Consumer theory
Lecture 2
The Consumer
We now turn to the neoclassical theory of consumer behavior. We suppose that an individual
consumer seeks to maximize the function
u(x1, x2, ... , xn)
1)
where (x1, x2, ... , xn) represents the levels of goods (1, 2, ... , n) that the consumer actually
consumes, and u(x1, x2, ... , xn) is a function that fully describes the consumer's subjective
valuation (utility) of consuming these goods.
Though consumers seek to get as much satisfaction from consuming the n goods as possible,
each is constrained by the amount of money they have to spend. We assume that each consumer
faces a budget constraint
n
m =
∑px ,
2)
i i
i=1
where m is the consumer's budget (or income), and pi is the unit price of good i. Consumers are
assumed to be competitive in the sense that they take each price as given.
The mathematical statement of the behavioral assertion that consumers maximize utitlity subject
to a budget constraint is
max
(x1, x2, ... , xn)
subject to
u(x1, x2, ... , xn)
3)
n
m =
∑px .
i i
i=1
Preferences and utility functions
Although the concept of a budget constraint is straightforward, the concept of a utility function is
not. Essentially, we require that consumers be rational in the sense that their preferences exhibit
the following characteristics: Take any two consumption bundles A and B. Then;
1) (Completeness) A particular consumer prefers A to B, or B to A, or is indifferent between
them. Only one of these three possibilities is true for any pair of consumption bundles.
2) (Transitivity) If a consumer prefers A to B, and B to another bundle C, then the consumer
prefers A to C.
These postulates of rationality only require that the consumer be able to rank every possible
consumption bundle in order of preference. It does not require that the consumer be able to
make non-sensical statements like "I like bundle A twice as much as bundle B." That a
consumer need only rank bundles means that he/she only possesses an ordinal measure of
2.1
Consumer theory
satisfaction. Requiring that a consumer be able to judge the degree to which he/she prefers one
bundle over another would require a cardinal measure of satisfaction.
It is convenient to summarize the preferences of a consumer by a utility function. Because of the
ordinal character of preferences, a utility function is only a ranking. That is, all we require of
utility functions is that if a consumer prefers A to B, then u(A) > u(B), and if he is indifferent
between them, u(A) = u(B).
Other assumptions about utility functions, u(x1, x2, ... , xn).
Differentiability: The utility function is twice continuously differentiable.
Nonsatiation: The utility function is monotonically increasing in the consumption of each good.
That is, ui = ∂ u / ∂ x i > 0, ∀i = 1,2,..., n. This assumption is often referred to as "more is
preferred to less."
Strict quasi-concavity: The utility function is strictly quasi-concave. Suppose that a consumer
gets satisfaction from consuming only two goods, 1 and 2. Her utility function is u(x1, x2). The
bordered Hessian of u(x1, x2) is
0
BH =  u1
u2
u1
u11
u21
u2 
u12  .
u22 
Strict quasi-concavity requires that the determinant of BH be greater than zero. That is,
0
u1
u2
BH = u1
u11
u12 = 2 u1u2 u21 − u1 u22 − u2 u11 > 0.
u2
2
2
4)
u21 u22
In a moment you'll see that this says something very important about individual preferences.
Indifference curves are downward sloping: An indifference curve is the collection of all
consumption bundles that give a consumer the same level of utility. An explicit function of an
indifference curve is x2 = x2(x1, u0). Substitute this relation into the utility function and note that
u(x1, x2(x1, u0)) ≡ u0.
5)
2.2
Consumer theory
Differentiate 5) with respect to x1:
∂u
∂ u ∂ x2
+
≡ 0.
∂ x1
∂ x2 ∂ x1
Rearrange to obtain
u
∂ x2
∂ u ∂ x1
≡ −
≡ − 1.
∂ x1
∂ u ∂ x2
u2
6)
Since u1 and u2 are greater than zero, 6) reveals that indifference curves are downward sloping.
[Note that downward sloping indifference curves is implied by the assumption of "more is
preferred to less."] We can interpret the slope of an indifference curve as the trade-off between
goods 1 and 2 that the consumer is just willing to make. Put another way, the slope of an
indifference curve tells us how much extra of good 2 is needed to compensate the consumer for
giving up a unit of good 1. Still another way: The slope is the consumer's subjective marginal
valuation of consumption of good 1 in terms of good 2.
The absolute value of the slope of an indifference curve at a point is often referred to as the
marginal rate of substitution (MRS). That is, MRS = u1/u2.
Indifference curves are strictly convex: This implies that MRS is decreasing as more of good 1
is consumed. Intuitively; the subjective marginal valuation of any good decreases as more of
that good is consumed. Graphically,
x2
u0
x1
To check the convexity of indifference curves, differentiate 6) with respect to x1:
∂ 2 x2
1
= 3 ( 2u12 u1 u2 − u11 u22 − u22 u12 ) .
2
u2
∂ x1
2.3
Consumer theory
Convexity of indifference curves requires that the term in parentheses be positive. Note that
strict quasi-concavity of the utility function assures us that indifference curves will be strictly
convex [see equation 4)].
Indifference curves cannot cross: Consider the following graph.
x2
A
C
B
x1
Since bundles A and B lie on the same indifference curve, u(A) = u(B). Likewise, since A and C
lie on the same indifference curve, u(A) = u(C). By transitivity, it must be true that u(B) = u(C).
However, bundle C represents more consumption of both goods than bundle B. The assumption
that more is preferred to less implies that u(C) > u(B). This contradiction implies that
indifference curves cannot cross.
We can fully describe a utility function (and hence, an individual's preferences) in the (x2, x1)
space by an indifference map.
x2
0
u
2.4
u1
u2
x1
Consumer theory
Utility Maximization
Suppose a consumer gets utility from consuming two goods, 1 and 2. The consumer's problem is
u(x1, x2)
max
(x1, x2)
subject to
7)
m = p1x1 + p2x2.
