Microeconomic Theory:
Basic Principles and Extensions
12th Edition
Solutions Manual
Walter Nicholson & Christopher Snyder
Preface
This Solutions Manual for the 12th edition of Microeconomic Theory: Basic Principles and
Extensions provides answers to all of the end-of-chapter problems, and the answers are provided
in more detail than the Brief Answers in the back of the textbook. It also includes a brief
introductory statement about the nature of the problems in each chapter and brief notes on the
economic point of each problem. This additional descriptive material may help to focus
discussions about the problems and serve to integrate them more completely with the theoretical
material in the text. That is especially important for problems that incorporate theoretical
concepts which, while not sufficiently central to warrant explicit treatment in the text, contain
ideas that should be understood by most students of economic theory.
Problems that introduce such basic results are explicitly highlighted in the commentary.
In general, the problems in the text are arranged from the least to most difficult. Particularly
difficult problems are separately identified here together with some hints on how to get students
started on them. All of the text is set using Word 2010 and most of the in-line and displayed
math is typeset using Math Type 6.7.
For the end-of-chapter problems we have assigned in our own classes, we have found
several methods of cutting and pasting from this manual into a problem set answer key for the
class to be particularly convenient. One way is simply to copy the answer to the relevant
problem from the Word files provided into a new Word file for the class answer key. Another is
to use the “snapshot tool” in Adobe Acrobat Professional to select and copy the relevant block
from the PDF files provided into a PowerPoint or other file format containing the class answers.
Blowing the PDF file up to around 200% before applying the snapshot tool helps maintain a
readable resolution.
The authors are grateful to Henry Senkfor and Paulina Karpis, undergraduate students at
Dartmouth College, for their extensive work on various editions of this manual. We are indebted
to Kory Hirak for administrative assistance, Joseph Malcolm and his colleagues at Lumina
Datamatics for excellent copyediting and production management, and Anita Verma for content
development. Christopher Snyder thanks his wife, Maura Doyle, for numerous suggestions
compiled from her use of the book in her intermediate micro classes at Dartmouth. Walter
Nicholson thanks the many generations of Amherst College students who have sometimes
struggled through these problems, often offering good advice on how to improve them.
Every attempt has been made to purge the errors remaining from previous editions, but
errors and confusing passages may remain. The solutions to the problems new in this edition
will invariably contain some new errors as well. We encourage instructors to email us
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated,
or posted to a publicly accessible website, in whole or in part.
(chris.snyder@dartmouth.edu) with comments and corrections. We are grateful to these
instructors, a number of whom are thanked by name in text’s preface, for their help in
continually improving the quality of the book.
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated,
or posted to a publicly accessible website, in whole or in part.
CHAPTER 2:
Mathematics for Microeconomics
The problems in this chapter are primarily mathematical. They are intended to give students
some practice with the concepts introduced in Chapter 2, but the problems in themselves offer
few economic insights. Consequently, no commentary is provided. Results from some of the
analytical problems are used in later chapters, however, and in those cases the student will be
directed back to this chapter.
Solutions
2.1
f ( x, y ) = 4 x 2 + 3 y 2 .
a.
f x = 8 x, f y = 6 y.
b.
Constraining f ( x, y ) = 16 creates an implicit function between the variables. The
f
dy
−8 x
slope of this function is given by
=− x =
for combinations of x and y
dx
fy
6y
that satisfy the constraint.
c.
Since f (1, 2) = 16 , we know that at this point
d.
The f ( x, y ) = 16 contour line is an ellipse centered at the origin. The slope of the
line at any point is given by dy dx = − 8 x 6 y . Notice that this slope becomes
more negative as x increases and y decreases.
dy
8 ⋅1
2
=−
=− .
dx
6⋅2
3
2.2
a.
Profits are given by π = R − C = −2q 2 + 40q − 100. The maximum value is found
by setting the derivative equal to 0:
dπ
= − 4q + 40 = 0 ,
dq
implies q* = 10 and π * = 100.
b.
Since d 2π dq = − 4 < 0, this is a global maximum.
2
c.
MR = dR dq = 70 − 2q. MC = dC dq = 2q + 30. So, q* = 10 obeys
MR = MC = 50.
2
2.3
Chapter 2: Mathematics for Microeconomics
First, use the substitution method. Substituting y = 1 − x yields
f% ( x) = f ( x,1 − x) = x(1 − x) = x − x 2 . Taking the first-order condition, f% ′ ( x) = 1 − 2 x = 0,
and solving yields x* = 0.5, y* = 0.5 , and f% ( x* ) = f ( x* , y* ) = 0.25. Since
f% ′′( x* ) = −2 < 0, this is a local and global maximum.
Next, use the Lagrange method. The Lagrangian is L = xy + λ (1 − x − y ). The
first-order conditions are
Lx = y − λ = 0,
Ly = x − λ = 0,
Lλ = 1 − x − y = 0.
Solving simultaneously, x = y. Using the constraint gives x* = y* = 0.5, λ = 0.5, and
x* y* = 0.25.
2.4
Setting up the Lagrangian, L = x + y + λ (0.25 − xy). The first-order conditions are
Lx = 1 − λ y ,
L y = 1 − λ x,
Lλ = 0.25 − xy = 0.
So x = y. Using the constraint ( xy = x 2 = 0.25) gives x* = y * = 0.5 and λ = 2. Note that
the solution is the same here as in Problem 2.3, but here the value for the Lagrangian
multiplier is the reciprocal of the value in Problem 2.3.
2.5
a.
The height of the ball is given by f (t ) = −0.5 gt 2 + 40t. The value of t for which
height is maximized is found by using the first-order condition: df dt = − gt + 40 = 0, implying
t * = 40 g .
b.
Substituting for t * ,
2
 40 
 40  800
f (t ) = −0.5 g   + 40   =
.
g
 g 
 g 
*
Hence,
df (t * )
800
=− 2 .
dg
g
c.
Differentiation of the original function at its optimal value yields
df (t * )
= −0.5(t * )2 .
dg
Because the optimal value of t depends on g ,
Chapter 2: Mathematics for Microeconomics
2
 40 
df (t * )
−800
= − 0.5(t * ) 2 = −0.5   =
,
2
dg
g
 g 
as was also shown in part (c).
d.
If g = 32, t * = 5 4. Maximum height is 800 32 = 25. If g = 32.1, maximum
height is 800 32.1 = 24.92, a reduction of 0.08. This could have been predicted
from the envelope theorem, since
 −800 
 −25 
df (t * ) =  2  dg = 
 (0.01) ≈ −0.08.
 32 
 32 
2.6
a.
This is the volume of a rectangular solid made from a piece of metal, which is x
by 3x with the defined corner squares removed.
b.
The first-order condition for maximum volume is given by
∂V
= 3x 2 − 16 xt + 12t 2 = 0.
∂t
Applying the quadratic formula to this expression yields
16 x ± 256 x 2 − 144 x 2 16 x ± 10.6 x
=
= 0.225 x.
24
24
The second value given by the quadratic (1.11x) is obviously extraneous.
t=
c.
If t = 0.225 x, V ≈ 0.67 x3 − 0.04 x3 + 0.05 x3 ≈ 0.68 x3.
So volume increases without limit.
d.
This would require a solution using the Lagrangian method. The optimal solution
requires solving three nonlinear simultaneous equations, a task not undertaken
here. But it seems clear that the solution would involve a different relationship
between t and x than in parts (a–c).
2.7
a.
Set up the Lagrangian: L = x1 + 5ln x2 + λ (k − x1 − x2 ). The firstorder conditions are
Lx1 = 1 − λ = 0,
5
− λ = 0,
x2
Lλ = k − x1 − x2 = 0.
Lx2 =
Hence, λ = 1 = 5 x2 . With k = 10, the optimal solution is x1* = x2* = 5.
b.
With k = 4, solving the first-order conditions yields x1* = −1 and x2* = 5.
3
4
Chapter 2: Mathematics for Microeconomics
c.
If all variables must be nonnegative, it is clear that any positive value
for x1 reduces y. Hence, the optimal solution is x1* = 0, x2* = 4, and y* = 5ln 4.
d.
If k = 20, optimal solution is x1* = 15, x2* = 5, and y* = 15 + 5ln 5. Because
x2 provides a diminishing marginal increment to y as its value increases,
whereas x1 does not, all optimal solutions require that once x2 reaches 5, any
extra amounts be devoted entirely to x1 . In consumer theory, this function can be
used to illustrate how diminishing marginal usefulness can be modeled in a very
simple setting.
2.8
a.
Because MC is the derivative of TC , TC is an antiderivative of
MC . By the fundamental theorem of calculus,
q
∫ MC ( x) dx = TC (q) − TC (0),
0
where TC (0) is the fixed cost, which we will denote TC (0) = K for short.
Rearranging,
q
TC (q ) = ∫ MC ( x) dx + K
0
q
= ∫ ( x + 1)dx + K
0
x =q
 x2

=  + x + K
 2
 x =0
=
b.
q2
+ q + K.
2
For profit maximization, p = MC (q) = q + 1, implying q = p − 1. But p = 15
implies q = 14. Profit are
TR − TC = pq − TC (q)
 142

= 15 ⋅14 − 
+ 14 + K 
 2

= 98 − K .
If the firm is just breaking even, profit equals 0, implying fixed cost is K = 98.
c.
When p = 20 and q = 19, follow the same steps as in part (b), substituting fixed
cost K = 98. Profit are
Chapter 2: Mathematics for Microeconomics
5
TR − TC = pq − TC (q)
 192

= 20 ⋅19 − 
+ 19 + K 
 2

= 180.5 − 98
= 82.5.
d.
Assuming profit maximization, we have
π ( p) = pq − TC (q)
 ( p − 1) 2

= p( p − 1) − 
+ ( p − 1) + 98
 2

2
( p − 1)
=
− 98.
2
e.
i.
Using the above equation, π ( p = 20) − π ( p = 15) = 82.5 − 0 = 82.5.
ii.
The envelope theorem states that d π dp = q* ( p ). That is, the derivative of
the profit function yields this firm’s supply function. Integrating over p shows
the change in profits by the fundamental theorem of calculus:
20
dπ
π (20) − π (15) = ∫
dp
dp
15
20
= ∫ ( p − 1)dp
15
p = 20
 p2

=
− p
 2
 p =15
= 180 − 97.5
= 82.5.
Analytical Problems
2.9
Concave and quasi-concave functions
The proof is most easily accomplished through the use of the matrix algebra of quadratic forms.
See, for example, Mas Colell et al.,1995, pp. 937–939. Intuitively, because concave functions lie
below any tangent plane, their level curves must also be convex. But the converse is not true.
Quasi-concave functions may exhibit “increasing returns to scale”; even though their level
curves are convex, they may rise above the tangent plane when all variables are increased
together.
6
Chapter 2: Mathematics for Microeconomics
A counter example would be the Cobb–Douglas function, which is always quasiconcave, but convex when α + β > 1.
2.10
The Cobb–Douglas function
a.
f1 = α x1α −1 x2β > 0,
f 2 = β x1α x2β −1 > 0,
f11 = α (α − 1) x1α − 2 x1β < 0,
f 22 = β ( β − 1) x1α x2β − 2 < 0,
f12 = f 21 = αβ x1α −1 x2β −1 > 0.
Clearly, all the terms in Equation 2.114 are negative.
b.
A contour line is found by setting the function equal to a constant: y = c = x1α x2β ,
implying x2 = c1 β x1−α β . Hence,
dx2
< 0.
dx1
Further,
d 2 x2
< 0,
dx12
implying the countour line is convex.
c.
2.11
Using Equation 2.98, f11 f 22 − f122 = αβ (1 − β − α ) x12α −2 x22 β − 2 , which is negative for
α + β > 1.
The power function
a.
Since y′ > 0 and y′′ < 0, the function is concave.
b.
Because f11 , f 22 < 0 and f12 = f 21 = 0, Equation 2.98 is satisfied, and the function
is concave. Because f1 , f 2 > 0, Equation 2.114 is also satisfied, so the function is
quasi-concave.
c.
y is quasi-concave as is yγ . However, y is not concave for γδ > 1. This can be
shown most easily by f (2 x1 , 2 x2 ) = 2γδ f ( x1 , x2 ).
2.12
Proof of envelope theorem
Chapter 2: Mathematics for Microeconomics
a.
7
The Lagrangian for this problem is
L ( x1 , x2 , a) = f ( x1 , x2 , a ) + λ g ( x1 , x2 , a ).
The first-order conditions are
L1 = f1 + λ g1 = 0,
L2 = f 2 + λ g 2 = 0,
Lλ = g = 0.
b., c. Multiplication of each first-order condition by the appropriate deriviative yields
dx
dx
dx 
 dx
f1 1 + f 2 2 + λ  g1 1 + g2 2  = 0.
da
da
da 
 da
d.
The optimal value of f is given by f ( x1 (a ), x2 ( a ), a ) . Differentiation of this
with respect to a shows how this optimal value changes with a :
df *
dx
dx
= f1 1 + f 2 2 + f a .
da
da
da
e.
Differentiation of the constraint g ( x1 ( a ), x2 (a ), a ) = 0 yields
dg
dx
dx
= 0 = g1 1 + g 2 2 + g a .
da
da
da
2.13
f.
Multiplying the results from part (e) by λ and using parts (b) and (c) yields
df *
= f a + λ g a = La .
da
This proves the envelope theorem.
g.
In Example 2.8, we showed that λ = P 8. This shows how much an extra unit of
perimeter would raise the enclosed area. Direct differentiation of the original
Lagrangian shows also that
dA*
= LP = λ .
dP
This shows that the Lagrange multiplier does indeed show this incremental gain in
this problem.
Taylor approximations
a.
A function in one variable is concave if f ′′( x) < 0. Using the quadratic Taylor
formula to approximate this function at point a :
f ( x) ≈ f (a ) + f ′(a)( x − a) + 0.5 f ′′(a )( x − a) 2
≤ f (a ) + f ′(a )( x − a ).
The inequality holds because f ′′(a) < 0. But the right-hand side of this equation is
8
Chapter 2: Mathematics for Microeconomics
the equation for the tangent to the function at point a. So we have shown that any
concave function must lie on or below the tangent to the function at that point.
b.
A function in two variables is concave if f11 f 22 − f122 > 0.
Hence, the quadratic form ( f11dx 2 + 2 f12 dx dy + f 22 dy 2 ) will also be
negative. But this says that the final portion of the Taylor expansion will be
negative (by setting dx = x − a and dy = y − b ), and hence the function will be
below its tangent plane.
2.14
More on expected value
a.
The tangent to g ( x) at the point E ( x) will have the form c + dx ≥ g ( x) for all
values of x and c + dE ( x) = g ( E ( x)). But, because the line c + dx is above the
function g ( x) , we know
E ( g ( x)) ≤ E (c + dx) = c + dE ( x) = g ( E ( x)).
This proves Jensen’s inequality.
b.
Use the same procedure as in part (a), but reverse the inequalities.
c.
Let u = 1 − F ( x), du = − f ( x), x = v, and dx = dv.
∞
∫ [1 − F ( x)] dx = [(1 − F ( x)) x ]
0
x =∞
x =0
∞
− ∫ [ − f ( x) ] xdx
0
= 0 + E ( x)
= E ( x).
d.
Use the hint to break up the integral defining expected value:
t
∞

E ( x) −1 
= t  ∫xf ( x)dx + ∫xf ( x)dx 
t
t
0

∞
≥ t −1 ∫xf ( x)dx
t
∞
≥ t −1 ∫tf ( x)dx
t
∞
= ∫ f ( x)dx
t
= P( x ≥ t ).
e.
1.
Show that this function integrates to 1:
Chapter 2: Mathematics for Microeconomics
∞
∞
−∞
1
−3
−2
∫ f ( x)dx = ∫ 2 x dx = − x
2.
x =∞
x =1
9
= 1.
Calculate the cumulative distribution function:
x
F ( x) = ∫ 2t −3dt = −t −2
t=x
= 1 − x −2 .
t =1
1
3.
Using the result from part (c):
∞
∞
1
1
E ( x) = ∫ [1 − F ( x)] dx = ∫x −2 dx = − x −1
4.
1.
x =1
= 1.
To show Markov’s inequality use
P ( x ≥ t ) = 1 − F (t ) = t −2 < t −1 =
f.
x =∞
E ( x)
.
t
Show that the PDF integrates to 1:
x =2
x2
x3
8  1
dx
=
= −  −  = 1.
∫−1 3
9 x =−1 9  9 
2
2.
Calculate the expected value:
x =2
2
x3
x4
15 5
E ( x) = ∫ dx =
=
= .
12 x =−1 12 4
−1 3
3.
Calculate P(−1 ≤ x ≤ 0 ):
x =0
0
x2
x3
1
dx
=
= .
∫−1 3
9 x =−1 9
4.
All we must do is adjust the PDF so that it now sums to 1 over the new,
smaller interval. Since P ( A) = 8 9,
f ( x | A) =
5.
f ( x) 3 x 2
=
defined on 0 ≤ x ≤ 2.
89
8
The expected value is again found through integration:
2
6.
2.15
x =2
3x3
3x 4
3
E ( x | A) = ∫
dx =
= .
8
32 x =0 2
0
Eliminating the lowest values of x increases the expected value of the
remaining values.
More on variances
a.
This is just an application of the definition of variance:
10
Chapter 2: Mathematics for Microeconomics
Var( x) = E [ x − E ( x)]
2
= E  x 2 − 2 xE ( x) + [ E ( x)]2 
= E ( x 2 ) − 2[ E ( x)]2 + [ E ( x)]2
= E ( x 2 ) − [ E ( x)]2 .
b.
Here, we let y = x − µ x and apply Markov’s inequality to y and remember that x
can only take on positive values.
E ( y 2 ) σ x2
P( y ≥ k ) = P( y 2 ≥ k 2 ) ≤
= 2.
k2
k
c.
Let xi , i = 1,…, n be n independent random variables each with expected value
µ and variance σ 2 .
 n 
E  ∑xi  = µ + L + µ = nµ .
 i=1 
 n 
Var  ∑xi  = σ 2 + L + σ 2 = nσ 2 .
 i=1 
n
Now, let x = ∑ i =1 ( xi n ).
nµ
= µ.
n
nσ 2 σ 2
Var( x ) = 2 =
.
n
n
E( x ) =
d.
Let X = kx1 + (1 − k ) x2 and E ( X ) = k µ + (1 − k ) µ = µ.
Var( X ) = k 2σ 2 + (1 − k ) 2 σ 2 = (2k 2 − 2k + 1)σ 2 .
dVar( X )
= (4k − 2)σ 2 = 0.
dk
Hence, variance is minimized for k = 0.5. In this case, Var( X ) = 0.5σ 2 . If
k = 0.7, Var( X ) = 0.58σ 2 (not much of an increase).
e.
Suppose that Var( x1 ) = σ 2 and Var( x2 ) = rσ 2 . Now
Var( X ) = k 2σ 2 + (1 − k )2 rσ 2 = (1 + r )k 2 − 2kr + r  σ 2 .
dVar( X )
= [ 2(1 + r )k − 2r ]σ 2 = 0.
dk
r
k=
.
1+ r
For example, if r = 2, then k = 2 3 , and optimal diversification requires that the
lower risk asset constitute two-thirds of the portfolio. Note, however, that it is still
Chapter 2: Mathematics for Microeconomics
11
optimal to have some of the higher risk asset because asset returns are
independent.
2.16
More on covariances
a.
This is a direct result of the definition of covariance:
Cov( x, y ) = E [ ( x − E ( x))( y − E ( y )) ]
= E[ xy − xE ( y ) − yE ( x) + E ( x) E ( y )]
= E ( xy ) − E ( x) E ( y ) − E ( y ) E ( x) + E ( x) E ( y )
= E ( xy ) − E ( x) E ( y ).
b.
Var(ax ± by ) = E[(ax ± by ) 2 ] − [ E (ax ± by )]2
= a 2 E ( x 2 ) ± 2abE ( xy ) + b 2 E ( y 2 ) − a 2 [ E ( x)]2
± 2abE( x)E( y ) − b 2 [E( y )]2
= a 2 Var( x) + b 2 Var( y ) ± 2abCov( x, y ).
The final line is a result of Problems 2.15a and 2.16a.
c.
The presence of the covariance term in the result of Problem 2.16b suggests that
the results would differ. In the two-variable case, however, this is not necessarily
the situation. For example, suppose that x and y are identically distributed and that
Cov( x, y ) = rσ 2 . Using the prior notation,
Var( X ) = k 2σ 2 + (1 − k ) 2 σ 2 + 2k (1 − k )rσ 2 .
The first-order condition for a minimum is
(4k − 2 + 2r − 4rk )σ 2 = 0,
implying
2 − 2r
k* =
= 0.5.
4 − 4r
Regardless of the value of r. With more than two random variables, however,
covariances may indeed affect optimal weightings.
d.
If x1 = kx2 , the correlation coefficient will be either + 1 (if k is positive) or − 1 (if
k is negative), since k will factor out of the definition leaving only the ratio of
the common variance of the two variables. With less than a perfect linear
0.5
relationship | Cov( x, y ) |< [ Var( x)Var( y ) ] .
e.
If y = α + β x,
Cov( x, y ) = E [ ( x − E ( x))( y − E ( y ))]
= E[( x − E ( x))(α + β x − α − β E ( x))]
= β Var( x).
12
Chapter 2: Mathematics for Microeconomics
Hence,
β=
Cov( x, y )
.
Var( x)
13
CHAPTER 3:
Preferences and Utility
These problems provide some practice in examining utility functions by looking at
indifference curve maps and at a few functional forms. The primary focus is on
illustrating the notion of quasi-concavity (a diminishing MRS) in various contexts. The
concepts of the budget constraint and utility maximization are not used until the next
chapter.
Comments on Problems
3.1
This problem requires students to graph indifference curves for a variety of
functions, some of which are not quasi-concave.
3.2
This problem introduces the formal definition of quasi-concavity (from Chapter
2) to be applied to the functions studied graphically in Problem 3.1.
3.3
This problem shows that diminishing marginal utility is not required to obtain a
diminishing MRS. All of the functions are monotonic transformations of one
another, so this problem illustrates that diminishing MRS is preserved by
monotonic transformations but diminishing marginal utility is not.
3.4
This problem focuses on whether some simple utility functions exhibit convex
indifference curves.
3.5
This problem is an exploration of the fixed-proportions utility function. The
problem also shows how the goods in such problems can be treated as a
composite commodity.
3.6
This problem asks students to use their imaginations to explain how advertising
slogans might be captured in the form of a utility function.
3.7
This problem shows how utility functions can be inferred from MRS segments. It
is a very simple example of “integrability.”
3.8
This problem offers some practice in deriving utility functions from indifference
curve specifications.
13
14
Chapter 3: Preferences and Utility
Analytical Problems
3.9
Initial endowments. This problem shows how initial endowments can be treated
in simple indifference curve analysis.
3.10
Cobb–Douglas utility. This problem provides some exercises with the Cobb–
Douglas function, including how to integrate subsistence levels of consumption
into the functional form.
3.11
Independent marginal utilities. This problem shows how analysis can be
simplified if the cross-partials of the utility function are zero.
3.12
CES utility. This problem shows how distributional weights can be incorporated
into the CES form introduced in the chapter without changing the basic
conclusions about the function.
3.13
The quasi-linear function. This problem provides a brief introduction to the
quasi-linear form, which (in later chapters) will be used to illustrate a number of
interesting outcomes.
3.14
Preference relations. This problem provides a very brief introduction to how
preferences can be treated formally with set-theoretic concepts.
3.15
The benefit function. This problem introduces Luenberger’s notion of reducing
preferences to a cardinal number of replications of a basic bundle of goods.
Solutions
3.1
Here we calculate the MRS for each of these functions:
a.
MRS =
Ux 3
= . MRS is constant.
Uy 1
b.
MRS =
U x 0.5( y x ) 0.5
y
=
= . Convex; MRS is diminishing.
−0.5
U y 0.5( y x )
x
c.
MRS =
U x 0.5 x −0.5
=
. MRS is diminishing.
Uy
1
d.
MRS =
Ux
2x
x
= 0.5( x 2 − y 2 ) −0.5 ⋅
⋅ 2 y = . MRS is increasing.
2
2 −0.5
Uy
0.5( x − y )
y
Chapter 3: Preferences and Utility
e.
3.2
3.3
MRS =
15
U x ( y ( x + y ) − xy ) ( x + y ) 2 y 2
=
= . Convex; MRS is diminishing.
U y ( x ( x + y ) − xy ) ( x + y ) 2 x 2
Because all of the first-order partials are positive, we must only check the secondorder partials.
a.
U xx = U yy = U xy = 0. Not strictly quasi-concave.
b.
U XX , U yy < 0, U xy > 0. Strictly quasi-concave.
c.
U xx < 0, U yy = 0, U xy = 0. Strictly quasi-concave.
d.
Even if we only consider cases where x ≥ y, both of the second-order
partials are ambiguous, and therefore the function is not necessarily
strictly quasi-concave.
e.
U xx , U yy < 0, U xy > 0. Strictly quasi-concave.
a.
U x = y, U xx = 0, U y = x, U yy = 0, MRS = y x .
b.
U x = 2 xy 2 , U xx = 2 y 2 > 0, U y = 2 x 2 y, U yy = 2 x 2 > 0, MRS = y x .
c.
U x = 1 x , U xx = −1 x < 0, U y = 1 y , U yy = − 1 y 2 < 0, MRS = y x . This
shows that monotonic transformations may affect diminishing marginal
utility, but not the MRS .
3.4
a.
In the range in which the same good is limiting, the
indifference curve is linear. To see this, take the case in which both x1 ≤ y1
and x2 ≤ y2 . Then k = U ( x1 , y1 ) = x1 = U ( x2 , y2 ) = x2 , implying
 x + x y + y2  x1 + x2 k + k
U 1 2, 1
=
=k
=
2 
2
2
 2
as well.
In the range in which the limiting goods differ, we can show the
indifference curve is strictly convex. Take the case k = x1 < y1 and
k = y2 < x2 . Then ( x1 + x2 ) 2 > k and ( y1 + y2 ) 2 > k , implying
 x + x y + y2 
U 1 2, 1
 > k.
2 
 2
16
Chapter 3: Preferences and Utility
Hence, the indifference curve is convex.
b.
Again, in the range in which the same good is maximum, the indifference
curve can be shown to be linear. Consider a range in which different goods
are maximum, specifically, k = x1 > y1 and k = y2 > x2 . Then
( x1 + x2 ) 2 < k and ( y1 + y2 ) 2 < k , implying
 x + x y + y2 
U 1 2, 1
 < k.
2 
 2
Hence, the indifference curve is concave.
c.
Here,
 x + x y + y2 
( x1 + y1 ) = k = ( x2 + y2 ) = U  1 2 , 1
.
2 
 2
Hence, the indifference curve is linear.
3.5
a.
All four are perfect compliments, U (h, b, m, r ) = min(h, 2b, m, 0.5r ).
b.
A fully condimented hot dog.
c.
$1.60.
d.
$2.10, an increase of 31 percent.
e.
Price would increase only to $1.725, an increase of 7.8 percent.
f.
Raise prices so that a fully condimented hot dog rises in price to $2.60.
This could be accomplished by raising all prices by 62.5 percent. But,
because of the fixed proportions nature of the utility function, it could also
be accomplished by any combination of increases in single good prices.
For example, raising the price of the hot dog to $2 would accomplish this
goal. Or, one could increase the hot dog price to $1.50 and the bun price to
$1.50 and accomplish the same thing. Because of the fixed proportions
Chapter 3: Preferences and Utility
17
utility function, all such increases would be equivalent to a lump-sum
reduction in purchasing power of about 62 percent.
3.6
For all the suggested utility functions, let x represent some other good and the
good in question is represented by the appropriate letter:
a.
U( x , p) ≥ U( x , b) for p = b.
b.
Given U( x , c ), U xc = Ucx > 0 .
c.
Given U( x , p), U( x ,1) < U( x ,0) < U( x , p > 1).
d.
U( x , kk ) > U( x , dd ) for kk = dd .
e.
U ( x , m) < U ( x , mresponsible ) for m > mresponsible .
3.7
a.
MRS = 1 3 at both points. Since both the points lie on the
same indifference curve (as the utility at both points is the same), the slope
of the indifference curve is constant (i.e., straight line). So the goods are
perfect substitutes.
b.
We know that for a Cobb–Douglas utility function,
α y
MRS =
.
β x
Using this formula, yields:
1 α y α
= ⋅ =
→ α = 2β
4 β x 8β
Now use the fact that the two points yield equal utility:
(8)2β (1)β = (4)2β (4)β →β = 1, α = 2 .
The utility function is of the form U = x 2 y.
c.
Yes, there was a redundancy. We never used the information about the
second MRS. In fact, given that the function is assumed to be Cobb–
Douglas, only the information about the first MRS was needed to get the
ratio of the exponents. Since utility is invariant up to a monotonic
transformation, any Cobb–Douglas for which α = 2β would yield the
same behavior. For example, if the exponents sum to one, we have
α = 2/3, β = 1/3 and this function also satisfies the conditions of the
18
Chapter 3: Preferences and Utility
problem.
3.8
a.
Exponentiate the function: U = xα y β zδ .
b.
Move the term in x to the LHS: U = x 2 + xy + y 2 .
c.
Multiply by 2x, move y2 to the LHS, square, and simplify:
U = x 2 y + y 2 z + z 2 x.
Chapter 3: Preferences and Utility
19
Analytical Problems:
3.9
Initial endowments
a.
3.10
b.
Any trading opportunities that differ from the MRS at x , y will provide
the opportunity to raise utility (see figure).
c.
A preference for the initial endowment will require that trading
opportunities raise utility substantially. This will be more likely if the
trading opportunities are significantly different from the initial MRS (see
figure).
Cobb–Douglas utility
a.
∂U ∂x α xα −1 y β α y
=
= ⋅ .
∂U ∂y β xα y β −1 β x
This result does not depend on the sum α + β , which, contrary to
production theory, has no significance in consumer theory because utility
is unique only up to a monotonic transformation.
MRS =
b.
The mathematics follow directly from part (a). If α > β , the individual
values x relatively more highly. Hence, MRS > 1 for x = y.
c.
The function is homothetic in x − x0 and y − y0 , but not in x and y.
20
3.11
Chapter 3: Preferences and Utility
Independent marginal utilities
From Problem 3.2, U xy = U yx = 0 implies diminishing MRS providing
U xx , U yy < 0. Conversely, the Cobb–Douglas not only has U xy > 0 and
U xx , U yy < 0 , but also has a diminishing MRS (see Problem 3.10a).
3.12
CES utility with weights
1−δ
a.
b.
∂U ∂x α xδ −1 α  y 
MRS =
=
=  
∂U ∂y β yδ −1 β  x 
, so this function is homothetic.
If δ = 1, MRS = α β , a constant. If δ = 0,
α y
MRS = ⋅ ,
β x
This agrees with Problem 3.10.
c.
∂MRS
α
= (δ − 1) y1−δ xδ − 2 . This is negative if and only if δ < 1.
∂x
β
d.
Follows from part (a). If x = y , MRS = α β .
e.
With δ = 0.5,
α
α
(0.9)0.5 = 0.949 ,
β
β
α
α
MRS (1.1) = (1.1)0.5 = 1.05 .
β
β
With δ = −1,
α
α
MRS (0.9) = (0.9)2 = 0.81 ,
β
β
α
α
MRS (1.1) = (1.1)2 = 1.21 .
β
β
Hence, the MRS changes more dramatically when δ = −1
than when δ = 0.5 . The indifference curves are more sharply curved when
δ is lower. When δ = −∞ , the indifference curves are L-shaped, implying
fixed proportions.
MRS (0.9) =
3.13
The quasi-linear function
a.
MRS = y. The MRS depends only on the amount of y. It is independent of
x.
Chapter 3: Preferences and Utility
b.
21
Check U xxU y 2 − 2U xyU xU y + U yyU x 2 < 0. We have
U x = 1,
1
Uy = ,
y
U xx = 0,
U yy = −
1
,
y2
U xy = 0.
So,
U xxU y 2 − 2U xyU xU y + U yyU x 2 = 0 + 0 −
3.14
1
1
= − 2 < 0.
2
y
y
c.
y = e k −x , where k is the utility level.
d.
Since the marginal utility of x is a constant at 1 while that of y is
decreasing as y increases (as it is of the form 1 y ), we would expect
consumers to shift more toward x when they buy more of both goods. We
explore this in much more detail in the next chapter.
e.
Refer to Example 3.4. This function is usually used to describe the
consumption of one commodity with respect to all other commodities. So,
ln y could represent the commodity of interest, while x could represent
all the other goods consumed.
Preference relations
All of the suggested preference relations are complete, transitive, and continuous.
a.
Summation:
Complete: Clearly all bundles are ranked by the sum of items
contained.
Transitive: If bundle A has more items than B and B has more
items than C, clearly A has more items than C.
Continuous: If bundle A contains more items than bundle B, then
A is preferred to B and any bundle with slightly more items than B
(but fewer than A) is also preferred to B.
b.
Lexicographic:
Complete: All bundles can be ranked in this ordered way.
Transitive: If bundle A is preferred to bundle B with ties being
22
Chapter 3: Preferences and Utility
broken at the ith good and B is preferred to C with ties broken at
the (i + j)th good, then A will be preferred to C because it will
break the tie at the ith good also.
Continuous: Suppose bundle A is preferred to B with the tie break
occurring at the ith good. Then there exists a bundle C with
slightly more of this good than B but less than A, which will be
preferred to B. Note, however, that the idea of “closeness” here is
being defined with respect to the first tie-break good only. The
ranking is not continuous when more general notions of
“closeness” are used.
c.
Bliss
Complete: Clearly all bundles are ranked by the distance metric.
Transitive: The distance metric itself imposes a cardinal ranking,
which is clearly transitive.
Continuous: If bundle A is any positive distance from bliss, there
will exist another bundle slightly closer since any single good that
is not at bliss can be made closer to it.
3.15
The benefit function
a.
U * = x1β y11−β = α β α1−β = α , hence b(U * ) = U * .
b.
In this case, the benefit function cannot be computed because the Cobb–
Douglas requires positive quantities of both goods to take a nonzero value.
c.
In the graph below, the benefit associated with any initial endowment is
the length of the vector from the initial endowment to the utility target
where the direction of the vector is given by the composition of the
elementary bundle.
d.
In the graph below, two initial endowments are shown (E1 and E2 ) . The
benefit for each endowment is also shown by the vectors in the graph. The
benefit is also shown for an initial endowment given by (E1 + E2 ) 2 . By
completing the parallelogram, it is clear that the convexity of the
indifference curve implies that
b(U * , E1 ) > b(U * , E1 + E 2 2) < b(U * , E 2 ).
Hence the benefit function is concave in the initial endowments.
Chapter 3: Preferences and Utility
23
y
E1
U
y0
E2
x0
*
x
CHAPTER 4:
Utility Maximization and Choice
The problems in this chapter focus mainly on the utility maximization assumption. Relatively
simple computational problems (mainly based on Cobb–Douglas and CES utility functions)
are included. Comparative statics exercises are included in a few problems, but for the most
part, introduction of this material is delayed until Chapters 5 and 6.
Comments on Problems
4.1
This problem is a simple Cobb–Douglas example. Part (b) asks students to compute
income compensation for a price rise and may prove difficult for them. As a hint,
they might be told to find the correct bundle on the original indifference curve first,
and then compute its cost.
4.2
This problem uses the Cobb–Douglas utility function to solve for quantity demanded
at two different prices. Instructors may wish to introduce the expenditure shares
interpretation of the function’s exponents (these are covered extensively in the
Extensions to Chapter 4 and in a variety of numerical examples in Chapter 5).
4.3
This problem starts as an unconstrained maximization problem—there is no income
constraint in part (a) on the assumption that this constraint is not limiting. In part (b),
there is a total quantity constraint. Students should be asked to interpret what
Lagrangian multiplier means in this case.
4.4
This problem shows that with concave indifference curves, first-order conditions do
not ensure a local maximum.
4.5
This problem is an example of a “fixed proportion” utility function. The problem
might be used to illustrate the notion of perfect complements and the absence of
relative price effects for them. Students may need some help with the min ( )
functional notation by using illustrative numerical values for v and g and showing
what it means to have “excess” v or g.
4.6
This problem introduces a third good to the Cobb–Douglas case for which optimal
consumption is zero until income reaches a certain level.
4.7
This problem repeats the lessons of the lump-sum principle for the case of a subsidy.
Numerical examples are based on the Cobb–Douglas expenditure function.
4.8
This problem uses two very simple utility functions to show how all of the major
functions derived from them can be stated in simple forms. This also illustrates how
the indirect utility functions (of prices and incomes) often have forms that are mirror
images of the underlying utility functions.
23
24
Chapter 4: Utility Maximization and Choice
4.9
This problem asks students to construct the expenditure function for a linear utility
function. Notice that this problem cannot be solved with calculus—rather students
must work through the various possibilities logically.
Analytical Problems
4.10
Cobb–Douglas utility. This problem provides a simple example of the Cobb–
Douglas expenditure function and seeks to build some intuition about how a good’s
relative importance affects that function.
4.11
CES utility. This problem provides some practice with the CES function. Parts (a–c)
are relatively straightforward but part (d) is computationally difficult. A somewhat
different form for this function is examined in Problem 4.13.
4.12
Stone–Geary utility. This problem introduces a simple two-good Stone–Geary
function in which a certain amount must be devoted to x consumption before any y
consumption occurs. More detail on this functional form is provided in the Extensions
to the chapter.
4.13
CES indirect utility and expenditure functions. This problem uses a more standard
form for the CES utility function and asks students to delve more deeply into the
characteristics of that function’s indirect utility function and expenditure function
analogs.
4.14
Altruism. This problem shows a simple way in which altruism can be incorporated
into a standard Cobb–Douglas utility function.
Solutions
4.1
a.
To find maximum utility given a fixed budget, set up the Lagrangian:
L = ts + λ (1.00 − 0.10t − 0.25s ).
The first-order conditions are
0.5
dL  s 
=   − 0.10λ = 0,
dt  t 
0.5
dL  t 
=   − 0.25λ = 0,
ds  s 
dL
= 1.00 − 0.10t − 0.25s = 0.
dλ
The ratio of first two equations implies t s = 2.5. Hence t = 2.5s and
1.00 = 0.10t + 0.25s = 0.50 s. Solving the equation yields s* = 2 and t * = 5.
Substituting into the objective function results in a utility of 10.
b.
First, find the utility maximizing conditions with the new ratio of prices.
Chapter 4: Utility Maximization and Choice
25
t 0.25 5
=
= ,
s 0.40 8
implying
5s
t= .
8
Substituting into the utility function yields
5s 2
= 10,
8
implying s* = 4 and t * = 2.5. Paul needs another dollar because the bundle
costs $2.
4.2
Use a simpler notation for this solution: U ( f , c) = f 2 3c1 3 and I = 600.
a.
Setting up the Lagrangian:
L = f 2 3c1 3 + λ (600 − 40 f − 8c).
The first-order conditions are
13
2 c 
L f =   − 40λ = 0,
3 f 
23
1 f 
Lc =   − 8λ = 0.
3 c 
Hence, 5 = 2c f , implying 5 f = 2c. Substituting into budget constraint
yields f * = 10 and c* = 25.
b.
With the new constraint, f * = 20 and c* = 25.
Note: This person always spends 2 3 of income on f and 1 3 on c.
Consumption of California wine does not change when the price of French
wine changes. These are two unique characteristics of the Cobb–Douglas
function and would not be expected to hold for more complicated forms of the
utility function.
c.
In part (a),
U ( f , c) = f 2/3c1 3 = 102/3251 3 = 13.5.
In part (b),
U ( f , c) = 202/3251 3 = 21.5.
This person will need more income to achieve the part (b) utility with the part
(a) prices. Setting the value of indirect utility to the utility level in part (b):
23
13
 2 1
21.5 =     I p −f 2 3 pc−1 3
 3 3
23
13
 2 1
=     I (40) −2 3 (8) −1 3 .
 3 3
26
Chapter 4: Utility Maximization and Choice
Solving this equation for the required income gives I = 964. With such an
income, this person would purchase f = 16.1 and c = 40.2, and, by
construction, would obtain utility of U = 21.5.
4.3
Given U (c, b) = 20c − c 2 + 18b − 3b2 .
a.
The first-order conditions are
∂U
= 20 − 2c = 0,
∂c
∂U
= 18 − 6b = 0.
∂b
Solving, c* = 10, b* = 3, and U * = 127.
b.
The constraint is b + c = 5. Set up the Lagrangian:
L = 20c − c2 + 18b − 3b2 + λ (5 − c − b).
The first-order conditions are
Lc = 20 − 2c − λ = 0,
Lb = 18 − 6b − λ = 0,
Lλ = 5 − c − b = 0.
Solving the first two equations yields c = 3b + 1. So b + 3b + 1 = 5, implying
b* = 1, c* = 4, and U * = 79.
4.4
Given U ( x, y) = ( x 2 + y 2 )0.5 . Note that maximizing U 2 will also maximize U .
a.
The Lagrangian is
L = x 2 + y 2 + λ (50 − 3x − 4 y).
The first-order conditions are
Lx = 2 x − 3λ = 0,
L y = 2 y − 4λ = 0,
Lλ = 50 − 3 x − 4 y = 0.
The first two equations give y = 4 x 3. Substituting in budget constraint gives
x* = 6, y* = 8, and U * = 10.
b.
4.5
This is not a local maximum because the indifference curves do not have a
diminishing MRS; they are in fact concentric circles. Hence, we have
necessary but not sufficient conditions for a maximum. In fact, the calculated
allocation is a minimum utility. He would get a much higher utility if he
spends all of his income on x ( U * = 50 3 = 16.7 ).
Given U (m) = U ( g , v) = min ( g 2, v ) .
Chapter 4: Utility Maximization and Choice
27
a.
No matter what the relative prices are (i.e., the slope of the budget constraint),
the maximum utility intersection will always be at the vertex of an
indifference curve where g = 2v.
b.
Substituting g = 2v into the budget constraint yields 2 p g v = pv v = I , or
I
.
2 pg + pv
Furthermore,
2I
g=
.
2 pg + pv
It is easy to show that these two demand functions are homogeneous of degree
zero in p g , pv , and I .
v=
c.
Since U = g 2 = v, indirect utility is
I
V ( pg , pv , I ) =
.
2 pg + pv
d.
The expenditure function is found by interchanging I (= E ) and V ,
E ( p g , pv , V ) = (2 p g + pv )V .
4.6
a.If x = 4 and y = 1, then U ( z = 0) = 2. If z = 1, then U = 0 , since x = y = 0.
If z = 0.1, then x = 0.9 0.25 = 3.6 and y = 0.9. Hence, for z = 0,
maximization of utility results in the same optimal choices as in Example 4.1.
Any choice that results in z > 0 reduces utility from this optimum, since
U = (3.6)0.5 (0.9)0.5 (1.1)0.5 = 1.89, which is less than U ( z = 0).
b.
At x = 4, y = 1 , and z = 0.
MU x MU y
=
= 1.
px
py
MU z
= 0.5.
pz
So even at z = 0, the marginal utility from z is “not worth” the good’s price.
Notice here that the “1” in the utility function causes this individual to incur
some diminishing marginal utility for z before any of it is bought. Good z
illustrates the principle of “complementary slackness” discussed in Chapter 2.
c.
If I = 10, optimal choices are x = 16, y = 4, and z = 1. A higher income
makes it possible to consume z as part of a utility maximum. To find the
minimal income at which any (fractional) z would be bought, use the fact that
this person will spend equal amounts on x, y, and (1 + z ) with the Cobb–
Douglas:
p x x = p y y = p z (1 + z ).
Substituting this into the budget constraint yields
28
Chapter 4: Utility Maximization and Choice
2 pz (1 + z) + pz z = I ,
implying
3 pz z = I − 2 p z .
Hence, it must be the case that I > 2 pz or I > 4 for z > 0.
Chapter 4: Utility Maximization and Choice
4.7
29
a.
E ( px , p y , U ) = 2 px0.5 p y0.5U . With px = 1 and p y = 4, we have
U = 2 and E = 8. To raise utility to 3 would require E = 12, that is, an
income subsidy of 4.
b.
c.
Now we require E = 8 = 2 ⋅ p x0.5 ⋅ 4 0.5 ⋅ 3 or px0.5 = 8 12 = 2 3. So
px = 4 9; that is, each unit must be subsidized by 5 9. At the subsidized
price, this person chooses to buy x = 9. So total subsidy is 5, one dollar
greater than in part (c).
d.
E ( px , p y , U ) = 1.84 px0.3 p y0.7U . With px = 1 and p y = 4, we
have U = 2 and E = 9.71. Raising U to 3 would require extra expenditures
of 4.86. Subsidizing good x alone would require a price of px = 0.26, that is,
a subsidy of 0.74 per unit. With this low price, the person would choose
x = 11.2, so the total subsidy would be 8.29.
e.
In the fixed proportions case, an income grant and a price subsidy would cost
the same. If E = ( p x + 0.25 p y )U . If p x = 1, p y = 4,U = 4 then E = 8. To
increase utility to 5 would require E = 10 —an increase of 2. If good x were
subsidized, its price would have to fall to 0.6 to reach a utility level of 5 with
an expenditure of 8. In this case, consumption would be x = 5, y = 1.25 and
the cost of the subsidy would be 5 ⋅ (0.4) = 2
4.8
a.
If U ( x, y ) = min( x, y ) , utility maximization requires x = y . Substitution into
the budget constraint yields x = I ( p x + p y ) = y . Hence,
V ( p x , p y .I ) =
I
,
px + p y
E ( px , p y , V ) = ( px + p y )V .
If U ( x, y) = x + y , utility maximization requires the purchase of whichever of
30
Chapter 4: Utility Maximization and Choice
these two perfect substitutes has the lower price. So, if
p x > p y , x = 0, y = I p y . If p x < p y , x = I p x , y = 0. Given these results,
V ( px , p y .I ) =
I
,
min( px , p y )
E ( px , p y ,V ) = min( px , p y )V .
b.
4.9
It is interesting that the discontinuous utility function has continuous indirect
utility and expenditure functions, whereas the linear utility function has
discontinuous indirect utility and expenditure functions. Similar dualities
occur in many maximization problems.
Given U ( x, y ) = ax + by. There are two cases to consider. First, assume
a
b
>
,
px p y
implying
px a
(1)
> .
py b
Then x* = I p x and y* = 0. Second, assume
px a
(2)
< .
py b
Then y* = I p y and x* = 0. Hence, E = pxU a for condition (1) and E = p yU b for
condition (2). For the knife-edged case of equality,
px a
(3)
= ,
py b
we have E = p xU a = p yU b .
Analytical Problems:
4.10
Cobb–Douglas utility
a.
The demand functions in this case are
αI
x=
,
px
y=
(1 − α ) I
.
py
Substituting these into the utility function gives
α
 α I   (1 − α ) I 
−α − (1−α )
V ( px , p y , I ) = 
V,
 = BIp x p y
 
 px   p y 
where B = α α (1 − α )1−α .
Chapter 4: Utility Maximization and Choice
4.11
31
b.
Interchanging I and V yields E ( px , p y ,V ) = B −1 pαx p1y−αV .
c.
As for all exponential equations, the exponent α gives the elasticity of
expenditures with respect to px . That is, the more important x is in the utility
function, the greater the proportion that expenditures must be increased to
compensate for a proportional rise in the price of x.
CES utility
a.
For utility maximization,
∂U ∂x  x 
MRS =
= 
∂U ∂ y  y 
δ−1
=
px
.
py
Hence,
1
−σ
x  px  δ−1  px 
=  =  ,
p 
y  p y 
 y
where
σ=
b.
1
.
1− δ
If δ = 0,
x py
=
,
y px
implying p x x = p y y.
c.
Part (a) shows
1−σ
px x  px 
= 
p y y  p y 
.
Hence, for σ < 1, the relative share of income devoted to good x is positively
correlated with its relative price—a sign of low substitutability. For σ > 1, the
relative share of income devoted to good x is negatively correlated with its
relative price—a sign of high substitutability.
d.
The algebra is a bit tricky here, but worth doing once. Let’s solve for indirect
utility:
x  px 
= 
y  p y 
−σ
,
or
−σ
p 
x = y x  .
p 
 y
Substituting into the budget constraint yields
32
Chapter 4: Utility Maximization and Choice
yp1x−σ
+ p y y or Ip y−σ = y ( p1x−σ + p1y−σ )
−σ
py
I = px x + p y y =
or
y=
Ip −y σ
p1x−σ + p1y−σ
.
Similarly,
Ipx−σ
x = 1−σ
.
px + p1y−σ
Hence,
δ
δ




p y−σ
px−σ
δ
δ U = x + y = I  1−σ
+ I  1−σ
.

