# The Theory of Consumer Behavior The postulate of rationality

Document Sample

```					Advanced Duality Theory                                                                            1

The Theory of Consumer Behavior

The postulate of rationality is the philosophical, or metaphysical, point of departure for the eco-

nomic theory of individual choice. The consumer, or individual decision maker, is presumed to

choose among available alternatives so as to maximize the satisfaction (utility) derived from the

choices that he or she makes. This implies that consumers are:

(a) aware of the alternatives; and

(b) capable of evaluating those alternatives.

All information relevant to the satisfaction that an individual derives from his choices is con-

tained in the utility function.

In general, a consumer will not add to the consumption of a commodity if an additional unit in-

volves a net utility loss. Consumption will be increased only if a net gain of utility is realized

form it.

The postulate of rationality is equivalent to the following axioms:

1. Completeness. For all possible pairs of alternatives, A and B, the individual either prefers

A at least as much as B, denoted A     B, or else he prefers B at least as much as A, B   A.

2. Reflexivity. For all possible alternatives A, we have A    A.

3. Transitivity. If a consumer prefers A to B and B to C, then he will prefer A to C. A        B

and B    C⇒A          C.

4. Closure or Continuity. Let the set of all consumption alternatives be connected – that is,

if A and B are available to the consumer – then there is a continuous path of available alterna-

tives connecting A and B. Given any alternative A, we may consider the set of all alternatives

1
Jeffrey T. LaFrance                                                                                                       2

at least as well liked as A, {B: B         A}, and the set of all alternatives not more liked than A,

{B: A    B}.

These two sets are closed -- that is, they contain all their limit points.

Postulates 1-4 are equivalent to the existence of an ordinal, continuous utility function

when the commodity space is a subset of n–dimensional Euclidean space.

That is, the preferences of the individual are represented mathematically by a utility function

mapping points in the space of available alternatives to points on the real number line such that

z0    z1 iff u(z 0 ) ≥ u( z1 ) for any pair of alternatives z0 and z1. The nature of the utility function

2
for the simple two commodity case in two-dimensional Euclidean space, denoted                                           , is

u = u( x, y ) , where x and y are quantities of the two commodities consumed. We will assume that

u(x,y) is twice continuously differentiable.

The utility function is not unique. In general, any single-valued function of x and y could serve

as a utility function. More specifically, given a preference relation,                  , suppose that u(x,y) repre-

sents the individual's preferences. Then, by definition, (x0,y0)                    (x1,y1) iff u(x 0 ,y 0 ) ≥ u(x1 ,y1 ) .

Now let v(u) be an increasing, continuous function of u, so that v 0 ≥ v1 iff u 0 ≥ u1 . Then we have

v 0 ≡ v (u 0 ) ≡ v (u( x 0 , y 0 )) ≥ v (u( x1 , y1 )) ≡ v(u1 ) ≡ v1

iff (x0,y0)   (x1,y1). Therefore, v represents the preference relation                 just as well as u.

The non-uniqueness of utility functions is the result of the ordinality of preferences –

preference relations and utility functions only indicate ordinal rankings of alternatives.

We require two additional assumptions to ensure that preference maps and observed choices are

consistent with one another.

2

5. Strict Monotonicity. If (x0,y0) ≥ (x1,y1) and (x0,y0) ≠ (x1,y1) then (x0,y0)                      (x1,y1), where

means "is strictly preferred to". This hypothesis simply states that more is preferred to less. A

weaker assumption is called local nonsatiation. Given any (x0,y0) and ε > 0, there exists an

(x1,y1) such that

( x1 , y1 ) − ( x 0 , y 0 ) ≡ ( x1 − x 0 )2 + ( y1 − y 0 ) 2 < ε

and (x1,y1)     (x0,y0). One can always do a little better. Either assumption rules out bliss points -

points in consumption space that are local maxima for preferences or utility.

6. Strict Convexity. Given (x0,y0) ≠ (x1,y1) and (x0,y0) ≠ (x2,y2), if (x0,y0)                    (x2,y2) and (x1,y1)

(x2,y2), then

{( tx   0
)(
+ (1 − t ) x1 , ty 0 + (1 − t ) y1   )}   ( x2 , y2 )

for all 0 < t < 1. An agent prefers averages to extremes. This is a generalization of the assump-

tion of diminishing marginal rates of substitution.

Properties 1-6 for the binary preference relation                     are equivalent to the existence of a

continuous, strictly increasing, and strictly quasi-concave utility function, u, such that

(x0,y0)     (x1,y1) iff u(x0,y0) ≥ u(x1,y1).

Substitution

A particular level of utility, say u0, can be derived from many different combinations of x and y.

At any point, the consumer is willing to give up some of one good to get an additional increment

of some other good. For a given level of utility, u0, the equation u 0 = u( x, y ) , is satisfied by an

infinite number of combinations of x and y because u is continuous. The locus of all commodity

combinations from which the individual derives the same level of satisfaction forms an indiffer-

3
Jeffrey T. LaFrance                                                                                                              4

ence curve. An indifference map is a collection of indifference curves corresponding to different

levels of satisfaction. The set of all commodity combinations on or above an indifference curve

is an upper contour set.

Marginal Rate of Commodity Substitution

Along an indifference curve, the level of utility remains constant, u(x,y) = u0. Consequently, the

total differential of u(x,y) must be zero,

∂u ( x, y )      ∂u ( x, y )
0 = du =                dx +             dy
∂x               ∂y

Solving for dy/dx for fixed u0 gives the slope of the indifference curve,1

dy      ∂u ( x, y ) ∂x
=−                .
dx u  0 ∂u ( x, y ) ∂y

The partial derivatives ∂u/∂x and ∂u/∂y are defined as the marginal utilities of x and y, respec-

tively. The numerical magnitudes of individual marginal utilities are without meaning (due to

ordinal preferences), but the signs and the ratios of marginal utilities are meaningful in an ordi-

nal analysis.

Totally differentiating dy dx u0 with respect to x, recognizing that y is implicitly a function of x

and u along an indifference curve,

1
The standard notation for the steps in this section are misleading. In reality, what we are doing is fixing u and find-
ing the solution, y ( x, u ) , to the implicit function u ( x, y ( x, u )) − u ≡ 0 , thereby artificially making y, rather thatn u,
2
the dependent variable. Viewed in this way, we calculate the partial derivatives ∂y ( x, u ) ∂x and ∂ 2 y ( x, u ) ∂x ,
which identify the slope and curvature of the curve that is the projection onto the (x,y) plane of the surface y ( x, u )
for a fixed value of u. It is much cleaner to think about these questions in this manner, which extends readily and
logically to goods spaces of any dimension. This extension has many powerful uses in economic analysis.