The Lagrange equation is
L = u(x1, x2) + λ(m - p1x1 - p2x2).
8)
The first-order conditions are
L1 = u1 - λp1 = 0
L2 = u2 - λp2 = 0
Lλ = m - p1x1 - p2x2 = 0.
9)
The second-order condition is that the determinant of the bordered Hessian of L be positive:
L11
BH = L21
Lλ1
L12
L22
Lλ 2
L1λ
u11
L2λ = u21
Lλλ
− p1
u12
u22
− p2
− p1
− p2 > 0 .
0
10)
This determinant is
|BH | =
2
2
2 p1 p2 u21 − p1 u22 − p2 u11 .
11)
From the first order conditions p1 = u1/λ, and p2 = u2/λ. Substitute these into 11) to obtain
BH =
( 2u u u
1 2
2
21
2
− u1 u 22 − u 2 u11
λ2
).
12)
This is positive if and only if the numerator is positive. The assumption of strict quasi-concavity
guarantees that |BH | is positive, and that the first-order conditions 9) identify unique optimal
solutions for the utility maximization problem 7).
Given satisfaction of the second-order condition, the first-order conditions implicitly define
explicit functions
x1 = x1m ( p1 , p2 , m)
x2 = x2m ( p1 , p2 , m)
λ = λm ( p1 , p2 , m) .
13)
m
m
x1 ( p1 , p2 , m) and x2 ( p1 , p2 , m) are utility-maximizing demand equations. Sometimes these are
referred to as money-income-held-constant demand functions. They are also called Marshallian
demand functions.
2.5
Consumer theory
We can illustrate utility-maximizing consumption choices in the (x1, x2) space by considering the
first order conditions 9). Consider first the condition Lλ = m - p1x1 - p2x2 = 0. Rearrange this
equation to obtain
p 
m
14)
x2 =
−  1  x1
p2
 p2 
Equation 14) gives us the budget constraint in the (x1, x2) space. Note that with fixed m, p1, and
p2, 14) is a linear function of x1 with slope -p1/p2. Equation 14) is called the budget line.
Now consider the first two first-order conditions from 9). Eliminating the Lagrange multiplier
and rearranging gives us
u1
p1
=
15)
p2
u2
15) tells us that at a utility-maximizing consumption choice, the ratio of the marginal utilities
must be equal to the ratio of the prices. Note that the ratio of the marginal utilities [the left-hand
side of 15)] is the absolute value of the slope of an indifference curve; that is, it is the marginal
rate of substitution. The ratio of the prices [the right-hand side of 15)] is the absolute value of
the slope of the budget line. That a utility-maximizing consumption choice must satisfy 15) and
the budget constraint implies that at an optimal consumption choice an indifference curve is
tangent to the budget line.
In the graph below, the budget line is fixed because income and prices are fixed. The utilitymaximizing consumption choice is found by choosing the highest indifference curve that has a
point in common with the budget line. This implies a tangency, and this tangency determines the
value of the Marshallian demands at this particular set of income and prices.
x2
m/p2
x2 m
u*
u0
x1
m
m/p1
2.6
x1
Consumer theory
Elasticities from the Marshallian demand equations
Own-price elastiticites:
The elasticity of the Marshallian demand for good i with respect to its price is defined to be
ε (mx , p ) = ε im,i =
Definitions:
i
i
If ε
m
i ,i
 < − 1,

 = − 1,
 ∈( -1,0),

∂ x im pi
∗ .
∂ pi x i
demand for good i is said to be elastic.
demand for good i is said to be unit elastic.
demand for good i is said to be inelastic.
Cross-price elastiticites:
The elasticity of the Marshallian demand for good i with respect to the price of good j is defined
to be
∂ x im p j
ε (mx , p ) = ε im, j =
∗
.
∂ p j xi
i
Definitions:
j
 > 0,
If ε im, j 
 < 0,
goods i and j are said to be substitutes.
goods i and j are said to be complements.
Income elasticities
The elasticity of the Marshallian demand for good i with respect to the consumer's income is
defined to be
ε
Definitions:
m
( xi , m )
= ε
m
i ,m
∂ x im m
=
∗ .
∂ m xi
 > 0,
If ε im,m 
 < 0,
good i is said to be a normal good.
good i is said to be an inferior good.
2.7
Consumer theory
Some characteristics of Marshallian demand functions
Proposition: Marshallian demands are invariant to any positive monotonic transformation of the
utility function.
Proof: Recall the optimization problem
u(x1, x2)
max
(x1, x2)
7)
m = p1x1 + p2x2.
subject to
The first-order conditions for a solution to this problem 9) imply that
u1
u2
=
p1
p2
and
m - p1x1 - p2x2 = 0.
16)
Consider F, an arbitrary positive monotonic transformation of u. Define a new utility function V
= F(u). Of course, Fu > 0. Note that V(x1, x2) = F[u(x1, x2)].
Consider the optimization problem
V(x1, x2)
max
(x1, x2)
subject to
7')
m = p1x1 + p2x2.
The first-order conditions for this problem imply
V1
V2
=
p1
p2
and
m - p1x1 - p2x2 = 0.
16')
However, V1 = Fuu1 and V2 = Fuu2. Therefore,
V1
V2
=
Fu u1
Fu u2
=
u1
.
u2
This last implies that 16') and 16) are equivalent, which implies further that the consumption
choice that solves 7) will be the consumption choice that solves 7'). Q.E.D.