1
−
σ
1
−
σ
p +p 
 p + p 
y
y
 x

 x

Now −δσ = 1 − σ , so
δ
δ
δ


1
δ U = I δ  1−σ
,
 ( p + p1−σ )δ −1 
y
 x

or
−1
)
V = I ( p1x−σ + p1y−σ )1−σ ,
)
where V = (δU )1 δ . This is the indirect utility function. Clearly, it is
homogeneous of degree zero in income and prices. Inverting the expression
yields the expenditure function:
1
)
E = I = V ( p1x−σ + p1y−σ )1−σ .
Clearly, this is homogeneous of degree one in the prices. Note that the odd
)
form for V here suggests the use of the CES form given in Problem 4.13 in
applications involving these functions.
4.12
Stone–Geary utility
a.
For x < x0 , utility is negative, so the individual will spend px x0 first. With
I − px x0 remaining income, this is a standard Cobb–Douglas problem:
px ( x − x0 ) = α ( I − px x0 ),
py y = β ( I − px x0 ).
b.
Calculating budget shares from part (a) yields
px x
(1 − α ) px x0
=α +
,
I
I
py y
β px x0
=β−
.
I
I
Thus,
Chapter 4: Utility Maximization and Choice
px x
= α,
I →∞ I
p y
lim y = β .
I →∞ I
lim
4.13
CES indirect utility and expenditure functions
a.
Given U = ( xδ + y δ )1 δ . The Lagrangian is
L = ( xδ + yδ )1 δ + λ ( I − xpx − yp y ).
One first-order condition is
1−δ
1
Lx = ( x δ + y δ ) δ (δx δ−1 ) − λpx = 0,
δ
implying
λ=
δ
δ
(x + y )
px
1−δ
δ
xδ −1
.
Similarly,
1−δ
( xδ + yδ ) δ yδ −1
λ=
.
py
Equating the λ yields either
x δ−1 y δ−1
(1)
=
px
py
or
1−δ
(2) ( xδ + y δ ) δ = 0.
Since we assume U ≠ 0, equation (2) cannot be a solution. Therefore, the
solution satisfies (1), which upon rearranging gives
1
 p  δ −1
x = y x  .
 py 
 
Substituting this value of x into the budget constraint yields
1
 p  δ−1
I =  x  ypx + yp y .
p 
 y
Solving for y,
Ip1 ( δ−1)
y* = δ ( δ−1) y δ ( δ−1) ,
px
+ py
implying
Ip1 ( δ−1)
x* = δ ( δ−1) x δ ( δ−1) .
px
+ py
Further,
33
34
Chapter 4: Utility Maximization and Choice
V = ( xδ + y δ )1 δ
1δ
δ
δ

 
 
Ip1y ( δ−1)
Ip1x ( δ−1)
=  δ ( δ−1)
 +
 
+ p yδ ( δ−1)   pxδ ( δ−1) + p δy (δ−1)  
 px


1δ
 pr + pr 
= I  rx ry δ 
(p + p ) 
y
 x

= I( p + p )
r
x
y
r
1−δ
δ
= I ( pxr + p yr ) −1 r ,
where
r=
δ
.
δ −1
b.
Scale all variables by t and the function is unchanged.
c.
The partial derivative of V with respect to I is positive as the prices are
positive.
d.
Again, the partial derivatives of V with respect to the prices are both negative.
For example,
− (1+ r )
∂V
= − Ipxr −1 ( pxr + p yr ) r < 0.
∂px
e.
Simply reversing the positions of V and I in the indirect utility function
yields
E = V ( pxr + p yr )1 r .
f.
Multiplying prices by any factor t multiplies expenditures by t.
g.
For example,
1− r
∂E
= Vpxr −1 ( pxr + p ry ) r > 0.
∂px
h.
Differentiating the expression from part (g),
1− 2 r
1− r
∂2 E
2r −2
r −2
r
= (1 − r )Vpx K
+ (r − 1)Vpx K r ,
∂px2
1− r
where K = pxr + p yr . Division of this expression by Vp xr − 2 k r
yields
(r − 1)(1 − k ) < 0, where k = px K < 1.
r
4.14
Altruism
a.
When
= 0,
=
, so Michele is completely self-interested. When
= 1,
Chapter 4: Utility Maximization and Choice
35
= , so she cares only about others, not herself. Definitions of a “perfect
altruist” may vary. According to the “Golden Rule” standard (“Regard others
as you would have them regard you”), Michele would have a symmetric
regard for the two consumption levels, corresponding to = 1/2.
b.
The choice problem is to maximize
subject to the budget constraint
+ = . This is a standard Cobb–Douglas utility-maximization problem,
having solutions ∗ = (1 − ) and ∗ = . Michele’s charity ∗ is directly
proportional to her altruism, .
c.
A proportional income tax just reduces her net income from to (1 − ) .
Substituting this new income into the solutions from part (b), ∗ =
(1 − )(1 − ) and ∗ = (1 − ) . Allowing a charitable deduction reduces
the relative “price” of Sofia’s consumption:
is still 1 but falls to 1 − .
Solving the utility maximization problem with these new prices and income
yields
c1* =
(1 − α )(1 − t )2 I
α (1 − t ) I
, c2* =
.
α + (1 − α )(1 − t )
α + (1 − α )(1 − t )
Charitable contributions still fall compared to the no-tax case because of the
income effect, but they rise relative to Michele’s own consumption because of
the change in relative prices.
d(1).
Substituting Sofia’s utility function into Michele’s and solving for
yields
U 1 (c1 , c2 ) = c11 (1+α ) c2α (1+α ) .
Solving the utility-maximization problem yields
c1* =
1
α
I , c2* =
I.
1+ α
1+ α
For a given , Michele reduces her charitable contributions compared to part
(b) because she takes into account Sofia’s benefit from Michele’s
consumption, leading Michele to keep her own consumption higher.
d(2).
Substituting Sofia’s utility into Michele’s and solving for
function as in part (b).
gives the same
CHAPTER 5:
Income and Substitution Effects
Problems in this chapter focus on comparative statics analyses of income and own-price
changes. Many of the problems are fairly easy so that students can approach the ideas
involved in shifting budget constraints in simplified settings. Theoretical material is
confined mainly to the analytical problems that stress various elasticity measures and
introduce the almost ideal demand system.
Comments on Problems
5.1
This problem is an example of perfect substitutes. Solving this problem is easy
with intuition, but students should not try to use calculus because of the “knifeedge” nature of demand with perfect substitutes.
5.2
This problem is a fixed-proportions example. The problem illustrates how the
goods used in fixed proportions (peanut butter and jelly) can be treated as a
single good looking at utility-maximizing choices.
5.3
An exploration of the notion of homothetic functions. This problem shows that
Giffen’s paradox cannot occur with homothetic functions.
5.4
This problem asks students to pursue the analysis of Example 5.1 to obtain
compensated demand functions. The analysis essentially duplicates Examples
5.3 and 5.4.
5.5
This problem is another utility-maximization example. In this case, utility is not
separable and cross-price effects are important.
5.6
This problem is in revealed preference theory. The bundles here violate the
strong axiom.
5.7
This problem is an example with no substitution effects. It shows how price
elasticities are determined only by income effects, which, in turn, depend on
income shares.
5.8
This problem shows the convenient result that budget shares can be computed
from expenditure functions through logarithmic differentiation.
Analytical Problems
35
Chapter 5: Income and Substitution Effects
36
5.9
Share elasticities. This problem shows that many conventional elasticity
measures can be derived from “share elasticities.” This is useful because many
budgetary studies proceed mainly by focusing on expenditure shares.
5.10
More on elasticities. This problem shows how the elasticity of substitution
affects the sizes of price elasticities.
5.11
Aggregation of elasticities for many goods. This problem shows how the
aggregation relationships introduced in Chapter 5 for two goods can be
generalized to any number of goods.
5.12
Quasi-linear utility (revisited). This problem extends Problem 3.13 to consider
the special form of the Slutsky equation for the quasi-linear function.
5.13
The almost ideal demand system. This problem introduces a parametrization
of the expenditure function that is widely used in empirical studies of demand.
The connections between this problem and Problem 5.9 are quite important in
the interpretation of many empirical studies.
5.14
Price indifference curves. This problem introduces a graphical concept that is
sometimes used to illustrate theoretical points.
5.15
The multiself model. This behavioral economics problem illustrates how a
model in which the individual has two different utility functions can be used to
examine: (1) situations where the utility function used to make decisions differs
from the true function and (2) situations where the person does not know what
his or her precise preferences are.
Solutions
5.1
a.
Utility = Quantity of water = 0.75 x + 2 y.
b.
The 0.75 liter bottles contain 3/8 the water of the 2 liter bottles. Hence,
if p x < 3 p y 8, then x* = I px and y* = 0. If p x > 3 p y 8, then x* = 0
and y* = I p y .
c.
Chapter 5: Income and Substitution Effects
d.
37
Increases in I shift demand for x outward. Reductions in p y do not
affect demand for x until p y < 8 p x 3 . Then the demand for x falls to
zero.
e.
The compensated demand curve for good x is a vertical line so
long as px < 3 p y 8. If the person buys only x, holding utility constant
requires that x = U 0.75 no matter what the price of x is.
5.2
a.
To avoid confusing goods’ names with prices, let b stand
for peanut butter. Utility maximization requires b = 2 j. The budget
constraint is 0.05b +0.1j = 3. Substitution gives b* = 30 and j * = 15.
b.
If p j = $0.15, substitution now yields j* = 12 and b* = 24.
c.
To continue buying j * = 15, b* = 30. David would need to buy 3 more
ounces of jelly and 6 more ounces of peanut butter. This would require
an increase in income of 3(0.15) + 6(0.05) = 0.75.
d.
e.
Because David uses only peanut butter and jelly to make sandwiches (in
fixed proportions), and because bread is free, it is just as though he buys
the good “sandwiches,” where ps = 2pb + p j .
In part (a), ps = 0.20 and qs = 15.
Chapter 5: Income and Substitution Effects
38
In part (b), ps = 0.25 and qs = 12.
In general, q s = 3 p s , so the demand curve for sandwiches is a
hyperbola.
5.3
f.
There is no substitution effect due to the fixed proportion. A change in
price results in an income effect only.
a.
As income increases, the ratio px p y stays constant, and the utilitymaximization conditions therefore require that MRS stay constant. Thus,
if MRS depends on the ratio y x , this ratio must stay constant as
income increases. Therefore, since income is spent only on these two
goods, both x and y are proportional to income.
5.4
b.
Because of part (a), ∂x ∂I > 0; Giffen’s paradox cannot arise.
a.
Since
x=
0.3I
0.7 I
and y =
,
px
py
we have
U = 0.30.3 0.7 0.7 I px−0.3 p y−0.7 = BIpx−0.3 p y−0.7 ,
where B = 0.30.30.7 0.7. The expenditure function is then E = B −1Up.3x p.7y .
b.
The compensated demand function is xc = ∂E ∂px = 0.3B −1Upx−0.7 p 0.7
y .
c.
It is easiest to show the Slutsky equation in elasticities by just reading
exponents from the various demand functions: ex, px = −1, ex , I = 1,
exc , p = −0.7, sx = 0.3. Hence, ex , px = exc , p − sx ex , I implies
x
x
−1 = −0.7 − (0.3)(1).
5.5
a.
The Lagrange method yields
p
y
= x,
x + 1 py
or p y y = px x + px . Substitution into the budget constraint yields
x=
I − px
I + px
and y =
.
2 px
2py
Hence, changes in p y do not affect x, but changes in px do affect y.
b.
The indirect utility function is
Chapter 5: Income and Substitution Effects
39
( I + p x )2
,
4 p x py
which yields an expenditure function of
E = V 4 px p y − px .
V=
c.
Clearly, the compensated demand function for x depends on p y ,
whereas the uncompensated function did not. By Shephard’s lemma:
∂E
xc =
= V 0.5 px−0.5 p 0.5
y − 1.
∂px
5.6
Year 2’s bundle is revealed preferred to year 1’s since both cost the same in
year 2’s prices. Year 2’s bundle is also revealed preferred to year 3’s because it
was chosen when year 3’s bundle cost less. But in year 3, year 2’s bundle costs
less than year 3’s but is not chosen. Hence, these violate the axiom.
5.7
a.
Because of the fixed proportions between h and c, we know that the
demand for ham is h = I ( ph + pc ). Hence,
p ( p + pc )
− ph
∂h ph
−I
eh , ph =
⋅
=
⋅ h h
=
.
2
∂ph h ( ph + pc )
I
( ph + pc )
Similar algebra shows
− pc
eh , pc =
.
( ph + pc )
So, ph = pc , eh, ph = eh, pc = −0.5.
5.8
b.
With fixed proportions, there are no substitution effects. Here, the
compensated price elasticities are zero, so the Slutsky equation shows
that ex, px = 0 − sx = −0.5.
c.
With ph = 2 pc , part (a) shows
d.
If this person consumes only ham and cheese sandwiches, the price elasticity of
demand for those must be −1 . Price elasticity for the components reflects the
proportional effect of a change in the price of the component on the price the
whole sandwich. In part (a), for example, a 10% increase in the price of ham
will increase the price of a sandwich by 5% and that will cause quantity
demanded to fall by 5%.
that
eh , ph = −2 3 and eh, pc = −1 3.
p xc
d ln E d ln E dE d px
1
1
=
⋅
⋅
= ⋅ xc ⋅
= x = sx .
d ln px
d E dpx d ln px E
1/ px
E
Analytical Problems
Chapter 5: Income and Substitution Effects
40
5.9
Share elasticities
a.
b.
∂ ( px x I )
Ip ∂x ∂I − px x I 2
I
⋅
= x
⋅
= ex , I − 1.
∂I
px x I
I2
px x
For example, if ex , I = 1.5, then esx , I = 0.5.
esx , I =
∂ ( px x I ) px
p ∂x ∂px + x I
⋅
= x
⋅ = ex , px + 1.
∂px
px x I
I
x
For example, if ex , px = −0.75, then esx , px = 0.25.
esx , px =
c.
Because I may be cancelled out of the derivation in part (b),
e p x , px = ex , px + 1.
x
p ∂x ∂p y p y I
∂ ( px x I ) p y
∂x p y
⋅
= x
⋅
=
⋅
= ex , p y .
∂p y
px x I
I
px x ∂p y x
d.
esx , p y =
e.
Use part (b):
kp yk px− k −1
kp yk px− k
k −k
esx , px =
⋅
p
(1
+
p
p
)
=
.
x
y x
(1 + p yk px− k )2
1 + p yk px− k
To simplify algebra, let d = p yk px− k . Then,
kd
kd − 1 − d
−1 =
.
1+ d
1+ d
Now, use the Slutsky equation (remembering that ex , I = 1 ):
ex , px = esx , px − 1 =
exc , p = ex, px + sx =
x
5.10
kd − d − 1
1
d (k − 1)
+
=
= (1 − sx )(−σ ).
1+ d
1+ d
1+ d
More on elasticities
a.
Since ex , px = −(1 − sx )σ − sx and ey , py = − sxσ − s y , we have
ex , px + ey , py = −σ − 1. The sum equals −2 (trivially) in the Cobb–
Douglas case.
b.
Result follows directly from part (a). Intuitively, price elasticities are
large when σ is large (in absolute value) and small when σ is small.
c.
A generalization from the multivariable CES function is possible, but the
constraints placed on behavior by this function are probably not tenable.
5.11
Aggregation of elasticities for many goods
Chapter 5: Income and Substitution Effects
41
a.
Because the demand for any good is homogeneous of degree zero,
Euler’s theorem states
n 
∂x 
∂x
 p j i  + I i = 0.
∑
∂p j 
∂I
j =1 
Multiplication by 1 xi yields the desired result.
b.
Parts (b) and (c) are based on the budget constraint: ∑ i =1 pi xi = I .
n
Differentiation with respect to I yields ∑ i =1 pi ( ∂xi ∂I ) = 1.
n
Multiplication of each term by xi I xi I yields ∑ i =1 si ei , I = 1.
n
c.
Differentiation of the budget constraint with respect to
p j produces ∑ i =1 ( pi ∂xi ∂p j ) + x j = 0. Multiplication by
n
( p I ) ( x x ) yields ∑ s e = −s .
j
i
i
i i, j
j
i
5.12
Quasi-linear utility (revisited)
a.
First, we need to find the demand functions for both the goods. This is
done by straightforward application of the Lagrange method to give:
I − px
p
x=
and y = x .
px
py
The income effect for x is
∂x p − I
−x = x 2 .
∂I
px
The income elasticity for x is
∂x I
I
ex, I = . =
.
∂I x I − px
The income effect for y is
∂y
−y
= 0.
∂I
The income elasticity for y is
∂y I
ey , I = . = 0.
∂I y
b.
Now, we need to find the compensated demand functions for both the
goods by first finding the indirect utility function:
I − px
V=
+ ln px − ln p y .
px
So, the expenditure function is
E = px (V − ln px + ln p y + 1),
implying
Chapter 5: Income and Substitution Effects
42
xc =
∂E
= V − ln px + ln p y
∂px
and
px
.
py
The substitution effect for x is
∂x c
1
=−
.
∂px
p xx
yc =
The compensated own-price elasticity for x is
∂x c p x
1
ex c , p =
. c =
.
x
∂px x
ln px − ln p y − V
The substitution effect for y is
p
∂y c
= − x2 .
∂p y
py
The compensated own-price elasticity for y is
∂y c p y
eyc , p =
.
= −1.
y
∂p y y c
c.
For the Slutsky equation, the own-price effects of the demand functions
are also needed:
∂x
I
p
∂y
= − 2 and
= − x2 .
∂p y
py
∂px
px
Now, put all the terms into the Slutsky equation. For x,
∂x
I
(1)
=− 2
∂px
px
and
∂x c
∂x
1 p −I
I
(2)
−x =− + x 2 =− 2
∂px
∂I
px
px
px
imply
∂x ∂x c
∂x
=
−x .
∂px ∂px
∂I
For y,
p
∂y
(3)
= − x2
∂p y
py
and
p
∂y c
∂y
(4)
−y
= − x2 − 0
∂p y
∂I
py
imply
∂y ∂y c
∂y
=
−y .
∂p y ∂p y
∂I
Thus, the Slutsky equation holds for both goods.
Chapter 5: Income and Substitution Effects
43
For the elasticity version of the equation, we need the own-price
elasticity of each good:
I
ex , px =
and ey , py = −1.
px − 1
Put all this into the elasticity version of the Slutsky equation. For x,
I
ex , px =
px − I
and
p  I
1

ex c , p − s x e x , I =
− 1 − x 
x
ln p x − ln p y − V 
I  I − px
=
px
−1
px − I
I
px − I
imply ex , px = exc , p − sx ex , I . For y,
=
x
ey , py = −1
and
eyc , p − s y e y , I = −1 − 0 = −1
y
imply ey , p y = e yc , p − s y ey , I . This confirms the elasticity version of the
y
Slutsky equation.
5.13
The almost ideal demand system
a.
1
1
ln E ( p1 , p2 , u ) = a0 + α1 ln p1 + α 2 ln p2 + γ 11 (ln p1 ) 2 + γ 22 (ln p2 ) 2 If
2
2
β1 β 2
+ γ 12 ln p1 ln p2 + u β 0 p1 p2 .
the function is homogeneous to degree 1 in prices,
E (kp1 , kp2 , u ) = kE ( p1 , p2 , u ),
where k is a scalar.
b.
ln E ( kp1 , kp2 , u )
1
1
= a0 + α1 ln kp1 + α 2 ln kp2 + γ 11 (ln kp1 ) 2 + γ 22 (ln kp2 ) 2
2
2
β1
β2
+ γ 12 ln kp1 ln kp2 + u β 0 ( kp1 ) (kp2 )
Chapter 5: Income and Substitution Effects
44
1
= a0 + α1 ln k + α1 ln p1 + α 2 ln k + α 2 ln p2 + γ 11 (ln k ) 2
2
1
1
+ γ 11 ln k ln p1 + γ 11 (ln p1 ) 2 + γ 22 (ln k ) 2 + γ 22 ln k ln p2
2
2
1
+ γ 22 (ln p2 ) 2 + γ 12 (ln k ) 2 + γ 12 ln k ln p1 + γ 12 ln k ln p2
2
+ γ 12 ln p1 ln p2 + k ( β1 + β 2 )u β 0 p1β1 p2 β2 .
Next, we gather together the terms with k in them. From the constraints,
we can see that
α1 + α 2 = 1,
γ 11 + γ 12 = 0,
γ 21 + γ 22 = 0,
β1 + β 2 = 0.
This simplifies the mess to a great extent to give:
1
ln E (kp1 , kp2 , u ) = ln k + a0 + α1 ln p1 + α 2 ln p2 + γ 11 (ln p1 ) 2
2
1
+ γ 22 (ln p2 ) 2 + γ 12 ln p1 ln p2 + u β 0 p1β1 p2β2
2
= ln k + ln E ( p1 , p2 , u )
= ln kE ( p1 , p2 , u ).
Thus, E (kp1 , kp2 , u ) = kE ( p1 , p2 , u ), and so, the function is homogeneous
to degree 1 in prices.
c.
From Problem 5.8, sx = ∂ ln E ∂ ln px . Hence,
s1 = α1 + γ 11 ln p1 + γ 12 ln p2 + u β 0 β1 p1β1 p2β2 .
Similarly,
s2 = α 2 + γ 22 ln p2 + γ 12 ln p1 + u β 0 β 2 p1β1 p2β2 .
So, neglecting the utility-related terms, the shares are simple functions
of the logarithms of prices.
5.14
Price indifference curves
a.
From Example 4.3, V = 0.5Ipx−0.5 p y−0.5 . Hence,
py =
I2
.
4V 2 px
b.
The slope shows the rate at which the prices should move relative to one
another for the utility to remain constant, given I:
dp y
−I 2
=
< 0.
dp y 4V 2 px2
Chapter 5: Income and Substitution Effects
c.
5.15
45
With increasing levels of utility, that is, with increasing V , the curves
move progressively “inward,” that is, toward the origin.
The multiself model
(1)
U1 ( x, y ) = x + 2 ln y,
(2)
U 2 ( x, y ) = x + 3ln y.
a.
Decision utility
i.
If px = p y = 1 and I = 10 , constrained maximization of function (1)
yields x = 8, y = 2, U 2 = 10.08 . Note that actual experienced utility is
calculated from function (2) even though the person acts to maximize
function (1)
ii.
Maximization of function (2) yields x = 7, y = 3, U 2 = 10.30 . Hence,
there is a loss of utility of 0.22 from using the decision utility function
rather than the true one.
iii.
This problem illustrates that with two targets (a utility of 10.30 and
consumption of y=3), a solution will generally require two
“instruments.” First, consider a subsidy on good y. Since optimization
requires y 2 = 1 p y , achieving y = 3 requires a price of p y = 2 / 3. With
this price, this person chooses x = 8, y = 3, U 2 = 8 + 3ln 3 = 11.30 , so this
subsidy would have to be accompanied by an income tax of 1 to arrive at
the same bundle as in part (ii). The cost of the subsidy is 1, but if
accompanied by the income tax its net cost would be 0.
Arriving at the bundle specified in part (ii) could also be achieved by
both taxing good x and subsidizing good y. That solution would require
a unit tax of 1/9 on good x and a unit subsidy of 7/27 on good y. With
this scheme, the ratio px p y = (10 / 9) (20 / 27) = 3 / 2. That scheme
would cost 21/27 in subsidy and raise 7/9 in taxes, so it would also break
even.
iv.
Utility could also be raised to 10.30 (from the 10.08 calculated in part i)
with an income grant of 0.22, all of which would be used to purchase
good x. So, that would not address the problem of the underconsumption
of good y.
b.
Preference uncertainty
i.
With U ( x, y ) = x + 2.5 ln y , the optimal choices are
x = 7.5, y = 2.5,U1 = x + 2 ln y = 9.33,U 2 = x + 3ln y = 10.25 .
Chapter 5: Income and Substitution Effects
46
ii.
With perfect knowledge of preferences, however,
U1 = 8 + 2 ln 2 = 9.39,U 2 = 7 + 3ln 3 = 10.30 , so in each case there is a
utility loss of about 0.05.
iii.
As a result of part (ii), this person would pay up to about 0.05 to learn
what his or her preferences will actually be.
CHAPTER 6:
Demand Relationships among Goods
Two types of demand relationships are stressed in the problems to Chapter 6: cross-price effects
and composite commodity results. The general goal of these problems is to illustrate how the
demand for one particular good is affected by economic changes that directly affect some other
portion of the budget constraint. Several examples are introduced to show situations in which the
analysis of such cross-effects is manageable.
Comments on Problems
6.1
Another use of the Cobb–Douglas utility function that shows that cross-price effects are
zero. Explaining why they are zero helps to illustrate the substitution and income effects
that arise in such situations.
6.2
Shows how some information about cross-price effects can be derived from studying
budget constraints alone. In this case, Giffen’s paradox implies that spending on all other
goods must decline when the price of a Giffen good rises.
6.3
A simple case of how goods consumed in fixed proportion can be treated as a single
commodity (buttered toast).
6.4
An illustration of the composite commodity theorem. Use of the Cobb–Douglas utility
produces quite simple results.
6.5
An examination of how the composite commodity theorem can be used to study the
effects of transportation or other transactions charges. The analysis here is fairly
intuitive—for more detail consult the Borcherding–Silverberg reference or Problem 6.12.
6.6
Illustrations of some of the applications of the results of Problem 6.5. More extensive
answers are provided in the solutions to Problem 6.12.
6.7
This problem demonstrates a special case in which uncompensated cross-price effects are
symmetric.
45
46
6.8
Chapter 6: Demand Relationships among Goods
This problem looks at cross-substitution effects in a three-good CES function.
Analytical Problems
6.9
Consumer surplus with many goods. This illustrates how expenditure functions can
help to clarify consumer surplus ideas when several prices change.
6.10 Separable utility. This problem shows that many of the complications in a many good
utility function can be greatly simplified if utility is assumed to be separable.
6.11
Graphing complements. The problem draws on Samuelson’s famous paper on
complementarity. It shows that there is a graphical representation of complements in the
three-good case that accurately reflects the Hicks definition.
6.12
Shipping the good apples out. This repeats the analysis in the Borcherding–Silberberg
paper in a simplified form. It is mainly intended to show how the various properties of
utility and demand function can be used to sign derivatives in special cases.
6.13
Proof of the composite commodity theorem. This problem outlines two general
approaches to proving the composite commodity theorem. The first, using duality, is
probably the most preferred such method.
6.14
Spurious product differentiation. This behavioral problem shows how firms may be
able to receive higher prices for their products if they can convince (spuriously)
consumers that they are better.
Solutions
6.1
a.
As for all Cobb–Douglas applications, first-order conditions show
that pmm = ps s = 0.5I . Hence, s = 0.5I ps and ∂s ∂pm = 0.
b.
Because indifference curves are rectangular hyperboles (ms = constant)
own substitution and cross-substitution effects are of the same proportional
size, but in opposite directions. Because indifference curves are homothetic,
income elasticities are 1.0 for both goods, so income effects are also of same
proportionate size. Hence, substitution and income effects of changes in pm on s
are precisely balanced.
c.
We have the two conditions
Chapter 6: Demand Relationships among Goods
0=
∂s
∂s
∂s
=
−m
∂pm ∂pm U
∂I
0=
∂m ∂m
∂m
=
−s
.
∂p s ∂p s U
∂I
But
∂s
∂m
=
.
∂pm U ∂ps U
So
m
d.
6.2
∂s
∂m
=s
.
∂I
∂I
From part (a),
 0.5 
 0.5 
 0.5 
∂s
∂m
m = m
.
 = m
 = s
=s
∂I
∂I
 ps 
 pm m s 
 pm 
Since ∂r / ∂pr > 0 , a rise in pr implies that pr r definitely rises. Since p j j = I − pr r ,
must fall, j will fall. Hence, ∂j ∂pr < 0.
6.3
6.4
a.
Yes, pbt = 2 pb + pt .
b.
Since p cc = 0.5I , ∂c ∂pbt = 0.
c.
Since changes in pb or pt affect only pbt , these derivatives are also zero.
a.
The amount spent on ground transportation is

p 
pbb + pt t = pb  b + t ⋅ t  = pb g ,
pb 

where
p
g = b + t ⋅ t.
pb
b.
Maximize U ( b, t , p ) subject to p p p + pb b + pt t = I . This is equivalent to
maximizing U ( g , p) = g 2 p subject to p p p + pb g = I .
c.
p=
I
2I
, g=
.
3 pp
3 pb
47
48
6.5
6.6
6.7
Chapter 6: Demand Relationships among Goods
d.
Given pb g, choose pbb = pb g 2 and pt t = pb g 2.
a.
Composite commodity = p2 x2 + p3 x3 = p3 (kx2 + x3 ).
b.
The relative price equals
p + t kp + t
= 2 = 3 .
p3 + t p3 + t
The relative price is less than 1 for t = 0. The relative price → 1 as t → ∞.
Hence, increases in t raise the relative price of x2 .
c.
Although it might seem like increases in t would reduce expenditures on the
composite commodity, the theorem does not apply to this directly. As part (b)
shows, changes in t also change relative prices.
d.
Rise in t should reduce relative spending on x2 more than on x3 since this raises
its relative price. However, see the Borcherding and Silberberg analysis.
a.
Transport charges make low-quality produce relatively more expensive at distant
locations. Hence, buyers will have a preference for high quality.
b.
Increase in baby-sitting expenses raise the price cheap meals relative to expensive
ones.
c.
High-wage individuals have higher value of time and hence a lower relative price
of Concorde flights.
d.
Increasing search costs lower the relative price of expensive items.
Assume xi = ai I and x j = a j I . Hence,
∂x j
∂xi
= ai a j I = xi
.
∂I
∂I
So income effects (in addition to substitution effects) are symmetric.
xj
6.8
a.
Example 6.3 gives
x=
Clearly,
I
.
p x + px p y + px pz
Chapter 6: Demand Relationships among Goods
49
∂x ∂x
,
< 0.
∂p y ∂pz
So these are gross complements.
b.
The Slutsky equation shows
∂x
∂x
=
∂p y ∂p y
So ∂x ∂p y
U =U
−y
U =U
∂x
.
∂I
could be positive or negative. Because of the symmetry of y and
z here, Hick’s third law suggests that both ∂x ∂p y
U =U
> 0 and ∂x ∂pz U =U > 0.
Analytical Problems
6.9
Consumer surplus with many goods
a.
CV = E ( p1′, p2′ , p3 ,K pn , U ) − E ( p1 , p2 , p3 ,K pn , U ).
b.
Notice that the rise in p1 shifts the compensated demand curve for x2 .
6.10
c.
Symmetry of compensated cross-price effects implies that order of calculation is
irrelevant.
d.
The figure in part (a) suggests that compensation should be smaller for net
complements than for net substitutes.
Separable utility
a.
This functional form assumes U xy = 0. That is, the marginal utility of x does not
depend on the amount of y consumed. Though unlikely in a strict sense, this
independence might hold for large consumption aggregates such as “food” and
50
Chapter 6: Demand Relationships among Goods
“housing.”
b.
Because utility maximization requires MU x p x = MU y p y , an increase in
income with no change in px or p y must cause both x and y to increase to
maintain this equality (assuming U i > 0 and Uii < 0 ).
c.
Again, using MU x p x = MU y p y , a rise in px will cause x to fall and MU x to
rise. So the direction of change in MU x px is indeterminate. Hence, the change
in y is also indeterminate.
d.
If U = xα y β , then MU x = α xα −1 y β .
But, ln U = α ln x + β ln y; and so MU x = α x . Hence, the first case is not
separable; the second is.
6.11
Graphing complements
a,b.
The figure shows that the loss in x1 can be compensated for by an additional j of
x3 or k of x2 .
x3
j
k
U( x10 - h)
U( x10 )
x2
c.
The new indifference curve is given by U 2 .
Chapter 6: Demand Relationships among Goods
x3
j
U2
k
U0
x2
51
52
Chapter 6: Demand Relationships among Goods
d.
The three cases are shown in the next three graphs:
x3
j
U2= U( x10 - 2h)
k
U0
x2
(i) independent
x3
j
U2
k
U= U(x1 - 2h)
U0
x2
(ii) complements
x3
j
U( x10 - 2h)
k
U0
U2
x2
(iii) substitutes
Chapter 6: Demand Relationships among Goods
53
Symmetry is shown by the fact that it does not matter in which order the j and k
compensations are made.
e.
Samuelson suggests the following proof. Consider the implicit equation:
x1 = f ( x2 , x3 , U ).
Holding U constant we can examine the sign of
∂2 f
.
∂x2 ∂x3
It is clear (from the graphical argument) that this second-order partial should be
negative for complements and positive for substitutes. Now, consider how the
derivative relates to the Hicks definition. The first partial is
∂x1
.
∂x2 U
This is the negative of the MRS. The MRS will remain constant since p1 and p2
remain constant. We wish to know how a change in x3 will change the levels of
the other goods that yield this MRS. It is also clear that
∂x3
< 0.
∂p3 U
Consider a fall in p3 . x3 will rise and if the second-order partial is positive, the
MRS will rise for given levels of x1 and x2 . To restore utility maximization, x2
will fall (and x1 will rise), and x2 and x3 would be declared substitutes by the
Hicks definition. Hence, the graphical and mathematical definitions agree.
f.
6.12
The mathematical ideas will always be relevant since they are in principle
measurable. However, it seems unlikely that these derivatives could fully capture
complicated relationships between actual goods, especially in models with many
narrowly defined goods.
Shipping the good apples out
a.
b.
∂(x x )
c
2
c
3
∂t
∂(x x )
c
2
∂t
c
3
=
=
x3c
x3c
∂x c
∂x2c
− x2c 3
∂t
∂t .
c 2
( x3 )
∂x c
∂x2c
− x2c 3
∂t
∂t
c 2
( x3 )
54
Chapter 6: Demand Relationships among Goods
 ∂x c ∂x c 
 ∂x c ∂x c 
x3c  2 + 2  − x2c  3 − 3 
∂p ∂p3 
 ∂p2 ∂p3 
=  2
2
( x3c )
c.
=
x2c  1 ∂x2c 1 ∂x2c 1 ∂x3c 1 ∂x3c 
+ ⋅
− ⋅
− ⋅
 ⋅

x3c  x2c ∂p2 x2c ∂p3 x3c ∂p2 x3c ∂p3 
=
x2c  s22 s23 s32 s33 
+
−
− .