4

⎡⎛ ∂ 2u ⎞ ⎛ ∂u ⎞ 2 ⎛ ∂ 2u ⎞ ⎛ ∂u ⎞2  ⎛ ∂ 2u ⎞ ⎛ ∂u ⎞ ⎛ ∂u ⎞ ⎤
⎢⎜ 2 ⎟ ⎜ ⎟ + ⎜ 2 ⎟ ⎜ ⎟ − 2 ⎜                ⎟ ⎜ ⎟ ⎜ ⎟⎥
d2y         ⎢⎝ ∂x ⎠ ⎝ ∂y ⎠ ⎝ ∂y ⎠ ⎝ ∂x ⎠         ⎝ ∂x∂y ⎠ ⎝ ∂x ⎠ ⎝ ∂y ⎠ ⎥
=−⎣                                                           ⎦
dx 2 u0                                 ⎛ ∂u ⎞
3

⎜ ∂y ⎟
⎝ ⎠

∂u         ∂u
Since strict monotonicity (Property 5 above) implies              > 0 and    > 0 , we have strictly convex
∂x         ∂y

indifference curves (Property 6 above) iff d 2 y dx 2 | u 0 > 0, iff

2
⎛ ∂ u ⎞ ⎛ ∂u ⎞ ⎛ ∂u ⎞ ⎛ ∂ 2u ⎞ ⎛ ∂u ⎞ ⎛ ∂ 2u ⎞ ⎛ ∂u ⎞ 2
2
2⎜      ⎟ ⎜ ⎟ ⎜ ⎟ − ⎜ 2 ⎟ ⎜ ⎟ − ⎜ 2 ⎟ ⎜ ⎟ > 0.
⎝ ∂x∂y ⎠ ⎝ ∂x ⎠ ⎝ ∂y ⎠ ⎝ ∂x ⎠ ⎝ ∂y ⎠ ⎝ ∂y ⎠ ⎝ ∂x ⎠

Note that the left-hand-side term is precisely the determinant of the bordered Hessian for u(x,y),

2
∂ u       ∂ u         ∂u
2

∂x 2      ∂x∂y        ∂x
2       2
2
∂ u
2
∂ u        ∂u    ⎛ ∂ 2u ⎞ ⎛ ∂u ⎞ ⎛ ∂u ⎞ ⎛ ∂ 2u⎞ ⎛ ∂u ⎞ ⎛ ∂ u ⎞ ⎛ ∂u ⎞ 2
= 2⎜      ⎟⎜ ⎟⎜ ⎟ − ⎜ 2 ⎟⎜ ⎟ − ⎜ 2 ⎟⎜ ⎟ .
∂x∂y       ∂y 2       ∂y    ⎝ ∂x∂y ⎠ ⎝ ∂x ⎠ ⎝ ∂y ⎠ ⎝ ∂x ⎠ ⎝ ∂y ⎠ ⎝ ∂y ⎠ ⎝ ∂x ⎠
∂u        ∂u
0
∂x        ∂y

Consequently, given strict monotonicity, strictly convex indifference curves – or equivalently,

diminishing marginal rates of substitution – are equivalent to u(x,y) strictly quasi-concave.

Utility Maximization

We take the typical problem faced by a consumer to be to maximize the utility obtained from

consumption of goods and services subject to the constraints he or she faces due to, for example,

a limited budget, finite amounts of time that can be allocated to various activities, and so forth.

In the simple case of a two good world and a single budget constraint, we represent this problem

by

5
Jeffrey T. LaFrance                                                                               6

maximize u( x, y ) subject to x ≥ 0, y ≥ 0, and p x x + p y y ≤ m , where px > 0 and py > 0 are the

prices of goods x and y, respectively, and m > 0 is the individual's total income.

We use the Kuhn-Tucker-Karush theory of constrained optimization to analyze the solution to

the consumer's choice problem. The Lagrangean function is given by

L = u ( x, y ) + λ ( m − p x x − p y y ) .

The first-order necessary conditions for an interior optimal solution are

∂L        ∂x = ∂u ∂x − λ p x = 0,

∂L        ∂y = ∂u ∂y − λ p y = 0,          ,

∂L        ∂λ = m − px x − p y y = 0.

The second-order sufficient condition for a unique constrained local optimum is

∂ 2L       ∂ 2L        ∂ 2L           ∂ 2u      ∂ 2u
− px
∂x 2       ∂x∂        ∂x∂λ           ∂x 2      ∂x∂y
∂ 2L       ∂ 2L        ∂ 2L   ∂ 2u              ∂ 2u
=                            − py
∂y∂x        ∂y 2       ∂y∂λ   ∂y∂x              ∂y 2
∂ 2L       ∂ 2L        ∂ 2L           − px      − py      0
∂λ∂x       ∂λ∂y         ∂λ 2

∂ 2u          ∂ 2u               ⎛ 1 ⎞ ∂u
−⎜ ⎟
∂x 2          ∂x∂y               ⎝ λ ⎠ ∂x
∂ 2u               ∂ 2u           ⎛ 1 ⎞ ∂u
=                                       −⎜ ⎟
∂y∂x               ∂y 2           ⎝ λ ⎠ ∂y
⎛ 1 ⎞ ∂u        ⎛ 1 ⎞ ∂u
−⎜ ⎟            −⎜ ⎟                   0
⎝ λ ⎠ ∂x        ⎝ λ ⎠ ∂y

⎛1⎞
2   ⎡ ⎛ ∂ 2u ⎞ ⎛ ∂u ⎞ ⎛ ∂u ⎞ ⎛ ∂ 2u ⎞ ⎛ ∂u ⎞2 ⎛ ∂ 2u ⎞ ⎛ ∂u ⎞2 ⎤
=⎜ ⎟       ⎢2 ⎜     ⎟ ⎜ ⎟ ⎜ ⎟ − ⎜ 2 ⎟ ⎜ ⎟ − ⎜ 2 ⎟ ⎜ ⎟ ⎥ > 0.
⎝λ⎠       ⎢ ⎝ ∂x∂y ⎠ ⎝ ∂x ⎠ ⎝ ∂y ⎠ ⎝ ∂x ⎠ ⎝ ∂y ⎠ ⎝ ∂y ⎠ ⎝ ∂x ⎠ ⎥
⎣                                                          ⎦

Note that a diminishing marginal rate of substitution – equivalently, a strictly quasi-concave util-

6

ity function – is sufficient for a unique constrained maximum.

Given the second-order condition, we know by the implicit function theorem that a unique solu-

tion for x, y, and λ exists and can be obtained from the first-order conditions. The optimal con-

sumption choices,

x* = h x ( p x , p y , m ) ,

y* = h y ( p x , p y , m ) ,

are ordinary or Marshallian market demands. The optimal value of the Lagrange multiplier,

λ* = λ ( px , p y , m) ,

is called the marginal utility of money.

Comparative Statics for Demand Models

{                                                       }
If we substitute h x ( p x , p y , m ), h y ( p x , p y , m ), λ ( px , p y m ) for x, y, and λ in the first-order condi-

tions (FOC's), then we obtain the identities,

(                                      )
∂u h x ( p x , p y , m ), h y ( p x , p y , m) ∂x − λ ( p x , p y , m ) p x ≡ 0 ,

(                                       )
∂u h x ( px , p y , m), h y ( p x , p y , m ) ∂y − λ ( p x , p y , m ) p y ≡ 0 ,

m − px h x ( px , p y , m) − p y h y ( px , p y , m) ≡ 0 .