Remark: This proposition highlights the ordinal nature of preferences. A positive monotonic
transformation of utility preserves the consumer's ranking of all the consumption bundles. And
since the consumer need only rank bundles (he/she doesn't have to assign a relative weight to
each bundle) a positive monotonic transformation of utility does not imply anything about the
behavior of the consumer.
2.8
Consumer theory
Proposition: Marshallian demand functions are homogeneous of degree zero in prices and
income. That is, for any positive constant t, xim ( tp1 , tp2 , tm) = xim ( p1 , p2 , m) , i = 1, 2.
Proof: For the optimization problem
u(x1, x2)
max
(x1, x2)
subject to
7)
m = p1x1 + p2x2,
the first-order conditions imply that at a utility-maximizing consumption choice
u1
u2
=
p1
p2
and
m - p1x1 - p2x2 = 0.
16)
Now consider the optimization problem when we multiply prices and income by a positive
constant t:
u(x1, x2)
max
(x1, x2)
subject to
7'')
tm = tp1x1 + tp2x2.
At a utility-maximizing consumption choice
u1
u2
=
tp1
tp2
and
t[m - p1x1 - p2x2] = 0.
16'')
16'') is obviously identical to 16). Therefore, the consumption choice that solves 7) will be the
consumption choice that solves 7''). Q.E.D.
Remarks:
1) The proposition implies that 'only relative prices matter' to the consumer when he/she makes
consumption choices. A practical implication of this is that consumption choices will not
change with general inflation where prices and wages increase at the same rate.
2) The reason Marshallian demands are homogeneous of degree zero is that consumption
opportunities do not change if prices and income change by the same proportion.
Consumption opportunities are described by the budget line. Consider the budget line given
by 14) with prices and income multiplied by t, and note the following:
x2 =
 tp 
tm
−  1  x1
tp2
 tp2 
=
p 
m
−  1  x1 .
p2
 p2 
Since the budget line does not change, opportunities are unaffected. Since opportunities do
not change, the utility-maximizing consumption choice will not change.
2.9
Consumer theory
Utility maximization and the envelope theorem
Recall the solutions to the utility-maximization problem 7):
x1 = x1m ( p1 , p2 , m)
x2 = x2m ( p1 , p2 , m)
λ = λm ( p1 , p2 , m) .
13)
By substituting the Marshallian demand equations into the utility function, we obtain the indirect
utility function:
u * ( p1 , p2 , m) = u x1m ( p1 , p2 , m), x 2m ( p1 , p2 , m) .
17)
Now, evaluate the Lagrange equation 8) at x1m , x 2m , and λm :
(
)
(
L x1m , x 2m , λm : p1 , p2 , m = u x1m , x 2m
)
(
)
+ λm m − p1 x1m − p2 x 2m .
18)
The envelope theorem applied to a constrained optimization problem tells us that the derivative
of the indirect utility function with respect to a parameter is equal to the derivative of the
Lagrange equation with respect to that parameter when the Lagrange equation is evaluated at
x1m , x 2m , and λm . That is,
∂u*
∂L
=
, where α ∈ ( p1 , p2 , m)
∂α
∂α
19)
Using 19) we have the following:
∂ u*
= − λm x1m ,
∂ p1
∂ u*
= − λm x 2m ,
∂ p2
and
∂u*
= λm .
∂m
20)
The Lagrange multiplier for the utility-maximization problem
As always we interpret the Lagrange multiplier to be the marginal effect on the optimal value of
the objective when the constraint varies. The relation from 20)
∂u*
= λm ,
∂m
makes it clear that in the utility-maximization problem the Lagrange multiplier is the marginal
utility of money. Note that it is not an observable value.
2.10
Consumer theory
Further characteristics of the indirect utility function
[1] The indirect utility function u*(p1, p2, m) is non-increasing in prices, and non-decreasing in
income.
To show this first consider the sign of the Lagrange multiplier. Since it is the marginal utility
of money it must be positive because of the assumption that 'more is preferred to less.'
[Another way to see that it must be positive is to consider the first order conditions given by
9), and note that the multiplier must be positive for these conditions to hold]. Given that λm
> 0, the relations given by 20) justify the assertion.
Remark:
This result is very intuitive. Individual consumers should be better-off when they have more
income, given that prices do not change. On the other hand, for a fixed income, they will be
worse-off when they face a higher price for some good.
[2] The indirect utility function u*(p1, p2, m) is homogeneous of degree zero in prices and
income.
To show this assertion note that
u * ( tp1 , tp2 , tm ) = u x1m ( tp1 , tp2 , tm ), x 2m ( tp1 , tp2 , tm) .
Since the Marshallian demand equations are homogeneous of degree zero,
u * ( tp1 , tp2 , tm ) = u x1m ( p1 , p2 , m), x 2m ( p1 , p2 , m) = u * ( p1 , p2 , m) .
This last completes the proof of [2].
Remark:
The reason the indirect utility function is homogeneous of degree zero is that consumption
opportunities (as described by the budget line ) do not change if prices and income change by
the same proportion. Since opportunities do not change, the maximal attainable utility will
not change.
[3] The indirect utility function u*(p1, p2, m) is quasi-convex in prices.
The proof of this assertion is a little too difficult at this stage. So, you'll just have to believe
me on this one.
2.11
Consumer theory
Roy's Identity
From 20) we have
∂ u*
= − λm xim .
∂ pi
21)
This relationship is known as Roy's Identity. Substituting
∂ u*
= λm
∂m
into 21) and rearranging gives us
xim = −
∂ u * ∂ pi
.
∂ u* ∂ m
22)
Expenditure Minimization
Suppose instead of assuming that consumers maximize utility subject to a budget constraint,
assume that they minimize the cost of achieving a given level of utility. That is,
min
e(x1, x2) = p1x1 + p2x2
(x1, x2)
subject to
23)
u(x1, x2) = u0.