x3c  x2c x2c x3c x3c 
Given
eijc =
Then
sij p j
∂ ( x2c x3c )
∂t
d.
xic
.
=
x2c  s22 s23 s32 s33 
+
−
− 

x3c  x2c x2c x3c x3c 
=
c
c
c
c

e23
e32
e33
x2c  e22
+
−
−
.
c 
x3  p2 p3 p2 p3 
Hicks’ third law is ∑ j =1 eij = 0 , for i = 1, 2,3. If we substitute for e23 and e33 in
3
the above equation, we get e23 = −e21 − e22 and e33 = −e31 − e32 . Thus,
e22 e23 e32 e33 e22 ( −e21 − e22 ) e32 ( −e31 − e32 )
+
−
−
=
+
−
−
p2 p3 p2 p3 p2
p3
p2
p3
 1
 1
1 
1  e −e
= e22  −  + e32  −
+  + 31 21
p3
 p2 p3 
 p2 p3 
 1
1  e −e
= (e22 − e32 ) 
−  + 31 21 .
p3
 p2 p3 
e.
The own-price elasticity of x2 , e22 , is negative. If goods 2 and 3 are substitutes,
then e32 should be positive. Therefore, e22 − e32 should be negative. Since good 2
is more expensive, 1 p2 < 1 p3 . Thus, the following product should be positive:
 1
1 
(e22 − e32 )  −  > 0.
 p2 p3 
We are almost done finding the sign of the last equation in part (d); we only need
to deal with the last term:
Chapter 6: Demand Relationships among Goods
55
e31 − e21
.
p3
This term is likely to be small if we assume that goods 2 and 3 have similar
relationships with 1: that is, e31 and e21 should have close values. Since goods 2
and 3 are close substitutes, such an assumption seems reasonable. Therefore,
overall, we can expect the expression to be positive.
f.
a)
In this example, high-quality apples and fresh oranges can
be represented as good 2 (using the above notation) and the low-quality
apples or oranges, respectively, as good 3. Clearly, high-quality apples (or
oranges) and low-quality apples (or oranges) are close substitutes, so the
term e22 − e32 should be negative (as explained above) and
 1
1 
(e22 − e32 )  −  > 0.
 p2 p3 
Moreover, we can assume that apples, whether of high or low quality,
have a similar relationship with the other commodities, thus e31 and e21
should have close values. Hence, transaction costs such as the
transportation costs are expected to increase the relative demand for the
more expensive fruits, making it harder to find such products.
b)
In this example, expensive restaurant meals and cheap restaurant
meals are close substitutes. The expensive meals can be represented as
good 2 (using the above notation) and the cheap meals as good 3. Since
goods 2 and 3 come from the same category, namely restaurant meals, we
can expect them to have similar relationships with the other categories of
goods, represented by x1. Therefore, e31 − e21 is likely to be small. And
since goods 2 and 3 are close substitutes, e22 − e32 should be negative,
implying
 1
1 
(e22 − e32 )  −  > 0.
 p2 p3 
Hence, a higher transaction cost (here, the higher baby-sitting expenses)
will increase the relative demand for the more expensive meals.
c)
We can assume that flying the Concorde falls in the category of expensive
flights, and these are close substitutes to cheaper flights. Thus, the
Concorde flights can be represented as good 2 (using the above notation)
and the cheaper flights as good 3. Since goods 2 and 3 come from the
same category, namely flights, we can expect them to have similar
relationships with the other categories of goods, represented by x1.
Therefore, e31 − e21 is likely to be small. And since goods 2 and 3 are close
substitutes, e22 − e32 should be negative, implying
56
Chapter 6: Demand Relationships among Goods
 1
1 
(e22 − e32 )  −  > 0.
 p2 p3 
Hence, a higher transaction cost (here, the value of the time lost flying
with a slower airplane) will increase the relative demand for the more
expensive Concorde flights.
d)
In this example, the value of the time spent searching is a transaction cost
that changes the relative price of the items. As above, we can represent the
expensive items by good 2 and the cheap ones as good 3. Since goods 2
and 3 come from the same category, we can expect them to have similar
relationships with the other categories of goods, represented by x1.
Therefore, e31 − e21 is likely to be small). And since goods 2 and 3 are
close substitutes, e22 − e32 should be negative, implying
 1
1 
(e22 − e32 )  −  > 0.
 p2 p3 
Hence, a higher transaction cost (here, the value of the time spent
searching) will decrease the relative price of expensive items.
6.13 Proof of the composite commodity theorem
a. Proof using duality
i.
Applying the envelope theorem to both minimization problems yields:
ii.
dE dE dp1 dE dp2 dE dp3
dE *
c 0
c 0
=
⋅
+
⋅
+
⋅
= 0 + x2 p2 + x3 p3 = y =
dt dp1 dt dp2 dt dp3 dt
dt
Again applying the envelope theorem to both problems yields:
dE
dE *
= x1c =
dp1
dp1
b. Proof using two-stage maximization
i.
ii.
Because neither the price of x2 or x3 changes, the maximum value for the function
V depends only on m. That is, there is a unique correspondence between m and the
utility it provides.
This equality is derived by repeated application of the envelope theorem to the
various optimization subproblems. The first-order conditions of Stage 2 require
∂V ∂m = δ . But, because V is the value function from Stage 1, the envelope
theorem also implies ∂V ∂m = µ . Hence, δ = µ . But the first-order conditions
for the original optimization problem require ∂U ∂x1 = λ p1 and the first-order
conditions for the Stage 2 problem require ∂V ∂x1 = ∂U ∂x1 = δ p1 . So, λ = δ .
Chapter 6: Demand Relationships among Goods
57
So, provided there is a unique solution to the original optimization problem, this
solution will be identical to the solution from the two-stage statement of the
problem.
6.14
Spurious product differentiation
a.
The first-order condition for utility maximization for brand 1 is p1y = 500 (1 + y1 ) .
Hence, the maximum price this person will pay for this brand ( y1 = 1) is 250. For
brand 2, the maximum price this person is willing to pay is 300.
b.
For brand 1, utility is 750 + 500 ln 2 = 1097. For brand 2, utility is
700 + 600 ln 2 = 1116 . Hence, this person will purchase brand 2.
c.
Assuming the utility from brand 2 is really that from brand 1, utility would be
700 + 500 ln 2 = 1047 , a loss of 50 from what could have been received if he/she
had purchased brand 1.
d.Spending funds to ascertain the quality of brand 2 (say by reading Consumer Reports)
would be equivalent to taking a gamble whose outcome depends on whether the
information reports that brand 2 is really better. Let p be the probability that
research determines that brand 2 really is better. Then this person will spend x on
gathering information provided the expected value of doing so exceeds what
he/she can receive from buying brand 1. Hence, we have
(1 − p)(750 − x + 500 ln 2) + p(700 − x + 600 ln 2) ≥ 1097 . Algebraic manipulation
of this condition yields: p(100 ln 2 − 50) ≈ 19 p ≥ x . This makes sense intuitively.
If p = 1 , this person would pay up to the 19 utility difference between the two
brands to know that brand 2 is better with certainty. On the other hand, if p = 0.5,
he or she would only pay half this amount because the expected value of the
information is less.
CHAPTER 7:
Uncertainty
Most of the problems in this chapter focus on illustrating the concept of risk aversion. They
assume that individuals have concave utility of wealth functions and therefore dislike variance in
their wealth. For some of these problems (especially the later ones), students will need to review
the material on mathematical statistics in Chapter 2.
Comments on Problems
7.1
This problem reverses the risk-aversion logic to show that observed behavior can be used
to place bounds on subjective probability estimates.
7.2
This problem provides a graphical introduction to the idea of risk-taking behavior. The
Friedman–Savage analysis of coexisting insurance purchases and gambling could be
presented here.
7.3
This is a nice, homey problem about diversification. The problem can be done
graphically, but instructors could introduce variances into the problem if desired.
7.4
This problem is a graphical introduction to the economics of health insurance that
examines cost-sharing provisions. Health insurance is discussed in more detail in Chapter
18.
7.5
This problem provides some simple numerical calculations involving risk aversion and
insurance when utility is logarithmic.
7.6
This is a rather difficult problem as written. It can be simplified by using a particular
utility function (e.g., U (W ) = ln W ). With the logarithmic utility function, one cannot use
the Taylor approximation until after differentiation, however. If the approximation is
applied before differentiation, concavity (and risk aversion) is lost. This problem can,
with specific numbers, also be done graphically, if desired. The notion that fines are more
effective can be contrasted with the criminologist’s view that apprehension of lawbreakers is more effective and some shortcomings of the economic argument (i.e., no
disutility from apprehension) might be mentioned.
7.7
This problem makes some numerical estimates of willingness to avoid specific risks. It
also shows how these values depend on wealth.
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly
accessible website, in whole or in part.
Chapter 7: Uncertainty and Information
57
7.8
This problem is an illustration of diversification. The problem also shows how insurance
provisions can affect diversification.
7.9
This is a new problem on diversification, here applied to investing in financial assets. The
problem illustrates a case in which it is optimal to diversify into an asset with obviously
lower expected returns. The problem shows that diversification can be beneficial with
independent asset returns and even more so with negatively correlated returns.
7.10
This problem covers option values. It is similar to Example 7.5, just tweaking some of the
functions. The similarity to the text example is useful to allow students to master the
fairly difficult concepts and calculations involved. The new functions do provide some
economic insight as well: an increase in the value of one of the choices can reduce option
value because just committing to the single enhanced choice provides a lot of utility.
Also, working through the case with risk aversion provides a somewhat surprising
example in which risk aversion reduces option value.
Analytical Problems
7.11
HARA utility. This problem shows that the harmonic absolute risk aversion utility
function is compatible with other frequently used forms. These other forms are just
special cases of the HARA function.
7.12
More on the CRRA function. This problem stresses the close connection between the
relative risk-aversion parameter and the elasticity of substitution. It is a good problem for
building an intuitive understanding of risk aversion in the state preference model. Part
(d) uses the CRRA utility function to examine the “equity-premium puzzle.”
7.13
Graphing risky investments. This problem provides an illustration of investment theory
in the state preference framework.
7.14
The portfolio problem with a Normally distributed risky asset. This problem shows
how the portfolio problem can be solved explicitly if asset returns are Normal.
Behavioral Problem
7.15
Prospect theory. A good problem to assign for instructors interested in integrating
behavioral economics into their course. It covers one of the most influential models in
behavioral economics, which Kahneman (Nobel Prize winner) and Tversky applied to
explain the results of their lab experiments. Actual experimental results are cited in the
problem.
Solutions
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58
7.1
Chapter 7: Uncertainty and Information
The expected utility with the bet must be greater than or equal to that without the bet. So,
p must satisfy:
p ln (1,100,000) + (1 − p ) ln ( 900,000) > ln (1,000,000) .
Solving,
13.9108p + 13.7102 (1 − p ) > 13.8155,
implying 2006p > 0.1053, or p > 0.525.
7.2
See graph.
This would be limited by the individual’s resources. Since unfair bets are continually
being accepted, he or she could run out of wealth.
7.3
a.
Strategy 1
Outcome
Probability
12 Eggs
0.5
0 Eggs
0.5
Expected value = 0.5 (12) + 0.5( 0) = 6.
Strategy 2
Outcome
12 Eggs
Probability
6 Eggs
0.5
0.25
0 Eggs
0.25
Expected value = 0.25 (12) + 0.5( 6) + 0.25(0) = 3 + 3 = 6.
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Chapter 7: Uncertainty and Information
59
b.
7.4
7.5
a.
The insurance company has a 50% chance of paying out $10,000. Its cost is thus
$5,000. The consumer has a certain wealth of $15,000 with fair insurance
compared to a 50–50 chance of wealth of $10,000 or $20,000 without insurance.
b.
Cost of the policy is 0.5 × 5, 000 = 2,500 . Hence, wealth is 17,500 with no illness
and 12,500 with the illness.
a.
Eno ins [U (Y )] = 0.75ln (10,000) + 0.25ln ( 9,000 ) = 9.1840.
b.
Eins [U (Y )] = ln ( 9,750 ) = 9.1850. Insurance is preferable.
c.
7.6
We have
ln(10, 000 − p ) = 9.1840.
Exponentiating,
10, 000 − p = e9.1840 = 9, 740,
implying p = 260.
Expected utility is
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60
Chapter 7: Uncertainty and Information
E[U (W )] = pU (W − f ) + (1 − p)U (W ) .
Computing elasticities,
∂E[U (W )]
p
p
⋅
= [U (W − f ) − U (W ) ]
e EU,p =
∂p
E[U (W )]
E[U (W )]
∂E[U (W )]
f
f
⋅
= − pU ′ (W − f ) ⋅
.
e EU, f =
∂f
E[U (W )]
E[U (W )]
Therefore,
eU,p U (W − f ) − U (W )
=
< 1,
− f U ′(W − f )
eU, f
where the last inequality follows from the formula given for the Taylor series
approximation. So, a fine is more effective.
The calculations are even more transparent in the special case of logarithmic
utility: U (W ) = ln W . Then expected utility is
E[U (W )] = p ln (W − f ) + (1 − p )ln W .
Computing elasticities,
p
− pf W
≈
,
e EU,p = ln (W − f ) − ln W  ⋅
E[U (W )] E[U (W )]
p
f
− pf (W − f )
⋅
=
.
U (W − f ) E[U (W )]
E[U (W )]
Following the logic from above,
eU,p W − f
=
< 1.
W
eU, f
e EU, f = −
2
7.7
a.
E (v 2 ) = ∑ p ( xi ) f ( xi ) = 0.5(−1) 2 + 0.5(1) 2 = 1.
i =1
b.
E (h 2 ) = 0.5(−k )2 + 0.5(k ) 2 = k 2 .
c.
If U (W ) = ln(W ), then for W > 0,
U ′′(W ) 1 W 2 1
r (W ) = −
=
= .
U ′(W ) 1 W W
d.
p = 0.5 E (h 2 )r (W ) = 0.5k 2
1
k2
=
.
W 2W
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Chapter 7: Uncertainty and Information
61
Calculate p when W = 10
k
p
0.5
0.0125
1
0.05
2
0.2
Calculate p when W = 100
k
p
0.5
0.00125
1
0.005
2
0.02
Risk premium is higher when the level of initial wealth is lower. The greater the
size of risk faced (larger the k ), the higher will be the risk premium. Because k
enters as a quadratic, increasing k and W in the same proportion will increase p.
7.8
a.
The farmer will plant corn since
Ewheat [U (Y )] = 0.5 ln ( 28,000 ) + 0.5 ln (10,000) = 9.7251.
Ecorn [U (Y )] = 0.5 ln (19,000 ) + 0.5 ln (15,000 ) = 9.7340.
b.
With half in each, YNR = 23,500 and YR = 12,500.
E50-50 mix [U (Y )] = 0.5 ln ( 23,500) + 0.5 ln (12,500) = 9.7491.
The farmer should plant a mixed crop. Diversification yields an increased
variance relative to corn only, but takes advantage of wheat’s high yield.
c.
Let α = percent in wheat.
Eα mix [U (Y )] = 0.5 ln ( 28, 000α + 19, 000(1 − α ) )
+ 0.5 ln (10, 000α + 15, 000(1 − α ) )
= 0.5 ln (19, 000 + 9, 000α ) + 0.5 ln(15, 000 − 5, 000α ).
Taking the first-order condition,
dEα mix [U (Y )]
4,500
2,500
=
−
= 0.
dα
19, 000 + 9, 000α 15, 000 − 5, 000α
Rearranging,
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62
Chapter 7: Uncertainty and Information
45(150 − 50α ) = 25 (190 + 90α ) ,
implying α = 0.444. Plugging α into the utility function yields
Eα*mix [U (Y )] = 0.5 ln ( 22,996) + 0.5 ln (12,780) = 9.7494.
This is a slight improvement over the 50–50 mix.
d.
If the farmer plants only wheat, YNR = 24,000 and YR = 14, 000.
Einsured wheat [U (Y )] = 0.5 ln ( 24,000) + 0.5 ln (14,000) = 9.8163.
Availability of this insurance will cause the farmer to forego
diversification.
7.9
a.
(1)
A has an expected return of 8 and B of 4.5. Maria’s expected utility from
investing the whole $1 in A is
1
1
E A [U (W )] =
16 +
0 = 2.
2
2
Dividing the investment equally between A and B leads to four possible
outcomes: the assets both turn out to yield a positive return, generating
utility 16 / 2 + 9 / 2 = 12.5; they both yield no return, generating utility
0 = 0; A yields a positive return and B does not, generating utility
16 / 2 + 0 / 2 = 8; and vice versa, generating utility
0 / 2 + 9 / 2 = 4.5. Each utility realization is equally likely, leading to
expected utility
1
1
1
1
Eequal split [U (W )] =
12.5 + ⋅ 0 +
8+
4.5 ≈ 2.121,
4
4
4
4
which is greater than the expected utility 2 from A alone.
(2)
Dividing the investment a in A and 1 − a in B leads to four possible
outcomes: the assets both turn out to yield a positive return, generating
utility 16a + 9(1 − a); they both yield no return, generating utility
0 = 0; A yields a positive return and B does not, generating utility
16a = 4 a ; and vice versa, generating utility 9(1 − a) = 3 1 − a . Each
realization is equally likely, leading to expected utility
1
1
Ea ,1−a split [U (W )] =
16a + 9(1 − a) + ⋅ 0
4
4
1
1
+ ⋅ 4 a + ⋅ 3 1− a.
4
4
One could try to maximize this with respect to a , but it is simpler to graph
it, as below, and see that the maximum is reached, restricting attention to
decimal values of a, for a = 0.8, at which point the expected utility is
2.185, higher than from an equal split. While Maria still diversifies, she
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Chapter 7: Uncertainty and Information
63
puts more in the asset ( A ) with the higher expected return.
b.
(1)
With perfect negative correlation, and half invested in each asset, there are
only two possible outcomes: A has a positive return and B nothing,
generating utility 16 / 2 = 8; and vice versa, generating utility
9 / 2 = 4.5. Each realization is equally likely, leading to expected
utility
1
1
Eequal split [U (W )] =
8+
4.5 ≈ 2.475.
2
2
This is greater than the expected utility from an equal split when asset
returns were independent from part (a1).
(2)
Dividing the investment a in A and 1 − a in B leads to two possible
outcomes when there is perfect negative correlation: A can yield a
positive return and B not, generating utility 16a = 4 a ; and vice versa,
generating utility 9(1 − a) = 3 1 − a . Each realization is equally likely,
leading to expected utility
1
1
Ea ,1−a split [U (W )] = ⋅ 4 a + ⋅ 3 1 − a .
2
2
Graphing this as in part (a2) shows that the optimal allocation (to the
nearest decimal) is a = 0.6, yielding expected utility 2.498. Perfect
negative correlation makes diversification even more appealing.
7.10
a.
As before, E (O1 ) = 1 2. But now
1
E (O2 ) = ∫ 2 x dx = 1
0
and
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64
Chapter 7: Uncertainty and Information
13
1
0
13
E  max ( O1 , O2 )  = ∫ (1 − x)dx + ∫ 2 x dx
x =1 3
x =1

x2 
=x− 
+ ( x2 )
x =1 3
2  x=0

1 1
1
= − +1−
3 18
9
7
= .
6
The difference between the best single choice and the flexible choice is the option
value: ( 7 6) − 1 = 1 6 ≈ 0.167.
b.
We have
1
E [U (O2 )] = ∫ 2 x dx
0
1
= 2 ∫ x1 2 dx
0
x =1
2 2 32
= 
x 
 3
 x =0
2 2
3
≈ 0.94
=
and
1
E  max (U (O1 ),U (O2 ) )  = ∫ max
( 1 − x − F , 2 x − F ) dx
0
13
1
0
13
= ∫ 1 − x − F dx + ∫ 2 x − F dx.
Let’s compute these integrals by substitution separately. To compute the first
integral, substitute r = 1 − x − F . Then
1− F
13
∫ 1 − x − F dx = ∫ r dr
12
( 2 3) − F
0
r =1− F
2

=  r3 2 
3
 r = ( 2 3) − F
32
2
32
2
 
1
−
F
−
−
F
) 
(
 .
3 
3
 
To compute the second integral, substitute r = 2 x − F . Then
=
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Chapter 7: Uncertainty and Information
1
∫
13
65
2− F
1
2 x − F dx =
r1 2 dr
∫
2 ( 2 3) − F
r =2− F
r3 2
=
3 r =( 2 3) − F
32
1
32
2
 
= ( 2 − F ) −  − F   .
3 
3
 
Putting these integrals together,
32
2
32
2
 
E  max (U (O1 ),U (O2 ) )  = (1 − F ) −  − F  
3 
3
 
32
1
32
2
 
+ ( 2 − F ) −  − F   .
3 
3
 
Setting this expression equal to the utility 0.94 from the best option gives the
option value. Unfortunately, this complicated equation is difficult to solve
analytically for F . The best approach is to graph the right-hand side of the
previous equation for a range of F and see where the height of the graph hits
0.94. Using the graph below, one can determine F ≈ 0.236.
Analytical Problems
7.11
HARA utility
a.
We have
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66
Chapter 7: Uncertainty and Information
r ( w) = −
U ′′( w)
U ′( w)
−1−γ

w  1
θ (1 − γ )(−γ )  µ +   
γ  γ 

=−
−γ

w
θ (1 − γ )  µ + 
γ 

2
−1

w
= µ +  .
γ 

The reciprocal, 1 r ( w), is linear in w since it is of the form a + bw, and
1
w
=µ+ .
r ( w)
γ
b.
When µ = 0 and
 1− γ 
θ =

 γ 
γ −1
,
then
 1− γ 
U ( w) = 

 γ 
Thus,
rr ( w) = − w
γ −1
1−γ

w
0+ γ 


=
w1−γ
.
(1 − γ )1−γ
U ′′( w)
U ′( w)
(1 − γ )(−γ ) w−1−γ
(1 − γ )1−γ
= −w
(1 − γ ) w−γ
(1 − γ )1−γ
w− γ
w− γ
=γ.
The function becomes a CRRA function.
=γ
c.
From part (a),
−1

w
r ( w) =  µ +  .
γ 

If γ → ∞, then
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Chapter 7: Uncertainty and Information
67
w
→0
γ
w
µ+ →µ
γ
1
r ( w) → .
µ
Thus, r ( w) is a constant as required if γ → ∞.
d.
Let
r ( w) →
1
= A.
µ
Then
U ′′( w) = − AU ′( w).
Solving this differential equation demonstrates
U ′( w) = ke− Aw .
Integration of this shows
U ( w) = −kA−1e − Aw .
This is indeed the equation in the text for k = A.
e.
If γ = −1,
w

U ( w) = θ  µ + 
−1 

= θ ( µ − w) 2
2
= θ ( µ 2 − 2µ w + w2 ).
Thus, we have a quadratic utility function.
f.
7.12
For certain values of the parameters, utility is still unbounded, so the St.
Petersburg paradox can be recovered. Similarly, the function exhibits many of the
shortcomings of the CARA and CRRA functions discussed in the text.
More on the CRRA function
a.
A high value for 1 − R implies a low elasticity of substitution between states of
the world. A very risk-averse individual is not willing to make trades away from
the certainty line except at very favorable terms.
b.
R = 1 implies the individual is risk-neutral. The elasticity of substitution between
wealth in various states of the world is infinite. Indifference curves are linear with
slopes of −1. If R = −∞, the individual has an infinite relative risk-aversion
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68
Chapter 7: Uncertainty and Information
parameter. His or her indifference curves are L-shaped implying an unwillingness
to trade away from the certainty line at any price.
7.13
c.
A rise in pb rotates the budget constraint counterclockwise about the W g
intercept. Both substitution and income effects cause Wb to fall. There is a
substitution effect favoring an increase in Wg but an income effect favoring a
decline. The substitution effect will increase as the elasticity of substitution
between states increases (while the degree of risk-aversion declines).
d.
(1)
Find the R that solves the equation:
(W0 ) R = 0.5(1.055W0 ) R + 0.5(0.955W0 ) R .
This yields an approximate value for R of –3, a number consistent with
some empirical studies.
(2)
A 2 percent premium roughly compensates for a ±10 percent gamble:
(W0 ) −3 ≈ (0.92W0 ) −3 + (1.12W0 ) −3 .
The “puzzle” is that the premium rate of return provided by equities seems
to be much higher than this.
Graphing risky investments
a.
See graph.
The risk-free option is R. The risky option is R′.
b.
Locus RR′ represents mixed portfolios.
c.
Risk aversion as represented by curvature of indifference curves will determine
equilibrium in RR′ (say E ).
d.
With constant relative risk aversion, the indifference curve map is homothetic.
Thus, the locus of optimal points for changing values of W will be along OE.
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Chapter 7: Uncertainty and Information
7.14
69
The portfolio problem with a Normally distributed asset
From Example 7.3,
A 2
σ w.
2
For the portfolio allocation, we are looking to allocate k to the risky asset and W0 – k to
the risk-free one. Since the risky asset r% is normally distributed with the distribution
N ( µr , σ r ) and final wealth is given by
W = W0 (1 + rf ) + k ( r% − rf )
E [U (W ) ] = µW −
(see Equation 7.48), final wealth is distributed as
E (W ) = W0 (1 + rf ) + k ( µ r − rf )
and
σ W2 = k 2σ r2 .
Expected utility is given by
A
E [U (W ) ] = W0 (1 + rf ) + k ( µ r − rf ) − k 2σ r2 .
2
Maximizing this with respect to k yields
µr − rf − Akσ r2 = 0.
Hence,
k=
ur − rf
.
Aσ r2
This makes sense intuitively:
∂k
∂k
∂k
> 0,
< 0,
< 0.
2
∂µ r
∂σ r
∂A
In words, a higher expected rate of return on the risky assets causes more to be invested
in it, whereas either a greater variance of return of a higher level of risk aversion causes
less to be invested in it.
Behavioral Problem
7.15
Prospect theory
Scenario 1
Gamble
Expected Wealth
A
1,000 + (1 2) (1,000 + 0) = 1,500
B
1, 000 + 500 = 1, 500
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70
Chapter 7: Uncertainty and Information
Scenario 2
Gamble
Expected wealth
C
2,000 − (1 2 ) (1,000 + 0) = 1,500
D
2, 000 − 500 = 1, 500
a.
Scenarios ( A )–( D ) provide the same expected wealth—$1,500—so a riskneutral Stan should be indifferent among them.
b.
The expected wealth levels are the same. Thus, a risk-averse Stan should select
the safe option B in Scenario 1 and D in Scenario 2.
c.
It is natural to suppose subjects are risk averse, so more should choose option D
in Scenario 2.
d.
(1)
Pete should make the same choices as the majority of experimental
subjects. Scenario 1 involves gains, so Pete behaves as predicted by
expected utility theory there. Scenario 2 involves losses. The certainty of a
small loss may be worse than a smaller chance of a large loss, so option C
may be preferred.
(2)
Utility
Scenario 2
•
Scenario 1
•
1,000 2,000
Wealth
The utility curve has to shift because of the kink at the anchor point.
Prospect Pete’s curve changes from convex to concave at the anchor point;
Standard Stan’s is linear or concave everywhere, so doesn’t have to shift.
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70
CHAPTER 8:
Strategy and Game Theory
These problems cover a variety of different concepts introduced in the chapter. They
range in difficulty from the simplest exercise of finding the Nash equilibrium in a twoby-two matrix to characterizing equilibrium when players have continuous actions and
payoffs with general functional forms. Practice with problems may be the primary way
for students to master the material on game theory.
Comments on Problems
8.1
This problem provides practice in finding pure- and mixed-strategy Nash
equilibria using a simple payoff matrix. The three-by-three payoff matrix makes
the problem slightly harder than the simplest case of a two-by-two matrix.
Although this problem points the student where to look for the mixed-strategy
equilibrium, in other cases there may be many possibilities that need to be
checked for mixed-strategy equilibria. In a game represented by a three-by-three
matrix, each player has four combinations of two or more actions, and so there
are 16 possible types of mixed-strategy equilibria to check. Software, called
Gambit, has been developed that can solve for all the Nash equilibria of games
the user specifies in extensive or normal form. Gambit is freely available on the
Internet. It is easy to use, almost functioning as a “game-theory calculator.” One
useful classroom exercise would be to have students solve some of the problems
on a game-theory problem set using Gambit, either alone or in teams.
McKelvey, R. D., A. M. McLennan, and T. L. Turocy (2007) Gambit: Software
Tools for Game Theory, Version 0.2007.01.30.
http://econweb.tamu.edu/gambit
8.2
A slight generalization of payoffs in the Battle of the Sexes provides students
with further practice in computing mixed-strategy Nash equilibria.
8.3
This problem provides practice in converting the payoff matrix for a
simultaneous game into one for a sequential game. Illustrates the application of
subgame-perfect equilibrium in the simple case of the famous Chicken game.
8.4
This problem provides practice in computing the Nash equilibrium in a game
with continuous actions (similar to the Tragedy of the Commons in this chapter
and in Chapter 15 with the Cournot game, except in this problem the bestresponse functions are upward-sloping). Players’ best responses are computed
using calculus, and the resulting equations are then solved simultaneously.
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71
Chapter 8: Strategy and Game Theory
8.5
This problem asks students to solve for the mixed-strategy Nash equilibrium
with a general number of players n . The “punchline” to the problem that the
blond is less likely to be approached as the number of males increases is a
paradoxical result characteristic of such games. The problem is based on a scene
in the Academy Award winning movie, A Beautiful Mind, about the life of John
Nash, in which the Nash character discovers his equilibrium concept (the one
scene in the movie that involves any game theory). If the classroom facilities
allow, it is worthwhile to show students this scene (Scene 5: “Governing
Dynamics”) when covering this problem.
8.6
This problem gives the student practice with the repeated version of the
Prisoners’ Dilemma, adjusting the payoffs in the version given in the text.
8.7
A simultaneous game of incomplete information providing practice in finding
the Bayesian–Nash equilibrium. Similar to the Tragedy of the Commons in
Example 8.6.
8.8
This problem asks students to solve for a hybrid perfect Bayesian equilibrium.
Students may find the application interesting given the growth in popularity of
poker on television, in particular Texas Hold ‘Em (to which the name “Blind
Texan” in the problem is meant to be a tongue-in-cheek reference). In typical
intermediate microeconomics courses, instructors will have only a short time to
cover signaling games, and in such courses it would be perfectly reasonable to
omit this problem, focusing exclusively on the simpler computations associated
with separating and pooling equilibria. Two reasons to delve into hybrid
equilibria if there is sufficient time—say in an advanced course with an
extensive game theory component—are, first, that games like Blind Texan do
not have separating and pooling equilibria, only a hybrid one and, second, that
the full power of Bayes Rule in a signaling game is only apparent with a hybrid
equilibrium since the application of the rule with separating and pooling
equilibria is fairly trivial.
Analytical Problems
8.9
Alternatives to Grim Strategy. This problem provides further practice with the
discounting calculations associated with infinitely repeated games. Demonstrates
the value of harsh punishments in sustaining cooperation by examining the
difficulty in sustaining cooperation with less than grim-strategy punishments.
8.10
Refinements of perfect Bayesian equilibrium. Part (a) is standard. Given it is
the simplest problem on signaling games, all instructors who cover the topic
should consider including it in the problem set. Part (b) goes beyond the material
in the chapter in exploring the intuitive criterion, a refinement of perfect Bayesian
equilibrium, which restricts posterior beliefs to be “reasonable.”
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72
Chapter 8: Strategy and Game Theory
Behavioral Problems
8.11
Fairness in the Ultimatum Game. This problem was added for professors
interested in including some behavioral economics in their course. The problem
covers the canonical Fehr–Schmidt fairness model, one of the simplest models in
which players care not just about their own payoffs but about the payoffs of
others. It applies the analysis to help understand experimental results from the
Ultimatum and Dictator games. The absolute value sign in the payoff function
gives some students trouble, so the answers dwell on how to deal with that in the
calculations.
8.12
Rotten Kid Theorem. This problem analyzes altruism, included in the
behavioral problems because it departs from the standard, selfish preferences.
Perhaps the most challenging problem in the chapter since it works with general
functional forms, so requires the application of the implicit function theorem
rather than the computation of explicit derivatives. Shows how subgame-perfect
equilibrium concept can be used to derive one of Nobel-prize winner Gary
Becker’s famous results. The parent-child application may hold interest for
students.
Solutions
8.1
a.
(C , F ).
b.
Let α and 1 − α be the probabilities that player 1 plays A and B ,
respectively. Player 2’s expected payoff from playing D then is
6α + 8(1 − α ) and from playing E is 8α + 6(1 − α ). For player 2 to be
indifferent between D and E and thus willing to randomize, these two
expressions must be equal, implying α * = 1 2. Similar calculations show
that player 2 randomizes with equal probability between D and E.
c.
Players each earn 4 in the pure-strategy equilibrium. Player 2 earns
6α * + 8(1 − α * ) = 6(1 2) + 8(1 2) = 7 in the mixed-strategy equilibrium.
Similar calculations show player 1 earns 6 in this equilibrium.
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73
Chapter 8: Strategy and Game Theory
d.
2
●
A
1●
B
2
●
C
2
●
8.2
D
E
7, 6
F
0, 0
5, 8
D
E
5, 8
F
1, 1
D
E
0, 0
F
4, 4
7, 6
1, 1
Let α and 1 − α be the wife’s probabilities, respectively, of playing ballet and
boxing. The husband’s expected payoff from ballet then is
(1)(α ) + (0)(1 − α ) = α
and from boxing is
(0)(α ) + ( K )(1 − α ) = K − K α .
For the husband to be indifferent, and thus willing to randomize, these two
expressions must be equal, implying α * = K (1 + K ). Similar calculations show
the husband plays boxing with probability K (1 + K ) and ballet with the
complementary probability. Substituting K = 2 allows us to recover the mixedstrategy equilibrium found in Example 8.3.
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74
Chapter 8: Strategy and Game Theory
8.3
a.
2, 2
Veer
2
●
Veer
Don’t
veer
Don’t
veer
Veer
1, 3
1●
3, 1
2
●
Don’t
veer
0, 0
b.
(Don’t veer, veer) and (veer, don’t veer).
c.
Let α and 1 − α be teen 1’s probabilities, respectively, of veering and not.
Teen 2’s expected payoff from veering then is ( 2)(α ) + (1)(1 − α ) = 1 + α
and from not veering is (3)(α ) + (0)(1 − α ) = 3α . For teen 2 to be
indifferent, and thus willing to randomize, these two expressions must be
equal, implying α * = 1 2. Symmetrically, teen 2 randomizes with equal
probabilities over the two actions.
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75
Chapter 8: Strategy and Game Theory
Teen 2’s
probability
of veering
1●
E1
BR2
●
1/2
BR1
Mixed-strategy
equilibrium
E
0
d.
●2
1
1/2
Teen 1’s
probability
of veering
Teen 2 has four contingent strategies: always veer, take the same action as
teen 1, do the opposite of teen 1, never veer. The normal and extensive
forms for the game are as follows:
Teen 2
(Veer | Veer
Veer | Don’t)
(Veer | Veer
Don’t | Don’t)
(Don’t | Veer
Veer | Don’t)
(Don’t | Veer
Don’t | Don’t)
2, 2
2, 2
1, 3
1, 3
3, 1
0, 0
3, 1
0, 0
Teen 1
Veer
Don’t
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76
Chapter 8: Strategy and Game Theory
2, 2
Veer
2
●
Veer
Don’t
veer
Don’t
veer
Veer
1, 3
1●
3, 1
2
●
Don’t
veer
8.4
0, 0
d.
There are three Nash equilibria: 1 veers and 2 never veers, 1 doesn’t veer
and 2 always does, and 1 doesn’t veer and 2 does the opposite of 1.
e.
The game has three subgames: the game itself and the subgames starting
from the node at which teen 2 moves. The Nash equilibrium following 1’s
having veered is for 2 not to and following 1’s having not veered for 2 to
veer. Thus, 2’s strategy must be to do the opposite of 1 in a subgameperfect equilibrium. Teen 1 thus would choose not to veer. The Nash
equilibrium in which 2 always veers is unreasonable because 2 would
prefer not to veer if he sees 1 has first; the Nash equilibrium in which 2
never veers is unreasonable because 2 would prefer to veer if he sees 1 has
not.
a.
Homeowner 1’s objective function is
l1 (10 − l1 + l2 2) − 4l1.
Taking the first-order condition with respect to l1 and rearranging yields
the best-response function l1 = 3 + l2 4. Symmetrically, Homeowner 2’s
best-response function is l2 = 3 + l1 4. Solving simultaneously yields
l1* = l2* = 4.
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77
Chapter 8: Strategy and Game Theory
b.
8.5
c.
The change is indicated by the shift (following the arrow) in Homeowner
1’s best-response function. In the new Nash equilibrium, 1 mows a lot less
and 2 mows a little less.
a.
If all play blond, then one would prefer to deviate to brunette to obtain a
positive payoff. If all play brunette, then one would prefer to deviate to
blond for the higher payoff of a rather than b.
b.
Playing brunette provides the male with a certain payoff of b. Playing
blond provides a payoff of a with probability (1 − p) n −1. This is the
probability no other player approaches the blond. Equating the two
payoffs yields:
1
 b  n −1
p* = 1 −   .
a
c.
The probability the blond is approached by at least one male equals 1
minus the probability no males approach her:
n
 b  n−1
1 − (1 − p* ) n = 1 −   .
a
This expression is decreasing in n because the exponent n ( n − 1) is
decreasing in n and the base of the exponent, b a , is a fraction.
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78
Chapter 8: Strategy and Game Theory
8.6
a.
Using the underlining algorithm or other method, one can verify that
finking is still a dominant strategy for the players and that the Nash
equilibrium is (fink, fink).
b.
Cooperation on silent is best sustained using grim strategies as described
in the text. In this cooperative equilibrium, each player earns present
discounted value of 1 each period:
V eq = 1 + δ (1) + δ 2 (1) + L
= 1(1 + δ + δ 2 + L)
 1 
= 1

 1− δ 
1
=
.
1− δ
The most a player can earn from deviating is a present discounted value of
V dev = 3 + δ (0) + δ 2 (0) + L
= 3.
The player earns 3 in the deviation period from his/her surprise fink, but
then players revert to the static Nash equilibrium of (fink, fink) from then
on. Cooperation is sustainable if
V eq ≥ V dev
1
⇒
≥3
1− δ
⇒ 1 ≥ 3(1 − δ ),
implying δ ≥ 2 3.
8.7
a.
The best-response function is lLC = 3.5 + l2 4 for the low-cost type of
Player 1, lHC = 2.5 + l2 4 for the high-cost type, and l2 = 3 + l1 4 for
Player 2, where l1 is the average for Player 1. Solving these equations
*
*
yields lLC
= 4.5, lHC
= 3.5, and l2* = 4.
b.
Player 2 best responds to the average best response across the two types of
player 1, given by the dashed line between the two best responses, and
resulting in a choice of landscaping level given by the dotted horizontal
line. The two types of player 1 best respond to the equilibrium landscaping
effort of player 2, resulting in the outcome labeled HC if player 1 is the
high-cost type and LC if player 1 is the low-cost type.
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79
Chapter 8: Strategy and Game Theory
l2
BRHC(l2)
BRLC(l2)
BR2(l1)
HC
LC
l1
c.
The low-cost type of player 1 earns 20.25 in the Bayesian–Nash
equilibrium and 20.55 in the full-information game, so would prefer to
signal its type if it could. Similar calculations show that the high-cost
player would like to hide its type.
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publicly accessible website, in whole or in part.
80
Chapter 8: Strategy and Game Theory
8.8
a.
Stay
-100, 100
Fold
50, -50
Stay
100, -100
Fold
50, -50
2
●
Stay
n1
1
●
Pr(H)
= 6/13
2
Fold
Stay
Pr(L)
= 7/13
●
n2
-50, 50
1
●
Fold
-50, 50
b.
In a hybrid equilibrium, at least some type of some player plays a mixed
strategy. If player 1 sees the low card, she prefers the pure strategy of
staying. So it must be that player 1 randomizes after seeing a high card.
(Verify that if player 1 plays a pure strategy of either folding or staying,
player 2’s best response is also a pure strategy, so the equilibrium would
not be a hybrid one.) For brevity, we will say that player 1 is the “high
type” if she sees a high card drawn and a “low type” if she sees a low card
drawn.
Let α and 1 − α be the probabilities that the high type stays and
folds, respectively. In order for the high type to be willing to randomize, it
must be that player 2 randomizes as well. (Verify that this is the case.) Let
β and 1 − β be the probabilities that the high type stays and folds,
respectively. Then β must be such that the high type is indifferent
between staying and folding for her to be willing to randomize. Staying
provides the high type with an expected payoff of β ( −100) + (1 − β )(50),
and folding provides her with a payoff of –50. Equating these two
expressions and solving yields β * = 2 3.
In order for player 2 to be willing to randomize, he must be
indifferent between staying and folding. His expected payoff from staying
is
Pr( H | stay)(100) + [1 − Pr( H | stay)]( −100),
where Pr( H | stay) is the posterior probability that player 1 is the high
type conditional on her staying. Player 2’s payoff from folding is –50.
Equating the two expected payoffs yields Pr( H | stay) = 1 4. Pr( H | stay)
must also satisfy Bayes’ rule:
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81
Chapter 8: Strategy and Game Theory
Pr(stay | H ) Pr( H )
Pr(stay | H ) Pr( H ) + Pr(stay | L) Pr( L)
α (6 13)
=
α (6 13) + (1)(7 13)
Pr( H | stay) =
6α
.
6α + 7
Equating this last expression with Pr( H | stay) = 1 4 and solving yields
α * = 7 18.
To summarize, in the hybrid equilibrium, the low type always
stays, the high type mixes between staying and folding with probabilities
7 18 and 11 18, and player 2 randomizes between staying and folding
with probabilities 2 3 and 1 3. Player 2’s posterior beliefs are that player
1 is the high type with certainty if she folds; if she stays she is the high
type with probability 1 4 and the low type with probability 3 4.
=
c.
The low type’s expected payoff is
100 ( 2 3) + 50 (1 3) = 83.3.
The high type’s expected payoff is –50 (she is indifferent between staying
and folding in equilibrium, and earns –50 from folding). Given the prior
probabilities of being a high and low type, player 1’s expected payoff
from the game (prior to learning her type) is
83.3 ( 7 13) + ( −50 )( 6 13) = 21.8.
Player 2’s expected payoff is –50 (he is indifferent between staying and
folding in equilibrium and earns –50 from folding). The game is clearly
tilted toward player 1.
Analytical Problems
8.9
Alternatives to Grim Strategy
a.
Cooperating gives a stream of per-period payoffs of 2, for a present
discounted value of 2 (1 − δ ). If players use tit-for-tat strategies, the
present discounted value from deviating to fink at the start of the game is
 2 
3 + δ ⋅1 + δ 2 ⋅ 
.
1−δ 
The deviator earns 3 in the first period, followed by a period in which both
fink and earn 1, followed by a return to cooperating in the third period and
thereafter. For the displayed payoff not to exceed 2 (1 − δ ), δ ≥ 1. The
players must be infinitely patient.
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82
Chapter 8: Strategy and Game Theory
If players use two periods of punishment, the present discounted
value from deviating is
 2 
3 + δ ⋅1 + δ 2 ⋅1 + δ 3 
.
1−δ 
For the displayed payoff not to exceed 2 (1 − δ ) , we see, upon multiplying
by 1 − δ and simplifying, the required condition is 2δ − δ 3 − 1 ≥ 0.
Factoring,
2δ − δ 3 − 1 = (1 − δ )(δ 2 + δ − 1).
Hence, the required condition can be written
δ 2 + δ − 1 ≥ 0.
Using the quadratic formula to obtain the roots of this quadratic, we have
δ ≥ 0.62.
b.
The required condition is that the present discounted value of the payoffs
from cooperating, 2 /(1 − δ ) , exceed that from deviating,
δ (1 − δ 10 ) 2δ 11
+
.
1−δ
1−δ
Simplifying,
2δ − δ 11 − 1 ≥ 0.
As the graph below shows, the expression 2δ − δ 11 − 1 crosses the x-axis
very slightly to the left of 0.5. Using numerical methods or a more precise
graph, it can be shown that the condition is δ ≥ 0.50025 . The resulting
condition is very close to the condition for cooperation with infinitely
many periods of punishment ( δ ≥ 1 2 ).
3+
0.03
0.02
0.01
0
0.49
0.495
0.5
0.505
0.51
-0.01
-0.02
-0.03
8.10
Refinements of perfect Bayesian equilibrium
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83
Chapter 8: Strategy and Game Theory
a.
The key condition is for the firm to be willing to offer a job to an
uneducated worker. (Regarding the other player, the worker, all worker
types obtain the highest payoffs possible, since they are hired and don’t
have to expend the cost of education.) The firm’s expected payoff from J
is
Pr( H | NE )(π − w) + [1 − Pr( H | NE )]( − w)
and from NJ is 0. The displayed expression exceeds 0 if
Pr( H | NE ) ≥ w π . According to Bayes’ rule, along the equilibrium path,
posterior beliefs are the same as prior beliefs in a pooling equilibrium.
Therefore, Pr( H | NE ) = Pr( H ) . The required condition for the specified
pooling equilibrium, thus, is Pr( H ) ≥ w π . All out-of-equilibrium beliefs
and strategies are consistent with this pooling equilibrium. If
Pr( H | E ) ≥ w π , then the firm would choose J conditional on observing
E. On the other hand, if Pr( H | E ) ≤ w π , then the firm would choose NJ
conditional on observing E.
b.
For the firm to prefer not to offer a job to an uneducated worker,
calculations similar to those in Part (a) (but with the inequalities reversed)
imply Pr( H ) ≤ w π . A high-skilled worker would deviate to E unless the
firm chooses NJ conditional on E. The firm prefers NJ to J conditional on
E when the out-of-equilibrium posterior beliefs satisfy Pr( H | E ) ≤ w π or
equivalently Pr( L | E ) ≥ 1 − w π . Suppose cH < w < cL . Then it would be
unreasonable to think that type L would ever deviate to E. Regardless of
what strategy the firm plays, type L’s payoff would be negative from E
and non-negative from NE. (By contrast, type H may have an incentive to
deviate: he or she earns a positive payoff if the firm plays J conditional on
E.) The Cho-Kreps intuitive criterion restricts the out-of-equilibrium
posterior belief Pr( L | E ) = 0. Since Pr( L | E ) = 0 is inconsistent with the
required condition Pr( L | E ) ≥ 1 − w π , the Cho-Kreps intuitive criterion
rules out the pooling equilibrium specified in this part, leaving only the
one specified in part (a).
Behavioral Problems
8.11
Fairness in the Ultimatum Game
a.
Solve using backward induction, starting with the responder. The
responder certainly accepts any offer of r > 0. The remaining question is
how he/she responds to an offer of r = 0. It turns out that the responder
must also accept an offer of 0 in equilibrium. If he/she rejected this, there
would be no equilibrium because whatever positive offer might be offered
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84
Chapter 8: Strategy and Game Theory
in equilibrium, there is always another lower positive offer that would also
be accepted but that would give the proposer a higher payoff. So in sum
the responder must accept all offers. Given the responder’s strategy, the
proposer’s equilibrium offer is r * = 0.
b.
The outcome in the Dictator Game is the same, also involving equilibrium
offer r * = 0.
c.
(1)
The answer here is a bit technical because of the absolute value
sign in the utility function, requiring the analysis of two cases. We
will avoid this technicality by noting that the proposer would never
offer r > 1 2. The responder obtains utility 1 2 > 0 from offer
r = 1 2 and would certainly accept it. Higher offers just reduce the
proposer’s utility (in two ways, reducing the money the proposer
obtains and reducing the fairness of the outcome—skewing things
toward the responder) without increasing the chance of acceptance.
So assume r ≤ 1 2 for the rest of the question.
The responder accepts any offer r providing a positive
payoff. In fact, we can say more: for the same reasons as in part
(a), the responder accepts any offer r providing him/her with a
non-negative payoff. The responder’s utility is
U 2 (r ,1 − r ) = r − a r − (1 − r )
= r − a (1 − 2r ) .
The first line follows from substituting into the utility formula
given in the text. The second line, removing the absolute value
signs, follows from r ≤ 1 2. The responder accepts r if
U 2 (r ,1 − r ) = r − a (1 − 2r ) ≥ 0,
implying
a
r≥
.
1 + 2a
(2)
The proposer obtains payoff
U1 (1 − r , r ) = (1 − r ) − a (1 − r ) − r
= 1 − r − a (1 − 2r ) .
For a < 1 2 , this payoff is higher the lower is r. Thus, the
equilibrium offer is the lowest r that would be accepted, that is,
the r for which the last condition in part (1) holds with equality:
a
r* =
.
1 + 2a
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85
Chapter 8: Strategy and Game Theory
(3)
8.12
In the Dictator Game, for a < 1 2 , the proposer makes the lowest
possible offer because, as just argued, his/her payoff is decreasing
in r but now there is no danger of rejection. The equilibrium is
r * = 0.
We conclude that with fairness preferences, the equilibrium
offer is higher and the division of money more even in the
Ultimatum than the Dictator Game. How even can the division be
in the Ultimatum Game? In the limit as a → 1 2 from below,
r * → 1 4, as one can see using the formula from part (2).
For a ≥ 1 2 , the equilibrium offer jumps to r * = 1 2
because the proposer prefers an even split independent of rejection.
But this is also the equilibrium offer in the Dictator Game. So the
biggest gap between the outcomes in the Ultimatum and Dictator
Games is observed in the limit as a → 1 2.
Rotten Kid Theorem
In the second stage, the parent chooses L to maximize
U 2 (Y2 (r ) − L) + αU1 (Y1 (r ) + L),
yielding first-order condition:
− U 2′ (Y2 ( r ) − L ) + αU 1′ (Y1 ( r ) + L ) = 0 .
Even though the preceding equation cannot be solved explicitly for L* (r ), we can
still use the implicit function rule to find the derivative:
dL* U 2′′Y2′(r ) − αU1′′Y1′(r )
=
.
dr
−U 2′′ + αU1′′
In the first state, the child maximizes U1 (Y1 ( r ) − L* ( r )), yielding first-order
condition