Differentiating each of these three identities with respect to px, we have in matrix form,

⎡ ∂ 2u ∂x 2 ∂ 2u ∂x∂y − px ⎤ ⎡ ∂h x ∂p x ⎤ ⎡ λ ⎤
⎢ 2                          ⎥⎢           ⎥ ⎢ ⎥
⎢ ∂ u ∂x∂y ∂ u ∂y      − p y ⎥ ⎢∂h y ∂p x ⎥ = ⎢ 0 ⎥
2     2

⎢ −p            − py    0 ⎥ ⎢ ∂λ ∂p x ⎥ ⎢ h x ⎥
⎢
⎣      x                     ⎥⎢
⎦⎣           ⎥ ⎣ ⎦
⎦

Similarly, differentiating the above identities with respect to py and m gives

7
Jeffrey T. LaFrance                                                                                   8

⎡ ∂ 2u ∂x 2 ∂ 2u ∂x∂y − p x ⎤ ⎡ ∂h x ∂p y ⎤ ⎡ 0 ⎤
⎢ 2                          ⎥⎢           ⎥ ⎢ ⎥
⎢ ∂ u ∂x∂y ∂ u ∂y
2     2
− p y ⎥ ⎢∂h y ∂p y ⎥ = ⎢ λ ⎥ ,
⎢ −p            − py    0 ⎥ ⎢ ∂λ ∂p y ⎥ ⎢ h y ⎥
⎢
⎣      x                     ⎥⎢
⎦⎣           ⎥ ⎣ ⎦
⎦

⎡ ∂ 2u ∂x 2 ∂ 2u ∂x∂y − p x ⎤ ⎡ ∂h x ∂m ⎤ ⎡ 0 ⎤
⎢ 2                          ⎥⎢         ⎥
⎢ ∂ u ∂x∂y ∂ u ∂y
2     2
− p y ⎥ ⎢∂h y ∂m ⎥ = ⎢ 0 ⎥ .
⎢ ⎥
⎢ −p            − py    0 ⎥ ⎢ ∂λ ∂m ⎥ ⎢ −1⎥
⎢
⎣      x                     ⎥⎢
⎦⎣         ⎥ ⎣ ⎦
⎦

The role of the implicit function theorem is clear from each of these sets of three (linear) equa-

tions in three unknowns. The rates of change in the choice variables ( x , y , λ ) in response to

changes in the parameters ( p x , p y , m ) exist uniquely for an arbitrary set of parameter changes iff

|H| ≠ 0, where H is the Hessian matrix of the Lagrangean with respect to ( x, y , λ ) ,

⎡ ∂2L      ∂2L       ∂ 2 L ⎤ ⎡ ∂ 2u      ∂ 2u          ⎤
⎢ 2                        ⎥ ⎢                    − px ⎥
⎢ ∂x       ∂x∂y     ∂x∂λ ⎥ ⎢ ∂x 2        ∂x∂y          ⎥
⎢ ∂2L       ∂2L      ∂ 2 L ⎥ ⎢ ∂ 2u       ∂ 2u         ⎥
H =⎢                          ⎥=⎢                    − py ⎥ .
⎢ ∂y∂x      ∂y 2    ∂y∂λ ⎥ ⎢ ∂y∂x         ∂y 2         ⎥
⎢ ∂2L       ∂2L      ∂2L ⎥ ⎢                           ⎥
⎢                          ⎥ ⎢ − px       − py     0 ⎥
⎢ ∂λ∂x
⎣          ∂λ∂y      ∂λ 2 ⎥ ⎢
⎦ ⎣                         ⎥
⎦

Before we proceed to actually calculate the comparative statics results, it is worthwhile to write

the full system of partial derivatives in a single set of matrix equations as

⎡ ∂ 2u               ⎤ ⎡ ∂h         ∂h x ∂h x ⎤
x
∂ 2u
⎢ 2             − px ⎥ ⎢                      ⎥
∂x∂y            ∂p         ∂p y ∂m ⎥
⎢ ∂x                 ⎥⎢ x                       ⎡λ    0    0⎤
⎢ ∂ 2u   ∂ 2u        ⎥ ⎢ ∂h y       ∂h y ∂h y ⎥ ⎢            ⎥
⎢               − py ⎥ ⎢                      ⎥= 0    λ    0 ⎥.
⎢ ∂y∂x   ∂y 2        ⎥ ⎢ ∂p x       ∂p y ∂m ⎥ ⎢ x
⎢     hy   −1⎥
⎢                    ⎥ ⎢ ∂λ         ∂λ   ∂λ ⎥ ⎣ h            ⎦
⎢ − px   − py    0 ⎥⎢                         ⎥
⎢
⎣                    ⎦⎢
⎥ ⎣ ∂p x       ∂p y ∂m ⎥
⎦

8

⎡h x ⎤    ⎡ x⎤          ⎡ px ⎤
Now, letting h = ⎢ ⎥ , z = ⎢ ⎥ , and p = ⎢ ⎥ , this condenses to the system
⎢h ⎥      ⎣ y⎦          ⎣ py ⎦
y
⎣ ⎦

⎡ ∂ h ∂ h⎤
⎡ ∂ 2u         ⎤⎢
⎢          − p ⎥ ∂ p′ ∂m ⎥ ⎡λ I     0⎤
∂ z∂ z ′       ⎢        ⎥=            .
⎢              ⎥ ⎢ ∂λ ∂λ ⎥ ⎢ h′
⎣      −1⎥
⎦
⎢ − p′
⎣           0 ⎥⎢
⎦ ∂ p′ ∂m ⎥
⎣        ⎦

The advantage of this is that we can use the matrix algebra results in the section on linear algebra

to derive explicit expressions for the comparative statics of demand systems of any dimension, as

we will see from developments presented below.

However, at this point it is preferable to return to the simple case of two goods and derive the

results directly. How do we find, e.g., ∂hx/∂px? One obvious method is to apply Cramèr's rule to

the set of equations first obtained by differentiating the first-order conditions with respect to px,

∂2 u
λ            − px
∂x∂y                    ⎛ ∂ 2u        ∂ 2u ⎞ x
−λ p 2 + ⎜ p x 2 − p y      ⎟h
⎝ ∂y          ∂x∂y ⎠
y
∂h x   1         ∂ 2u
=   0              − py =
∂p x   H         ∂y 2                         H
hx    − py     0

Recall that |H| > 0 by the second-order sufficient condition (strict quasi-concavity of u), so that

the sign of ∂hx/∂px is the same as the sign of the numerator in the right-hand-side term. Also note

that the first term in the numerator, −λ p 2 , is unequivocally negative, but without more assump-
y

tions on the structure of the model, which are ad hoc, we cannot sign the second term,

⎛ ∂ 2u       ∂ 2u ⎞ x
⎜ px 2 − p y      ⎟h .
⎝   ∂y       ∂x∂y ⎠

This is because the first term is the substitution effect on the demand for x due to a change in the

9
Jeffrey T. LaFrance                                                                              10

price of x, while the second term is the income effect. To see this, apply Cramèr's rule to the

third set of matrix equations obtained by differentiating the FOC's with respect to m,

∂ 2u
0              − px
∂x∂y              ⎛ ∂ 2u       ∂ 2u ⎞
− ⎜ px 2 − p y      ⎟
∂h x   1          ∂ 2u              ⎝ ∂y         ∂x∂y ⎠
=   0                 − py =                       .
