1
(α u − α 2 u12 )
λ* α 22
=−
+ x1* 1 22
∂α 1
D
D
=−
λ* α 22
D
∂x1
− x1*
∂α 3
as required. From the second-order conditions we have that D > 0. The first term
− λ* α 22 / D < 0, since λ* > 0. However, if we have no restrictions on the signs or relative
magnitudes of u12, u22 (or of α1, α2) then we cannot sign the term (a1u22 − α2u12). Thus
neither ∂x1/∂α1 nor ∂x1/∂α3 can be signed unambiguously.
If u is a utility function, α1, α2 prices, and α3 income then the above gives the Slutsky
equation for good x1. The first term is the substitution effect and we see that this is
negative. The second term is the income effect, and because its sign is ambiguous so is
that of the derivative ∂x1/∂α1, the slope of the Marshallian demand curve.
2. We wish to solve min f(x1, x2) s.t. g(x1, x2) = b, with fi > 0, i = 1, 2. In text Fig. I.2,
interchange f and g: the contour of f now becomes the one that is more concave or less
convex to the origin. The idea is that moving away from the optimal point x* along the
constraint contour always increases the value function, i.e. gives points on contours
further away from the origin. Thus the counterpart of [I.18] is
dx
d dx 2
− 2 <0
dx1 dx1 f dx1 g
at x = x*.
The rest of the analysis proceeds just as before, except that in [I.19] we set φ ′′( x1* ) > 0
as the second-order condition for a minimum.
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