dL* 
U1′ Y1′(r ) +

dr 



U1′
=
 [Y1′(r )(−U 2′′ + αU1′′) + U 2′′Y2′(r ) − αU1′′Y1′(r )]
 −U 2′′ + αU1′′ 
 U1′U 2′′ 
=
 [Y1′(r ) + Y2′(r )] = 0.
 −U 2′′ + αU1′′ 
This equation implies Y1′(r ) + Y2′(r ) = 0, the first-order condition for maximizing
their joint incomes.
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CHAPTER 9:
Production Functions
Because the problems in this chapter do not involve optimization (cost minimization principles
are not presented until Chapter 10) they tend to have a rather uninteresting focus on functional
form. Computation of marginal and average productivity functions is stressed along with a few
applications of Euler’s theorem. Instructors may want to assign one or two of these problems for
practice with specific functions, but the focus for Part (4) problems should probably be on those
in Chapters 10 and 11.
Comments on Problems
9.1
This problem illustrates the isoquant map for fixed proportions production functions.
Parts (c) and (d) show how variable proportions situations might be viewed as limiting
cases of a number of fixed proportions technologies.
9.2
This problem provides some practice with graphing isoquants and marginal productivity
relationships.
9.3
This problem explores a specific Cobb–Douglas case and begins to introduce some ideas
about cost minimization and its relationship to marginal productivities.
9.4
This problem involves production in two locations and develops the equal marginal
products rule.
9.5
This problem is a thorough examination of most of the properties of the general two-input
Cobb–Douglas production function.
9.6
This problem is an examination of the marginal productivity relations for the CES
production function.
9.7
This problem illustrates a generalized Leontief production function and provides a twoinput illustration of the general case, which is treated in the extensions.
9.8
Application of Euler’s theorem to analyze what are sometimes termed the “stages” of the
average–marginal productivity relationship. The terms “extensive” and “intensive”
margin of production might also be introduced here, although that usage appears to be
archaic.
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88
Chapter 9: Production Functions
Analytical Problems
9.9
Local returns to scale. This problem introduces the local returns-to-scale concept and
presents an example of a function with variable returns to scale.
9.10
Returns to scale and substitution. This problem shows how returns to scale can be
incorporated into a production function while retaining its input substitution features.
9.11
More on Euler’s theorem. This problem shows how Euler’s theorem can be used to
study the likely signs of cross-productivity effects.
Solutions
9.1
a, b.
k per
period
Large-mower
technology
10
Small-mower
technology
8
5
8
l per
period
With the small-mower technology, k = 1 and l = 1 are needed to mow 5,000 sq.
ft. To mow 40,000 = 8 ⋅ 5,000, need to scale inputs up by 8, that is, k = 8 and
l = 8. Similar calculations show that with the large-mower technology, use
k = 10 and l = 5.
c.
To mow half of the total 40,000 with each technology, use half of the inputs from
part (b), so allocate k = 4 and l = 4 to the small-mower technology and k = 5 and
l = 2.5 to the large-mower technology, for a total of k = 9 and l = 6.5.
To mow only a quarter of 40,000 with the small-mower technology,
allocate a quarter of the inputs from part (b) ( k = 2 and l = 2 ) to production using
the small-mower technology and 3 4 of the inputs from part (b) ( k = 7.5 and
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Chapter 9: Production Functions
89
l = 3.75 ) to the large-mower technology for a total of k = 9.5 and l = 5.75.
We can interpret fractions of k and l as use of the input for only part of
an hour.
d.
We know from part (c) that the combinations ( k = 9, l = 6.5) and
( k = 9.5, l = 5.75) can be used to mow at least 40,000 sq. ft. Let’s take for
granted for now that 40,000 is in fact the most that can be mowed with these
combinations. Then both these combinations are on the q = 40, 000 isoquant, as
pictured in this graph:
Assuming that the isoquant is linear, one can use the point-slope form of a line to
show that the equation for the line between the two combinations is
40 2l
(1) k =
− .
3 3
We made some assumptions to enable us to graph this isoquant, namely
that 40,000 was the highest quantity that could be produced by the input
combinations and that the isoquant was a line between the two points, not some
other curve. The formal proof that Equation 1 is the q = 40, 000 isoquant is hard.
Let (k1 , l1 ) be the inputs used in the small-mower technology and (k2 , l2 ) in the
large-mower technology. Letting ( k , l ) be the total input combination, we have
(2) k = k1 + k2 ,
(3) l = l1 + l2 .
Total output of 40,000 is the sum of the output from technologies 1 and 2, both
Leontief:
k

(4) 40, 000 = 5, 000 min ( k1 , l1 ) + 8, 000 min  2 , l2  .
 2

For efficient production, there should be no wastage, so
(5) min ( k1 , l1 ) = k1 = l1 ,
k
 k
(6) min  2 , l2  = 2 = l2 .
 2  2
Equations 2, 3, 5, and 6 are four equations that can be used to solve for the four
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90
Chapter 9: Production Functions
unknowns k1 , l1 , k2 , l2 in terms of k and l. Sparing the details, the solution is
k1 = 2l − k ,
l1 = 2l − k ,
k2 = 2l − 2k ,
l2 = k − l .
Substituting these solutions into Equations 5 and 6,
min ( k1 , l1 ) = 2l − k ,
k

min  2 , l2  = k − l.
 2 
Substituting these expressions back into Equation 4,
40, 000 = 5, 000 ( 2l − k ) + 8, 000 ( k − l ) .
One can rearrange this equation into Equation 1.
9.2
Given production function q = kl − 0.8k 2 − 0.2l 2 .
a.
When k = 10, total labor productivity is
TPl = 10l − 0.2l 2 − 80,
and average labor productivity is
q
80
AP l = = 10 − 0.2l − .
l
l
To find where APl reaches a maximum, take the first-order condition:
dAP l 80
=
− 0.2 = 0.
dl
l
The maximum is at l = 20. When l = 20, q = 40. The graph is provided after
part (b).
b.
Marginal labor productivity is
dq
MPl =
= 10 − 0.4l.
dl
To find where this is 0, set MPl = 10 − 0.4l = 0, implying l = 25.
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Chapter 9: Production Functions
c.
91
If k = 20,
TPl = 20l − 0.2l 2 − 320 = q,
AP l = 20 − 0.2l −
320
,
l
MPl = 20 − 0.4l.
APl reaches a maximum at l = 40, q = 160. At l = 50, MPl = 20 − 0.4l = 0.
d.
9.3
Doubling of k and l here multiplies output by 4 (compare parts (a) and (c)).
Hence, the function exhibits increasing returns to scale.
Given production function q = 0.1k 0.2l 0.8 .
a.
Given Sam spends $10,000 in total and equal amounts on both inputs, he spends
$5,000 on each. At the $50 per hour, he uses inputs k = 100, l = 100, and
produces output q = 10. Total cost is 10,000 (by design).
b.
We have
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92
Chapter 9: Production Functions
∂q
l
= 0.02   ,
∂k
k
0.8
MPk =
0.2
k
MPl = 0.08   .
l
Setting these equal yields l k = 4. Substituting into the production function,
q = 10 = 0.1k 0.2 (4k )0.8 = 0.303k .
Solving, k ≈ 33 and l ≈ 132. Total cost is 8,250.
c.
The cost savings in part (b) is 1,750. We saw in part (b) that $8,250 used in the
way Norm suggested produced 10 stools. Because the production function
exhibits constant returns to scale, if the full $10,000 were spent to produce stools
following Norm’s suggestion, more stools can be produced in proportion:
10, 000
×10 = 12.12,
8, 250
a little more than two extra stools.
9.4
d.
Carla’s ability to influence the decision depends on whether she provides a unique
input for Cheers. If she does, she can either resist increasing the number of stools
or can allow an increase in stools in exchange for a higher salary.
a.
The firm should allocated labor such that MPl1 =MPl2 . Thus,
∂q1 ∂q2
=
∂l1 ∂l2
⇒
∂ (10l10.5 ) ∂ (50l20.5 )
=
∂l1
∂l2
⇒ 5l1−0.5 = 25l2−0.5
l
1
⇒ 1 = .
l2 25
b.
In addition to the previous equation, we have l1 + l2 = l. Solving these two
equations for l1 and l2 as functions of l yields
l
25
l1 =
, l2 =
l.
26
26
Substituting into the production function and then summing over the two
locations, total output is
q = q1 + q2 = 10l10.5 + 50l20.5 = 10 26l .
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Chapter 9: Production Functions
9.5
93
Given production function q = Ak α l β .
a.
f k = α Ak α −1l β > 0,
f l = β Ak α l β −1 > 0,
f kk = α (α − 1) Ak α − 2l β < 0,
f ll = β ( β − 1) Ak α l β − 2 < 0,
f kl = αβ Ak α −1l β −1 > 0.
∂q k
k
⋅ = α Ak α −1l β ⋅ = α ,
∂k q
q
∂q l
k
eq ,l =
⋅ = β Ak α l β −1 ⋅ = β .
∂l q
q
b.
eq ,k =
c.
f (tk , tl ) = t α + β Ak α l β ,
∂q t
t
eq ,t = lim ⋅ = lim(α + β )t α + β −1q ⋅ = α + β .
t →1 ∂t q
t →1
q
d.
Quasi-concavity follows from the signs in part (a).
e.
Concavity looks at:
f kk f ll − f kl2 = α (α − 1) β ( β − 1) A2 k 2α − 2l 2 β − 2 − α 2 β 2 A2 k 2α − 2l 2 β − 2
= A2 k 2α − 2l 2 β − 2αβ (1 − α − β ).
This expression is positive (and the function is concave) only if α + β < 1.
9.6
a.
We have
MPk =
∂q 1 ρ ρ 1−ρρ
=  k + l  ⋅ ρ k ρ −1
∂k ρ
=q
1− ρ
⋅ k ρ −1
1− ρ
q
=  .
k
Similar manipulations yield
q
MPl =  
l
b.
1− ρ
.
Using the results from part (a),
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94
Chapter 9: Production Functions
1− ρ
RTS =
MPk  l 
= 
MPl  k 
.
Inverting,
1
l
= RTS 1− ρ ,
k
implying
1
k
= RTS ρ −1 ,
l
in turn implying
1
k
ln   =
ln RTS .
 l  ρ −1
From Equation 9.32,
d ln ( k l )
1
σ=
=
.
d ln RTS ρ − 1
c.
Computing elasticities,
−ρ
∂q k  q 
1
eq ,k =
⋅ =  =
,
∂k q  k 
1+ (l k ) ρ
q
e q, l =  
l
−ρ
=
1
1+ ( k l )
ρ
=
1
1+ ( l k )
−ρ
.
Putting these over a common denominator yields eq ,k + eq ,l = 1, which shows
constant returns to scale.
d.
9.7
The result follows directly from part (a) since
1
σ=
.
1− ρ
Given production function f (k , l ) = β 0 + β 1 kl + β 2 k + β 3 l.
a.
For constant returns to scale, f (tk , tl ) = tf (k , l ). But
f (tk , tl ) = β 0 + β 1 tk ⋅ tl + β 2tk + β 3tl
(
)
= β 0 + t β 1 kl + β 2k + β 3 tl ,
while
(
)
tf (tk , tl ) = t β 0 + t β 1 kl + β 2k + β 3 tl .
For these two equations to be equal, β 0 = 0.
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Chapter 9: Production Functions
b.
95
Assume β 0 = 0 to ensure constant returns to scale. Then
k
MPl = 0.5β1  
l
0.5
+ β3 ,
0.5
l
MPk = 0.5β1   + β 2 .
k
Both are homogeneous of degree zero with respect to ( k , l ) and exhibit
diminishing marginal productivities.
c
Footnote 6 provides the key formula in the special case of constant returns to
scale:
f f
σ= l k
f ⋅ f kl
( β1 2 )( k l )1 2 + β 3  ( β1 2 )( l k )1 2 + β 2 

.
=
 β1 ( kl )1 2 + β 2 k + β3l  ( β1 4 )( kl ) −1 2 



σ
=
0,
For
one of the factors in the numerator has to be 0. For this to be
true for all ( k , l ) , either β1 = β 3 = 0 or β1 = β 2 = 0. In other words, β1 = 0 and
either β 3 = 0 or β 2 = 0.
For σ = 1, the numerator has to equal the denominator. Expanding out the
numerator gives
β12 β1β 3
ββ
12
12
+
( l k ) + 1 2 ( k l ) + β 2 β3
4
2
2
and the denominator gives
β12 β1β 3
ββ
12
12
+
(l k ) + 1 2 ( k l ) .
4
4
4
For these two expressions to be equal for all ( k , l ) requires
β1β3 = β1β 2 = β 2 β 3 = 0.
This condition holds if any two of the three parameters β1 , β 2 , β 3 are 0.
For σ = ∞, the denominator must be 0. This only holds for all ( k , l ) if the
second factor is 0, that is,
−1 2
( β1 4 )( kl ) = 0.
For this condition to hold for all ( k , l ) requires β1 = 0.
9.8
Because q = f ( k , l ) exhibits constant returns to scale, it is homogeneous of degree 1. In
this special case, Euler’s theorem (Equation 2.109) states
q = f ( k , l ) = f k k + f l l.
Rearranging,
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96
Chapter 9: Production Functions
q
k
= fl + f k   .
l
l
If f l > q l , then f k < 0. But no firm would ever produce in such a range.
Analytical Problems
9.9
Local returns to scale
a.
If f (tk , tl ) = tf ( k , l ), then
∂f (tk , tl )
t
eq ,t = lim
⋅
t →1
∂t
f (tk , tl )
f (k , l )
= lim
t →1 f ( k , l )
= 1.
b.
eq ,t = lim
∂f (tk , tl )
t
⋅
t →1
∂t
f (tk , tl )
∂f  t
 ∂f
= lim  ⋅ k + ⋅ l  ⋅
t →1 ∂k
∂l  f

= eq , k + eq ,l .
c.
We have
 ∂ (1 + t −2 k −1l −1 ) t 
eq ,t = lim 
⋅ 
t →1
∂t
q


t
= lim  q 2 2t −3k −1l −1 ⋅ 
t →1
q

= 2qk −1l −1
1 
= 2q  − 1
q 
= 2 − 2q.
Hence, eq ,t > 1 for q < 0.5, and eq ,t < 1 for q > 0.5.
d.
The intuitive reason for the changing scale elasticity is that this function has an
upper bound of q = 1 and gains from increasing the inputs decline as q
approaches this bound.
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Chapter 9: Production Functions
9.10
97
Returns to scale and substitution
a.
 the marginal rate of technical
Let σ% denote the elasticity of substitution and RTS
substitution associated with F . Then
Fl γ f γ −1 fl
f

RTS =
=
= l = RTS .
γ −1
Fk γ f f k
fk
Because f is homogeneous of degree 1, it is homothetic, and so RTS is a
function only of the input ratio: that is, RTS = B(k l ). Hence, k l = b( RTS ),
where b = B −1. We have
d ( k l ) RTS
σ=
d RTS k l
= b′( RTS )
 )
= b′( RTS
RTS
k l

RTS
k l

d ( k l ) RTS
=

k l
d RTS
= σ% ,
 = RTS . Hence, the elasticity of
where the last set of equalities follow from RTS
substitution is the same for both functions.
In view of the formula for σ in the statement of the question, we can also
write
f f
σ% = σ = k l .
f f kl
b.
A general Cobb–Douglas for arbitrary returns to scale can be written
F (k , l ) = k a l b
 a b 
=  k a +b l a +b 


= ( k α l 1−α )
a +b
a +b
= f (k , l ) a +b ,
where
a
.
a+b
The formula
α=
(1) σ =
f k fl
f f kl
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98
Chapter 9: Production Functions
can be applied to show the elasticity of substitution for f equals 1. Since F is
just f scaled by the exponent a + b, the preceding results imply that the elasticity
of substitution for F also equals 1.
A general CES function can be written
F (k , l ) = ( k ρ + l ρ )
γ ρ
= f (k , l ) γ ,
where
f (k , l ) = ( k ρ + l ρ ) .
1ρ
The formula in Equation 1 can be used to show that the elasticity of substitution
for this f is
1
σ=
.
1− ρ
Hence, the elasticity of substitution for general CES function F is the same as
this σ .
9.11
More on Euler’s theorem
a.
By Euler’s theorem: ∑ xi fi = kf .
i
Differentiating with respect to x j :
f j + ∑ xi f ij = kf j .
i
Multiplying by x j :
x j f j + x j ∑ xi f ij = kx j f j ,
i
implying
∑ (x f + x ∑ x f ) = k∑ x f
j
j
j
j
i ij
j
i
j
j
⇒ ∑∑ xi x j fij =(k − 1)∑ x j f j
i
j
j
⇒ ∑∑ xi x j fij =(k − 1)k f .
i
b.
j
Using Young’s theorem, we have f12 = f 21. For n = 2 and k = 1, the above
expression becomes
x12 f11 + 2 x1 x2 f12 + x22 f 22 = 0.
If we assume diminishing marginal productivity, then f11 , f 22 < 0. Since x1 and
x2 denote quantities, they are always positive numbers. Thus, in order for the
above relation to hold, f12 > 0. Therefore, with two inputs and constant returns to
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Chapter 9: Production Functions
99
scale, increasing one input must increase the marginal product of the other input.
If k > 1, we have increasing returns to scale and
x12 f11 + 2 x1 x2 f12 + x22 f 22 = k ( k − 1) f .
Since k > 1, k ( k − 1) f > 0. As described above, under the assumption of
diminishing marginal productivity, f11 , f 22 < 0. Thus, f12 > 0.
If k < 1, we have decreasing returns to scale and
x12 f11 + 2 x1 x2 f12 + x22 f 22 = k ( k − 1) f .
Since k < 1, k ( k − 1) f < 0. Under the assumption of diminishing marginal
productivity, f11 , f 22 < 0. Therefore, the left-hand side of the expression must be
negative, and this could happen if f12 < 0 or f12 > 0. Thus, in this case, the sign
of f12 cannot be determined without knowing the form of the production
function.
c.
Under the assumption of diminishing marginal productivity, fii < 0 for i = 1,..., n.
Thus, we cannot infer the sign of any one of the cross-partial derivatives. But, for
k > 1, we know that
∑ ∑ xi x j fij > 0
i
j
for the expression in part (a) to hold. Therefore, at least one cross-partial
derivative must be positive. For the case of k < 1, like in the case of only two
inputs, the sign for the sum of the cross-partial derivatives is ambiguous.
d.
Given
n
f ( x1 , x2 ,..., xn ) = ∏ xiα i .
i =1
we have
n
f (tx1 , tx2 ,..., txn ) = t α1 +α 2 +L+α n ∏ xiα i .
i =1
Hence, returns to scale will be determined by k = α1 + α 2 + L + α n . Further,
f i = α i xiαi −1 ∏ x j j ,
α
j ≠i
f ii = α i (α i − 1) xiαi − 2 ∏ x j j ,
α
j ≠i
α −1
f ij = α iα j xiαi −1 x j j
∏x .
k ≠ j ≠i
αk
k
Using the results from part (a) yields
∑∑ xi x j fi j = ∑∑ α iα j f − ∑ α i f = k (k − 1) f .
i
j
i
j
i
So, the sign of
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100
Chapter 9: Production Functions
∑∑ α α − ∑ α
i
i
j
j
i
i
is given by the sign of k .
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accessible website, in whole or in part.
CHAPTER 10:
Cost Functions
The problems in this chapter focus mainly on the relationship between production and
cost functions. Most of the examples developed are based on the Cobb–Douglas function
(or its CES generalization), although a few of the easier ones employ a fixed proportions
assumption. Two of the problems (10.7 and 10.8) make use of Shephard’s lemma since it
is in describing the relationship between cost functions and (contingent) input demand
that this envelope-type result is most often encountered. The analytical problems in this
chapter focus on various elasticity concepts, including the introduction of the Allen
elasticity measures.
Comments on Problems
10.1
An introduction to the concept of “economies of scope.” This problem illustrates
the connection between that concept and the notion of increasing returns to scale.
10.2
A simplified numerical Cobb–Douglas example in which one of the inputs is held
fixed.
10.3
A fixed proportion example. The very easy algebra in this problem may help to
solidify basic concepts.
10.4
This problem derives cost concepts for the Cobb–Douglas production function
with one fixed input. Most of the calculations are very simple. Later parts of the
problem illustrate the envelope notion with cost curves.
10.5
Another example based on the Cobb–Douglas with fixed capital. Shows that in
order to minimize costs, marginal costs must be equal at each production facility.
Might discuss how this principle is applied in practice by, say, electric companies
with multiple generating facilities.
10.6
This problem focuses on the Cobb–Douglas cost function and shows, in a simple
way, how underlying production functions can be recovered from cost functions.
10.7
This problem shows how contingent input demand functions can be calculated in
the CES case. It also shows how the production function can be recovered in such
cases.
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publicly accessible website, in whole or in part.
Chapter 10: Cost Functions
10.8
101
Famous example of Viner’s draftsman. This may be used for historical interest or
as a way of stressing the tangencies inherent in envelope relationships.
Analytical Problems
10.9
Generalizing the CES cost function. Shows that the simple CES functions used
in the chapter can easily be generalized using distributional weights.
10.10 Input demand elasticities. Develops some simple input demand elasticity
concepts in connection with the firm’s contingent input demand functions (this is
demand with no output effects).
10.11 The elasticity of substitution and input demand elasticities. Ties together the
concepts of input demand elasticities and the (Morishima) partial elasticity of
substitution concept developed in the chapter. A principle result is that the
definition is not symmetric.
10.12 The Allen elasticity of substitution. Introduces the Allen method of measuring
substitution among inputs (sometimes these are called Allen/Uzawa elasticities).
Shows that these do have some interesting properties for measurement, if not for
theory.
Solutions
10.1
a.
By definition, total costs are lower when both q1 and q2 are produced by
the same firm than when the same output levels are produced by different
firms. C (q1 , 0) simply means that a firm produces only q1.
b.
Let q = q1 + q2 , where q1 , q2 > 0. By assumption,
C ( q1 , q2 ) C ( q1 , 0)
<
,
q
q1
implying
q1C (q1 , q2 )
< C (q1 , 0).
q
Similarly,
q2C (q1 , q2 )
< C (0, q2 ).
q
Summing yields
C (q1 , q2 ) < C (q1 , 0) + C (0, q2 ).
This proves economies of scope.
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102
10.2
Chapter 10: Cost Functions
a.
Substituting into the production function,
q = 9000.5 J 0.5 = 30 J .
Setting q = 150 and solving yields J = 25. Similarly, J = 100 when
q = 300, and J = 225 when q = 450.
b.
Because Smith’s effort is sunk, to compute marginal cost we only need to
consider Jones’ effort in the cost function. To produce q pages requires
q2
J=
900
hours of work. At $12 per hour, this leads to the cost function
12q 2
C=
.
900
Thus,
∂C 24q 2q
MC =
=
=
.
∂q 900 75
We have
q = 150 ⇒ MC = 4,
q = 300 ⇒ MC = 8,
q = 450 ⇒ MC = 12.
10.3
Given q = min ( 5k ,10l ) .
a.
In the long run, no input should be wasted. Hence, 5k = 10l = q, implying
k = 2l = q 5. Thus,
C = vk + wl
= v(2l ) + wl
q
 q
= v   + w 
5
 10 
q
= ( 2v + w ) .
10
Therefore,
C 2v + w
AC = =
q
10
∂C 2v + w
MC =
=
.
∂q
10
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publicly accessible website, in whole or in part.
Chapter 10: Cost Functions
b.
c.
10.4
103
q = min ( 50,10l ) , when k = 10. There are two cases to consider. First, if
l < 5, then q = 10l , implying q < 50. Hence,
 q 
STC = 10v +   w,
 10 
implying
STC 10v w
SAC =
=
+ ,
q
q 10
∂STC w
SMC =
= .
∂q
10
If l ≥ 5, then q = 50. It is impossible to produce more than 50 in
the short run. Hence, STC = SAC = SMC = ∞ for q > 50.
Finally, right at q = 50, we have the same formula for total cost as
above:
 q 
STC = 10v +   w.
 10 
SMC is technically not defined because the STC has different derivatives
to the right and left of q = 50. However, SAC is well-defined, and is the
same as the previous formula:
STC 10v w
SAC =
=
+ .
q
q 10
Substituting v = 1 and w = 3 into the formulae from the previous parts, in
the long run, AC = MC = 1 2. In the short run, for q < 50,
STC 10 3
SAC =
= + ,
q
q 10
∂STC 3
SMC =
= .
∂q
10
Given q = 2 kl , k = 100.
a.
Since q = 2 100 l , q = 20 l . Rearranging,
q
l= ,
20
implying
2
q
l=
.
400
Hence,
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104
Chapter 10: Cost Functions
2
 q2 
q
SC = vk + wl = 1(100) + 4
=
100
+
.

100
 400 
STC 100
q
SAC =
=
+
.
q
q 100
b.
We have
SMC =
∂SC q
=
.
∂q 50
If q = 25,
 252 
SC = 100 + 
 = 106.25,
 100 
100 25
SAC =
+
= 4.25,
25 100
25 1
SMC = = .
50 2
If q = 50,
 50 2 
SC = 100 + 
 = 125,
 100 
100 50
SAC =
+
= 2.5,
50 100
50
SMC = = 1.
50
If q = 100,
 100 2 
SC = 100 + 
 = 200,
 100 
100 100
SAC =
+
= 2,
100 100
100
SMC =
= 2.
50
If q = 200,
 200 2 
SC = 100 + 
 = 500,
 100 
100 200
SAC =
+
= 2.5,
200 100
200
SMC =
= 4.
50
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Chapter 10: Cost Functions
105
c.
d.
As long as the marginal cost of producing one more unit is below the
average-cost curve, average costs will be falling. Similarly, if the marginal
cost of producing one more unit is higher than the average cost, then
average costs will be rising. Therefore, the SMC curve must intersect the
SAC curve at its lowest point.
e.
Since q = 2 k1l , q 2 = 4k1l , implying
2
q
.
4k1
Substituting,
l=
wq 2
SC = vk1 + wl = vk1 +
.
4k1
f.
Deriving the first-order condition from the previous expression,
∂SC
wq 2
= v − 2 = 0.
∂k1
4k1
Rearranging,
q w
k1 =
.
2 v
g.
Substituting first for l and then for k1 into the cost function,
C = vk1 + wl (k1 )
= vk1 + w
q2
4k1
 q w  wq 2  2 v 
= v 
 +


4  q w 
2 v 
= q vw ,
(a special case of Example 10.2).
h.
If w = 4 and v = 1, in the long run, C = 2q.
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106
Chapter 10: Cost Functions
Let’s examine the short run in different cases. Fixing k1 = 100 in
the short run,
2
q
SC (k1 = 100) = 100 +
.
100
This is tangent to the long-run cost function for q = 100, as one can verify
SC = 200 = C.
Fixing k1 = 200 in the short run
2
q
.
200
This is tangent to the long-run cost function for q = 200, as one can verify
SC = 400 = C.
Finally, fixing k1 = 400 in the short run,
SC (k1 = 200) = 200 +
2
q
.
400
This is tangent to the long-run cost function for q = 400, as one can verify
SC = 800 = C .
SC (k1 = 400) = 400 +
10.5
a.
Total output is q = q1 + q2 , with
q1 = 25l 1 = 5 l 1
q 2 = 10 l 2 .
Thus,
2
SC1 = 25 +l1 = 25 +
q1
,
25
2
q2
,
100
implying total short-run cost is
SC2 = 100 +
2
2
q
q
SC = SC1 + SC2 = 125 + 1 + 2 .
25 100
To minimize cost, set up Lagrangian:
L = SC + λ (q − q1 − q2 ).
Taking the first-order conditions,
2q
Lq1 = 1 − λ = 0,
25
2q
Lq2 = 2 − λ = 0.
100
Therefore, 4q1 = q2 .
b.
Since 4q1 = q2 , we have q1 = q 5 and q2 = 4q 5. Therefore,
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publicly accessible website, in whole or in part.
Chapter 10: Cost Functions
SC = 125 +
107
2
q
,
125
2q
,
125
125 q
SAC =
+
.
q 125
SMC =
Substituting various quantities, SMC (100) = $1.60, SMC (125 ) = $2.00,
and SMC ( 200 ) = $3.20.
10.6
c.
In the long run, given constant returns to scale, location doesn’t really
matter because one can change k . The entrepreneur could split evenly or
produce all output in one location, etc.
C = k + l = 2q.
AC = 2 = MC.
d.
If there are decreasing returns to scale with identical production functions,
then the entrepreneur should let each firm have equal share of production.
AC and MC not constant anymore, becoming increasing functions of q.
a.
From Shephard’s lemma,
1
∂C 2  v  3
l=
= q  ,
∂w 3  w 
2
∂C 1  w  3
k=
= q  .
∂v 3  v 
10.7
b.
Eliminate the w / v from these equations:
23
q = ( 3 2 ) 31 3 l 2 3 k 1 3 = Bl 2 3 k 1 3 .
This is a Cobb–Douglas production function.
a.
As for many proofs involving duality, this one can be algebraically messy
unless one sees the trick. Here the trick is to let
B = ( v 0.5 + w0.5 ) .
With this notation, C = B 2 q. Using Shephard’s lemma,
∂C
k=
= Bv −0.5 q,
∂v
∂C
l=
= Bw−0.5 q.
∂w
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108
10.8
Chapter 10: Cost Functions
b.
From part (a),
q v 0.5
=
,
k
B
q w0.5
=
.
l
B
Thus,
q q
+ = 1,
k l
k −1 + l −1 = q −1.
The production function then is q = (k −1 + l −1 ) −1 .
c.
This is a CES production function with r = −1. Hence,
1
s=
= 0.5.
1− r
Comparison to Example 8.2 shows the relationship between the
parameters of the CES production function and its related cost function.
Support the draftsman. It is geometrically obvious that SAC cannot be at
minimum because it is tangent to AC at a point with a negative slope. The only
tangency occurs at minimum AC.
Analytical Problems
10.9
Generalizing the CES cost function
1
a.
 v 1−σ  w 1−σ  1−σ
C = q   +    .
 β  
 α 
b.
C = qα −α β − β vα wβ .
c.
wl β
= .
vk α
d.
Using Shephard’s lemma with the cost function derived in part (a),
1
γ
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Chapter 10: Cost Functions
l=
∂C
 1   v 
=q 
  
∂w
 1 − σ   α 
1−σ
1
γ
109
1−σ
 w
+ 
β 



1
−1
1−σ
1
w−σ β σ −1 ,
−1
1−σ
1−σ
 w   1−σ −σ σ −1
∂C
 1   v 
k=
=q 
v α .
   +   
∂v
 1 − σ   α 
 β  
Thus, labor’s relative share is
1
γ
1
γ
1
1−σ
1−σ 1−σ −1 1−σ σ −1
 1 

q 
v
α
+
w
β
w β
(
)
(
)


wl
1−σ  

= 1
1
vk
1−σ
1−σ 1−σ −1 1−σ σ −1
 1 

qγ 
v
α
+
w
β
) ( )  v α
 (
 1−σ  
1−σ
w α 
= ⋅  .
v β
Labor’s relative share depends on σ . If σ > 1, labor’s share moves in the
opposite direction as w v and the same direction as β α . If σ < 1, the opposite
is true. This accords with intuition on how substitutability should affect shares.
10.10 Input demand elasticities
a.
The elasticities can be read directly from the contingent demand functions
in Example 10.2. For the fixed proportions case,
el c ,w = ek c ,v = 0
This is because q is held constant. For the Cobb–Douglas,
α
el c ,w = − + β ,
α
−β
ek c ,v =
+ β.
α
Evidently, the CES in this form has nonconstant elasticities.
b.
Because cost functions are homogeneous of degree 1 in input prices,
contingent demand functions are homogeneous of degree 0 in those prices
as intuition suggests. Using Euler’s theorem gives lwc w + lvc v = 0. Dividing
by l c gives the result.
c.
Use Young’s theorem:
∂l c ∂ 2C
∂ 2C ∂k c
=
=
=
.
∂v ∂v∂w ∂w∂v ∂w
Now multiply the left side by
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110
Chapter 10: Cost Functions
vwl c
l cC
and the right side by
vwk c
.
k cC
d.
Multiplying by shares in part (b) yields
sl el c , w + sl el c ,v = 0.
Substituting from part (c) yields
sl el c ,w + sk ek c ,w = 0 .
e.
All of these results give important checks to be used in empirical work.
10.11 The elasticity of substitution and input demand elasticities
a.
If wi does not change,
si , j =
∂ ln( xic x cj )
∂ ln( w j wi )
=
∂ ln( xic x cj )
∂ ln( w j )
,
∂ ln xic
exc , w =
,
i
j
∂ ln w j
exc , w =
j
j
∂ ln x cj
∂ ln w j
,
c
c
c
∂ ln xic ∂ ln x j ∂ ln( xi x j )
exc , w − exc , w =
−
=
= si , j .
i
j
j
j
∂ ln w j ∂ ln w j
∂ ln w j
b.
If w j does not change,
s j ,i =
∂ ln( x cj xic )
∂ ln( wi w j )
=
∂ ln( x cj xic )
∂ ln( wi )
,
exc , w = ∂ ln x cj ∂ ln wi ,
j
i
exc , w = ∂ ln xic ∂ ln wi ,
i
i
exc , w − exc , w =
j
c.
i
i
i
∂ ln x cj
∂ ln wi
−
c
c
∂ ln xic ∂ ln( x j xi )
=
= s j ,i .
∂ ln wi
∂ ln wi
The cost function will be (similar to Equation 10.29)
 n

C ( w1, w2 ,..., wn , q) = q  ∑ wkρ (ρ−1) 
 k =1

1/γ
(ρ −1)/ρ
.
Let
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Chapter 10: Cost Functions
111
n
B = ∑ wkρ ( ρ −1) .
k =1
By Shephard’s lemma,
xic ( w1 , w2 ,..., wn , q ) =
∂C ( w1 , w2 ,..., wn , q)
= q1/γ B −1 ρ wi1 ( ρ −1) ,
∂wi
x cj ( w1 , w2 ,..., wn , q) =
∂C ( w1 , w2 ,..., wn , q)
= q1/γ B −1 ρ w1j ( ρ −1) .
∂w j
exc , w =
i
j
∂xic w j
−1 −1 ρ ( ρ −1)
⋅ c =
B wj
,
∂w j xi
ρ −1
∂x cj w j
 1

1
exc , w =
⋅ c =
−
B −1wρj ( ρ −1)  = −σ + exc ,w ,
j
j
i
j
∂w j x j ρ − 1  ρ − 1

∂x cj wi
−1 −1 ρ ( ρ −1)
exc , w =
⋅ c =
B wj
,
j
i
∂wi x j ρ − 1
exc , w =
i
i
∂xic wi
 1