∂m     H          ∂ y2                     H
−1     − py         0

Therefore, substituting ∂hx/∂m into the expression for ∂hx/∂px gives

∂h x −λ p y ∂h x x
2
=      −    h ,
∂p x   H      ∂m

or equivalently,

∂h x ∂h x x −λ p y
2

+    h =       ≤ 0.
∂p x ∂m      H

The far left-hand-side term is called the Slutsky own-price substitution term for x. Ultimately,

we will see that it is precisely the change in the demand for x due to a change in the price of x as

income is compensated to maintain a constant level of utility – that is, as we move along an in-

difference curve. Using precisely the same procedures, we obtain the following set of results:

⎛ ∂ 2u       ∂ 2u ⎞ y
λ px p y + ⎜ px 2 − p y      ⎟h
∂h x              ⎝ ∂y         ∂x∂y ⎠
=                                  ;
∂p y                   H

⎛       ∂ 2u      ∂ 2u ⎞ x
λ px p y + ⎜ p y        − px      ⎟h
∂h y                ⎝       ∂x 2      ∂x∂y ⎠
=                                         ;
∂p x                         H

10

⎛ ∂ 2u        ∂ 2u ⎞ y
−λ p x + ⎜ p y 2 − p x
2
⎟h
∂h y            ⎝ ∂x          ∂x∂y ⎠
=                                 ;
∂p y                  H

⎛ ∂ 2u       ∂ 2u ⎞
− ⎜ p y 2 − px      ⎟
∂h y     ⎝ ∂x         ∂x∂y ⎠
=                       .
∂m              H

Combining all of these conditions, it follows that

∂h x ∂h x y λ p x p y ∂h y ∂h y x
+    h =         =     +   h ;
∂p y ∂m       H        ∂p x ∂m

∂h y ∂h y y −λ p x
2
+    h =       ≤ 0.
∂p y ∂m      H

If we arrange the Slutsky substitution terms into a 2×2 matrix, we have

⎡ ∂h x ∂h x x            ∂h x ∂h x y ⎤
⎢     +     h                 +     h ⎥
⎢ ∂p x ∂m                ∂p y ∂m ⎥ ⎛ λ ⎞ ⎡ − p 2 y        px p y ⎤
S=                                         =
⎢ ∂h y ∂h y              ∂h     ∂h y  ⎥ ⎜ H ⎟⎢p p         − px ⎥
2
⎥.
⎠⎢ x y
y     y
⎢     +     hx                +     h ⎥ ⎝    ⎣                   ⎦
⎢ ∂p x ∂m
⎣                        ∂p y ∂m ⎥    ⎦

Taking the determinant of S, we find that

2
⎛ λ ⎞ ⎡
(   )( − p ) − ( p p ) ⎤ = 0.
2
S =⎜   ⎟ ⎢ − py
2            2
⎥
⎝ H ⎠ ⎣                              ⎦
x     x   y

This means that S, known as the matrix of Slutsky substitution effects, is symmetric, negative

semi-definite, and singular.

Properties of the Marshallian ordinary demands:

1. h x , h y are positive valued;

11
Jeffrey T. LaFrance                                                                                          12

2. h x , h y are 0° homogeneous in ( p x , p y , m ) (px, py, m);

3. h x , h y satisfy the budget identity, p x h x (p x , p y , m) + p y h y (p x , p y , m ) ≡ m ; and

4. the Slutsky substitution matrix is symmetric, negative semi-definite, and satisfies

⎡ p x ⎤ ⎡ 0⎤
S ⎢ ⎥≡⎢ ⎥
⎣ p y ⎦ ⎣ 0⎦

∀ (p x , p y , m ) ∈   3
+   for which the demands are well-behaved.

However, it is important and useful to understand the following simple fact. In a two-good

world, all we need to know is the demand for one good, say x. The reason is that if we know

x = h x ( p x , p y , m ) , then we can use the budget identity to identify the demand for y,

⎛m⎞ ⎛p ⎞
y = h y ( p x , p y , m) ≡ ⎜ ⎟ − ⎜ x ⎟ h x ( p x , p y , m ).
⎜ py ⎟ ⎜ py ⎟
⎝ ⎠ ⎝ ⎠

Thus, if hx is homogeneous of degree zero in prices and income, then hy is as well. Second, defin-

ing hy in this way automatically ensures the adding up condition (budget identity).

⎡⎛ m ⎞ ⎛ p ⎞ ⎤
p x h x + p y h y ≡ p x h x + p y ⎢⎜ ⎟ − ⎜ x ⎟ h x ⎥ ≡ m.
⎢⎜ p y ⎟ ⎜ p y ⎟ ⎥
⎣⎝ ⎠ ⎝ ⎠ ⎦

Third, 0 ≤ h x ≤ m px iff 0 ≤ h y ≤ m p y ; that is, 0 ≤ h x iff h y = ( m − p x h x ) p y ≤ m p y , while

h x ≤ m /p x iff 0 ≤ ( m − p x h x ) p y = h y .

Finally, we now show that the Slutsky matrix S in the two-good case is completely identified by

sxx, and that s xx = ∂h x ∂p x + (∂h x ∂m )h x ≤ 0 is necessary and sufficient for S to be symmetric,

negative semi-definite. From the definition of hy we have

12

∂h y     ⎛ m ⎞ ⎛ px ⎞       ⎛ p ⎞ ∂h x
= − ⎜ 2 ⎟ + ⎜ 2 ⎟ hx − ⎜ x ⎟        ,
∂p y     ⎜ py ⎟ ⎜ py ⎟      ⎜ p y ⎟ ∂p y
⎝ ⎠ ⎝ ⎠            ⎝ ⎠

∂h y     ⎛ 1 ⎞    ⎛ p ⎞ ∂h x
= − ⎜ ⎟ hx − ⎜ x ⎟        ,
∂p x     ⎜ py ⎟   ⎜ p y ⎟ ∂p x
⎝ ⎠      ⎝ ⎠

∂h y ⎛ 1 ⎞ ⎛ p x ⎞ ∂h x
=⎜ ⎟−⎜ ⎟            .
⎜ ⎟ ⎜ ⎟
∂m ⎝ p y ⎠ ⎝ p y ⎠ ∂m

Combining the first and last of these with the definition for hy gives

∂h y ∂h y y   ⎛ m ⎞ ⎛ p x ⎞ x ⎛ p x ⎞ ∂h x
s yy   =     +    h = −⎜ 2 ⎟ + ⎜ 2 ⎟h − ⎜ ⎟
∂p y ∂m       ⎜ py ⎟ ⎜ py ⎟    ⎜ p y ⎟ ∂p y
⎝ ⎠ ⎝ ⎠          ⎝ ⎠

⎡ ⎛ 1 ⎞ ⎛ p ⎞ ∂h x ⎤ ⎡⎛ m ⎞ ⎛ p ⎞ x ⎤
+ ⎢⎜ ⎟ − ⎜ x ⎟                   − x h
⎜ p y ⎟ ⎜ p y ⎟ ∂m ⎥ ⎢⎜ p y ⎟ ⎜ p y ⎟ ⎥
⎢⎝ ⎠ ⎝ ⎠                ⎜ ⎟ ⎜ ⎟ ⎥
⎥ ⎢⎝ ⎠ ⎝ ⎠ ⎦
⎣                    ⎦⎣
⎛ p x ⎞ ⎛ ∂h   x
∂h x          ∂h x ⎞
= − ⎜ 2 ⎟ ⎜ py        +m       − px h x      ⎟.