The Lagrange equation for this problem is
L = p1x1 + p2x2 + λ[u0 - u(x1, x2)].
The first order conditions are
L1 = p1 - λu1 = 0
L2 = p2 - λu2 = 0
Lλ = u0 - u(x1, x2) = 0.
24)
The second-order condition is that the determinant of the bordered Hessian of L be negative
[Remember that we are looking for a constrained minimum]:
L11
BH = L21
Lλ1
L12
L22
Lλ 2
− λu11
L1λ
L2λ = − λu21
Lλλ
− u1
− λu12
− λu22
− u2
− u1
− u2 < 0 .
0
25)
This determinant is
|BH | =
(
)
− λ 2u1 u 2 u 21 − u1 2 u 22 − u 2 2 u11 .
2.12
26)
Consumer theory
Since the first-order conditions imply that the Lagrange multiplier is positive, the determinant is
negative if and only if the term in parentheses is positive. But, this term is positive if the utility
function is strictly quasi-concave.
Given satisfaction of the second-order condition, the first-order conditions implicitly define
explicit functions
x1 = x1u ( p1 , p2 , u)
x2 = x2u ( p1 , p2 , u)
λ = λu ( p1 , p2 , u) .
27)
x1u ( p1 , p2 , u ) and x 2u ( p1 , p2 , u ) are known as Hicksian demand functions. The difference
between Hicksian demands and Marshallian demands is that utility is a constant along any
Hicksian demand function while it is income that is held constant along any Marshallian demand
function. Also, Marshallian demands are observable while Hicksian demands are not.
The first-order conditions for the expenditure minimization problem [equation 24)] imply that at
an optimal consumption bundle
u1
u2
=
p1
p2
and
m - p1x1 - p2x2 = 0.
These conditions are those that must be satisfied for a utility-maximizing consumption choice.
However, the problems are very different. Graphically, for a particular set of prices and some
level of utility, Hicksian demands are found by 'choosing' the lowest budget line that has a point
in common with the indifference curve that is drawn for the fixed level of utility.
x2
x2 u
u
x1 u
2.13
x1
Consumer theory
The comparative statics of the utility-maximization problem and the expenditure minimization
problem are also very different. Suppose initially that the price of good 1 is p1'. The utilitymaximizing consumption choice is A on the budget line with intercepts m/p2 and m/p1', and slope
-p1'/p2. Suppose that the price of good 1 falls to p1''. With income constant the new budget line
has intercepts m/p2 and m/p1'', and slope -p1''/p2. The new utility-maximizing choice of
consumption will be a point like B.
Now, since utility is held constant for the expenditure-minimization problem, a decrease in the
price of good 1 slides the budget line down the indifference curve. The new expenditureminimization consumption choice will be point C where a budget line with slope -p1''/p2 is
tangent to the indifference curve.
x2
m/p2
A
B
slope = -p1′′/p2
C
u**
u*
slope = -p1′/p2
x1
m/p1′′
m/p1′
The graph above suggests an inverse relationship between the Hicksian demand for a good and
its price. It turns out that this is a general result that is shown by obtaining the comparative static
∂ x1u ∂ p1 . To find the comparative static, first substitute equations 27) into the first-order
conditions 24):
[
]
− λ ( p , p , u ) u [ x ( p , p , u ), x ( p , p , u ) ] ≡
u − u[ x ( p , p , u), x ( p , p , u)] ≡
p1 − λu ( p1 , p 2 , u)u1 x1u ( p1 , p2 , u), x 2u ( p1 , p2 , u) ≡ 0.
p2
u
1
2
0
u
1
2
u
1
1
1
u
2
2
2
2.14
u
2
1
1
2
2
0.
0.
28)
Consumer theory
Now differentiate each equation with respect to the price of good 1:
1 − u1
∂ xu
∂ x 2u
∂ λu
− λu u11 1 − λu u12
= 0
∂ p1
∂ p1
∂ p1
−u2
∂ xu
∂ x 2u
∂ λu
− λu u21 1 − λu u22
= 0
∂ p2
∂ p1
∂ p1
− u1
∂ x1u
∂ x 2u
− u2
= 0.
∂ p1
∂ p1
29)
Rearrange equations 29) and write them in matrix form:
 − λu u11
 u
− λ u21
 − u1

− λu u12
− λu u22
− u2
− u1 

− u2 
0 
∂ x1u ∂ p1 
-1
 u

 
∂ x 2 ∂ p1  =  0  .
∂ λu ∂ p1 
 0 


30)
Note that the matrix on the left-hand side of 30) is the bordered Hessian of the Lagrange
equation for this problem. Its determinant, |BH|, is given by 25) and is negative.
Apply Cramer's rules to 30) to obtain
∂x
∂ p1
u
1
=
−1 − λu u12
− u1
0
u
− λ u22
− u2
0
− u2
0
BH
=
u22
BH
< 0 .
Thus, there is an inverse relationship between the Hicksian demand for a good and its price. It is
also easy to see that
∂ x 2u
∂ p1
=
− u1u2
BH
> 0.
However, this result holds only in the two-good case.
2.15
Consumer theory
Further characteristics of Hicksian demand equations
[1] Hicksian demand functions are invariant to any positive monotonic transformation of the
utility function.
[2] Hicksian demand functions are homogeneous of degree zero in prices.
These assertions are very easy to prove so we won't pursue them further.
Expenditure minimization and the envelope theorem
By substituting the Hicksian demand equations into the expenditure function, we obtain the
indirect expenditure function:
e * ( p1 , p2 , u ) = p1 x1u ( p1 , p2 , u ) + p2 x 2u ( p1 , p2 , u ) .