1
⋅ c =
−
B −1wiρ ( ρ −1)  = −σ + exc , w ,
j
i
∂wi xi ρ − 1  ρ − 1

sij = s ji = exc , w − exc ,w = exc , w − exc , w = σ .
i
j
j
j
j
i
i
i
10.12 The Allen elasticity of substitution
a.
By Shephard’s lemma:
∂C
xic =
= Ci .
∂wi
Thus,
w
w
∂x c w
∂C
exc , w = i ⋅ cj =
⋅ cj = Cij ⋅ j ,
i
j
∂w j xi ∂wi ∂w j xi
Ci
w j x cj
sj =
ex c , w
i
sj
b.
C
j
=
= Cij ⋅
wjC j
C
wj
Ci
⋅
,
CC
C
= ij = Aij .
w j C j Ci C j
We have
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112
Chapter 10: Cost Functions
esi , p j =
=
∂si p j
⋅
∂p j si
pj
∂ ( pi Ci C )
⋅
∂p j
piCi C
= pi ⋅
= pi ⋅
∂ ( Ci C ) p j C
⋅
∂p j
pi Ci
C ji C − Ci C j p j C
⋅
C2
pi Ci
pj
= (C ji C − CiC j )
=
C ji C
C j Ci − 1
Ci C
p jC j
⋅
C
= s j ( Aij − 1).
c.
In the Cobb–Douglas case,
α
1
β
C = q α + β Bv α + β wα + β ,
where
B = (α + β )α
−α
α +β
β
−β
α +β
.
Then,
α
−α
−β
β
 β  α +β α +β
∂C
Cl =
= q α +β B 
v w ,
∂w
α + β 
1
 α  α +β α +β
∂C
Ck =
= qα +β B 
v w ,
∂v
α + β 
1
−β
−α
 αβ  α + β α + β
∂C
Ckl = k = q α + β B 
v w ,
2 
∂w
 (α + β ) 
1
Ckl C
= 1.
Ck Cl
In the CES case,
Akl =
1
γ
1−σ
C = q (v
1
1−σ 1−σ
+w
)
.
Then
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Chapter 10: Cost Functions
113
σ
∂C
= q γ (v1−σ + w1−σ )1−σ w−σ ,
∂w
1
Cl =
σ
∂C
Ck =
= q γ (v1−σ + w1−σ )1−σ v −σ ,
∂v
1
2σ −1
∂C
Ckl = k = q γ σ (v1−σ + w1−σ ) 1−σ v −σ w−σ ,
∂w
C C
Akl = kl = σ .
Ck Cl
1
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publicly accessible website, in whole or in part.
CHAPTER 11:
Profit Maximization
Problems in this chapter consist mainly of applications of the P = MC rule for profit
maximization by a price-taking firm and some examination of the firm’s derived demand for
inputs. A few of the problems (13.2–13.5) ask students to work through derivations related to
marginal revenue, but this concept is not really used in the monopoly context until Chapter 14.
The last problem provides practice with the new material on the “theory of the firm.”
Comments on Problems
11.1
A very simple application of the P = MC rule. Results in a linear supply curve.
11.2
Uses the MR–MC condition to illustrate third degree price discrimination. Instructors
might point out the general result here (which is discussed more fully in Chapter 13) that,
assuming marginal costs are the same in the two markets, marginal revenues should also
be equal and that implies price will be higher in the market in which demand is less
elastic.
11.3
An algebraic example of a profit function with one input. The problem asks the student
to derive the supply and input demand functions from this profit function using
Shephard’s lemma.
11.4
A problem in the theory of supply under uncertainty. This example shows that setting
expected price equal to marginal cost does indeed maximize expected revenues, but that,
for risk-averse firms, this may not maximize expected utility. Part (d) asks students to
calculate the value of better information.
11.5
A simple use of the profit function with fixed proportions technology.
11.6
Easy problem that shows that a tax on profits will not affect the profit-maximization
output choice unless it affects the relationship between marginal revenue and marginal
cost.
11.7
Practice with calculating the marginal revenue curve for a variety of demand curves.
11.8
This is a conceptual examination of the effect of changes in output price on input
demand.
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114not be scanned, copied or duplicated, or posted to a
Chapter 11: Profit Maximization
115
Analytical Problems
11.9
A CES profit function. A very brief introduction to the CES profit function. Deriving
the function involves a lot of algebra, but seeing how the parameters of the underlying
production function enter this profit function is quite instructive.
11.10 Some envelope results. This problem describes some additional mathematical
relationships that can be derived from the profit function.
11.11 Le Châtelier’s principle. This problem demonstrates this central principle of economics
in various contexts. The principle compares long-run to short-run changes. The logic
behind the principle is that in the long-run, there are more margins to adjust, so a “better”
outcome can be produced than in the short run. Whether this “better” outcome involves a
bigger or smaller change in the variable of interest depends on the nature of the
optimization, whether maximization or minimization. In maximization problems (as in
parts (a) and (b)), the long-run change will generally be bigger. In minimization
problems [as in parts (c) and (d)], the change will generally be smaller in the long run.
11.12 More on derived demand with two inputs. This problem shows how an industry’s
demand for an input can be computed and why that demand will depend on the elasticity
of demand for the good being produced. This is a nice problem therefore for tying
together input and output markets.
11.13 Cross-price effects in input demand. This is a continuation of Problem 11.11 to
consider cross-price effects. The problem attempts to clarify how input cost shares enter
into input demand elasticities.
11.14 Profit functions and technical change. Applies the envelope theorem to derive a result
useful for empirical work on the measuring the impact of technical progress.
11.15 Property rights theory of the firm. The material from the Extensions on “theories of
the firm” is somewhat more philosophical than most of the rest of the book, so the
numerical example in that part of the text can be quite instructive. This problem has
students work through a simple tweak of that numerical example. The tweak has
independent interest, showing that vertical integration between the car body and assembly
can be beneficial if the assembly’s investment is important enough.
Solutions
11.1
a.
MC = ∂C ∂q = 0.2q + 10. Setting MC = P = 20, yields q* = 50.
b.
π = Pq − C = 1000 − 800 = 200.
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116
Chapter 11: Profit Maximization
c.
11.2
Total cost is
2
C = 0.25q 2 = 0.25 ( qA + qL ) ,
and demands are
qA = 100 − 2PA ,
qL = 100 − 4PL .
Inverting demands,
q
PA = 50 − A ,
2
qL
PL = 25 − .
4
Revenues are
2
qA
,
2
2
qL
RL = PL qL = 25q L −
.
4
Hence, marginal revenues are
MRA = 50 − q A ,
RA = P A q A = 50q A −
qL
.
2
Differentiating the total cost function,
MC A = 0.5 ( q A + qL ) ,
MRL = 25 −
MCL = 0.5 ( q A + qL ) .
Setting MRA = MC A and MRL = MCL yields
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Chapter 11: Profit Maximization
117
50 − q A = 0.5q A + 0.5qL ,
qL
= 0.5q A + 0.5qL .
2
Solving these simultaneously gives
25 −
qA = 30 , PA = 35,
qL = 10, PL = 22.5.
Also,
11.3
a.
π = 1,050 + 225 − 400 = 875.
Since q = 2 l , q 2 = 4l.
2
wq
C = wl =
.
4
b.
2P 2 P 2 P 2
−
=
.
w
w
w
This is homogeneous of degree 1 in P and w.
π = Pq − TC =
c.
Profit maximization requires
2wq
P = MC =
.
4
Solving for q yields q = 2 P w .
The result can also be derived from Shephard’s lemma:
∂π 2 P
q=
=
.
∂P w
d.
From the production function, l = q 2 4. Replacing q from the supply function,
we get l = P 2 w2 . Shephard’s lemma gives the same result:
l=−
e.
∂π P 2
=
.
∂w w2
Intuitive properties include the following:
•
Total cost increases with wages and output.
•
Profits increase with output price and decrease with wages. The function
is homogeneous of degree 1 in the prices.
•
Supply increases with output price and decreases with the wage. The
function is homogeneous of degree 0 in input and output prices.
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118
Chapter 11: Profit Maximization
•
11.4
a.
Labor demand increases with output price and decreases with the wage.
Input demand is homogeneous of degree 0 with respect in output and input
prices.
Expected profits are
E (π ) = 0.5 [30q − C (q) ] +0.5  20q − C ( q ) 
= 25q − C ( q ) .
Notice that 25 = E ( P ) determines expected profits. For profit maximum, set
E ( P ) = MC = q + 5.
So q = 20. Further,
E (π ) = E ( P ) q − C ( q ) = 500 − 400 = 100.
b.
In the two states of the world, profits are
π = 600 − 400 = 200.
P = 30,
π = 400 − 400 = 0.
P = 20 ,
Expected utility is given by E (U ) = 0 .5 200 + 0.5 0 = 7.1.
c.
Output levels between 13 and 19 all yield greater utility than does q = 20.
Reductions in profits from producing less when P is high are compensated for
(in utility terms) by increases in profits when P is low. Calculating true
maximum expected utility is difficult; it is approximately q = 17.
d.
If the firm can predict P , set P = MC in each state of the world.
When P = 30, q = 25 and π = 212.5.
When P = 20, q = 15 and π = 12.5.
E (π ) = 112.5.
E (U ) = 0.5 212.5 + 0.5 12.5 = 9.06.
This represents a substantial improvement.
11.5
a.
In order for the second-order condition for profit maximization to be satisfied,
marginal cost must be increasing which, in this case, requires diminishing returns
to scale.
b.
Since q = 10k 0.5 = 10l 0.5 ,
q2
k =l =
,
100
C = vk + wl =
q 2 (v + w)
.
100
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Chapter 11: Profit Maximization
119
Profit maximization requires
q(v + w)
P = MC =
.
50
Rearranging,
50 P
q=
.
v+w
Thus,
Π (v, w, P ) = Pq − C
=
50 P 2  50 P   v + w 
−
 

v + w  v + w   100 
=
25 P 2
.
v+w
2
c.
If v = 1, 000, w = 500, and P = 600, then q = 20 and π = 6, 000.
d.
If v = 1, 000, w = 500, and P = 900, then q = 30 and π = 13,500.
e.
11.6
Without any tax, π ( q ) = R ( q ) − C ( q ). With a lump sum tax T ,
π ( q ) = R (q ) − C ( q ) − T .
The first-order condition is
∂π ∂R ∂C
=
−
− 0 = 0.
∂q ∂q ∂q
Since MR = MC , there is no change.
With a proportional tax,
π ( q ) = (1 − t )[ R ( q ) − C ( q )].
The first-order condition is
∂π
= (1 − t )( MR − MC ) = 0.
∂q
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120
Chapter 11: Profit Maximization
Since MR = MC , there is no change.
With a tax per unit,
π ( q ) = R ( q ) − C ( q ) − tq.
The first-order condition is
∂π
= MR − MC − t = 0.
∂q
Since MR = MC + t , q is changed. A per unit tax does affect output.
11.7
a.
If q = a + bP,
MR = P + q
dP q − a
 1  2q − a
=
+ q =
.
dq
b
b
b
Hence,
a + bMR
.
2
Because the distance between the vertical axis and the demand curve is
q = a + bP, the marginal revenue curve must bisect this distance for any line
parallel to the horizontal axis.
q=
b.
If q = a + bP, b < 0, and P = (q − a) b, then
2q − a
MR =
,
b
1
P − MR = − q.
b
c.
The constant elasticity demand curve is q = aP b , where b is the price elasticity
of demand.
1b
 (q a)1 b 
∂P  q 
MR = P + q
=  +
.
∂q  a 
 b 
Thus, vertical distance is
1
−(q a ) b − P
P − MR =
=
.
b
b
This is positive because b < 0.
d.
If eq , P < 0 (downward sloping demand curve), then marginal revenue will be less
than price. Hence, vertical distance will be given by P − MR.
Since MR = P + q
dP
, the vertical distance is
dq
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Chapter 11: Profit Maximization
121
dP
.
dq
Since dq dP = b is the slope of the tangent linear demand curve, the distance
becomes − q b as in part (b).
−q
e.
11.8
a.
With marginal cost increasing, an increase in P will be met by an increase in q.
To produce this extra output, more of each input will be hired (unless an input is
inferior).
b.
The Cobb–Douglas case is best illustrated in two of the examples in Chapter 11.
In Example 11.4, the short-run profit function exhibits a positive effect of P on
labor demand. A similar result occurs in Example 11.5, where holding a third
input constant leads to increasing marginal cost.
c.
Differentiating Equation 11.52 with respect to P :
∂l ∂l c ∂q
=
.
∂P ∂q ∂P
The sign of the second factor can be pinned down as follows:
∂q ∂ 2 Π
=
>0
∂P ∂P 2
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122
Chapter 11: Profit Maximization
because of the convexity of the profit function. The sign of the first factor,
∂l c ∂q , depends on whether labor is an inferior input (negative if so).
∂l ∂ ( − ∂Π ∂w ) −∂ 2 Π −∂q
=
=
=
.
∂P
∂P
∂P∂w ∂w
The sign of the final term may be negative if l is an inferior input.
Analytical Problems
11.9
A CES profit function
b.
Diminishing returns is required if MC is to be increasing—the required secondorder condition for profit maximization.
c.
σ determines how easily firms can adapt to differing input prices. The higher is
this elasticity, the easier it is to offset increases in the price of any one input as
shown by the exponents of input prices.
γ
d.
e.
γ
∂Π
q=
= K (1 − γ ) −1 P 1−γ (v1−σ + w1−σ ) (1−σ )(γ −1) .
∂P
This supply function shows that σ does not affect the supply elasticity (the
exponent of price) directly, but it does affect the shift term that involves input
prices. Larger values for σ imply smaller shifts in the supply relationship for
given changes in input prices.
See the results provided in Sydsaeter, Strom, and Berck’s book.
11.10 Some envelope results
a.
We have
∂l ∂ 2 Π
∂ 2 Π ∂k
=
=
=
.
∂v ∂v∂w ∂w∂v ∂w
This shows that cross-price effects in input demand are equal. The result is
similar to the equality of compensated cross-price effects in demand theory.
b.
The direction of effect depends on whether capital and labor are substitutable or
complementary inputs.
c.
We have
∂q
∂ 2Π
∂ 2Π
∂l
=
=
=− .
∂w ∂w∂P ∂P∂w
∂P
This shows that increases in wages have the same effect on reducing output that a
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Chapter 11: Profit Maximization
123
fall in the product price has on reducing labor demand. That is, the effects of
wages and prices are in some ways symmetrical.
d.
Because it seems likely that ∂l ∂P > 0 (see Problem 11.8), we can conclude that
∂q ∂w < 0. A tax on labor input should reduce output:
∂l ∂ ( − ∂Π ∂w ) −∂ 2 Π −∂q
=
=
=
.
∂P
∂P
∂P∂w ∂w
11.11 Le Châtelier’s principle
a., b.
Totally differentiate both sides of the definitional relation with respect to P :
∂q* ∂q s ∂q s ∂k *
=
+
⋅
.
∂P ∂P ∂k ∂P
Now totally differentiate the definitional relation with respect to v :
∂q* ∂q s ∂k *
=
⋅
,
∂v
∂k ∂v
implying
∂q s ∂q* ∂v
=
.
∂k ∂k * ∂v
By analogy to part (c) of Problem 11.10,
∂q*
∂k *
=−
.
∂v
∂P
Substituting the preceding equations in succession into the initial derivative,
*
*
∂q* ∂q s ( ∂q ∂v )( ∂q ∂P )
=
+
∂P ∂P
∂k * ∂v
*
∂q s ( ∂k ∂P )
=
−
∂P
∂k * ∂v
2
∂q s
.
∂P
The last step holds because ∂k * ∂v ≤ 0 (see the discussion of Equation 11.53 in
the text).
≥
c.
Totally differentiate the definitional relation with respect to w :
∂l * ∂l s ∂l s ∂k *
=
+
⋅
.
∂w ∂w ∂k ∂w
Totally differentiate it with respect to v :
∂l * ∂l s ∂k *
=
⋅
,
∂v ∂k ∂v
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124
Chapter 11: Profit Maximization
implying
∂l s ∂l * ∂v
=
.
∂k ∂k * ∂v
Separately, we have
∂k * ∂ 2 Π ∂l *
=
=
.
∂w ∂v∂w ∂v
Substituting in succession into the initial derivative,
*
*
∂l * ∂l s ( ∂l ∂v )( ∂k ∂w )
=
+
∂w ∂w
∂k * ∂v
*
∂l s ( ∂l ∂v )
=
+
∂w
∂k * ∂v
≤
d.
2
∂l s
.
∂w
It is difficult to use the methods from parts (a)–(c) here. Let’s see what happens
when we try. Start from the definitional relation
C (v, w, q) = SC (v, w, q, k c (v, w, q)).
Totally differentiate with respect to w :
∂C ∂SC ∂SC ∂k c
=
+
⋅
.
∂w ∂w
∂k ∂w
But, by the envelope theorem (see Equation 10.60 for the application to the
present context),
∂SC
= 0.
∂k
Hence,
∂C ∂SC
=
,
∂w ∂w
that is, the long- and short-run effects from a local (infinitesimal) change in the
wage are the same.
It turns out that the long- and short-run cost changes can differ for a larger
(discrete) change in the wage, and the two can be ranked. But because derivative
methods don’t work for large changes, we will resort to a different technique in
the proof (sometimes called a “revealed preference” approach, although the
technique is applied beyond the revealed-preference analysis of consumer theory).
Consider a discrete wage increase from w′ to w′′. We gave the
definitional relations
C (v, w′, q ) = SC (v, w′, q , k ′)
C (v , w′′, q ) = SC (v, w′′, q, k ′′),
where
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Chapter 11: Profit Maximization
125
k ′ = k c (v, w′, q),
k ′′ = k c (v, w′′, q)
are the conditional demands for capital from the cost-minimization problem.
Now
SC (v, w′′, q, k ′′) ≤ SC (v, w′′, q, k ′).
This holds because k ′′ is optimal given input price w′′, so any other alternative
capital level, including k ′, must lead to a (weakly) higher value of short-run cost.
Putting these results together,
C (v, w′′, q ) − C (v, w′, q ) = SC (v , w′′, q , k ′′) − SC (v, w′, q , k ′)
≤ SC (v, w′′, q, k ′) − SC (v, w′, q , k ′).
We have thus shown that a wage increase causes a bigger rise in short-run than
long-run cost. Intuitively, the firm has more margins to adjust to the wage
increase in the long run, so can maintain lower costs.
11.12 More on derived demand with two inputs
a.
By Shephard’s lemma, each partial derivative gives the quantity of input
demanded to produce one unit of output. Multiplication by Q gives total industry
demand.
b.
Under the assumption of constant returns to scale,
P = MC = C ( v, w,1) .
So in equilibrium,
Q = D ( MC ) = D (C (v, w,1)).
Furthermore,
k = QCv = D (C (v, w,1))Cv ,
l = QCw = D(C (v, w,1))Cw ,
implying
∂k
= D′Cv2 + CvvQ,
∂v
∂l
= D′Cw2 + CwwQ.
∂w
c.
Because costs are homogenous of degree 1, the derivatives of C are
homogeneous of degree 0. Hence,
vCvv + wCwv = 0,
implying
 −w 
Cvv = 
 Cwv .
 v 
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126
Chapter 11: Profit Maximization
Similarly,
 −v 
C ww =   C wv .
 w
d.
We have
CC wv
.
C wC v
Rearranging,
σ CwCv
Cwv =
.
C
Replacing Cwv in the relation proved in part (c) yields
σ =
− w σ CwCv
⋅
,
v
C
−v σ CwCv
Cww =
⋅
.
w
C
Replacing Cvv and Cww with the above expressions, and Cv and Cw with the
expressions from part (a), we rewrite the equations from part (b) as
∂k
σ wkl D′k 2
=−
+ 2 ,
∂v
QvC
Q
Cvv =
∂l
σ vkl D′l 2
=−
+
.
∂w
QwC Q 2
∂l w −σ vk D′wlP
⋅ =
+ 2 = −σ sk + sl eQ , P ,
∂w l
QC
Q P
∂k v −σ wl D′vkP
ek ,v =
⋅ =
+ 2 = −σ sl + sk eQ , P .
∂v k
QC
Q P
e.
el , w =
f.
The terms −σ sl and −σ sk are a mathematical representation of the substitution
effect. Because the sign of σ is positive and sl is positive, the overall sign of the
substitution effect will be negative. The size of the effect increases when the
goods are closer substitutes (when σ is larger) and when we have a larger share
of the other input (which makes it more easily replaceable).
The terms sk eQ , P and sl eQ , P are the mathematical representation of the
output effect. Assuming the demand elasticity is negative, the output effect will be
negative and resulting own-price elasticity of the inputs will be negative. The
output effect will increase the more elastic the demand for the output and the
larger the share of the input (a larger share implies that a price increase for the
input will have a larger effect on marginal cost and on price).
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Chapter 11: Profit Maximization
127
11.13 Cross-price effects in input demand
a.
Similarly to 11.11 part (b),
k = QCv = D (C (v, w,1))Cv ,
l = QCw = D(C (v, w,1))Cw ,
implying
∂k
= D′Cv Cw + CvwQ,
∂w
∂l
= D′CwCv + Cwv Q.
∂v
From 11.11 part (c),
CC wv
σ =
,
C wCv
where
σ CwCv
Cwv =
.
C
Thus,
Qσ Cv Cw
∂k
= D′Cv Cw +
∂w
C
 k  l 
( k Q )( l Q )
= D′    + Qσ
C
 Q  Q 
D′kl σ kl
= 2 +
Q
QC
and
 ∂k  w 
ek , w = 
 
 ∂w  k 
D′wlP σ wl
= 2 +
Q P
QC
 D′P 
=
 sl + σ sl
 Q 
= eQ , P sl + σ sl
= sl (eQ , P + σ ).
Similarly,
∂l D′kl σ kl
=
+
∂v Q 2 QC
and
el ,v = sk (eQ , P + σ ).
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128
Chapter 11: Profit Maximization
b.
In the formula for cross-price elasticity of input demand, both the elasticity of
substitution and the elasticity of demand for the output are weighted by the share
of the other input. This happens because the effect of a change in the price of the
other input will depend primarily on the importance of this other input. In the
case of ek , w , for example, the degree to which capital can be substituted for labor
will be greater, the greater in labor’s share. Similarly, the output effect of an
increase in the wage will be greater the greater is labor’s share.
c.
From Euler’s theorem,
v
Cww = − Cwv .
w
Hence,
C C
vC wv C
All = ww = −
.
CwCw
wC wC w
Multiplying this by Cv Cv and using Shephard’s lemma yields
vkC wv C
s
All = −
= − k Akl .
wlC wCv
sl
11.14 Profit functions and technical change
The profit function is
Π ( P, v, w, t ) = max { Pf (k , l , t ) − vk − wl} .
k ,l
By the envelope theorem,
∂Π
= Pf t .
∂t
That is, we only need to consider the direct effect of t on Π ; the indirect effects through
the input demands are zero because of the first-order conditions. Now,
∂ ln Π ∂Π 1 Pft
=
⋅ =
∂t
∂t Π Π
∂ ln Π
⇒ Π⋅
= Pft
∂t
Π ∂ ln Π Pf t f t f t
⇒
⋅
=
= = .
Pq ∂t
Pq q
f
11.15 Property rights theory of the firm
First, consider keeping the assets separate. Fisher Body maximizes
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Chapter 11: Profit Maximization
129
1 1/ 2
xF + axG1/2 ) − xF .
(
2
The first-order condition is
1 −1/2
xF − 1 = 0,
4
implying xF* = 1 16. GM maximizes
1 1/ 2
( xF + axG1/2 ) − xG .
2
The first-order condition is
1 −1/2
axG − 1 = 0,
4
implying xG* = a 2 16. Joint surplus is
2
1
3
a 1 a
+ a   − − = (1 + a 2 ) .
4
 4  16 16 16
Next, consider GM ownership. Fisher Body of course sets x **
F = 0. GM maximizes
x1/F 2 + axG1/ 2 − xG .
The first-order condition is
1 −1/2
axG − 1 = 0,
2
implying xG** = a 2 4. Joint surplus becomes
2
a2
a a
a  − = .
4
2 4
Comparing the joint surpluses under the two ownership structures, GM ownership is
better if
a2 3
> (1 + a 2 )
4 16
⇒ 4a 2 > 3 + 3a 2
⇒ a2 > 3
⇒ a > 3.
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CHAPTER 12:
The Partial Equilibrium Competitive Model
The problems in this chapter focus on competitive supply behavior in both the short and long
runs. For short-run analysis, students are usually asked to construct the industry supply curve (by
summing firms' marginal cost curves) and then to describe the resulting market equilibrium. The
long-run problems (12.4–12.7), on the other hand, make extensive use of the equilibrium
condition P = MC = AC to derive results. In most cases, students are asked to graph their
solutions because such graphs provide considerable intuition about what is going on. The
analytical problems here mainly involve taxation. Problem 12.9 shows that many of the results
for per-unit taxes introduced in the chapter carry over for ad valorem taxes. Problem 12.10
introduces the Ramsey formula for optimal taxation.
Comments on Problems
12.1
This problem asks students to construct a marginal cost function from a cubic cost
function and use this to derive a supply curve and a supply–demand equilibrium. The
math is rather easy, so this is a good starting problem.
12.2
This problem illustrates “interaction effects.” As industry output expands, the wage for
diamond cutters rises, thereby raising costs for all firms.
12.3
This problem shows that, with simple linear demand and supply curves, equilibrium
solutions can be found either through substitution or through the comparative statics
procedures illustrated in the chapter.
12.4
This is a simple problem in the interaction between short-run and long-run analysis. The
long-run equilibrium price is always $10. But the price may diverge from this in the short
run.
12.5
This problem introduces the concept of increasing input costs into long-run analysis by
assuming that entrepreneurial wages are bid up as the industry expands. Solving part (b)
can be a bit tricky; perhaps an educated guess is the best way to proceed.
12.6
This is a problem in (short-run) tax incidence. The final part of the problem concerns the
change in short-run producer surplus as a result of the tax.
12.7
This is a problem in long-run producer surplus. It makes the point that the producer’s
share of any tax is ultimately borne by that input that is in inelastic supply. Here, it is the
film studio that incurs all of the producer’s share of the tax burden.
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132
Chapter 12: The Partial Equilibrium Competitive Model
12.8
This is a simple partial equilibrium problem in trade theory.
12.9
A simple algebraic model that shows how general parameters for the demand curve and
firms’ cost curves interact to determine the equilibrium price.
Analytical Problems
12.10 Ad valorem taxes . This problem shows that the comparative statics results for ad
valorem taxes are very similar to the results for per-unit taxes shown in Chapter 12. The
problem provides another illustration of why the comparative statics approach taken here
is useful.
12.11 The Ramsey formula for optimal taxation . This problem shows how to compute
optimal rates of ad valorem taxation that minimize the excess burden of these taxes
subject to a total revenue constraint.
12.12 The Cobweb model . This is a simple algebraic model where a lagged supply response
leads to fluctuating prices.
12.13 More on the comparative statics of supply and demand . This exercise contains three
subproblems. The first just asks the student to repeat the analysis in the chapter for a shift
in supply rather than demand. The second examines the effects of a “quantity wedge.”
This yields results very similar to the “tax wedge” analysis in the chapter. Finally, the
problem provides a brief introduction to the identification problem in econometrics as
applied to models of supply and demand.
12.14 The Le Chatelier principle . This introduces Samuelson’s Le Chatelier principle, which
in the supply–demand context simply, shows that any effect of a shift in demand on
prices may set in motion forces (i.e., entry) that tend to reduce the initial price increase.
Such moderation does not operate in the quantity dimension where initial effects become
larger over time.
Solutions
12.1
Given the cost function
1 3
2
C (q) =
q + 0.2q + 4q + 10.
300
Differentiating,
2
MC (q ) = 0.01q + 0.4 q + 4.
a.
Setting P = MC yields:
P = 0.01q 2 + 0.4q + 4.
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Chapter 12: The Partial Equilibrium Competitive Model
133
Solving,
100 P = q 2 + 40q + 400
⇒ ( q + 20 ) = 100 P
2
⇒ q + 20 = 10 P
⇒ q = 10 P − 20.
b.
Q = 100 q = 1, 000 P − 2, 000.
c.
Demand is Q = −200 P + 8, 000. Equating quantity demanded and supplied,
− 200 P + 8,000 = 1, 000 P − 2, 000
⇒ 1, 000 P + 200 P = 10, 000
⇒ 5 P + P = 50.
This is a quadratic equation in P . Solving yields P* = 25, Q* = 3, 000. For each
firm, q * = 30, C = 400, AC = 13.3, π = 351.
12.2
Given the cost function
C = q 2 + wq.
a.
Differentiating total cost gives marginal cost:
MC = 2 q + w.
Substituting w = 10,
C = q 2 + 10q.
MC = 2q + 10.
Set MC = P and solve for the firm’s (short-run) supply curve:
2q + 10 = P
⇒ q = 0.5 P − 5.
The industry’s supply curve is
1000
Q = ∑ q = 500 P − 5, 000.
1
At P = 20, Q = 5,000. At P = 21, Q = 5,500.
b.
Here, MC = 2q + 0.002Q. For profit maximum, set MC = P :
q = 0.5 P − 0.001Q.
The industry supply curve is
1000
Q = ∑ q = 500 P − Q.
1
Q = 250 P.
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134
Chapter 12: The Partial Equilibrium Competitive Model
If P = 20, Q = 5,000. If P = 21, Q = 5,250. Supply is more steeply sloped in the
case where expanded output bids up wages.
12.3
Demand: Q = a + bP + cI or a + bP + cI − Q = 0 (b < 0),
Supply: Q = d + gP or d + gP − Q = 0 ( g > 0).
a.
Equating quantity demanded to quantity supplied yields:
a−d
c
a + bP + cI = d + gP or P* =
+
I,
g −b g −b
g (a − d )
cg
Q* = d + gP* = d +
+
I.
g −b
g −b
b.
dP*
c
=
> 0,
dI
g −b
dQ*
cg
=
> 0.
dI
g −b
c.
Differentiation of the demand and supply equations yields:
dP
dQ
+c−
= 0,
dI
dI
dP dQ
g
−
= 0.
dI dI
b
Putting this into matrix notation:
b − 1   dP dI   −c 
 g − 1 ⋅  dQ dI  =  0  and applying Cramer’s rule gives

 
  
−c − 1
0 −1
dP
c
=
=
b −1
dI
g −b
g −1
*
d.
b −c
*
g 0
dQ
cg
=
=
.
b −1 g − b
dI
g −1
Suppose
a = 10, b = −1, c = 0.1, d = −10, g = 1, I = 100 : P* = 10 + 0.05 ⋅100 = 15, Q* = 5.
an increase of income of 10 would increase quantity demanded by 1 if price were
held constant. This would create an excess demand of 1 that must be closed by a
price rise. Because the demand and supply relations have price coefficients that
are equal and opposite in sign, a price rise of 0.5 will reduce quantity demanded
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Chapter 12: The Partial Equilibrium Competitive Model
135
by 0.5 and increase quantity supplied by the same amount. Equilibrium quantity
would also increase by 0.5. Hence, the new equilibrium is P* = 15.5, Q* = 5.5 as
would have been predicted by the multipliers from parts (b) and (c).
12.4
a.
The long-run supply curve is horizontal at P * = MC = AC = 10.
b.
Given demand Q = 1,500 − 50 P , Q* = 1, 000 when P * = 10. For each firm:
qi* = 20, π = 0. There are 50 firms.
c.
MC = qi − 10.
200
.
qi
AC is at a minimum when AC = MC :
200
0.5qi =
,
qi
implying qi = 20.
AC = 0.5qi − 10 +
d.
The profit-maximization condition is P = MC = qi − 10. Rearranging yields firm
supply qi = P + 10. The industry supply curve is
50
Q = ∑ qi = 50 P + 500.
i =1
e.
New demand is Q = 2,000 − 50 P . With Q = 1,000 in the very short run, P * = 20.
For each firm: qi* = 20,
π = 20 ( 20 − 10 ) = 200.
f.
Solving 50 P + 500 = 2,000 − 50 P yields P* = 15, Q* = 1,250. For each firm,
qi* = 25 and
π = 25 (15 − AC ) = 25 (15 − 10.5) = 112.5.
g.
12.5
P * = 10 again. Further, Q* = 1,500, 75 firms produce 20 each, π = 0.
Given cost function
C (q, w) = 0.5q 2 − 10q + w.
a.
Equilibrium in the entrepreneur market requires
QS = 0.25w = QD = n,
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136
Chapter 12: The Partial Equilibrium Competitive Model
or w = 4n. Hence,
C (q, w) = 0.5q 2 − 10q + 4n.
Thus,
MC = q − 10,
4n
AC = 0.5q − 10 + .
q
In long-run equilibrium, AC = MC. Substituting,
4n
q − 10 = 0.5q − 10 +
q
4n
⇒ 0.5q =
q
⇒ q = 8n .
Total output is given in terms of the number of firms by
Q = nq = n 8n .
Find the conditions in supply–demand equilibrium:
QD = 1,500 − 50 P.
P = MC = q − 10.
Solve for q and then Qs :
q = P + 10.
QS = nq = n( P + 10).
You are left with three equations in Q, n, P. Since Q = n 8n and
Q = n ( P + 10), we have
n 8n = n( P + 10)
⇒ P = 8n − 10.
Substitute P into QD :
Q D = 1,500 − 50 P
= 1,500 − 50 8n + 500.
Since QD = QS ,
2, 000 − 50 8n = Q S
⇒ 2, 000 − 50 8n = n 8n
⇒ (n + 50) 8n = 2, 000.
Since the number of entrepreneurs is n = 50,
Q* = n 8n = 1, 000,
Q*
= 20,
n
P* = q* − 10 = 10,
q* =
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Chapter 12: The Partial Equilibrium Competitive Model
137
w* = 4n = 200.
b.
Following the same algebraic calculations as before yields (n + 50) 8n = 2,928.
Since n = 72,
Q* = n 8n = 1, 728,
Q*
= 24,
n
P* = q* − 10 = 14,
w* = 4n = 288.
q* =
c.
This curve is upward sloping because as new firms enter the industry, the cost
curves shift up:
4n
AC = 0.5q − 10 + .
q
As n increases, AC increases.
Using linear approximations, the increase in PS from the supply curve is
given by
4 ⋅1, 000 + 0.5 ⋅ 728 ⋅ 4 = 4, 000 + 1, 456 = 5, 456.
If we use instead the supply curve for entrepreneurs, the area is
88 ⋅ 50 + 0.5 ⋅ 88 ⋅ 22 = 4, 400 + 968 = 5,368.
These two numbers agree roughly. To get exact agreement would require
recognizing that the long-run supply curve here is not linear; it is slightly concave.
12.6
a.
Short-run supply is q = P − 10. Market supply is 100q = 100 P − 1,000.
b.
Equilibrium where 100 P − 1,000 = 1,100 − 50 P. So, P* = 14, Q* = 400.
c.
Since QS = 0 when P = 10, producer surplus = 0.5 (14 − 10 )( 400 ) =800.
d.
Total industry fixed cost = 500. For a single firm,
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138
Chapter 12: The Partial Equilibrium Competitive Model
π = 14 ⋅ 4 −  0.5(4) 2 + 40 + 5 = 56 − 5 = 3.
Total industry profits = 300. Finally, we have
Short-run profits + fixed cost = 800 = producer surplus.
e.
With tax, PD = PS + 3. Equating supply and demand,
1,100 − 50 PD = 100 PS − 1,000
⇒ 1,100 − 50 PD = 100 ( PD − 3) − 1,000
⇒ 150 PD = 2,400.
Thus,
PD* = 16
PS* = 16 − 3 = 13
Q* = 1,100 − (50 ⋅16) = 300.
Total tax = 900.
f.
Consumers pay 300 (16 − 14 ) = 600. Producers pay 300 (14 − 13) = 300.
g.
PS = 0.5 ( 300 )(13 − 10 ) = 450, a loss of 350 from Problem 11.2, part (d). Shortrun profits equal 13 ( 300 ) − 100C. But
C = 0.5 ( 3) + 30 + 5 = 39.5.
2
Hence,
π = 3,900 − 3,950 = −50.
Since total profits were 300, this is a reduction of 350 in short-run profits.
12.7
a.
The long-run equilibrium price is
10 + r = 10 + 0.002Q.
So,
Q = 1,050 − 50 (10 + 0.002Q )
= 550 − 0.1Q.
Solving yields Q* = 500, P* = 11, r * = 1.
b.
Now,
Q = 1,600 − 50 (10 + 0.002Q )
= 1,100 − 0.1Q.
Solving yields Q* = 1,000, P* = 12, r * = 2.
c.
The change in producer surplus is
∆PS = 1( 500 ) + 0.5 (1)( 500 ) = 750.
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Chapter 12: The Partial Equilibrium Competitive Model
d.
The change in rents is
1( 500 ) + 0.5 (1)( 500 ) = 750.
The areas are equal.
e.
With tax, PD = PS + 5.5. Supply is PS = 10 + .002Q. In terms of the consumer
price, PD = 15.5 + .002Q. Equating demand and supply,
139
Q = 1,050 − 50 (15.5 + 0.002Q ) = 275 − 0.1Q.
Solving 1.1Q = 275 yields Q* = 250, PD* = 16, r * = 0.5. The total tax is
5.5 ( 250 ) = 1,375.
Demanders pay 250 (16 − 11) = 1, 250. Producers pay 250 (11 − 10.5 ) = 125.
f.
CS originally = 0.5 ( 500 )( 21 − 11) = 2,500.
CS now = 0.5 ( 250 )( 21 − 16 ) = 625.
PS originally = 0.5 ( 500 )(11 − 10 ) = 250.
PS now = 0.5 ( 250 )(10.5 − 10 ) = 62.5.
g.
12.8
Loss of rents = 0.5 ( 250 ) + 0.5 ( 250 )( 0.5) = 187.5. This is the total loss of PS in
part (b). It occurs because the only reason for upward sloping supply is upward
slope of film royalties’ supply.
a.
Solve 150 P = 5,000 − 100 P for domestic equilibrium. This yields P* = 20,
Q* = 3,000 (i.e., 3 million).
b.
If the price drops to 10, QD* = 4,000.
Domestic production = 150 ⋅10 = 1,500.
Imports = 2,500.
c.
If price rises to 15, QD* = 3,500.
Domestic production = 150 ⋅15 = 2,250.
Imports = 1,250.
Tariff revenues = 6,250.
CS with no tariff = 0.5 ( 4,000 )( 50 − 10 ) = 80, 000.
CS with tariff = 0.5 ( 3,500 )( 50 − 15 ) = 61, 250.
Loss = 18, 750.
Transfer to producers = 5 (1,500 ) + 0.5(2,250 − 1,500) (15 − 10 ) = 9,375.
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140
Chapter 12: The Partial Equilibrium Competitive Model
Deadweight loss = Total loss – Tariffs – Transfer = 3,125.
Check results using triangles.
Loss = 0.5(2,250 − 1,500) ( 5 ) + 0.5 ( 4,000 − 3,500 )( 5 ) = 1,875 + 1, 250 = 3,125.
12.9
d.
With quota of 1,250, results duplicate part (c) except no tariff revenues are
collected. Now, 6,250 can be obtained by rent seekers.
a.
Long-run equilibrium requires P = AC = MC.
k
AC = + a + bq = MC = a + 2bq,
q
k
b
hence, q =
b.
P = a + 2 kb .
Want supply = demand nq = n
Hence, n =
k
= A − BP = A − B( a + 2 kb )
b
A − B ( a + kb )
.
k b
c.
A has a positive effect on n. That makes sense since A reflects the “size” of the
market. If a > 0, the effect of B on n is clearly negative.
d.
Fixed costs (k) have a negative effect on n. Higher marginal costs raise price and
therefore reduce the number of firms.
Analytical Problems
12.10. Ad valorem taxes
D[ P (1 + t )] − Q = 0,
S ( P ) − Q = 0,
a.
DP P + DP (1 + t )
SP
dP dQ
dP dQ
−
= 0 ≈ DP
−
= − D p P,
dt dt
dt dt
dP dQ
−
= 0.
dt dt
Writing this in matrix notation:
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Chapter 12: The Partial Equilibrium Competitive Model
141
 DP − 1  dP dt   − DP P 
 S − 1  ⋅  dq dt  = 
 and applying Cramer’s rule:
  0 
 P
 
− DP P − 1
*
dP
=
dt
b.
0
DP
eD , P
−1
DP P
1 dP* d ln P*
DP
=
or ⋅
=
=
=
.
−1
S P − DP
P dt
dt
S P − DP eS , P − eD , P
SP
−1
DW is given as the area of the shaded region in the graph below. For a small tax
increase starting from t = 0, DW can be approximated using the formula for the
area of a triangle. (This is only an approximation because the supply and demand
curves may not be straight lines.) Thus,
1
DW = [ P(1 + t ) − P ] ( Q* − Q0 )
2
tP
= ( Q* − Q0 )
2
tP  dP
≈ SP
( −t ) ,
2  dt

or, rearranging,
t2
dP
DW = − ( S P P ) .
2
dt
As shown in part (a),
d ln P
eD
=
dt
eS − eD
dP
eD
eD
⇒
= P⋅
≈ P0 ⋅
.
dt
eS − eD
eS − eD
The last approximation is good for a small tax increase above 0, implying P ≈ P0 .
Further, manipulating the expression to have an elasticity show up,
 Q* 
SP P = SP P  * 
Q 
= eS , P Q*
≈ eS , P Q0 ,
where the approximation Q * ≈ Q0 is again good for a small tax increase above 0.
Substituting these results into the expression for DW ,
t 2 eD eS
DW ≈ −
P0Q0 ,
2 eS − eD
as was to be shown.
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142
Chapter 12: The Partial Equilibrium Competitive Model
P
Supply
PS (1+t)
PS
Demand
QS
c.
Q0
Q
Under perfect competition the tax “wedge” diagram shows that if a unit tax and an
ad valorem tax collect the same amount in total tax revenue, then the size of the
tax wedge between demand and supply prices must be the same. Hence, the two
equal revenue taxes will have identical effects on final prices. In more complex
models, even in competitive conditions, the two taxes may have differential
effects on the characteristics of goods. The equality of the two taxes does not
persist under alternative market structures. Specifically, under monopoly the ad
valorem tax has a small distortionary effect than a unit tax that collects the same
amount of revenue. See Problem 14.10.
12.11 The Ramsey formula for optimal taxation
a.
Use the deadweight loss formula from Problem 12.9:
n
n


L = ∑ DW (ti ) + λ  T − ∑ ti pi xi .
i =1

i =1

 e e 
∂L
= 0.5  D S  2 ti pi xi − λ pi xi = 0.
∂ti
 eS − eD 
n
∂L
= T − ∑ ti pi xi = 0.
∂λ
i =1
Thus,
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Chapter 12: The Partial Equilibrium Competitive Model
ti =
143
1 1 
−λ (eS − eD )
= λ  − .
eS eD
 eS eD 
b.
The above formula suggests that higher taxes should be applied to goods with
more inelastic supply and demand. A tax on a good discourages the consumption
and production of that good. Thus, taxing a good with more inelastic supply and
demand would result in less change in the consumption of the good. Therefore,
the tax would produce smaller distortions: DW would be smaller.
c.
This result was obtained under a set of very restrictive assumptions. First, it was
obtained under partial equilibrium (the welfare analysis is undertaken in each
market separately), ignoring the general equilibrium interactions between
markets. Also, income effects and cross-price elasticities are not taken into
account.
12.12 Cobweb models
a.
a − bP* = c + dP* ⇒ P* = (a − c) (b + d ).
b.
a − bP1 = c + dP0 ⇒ P1 =
c.
a−c d
− P0 .
b
b
Repeated substitution yields
a−c  d  d 
 −d 
1 − +   + ... + 

b  b  b 
 b 
2
Pt =
d.
t −1
  − d t
+
 P0 .
  b 
Use
t −1
 d  −d 2
1
( − d b)t
b
 −d  
1
−
+
+
...
+
=
−
=
1 − ( − d b)t )
(




 
 b   1 + d b 1 + d b (b + d )
 b  b 
So,
Pt = P * (1 − ( − d b)t ) − ( − d b)t P0 .
e.
Clearly, for d < b, Pt → P * as t → ∞.
f.
The graph has a “cobweb” shape for d < b.
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144
Chapter 12: The Partial Equilibrium Competitive Model
12.13 More on the comparative statics of supply and demand
a.
Shifts in supply: Assume demand is given by D ( P ) − Q = 0 and supply by
S ( P , β ) − Q = 0 . Differentiation of these yields
dP dQ
DP
−
= 0,
dβ dβ
dP
dQ
SP
+ Sβ −
= 0.
dβ
dβ
In matrix notation
 DP − 1  dP d β   0 
 S − 1  ⋅  dQ d β  =  − S 
  β
 P
 