⎜ p y ⎟ ⎜ ∂p y       ∂m            ∂m ⎠ ⎟
⎝ ⎠⎝

By Euler's theorem, hx is 0º homogeneous in (px,py,m) iff

∂h x      ∂h x    ∂h x                           ∂h x    ∂h x        ∂h x
px        + py      +m      ≡ 0, or equivalently, p y      +m      ≡ − px      .
∂p x      ∂p y    ∂m                             ∂p y    ∂m          ∂p x

Substituting this into the above expression for syy gives

2                            2
⎛ p x ⎞ ⎛ ∂h x ∂h x x ⎞ ⎛ p x ⎞
s yy   ≡⎜ ⎟ ⎜         +    h ⎟ ≡ ⎜ ⎟ s xx
⎜ ⎟                      ⎜ ⎟
⎝ p y ⎠ ⎝ ∂p x ∂m ⎠ ⎝ p y ⎠

so that syy ≤ 0 iff sxx ≤ 0. Using the same type of reasoning, we have

13
Jeffrey T. LaFrance                                                                                   14

∂h y ∂h y x
s yx =       +    h
∂px ∂m

⎛ 1 ⎞    ⎛ p ⎞ ∂h x ⎡⎛ 1 ⎞ ⎛ px ⎞ ∂h x ⎤ x
= − ⎜ ⎟ hx − ⎜ x ⎟       +        −
⎜ p y ⎟ ∂p x ⎢⎜ p y ⎟ ⎜ p y ⎟ ∂m ⎥
h
⎜ py ⎟                   ⎜ ⎟ ⎜ ⎟
⎢⎝ ⎠ ⎝ ⎠           ⎥
⎝ ⎠      ⎝ ⎠           ⎣                  ⎦

⎛ 1 ⎞ ⎛ ∂h x          ∂h x ⎞
= − ⎜ ⎟ ⎜ px     + px h x      ⎟
⎜ py ⎟
⎝ ⎠ ⎝ ∂p x            ∂m ⎠

⎛ 1 ⎞ ⎡⎛     ∂h x    ∂h x ⎞          ∂h x ⎤
= − ⎜ ⎟ ⎢⎜ − p y      −m      ⎟ + px h x      ⎥
⎜ p y ⎟ ⎢⎜   ∂p y    ∂m ⎟            ∂m ⎦ ⎥
⎝ ⎠ ⎣⎝                    ⎠

⎛ 1 ⎞⎡    ∂h x            x ∂h
x⎤
= − ⎜ ⎟ ⎢− py
⎜ py ⎟ ⎢  ∂p y
(
− m − px h
∂m ⎥
⎥   )
⎝ ⎠⎣                            ⎦
⎛ 1 ⎞⎛     ∂h x         ∂h x ⎞
= − ⎜ ⎟ ⎜ − py      − pyh y      ⎟
⎜ py ⎟ ⎜   ∂p y         ∂m ⎟
⎝ ⎠⎝                         ⎠

⎛ ∂h x ∂h x y ⎞
=⎜     +    h =s
⎜ ∂p y ∂m ⎟ xy .
⎟
⎝             ⎠

Thus, zero degree homogeneity of hx and the budget identity to define hy imply that symmetry is

automatically satisfied and that syy ≤ 0 iff sxx ≤ 0. It follows from the second line in the above de-

velopment of symmetry that

⎛p ⎞
s yx = s xy = − ⎜ x ⎟ s xx .
⎜ py ⎟
⎝ ⎠

Therefore, we have

⎛ p2 ⎞       ⎛ p2 ⎞
p x s xx + p y s xy ≡ 0 and p x s yx + p y s yy ≡ − ⎜ x ⎟ s xx + ⎜ x ⎟ s xx ≡ 0 ,
⎜ py ⎟       ⎜ py ⎟
⎝ ⎠          ⎝ ⎠

so that S has rank one, and is (identically) singular with null space defined by any vector in the

direction of the market prices (px,py).

14

Summarizing, in a two-good world, the demand for x satisfies

1. 0 ≤ hx ≤ m/px,

2. hx is 0 ° homogeneous in (px, py, m), and

3. ∂h x /∂p x +(∂h x /∂m )h x ≤ 0

iff the pair of demands for (x,y) satisfies all of the properties of ordinary Marshallian

demand functions.

******************************************************************************

A Digression: Utility Maximization and Comparative Statics: the n–good Case

⎡ x1 ⎤
⎢x ⎥
Let:    x = ⎢ 2⎥∈      n
+   be the (n×1) vector of consumption goods;
⎢ ⎥
⎢ ⎥
⎣ xn ⎦

⎡ p1 ⎤
⎢p ⎥
p = ⎢ 2⎥∈      n
++   the (n×1) vector of prices for x;
⎢ ⎥
⎢ ⎥
⎣ pn ⎦

m > 0 the consumer’s income level; and

u(x) the consumer’s utility function.

The consumer’s problem is to maximize u(x) subject to x ≥ 0 and p’x ≤ m. Let the Lagrangean

function for this problem be given by

L = u( x ) + λ (m − p′x ) .

We will assume throughout that an interior constrained maximum is obtained, and that u(x) is

15
Jeffrey T. LaFrance                                                                                         16

twice continuously differentiable, strictly increasing, and strictly quasi-concave in x. Then, in

matrix notation, the first-order conditions are as follows:

∂ L / ∂ x = ∂ u( x ) / ∂ x − λ p = 0 ;

∂ L / ∂λ = m − p′x = 0 .

Given the twice continuous differentiability and strict quasi-concavity of u(x), these n+1 equa-

tions in n+1 unknowns have a unique solution that is continuously differentiable in (p,m) for all

values of (p,m) such that x*              0 . We will denote this solution by

x* = h( p, m) and λ *=λ (p, m ) .

When we substitute the optimal solutions for x and λ into the first-order conditions, the n+1

equations in (1) and (2) become identities that can be differentiated with respect to p and m. In

matrix notation, the result of this can be written as

⎡ ∂h( p, m )     ∂h( p, m) ⎤
⎡ ∂ 2u( h( p, m))        ⎤⎢
⎢                     − p⎥      ∂p′            ∂m ⎥ ⎡λ ( p, m ) I           0n ⎤
∂x∂x′                ⎢                            ⎥=                       .
⎢                        ⎥ ⎢ ∂λ ( p, m )    ∂λ ( p, m ) ⎥ ⎢ h( p, m )′
⎣                −1⎥⎦
⎢
⎣       − p′           0 ⎥⎢
⎦                              ⎥
⎣ ∂p′               ∂m ⎦

The determinant of the far left-hand-side matrix can not be zero by the strict quasi-concavity of

the utility function, so we can invert it and rewrite (3) in the form

⎡ ∂h( p, m )      ∂h( p, m ) ⎤                               −1
⎢ ∂p′                           ⎡ ∂ 2u( h( p, m ))          ⎤
∂m ⎥ ⎢                             − p ⎥ ⎡ λ ( p, m ) I n    0n ⎤
⎢                            ⎥=       ∂x∂x′                                            .