31)
Now, evaluate the Lagrange equation for the expenditure-minimization problem at
x1u , x 2u , and λu :
(
[
)
(
)]
L x1u , x 2u , λu : p1 , p2 , u = p1 x1u + p2 x 2u + λu u − u x1u , x 2u .
32)
From the envelope theorem,
∂ e*
∂L
=
, where α ∈ ( p1 , p 2 , u)
∂α
∂α
33)
The Lagrange multiplier for the expenditure minimization problem
As always we interpret the Lagrange multiplier to be the marginal effect on the optimal value of
the objective when the constraint varies. Using 32) and 33) we have
∂ e*
= λu .
∂u
Thus, λu is the additional amount of money required to increase utility by one unit. Note that it
must be a positive value. Note further that it is not observable.
2.16
Consumer theory
Further characteristics of the indirect expenditure function
[1] The indirect expenditure function e*(p1, p2, u) is non-decreasing in prices and utility.
From 33) we have
∂ e*
= x1u ,
∂ p1
∂ e*
∂ e*
= x 2u , and
= λu
∂ p2
∂u
34)
All of these derivatives are positive.
[2] The partial derivative of the expenditure function e*(p1, p2, u) with respect to the ith price is
the Hicksian demand for the ith good.
This assertion is obvious from 34).
[3] The indirect expenditure function e*(p1, p2, u) is homogeneous of degree one in prices.
Note that
e * ( tp1 , tp2 , u ) = [ tp1 ] x1u ( tp1 , tp2 , u ) + [ tp2 ] x 2u ( tp1 , tp2 , u ) .
Since the Hicksian demand equations are homogeneous of degree zero,
e * ( tp1 , tp2 , u ) = tp1 x1u ( p1 , p2 , u ) + tp2 x 2u ( p1 , p2 , u ) = te * ( p1 , p2 , u )
[4] The indirect expenditure function e*(p1, p2, u) is concave in prices.
The proof of this assertion is exactly the one for concavity in prices of the indirect cost
function for the competitive firm.
2.17
Consumer theory
Relationships between utility-max and expenditure-min
From the utility-maximization problem we derived the indirect utility function u*(p1, p2, m) and
associated Marshallian demand equations x im ( p1 , p 2 , m) , i = 1, 2. From the expenditureminimization problem we have the indirect expenditure function e*(p1, p2, u) and associated
Hicksian demand equations x iu ( p1 , p 2 , u) , i = 1, 2. The following identities link the two
problems together
[1] e * ( p1 , p2 , u * ( p1 , p2 , m)) ≡ m.
The minimum expenditure necessary to reach utility u*(p1, p2, m) is m.
[2] u * ( p1 , p2 , e * ( p1 , p 2 , u)) ≡ u.
The maximum utility from income e*(p1, p2, u) is u.
[3] x im ( p1 , p 2 , m) ≡ xiu ( p1 , p2 , u * ( p1 , p2 , m)) .
The Marshallian demand for good i at income m is the same as the Hicksian demand at
utility u*(p1, p2, m).
[4] x iu ( p1 , p 2 , u) ≡ x im ( p1 , p2 , e * ( p1 , p 2 , u)) .
The Hicksian demand for good i at utility u is the same as the Marshallian demand at
income e*(p1, p2, u).
Perhaps the most important of these identities is the fourth one because it links the unobservable
Hicksian demand for good i to its observable Marshallian demand.
The Slutsky equation
Differentiate the fourth identity with respect to pi:
∂ xiu
∂ xim
∂ xim ∂ e *
=
+
∂ pi
∂ pi
∂ m ∂ pi
Note that
35)
∂ e*
= xiu = xim .
∂ pi
Substitute this into 35) and rearrange to obtain
∂ xiu
∂ xim
∂ xm
=
+ xim i .
∂ pi
∂ pi
∂m
36)
The left-hand side of 36) is how the Hicksian demand for good i changes when its price changes.
This is unobservable but it is equal to a function of observable values.
2.18
Consumer theory
Rearrange equation 36) to obtain
∂ xim
∂ xiu
∂ xm
=
− xim i .
∂ pi
∂ pi
∂m
37)
This equation is known as the Slutsky equation. It decomposes a change in demand (observable)
induced by a price change into a substitution effect and an income effect. The substitution effect
is captured by the first term on the right-hand side of 36) while the income effect is captured by
the second term on the right-hand side. In the graph below, this decomposition is illustrated for
a decrease in the price of good 1.
x10 → x11 substitution effect
x2
x11 → x12 income effect
x10 → x12 total effect
A
C
B
u**
u*
x10
x11
x12
x1
Remarks:
1) The Slutsky equation does not imply that a Marshallian demand is decreasing in its price.
However, note that if the good is a normal good, demand for it is decreasing in price. Note
also that if demand is increasing in price, the good must be an inferior good. However,
inferiority does not imply an upward sloping demand function.
2) Just because our theory of consumer behavior says that an upward sloping demand function
is possible does not mean that we have to treat that possibility very seriously. To my
knowledge, nobody has discovered a good for which demand is increasing in its price.
[These sorts of goods are called Giffen goods].
3) We can also derive a Slutsky equation for cross-price effects:
2.19
Consumer theory
∂ xim
∂ xiu
∂ xim
=
− x mj
∂ pj
∂ pj
∂m
Measuring changes in consumer welfare
In this section we consider measuring changes in consumer welfare when the economic
environment changes. This exercise is somewhat problematic because utility is not observable.
In fact, an exact measure of changes in consumers' welfare is not possible (except in special
cases), but we have some approximations.
Equivalent and compensating variation
Imagine a change in the economic environment that induces a change from the status quo prices
and income (p0, m0) to (p1, m1), where p = (p1, p2, ... , pn) The change in welfare for the
individual is the change in indirect utility
u*(p1, m1) - u*(p0, m0).