And Cramer’s rule shows that
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Chapter 12: The Partial Equilibrium Competitive Model
145
0 −1
−Sβ − 1
−Sβ
dP
=
=
,
d β DP − 1 S P − DP
SP − 1
DP
0
− DP S β
dQ S P − S β
=
=
.
DP − 1 S P − DP
dβ
SP −1
Hence, if S β > 0, then dP* d β < 0 and dQ* d β > 0. This is precisely the lesson
from introductory economics—a shift outward in the supply curve lowers price
and increases output.
b.
A quantity wedge
D ( P ) − Q = 0,
S ( P) − Q − Q = 0,
dP dQ
DP
−
= 0,
dQ dQ
dP
dQ
SP
−1−
= 0.
dQ
dQ
Applying Cramer’s rule
0 −1
1 −1
dP
1
=
=
> 0,
dQ DP − 1 S P − DP
SP
−1
DP
0
S 1
dQ
DP
= P
=
< 0.
DQ DP − 1 S P − DP
SP − 1
So, a (positive) quantity wedge increases price and reduces the quantity that goes
to meet demand. A negative wedge (such as an import quota) would have the
opposite effects (when starting from a no-trade equilibrium).
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146
Chapter 12: The Partial Equilibrium Competitive Model
c.
The identification problem
dQ* dα
i. The analysis in the chapter shows that
= S P . With sufficient
dP* dα
observations on the impact of differing values of α , one could identify the slope
of the supply curve.
dQ* d β
= DP . With sufficient observations
dP* d β
on the impact of differing values of β , one could identify the slope of the
demand curve.
ii. Part (a) of this problem shows that
iii.
If the same parameter shifts both curves it is not possible to identify the
slope of either of them.
12.14 The Le Chatelier Principle
a.
Here are Equations 12.24:
dP*
dQ*
dP* dQ*
+ Dα −
= 0 or DP
−
= − Dα
dα
dα
dα dα
dP* dQ*
SP
−
= 0.
dα dα
DP
Differentiation with respect to t yields
d 2 P d 2Q
−
= 0,
dα dt dα dt
d 2P
dP d 2Q
SP
+ S Pt
−
= 0.
dα dt
dα dα dt
DP
b.
Cramer’s rule can now be used to solve for the second-order partials:
−1
0
d 2P
=
dα dt
− S Pt
dP
dα
−1
DP
−1
SP
−1
dP
dα .
=
S P − DP
− S Pt
This expression shows that
d 2P
dP
and
are of opposite signs. That is, the
dα dt
dα
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Chapter 12: The Partial Equilibrium Competitive Model
147
effect of an outward demand shift on increasing price diminishes over time.
Similarly, a reduction in demand initially reduces price, but then price rises over
time back toward the old equilibrium. The Le Chatelier principle therefore
captures the way in which entry and exit affect price in the model of competitive
pricing developed in this chapter.
c.
Again, we use Cramer’s rule:
DP
2
d Q
=
dα dt
SP
0
dP
dP − DP S Pt ⋅ dQ
− DP S Pt
dα
dα =
dα = S P
,
S P − DP
S P − DP
S P − DP
− S Pt
where the final equation uses the combined results of Equations 12.26 and 12.27.
Because DP < 0 and SPt > 0 , this result shows that the effect of α on equilibrium
quantity is exaggerated over time. A shift outward in demand will have a greater
long-run effect on increasing equilibrium quantity than on short-run equilibrium
quantity. Similarly, a reduction in demand will reduce quantity more in the long
run than in the short run. Again these results mirror the analysis of entry and exit
in the chapter.
d.
See answers to parts (b) and (c) above.
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CHAPTER 13:
General Equilibrium and Welfare
WelfareExternalities and Public Goods
The problems in this chapter focus primarily on the simple two-good general equilibrium model
in which “supply” is represented by the production possibility frontier and “demand” by a set of
indifference curves. One shortcoming of this approach is that students do not see the interaction
between output and input markets. Problems 13.7 and 13.8 seek to remedy this by using the
computer general equilibrium model presented in the chapter. The Analytical Problems in the
chapter illustrate a few general equilibrium “theorems,” but no very formal proofs are intended.
Comments on Problems
13.1
This problem repeats an example from Chapter 1 in which the production possibility
frontier is concave (a quarter ellipse). It is a good starting problem because it involves
very simple computations.
13.2
This problem is a simple example of general equilibrium with linear production functions
and differing preferences among the two people in the economy.
13.3
This problem is a fixed-proportions example that yields a concave production possibility
frontier. This is a good initial problem although students should be warned that calculustype efficiency conditions do not hold precisely for this type of problem.
13.4
This problem uses a quarter-circle production possibility frontier and a Cobb–Douglas
utility function to derive an efficient allocation. The problem then proceeds to illustrate
the gains from trade. It provides a good illustration of the sources of those gains.
13.5
This problem provides a numerical example of an Edgeworth Box in which efficient
allocations are easy to compute because one individual wishes to consume the goods in
fixed proportions.
13.6
This provides an example of efficiency in the regional allocation of resources. The
problem could provide a good starting introduction to mathematical representations of
comparative versus absolute advantage or for a discussion of migration. To make the
problem a bit easier, students might be explicitly shown that the production possibility
frontier has a particularly simple form for both the regions here (e.g., for region A it is
x 2 + y 2 = 100 ).
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Chapter 13: General Equilibrium and Welfare
13.7
147
This problem uses the computer model to examine the consequences of various changes
in preferences or technology. Having students try to explain why things turn out the way
they do is a good way to build intuition.
Analytical Problems
13.8
Tax equivalence theorem. This problem uses the computer simulation model to shows
the formal equivalence between input and output taxes.
13.9
Returns to scale and the production possibility frontier. Here students are asked to use
Excel or some other software to illustrate the shape of production possibility frontiers
with varying degrees of returns to scale. One result is that frontiers can still be convex
with modest increasing returns providing input proportions are sufficiently different.
13.10 The trade theorems. This problem provides simple two-good graphical proofs of three
major trade theorems: (1) factor-price equalization; (2) the Stolper–Samuelson theorem;
and (3) the Rybczynski theorem. Although it requires only facility with the production
box diagram (and its underlying Edgeworth Box), it is a fairly difficult problem. Extra
credit might be given for the correct spelling of the discoverer of the third theorem.
13.11 An example of Walras’ law. This problem is a algebraic example of how Walras’ law
can be used to find the excess demand function for good 1.
13.12 Productive efficiency with calculus. This problem illustrates how the simple two-good
general equilibrium model of production can be solved for efficient allocations using
calculus. Especially important is to show how the tradeoffs implied by the calculus
results can be interpreted as providing equilibrium relative prices.
13.13 Initial endowments, equilibrium prices, and the first theorem of welfare economics.
This problem shows how initial endowments can constrain the possible prices that can
emerge from competitive bargaining. This would be a good opening to discussing the
concept of the “core” of a competitive economy, though that concept is not explicitly
covered in Chapter 13.
13.14 Social welfare functions and income taxation. This problem explores the complex
relationship between social welfare and the appropriate tax function.
Solutions
13.1
a.
The frontier is a quarter ellipse.
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148
13.2
Chapter 13: General Equilibrium and Welfare
b.
9 x 2 = 900, so, x = 10 and y = 20.
c.
The slope of the frontier is given by
f
2x
x
− x =−
=− .
fy
4y
2y
At x = 10, y = 20, the slope is −0.25. This can be approximated by showing that
when x = 9, y = 20.24, and when x = 11, y = 19.74. Hence, the slope is
−∆y ∆x = −0.5 2 = −0.25. The price ratio will therefore be px p y = 1 4.
a.
px p y = 3 2 .
b.
If the wage is 1, each person’s income is 10. Smith’s demand for x is x = 3 px
and for y is y = 7 p y . Similarly, Jones’ demands are x = 5 px , y = 5 p y . Hence,
total demands are x = 8 px , y = 12 p y . Supply is determined by total labor
available that requires
x y
+ = 20 .
2 3
Substituting the demands for x and y yields
8
12
20
+
=
= 20,
2 px 3 p y 2 px
or
1
1
px = , p y = .
2
3
Given these prices, Smith chooses x = 6, y = 21. Jones chooses x = 10, y = 15.
c.
Production is x = 16, y = 36 and the 20 total hours of labor must be devoted as 8
to production of x and 12 to production of y.
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Chapter 13: General Equilibrium and Welfare
13.3
13.4
149
Let f denotes food and c cloth.
a.
Labor constraint: f + c = 100 (see graph below).
b.
Land constraint: 2 f + c = 150 (see graph below).
c.
The heavy line in graph below satisfies both constraints.
d.
The frontier is concave because it must satisfy both constraints. Since the RPT = 1
for the labor constraint and 2 for the land constraint, the production possibility
frontier of part (c) exhibits an increasing RPT; hence it is concave.
e.
Constraints intersect at f = 50, c = 50. For f < 50, dc df = −1; so for f > 50,
dc df = −2; so, p f pc = 2.
f.
If MRS = dc df = 5 4 for consumers, then p f pc = 5 4. With any other price
ratio, only one of the goods would be consumed.
g.
Both price lines “tangent” to production possibility frontier at its kink. Any price
ratio between 1 and 2 would also be tangent at the kink (including p f pc = 5 4 ).
h.
The constraint is 0.8 f + 0.9c = 150. This constraint lies totally outside the PPF
and is nowhere binding. Hence, this constraint would not affect the relative prices
of the goods.
The PPF has the form f 2 + c 2 = 200. RPT = −dc df = 2 f 2c = f c .
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150
Chapter 13: General Equilibrium and Welfare
Consumer preferences imply MRS = U f U c = c f .
a.
For efficiency, set MRS = RPT . Substituting, we have f c = c f , implying
c = f . Using the PPF yields f = c = 10, U = 10, RPT = MRS = 1.
b.
Demand: p f pc = 2. For utility maximization, 2 = MRS = c f , implying c = 2 f .
Budget constraint is 2 f + c = 30 because 30 is the total value of production at the
new world prices. Substituting the condition for utility maximization into the
budget constraint yields c = 15, f = 7.5, and U = 112.5, which is a distinct
improvement from the no trade level.
c.
To adjust to world prices, set RPT = p f pc = 2 = f c , implying f = 2c. Using
the PPF yields c = 2 10, f = 4 10. The budget constraint is now
2 f + c = 10 10. Again using the utility-maximization condition yields c = 5 10,
f = 2.5 10, and U = 125. This is a further improvement from part (b) because
of the ability to specialize in production.
d.
13.5 a.
Contract curve is straight line with slope of 0.5. The only price
ratio in equilibrium is pc ph = MRS (for Jones) = 3 4.
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Chapter 13: General Equilibrium and Welfare
b.
151
An initial endowment for Smith of cS = 80, hS = 40 is on the contract curve. At
this point, Jones has an endowment of cJ = 120, hJ = 60, and no trading is
possible.
c.
An initial endowment of cS = 80, hS = 60 for Smith is not on the contract curve.
The equilibrium must be between cS = 80, hS = 40 —where Jones gets all the
gains from trade—and a trade that leaves Jones indifferent. This other equilibrium
must provide a utility of
U J (cJ , hJ ) = 3cJ + 4hJ = 3(120) + 4(40) = 520.
Because along the contract curve cS = 2hS , we can compute this new equilibrium
by
U J (cJ , hJ ) = 3cJ + 4hJ = 10hJ = 520,
Implying cJ = 104, hJ = 52. At this point, Smith gets cS = 96, hS = 48.
13.6
d.
Smith grabs everything; trading ends up at OJ on the contract curve.
a.
For region A, the PPF is x A2 + y A2 = 100. For region B, the PPF is xB2 + yB2 = 25.
b.
For efficiency, the RPTs should be equal.
c.
For both regions, the RPT is given by xi yi . So,
x A xB
=
y A yB
⇒
⇒
 y2 
y A2 = x A2  B2 
 xB 
xA xB
=
y A yB
 y2 
2
y A = x 2A  2B  .
 xB 
2
2
2
But x 2A + y A = 4( x 2B + y B ); so substituting for y A yields
⇒
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152
Chapter 13: General Equilibrium and Welfare
 y 2B 
 y 2B 
2
1
+
=
4
x 
x B 1 + 2  ,
2 
x
B


 xB 
implying xA = 2 xB and y A = 2 yB . Using the subscript t to denote total
2
A
production yields xt = x A + xB = 3 xB , implying x t2 = 9 x 2B . Similarly, yt2 = 9 yB2 .
Therefore, xt2 + yt2 = 9( x 2B + y B) = 9 ⋅ 25 = 225. If xt = 12, then
2
yt = 225 − 144 = 9.
Note: Can also show that more of both goods can be produced if labor
could move between regions.
13.7
a.
By changing the utility of household 1 to
(
U 1 = x10.6 y10.2 l 1 − l1
) ,
0.2
we increased household 1’s relative preference for x as opposed to y. By running
the simulation we obtain:
px = 0.3744,
p y = 0.2377,
pk = 0.1239,
pl = 0.2641.
The utility-maximizing choices for household 1 are x1 = 18.10, y1 = 9.10, and
l1 − l1 = 8.55, giving U1 = 13.69. The utility-maximizing choices for household 2
are x2 = 8.1, y2 = 12.75, and l2 − l2 = 5.74, giving U 2 = 9.06.
Because of its increased preference for good x, household 1 will demand
relatively more x and relatively less of y. The demand for good x will increase and
thus, the relative price of good x will also increase. Since y is relatively less
preferred, its price will decrease and thus, more of good y will be consumed by
both households. Because the relative price of y decreases and y is the capitalintensive good, the relative price of capital will also decrease. Since x is the laborintensive good, the increased demand for x will cause the price of labor to
increase and this will lead to more labor-hours being supplied, thus reducing the
leisure demanded by each household.
b.
After reversing the production functions, we obtain the following equilibrium
prices:
px = 0.2835,
p y = 0.3375,
pk = 0.1622,
pl = 0.2169.
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Chapter 13: General Equilibrium and Welfare
153
The utility-maximizing choices for household 1 are x1 = 20.62, y1 = 10.39, and
l1 − l1 = 10.78, giving U1 = 14.75. The utility-maximizing choices for household 2
are x2 = 9.63, y2 = 8.09, and l2 − l2 = 6.30, giving U 2 = 8.25.
Good x provides more utility for household 1 (the wealthy household), so
it is relatively more demanded than good y. Thus, compared to the initial case, the
relative price of capital will increase and the relative price of labor will decrease
because now x is capital-intensive. The falling relative price of x may seem
counter-intuitive here. In part this arises because total labor supply is reduced in
this simulation and now y is the labor-intensive good.
c.
If the utility functions are changed to
( ) ,
U = x y (l − l ) ,
U 1 = x10.4 y10.2 l1 − l1
2
0.3
2
0.3
2
0.4
0.4
2
2
the following equilibrium is obtained:
px = 0.4040,
p y = 0.1912,
pk = 0.0904,
pl = 0.3143.
The utility-maximizing choices for household 1 are x1 = 12.27, y1 = 6.48, and
l1 − l1 = 15.78, giving U1 = 9.91. The utility-maximizing choices for household 2
are x2 = 6.27, y2 = 13.25, and l2 − l2 = 10.75, giving U 2 = 9.74.
The greater utility gained from the labor-hours not sold on the market will
cause households to sell less labor. This will cause the relative price of labor to
increase and the relative price of the other input (capital) to decrease. Thus, the
labor-intensive good, good x, will become relatively more expensive and the
capital-intensive good, good y, will become relatively cheaper.
Analytical Problems
13.8
Tax equivalence theorem
Adding an ad valorem tax of 0.2 on goods x and y raises the same revenue (3.10) as an ad
valorem tax of 2.5 on capital and labor. Running the simulation with either tax yields the
same results:
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154
Chapter 13: General Equilibrium and Welfare
px = 0.3989,
p y = 0.2667,
pk = 0.1131,
pl = 0.2213.
13.9
Returns to scale and the production possibility frontier
We have never succeeded in deriving an analytical expression for all these cases. We
have, however, used computer simulations (e.g., with Excel) to derive approximations to
these production possibility frontiers. These tend to show that increasing returns to scale
is compatible with concavity providing factor intensities are suitably different (case [e]),
but convexity arises when factor intensities are similar (case [d]).
13.10 The trade theorems
For all of these proofs, draw the PPF and its underlying Edgeworth Box Diagram. The
world price ratio determines where production will occur on the PPF and where it will
occur in the Edgeworth Box. Given the assumption about factor intensities, the contract
curve in the Edgeworth Box will be concave (i.e., bowed upward).
a.
Factor price equalization theorem: Because productive technology is
everywhere the same, the world price ratio will identify the same point in every
country’s Edgeworth Production Box. The slope of the isoquants in the Box gives
the ratio of relative factor prices, which will be the same everywhere.
b.
Stolper–Samuelson theorem: The increase in p will increase x production and
decrease y production. This will cause a clockwise move along the PPF and a
northeast movement along the contract curve in the Edgeworth Box. Because of
the concave nature of the contract curve, this will decrease the k l ratio in the
production of both goods. This happens because x is capital intensive. Such a
decrease will raise the relative price of capital. So, assuming x is the exported
good, increasing trade increases the relative factor price of the input used
intensively in exports. According to another theorem (the Heckscher–Ohlin
theorem) this will be a country’s relatively “abundant” factor.
c.
The Rybczynski theorem: If the world price ratio stays constant, the ratio of
inputs used must stay constant. If capital increases, the only way this can happen
is if output of the capital-intensive good expands and of the labor-intensive good
falls.
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Chapter 13: General Equilibrium and Welfare
155
13.11 An example of Walras’ law
a.
Functions are obviously homogeneous of degree zero since doubling of p1 , p2 , p3
does not change ED2 or ED3 .
b.
Walras’s law states ∑ i pi EDi = 0. Hence, if ED2 = ED3 = 0, then p1 ED1 = 0,
implying ED1 = 0. One can calculate ED1 using
p1 ED1 = − p2 ED2 − p3 ED3 .
This yields
3 p 2 − 6 p2 p3 + 2 p32 + p1 p2 + 2 p1 p3
ED1 = 2
.
p12
Notice that ED1 is homogeneous of degree zero also.
c.
ED2 = 0 and ED3 = 0 can be solved simultaneously for p2 p1 and p3 p1 .
Simple algebra yields p2 p1 = 3, p3 p1 = 5. If arbitrarily set, p1 = 1 have p2 = 3
and p3 = 5 at these absolute prices. Furthermore, ED1 = ED2 = ED3 = 0.
13.12 Productive efficiency with calculus
a.
The problem for this society is to maximize utility subject to the technological
constraint on production. The Lagrangian for this problem is
L = U ( x, y) + λT ( x, y).
b.
First-order conditions for a maximum are
Lx = U x + λTx = 0,
L y = U y + λTy = 0,
Lλ = T ( x, y ) = 0.
These can be manipulated to yield the familiar
Ux
T
= MRS = x = RPT .
Uy
Ty
c.
A competitive equilibrium price ratio of p*x p*y would cause utility-maximizing
consumers to choose
Ux
px*
= MRS = * ,
Uy
py
and profit-maximizing firms to choose
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156
Chapter 13: General Equilibrium and Welfare
Tx
MCx px*
= RPT =
=
.
Ty
MC y p*y
Hence, the “invisible hand” will promote the efficiency condition in part (b).
d.
As discussed in the chapter, any factor that leads equilibrium prices to incorrectly
reflect true marginal costs could cause the theorem to fail. Such factors include
monopoly, externalities (including public good externalities), and imperfect
information about prices.
13.13 Initial endowments, equilibrium prices, and the first theorem of welfare economics
a.
The value of A’s initial endowment is pxA + y A . Hence, his or her demand for
good x is given by
2( px A + y A )
xA =
.
3p
Similarly, B’s demand for good x is given by
p (1, 000 − x A ) + 1,000 − y A
xB =
.
3p
b.
Setting total demand equal to total supply for good x yields
px + y A + 1,000 p + 1, 000
x A + xB = A
= 1, 000,
3p
implying
p( xA − 2,000) = − y A − 1,000,
in turn implying
y + 1, 000
p= A
.
2, 000 − xA
c.
With these initial endowments, p = 1. Person A demands
2, 000
1, 000
xA =
, yA =
,
3
3
and person B demands
1, 000
2, 000
xB =
, yB =
.
3
3
This solution is efficient because it obeys Equation 13.43.
d.
Part (b) shows that increase in the endowment of either good for person A will
raise the relative price of good x because that good is favored by this person.
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Chapter 13: General Equilibrium and Welfare
157
13.14 Social welfare functions and income taxation
a.
The Lagrange problem for a welfare optimum is
uγ
L = ∑ i + λ  ∑ T ( I i ) − R .
γ
The first-order condition for a maximum is that for each person the following
condition should hold: −uiγ −1T ′ + λT ′ = 0, or uiγ −1 = λ for all i. But this requires
after-tax income (adjusted for the utility costs of working) should be equal for all
individuals.
b.
A similar condition to that found in part (a) holds if the tax function is given by
T (ai ).
c.
If taxation is based on observed income we would need to model how ci responds
to such taxation. If taxation causes a person to incur a lower value for ci (and
hence a lower income) that would have to be taken into account in the calculation.
d.
If all individuals have the same relative weight in the social welfare function
(ki = 1 for all i ), Diamond’s formula would suggest a zero marginal tax rate for
the highest income person. If kmax = 0.5 and eL , w = 0.5, the formula suggests an
optimal marginal tax rate of
0.75
T ′( I max ) =
= 0.43.
1.75
Greater work disincentives would imply a lower tax rate. For example, if
kmax = 0.5 and eL, w = 1.0, the optimal marginal tax rate is
1
T ′( I max ) = .
3
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accessible website, in whole or in part.
CHAPTER 14:
Monopoly
The problems in this chapter deal primarily with marginal revenue-marginal cost calculations in
different contexts. For such problems, students’ primary difficulty is to remember that the
marginal revenue concept requires differentiation with respect to quantity. Often students choose
to differentiate total revenue with respect to price and then get very confused on how to set this
equal to marginal cost. Of course, it is possible to phrase the monopolist’s problem as one of
choosing a profit-maximizing price, but then the inverse demand function must be used to derive
a marginal cost expression.
The analytical and behavioral problems in this chapter introduce students to some stateof-the-art research on monopoly reflected in recent academic articles.
Comments on Problems
14.1
This problem is a simple marginal revenue-marginal cost and consumer surplus
computation.
14.2
This problem is an example of the MR = MC calculation with three different types of cost
curves.
14.3
This problem is an example of the MR = MC calculation with three different demand and
marginal revenue curves. The problem also illustrates the “inverse elasticity” rule.
14.4
This problem examines graphically the various possible ways in which shift in demand
may affect the market equilibrium in a monopoly.
14.5
This problem introduces advertising expenditures as a choice variable for a monopoly.
The problem also asks the student to view market price as the decision variable for the
monopoly.
14.6
Note: This problem has been subtly revised from the previous edition; the numbers for
production and transportation cost are now different, helping students see where each
distinctly shows up in the calculations. This is a price-discrimination example in which
markets are separated by transport costs, showing how the price differential is
constrained by the extent of those costs. Part (d) asks students to consider a simple twopart tariff.
14.7
This problem shows how the welfare cost of monopoly may be larger than in the
traditional case if the monopoly has higher costs.
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160
Chapter 14: Monopoly
14.8
This problem examines some issues in the design of subsidies for a monopoly.
14.9
This problem involves quality choice. The result shows that, in this case, the monopoly
and competitive choices are the same (though output levels differ).
Analytical Problems
14.10 Taxation of a monopoly good. This problem focuses mainly on ad valorem taxes on a
monopoly good. The final part of the problem compares ad valorem and specific taxes.
14.11 Flexible functional forms. This problem has students run through the standard
monopoly analysis but for a class of flexible functional forms introduced in a recent
influential paper by Fabinger and Weyl (2015). While slightly complicated, the
functional forms allow for U-shaped average cost curves and realistic demand shapes.
14.12 Welfare possibilities with different market segmentations. This problem illustrates
extreme possibilities for price discrimination to create or destroy welfare identified in the
important recent paper by Bergemann, Brooks, and Morris (2015). To make their results
accessible, takes the simplest case of two consumer types, but the analysis of this case is
done in full generality.
Behavioral Problem
14.13 Shrouded prices. This problem introduces students to the problem of shrouded prices, a
topic that has received wide attention in behavioral economics. See for example, D.
Laibson and X. Gabaix, “Shrouded Attributes, Consumer Myopia, and Information
Suppression in Competitive Markets,” Quarterly Journal of Economics (May 2006):
505–540. More on whether competition uncovers shrouding to come in the next chapter.
Solutions
14.1
a.
Given P = 53 − Q. Then TR = PQ = 53Q − Q 2 , implying MR = 53 − 2Q. Profit
maximization yields MR = 53 − 2 q = MC = 5, implying Qm = 24, Pm = 29, and
π m = ( P − AC ) Q = 576.
b.
MC = P = 5 implies Pc = 5 and Qc = 48.
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accessible website, in whole or in part.
Chapter 14: Monopoly
c.
14.2
161
Consumer surplus under competition is 2(48) 2 = 1,152. See the graph for
monopoly.
Given market demand is Q = 70 − P , marginal revenue is MR = 70 − 2Q.
a.
Given AC = MC = 6. To maximize profit, set MC = MR. We have 6 = 70 − 2Q,
implying Qm = 32, Pm = 38, π m = ( P − AC )Q = (38 − 6)32 = 1, 024.
b.
C = 0.25Q 2 − 5Q + 300 implies MC = 0.5Q − 5. Setting MC = MR gives
0.5Q − 5 = 70 − 2Q, implying Qm = 30, Pm = 40, and
π m = TR − TC
= 30 ⋅ 40 − ( 0.25 ⋅ 302 − 5 ⋅ 30 + 300 )
= 825.
c.
C = 0.0133Q3 − 5Q + 250 implies MC = 0.04Q 2 − 5. Setting MR = MC yields
0.04Q 2 − 5 = 70 − 2Q, or 0.04Q 2 + 2Q − 75 = 0. Applying the quadratic formula,
Qm = 25. Solving for the other equilibrium variables, Pm = 45, Rm = 1,125,
Cm = 332.8, MCm = 20, π m = 792.2.
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162
14.3
Chapter 14: Monopoly
a.
Given AC = MC = 10 and Q = 60 − P , implying MR = 60 − 2Q. For profit
maximum, MC = MR ⇒ 10 = 60 − 2Q ⇒ Qm = 25. Solving for the other
equilibrium variables, Pm = 35 and π m = TR − TC = 25 ⋅ 35 − 25 ⋅10 = 625.
b.
Given AC = MC = 10 and Q = 100 − 2 P , implying MR = 90 − 4Q. For profit
maximum, MC = MR ⇒ 10 = 90 − 4Q ⇒ Qm = 20. Solving for the other
equilibrium variables, Pm = 50 and π m = 40 ⋅ 30 − 40 ⋅10 = 800.
c.
Given AC = MC = 10 and Q = 100 − 2 P , implying MR = 50 − Q. For profit
maximum, MC = MR ⇒ 10 = 50 − Q ⇒ Qm = 40. Solving for the other
equilibrium variables, Pm = 30 and π m = 40 ⋅ 30 − 40 ⋅10 = 800. π = (40)(30) –
(40)(10) = 800.
Note: Here the inverse elasticity rule is clearly illustrated:
Problem part
eQ , P =
∂Q P
⋅
∂P Q
−1
eQ,P
=
P − MC
P
(a)
−1( 35 25) = −1.4
0.71 = ( 35 − 10 ) 35
(b)
−0.5 ( 50 20 ) = −1.25
0.80 = ( 50 − 10 ) 50
(c)
−2 ( 30 40 ) = −1.5
0.67 = ( 30 − 10 ) 30
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Chapter 14: Monopoly
163
d.
The supply curve for a monopoly is a single point, namely, that quantity–price
combination that corresponds to the quantity for which MC = MR. Any attempt to
connect equilibrium points (price–quantity points) on the market demand curves
has little meaning and brings about a strange shape. One reason for this is that as
the demand curve shifts, its elasticity (and its MR curve) usually changes bringing
about widely varying price and quantity changes.
14.4
a.
b.
There is no supply curve for monopoly; have to examine MR = MC intersection
because any shift in demand is accompanied by a shift in MR curve. Cases (1) and
(2) above show that P may rise or fall in response to an increase in demand.
c.
Can examine this using inverse elasticity rule:
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164
Chapter 14: Monopoly
P
P
=
.
P − MC P − MR
As −e falls toward 1 (becomes less elastic), P − MR increases.
−e =
•
Case 1 MC constant, so profit-maximizing MR is constant
o If −e ↓, then P − MR ↑ ⇒ P ↑ .
o If −e constant, then P − MR constant ⇒ P constant.
o If −e ↑, then P − MR ↓ ⇒
14.5
P ↓.
•
Case 2 MC falling, so profit-maximizing MR falls
o If −e ↓, then P − MR ↑ ⇒ P may rise or fall.
o If −e constant, then P − MR constant ⇒ P ↓ .
o If −e ↑, then P − MR ↓ ⇒ P ↓ .
•
Case 3 MC rising, so profit maximizing MR must increase
o If −e ↓, then P − MR ↑ ⇒ P ↑ .
o If −e constant, then P − MR constant ⇒ MR ↑ ⇒ P ↑ .
o If −e ↑, then P − MR ↓ ⇒ P may rise or fall.
Given Q = ( 20 − P ) (1 + 0.1A − 0.01A2 ) . Let K = 1 + 0.1 A − 0.01 A2 . Then
dK dA = 0.1 − 0.02 A and
π = PQ − C
= ( 20 P − P 2 ) K − ( 200 − 10 P ) K − 15 − A.
The first-order condition with respect to price is
∂π
= ( 20 − 2 P ) K + 10 K = 0.
∂P
Solving, 20 − 2 P + 10 = 0 ⇒ Pm = 15, regardless of K or A.
a.
If A = 0, Qm = 5, Cm = 65, π m = 10.
b.
Substituting P = 15 into the profit function,
π = 75K − 50 K − 15 − A
= 25K − 15 − A
= 10 + 1.5 A − 0.25 A2 .
The first-order condition with respect to A is
dπ
= 1.5 − 0.5 A = 0,
dA
implying A = 3, Qm = 5(1 + 0.3 − 0.09) = 6.05, Rm = 90.75,
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Chapter 14: Monopoly
165
Cm = 60.5 + 15 + 3 = 78.5, and π m = 12.25; this represents an increase over the
case A = 0.
14.6
a.
In the first market, Q1 = 55 − P1
⇒ R1 = (55 − Q1 )Q1 = 55Q1 − Q12
⇒ MR1 = 55 − 2Q1. Setting MR1 = MC = 5 yields Q1* = 25 and P1* = 30. In the
second market, Q2 = 70 − 2 P2
⇒ R2 = [(70 − Q2 ) / 2]Q2 = (70Q2 − Q22 ) / 2
⇒ MR2 = 35 − Q2 . Setting MR2 = MC = 5 yields Q2* = 30 and P2* = 20. Profits
across both markets are π = (30 − 5) ⋅ 25 + (20 − 5) ⋅ 30 = 1, 075.
b.
If the producer ignores the problem of arbitrage among consumers, the price
differential between the two markets found to be optimal in the previous part
($10) induces arbitrage. The producer does better by preventing arbitrage by
keeping the price differential to $4, that is, P1 = P2 + 4. We can solve this as a
constrained maximization problem. Setting up the associated Lagrangian,
L = ( P1 − 5)( 55 − P1 ) + ( P2 − 5)( 70 − 2 P2 ) + λ ( 4 − P1 + P2 ) .
Taking the first-order conditions,
LP1 = 60 − 2 P1 − λ = 0,
LP2 = 80 − 4 P1 + λ = 0,
Lλ = 4 − P1 + P2 = 0.
This yields two equations in two unknowns 60 − 2 P1 = 4 P2 − 80 and P1 = P2 + 4.
Solving, 60 − 2 ( P2 + 4 ) = 4 P2 − 80, or P2* = 22. Further, P1* = 26 and π * = 1, 051.
(The same answer can be obtained by substituting P1 = P2 + 4 into profits from the
two markets and solving as a single-variable, unconstrained maximization
problem.)
c.
Now P1 = P2 = P. So π = 140 P − 3 P 2 − 625. Taking the first-order condition,
dπ dP = 140 − 6 P = 0, implying P* = 140 / 6 = 23.33, Q1* = 31.67, Q2* = 23.33,
and π * = 1, 008.33.
d.
If the firm adopts a linear tariff of the form T (Qi ) = α i + mQ i, it can maximize
profit by setting m = 5,
α1 = 0.5(55 − 5)(50) = 1, 250,
α 2 = 0.5(35 − 5)(60) = 900,
earning π * = 2,150. Notice that in this problem neither market can be uniquely
identified as the “least willing” buyer, so a solution similar to Example 14.5 is not
possible. If the entry fee were constrained to be equal in the two markets, the firm
could set m = 0 and charge a fee of 1,225 (the most buyers in market 2 would
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166
Chapter 14: Monopoly
pay). This would yield profits of 2, 450 − 125 ⋅ 5 = 1,825, which is inferior to
profits obtained with T (Qi ).
14.7 a.
Under perfect competition, MC = 10. Under monopoly, MC = 12.
Demand is QD = 1, 000 − 50 P. The competitive equilibrium is Pc = MC = 10,
implying Qc = 500. To solve for the monopoly outcome, P = 20 − Q 50
⇒ R = 20Q − Q 2 50 ⇒ MR = 20 − Q 25. Profits are maximized where
MR = MC ⇒ 20 − Q 25 = 12 ⇒ Qm = 200. Further, Pm = 16.
b.
Calculations are aided by the graph below. Loss of consumer
surplus equals CSc − CSm = 2,500 − 400 = 2,100. Of this 2,100 loss, 800 is a
transfer into monopoly profit, 400 is a loss from increased costs under monopoly,
and 900 is a “pure” deadweight loss.
c.
The new feature of the analysis is that costs are not given, but vary
with the market structure, rising under monopoly. The possibility of higher costs
under monopoly was dubbed “X-inefficiency.”
14.8
a.
The government wishes the monopoly to expand output toward P = MC. A lumpsum subsidy will have no effect on the monopolist's profit maximizing choice, so
this will not achieve the goal.
b.
A subsidy per unit of output will effectively shift the MC curve downward. The
figure illustrates this for the constant MC case.
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Chapter 14: Monopoly
167
c.
A subsidy ( t ) must be chosen so that the monopoly chooses the socially
optimal quantity, given t . Since social optimality requires P = MC and profit
maximization requires
 1
MC − t = MR = P 1 +  ,
 e
substitution yields
t
1
=− ,
P
e
as was to be shown. Intuitively, the monopoly creates a gap between price and
marginal cost and the optimal subsidy is chosen to equal that gap expressed as a
ratio to price.
14.9
Since consumers only value XQ, firms can be treated as selling that commodity
(i.e., batteries of a specific useful life). Firms seek to minimize the cost of
producing XQ for any level of that output. Setting up the Lagrangian,
L = C ( X )Q + λ ( K − XQ )
yields the following first-order conditions for a minimum:
LX = C ′( X )Q − λQ = 0,
LQ = C ( X ) − λ X = 0,
Lλ = K − XQ = 0.
Combining the first two shows that C ( X ) − C ′( X ) X = 0, or
C( X )
X=
.
C ′( X )
Hence, the level of X chosen is independent of Q (and of market structure). The
nature of the demand and cost functions here allows for the durability decision to
be separated from the output-pricing decision. (This may be the most general case
for which such a result holds.)
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168
Chapter 14: Monopoly
Analytical Problems
14.10 Taxation of a monopoly good
The inverse elasticity rule is
MC
P=
.
1+1 e
When the monopoly is subject to an ad valorem tax, t, this becomes
MC
1
P=
⋅
.
1− t 1+1 e
a.
With linear demand, e falls (becomes more elastic) as price rises. Hence,
MC
1
Pafter tax =
⋅
1 − t 1 + 1 eafter tax
<
MC
1
⋅
1 − t 1 + 1 epre tax
=
Ppre tax
1− t
.
b.
With constant elasticity demand, the inequality in part (a) becomes an equality so
P
Pafter tax = pre tax .
1− t
c.
If the monopoly operates on a negatively sloped portion of its marginal cost curve
we have (in the constant elasticity case)
MCafter tax
1
Pafter tax =
⋅
1− t
1+1 e
MCpre tax
1
>
⋅
1− t 1+1 e
P
= pre tax .
1− t
d.
The key part of this question is the requirement of equal tax revenues. That is
tPa Qa = τ Qs , where the subscripts refer to the monopoly’s choices under the two
tax regimes. Suppose that the tax rates were chosen so as to raise the same
revenue for a given output level, say Q. Then τ = tPa , hence τ > tMRa . But in
general under an ad valorem tax MRa = (1 − t ) MR = MR − tMR, whereas under a
specific tax, MRs = MR − τ . Hence, for a given Q, the specific tax that raises the
same revenue reduces MR by more than does the ad valorem tax. With an
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Chapter 14: Monopoly
169
upward sloping MC , less would be produced under the specific tax, thereby
dictating an even higher tax rate. In all, a lower output would be produced, at a
higher price than under the ad valorem tax. Under perfect competition, the two
equal-revenue taxes would have equivalent effects.
14.11 Flexible functional forms
a.
Writing the monopoly profit function as π (Q ) = [ P (Q ) − AC (Q )]Q , substituting
the given functional forms yields, after rearranging,
π (Q) = (a0 − c0 ) + (a1 − c1 )Q − s  Q.


The first-order condition with respect to Q is
(a0 − c0 ) + (1 − s )(a1 − c1 )Q − s = 0.
It is straightforward to solve this equation directly for the optimal quantity:
1/ s
 ( s − 1)(a1 − c1 ) 
Qm = 
 .
 a0 − c0

Looking ahead to part (c), where it will be important to simplify, we could have
alternatively made the substitution x = Q s in the first-order condition and solved
for x. Let’s try that, as well as substituting di = ai − ci to further simplify. The
first-order condition becomes
1
d 0 + (1 − s ) d1 ⋅ = 0,
x
yielding xm = ( s − 1)d1 / d0 , or Qm = [ ( s − 1) d1 / d 0 ]
1/ s
b.
, the same solution as above.
Constant average and marginal cost corresponds to c1 = 0. Substituting into the
solution from part (a) gives
1/ s
 ( s − 1)a1 
Qm = 

 a0 − c0 
c.
.
Monopoly profit with this yet more flexible specification is
π (Q ) = (a0 − c0 ) + (a1 − c1 )Q − s + (a2 − c2 )Q s  Q


−
s
s
=  d 0 + d1Q + d 2Q  Q.