⎢ ∂λ ( p, m)      ∂λ ( p, m) ⎥ ⎢                            ⎥ ⎢ h( p, m)′         −1⎥
⎥ ⎢        − p′             0 ⎥ ⎣                       ⎦
⎢
⎣ ∂p′                ∂m ⎦ ⎣                                 ⎦

Now, since u is strictly quasi-concave, the rank of ∂ 2u / ∂ x∂ x ′ is at least n–1. The reasoning be-

hind this claim is the fact that u is strictly quasi-concave iff

16

∂ 2u( x )                                   ∂u( x )
z′             z < 0 for all z ≠ 0 such that z ′         = 0.
∂ x∂ x′                                      ∂x

Note that there is only one restriction on the choice of z, and that this restriction is a linear func-

tion of z. Since ∂ u( x ) / ∂ x ≠ 0 , the coefficients for this linear restriction do not depend on the

vector z and span exactly one dimension. Therefore, there are n–1 degrees of freedom in the

choice of z. Also, for the quadratic form to be strictly less than zero, it must be the case that

∂ 2u( x )
z ≠0.
∂ x∂ x′

This is because the vector product of any vector z with 0 must be zero, z′0 = 0 .

Therefore, the Hessian matrix for a strictly quasi-concave utility function must span at least n–1

dimensions. It could in fact span n dimensions, and if u(x) were in fact strictly concave, then this

would certainly be the case. However, the Hessian also could be of full rank without concavity;

for example, u( x1 , x2 ) = x1 x2 has a full rank 2 Hessian but is not concave. It is, however, strictly

quasi-concave throughout the positive quadrant for ( x1 , x2 ) . We will assume that the Hessian

matrix for u(x) is nonsingular, that is, has rank equal to n, or equivalently, spans n dimensions. It

can be shown that this is true for almost all twice continuously differentiable, strictly quasi-

concave utility functions, so there is not much loss in generality in assuming this.

The assumption that the Hessian matrix for u(x) is nonsingular makes it easier for us to derive

the comparative statics results for the demands for x and the marginal utility of money λ. This is

accomplished by using the partitioned nature of the Hessian matrix for the Lagrangean to con-

struct its inverse:

17
Jeffrey T. LaFrance                                                                                             18

−1
⎡ ∂ 2u     ⎤
⎢       − p⎥                ⎡ A b⎤
=⎢
⎢ ∂x∂x′    ⎥                       ⎥,
⎣ b′ c ⎦
⎢ − p′
⎣        0 ⎥
⎦

where A is an n×n symmetric matrix. In (5) we have made use of the fact that the inverse of a

symmetric matrix is symmetric. From the definition of an inverse, we obtain the following set of

equations that define the unknowns, A, b, and c:

∂ 2u
A − pb′ = I ;
∂ x∂ x′

∂ 2u
b − pc = 0 ;
∂ x∂ x′

− p′A + 0b′ = 0′ ;

− p′b + 0c = 1.

−1
⎡ ∂ 2u ⎤
Now, because ⎢         ⎥        exists, we can use it to solve for A, b, and c as follows. First, we find
⎣ ∂ x∂ x ′⎦

that

−1
⎛ ∂ 2u ⎞
A=⎜          ⎟         ( I + pb′) ,
⎝ ∂ x∂ x ′ ⎠

from the first expression. Then, from the third expression, if we pre-multiply A by p' then we

find that

−1                                −1                  −1
⎛ ∂ 2u ⎞                           ⎛ ∂ 2u ⎞          ⎛ ∂ 2u ⎞
0′ = p′A = p′ ⎜         ⎟        ( I + pb′) = p′ ⎜          ⎟ + p′ ⎜          ⎟        pb′ ,
⎝ ∂ x∂ x′ ⎠                        ⎝ ∂ x∂ x ′ ⎠      ⎝ ∂ x∂ x ′ ⎠

which in turn can be solved for b’ (or equivalently, for b) to get

18

−1
⎡ ⎛ ∂ 2u ⎞ −1 ⎤              ⎛ ∂ 2u ⎞
−1

b′ = − ⎢ p′ ⎜      ⎟ p⎥          p′ ⎜         ⎟ ,
⎢ ⎝ ∂ x∂ x′ ⎠  ⎥             ⎝ ∂ x∂ x′ ⎠
⎣              ⎦

−1
⎡ ⎛ ∂ 2u ⎞ −1 ⎤ ⎛ ∂ 2u ⎞ −1
or equivalently,               b = − ⎢ p′ ⎜       ⎟ p⎥ ⎜         ⎟ p.
⎢ ⎝ ∂ x∂ x ′ ⎠  ⎥ ⎝ ∂ x∂ x′ ⎠
⎣               ⎦

Next, substituting this solution for b' into the expression for A as a function of b' just above, we

obtain

−1                   −1
⎛ ∂ 2u ⎞       ⎡ ⎛ ∂ 2u ⎞ −1 ⎤ ⎛ ∂ 2u ⎞ −1      ⎛ ∂ 2u ⎞
−1

A=⎜         ⎟ − ⎢ p′ ⎜       ⎟ p⎥ ⎜         ⎟ pp′ ⎜         ⎟ .
⎝ ∂ x∂ x′ ⎠    ⎢ ⎝ ∂ x∂ x′ ⎠
⎣
⎥ ⎝ ∂ x∂ x′ ⎠
⎦                 ⎝ ∂ x∂ x′ ⎠

Note that throughout our derivation for A and b we have made use of the fact that

−1
⎡ ⎛ ∂ 2u ⎞ −1      ⎤
⎢ p′ ⎜      ⎟     p⎥
⎢ ⎝ ∂ x∂ x′ ⎠      ⎥
⎣                  ⎦

is a scalar (i.e., a 1×1 matrix) whose place can be shifted around relative to any matrix or vector

that it is a multiplicative factor for (recall the commutative law of scalar multiplication for linear

algebra).

The only unknown term remaining to be found is c. From the last equation in our system of

equations for calculating this matrix inverse, we find that p′b = −1 , while from the second set of

relationships we have that

−1
⎛ ∂ 2u ⎞
b=⎜          ⎟        pc.
⎝ ∂ x∂ x ′ ⎠

Therefore, pre-multiplying the latter expression by p′ , we obtain

19
Jeffrey T. LaFrance                                                                                                20

−1
⎛ ∂ 2u ⎞
−1 = p′b = p′ ⎜          ⎟        pc .
⎝ ∂ x∂ x ′ ⎠

As long as the quadratic form on the right-hand-side is not zero (which is guaranteed by a non-

singular Hessian matrix), this can be solved for the last unknown, c, to get

−1
⎡ ⎛ ∂ 2 u ⎞ −1         ⎤
c = − ⎢ p′ ⎜    ⎟           p⎥ .