Since this is in terms of utility it is unobservable. Let's try to get some measure in terms of
money. Consider e(q, u*(p0, m0)), where q is some vector of prices. This is how much money
the consumer would need at prices q to be as well off as at prices and income (p0, m0). Now
consider the following measure of the change in the consumer's welfare:
e(q, u*(p1, m1)) - e(q, u*(p0, m0)).
38)
If the change from (p0, m0) to (p1, m1) is beneficial, this is how much extra money the consumer
needs to reach utility level u*(p1, m1) from u*(p0, m0) if she faces prices q. If she prefers (p0, m0)
to (p1, m1), 38) is how much less money she needs to reach utility level u*(p1, m1) than what she
requires to reach u*(p0, m0) if she faces prices q.
Equivalent variation
Let q = p0. Then, the equivalent variation measure of a welfare change is
EV = e(p0, u*(p1, m1)) - e(p0, u*(p0, m0)).
39)
The equivalent variation is the difference between the amount of money the consumer needs to
reach utility u*(p1, m1) and the amount needed to reach u*(p0, m0) evaluated at the status quo
(p0, m0).
Compensating variation
If we let q = p1. Then, the compensating variation is
CV = e(p1, u*(p1, m1)) - e(p1, u*(p0, m0)).
2.20
40)
Consumer theory
The compensating variation is the difference between the amount of money the consumer needs
to reach utility u*(p1, m1) and the amount needed to reach u*(p0, m0) evaluated at the proposed
change (p1, m1).
In general the equivalent and compensating variations will give different answers for the welfare
effect of some change. In the graph below I suppose that m = m0 = m1. There are just two
goods. Let p2 = 1 so that money is measured on the vertical axis. Let p be the price of good 1.
The graph gives the compensating and equivalent variation measures for the change in the
consumer's welfare when the price of good 1 decreases from p0 to p1.
x2
e(p0, u*(p1, m))
e(p0, u*(p0, m)) = e(p1, u*(p1, m))
EV
e(p1, u*(p0, m))
CV
B
slope = -p1
A
u*(p1, m)
u*(p0, m)
x1
slope = -p0
Comparison of equivalent and compensating variations to consumer surplus
Imagine the same comparison as in the graph above. The consumer surplus measure of the
welfare change when the price of good 1 falls is
CS =
∫
p0
p1
x1m ( p, m)dp .
41)
The consumer surplus measure is the shaded area in the following graph.
2.21
Consumer theory
p
p0
p1
x1m(p, m)
x1
As before, the equivalent and compensating variations are
EV = e(p0, u1) - e(p0, u0)
CV = e(p1, u1) - e(p1, u0).
42)
where, to simplify the notation, we let u0 = u*(p 0, m) and u1 = u*(p1, m).
Now, since e(p0, u*(p0, m)) = e(p1, u*(p1, m)) = m, it follows that e(p0, u0) = e(p1, u1) and we can
write 42) as
EV = e(p0, u1) - e(p1, u1)
CV = e(p0, u0) - e(p1, u0).
43)
Recall from Shephard's Lemma that
∂ e( p, u)
= x1u ( p, u) .
∂p
From this and the Fundamental Rule of Calculus,
(
e p, u
1
)
=
∫
∂ e( p, u 1 )
∂p
dp =
∫ x ( p, u )dp .
u
1
1
Taking differences we have
EV = e(p0, u1) - e(p1, u1) =
∫
p0
p1
(
)
44)
(
)
45)
x1u p, u 1 dp .
Similarly,
CV = e(p0, u0) - e(p1, u0) =
∫
p0
p1
x1u p, u 0 dp .
2.22
Consumer theory
Recall,
CS =
∫
p0
p1
x1m ( p, m)dp .
46)
To compare 44), 45) and 46), we need the relationship between Hicksian and Marshallian
demands which is given by the Slutsky equation:
∂ x1m
∂ x1u
∂ xm
=
− x1m 1 .
∂ p1
∂ p1
∂m
47)
Assume that good 1 is a normal good. Then,
∂ x1m
∂ x1u
<
.
∂ p1
∂ p1
Hence, the Marshallian demand is steeper than the Hicksian demand in the (price, quantity)
space. The reverse is true in the (quantity, price) space. Using this relationship for normal
goods, we can illustrate the differences among the compensating and equivalent variations, and
consumer surplus using the following areas in the graph below.
CS =
EV =
CV =
∫
p0
∫
p0
p1
p1
∫
p0
p1
x1m ( p, m)dp = area(p1p0ac)
(
)
(
)
x1u p, u 1 dp = area(p1p0bc)
x1u p, u 0 dp = area(p1p0ad)
p
x1u(p, u0)
p0
p1
x1u(p, u1)
a
b
d
c
x1m(p, m)
x1
2.23
Consumer theory
Note that CV < CS < EV. Now, the problem with finding the compensating and equivalent
variations is that Hicksian demands are not observable. However, this exercise tells us that we
can approximate the compensating and equivalent variations by consumer surplus which is
calculable. Furthermore, if we know whether a good is normal or inferior, we know which way
the bias runs.
Expected utility theory
State-contingent consumption and utility
A state-contingent commodity is a good that can be consumed only if a specific state occurs. In
this set-up an umbrella on a sunny day is different from an umbrella on a rainy day.
Now suppose that there are just two states of the world, 1 and 2, and that in state 1 a consumer’s
wealth is w1 and in state 2 it is w2. Suppose further that the two states are mutually exclusive and
collectively exhaustive. Then, assuming that π is the probability that state 1 occurs, (1 - π) is the
probability that state 2 occurs. The consumer treats these probabilities as exogenous. Then we
can define the consumer’s state-contingent utility as
U(w1, w2; π).