The first-order condition with respect to Q is
d0 + (1 − s )d1Q − s + (1 + s )d 2Q s = 0,
or, substituting x = Q s ,
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170
Chapter 14: Monopoly
1
d 0 + (1 − s ) d1 ⋅ + (1 + s ) d 2 x = 0.
x
Multiplying both sides by x turns the first-order condition into the quadratic
equation
(1 + s )d 2 x 2 + d0 x + (1 − s )d1 = 0.
The quadratic formula yields two solutions, one of which will be negative in what
is probably the leading case of positive d0 , d 2 . The other solution is
xm =
d02 + 4d1d 2 ( s 2 − 1) − d0
2(1 + s)d 2
.
s
Using the relationship x = Q s , we can solve for quantity as Qm = x1/
m .
d.
Here is a graph showing possible shapes for the average cost curve.
14.12 Welfare possibilities with different market segmentations
a.
The perfectly competitive outcome with marginal-cost pricing ( pc = 0 ) yields the
socially efficient outcome. Both types of consumers purchase, implying
qc = q + q . Welfare is the sum of values added up over consumers Wc = qv + q v .
b.
With just two consumer types, the monopolist can achieve perfect price
discrimination by segmenting each type into one of two markets, charging v on
the low-value market and v on the high-value market. Social welfare is the same
Wc = qv + q v as under perfect competition, but now the monopolist appropriates
it all as profit; consumers obtain zero surplus.
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Chapter 14: Monopoly
c.
171
The single-price monopolist can choose from one of two pricing strategies, either
selling at the high types’ willingness-to-pay and just serving them, earning profit
q v , or selling at the low types’ willingness-to-pay and serving all consumers,
earning profit (q + q ) v . The assumed inequality means that the high-price
strategy is more profitable.
i.
The monopoly price is v , quantity is q , and profit is q v . There is no
consumer surplus because the whole valuation is extracted from the highvalue consumers who end up buying. Welfare equals the profit, q v .
ii.
The profit from serving just high-value consumers in segment B is bq v ,
and from serving all consumers in that segment is (bq + q ) v . At b* , these
profits are equal: b*q v = (b*q + q ) v , or solving,
b* =
qv
.
q (v − v )
This is obviously positive because v > v. The assumed inequality in part
(c) ensures b* < 1.
The discriminatory price is v in segment A and (by assumption
when it is indifferent) v in segment B. Monopoly profit is
(1 − b* ) q v + (b*q + q ) v = (1 − b* ) q v + b*q v = q v ,
where the first equality follows from substituting from the indifference
condition on profits and the second equality from simplifying. The only
consumer surplus comes from the b*q high-value consumers in segment
B , who obtain surplus v − v each for total consumer surplus of
b*q (v − v ) = qv . Profit is the same as under a single price. Welfare equals
qv + qv , the same as under perfect competition. The gain in welfare from
a single-price monopolist to perfect competition all accrues to consumers.
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172
Chapter 14: Monopoly
iii.
To get to a point somewhere in the middle of the base of the triangle—in
other words, to move only part way from part (c.i) to part (c.ii)—one
could imagine subdividing market B in two segments, keeping the same
proportion of the two types in each. In one segment, the monopolist, still
indifferent between a high and low price, could charge the low price but in
the other it could charge the high price. The monopolist’s profit stays the
same, but the increase in consumer surplus would only be a fraction of
what was seen in part (c.ii).
To get to a point somewhere above the base of the triangle, one
could imagine carving new segments from existing ones containing single
types across which the monopolist can perfectly price discriminate, raising
the monopolist’s profit.
d.
The inequality assumed in this part means that the profit-maximizing single price
for the monopolist now equals v .
i.
The monopoly price is v , quantity is q + q, profit is (q + q ) v , consumer
surplus is q (v − v ), and welfare is Wc = qv + q v .
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Chapter 14: Monopoly
ii.
173
The profit from serving just high-value consumers in segment A is q v ,
and from serving all consumers in that segment is (q + aq ) v . At a* , these
profits are equal: q v = ( q + a* q ) v , or upon solving,
a* =
q (v − v )
,
qv
the reciprocal of b* . This can be shown to be in the interval (0,1) in the
same way we showed this for b* in part (c).
The discriminatory price is v in segment A (by assumption when
it is indifferent) and v in segment B. Monopoly profit is
q v + (1 − a* ) qv = ( q + a* q ) v + (1 − a* ) qv = ( q + q ) v .
There is no consumer surplus, and welfare equals profit. Relative to the
single-price case, profit stays the same but all consumer surplus is
destroyed, and welfare falls.
iii.
The graph is identical to that in part (c) except that the labels on the
corners have been swapped because price discrimination across this
segmentation destroys rather than creates consumer surplus.
14.13 Shrouded prices
a.
Monopoly profit is π = Q ( P − AC ) = (10 − P )( P − 6). Solving the first-order
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174
Chapter 14: Monopoly
condition 16 − 2P = 0 yields Pm = 8. Thus, Qm = 2, π m = 4,
CS m = (1/ 2)(10 − 8) ⋅ 2 = 2, and Wm = π m + CSm = 4 + 2 = 6.
b.
Monopoly profit is π = Q ( P + s − AC ) = (10 − P )( P + s − 6). Solving the first-order
condition 16 − 2P − s = 0 yields Pm = 8 − s / 2. Thus, Qm = 2 + s / 2 and
π m = (2 + s / 2) 2 . The monopolist would like the shrouded price to be as high as
possible (infinite) because this allows it to extract a higher price per unit without
the decline in quantity demanded that accompanies the usual price increase.
c.
Gross consumer surplus can be computed as the area of the trapezoid under the
demand curve up to the quantity sold:
1
1
s 
s
GCSm = (10 + Pm )Qm = 18 −   2 +  .
2
2
2 
2
Consumers’ expenditure equals
s 
s

( Pm + s)Qm =  8 +  2 +  .
2 
2

Subtracting,
1
s 
s 
s 
s 1
CSm = 18 −   2 +  −  8 +   2 +  = (4 − 3s)(4 + s ).
2
2 
2 
2 
2 8
d.
Welfare is
2
s 1
1

Wm = π m + CSm =  2 +  + ( 4 − 3s )( 4 + s ) = (12 − s )(4 + s ),
2 8
8

a quadratic function, maximized for s* = 4. While shrouded prices distort
consumer behavior, this distortion counteracts the monopoly distortion to some
extent, so a positive amount of shrouding can be good for welfare in a monopoly
market. Notice that this level of shrouding induces the monopolist to reduce
perceived price Pm down to marginal cost.
e.
The solution for the monopoly price is exactly as in part (b). The difference here
is that the subsidy expenditure sQm comes from the government, whereas the
shrouded expenditure comes from consumers, so these parties’ surpluses must be
adjusted accordingly. To the extent that the subsidy is funded by a tax that
ultimately comes from citizens = consumers, the distributional consequences of
subsidies and shrouding could be quite similar.
A moderate, positive subsidy improves welfare in a monopoly market
because it induces the monopolist to lower price (similar to a reduction in
marginal cost). This is not true under perfect competition; a subsidy induces
overconsumption and introduces a deadweight loss. By analogy to shrouding,
while some shrouding can improve welfare in a monopoly market, any positive
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Chapter 14: Monopoly
175
shrouding will lower welfare under perfect competition, again because of the
overconsumption that is induced.
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CHAPTER 15:
Imperfect Competition
The problems in this chapter provide the student with some practice with many of the different
models of imperfect competition introduced in the text. Space considerations forced us to omit
problems on search, advertising, and innovation. The instructor may wish to supplement the
problem set with exercises from these areas depending on interest.
Comments on Problems
15.1
This problem compares the monopoly outcome to the outcome under Cournot and
Bertrand competition in a simple example with perfect substitutes and linear demand.
15.2
This problem generalizes the previous problem to general linear demand functions and
arbitrary numbers of firms.
15.3
This problem analyzes Cournot competition when firms have different marginal costs.
This departure from identical firms allows the student to shift around firm’s best
responses independently on a diagram.
15.4
This problem analyzes Bertrand competition when firms have different marginal costs. If
we adopt the usual assumption that demand is allocated evenly to equal-priced firms, then
we encounter a technical problem, called the open-set problem, in this setting. It would
not be an equilibrium for firms to charge 10 (the high marginal cost). The firm 2 would
profit from undercutting slightly and capturing all demand. The problem is that firm 2 has
no “best” undercutting price. For any price just below 10, say 9.999, firm 2 could earn
more by increasing price slightly but keeping it below 10, say 9.9999. One way to avoid
this problem is to assume prices are denominated in discrete units, say pennies, and
fractions of pennies are not allowed. The solution to the open-set problem suggested here
is to assume that the low-cost firm gets all the demand at equal prices.
15.5
This problem analyzes Bertrand competition with differentiated products. The problem
gives students practice in drawing diagrams with upward-sloping best responses (as
opposed to downward-sloping with Cournot).
15.6
This problem exercises in collusion in infinitely repeated games.
15.7
This problem analyzes the Stackelberg game both with and without the possibility of
entry-deterring investment.
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Chapter 15: Imperfect Competition
15.8
173
This problem analyzes entry deterrence in a Hotelling model.
Analytical Problems
15.9
Herfindahl index of market concentration. Many economists subscribe to the
conventional wisdom that increases in concentration are bad for social welfare. This
problem leads students through a series of calculations showing that the relationship
between welfare and concentration is not this straightforward.
15.10 Inverse elasticity rule. This problem derives alternatives form of the inverse elasticity
rule for a Cournot firm that are related to the one derived for a monopoly.
15.11 Competition on a circle. This problem is a useful twist on the Hotelling model that has
been used in a wide variety of applications because the symmetry of a circle makes the
analysis of entry easier.
15.12 Signaling with entry accommodation. In this problem, the incumbent must
accommodate entry because the fixed cost is low enough that entry cannot be deterred.
Given that best responses are upward-sloping (strategic complements), the incumbent
pursues a “puppy dog” strategy of trying to convince its rival it is a weak (high-cost)
competitor, with the hope of inducing its rival to charge a high price.
Behavioral Problem
15.13 Can competition unshroud prices? This new behavioral problem shows that market
forces (competition, advertising) may not be sufficient to overcome consumers’
behavioral biases.
Solutions
15.1
a.
The monopolist maximizes profit Q (150 − Q ) yielding first-order
condition 150 − 2Q = 0 and monopoly outcome P m = Q m = 75 and
Π m = 5, 625.
b.
Cournot firm 1 maximizes profit q1 (150 − q1 − q2 ) yielding first-order
condition 150 − 2q1 − q2 = 0 and best-response function q1 = 75 − q2 2.
Symmetrically, firm 2’s best-response function is q2 = 75 − q1 2. Solving
simultaneously, qic = 50 = P c and π ic = 2, 500.
c.
The Nash equilibrium of the Bertrand game is for both firms to charge marginal
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174
Chapter 15: Imperfect Competition
cost (here zero). Thus, P b = 0, Q b = 150, and π ib = 0.
d.
P
D
Monopoly
Cournot
MC = 0
15.2
Bertrand
Q
a.
A monopolist maximizes Q ( a − bQ − c ), yielding first-order condition
a − 2bQ − c = 0 and the monopoly outcome
a−c
Qm =
,
2b
a+c
Pm =
,
2
(a − c) 2
Πm =
.
4b
b.
Cournot firm 1 maximizes q1[a − b(q1 + q2 ) − c], yielding first-order condition
a − 2bq1 − bq2 − c = 0 and best-response function
a − bq2 − c
q1 =
.
2b
Symmetrically for firm 2,
a − bq1 − c
q2 =
.
2b
The Nash equilibrium outcome is
a −c
qic =
,
3b
a 2c
Pc = + ,
3 3
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accessible website, in whole or in part.
Chapter 15: Imperfect Competition
π ic =
175
(a − c) 2
.
9b
c.
The Nash equilibrium of the Bertrand game involves marginal-cost pricing:
P b = c,
a−c
Qb =
,
b
π ib = 0.
d.
Cournot firm i maximizes qi [a − b(Q− i + qi ) − c], yielding first-order condition
a − bQ− i − 2bqi − c = 0. Once the first-order condition has been taken, we can
apply the fact that firms are symmetric and so the equilibrium will be symmetric.
Substituting Q−c i = (n − 1)qic into the first-order condition and solving for qic
yields
a−c
qic =
.
b(n + 1)
Therefore,
n a−c
Qc =
⋅
,
n +1 b
a + nc
Pc =
,
n +1
(a − c ) 2
c
πi =
,
b(n + 1)2
n( a − c ) 2
Πc =
.
b(n + 1) 2
15.3
e.
It is easy to verify the answers to parts (a)–(c) by making the indicated
substitutions for n.
a.
Skipping preliminary calculations, firm 1’s best-response function is
1 − q2 − c1
q1 =
.
2
Firm 2’s is
1 − q1 − c2
q2 =
.
2
Solving simultaneously,
1 − 2c1 + c2
q1c =
,
3
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176
Chapter 15: Imperfect Competition
q2c =
Further,
1 − 2c2 + c1
.
3
2 − c1 − c2
,
3
1 + c1 + c2
Pc =
,
3
(1 − 2ci + c j ) 2
π ic =
,
9
Π c = π 1c + π 2c ,
Qc =
(2 − c1 − c2 ) 2
,
18
W c = Π c + CS c .
CS c =
b.
The reduction in firm 1’s marginal cost shifts its best response out and shifts the
equilibrium from E to E ′.
q2
BR1(q2)
●
E
E’
●
BR2(q1)
15.4
a.
q1
The most reasonable Nash equilibrium is for both firms to charge the high
marginal cost: p1* = p2* = 10. (The are other Nash equilibria in which both firms
charge equal prices somewhere between 8 and 10, but these equilibria involve
weakly dominated strategies for the high-cost firm. See Chapter 18 for further
discussion of weakly dominated strategies. Charging a price of, for example, 9 is
weakly dominated for firm 1. Charging a price of 10 weakly dominates charging
lower prices: firm 1 earns 0 by charging 10 but can earn negative profit if it
charges 9 and firm 2 charges a higher price.)
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Chapter 15: Imperfect Competition
b.
177
Firm 1 earns zero and produces zero. Firm 2 produces 500 − ( 20 ⋅10 ) = 300 and
earns (10 − 8)(300) = 600.
15.5
c.
No. The welfare-maximizing outcome is for firm 2 to charge its marginal cost (8).
Social welfare in the Nash equilibrium from part (a) can be shown to be 2,850
(the 600 in profit plus 2,250 in consumer surplus). Social welfare in the welfare
maximum is 2,890. Deadweight loss equals the difference, 40.
a.
Firm 1 maximizes p1 (1 − p1 + bp2 ) with respect to p1 , yielding the firstorder condition 1 − 2 p1 + bp2 = 0. The best-response function is
1 + bp2
p1 =
.
2
Symmetrically,
1 + bp1
p2 =
.
2
Solving simultaneously,
1
p1* = p2* =
.
2−b
b.
qi* =
c.
An increase in b pivots the best-response functions, shifting the equilibrium from
E to E ′.
1
1
, π i* =
.
2−b
(2 − b) 2
p2
BR1(p2)
BR2(p1)
●E’
E●
p1
15.6
a.
The present discounted value of the stream of payoffs from colluding is
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178
Chapter 15: Imperfect Competition
Vc =
Πm 1
⋅
.
n 1− δ
and from deviating is V d = Π m + δΠ m (the deviator can undercut and obtain
virtually all the monopoly profit for two periods, but then earns no profit when the
grim strategy kicks in in periods three and after). Rearranging the condition
V c ≥ V d implies
1
δ ≥ 1− .
n
15.7
b.
From Example 15.7, we know that δ ≥ 1 − 1 n for collusion to be sustainable, or
rearranging to get a condition on n , we have
1
n≤
.
1−δ
Our second condition is that the present value of the stream of a firm’s profits
from collusion, which at best equals
Πm 1
⋅
.
n 1−δ
This is greater than the initial investment cost K . Rearranging to get a condition
on n , we have
Πm
n≤
.
K (1 − δ )
In sum, n must be lower than the smaller of both:
 1
Πm 
n ≤ min 
,
.
1 − δ K (1 − δ ) 
a.
Solve the game using backward induction starting with firm 2’s action. We saw
from Problem 15.1 part (b) that firm 2’s best-response function is
q
q2 = 75 − 1 .
2
Substituting this back into firm 1’s profit function gives

q 

π 1 = q1[150 − (q1 + q2 )] = q1 150 − q1 −  75 − 1   .
2 


Taking the first-order condition with respect to q1 and solving yields q1* = 75.
Substituting this into firm 2’s best-response function yields q2* = 37.5.
b.
If firm 1 accommodates 2’s entry, the outcome in part (a) arises, and 1 earns
2,812.5. To deter 2’s entry, 1 needs to produce q1 sufficiently high that even if 2
best responds to q1 , generating profit
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Chapter 15: Imperfect Competition
179
(150 − q1 )2
.
4 − K2
This profit is less than or equal to 0. The threshold value of q1 is
q1 = 150 − 2 K 2 . Firm 1’s profit from operating alone in the market and
producing this output is (150 − 2 K 2 )(2 K 2 ), which exceeds 2,812.5 if
K 2 ≥ 120.6 (as can be shown by graphing both sides of the inequality).
15.8
a.
The two firms engage in Bertrand competition in homogeneous products at the
right end of the beach, leading to prices equal to marginal cost (here zero). Firm
A’s demand at the left end of the beach from Example 15.5 is
L p − pA L pA
qA = + B
= −
.
2
2tL
2 2tL
Maximizing profit (the displayed quantity times p A ) yields the first-order
condition
L pA
−
= 0.
2 tL
This implies
tL2
*
pA =
.
2
A’s profit is
tL3
π *A =
.
8
b.
No. B earns zero profit, so would not sink any positive investment cost to enter.
c.
A’s entry-deterring strategy is not credible. From its strategy in part (a), it earns
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180
Chapter 15: Imperfect Competition
tL3
.
8
Substituting firms locations a = 0 and b = L into Equation 15.42 in Example
15.5 shows that if A exits the right-hand side of the beach, it earns
tL3
.
2
π *A =
Analytical Problems
15.9
Herfindahl index of market concentration
a.
Reprising the analysis from Problem 15.2, firm i’s profit is qi (a − bqi − bQ− i − c)
with associated first-order condition a − 2b − bQ− i − c = 0. Imposing symmetry
[Q−*i = ( n − 1) qi* ] and solving,
a−c
qi* =
.
(n + 1)b
Further,
n ( a − c)
Q* =
,
(n + 1)b
a + nc
P* =
,
n +1
n  a−c 
Π = nπ = ⋅ 
 ,
b  n +1 
2
*
*
i
n2  a − c 
CS = ⋅ 
 ,
2b  n + 1 
2
*
n(n + 2)  a − c 
⋅
 .
2b
 n +1 
Because firms are symmetric, si = 1 n, implying
2
W* =
2
1 1
H = n ⋅  = .
n
n
b.
We can obtain a rough idea of the effect of merger by seeing how the variables in
part (a) change with a reduction in n. Per-firm output, price, industry profit, and
the Herfindahl index increase. Total output, consumer surplus, and welfare
decrease.
c.
Substituting c1 = c2 = 1 4 into the answers for 15.3, we have qi* = 1 4 , Q* = 1 2 ,
P* = 1 2, Π * = 1 8, CS * = 1 8, W * = 1 4, H = 1 2.
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Chapter 15: Imperfect Competition
d.
181
Substituting c1 = 0 and c2 = 1 4 into the answers for 15.3, we have q1* = 5 12,
q1* = 2 12, Q* = 7 12, P* = 5 12, Π * = 29 144, CS * = 49 288, W * = 107 288,
H = 29 49.
e.
Comparing part (a) with part (b) suggests that increases in the Herfindahl index
are associated with lower welfare. The opposite is evidenced in the comparison of
parts (c)–(d): welfare and the Herfindahl increase together. General conclusions
are thus hard to reach.
15.10 Inverse elasticity rule
Equation 15.2 can be rearranged as follows:
P − C ′ − P ' qi −dP / dqi ⋅ qi
1
=
=
=
,
P
P
P
| ε qi , P |
where ε qi , P is the elasticity of demand with respect to firm i ’s output. The second
equality uses the fact that
dP dP
P′ =
=
.
dQ dqi
Using this same fact, we can also rearrange Equation 15.2 as
si
P − C ′ − P ' qi −dP / dQ ⋅ qi  −dP / dQ ⋅ Q   qi 
=
=
=
.
  =
P
P
P
P

  Q  | ε Q,P |
15.11 Competition on a circle
a.
This is the indifference condition for a consumer located distance x from firm i:
the generalized cost (price plus transportation cost) of buying from I equals the
generalized cost of buying from the closest alternative firm.
b.
Solving the displayed equation in part (a) of the statement of the problem for x,
we obtain
1 p* − p
x=
+
.
2n
2t
The firm’s profit equals ( p − c )2 x. Substituting for x, taking the first-order
condition with respect to p , and solving for p gives the best response
p=
c.
p* + c + t n
.
2
Setting p = p * and solving for p* gives the specified answer. Equilibrium price
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182
Chapter 15: Imperfect Competition
is increasing in cost and the degree of differentiation, given by the transportation
cost and the spacing between firms (depending on their numbers).
d.
Substituting p = p* = c + t n into the profit function gives the specified answer.
e.
Setting t n 2 − K = 0 and solving for n yields n* = t K .
f.
Total transportation costs equal the number of half-segments between firms, 2n,
times the transportation costs of consumers on the half segment,
1 2n
t
∫0 tx dx = 8n2 .
Total fixed cost equals nF . The number of firms minimizing the sum of the two
is
n** =
1 t
.
2 K
15.12 Signaling with entry accommodation
a.
The high type’s best-response function is
p
p1H = 1 + 2 .
2
Firm 2’s is
1 pH
p2 = + 1 .
2 4
Solving simultaneously yields p1H * = 6 5 and p2* = 4 5. Profits are
π 1H * = (6 5) 2 = 1.44 and π 2* = (4 5) 2 = 0.64.
b.
This is similar to part (a) except the relevant best-response function for firm 1 is
that for the low type:
1 p
p1L = + 2 .
2 4
The equilibrium is p1L* = p2* = 2 3. Profits are π 1L* = π 2* = (2 3) 2 = 0.444.
c.
The best response for the high type is given in part (a) and for the low type in part
(b). The best response for firm 2 comes from maximizing its expected profit:
p1L  1 
p1H 
p1 
1 

p2 1 − p 2 +
 + p2 1 − p2 +
 = p2 1 − p2 +  .
2 
2  2 
2 
2

Note:
p L + p1H
p1 = 1
.
2
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Chapter 15: Imperfect Competition
183
Solving these three best responses simultaneously yields
p1L* = 41 60 = 0.683,
p1H * = 71 60 = 1.183,
p2* = 44 60 = 0.733.
d.
Firm 2 earns an expected payoff of about 0.542 under complete information and
0.537 under incomplete information, and thus would prefer complete information.
e.
We need to check that the low type would prefer its equilibrium profit to the
profit from charging the high type’s price in the first period and then having firm
2 believe it has high costs in the second period. The low type earns about 0.4669
from pricing low and 0.2169 from pricing high in the first period. In the second
period, the low type earns 0.444 in equilibrium. If firm 2 believes it to be the high
type, firm 1 earns (7 10) 2 = 0.49. The low type’s first-period loss from pooling,
0.4669 − 0.2169 = 0.25, exceeds its second-period gain from pooling,
0.49 − .0444 = 0.046. Hence, the low type has no incentive to deviate.
Behavioral Problem
15.13 Can competition unshroud prices?
a.
Sophisticated consumers will only pay si if it is less than e, their personal
avoidance cost. If e is very small, firms will find it more profitable to try to
exploit myopic consumers instead, charging the highest value that keeps them
from voiding the purchase. They do not void if pi + si ≤ v. Hence, si = v − pi .
Suppose the firm posting the weakly lower price earned positive profit.
This cannot be a Nash equilibrium because its rival (which at best only makes half
the sales and indeed makes no sales if its price is strictly higher) strictly gains by
undercutting slightly. Hence, the expected margin on each consumer must be zero
at the posted prices:
(1 − α )(v − c) + α ( pi − c) = 0.
The first term is the share of myopic consumers times the margin
pi + si − c = v − c earned on each of them. The second term is the share of
sophisticated consumers times the margin earned on each of them; this margin is
only pi − c since sophisticated consumers avoid si . Solving the displayed
equation yields
1− α
1
pi* = c −
(v − c) = [c − (1 − α )v].
α
α
implying si* = (v − c ) / α . It is easy to see from the first equality in the displayed
equation that equilibrium posted prices are less than c. It is easy to see from the
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184
Chapter 15: Imperfect Competition
second equality in the displayed equation that posted prices can be negative (if v
is much bigger than c ).
b.
The posted prices of color laser printers is close to their monochrome counterparts
even though they must be considerably more complicated to manufacture (and
come with four or five separate toner cartridges, which must be costlier than the
single black one). A look at the price of toner cartridges may explain why. Toner
prices can be $125 each for a total of $600 for those color printers requiring five
separate cartridges. These prices are shrouded as they rarely show up in printer
catalog entries.
c.
If firm 1 advertises, the net surplus consumers (who are all sophisticated now)
obtain from firm 2 is
 1− a
 1
v − e − p2* = v − e −  c −
(v − c )  = (v − c ) − e.
α

 α
Since firm 1 is transparent about its shrouded price, it may as well set this to zero
and earn revenue just through the posted price. The highest posted price it can
charge, p1d , must leave consumers with at least as much surplus as in the
previous equation:
1
1
v − p1d = (v − c) − e ⇒ p1d = v − (v − c) − e.
α
α
This gives it a profit margin of
1−α 
p1d − c = e − 
 (v − c ),
 α 
which is negative when the condition stated in part (c) holds.
d.
By “de-biasing” myopic consumers, advertising educates them about how to buy
at the posted price from the rival firm, which is very difficult for the advertising
firm to beat and still break even. The only way it can is if the inconvenience costs,
which the advertising firm can promise that consumers save, is sufficiently high.
e.
Let pia be the equilibrium posted price from part (a):
1
pia = [c − (1 − α )v].
α
If pia ≥ 0, then the nonnegativity constraint does not bind, and the equilibrium
remains unchanged from part (a). So suppose pia < 0. Then the equilibrium price
with when posted prices must be nonnegative is pi* = 0. The expected margin on
each consumer can be found by substituting pi* = 0 into the first displayed
equation from part (a):
(1 − α )(v − c) + α (0 − c) = (1 − α )v − c,
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Chapter 15: Imperfect Competition
185
which is positive exactly when pia < 0.
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accessible website, in whole or in part.
CHAPTER 16:
Labor Markets
Because the subject of labor demand was extensively treated in Chapter 11, the problems in this
chapter focus primarily on labor supply and on equilibrium in the labor market. Most of the labor
supply problems (16.1–16.3) start with the specification of a utility function and then ask
students to explore the labor supply behavior implied by the function. The primary focus of most
of the problems that concern labor market equilibrium is on monopsony and the marginal
expense concept (problems 16.5–16.7). Analytical problems are concerned with generalizing the
labor supply problems to consider risk, family labor supply, and intertemporal labor supply.
Comments on Problems
16.1
This problem is an algebraic example of labor supply that is based on a Cobb−Douglas
(constant budget shares) utility function. Part (b) shows in a simple context the work
disincentive effects of a lump-sum transfer. Three-fourths of the extra 4,000 is “spent” on
leisure which, at a price of $5 per hour, implies a 600-hour reduction in labor supply. Part
(c) then illustrates a positive labor supply response to a higher wage since the $3,000
spent on leisure will now only buy 300 hours. Notice that a change in the wage would not
affect the solution to part (a), because, in the absence of nonlabor income, the constant
share assumption assures that the individual will always choose to consume 6,000 hours
(=3/4 of 8,000) of leisure.
16.2
This problem uses the expenditure function approach to study labor supply. It shows why
income and substitution effects are precisely off-setting in the Cobb–Douglas case.
16.3
This problem is an application of labor supply theory to the case of means-tested income
transfer programs. The problem results in a kinked budget constraint. Reducing the
implicit tax rate on earnings (parts (f) and (g)) has an ambiguous effect on H since
income and substitution effects work in opposite directions.
16.4
This problem is a simple supply–demand example that asks students to compute various
equilibrium outcomes.
16.5
This problem is an illustration of marginal expense calculation. The problem also shows
that imposition of a minimum wage may actually raise employment in the monopsony
case.
16.6
This problem is an example of monopsonistic discrimination in hiring. The problem
shows that wages are lower for the less elastic supplier. The calculations are relatively
simple if students calculate marginal expense correctly.
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Chapter 16: Labor Markets
185
16.7
This is a bilateral monopoly problem for an input (here, pelts). Students may get confused
on what is required here, so they should be encouraged first to take an a priori graphical
approach and then try to add numbers to their graph. In that way, they can identify the
relevant intersections that require numerical solutions.
16.8
This problem is a numerical example of the union–employer game illustrated in Example
16.5.
Analytical Problems
16.9
Compensating wage differentials for risk. This problem develops the idea of a
certainty-equivalent wage rate.
16.10 Family labor supply. This problem introduces (in part (b)) the concept of “home
production.” The functional forms specified here are so general that this problem should
be regarded primarily as a descriptive one that provides students with a general
framework for discussing various possibilities.
16.11 A few results from demand theory. This problem shows how many problems in labor
supply theory can be addressed using demand theory concepts from Part 2 of the text.
16.12 Intertemporal labor supply. This problem is an introduction to some general concepts
in the theory of multiperiod labor supply. Because time has not yet been explicitly
introduced, however, the results pertain only to a situation with no discounting.
Solutions
16.1 a.
With 8,000 hours/year, full income is $40,000. If 75 percent of this is
devoted to eisure, this $30,000 will “buy” 6,000 hours of leisure at $5 per hour. Hence,
work time will be 2000 hours.
b.
Full income is now $44,000, so this person will devote $33,000 to leisure. This
will buy 6,600 hours of leisure, so labor supply will fall to 1,400 hours.
c.
With the higher wage, full income is $84,000, $63,000 of which will be devoted
to leisure. Hence, leisure time is 6,300 hours and work time is 1,700 hours. In this
case, therefore the higher wage promotes a greater labor supply even in the
presence of nonlabor income.
Leisure = 6,300 hours; work = 1,700 hours. Hence, higher wage leads to
more labor supply. Note that in part (a) labor supply is perfectly inelastic at 2,000
hours.
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186
Chapter 16: Labor Markets
d.
Labor supply is given by
0.75(8, 000 w + 4, 000)
3, 000
l = 8, 000 −
= 2, 000 −
.
w
w
The supply curve is therefore asymptotic to 2,000 hours.
a.
Setting up the Lagrangian,
L = c + wh − 24 w + λ (U − cα h1−α ),
yields the following first-order conditions for a minimum:
Lc = 1 − λα cα −1h1−α = 0,
16.2
Lh = w − λ (1 − α )cα h −α = 0,
Lλ = U − cα h1−α = 0.
Combining the first two equations gives the familiar result:
1
αh
=
.
w (1 − α )c
Manipulation of this condition and substitution into the utility function yields the
results that
h = Uk −α w−α ,
c = Uk 1−α w1−α ,
where
α
.
1−α
Substituting for expenditures gives
E = c + wh − 24w = KUw1−α − 24 w,
where K = k −α + k 1−α .
k=
b.
hc = ∂E ∂w = (1 − α )Uw−α K − 24.
c.
l c = 24 − h c = 48 − (1 − α )Uw−α K . Clearly, ∂l c ∂w = α (1 − α )UKw−α −1 > 0 .
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Chapter 16: Labor Markets
d.
187
The algebra is considerably simplified here by assuming α = 0.5, K = 2 and using
a period of 1.0 rather than 24. With these simplifications,
∂l c
l c = 2 − Uw−0.5 l = 0.5 − 0.5nw−1 .
= 0.5Uw−1.5 .
∂w
Now, letting n = E in the expenditure function and solving for utility gives
U = 0.5 w0.5 + 0.5nw −0.5 .
For n = 0, substitution yields
∂l c
= 0.25w−1 .
∂w
This is the substitution effect in the labor supply function.
To calculate the income effect, use the uncompensated function:
−1
∂l
l
= (0.5 − 0.5nw−1 )( −0.5w−1 ) = −0.25w ,
∂n
when n = 0. Hence, the substitution and income effects cancel out. (Note: In
working this problem, it is important not to impose the n = 0 condition until after
taking all derivatives.)
16.3
a.
Grant = G = 6000 − 0.75I .
I
G
0
6, 000
2, 000
4, 500
4, 000
3, 000
b.
G = 0 when I = 6, 000 0.75 = 8, 000.
c.
Assuming there are 8,000 hours in the year, budget constraint is
32,000 = c + 4h.
d.
Budget constraint is now
32, 000 + G = 38, 000 − 0.75(32, 000 − 4h)
= 14, 000 + 3h
= c + 4h
for h ≥ 6,000. Hence, the budget constraint is kinked at h = 6,000. Its
mathematical form is
14, 000 = c + h for h ≥ 6, 000,
32, 000 = c + 4h for h < 6,000.
Leisure is inexpensive for h ≥ 6,000 , expensive when h < 6,000.
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188
Chapter 16: Labor Markets
e.
16.4
f.
New budget constraint is 23, 000 = c + 2h for h > 5, 000.
g.
Income and substitution effects of law change work in opposite directions (see
graph). Substitution effect favors more work (new budget constraint is steeper);
income effect favors less work (person has more income for h ≥ 5,000 ).
Labor demand is L = −50 w + 450, and labor supply is L = 100 w.
a.
Setting labor demand equal to supply yields w = 3, L = 300.
b.
With the subsidy, demand becomes L = −50( w − s ) + 450. Setting w = 4 and
equating supply and demand yields
400 = −50(4 − s ) + 450,
implying s = 3. Total cost of subsidy is 1,200.
c.
With a minimum wage of w = 4, labor demand = 250, labor supply = 400, and
unemployment = 150.
d.
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Chapter 16: Labor Markets
16.5
189
Given the supply curve for labor, marginal expense is computed as
l = 80 w,
⇒ w = l 80,
l2
,
80
d ( wl )
l
⇒ MEl =
= .
dl
40
⇒ wl =
a.
For monopsonist, profit maximization required MEl = MRPl :
l
l
MEl =
= MRPl = 10 − ,
40
40
implying l = 200 and w = l 80 = 2.5.
b.
For Carl, the marginal expense of labor now equals the minimum wage, and in
equilibrium the marginal expense of labor will equal the marginal revenue
product of labor.
l
MEl = wmin = 4 = MRPi = 10 − ,
40
implying l = 240. With this wage, supply will be 320. Hence there will be
unemployment of 80. But employment has increased from 200 to 240.
c.
d.
16.6
Under perfect competition, a minimum wage means higher wages but fewer
workers employed. Under monopsony, a minimum wage may result in higher
wages and more workers employed.
First, look at the case of males:
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190
Chapter 16: Labor Markets
wm = l m 9
2
⇒ w ml m =
3/2
lm
3
0.5
l
⇒ MElm = m = MRPl = 10,
2
implies lm = 400 and wm = 20 / 3.
For females, the calculation is
l
wf = f
100
l2
⇒ wf l f = f
100
l
⇒ MEl f = f = 10,
50
implies l f = 500 and w f = 5. The profit per hour on machinery equals
9, 000 − 5(500) − 6.66(400) = 3,833.
If same wage must be paid to men and women, w = MRPl = 10,
l = lm + l f = 900 + 1, 000 = 1, 900.
Furthermore,
π = 1, 900(10) − 900(10) − 1, 000(10) = 0.
16.7
a.
Since q = 240 x − 2 x 2 , TR = 5q = 1, 200 x − 10 x 2 . MRP for pelts is
dTR
= 1, 200 − 20 x.
dx
Production of pelts x = l , C = wl = 10 x 2 , MC = 20 x. Under competition, price
of pelts px = MC = 20 x and MRPx = px . Hence, 1, 200 − 20 x = 20 x, implying
x = 30 and px = 600.
b.
From Dan’s perspective, demand for pelts equals MRPx = 1, 200 − 20 x = px .
Hence,
TR = p x x = 1, 200 x − 20 x 2 ,
implying
dTR
MR =
= 1, 200 − 40 x.
dx
For profit maximization, use marginal revenue equals marginal cost:
1, 200 − 40 x = 20 x,
implying x = 20 and px = 800.
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Chapter 16: Labor Markets
c.
191
From UF’s perspective, the supply of pelts is reflected in the marginal cost curve
MC = 20 x = px . Total cost is given by C = p x x = 20 x 2 .
MEx = dC dx = 40 x.
For profit maximization set MEx = 40 x = MRPx = 1, 200 − 20 x, implying x = 20
and px = 400.
d.
Both the monopolist and monopsonist agree on x = 20, but they differ widely on
price to be paid (800 vs. 400). Bargaining will determine the result.
16.8 a.
As in Example 16.5, this is solved by backward induction. In the
second stage of the game, the employer chooses l to maximize 10l − l 2 − wl ,
implying l = (10 − w) 2. The union chooses w to maximize
10w − w2
= 5w − 0.5w2 ,
2
implying w* = 5, l * = 2.5, U * = 12.5, π * = 6.25.
wl =
b.
With w′ = 4 and l ′ = 4, we have U ′ = 16 and π ′ = 8, which is Pareto-superior to
the contract in part (a).
c.
For sustainability, one needs to focus on the employer who has incentive to cheat
if union chooses w′ = 4 (profit maximizing l is 3, not 4). Since π (l = 3) = 9, the
condition for sustainability is
8
6.25 δ
>9+
1−δ
1− δ
implying 1 < 2.75 δ , or δ > 1 2.75 = 0.36.
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192
Chapter 16: Labor Markets
Analytical Problems
16.9
Compensating wage differentials for risk
Considering the first (riskless) job, U ( y ) = 100 y − 0.5 y 2 and y = wl with w = 5 and
l = 8 implies U (40) = 3, 200. That is,
U Job 1 = 3, 200.
Considering the second (risky) job,
E (U Job 2 ) = 100 E ( y ) − 0.5 E ( y 2 )
= 800 w − 0.5  Var( y ) + E ( y ) 2 
= 800 w − 0.5 ( 36w2 + 64 w2 )
= 800 w − 50 w2 .
Hence, to take the second job it must be the case that
800w − 50w2 ≥ 3, 200 → ( w − 8)2 ≥ 0
Thus, the required wage is w ≥ 8.
16.10 Family labor supply
a.
∂h1 ∂w2 and ∂h2 ∂w1 are both probably positive because of the income effect.
b.
c1 = f (h1 ), so, optimal choice would be to choose h1 so that f ′ = w1. This would
probably lead person 1 to work less in the market. That may in turn lead person 2
to choose a lower level of h2 on the assumption that h1 and h2 are substitutes in
the utility function. If they were complements, the effect could go the other way.
Clearly, one can greatly elaborate on this theory by working out all of the firstorder conditions and comparative statics results.
16.11 A few results from demand theory
a.
Applying the envelope theorem to Equation 16.20,
∂V ( w, n) ∂L
=
= λ (1 − h) = λl ,
∂w
∂w
∂V ( w, n) ∂L
=
= λ.
∂n
∂n
Hence,
∂V ∂w
l=
.
∂V ∂n
With the Cobb–Douglas,
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Chapter 16: Labor Markets
193
β
n

V ( w, n) = α α ( w + n)α β β 1 +  .
 w
Therefore,
β −1
∂V ( w, n)
n  n 
α
α
β +1 
= α ( w + n) β  1 +   − 2 
∂w
 w  w 
+ β β (1 + n w) β α α +1 ( w + n)α −1
∂V ( w, n)
n

= α α ( w + n)α β β +1  1 + 
∂n
 w
β −1
1
 
 w
β
β α +1
+ β (1 + n w) α ( w + n)α −1 .
Dividing the first equation by the second yields (after some manipulation)
n
l = (1 − β ) − β .
w
This is the labor supply function given in Equation 16.24.
b.
Using the logic of the development of the Slutsky equation, for any consumption
good
∂xi ∂xi
∂x
=
+h i .
∂w ∂w U
∂I
Hence, for any normal good, the income effect in this expression will be positive.
This positive effect will be reinforced for goods that are Hicksian complements
with labor (substitutes for leisure). The substitution effect will be negative,
however, for goods that are Hicksian substitutes for labor (complementary with
leisure), which is probably the case for most ordinary consumption goods. Hence,
it seems that in most cases the sign of this derivative will be ambiguous.
c.
Marginal expense is the change in total labor costs for a change in hiring:

∂wl
∂w
l ∂w 
1 

MEl =
= w+l
= w 1 +
=
w
1
+
.


 e 
∂l
∂l
 w ∂l 
l ,w 

Notice that since el , w is likely to be positive, MEl > w. If el , w = ∞, then MEl = w.
16.12 Intertemporal labor supply
a.
The Lagrangian expression for this utility-maximization problem is
L = U (c1 , h1 ) + E[U (c2 , h2 )] + λ [W0 + w1 (1 − h1 ) − c1 + E ( w2 )(1 − h2 ) − c2 ] .
Notice that here the budget constraint holds in expected value terms.
The first-order conditions for a maximum are
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194
Chapter 16: Labor Markets
Lc1 = U c1 − λ = 0,
Lh1 = U h1 − λ w1 = 0,
Lc2 = E (U c2 ) − λ = 0,
Lh2 = E (U h2 ) − λ E ( w2 ) = 0,
Lλ = W0 + w1 (1 − h1 ) − c1 + E ( w2 )(1 − h2 ) − c2 = 0.
Combining the first two equations yields the familiar condition for a maximum:
Uh
MRS = 1 = w1.
U c1
An increase in initial wealth should increase both leisure and consumption
assuming they are normal goods.
b.
The equation just says that second-period indirect utility is a function of the
wealth available at the start of that period and the second-period wage (which is
uncertain).
c.
Because V is an optimized function we need to return to its original Lagrangian
expression to interpret derivatives. The indirect utility function arises from the
problem
max E[U (c2 , h2 )]
subject to
W * = w2 (1 − h2 ) − c2 .
Because the second-period wage is random, the Lagrange multiplier associated
with W * will also be random here (call this multiplier λ2 ). But the solution to this
optimization problem will require
E (U c2 ) = E (λ2 ).
Comparing this result to the original first-order conditions from part (a) shows
that λ = E (λ2 ). The implicit value of wealth must be the same in the two periods
(in expected value) or there will be an incentive to move wealth from a period
where λ is low to one where it is higher in expected value terms.
d.
A certain increase in second-period wages is similar to an increase in initial
wealth. The first-period effects therefore should be to increase both consumption
and leisure. The effects on second-period labor supply are uncertain because
income and substitution effects work in opposite directions.
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CHAPTER 17:
Capital and Time
The problems in this chapter are of two general types: (1) those that focus on
intertemporal maximization and (2) those that ask students to make fairly simple present
discounted value calculations. Before undertaking any of these, students should be sure to read
the Appendix in Chapter 17. The appendix is especially important for problems involving
continuous compounding because students may not have encountered that concept in earlier
courses.
Comments on Problems
17.1
This problem is a simple analysis of intertemporal choices. The problem illustrates the
indeterminacy of the sign of the effect of the real interest rate on current savings. Part (c)
concerns intertemporal allocation with initial endowments in both periods.
17.2
This is a present discounted value problem. I have found that the problem is most easily
solved using continuous compounding (see below), but the discrete approach is also
relatively simple. Instructors may wish to point out that the savings rate calculated here
(22.5%) is considerably above the personal savings rate in the United States.
17.3
This is a simple present discounted value problem that should be solved with continuous
compounding.
17.4
This is a traditional capital theory problem involving trees. Students seem to have
difficulty in seeing their way through this problem and in interpreting the results. Hence,
instructors may wish to allow some time for discussion of it.
17.5
This problem is a discussion question that asks students to explore the logic of the U.S.
corporate income tax. The case of accelerated depreciation is, I believe, a particularly
effective example of the time value of money.
17.6
This problem presents a discounted value example of life insurance sales tactics. Students
tend to like this problem and, I’m told, some have even used its results when approached
by actual salespeople.
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196
Chapter 17: Capital and Time
17.7
This problem is a simple numerical example of the “Hotelling rule” for natural resource
pricing developed in the text.
Analytical Problems
17.8
Capital gains taxation. This is a graphic problem that shows how changes in the interest
rate induce capital gains that might be taxed.
17.9
Precautionary saving and prudence. This is a simple example showing how uncertainty
can be incorporated into the saving model presented in the chapter. It shows that the third
derivative of the utility function matters.
17.10 Monopoly and natural resource prices. This is a resource economics problem that
shows, with a finite resource, monopoly pricing options are severely constrained.
17.11 Renewable timber economics. This is a continuation of Problem 17.4, which shows that
optimal timber harvesting rules may be a bit different once the possibility of replanting is
considered.
17.12 More on the rate of return on a risky asset. This problem pursues the asset pricing
material in the chapter with a more explicit focus on the expected rate of return. It
describes the Sharpe ratio and uses the bound on that ratio to provide a simple example of
the equity premium puzzle.
17.13 Hyperbolic discounting. This behavioral problem introduces Laibson’s hyperbolic
utility function and provides a relatively intuitive presentation of the intertemporal
behavior implied by this function.
Solutions
17.1
a.
The Lagrangian expression for this maximization problem is
c 