⎢ ⎝ ∂x∂x′ ⎠            ⎥
⎣                      ⎦

This completes our derivation, and the final solution is given by

⎡⎡                ⎛ ∂ 2u ⎞
−1
⎛ ∂ 2u ⎞ ⎤
−1                                        ⎤
⎢⎢             −1 ⎜          ⎟ pp    ′⎜           ⎟    ⎥                        −1            ⎥
⎢ ⎢⎛ ∂ 2u ⎞         ∂ x∂ x ′ ⎠           ∂ x∂ x ′ ⎠ ⎥      ⎡ ⎛ ∂ 2u ⎞ −1 ⎤ ⎛ ∂ 2u ⎞ −1 ⎥
⎝                   ⎝
⎡ ∂ 2u        ⎤
−1   ⎢ ⎢⎜ ∂ x∂ x′ ⎟ −                          −1           ⎥ − ⎢ p′ ⎜ ∂ x∂ x′ ⎟ p ⎥ ⎜ ∂ x∂ x′ ⎟ p ⎥
⎢⎢ ⎝         ⎠               ⎛ ∂ u ⎞
2
⎥   ⎢ ⎝            ⎠   ⎥ ⎝         ⎠   ⎥
⎢          − p⎥                               p′ ⎜                              ⎣                  ⎦
∂ x∂ x ′         = ⎢⎢                                      ⎟ p            ⎥                                      ⎥.
⎢             ⎥      ⎢⎣                           ⎝ ∂ x∂ x ′ ⎠              ⎦                                      ⎥
⎢             ⎥
⎣ − p′      0 ⎦      ⎢                             −1                                                      −1      ⎥
⎢       ⎡ ⎛ ∂ 2u ⎞ −1 ⎤             ⎛ ∂ u ⎞2    −1
⎡ ⎛ ∂ 2u ⎞ −1 ⎤           ⎥
⎢ − ⎢ p′ ⎜           ⎟ p ⎥ p′ ⎜               ⎟                  − ⎢ p′ ⎜        ⎟ p⎥         ⎥
⎢       ⎢ ⎝ ∂ x∂ x ′ ⎠     ⎥        ⎝ ∂ x∂ x′ ⎠                     ⎢ ⎝ ∂ x∂ x ′ ⎠  ⎥         ⎥
⎣       ⎣                  ⎦                                        ⎣               ⎦         ⎦

We are now in a position to complete our derivations of the comparative statics results in the

general n–good case. This is accomplished by recalling the original expression, making the ap-

propriate substitutions, and carrying out a bit of matrix algebra as follows:

⎡ ∂h( p, m )    ∂h( p, m ) ⎤                                    −1
⎢ ∂p′                          ⎡ ∂ 2u( h( p, m ))            ⎤
∂m ⎥ ⎢                                − p ⎥ ⎡ λ ( p, m ) I   0⎤
⎢                           ⎥=       ∂ x∂ x′
⎢ ∂λ ( p, m )   ∂λ ( p, m ) ⎥ ⎢                              ⎥ ⎢ h( p, m)′      −1⎥
⎢       − p′               0 ⎥ ⎣                   ⎦
⎢ ∂p′
⎣                  ∂m ⎥ ⎣   ⎦
⎦

⎡ A b ⎤ ⎡λ ( p, m) I       0 ⎤ ⎡λ ( p, m) A + bh( p, m)′ − b ⎤
=⎢      ⎥⎢                     =                                 .
⎣ b′ c ⎦ ⎣ h( p, m)′       −1⎥ ⎢ λ ( p, m)b′ + ch( p, m)′ − c ⎥
⎦ ⎣                              ⎦

Expanding the corresponding block of each set of terms gives

20

−1
⎧            −1
⎡ ⎛ ∂ 2u ⎞ −1 ⎤ ⎛ ∂ 2u ⎞ −1                 −1 ⎫
∂h( p, m )              ⎪⎛ ∂ 2u ⎞                                        ⎛ ∂ 2u ⎞ ⎪
= λ ( p, m ) ⎨ ⎜        ⎟ − ⎢ p′ ⎜       ⎟ p⎥ ⎜         ⎟ pp′ ⎜         ⎟ ⎬
∂p′                   ⎪⎝ ∂ x∂ x′ ⎠    ⎢ ⎝ ∂ x∂ x′ ⎠  ⎥ ⎝ ∂ x∂ x′ ⎠     ⎝ ∂ x∂ x′ ⎠ ⎪
⎩               ⎣              ⎦                               ⎭
−1
⎡ ⎛ ∂ 2u ⎞ −1 ⎤ ⎛ ∂ 2u ⎞ −1
− ⎢ p′ ⎜       ⎟ p⎥ ⎜         ⎟ ph( p, m)′,
⎢ ⎝ ∂ x∂ x ′ ⎠  ⎥ ⎝ ∂ x∂ x′ ⎠
⎣               ⎦

−1      −1
∂h( p, m ) ⎡ ⎛ ∂ 2u ⎞        ⎤ ⎛ ∂ 2u ⎞ −1
= ⎢ p′ ⎜      ⎟ p⎥ ⎜           ⎟ p,
∂m        ⎢ ⎝ ∂ x∂ x′ ⎠    ⎥ ⎝ ∂ x∂ x′ ⎠
⎣                ⎦

−1
∂λ ( p, m)     ⎡ ⎛ ∂ 2u ⎞ −1 ⎤         ⎡                          ⎛ ∂ 2u ⎞ ⎤
−1

= − ⎢ p′ ⎜      ⎟ p⎥        ⎢ h( p, m)′ + λ ( p, m) p′ ⎜         ⎟ ⎥,
∂p′         ⎢ ⎝ ∂ x∂ x′ ⎠  ⎥        ⎢                          ⎝ ∂ x∂ x′ ⎠ ⎥
⎣              ⎦        ⎣                                        ⎦

−1       −1
∂λ ( p, m) ⎡ ⎛ ∂ 2u ⎞         ⎤
= ⎢ p′ ⎜      ⎟    p⎥ .
∂m       ⎢ ⎝ ∂ x∂ x′ ⎠     ⎥
⎣                 ⎦

Note, in particular, that

∂h( p, m) ∂h( p, m)
S≡            +          h( p, m)′ =
∂p′       ∂m
−1
⎧             −1
⎡ ⎛ ∂ 2u ⎞ −1 ⎤ ⎛ ∂ 2u ⎞ −1                 −1 ⎫
⎪⎛ ∂ 2u ⎞                                         ⎛ ∂ 2u ⎞ ⎪
λ ( p, m ) ⎨⎜          ⎟ − ⎢ p′ ⎜       ⎟ p⎥ ⎜         ⎟ pp′ ⎜         ⎟ ⎬,
⎪⎝ ∂ x∂ x ′ ⎠    ⎢ ⎝ ∂ x∂ x′ ⎠
⎣
⎥ ⎝ ∂ x∂ x′ ⎠
⎦                 ⎝ ∂ x∂ x′ ⎠ ⎪
⎩                                                               ⎭

which is a symmetric (n×n) matrix. Moreover, post-multiplying S by p gives

21
Jeffrey T. LaFrance                                                                                               22

−1
⎧             −1
⎡ ⎛ ∂ 2u ⎞ −1 ⎤ ⎛ ∂ 2u ⎞ −1                 −1 ⎫
⎪⎛ ∂ 2u ⎞                                         ⎛ ∂ 2u ⎞ ⎪
Sp = λ ( p, m) ⎨⎜          ⎟ − ⎢ p′ ⎜       ⎟ p⎥ ⎜         ⎟ pp′ ⎜         ⎟ ⎬p
⎪ ⎝ ∂ x∂ x′ ⎠    ⎢ ⎝ ∂ x∂ x′ ⎠
⎣
⎥ ⎝ ∂ x∂ x′ ⎠
⎦                 ⎝ ∂ x∂ x′ ⎠ ⎪
⎩                                                               ⎭
−1
⎧            −1
⎡ ⎛ ∂ 2u ⎞ −1 ⎤ ⎛ ∂ 2u ⎞ −1                  −1 ⎫
⎪ ⎛ ∂ 2u ⎞                                        ⎛ ∂ 2u ⎞       ⎪
= λ ( p, m ) ⎨ ⎜        ⎟ p − ⎢ p′ ⎜      ⎟ p⎥ ⎜         ⎟ pp′ ⎜         ⎟ p⎬
⎪⎝ ∂ x∂ x′ ⎠     ⎢ ⎝ ∂ x∂ x′ ⎠
⎣
⎥ ⎝ ∂ x∂ x′ ⎠
⎦                 ⎝ ∂ x∂ x′ ⎠    ⎪
⎩                                                                ⎭
−1
⎧            −1
⎡ ⎛ ∂ 2u ⎞ −1 ⎤ ⎡ ⎛ ∂ 2u ⎞ −1 ⎤ ⎛ ∂ 2u ⎞ −1 ⎫
⎪⎛ ∂ 2u ⎞                                                             ⎪
= λ ( p, m) ⎨⎜         ⎟ p − ⎢ p′ ⎜         ⎟ p ⎥ ⎢ p′ ⎜         ⎟ p⎥ ⎜       ⎟ p ⎬ ≡ 0.