A gamble (or lottery) is a list of possible outcomes and their associated probabilities. Thus, in
our simple set-up, U is a function of the gamble (w1, w2; π).
Expected utility (Von Neumann/Morgenstern utility)
The consumer’s state-contingent utility defined above is very general -- too general to be useful.
However, under a certain set of conditions, U takes on a very useful form.
[1] Completeness and transitivity
Given two certain outcomes A and B, either A P B, B P A, or A I B. [completeness]
[P denotes ‘is preferred to’, and I denotes ‘is indifferent between’].
Also, Given three certain outcomes A, B and C, if A P B and B P C, then A P C.
[transitivity]
Recall that we required these for preferences under certainty.
[2] Continuity
Take three certain outcomes A, B and C such that A P B and B P C. Then there exists π
such that B I πA + (1 - π)C.
There exists a probability that A occurs such that the individual is indifferent between B
with certainty and the gamble (A, C; π).
2.24
Consumer theory
[3] Unequal probabilities
Suppose that A P B. Then, π1A + (1 - π1)B P π2A + (1 - π2)B if and only if π1 > π2.
Take two gambles (A, B; π1) and (A, B; π2). The individual prefers the gamble that has
the highest probability that its most-preferred outcome (A) occurs.
[4] Independence of irrelevant alternative
If A P B, then for any C, πA + (1 - π)C P πB + (1 - π)C.
[5] Reduction of compound lotteries
Consider an uncertain outcome A = πB + (1 - π)C. Then, the consumer is indifferent
between π0A + (1 - π0)C and ππ0B + (1 - ππ0)C.
The expected outcome of the gamble (A, C, π0) is the same as the expected outcome of the
gamble (B, C, ππ0), hence, the individual is indifferent between them. To see that the
expectations are the same, note that
π0A + (1 - π0)C = π0[πB + (1 - π)C] + (1 - π0)C
= ππ0B + π0C - ππ0C + C - π0C
= ππ0B + (1 - ππ0)C.
If preferences satisfy axioms [1] through [6], there exists a function u(w) such that
U(w1, w2; π) = πu(w1) + (1 - π)u(w2).
Note that U is expressed as the expected value of a utility function u. If we denote the
expectation operator as E,
U(w1, w2; π) = E[u(w)].
The theorem also holds for more complicated lotteries. Suppose that outcome wi is received with
probability πi for i = 1, ... , n. Then, the expected utility of this lottery is
n
∑ π i u(wi ) .
i =1
For continuous distributions, if π(w) is the probability density function defined on the continuous
outcomes w, expected utility is
∫ π (w)u(w)dw .
2.25
Consumer theory
Attitudes towards risk
Suppose that the consumer has a utility function u(w) that satisfies the six axioms; is a function
of wealth only, and is monotonically increasing in wealth.
Now consider a gamble (w1, w2, π) -- the expectation of this gamble is πw1 + (1 - π)w2. Take a
certain outcome wc. Define a fair gamble as one for which the expectation of the gamble (w1, w2, π)
is equal to the certain outcome wc; that is, E(w) ≡ πw1 + (1 - π)w2 = wc.
Given a fair gamble, we say that an individual is risk averse if the utility of expected wealth is
greater than the expected utility of wealth. That is,
u[E(w)] > E[u(w)],
or equivalently,
u[πw1 + (1 - π)w2] ≡ u(wc) > πu(w1) + (1 - π)u(w2).
A risk averse individual prefers wc with certainty to a gamble that gives him expected wealth wc.
Graphically,
utility
u(w)
u[E(w)] = u[πw1 + (1 - π)w2]
E[u(w)] = πu(w1) + (1 - π)u(w2)
w
1
wc
w
w
2
Note that the individual’s utility function u(w) is strictly concave. It turns out that if a consumer
is risk averse, his utility function is strictly concave.
Given a fair gamble, we say that an individual is risk neutral if the utility of expected wealth is
equal to the expected utility of wealth. That is,
u[E(w)] = E[u(w)].
A risk neutral individual is indifferent between wc with certainty to a gamble that gives him
expected wealth wc. The utility function of a risk neutral individual is linear.
Given a fair gamble, we say that an individual is risk loving if the utility of expected wealth is
less than the expected utility of wealth. That is,
2.26
Consumer theory
u[E(w)] < E[u(w)].
A risk neutral loving individual prefers a gamble that gives him expected wealth wc to wc with
certainty. The utility function of a risk loving individual is strictly convex.
State-contingent indifference curves and the degree of risk aversion
Now let’s think about indifference curves of the expected utility function in the (w1, w2) space.
These indifference curves will give us the individual’s subjective trade-off between wealth in
state 1 and wealth in state 2. An indifference curve is implicitly defined by
E[u(w)] - U0 = πu(w1) + (1 - π)u(w2) - U0 ≡ 0.
Totally differentiate this and rearrange to get
−πu' (w1 )
dw2
=
< 0.
dw1
(1 − π )u' (w2 )
1)
This reveals that the state-contingent indifference curves slope down. To get the shape of the
indifference curves take the second derivative and substitute 1) to get
d 2 w2
dw1
2
=
{
− [ (1 − π )u' (w2 )] πu' ' (w1 ) + (1 − π )u' ' (w2 )[πu' (w1 )]
2
[ (1 − π )u' (w2 )]
3
2
}.
Now evaluate this at w1 = w2 = w:
d 2 w2
dw1
2
=
−πu' ' (w)
(1 − π ) 2 u' (w)
.
Remarks:
1) The shape of the indifference curves depend on the shape of the utility function, and hence, on
the individual’s attitude toward risk. If the individual is risk averse, the indifference curves
are strictly convex; if risk neutral, the indifference curves are linear, and if risk loving, the
indifference curves are strictly concave.