L = U (c1 , c2 ) + λ  W − c1 − 2 
1+ r 

The first-order conditions for a maximum are
Lc1 = U c1 − λ = 0,
Lc2 = U c2 − λ (1 + r ) = 0,
Lλ = W − c1 − c2 (1 + r ) = 0,
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Chapter 17: Capital and Time
197
Division of the first two of these yields
MRS =
b.
U c1
= 1 + r.
U c2
∂c2 ∂r ≥ 0 because c2 is a normal good with price 1 (1 + r ) .
The sign of ∂c1 ∂r is ambiguous because the substitution effect is negative but the
income effect is positive. If ∂c1 ∂r < 0 a fall in the “price” of c2 raises total
spending on c2. Therefore, demand for c2 is elastic.
c.
Budget constraint has same slope as in part (a) and passes through the point
c1 = y1 , c2 = y2 . If, at utility-maximizing point c1* > y1 , the individual borrows in
period 1 and repays in period 2. If c1* < y1 , the individual saves in period 1 and uses
savings to increase consumption in period 2.
17.2
This problem can be most easily worked using continuous time:
y t = y 0e0.03t ,
y 40 = y 0e1.2 s t = sy t = sy 0e0.03t
Accumulated savings after 40 years
40
= ∫ ste
0
40
0.03(40 −t )
dt = sy 0 ∫ e
40
0.03t
e
0.03(40 −t )
0
dt = sy 0 ∫ e1.2 dt = sy 0 40e1.2.
0
Present value of spending in retirement
20
20
0
0
= ∫ 0.6 y 40 e −0.03t dt = 0.6 y 0e1.2 ∫ e −0.03t dt = 0.6 y0 e1.2
e −0.03t 20
= 9.02 y0e1.2 .
|
0
−0.03
For accumulated savings to equal the present value of dissavings, it must be the
case that
s=
9.02 y0 e1.2 9.02
=
= 0.225.
40 y0 e1.2
40
This seems a surprisingly high number given actual savings rates. How can one
explain the difference?
17.3
Using Equation 17.55 yields
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198
Chapter 17: Capital and Time
0.05 =
f ′(t ) ( t −0.5 − 0.15 ) ( e 2 t −0.15t )
=
⇒ t −0.5 = 0.2 ⇒ t = 25.
2 t − 0.15t
f (t )
e
17.4
a.
The present value of the wood in any tree is given by e − rt f (t ) . As before, to
optimize this value requires dPDV dt = e − rt f ′(t ) − re − rt f (t ) = 0 ⇒ r = f ′(t ) f (t ).
Because wages to tree-planters are paid currently, the zero-profit condition requires
− rt
w = e f (t * ) .
b.
The value of a u-year-old tree is = e− r (t −u ) f (t * ) = we ru ,
where weru grows at rate r, the tree grows faster than r except at t * . Because of
this, a u-year-old tree is worth more than the wood in the tree because it can be
allowed to continue to mature.
c, d.
The total value of a balanced woodlot is found by integration over all vintages of
trees:
t*
t*
e ru t*  rt*
1
ru
V = ∫ we du = w ∫ e ru du = w
=  we − w .
|
r 0
r
0
0
Hence, rV = f (t * ) − w . That is, the instantaneous foregone interest from investing
in the woodlot precisely equals the instantaneous “profits” earned from harvesting
a mature tree and planting a new one (but see Problem 17.11).
17.5
a.
Not at all, because there are no pure economic profits in the long run.
b.
In long-run equilibrium: v = PK(r + d). Government taxes opportunity cost of
capital. This raises v and provides an incentive to substitute labor for capital.
c.
Tend to increase use of capital since there is a tax advantage in early years. Total
taxes paid are equal, but timing of payments is different. Consequently, present
value of tax liabilities under accelerated depreciation is less than under straight line.
d.
If tax rate declines, tax benefits of accelerated depreciation are smaller. May reduce
investment.
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Chapter 17: Capital and Time
17.6
199
For the whole life policy, the present value of premiums paid is
4
∫ 2, 000e
−.1t
dt = 6, 304.
0
For the term policy, the present value of premiums is
35
∫ 400e
−0.1t
dt = 3879.
0
The salesman is wrong. The term policy represents a better value to this consumer.
17.7
Using Equation 17.71, we get
p(15) = e0.05⋅15 ( p0 − c0 ) + c0 e−0.3 ,
p(15) = e0.75 p0 − e0.75c0 + c0e −0.3 ,
125 = e0.75 p0 − 7e0.75 + 7e−0.3 ,
p0 = 63.6.
As of this writing, crude oil prices are well below this figure. But such low prices
may be a transient phenomenon related to the noncompetitive market structure of
the crude oil market. Where crude prices will go in the future is anyone’s guess.
Analytical Problems
17.8
Capital gains taxation
a.
Current savings are given by I − c0* .
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200
17.9
Chapter 17: Capital and Time
b.
Once the one-period bonds are purchased, fall in r causes budget constraint to rotate
to Iʹ. Increase in utility from U0 to U1 (point B) represents a capital gain.
c.
Accrued capital gains are measured by the total increase in ability to consume c0
(this is the “Haig–Simmons” definition of income) measured by distance IIʹ.
d.
Realized capital gains are given by distance c0* , cB that is the present value of oneperiod bonds that must be sold to attain the new utility-maximizing choice of cB.
e.
The “true” capital gain is given by the value, in terms of c0, of the utility gain. That
is measured by II″. Notice that this is smaller than either of the “gains” calculated
in parts (c) or (d). Hence, the current practice of taxing realized gains, while more
appropriate than full taxation of all accrued gains, still amounts to some degree of
overtaxation because it neglects the effects on c1 consumption opportunities.
Precautionary saving and prudence
a.
In the context of uncertainty, the person will aim to maximize the total expected
utility. Thus, if consumption is certain in the current period and uncertain in the
next period, utility maximization will be achieved when the current marginal utility
from consumption is equal the expected marginal utility of consumption in the next
period, that is, U ′(c0 ) = E[U ′(c1 )] .
b.
If U ′ is convex, Jensen’s inequality gives E[U ′(c1 )] > U '[ E (c1) ].
So, we know that E [U ' (c1 )] > U ' (c1p ). Using the fact that utility maximization
requires U ' (c0 ) = E [U ' (c1 )] , we get U ' (c0 ) > U ' (c1p ). If U ′ is decreasing in
consumption, this implies c0 < c1p .
Conversely, if the person opts for c0 < c1p , we know utility maximization requires
U ′( c0 ) = E [U ′( c1 )] so U ′[ E (c1 )] = U ′(c1p ) < U ′(c0 ) = E[U ′(c1 )] , so, by the converse
of Jensen’s inequality, U ′ is convex.
c.
The person with convex U ′ will opt for a higher scheduled level of consumption in
the next period. In other words, this person will save more in the current period, in
order to decrease the effect of any negative shock that may affect next period’s
consumption. Thus, this person can be described as “prudent,” since he or she has
a higher tendency to prepare himself or herself in the face of uncertainty.
d.
The above considerations imply that a faster consumption growth rate is optimal in
the presence of uncertainty. Hence, a lower real interest rate would be consistent
with observed consumption growth, in part explaining the low real rate.
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Chapter 17: Capital and Time
201
17.10 Monopoly and natural resource prices
a.
If the resource is owned by a single firm, then the firm sets the market price. Thus,
the price in Equation 17.63 would be a function of q.
b.
The Hamiltonian would be
dλ
 1

H = e− rt  1/ b q(t )1/ b q(t ) − c(t )q(t )  + λ ( −q(t ) ) + x(t )
.
dt
a

The profit-maximizing conditions would be
1 1
H q = (1 + ) 1/ b q(t )1/b e − rt − c(t )e − rt − λ = 0,
b a
dλ
Hx =
= 0.
dt
The first of these conditions can be simplified as [ MR (t ) − c(t )]e − rt = λ .
Differentiation with respect to t yields
λ& = −r( MR(t ) − c(t ))e − rt + ( MR& (t ) − c&(t ))e − rt = 0 . Diving by e − rt and using the
fact that MR = p(1 + 1 b)
MR& = p& (1 + 1 b) yields
•
c − cr
.
p = rp +
1+1 b
•
c.
This equation implies almost identical price dynamics as under competition. For
example, if c(t ) = 0 the price dynamics implied by part (b) are precisely the same
as in Equation 17.69.
17.11 Renewable timber economics
a.
Since
x
= x + x 2 + ... for x < 1
1− x
V = − w+
b.
( f (t ) − w)e− rt f (t ) − w
= rt
−w.
1 − e− rt
e −1
dV (e rt − 1) f ′(t ) − [ f (t ) − w]rert
=
=0
dt
(e rt − 1) 2
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202
Chapter 17: Capital and Time
So, for a maximum,
f ′(t ) =
*
[ f (t * ) − w]rert
e −1
rt *
*
= rf (t * ) + rV (t * )
c.
The condition implies that, at optimal t * , the increased wood obtainable from
lengthening t must be balanced by: (1) the delay in getting the first rotation's yield
and (2) the opportunity cost of a one-period delay in all future rotations' yield.
d.
f (t ) is asymptotic to 50 as t → ∞ .
e.
t * = 100. This is not “maximum yield” since tree continues growing after 100
years.
f.
Now t * = 104.1. A lower interest rate lengthens the growing period.
17.12 More on the rate of return on a risky asset
a.
Equation 17.37 is pi = E (m ⋅ xi ) = E (m) ⋅ E ( xi ) + Cov(m, xi ) =
E ( xi )
+ Cov(m, xi ) .
Rf
Multiplication by R f pi and rearranging terms yields
E ( Ri ) − R f = −
Rf
Cov(m, xi ) = − R f Cov(m, Ri ),
pi
where the final equation follows because pi is treated as certain at the time the
investment is made.
b.
This is just a direct application of the Cauchy–Schwartz inequality to the equation
derived in part (a). One way to see why the Cauchy–Schwartz inequality holds is
by looking at the correlation coefficient between any two random variables
Cov( x, y )
−1 ≤ ρ ( x, y ) ≡
≤ 1 . So, E ( Ri ) − R f ≤ R f σ mσ Ri
σ xσ y
c.
Division of the equation from part (b) by σ Ri yields
E ( Ri ) − R f
σ Ri
d.
≤ Rf σ m =
σm
= CV (m) .
E ( m)
The result uses the approximation that for a small x e x ≈ 1 + x. So,
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Chapter 17: Capital and Time
CV (m) ≈ γ 2σ ln2 ∆c = γσ ln ∆c . Hence,
e.
E ( Ri ) − R f
σ Ri
203
≤ CV (m) ≈ γσ ln ∆c .
The Sharpe ratio for common stocks is about 0.5—the long-run real rate of return
is about 0.09, the risk free rate about 0.01, and the standard deviation of stock
returns is about 0.16. So (0.09 − 0.01) 0.16 = 0.5 . With σ ln ∆c = 0.01 this implies a
value for γ of about 50—far above the value generally believed to characterize the
typical person’s attitude toward risk. A vast literature in finance attempts to explain
this paradox.
17.13 Hyperbolic discounting
a.
For the given utility function, the discount factors have the following values:
1, βδ , βδ 2 ,K .
For β = 0.6 and δ = 0.99 , the set of discount factor values are
1, 0.594, 0.594(0.99), 0.594(0.99) 2 . Thus, the factors drop significantly (from 1 to
0.594) for period t + 1 and then follow a slow and steady geometric rate of decline
of 0.99.
b.
The significant drop of the discount factors for period t + 1 means that preferences
at time t are inconsistent with preferences at time t + 1. Therefore, decisions
regarding the next periods made in period t may not be preferred in period t + 1 and
so, they are likely to be changed in order to fit the new set of preferences. In other
words, long-term plans made in the current period are likely to be changed in the
next period, leading to a shortsighted behavior.
c.
In period t, the MRS between ct+1 and ct+2 will be U ′(ct +1 ) / δ U ′(ct + 2 ).
At time t + 1, the MRS between ct+1 and ct+2 will be U ′(ct +1 ) / βδU ′(ct + 2 ).
From the perspective of the decision-maker, the MRS between ct+1 and ct+2 in period
t would not be equal to the MRS between ct+1 and ct+2 in period t + 1. This means
that effectively preferences would change between the two periods, prompting him
or her to change the consumption choices made in the previous period, in order to
reach the new MRS.
d.
Constraints are necessary so as to avoid changes in the consumption decision from
one period to the other. Constraining future consumption choices in period t would
avoid overconsumption in period t + 1. Self t would limit self t + 1’s ability to
consume the accumulated wealth, thus ensuring that consumption in the following
period will follow the utility-maximizing path.
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204
Chapter 17: Capital and Time
e.
Examples include retirement funds with penalties for early withdrawal of funds,
real estate, saving bonds, certificates of deposit. In general, illiquid assets provide
a form of commitment against future overconsumption.
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accessible website, in whole or in part.
CHAPTER 18:
Asymmetric Information
Most of the problems in this chapter focus on different applications of the principal–agent model.
Additional problems are provided on auctions and the lemons problem.
Problem 18.5 requires the solution to a complicated maximization problem that has to be
solved using numerical methods similar to Example 18.5 in the text. An Excel spreadsheet with
the solution method is provided on the textbook’s website. Other mathematical software can also
be used of course.
Comments on Problems
18.1
This problem studies the moral-hazard model in the context of shareholders inducing
effort from a manager using various contractual forms (profit sharing, bonuses, buyouts).
18.2
This problem applies the moral-hazard model to the relationship between a client (in the
role of principal) and a lawyer (in the role of agent).
18.3
This problem computes the optimal linear (i.e., per-unit) price for coffee to compare to
the optimal nonlinear tariff computed in Example 18.4. As a first step, the problem
requires students to convert representations of consumer utility functions into demand
functions.
18.4
This problem provides students with further practice on computing optimal nonlinear
tariffs by slightly changing the numbers used in Example 18.4.
18.5
This problem, similar to Example 18.2, provides students with further practice on moral
hazard in insurance.
18.6
This problem, similar to Example 18.5, provides students with further practice on adverse
selection in insurance. The tongue-in-cheek application, involving a higher accident rate
for left-handers, has an interesting history in the medical literature. Early studies
estimated a much higher mortality rate for left-handers by comparing mean ages at death
for left- and right-handers. Later studies suggested that the difference in mean death age
was an artifact of parents more recently abandoning the practice of training left-handed
children to be right-handed. The later parts of the problem treat competitive insurance as
in Example 18.6 in the text.
18.7
This problem is a simple version of Akerlof’s lemons problem.
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Chapter 18: Asymmetric Information
18.8
205
This problem has students work through a very simple model of a common-values
auction in which the winner’s curse arises.
Analytical Problems
18.9
Doctor–patient relationship. This problem works through a moral-hazard problem in
which the patient is the principal and the doctor is the agent. The problem is more
difficult than a standard one because the students are asked to work with general
functional forms.
18.10 Increasing competition in an auction. This problem provides students with practice
working through the calculation of optimal strategies in an auction by repeating the
analysis from the chapter except with n bidders rather than just two. The problem
establishes an interesting proposition about the benefits of competition (here, that
increasing competition in an auction increases the seller’s revenue).
18.11 Team effort. This problem works through the logic of Holmstrom’s famous “Moral
Hazard in Teams” article, showing that incentives are diluted in large teams. The problem
suggests that employee stock ownership plans may not provide good incentives to work
hard and so probably have other rationales.
Behavioral Problem
18.12 Nudging consumers into adverse selection. This problem shows that nudges that
improve individual consumers’ choices can exacerbate adverse selection and reviews a
recent empirical article by Benjamin Handel demonstrating this point for health
insurance.
Solutions
18.1 a.
With a half share,
E(Utility) = (0.5)(1,000/2) + (0.5)(400/2) – 100 = 250.
With a quarter share,
E(Utility) = (0.5)(1,000/4) + (0.5)(400/4) – 100 = 75.
She would accept either contract because either provides her with positive expected
utility. The lowest share s she would accept solves
(0.5)(1,000 s) + (0.5)(400 s) – 100 = 0,
implying s ≈ 14%.
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206
Chapter 18: Asymmetric Information
b.
The most she would pay equals (0.5)(1,000) + (0.5)(400) – 100 = 600.
c.
Clare would need to be offered a fixed salary solving (0.5)(100) + f – 10 = 0, or f
= 50.
d.
18.2
i.
For Clare to exert effort, her gross-profit share must solve
(0.5)(1,000 s) + (0.5)(400 s) – 100 ≥ 400 s, or s ≥ 1/3.
ii.
The bonus b that would induce her to work hard solves (0.5) b – 100 ≥ 0,
or b ≥ 200. She would not need an additional fixed wage since the bonus
also would give her at least as much expected utility as her outside option.
a.
The lawyer maximizes
l l2
− ,
3 2
yielding equilibrium effort l * = 1 3. His surplus is 1 18 and the plaintiff’s is
2 * 2
l = .
3
9
b.
The lawyer maximizes
l2
cl − ,
2
yielding equilibrium effort l * = c. His surplus is c 2 2 and the plaintiff’s is
c(1 − c).
c.
The optimal contingency fee for the plaintiff is c* = 1 2, maximizing her surplus
c(1 − c). Her surplus is 1/4 and the lawyers is 1/8.
d.
With a 100% contingency fee, the lawyer chooses l * = 1 and earns a surplus of
1/2, which the plaintiff can extract initially by selling the case to him. This
arrangement leads to lawyer effort that is first best from the joint perspective of
the plaintiff–lawyer team. Selling cases may be outlawed in some jurisdictions
because, although selling cases may maximize plaintiff–lawyer joint surplus, it
may not maximize overall social surplus, which also includes defendant surplus
and court costs. It may induce dishonest behavior on the part of the lawyer, who is
supposed to be an officer of the court and not take any means necessary to win
cases. Selling cases would be inefficient even for the plaintiff–lawyer team if
there is moral hazard on the plaintiff’s side, that is, if plaintiff effort is also
required to increase the probability of prevailing at trial. A plaintiff who sold a
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Chapter 18: Asymmetric Information
207
case to a lawyer would have no incentive to exert effort for the case.
18.3
First solve for type θ ’s demand. Given linear price p , this type will choose q to
maximize θ (2 q ) − pq. Taking the first-order condition with respect to q and solving
yields q = (θ p ) . Thus, q H = ( 20 p ) and q L = (15 p ) . Next, we solve for the optimal
linear price. The monopolist’s expected profit from the linear price is
2
2
2
2
2
 20  1
 15  625( p − c)
1
( p − c)   + ( p − c)   =
.
2
2 p2
 p 2
 p
The first-order condition with respect to p is
625(2c − p )
= 0,
2 p3
implying p* = 2c, or p* = 10 when c = 5. The monopolist’s expected profit at p* = 10
is 15.625.
18.4
By Equation 18.51, the low type’s second-best quantity satisfies
 1 
23
1
 = 5+
15 ⋅ 
(20 − 15)
,
**
**
 q 
13
q
L
L


**
implying 15 = 5 q**
L + 10, or q L = 1. This one-ounce cup is sold for
TL** = θ L v(1) = (15)(2 1) = 30 cents.
The high type’s cup has the same size as in the first best: q**
H = 16. The tariff is
(
)
( )
TH** = 20 2 16 − 2 1 + 15 2 1 = 150.
Compared to Example 18.4, the low type’s cup is distorted even further from the first
best because the cost of doing so has fallen since there are a smaller proportion of them.
The shop owner can then increase the high type’s tariff.
18.5
a.
The premium satisfies p = (0.5)(10, 000) = 5, 000.
b.
The premium satisfies p = (0.5)(5, 000) = 2,500. The individual’s utility is
0.5 ln(20, 000 − 2,500) + 0.5ln(20, 000 − 2,500 − 10, 000 + 5, 000)
= 9.6017
His utility from part (a) is
ln(20,000 − 5, 000) = 9.6158,
verifying that he prefers full to partial insurance.
c.
The premium satisfies p = (0.5)(7, 000 / 2) = 1, 750. The individual’s utility from
partial insurance now is
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208
Chapter 18: Asymmetric Information
0.5ln(20, 000 − 1, 750) + 0.5ln(20, 000 − 1, 750 − 7, 000 + 3,500)
= 9.7055,
so he now prefers partial to full insurance.
18.6
a.
Full insurance for left handers involves a certain payout of $500. They would be
unwilling to pay more than the $500 they receive. So full insurance for them is
equivalent to no insurance. The premium for full insurance for right handers
satisfies
1
1
ln(100 − pRH ) = ln(1, 000 − 500) + ln(1,000),
2
2
implying
ln(100 − pRH ) = ln( 500 ⋅1, 000),
in turn implying
p*RH = 1, 000 − 500 ⋅1, 000 ≈ 292.9.
b.
Left handers are fully insured. Hence, xLH = 500. The premium just satisfies
incentive compatibility, making them indifferent between their contract and the
one for right handers:
ln(1, 000 − pLH ) = ln(1, 000 − 500 − pRH + xRH ),
implying
(1)
pLH = 500 + pRH − xRH .
The premium for right handers reduces them to their outside option of no
insurance:
1
1
ln(1,000 − 500 − pRH + xRH ) + ln(1,000 − pRH )
2
2
1
1
= ln(1,000 − 500) + ln(1,000).
2
2
Solving for xRH ,
p (1,500 − pRH )
xRH = RH
.
(2)
1, 000 − pRH
The insurer’s profit is
x 

10( pLH − xLH ) + 100  pRH + LH  .
2 

Substituting the full-insurance payment x*LH = 500, using Equation 1 to substitute
for pLH , and using Equation 2 to substitute for xRH , we can rewrite profits
solely as a function of pRH :
50 pRH (400 − pRH )
.
1, 000 − pRH
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Chapter 18: Asymmetric Information
209
One can try to use calculus to find the maximum of this function, but perhaps a
simpler route is just to look at its graph.
The graph peaks at p*RH ≈ 225 yielding total profit π * ≈ 2,540. Substituting back
into Equations 1 and 2 gives x*RH ≈ 370 and p*LH ≈ 355. Profit per consumer is
*
*
π LH
= −145 and π RH
= 40.
At first it seems odd that the insurer would choose to serve left handers,
given it earns negative profit from them. Given left handers can always take the
contract meant for right handers, they cannot be excluded from the market, and
this ends up being the best the insurer can do. The adverse-selection problem
really hurts the insurer.
c.
The competitive equilibrium under full information results in full, fair insurance
for each type. Full, fair insurance for left-handers involves p*LH = x*LH = 500,
which is equivalent to no insurance. Full, fair insurance for right handers involves
x*RH = 500 and p*RH = x*RH / 2 = 250.
d.
The separating contract for the risky type involves full insurance. We saw that the
full insurance contract, if priced fairly as required under competition, was
equivalent to no insurance. On the other hand, any fairly priced contract attracting
only righties involves pRH = xRH 2. A lefty’s expected utility from that contract
is
ln (1, 000 − 500 − pRH + xRH )
x
x 



= ln  500 − RH + xRH  = ln  500 + RH  .
2
2 



This is greater than the expected utility (which is also certain) of ln(500) from no
insurance. So a lefty would always choose the righty’s contract for any xRH > 0.
The text showed why pooling can never be a competitive equilibrium. We
are left with only one possibility, that x*RH = 0, which means that righties receive
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210
Chapter 18: Asymmetric Information
no insurance. The full insurance offered to lefties is equivalent to no insurance. So
the equilibrium is equivalent to all types receiving no insurance.
18.7
18.8
a.
(1/ 2)(10, 000) + (1/ 2)(2, 000) = $6, 000 .
b.
If sellers value good cars at $8,000, they would not be willing to sell even at the
buyers’ maximum willingness to pay from part (a). So only the 50 lemons would
be offered for sale at a price of $2,000. If sellers value good cars at $6,000, then
they would be willing to offer them for sale at the price from part (a). All 100 cars
sell for $6,000.
a.
If a buyer’s signal is L, the object is certainly worth 0. If the buyer’s signal is H ,
it is equally likely that the other buyer gets either signal, so equally likely to be
worth 0 or 1, for an expected value of 1/2.
b.
If the buyers’ strategy is to bid 0 conditional on L and 1/2 conditional on H , a
buyer earns 0 conditional on L and
1

 1  1 
Pr(other sees L)  0 −  + Pr(other sees H )  1 − 
2

 2  2 
 1  1   1  1  1 
=   −  +    
 2  2   2  2  2 
1
=− .
8
conditional on H . To see this payoff, if the other sees L, the object is certainly
worthless. If the other sees H , he bids according to the specified strategies, and
the end up in a tie. One of them is randomly selected as the winner, so with
probability 1 2 ends up with net surplus 1 − 1 2 = 1 2. Each buyer’s overall
expected surplus is negative. The winner’s curse arises because whenever a buyer
wins the auction outright, he knows he just won a worthless object. In
equilibrium, buyers would shade their bids downward.
Analytical Problems
18.9
Doctor–patient relationship
The doctor chooses m to maximize U d ( I d ) + U p , where I d = pm m and
U p = U p ( m, x) = U p ( m, I c − pm m ).
Making these substitutions into the doctor’s objective function and taking the first-order
condition with respect to m yields
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Chapter 18: Asymmetric Information
pmU d′ +
∂U p
+
211
∂U p
(− pm ) = 0.
∂m
∂x
At this solution, the patient’s implied marginal rate of substitution between m and x is

∂U p ∂m
U d′ 
= pm  1 −
< pm .
 ∂U ∂x 
∂U p ∂x
p


A fully informed patient would choose m to maximize U p ( m, I c − pm m), yielding firstorder condition
∂U p ∂U p
+
(− pm ) = 0.
∂m
∂x
and an implied marginal rate of substitution of
∂U p ∂m
= pm .
∂U p ∂x
Graphically, the fully informed patient chooses a point of tangency A between his
indifference curve and the budget constraint. The doctor chooses a point B, which we
know must be on the same budget constraint and which we also know from the previous
analysis involves a lower marginal rate of substitution. The graph shows that point B is to
the right of A and thus involves more m.
x
Up
Ic
A
B
Ic/pm
m
18.10 Increasing competition in an auction
a.
Bidder 1 maximizes
Pr ( b1 > max(b2 ,K , bn ) ) (v1 − b1 ),
which, assuming rivals use linear bidding strategies, equals
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212
Chapter 18: Asymmetric Information
n
b 

(v1 − b1 )∏ i = 2 Pr  vi < 1  ,
k

which, assuming a uniform distribution of valuations, in turn equals
n −1
b 
(v1 − b1 )  1  .
k
Maximizing with respect to b1 yields b1 = v1 (n − 1) / n. Expected revenue is
E (v( n ) )( n − 1) / n. This equals (n − 1) / (n + 1) using the formula (Equation 18.71)
for the expected value of the maximum order statistic v( n ) .
b.
Expected revenue equals the expected value of the second highest bid. Since
buyers bid their valuations ( bi = vi ), the expected value of the second highest bid
equals the expected value of the second highest valuation: E (v( n −1) ). But E (v( n −1) )
equals (n − 1) / (n + 1) using the formula for expectation of an order statistic in
Equation 18.71.
c.
Yes.
d.
Bids converge to valuations in the first-price auction as the number of bidders
grows large. Bids remain at valuations in a second-price auction. In both cases,
expected revenue approaches the highest possible valuation in the range [0,1]:
n −1
lim
= 1.
n →∞ n + 1
18.11 Team effort
a.
Partner i obtains a 1 n share of the revenues. He chooses ei to maximize
ei2  1 
ei2
1
R
−
=
(
e
+
L
+
e
)
−
.
n
 
  1
2 n
2
n
Taking the first-order condition with respect to ei and solving yields ei* = 1 n .
Partner i ’s surplus is
* 2
2n − 1
 1  * (ei )
ne
−
=
.
  i
2
2n 2
n
b.
If worker i gets a 100% share, he chooses ei to maximize
ei2
ei2
R − = ei − .
2
2
*
His optimum is ei = 1. Partner i ’s surplus is 1 − 1 2 = 1 2. The average surplus
per partner (the average including the other n − 1 partners, who obtain no return)
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Chapter 18: Asymmetric Information
213
is 1 2n . For n = 2, the per-partner surplus is the same as in part (a). For n > 2,
the average surplus from dispersing the shares is greater.
c.
We have
d  2n − 1  1 − n
,

=
dn  2n 2  n3
which is negative for n > 1. Further,
 2n − 1 
lim 
= 0.
2 
n →∞
 2n 
d.
The analysis suggests it is unlikely that the stock plan provides incentives in a
rational model. There may be psychological effects on employee morale.
Alternatively, such plans may facilitate employee/management bargaining over
wage contracts. Tying compensation to firm performance avoids the problem of
the breakdown in bargaining that may follow, for example, if employees make
excessively demanding wage demands when the firm may be (unknown to the
employees) in a declining position. Avoiding bargaining breakdown allows the
firm and workers to avoid losses from strikes and bankruptcy.
Behavioral Problem
18.12 Nudging consumers into adverse selection
a.
Red-car owners are fully insured in equilibrium, leading to certain “endgame”
wealth 100, 000 − 5, 000 = 95, 000, which is their certainty equivalent. Gray-car
owners’ certainty equivalent satisfies
ln(CEG ) = 0.15ln(80,000 − 453 + 3,021) + 0.85ln(100,000 − 453),
giving CEG = 96,793 (rounded to the nearest digit). The weighted average over
certainty equivalents is
(0.1)(95, 000) + (0.9)(96, 793) = 96, 614.
b.
Fixing the separating contracts from the previous part, there are four certainty
equivalents to be averaged together, depending on the type of owner and whether
they happen to choose the right contract or not. We already computed types’
certainty equivalents for the right contract in the previous part. It remains to
choose their certainty equivalents for the wrong contract. For red-car owners who
choose the wrong contract,
ln(CERW ) = 0.25ln(80,000 − 453 + 3,021) + 0.75ln(100,000 − 453),
giving CERW = 95,000 (slightly less, actually, but rounds to this). Gray-car
owners who choose the wrong contract get the certainty equivalent of 95,000
from the full-insurance contract. The weighted average of these certainty
equivalents in the population is
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214
Chapter 18: Asymmetric Information
1
1

(0.1)(95, 000) + (0.9)  ⋅ 96, 614 + ⋅ 95, 000  = 95,807,
2
2

less than in part (a). Thus, a nudge to reduce the behavioral bias increases welfare.
c.
There exists a competitive equilibrium with a pooling contract offering full
insurance. The average risk for someone in the population is
(0.1)(0.25) + (0.9)(0.15) = 0.16.
Thus, the fair price for this full insurance is (0.16)(20, 000) = 3, 200. This gives
all consumers a certainty equivalent of 100, 000 − 3, 200 = 96,800, greater than in
part (a). Thus the nudge reduces welfare. The behavioral bias eliminates adverse
selection because consumers randomly choose policies. Nudging them reintroduces adverse selection, reducing welfare.
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accessible website, in whole or in part.
CHAPTER 19:
Externalities and Public Goods
The problems in this chapter illustrate how externalities in consumption or production can affect
the optimal allocation of resources and, in some cases, describe the remedial action that may be
appropriate. Many of the problems have specific, numerical solutions, but a few (Problems 19.4
and 19.5) are essay-type questions that require extended discussion and, perhaps, some
independent research. Because the problems in the chapter are intended to be illustrative of the
basic concepts introduced, many of the simpler ones may not do full justice to the specific
situation being described. One particular conceptual shortcoming that characterizes most of the
problems is that they do not incorporate any behavioral theory of government—that is, they
implicitly assume that governments will undertake the efficient solution (i.e., a Pigovian tax)
when it is called for. In discussion, students might be asked whether that is a reasonable
assumption and how the theory might be modified to take actual government incentives into
account.
Comments on Problems
19.1
This problem provides an example of a Pigovian tax on output. Instructors may wish to
supplement this with a discussion of alternative ways to bring about the socially optimal
reduction in output.
19.2
This problem provides a simple example of the externalities involved in the use of a
common resource. The allocational problem arises because average (rather than
marginal) productivities are equated on the two lakes. Although an optimal taxation
approach is examined in the problem, students might be asked to investigate whether
private ownership of Lake X would achieve the same result.
19.3
This is another example of externalities inherent in a common resource. This question
poses a nice introduction to discussing “compulsory unitization” rules for oil fields and,
more generally, for discussing issues in the market's allocation of energy resources.
19.4
This is a descriptive problem involving externalities, now in relation to product liability
law. For a fairly complete analysis of many of the legal issues posed here, see S. Shavell,
Economic Analysis of Accident Law.
19.5
This problem is an illustration of the second-best principle to the externality issue. It
shows that the ability of a Pigovian tax to improve matters depends on the specific way in
which the market is organized.
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Chapter 19: Externalities and Public Goods
215
19.6
This is an algebraic public-goods problem in which students are asked to sum demand
curves vertically rather than horizontally.
19.7
Another public-goods problem. In this case, the formulation is more general than in
Problem 19.6 because there are assumed to be two goods and many (identical)
individuals. The problem is fairly easy if students begin by developing an expression for
the RPT and for the MRS for each individual and then apply Equation 19.40.
Analytical Problems
19.8
More on Lindahl equilibrium. This problem asks students to generalize the discussions
of Nash and Lindahl equilibria in public goods demand to n individuals. In general,
inefficiencies are greater with n individuals than with only two.
19.9
Taxing pollution. This problem primarily focuses on the idea that a Pigovian tax must
tax the externality, not just the output of the externality-creating firm.
19.10 Vote trading. This problem shows that voluntary trading of votes may still be unable to
yield a sensible social welfare equilibrium.
19.11 Public choice of unemployment benefits. A simple problem focusing on an individual’s
choice for the parameters of an unemployment compensation policy.
19.12 Probabilistic voting. This problem shows how to introduce continuity into voting
decisions by specifying a probability of voting function. This process is similar to
developing the mixed strategy concept in game theory.
Solutions
19.1
a.
Given MC = 0.4 q and p = 20. Setting p = MC implies 20 = 0.4 q, in turn
implying q = 50.
b.
Given SMC = 0.5q. Setting p = SMC implies 20 = 0.5q, in turn implying
q = 40. At the optimal production level of q = 40, the marginal cost of
production is MC = 0.4q = 0.4 ⋅ 40 = 16, so the excise tax t = 20 − 16 = $4.
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216
Chapter 19: Externalities and Public Goods
c.
19.2
a.
Given F x = 10 lx − 0.5 lx2 and F y = 5 l y . First, show how total catch
depends on the allocation of labor. We have lx + l y = 20, implying l y = 20 − lx .
Further,
T
x
y
F =F +F
= 10 lx − 0.5 lx2 + 5(20 − lx )
= 5 lx − 0.5 lx2 +100.
Equating the average catch on each lake,
y
x
F =F ,
ly
lx
gives 10 − 0.5lx = 5, implying lx = l y = 10. Substituting,
F T = 50 − 0.5(100) + 100 = 100.
b.
Maximizing F T = 5 lx − 0.5 lx2 +100, the first-order condition is
dF T dlx = 5 − lx = 0, implying lx = 5, l y = 15, and F T = 112.5.
c.
In part (a), F x = 50, so the average catch = 50/10 = 5.
In part (b), F x = 37.5, so the average catch = 37.5/5 = 7.5.
Thus, the license fee on Lake X = 2.5.
d.
The arrival of a new fisher on Lake X imposes an externality on the fishers
already there in terms of a reduced average catch. Lake X is treated as common
property here. If the lake were private property, its owner would choose lx to
maximize the total catch less the opportunity cost of each fisher (the 5 fish he/she
can catch on Lake Y). So the problem is to maximize F x − 5l x , which yields
lx = 5, as in the optimal allocation.
19.3
Given AC = MC = 1, 000 per well.
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Chapter 19: Externalities and Public Goods
19.4
217
a.
Produce where revenue per well equals 1,000 = 10q = 5,000 − 10n. Solving,
n = 400. There is an externality here because drilling another well reduces output
in all wells.
b.
Produce where MVP = MC of well. Total value = 5,000n − 10n2 . Thus
MVP = 5, 000 − 20n − 1, 000 n, implying n = 200.
c.
Let the tax be x. Want revenue/well − x = 1, 000, when n = 200. At n = 200,
average revenue/well = 3,000. So charge x = 2, 000.
Under caveat emptor, buyers would assume all losses. The demand curve under
such a situation might be given by D. Firms, which assume no liability, might
have a horizontal long-run supply curve of S. A change in liability assignment
would shift both supply and demand curves.
Under caveat vendor, losses (of amount L) would now be incurred by
firms, thereby shifting the long-run supply curve to S ′.
Individuals now no longer have to pay these losses and their demand curve will
shift upward by L to D′ . In this example, then, market price rises from P1 to P2
(although the real cost of owning the good has not changed), and the level of
production stays constant at Q* . Only if there were major information costs
associated with either the caveat emptor or caveat vendor positions might the two
give different allocations.
It is also possible that L may be a function of liability assignment
(the moral hazard problem), and this would also cause the equilibria to
differ.
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218
Chapter 19: Externalities and Public Goods
19.5
In the diagram, the untaxed monopoly produces QM at a price of PM . If the marginal
social cost is given by MC ′, QM is, in fact, the optimal production level. A per-unit tax of
t would cause the monopoly to produce output QR which is below the optimal level.
Since a tax will always cause such an output restriction, the tax may improve matters
only if the optimal output is less than QM, and even then, in many cases it will not.
19.6
a.
To find the total demand for mosquito control, demand curves must be summed
vertically. Letting Q be the total quantity of mosquito control (which is equally
consumed by the two individuals), the individuals' marginal valuations are
P = 100 − Q (for a),
P = 200 − Q (for b).
Hence, the total willingness-to-pay is given by 300−2Q. Setting this equal
to MC (= 120) yields optimal Q = 90.
19.7
b.
In the private market, price will equal MC = 120. At this price (a) will demand 0,
(b) will demand 80. Hence, output will be less than optimal.
c.
A tax price of 10 for (a) and 110 for (b) will result in each individual demanding
Q = 90 and tax collections will exactly cover the per-unit cost of mosquito
control.
a.
In the general equilibrium model of Chapter 13, we saw that the perfectly
competitive equilibrium involved a tangency between the production possibility
frontier and individual’s indifference curve. In other words, RPT = MRSi . To find
RPT , the production possibility frontier can be rewritten as the implicit function
f ( x, y ) = 100 x 2 + y 2 − 5,000 = 0.
Using the implicit function rule to find the derivative,
 f  200 x 100 x
dy
RPT = −
= −− x  =
=
.
 f 
dx
2y
y
y 

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Chapter 19: Externalities and Public Goods
219
Next, use the implicit function rule to find MRSi :
MU x (1 2 )( xyi ) yi yi
MRSi =
=
= .
MU yi (1 2 )( xyi )−1/ 2 x
x
−1/ 2
Equating the two and substituting yi = y 100 yields
100 x yi
y
= =
.
y
x 100 x
Hence, y = 100 x. Substituting into the production possibility frontier and solving
yields x* = 0.704 and y* = 70.4. An individual’s utility in this equilibrium is
0.704.
b.
To find the social optimum, equate RPT with the sum of MRSi :
y
100 x 100
= ∑ MRSi =100 i .
y
x
i =1
Substituting yi = y 100 and solving yields y = 10x. Substituting into the
RPT =
production possibility frontier and solving yields x** = 5 and y** = 50. An
individual’s utility in this optimum is 1.58.
Next, compute the tax implementing this optimum. We want to induce
**
individuals to consume the optimum: x** = 5, yi = 0.5, at which point
yi** 0.5 1
=
= .
x**
5 10
Thus, we want the price ratio that includes tax t to satisfy
px
1
= .
p y + t 10
One can push the answer further. Based on the analysis from part (a), one can
show equilibrium prices must satisfy px = py . Substituting into the preceding
MRSi** =
expression, the optimal tax is t ** = 9 px .
Analytical Problems
19.8
More on Lindahl equilibrium
a.
The condition for efficiency is
n
∑ i =1 MRSi = RPT .
The fact that the MRS’s are summed captures the assumption that each person
consumes the same amount of the nonexclusive public good. The fact that the
RPT is independent of the level of consumers shows that the production of the
good is nonrival.
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220
Chapter 19: Externalities and Public Goods
b.
c.
19.9
In a Nash equilibrium, each person would opt for a share under which
MRSi = RPT , implying a much lower level of public good production than is
efficient.
Lindahl Equilibrium requires that αi = MRSi RPT and ∑ α i = 1. This would
seem to pose even greater informational difficulties than in the two-person case.
Taxing pollution
a.
The efficient allocation of resources would maximize total industry profits, given
the constraint that total pollution must equal K.
Assuming a fixed market price p and a wage w, the associated Lagrangian is
n
n
n


L = p ∑ fi (li ) − w∑ li + λ  K − ∑ g i (li )  ,
i =1
i =1
i =1


yielding first-order conditions
Li = pf i′− w − λ g i′ = 0,
n
Lλ = K − ∑ gi (li ) = 0.
i =1
Thus, for each firm, the optimal choice of labor input is given by pfi′− λ gi′ = w. In
words, the “net” marginal product of labor should equal the wage where “net” is
defined as gross marginal product minus the increased pollution generated
valuated at the shadow price λ .
b.
With a tax t on output, the profit function for each firm would be
π i = pfi (li ) − wli − tfi (li ).
Profit maximization yields
∂π i
= pf i′− w − tf i′ = 0,
∂li
implying
w
fi ′ =
.
p −t
A tax on output would reduce the quantity produced by each firm, but would only
equilibrate the gross marginal product across firms—it would pay no attention to
differences in pollution generation across firms.
c.
With a tax on pollution, the profit for each firm would be
π i = pfi (li ) − wli − tgi (li )
The first-order condition from profit maximization is
∂π i
= pfi ′− w − tgi′ = 0.
∂li
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Chapter 19: Externalities and Public Goods
221
The tax on pollution would lead to the efficient allocation described in part (a), if
t=λ.
d.
Pollution control strategies that involve a tax on pollution lead to an efficient
allocation of production among firms. A tax on output does not necessarily give
this result.
19.10 Vote trading
Suppose preferences are as follows:
Preference
Individual 1
C
A
B
Individual 2
A
B
C
Individual 3
B
C
A
a.
Under majority rule, APB (where P means “is socially preferred to”), BPC, but
CPA. Hence, social choices are not transitive.
b.
Suppose individual 3 is very averse to A and reaches an agreement with individual
1 to vote for C over B if individual 1 will vote for B over A. Now, majority rule
results in CPA, CPB, and BPA. The final preference seems to assign too important
a role to individual 3 since B is preferred to A only by this person. Arrow would
say this violates the principle of “non-dictatorship.”
c.
With point voting, each option would get six votes, so all the options would be
ranked the same. This could be overturned by introducing an “irrelevant
alternative,” say D.
19.11 Public choice of unemployment benefits
a.
So long as this utility function exhibits diminishing marginal utility of income,
this person will opt for parameters that yield y1 = y2 . Here, that requires
w(1 − t ) = b. Inserting this into the governmental budget constraint produces
uw(1 − t ) = tw(1 − u ), which requires t = u.
b.
A change in u will change the tax rate by an identical amount.
c.
The solutions in parts (a) and (b) are independent of the risk aversion parameter
δ.
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222
Chapter 19: Externalities and Public Goods
19.12 Probabilistic voting
a.
Candidate A will choose a set of θiA so as to maximize the expected probability of
his/her election. Thus, he/she will want to maximize
∑ π = ∑ f (U (θ ) − U (θ ) ) .
n
n
A
i =1
i
i =1
i
i
B
i
i
Similarly, candidate B will choose to maximize his/her expected votes:
∑ (1 − π ) = n − ∑ f (U (θ ) − U (θ ) ).
n
n
A
i
i =1
i =1
i
i
i
B
i
Setting up the Lagrangian for candidate A yields
(
)
L = ∑ f (U i (θiA ) − U i (θiB* ) ) + λA 0 − ∑ n θiA ,
n
i =1
where θiB* is the optimal choice for candidate B. The first-order condition is
∂L
= f ′U i′ − λA = 0.
∂θiA
Thus, f ′U 'i = λA , a constant.
Whether it seems likely that candidates would be bound by the zero net
value constraint in their promises is left up to the reader to decide.
For candidate B, a similar solution holds. The Lagrangian is
(
)
L = n − ∑ f (U i (θiA* ) − U i (θiB ) ) + λB 0 − ∑ n θiB ,
n
i =1
where θiA* is the optimal platform chosen by candidate A. Similar to above,
∂L
= f ′U i′ − λB = 0,
∂θiB
implying f ′Ui′ = λB .
b.
Yes. This is a consequence of Glicksberg’s (1952) extension of Nash’s existence
proof to games with continuous strategies. See the Extensions to Chapter 8, in
particular section E8.4.
c.
For each candidate, the optimal platform satisfies the condition that f ′U i′ is a
constant. Since f is the same for all voters, the above relation implies that Ui′
should be the same for all voters.
If instead we maximize social welfare against the constraint that
∑ n θiA =∑ n θiB =0 ,
we obtain Lagrangian
n
(
L = ∑ U i (θi ) + λ 0 − ∑ n θi
i =1
)
and first-order condition
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Chapter 19: Externalities and Public Goods
223
∂L
= U i′ − λ = 0.
∂θi
Thus, the socially optimal solution requires Ui′ to be the same for all voters. This
is the same policy adopted by each candidate seeking to maximize expected vote.
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accessible website, in whole or in part.