2) The shape of the indifference curves (degree of convexity) tells us something about the degree
of risk aversion, and, since π is constant, this is proportional to -u′′(w)/u′(w).
3) -u′′(w)/u′(w) is called the absolute (or Arrow/Pratt) measure of risk aversion. The larger it is,
the more risk averse the individual is.
2.27
Consumer theory
Exercises
[1] Consider a consumer with the utility function u(x1, x2) = x1 x2β , where β > 0.
[a] Provide the first-order conditions for the problem of maximizing u(x1, x2) = x1 x2β subject
to a budget constraint. Show that the second-order condition is satisfied. What does the
second-order condition imply about this utility function?
[b] Derive expressions for the Marshallian demand equations. Show that the demand
equations are homogeneous of degree zero in prices and income.
[c] For the Marshallian demand functions from b), show the following:
i) Demand for each good is unit elastic.
ii) Each good is a normal good.
iii) The cross-price elasticities are zero.
[d] Now consider the transformation of u(x1, x2) into F[u(x1, x2)] = ln[ x1 x2β ]. Rederive the
Marshallian demand equations to verify that they are invariant to this transformation.
[2] An important class of utility function in economics and game theory is the so-called quasilinear utility function. Quasi-linear utility functions are of the form u(x1, x2) = v(x1) + x2,
where v(x1) is monotonically increasing and strictly concave.
[a] Show that u(x1, x2) = v(x1) + x2 is strictly quasi-concave.
[b] State the first-order conditions that determine the Marshallian demand equations. Show
that at an optimal choice of (x1, x2), v' = p1/p2. What does this imply about the marginal
rate of substitution for quasi-linear utility functions?
[c] Show that the income elasticity for good 1 is zero, and that good 2 is a normal good.
Interpret this result.
[d] Show that ∂ x1u ∂ p1 = ∂ x1m ∂ p1 . Why is this true?
[3] Consider a consumer with the utility function u(x1, x2) = x1x2.
[a] Find the Marshallian demand equations for each good. Derive the indirect utility
function u*(p1, p2, m) and verify that it is homogeneous of degree zero. Interpret this last
result.
[b] Verify that u*(p1, p2, m) is decreasing in prices and increasing in income. Provide some
intuition about these results.
∂ u * ∂ pi
[c] Verify that xim = −
, i = 1, 2.
∂ u* ∂ m
[d] Find the Hicksian demand equations for each good and derive the indirect expenditure
function e*(p1, p2, u).
[e] Verify that e*(p1, p2, u) is increasing in prices and utility and provide some intuition
about these results. Verify that e*(p1, p2, u) is homogeneous of degree one in prices.
[f] Verify that xiu ( p1 , p2 , u) = ∂ e * ∂ pi , i = 1, 2.
[g] Verify that x iu ( p1 , p 2 , u) ≡ x im ( p1 , p2 , e * ( p1 , p 2 , u)) , i = 1, 2.
[h] Verify that x im ( p1 , p 2 , m) ≡ xiu ( p1 , p2 , u * ( p1 , p2 , m)) , i = 1, 2.
[i] Derive the Slutsky decomposition of the change in the Marshallian demand for good 1 when
its price changes. Explain.
2.28
Consumer theory
[4] Consider a consumer with the utility function u(x1, x2) = x11 2 + x 2 .
[a] Derive the Marshallian demand equations x im ( p1 , p 2 , m) , i = 1, 2, and the indirect utility
function u*(p1, p2, m). Provide a condition on the relationship between m, p1 and p2 that
guarantees that the consumer will purchase a strictly positive amount of good 2?
[b] Verify that u*(p1, p2, m) is increasing in income and decreasing in prices. Also show that
it is homogeneous of degree zero in prices and income. Provide some intuition about the
homogeneity result.
[c] Derive the Hicksian demand equations x iu ( p1 , p 2 , u) , i = 1, 2, and the indirect
expenditure function e*(p1, p2, u). Provide a condition on the relationship between u, p1
and p2 that guarantees that x2u is strictly positive.
[d] Verify that the Marshallian demand for good 2 at income m is the same as the Hicksian
demand for good 2 at utility u*(p1, p2, m).
[e] Verify that the Hicksian demand for good 2 at utility u is the same as the Marshallian
demand for good 2 at income e*(p1, p2, u).
[f] Derive the Slutsky decomposition of the change in the Marshallian demand for good 2
when its price changes. Explain
[5] In a two good commodity space, illustrate the equivalent and compensating variations for an
increase in the price of a good.
[6] Suppose that a consumer has a quasi-linear utility function u(x1, x2) = v(x1) + x2. Prove that
the equivalent and compensating variations, and consumer surplus measure of a change in
the price of good 1 are all equal to each other.
[7] Prove that the equivalent and compensating variations, and consumer surplus measure of a
change in the price of some good are such that EV < CS < CV if the good is inferior (but not
Giffen).
[8] Suppose that a consumer's utility function is given by u(x1, x2) = x11/ 3 x2 . Let p1, p2, and m
denote prices and income. Your task in this exercise is examine the relationships among the
various surplus measures. To begin, derive the consumer's Marshallian demand curves for
both goods, and find her indirect utility function. Suppose that m = $100 and that the price of
x falls from $1.50 to $1.00. Using the Marshallian demand curve for good 1, compute the
consumer surplus of this price decrease (you'll need to evaluate an integral here). Next, find
the Hicksian demand function for good 1.
[a] Compute the compensating variation for the price change. How does the compensating
variation compare in magnitude to consumer surplus?
[b] Compute the equivalent variation for the price change. How does it compare to consumer
surplus?
[c] Is good 1 a normal or inferior good?
2.29