⎪⎝ ∂ x∂ x′ ⎠     ⎢
⎣ ⎝
∂ x∂ x′ ⎠   ⎥ ⎢
⎦ ⎣ ⎝
∂ x∂ x′ ⎠  ⎥ ∂ x∂ x′ ⎠
⎦⎝            ⎪
⎩                                                                     ⎭

It follows that S has rank no greater than n–1. But we can show that in fact the rank of S is equal

to n–1. This is accomplished simply by taking the quadratic form, z’Sz, for any z ≠ αp for some

constant α. The result we obtain is

−1
⎧              −1
⎡ ⎛ ∂ 2u ⎞ −1     ⎤ ⎛ ∂ 2u ⎞ −1                   −1 ⎫
⎪⎛ ∂ 2u ⎞                                               ⎛ ∂ 2u ⎞ ⎪
z ′Sz = λ ( p, m ) z ′ ⎨⎜           ⎟ − ⎢ p′ ⎜        ⎟   p⎥ ⎜          ⎟ pp ′ ⎜          ⎟ ⎬z
⎪ ⎝ ∂ x∂ x ′ ⎠    ⎢ ⎝ ∂ x∂ x ′ ⎠
⎣
⎥ ⎝ ∂ x∂ x ′ ⎠
⎦                   ⎝ ∂ x∂ x ′ ⎠ ⎪
⎩                                                                      ⎭
−1
⎧                  −1
⎡ ⎛ ∂ 2u ⎞ −1        ⎤                 −1               −1 ⎫
⎪ ⎛ ∂ u ⎞                                        ⎛ ∂ 2u ⎞          ⎛ ∂ 2u ⎞ ⎪
2
= λ ( p, m ) ⎨ z ′ ⎜          ⎟ z − ⎢ p′ ⎜       ⎟      p⎥ z′ ⎜          ⎟ pp ′ ⎜          ⎟ z⎬
⎪ ⎝     ∂ x∂ x ′ ⎠     ⎢ ⎝ ∂ x∂ x ′ ⎠       ⎥    ⎝ ∂ x∂ x ′ ⎠      ⎝ ∂ x∂ x ′ ⎠ ⎪
⎩                      ⎣                    ⎦                                     ⎭
−1
⎧              −1
⎡ ⎛ ∂ 2u ⎞ −1            ⎤            ⎡ ⎛ ∂ 2u ⎞ −1     ⎤
2
⎫
⎪ ⎛ ∂ u ⎞                                                                          ⎪
2
= λ ( p, m ) ⎨ z ′ ⎜      ⎟ z − ⎢ p′ ⎜       ⎟          p⎥            ⎢ z′ ⎜       ⎟   p⎥       ⎬
⎪ ⎝ ∂ x∂ x ′ ⎠     ⎢ ⎝ ∂ x∂ x ′ ⎠
⎣
⎥
⎦
⎢ ⎝ ∂ x∂ x ′ ⎠
⎣
⎥
⎦       ⎪
⎩                                                                                  ⎭

⎧                 −1                  −1
⎡ ⎛ ∂ 2u ⎞ −1               ⎤
2
⎫
⎪ ⎛ ∂ 2u ⎞               ⎛ ∂ 2u ⎞                                             ⎪
⎨ p′ ⎜          ⎟ p × z′ ⎜          ⎟ z − ⎢ z′ ⎜       ⎟             p⎥       ⎬
⎪    ⎝ ∂ x∂ x ′ ⎠        ⎝ ∂ x∂ x ′ ⎠     ⎢ ⎝ ∂ x∂ x ′ ⎠
⎣
⎥
⎦       ⎪
= λ ( p, m ) ⎩                                         −1
⎭.
⎛ ∂ 2u ⎞
p′ ⎜          ⎟        p
⎝ ∂ x∂ x ′ ⎠

By the Cauchy-Schwartz inequality in n dimensions, the last term in braces, {⋅}, will not vanish

so long as z ≠ αp. Strict monotonicity of u(x) implies that λ(p,m) > 0. Therefore, z’Sz ≠ 0, for all

22

z ≠ αp, and the rank of S must be at least n–1. Since it can be no greater than n–1, its rank is ex-

actly n–1.

When we consider the dual problem of minimizing the expenditure necessary to obtain a given

level of utility, we find that the matrix S is the Hessian matrix for the expenditure function. Ex-

penditure functions are increasing, 1° homogeneous and concave in prices. This implies that S is

negative semi-definite.

However, it is useful and informative to prove this result directly by making use of some

straightforward results from linear algebra. We will do so using condensed subscript notation for

the vectors and matrices of first- and second-order derivatives of u with respect to x. Given

−1          −1          −1       −1
S = λ ⎡ u xx′ − ( p′u xx′ p) −1 u xx′ pp′u xx′ ⎤ ,
⎣                                        ⎦

we can write the singularity of S in the direction of prices, p, as Sp ≡ 0. The truth of this is easily

verified by post-multiplying S by p above. This means that if we post-multiply the price vector p

by any conformable matrix, say A, then we obtain a matrix of zeroes, i.e., SpA ≡ 0. It is conven-

−1             −1
ient to set A equal to the 1×n matrix, ( p′u xx′ p ) −1 p′u xx′ , so that we have the identity,

⎡ −1           −1          −1
S ≡ λ ⎣u xx′ − ( p′u xx′ p) −1 u xx′ pp′u xx′ ⎦ × ⎣ I − ( p′u xx′ p) −1 pp′u xx′ ⎦ .
−1
⎤ ⎡             −1             −1
⎤

In other words, we simply added an n×n matrix of zeroes to S to obtain the above expression.

We next exploit the fact that we can always write the identity matrix as an arbitrary n×n nonsin-

−1
gular matrix times its inverse, in this case, I = u xx′u xx′ . The identity matrix times any conform-

able matrix does not change the latter matrix, so we can introduce this form of I into the middle

of the right-hand-side expression for S, to get

23
Jeffrey T. LaFrance                                                                                                  24

⎡ −1           −1          −1       −1           −1
S ≡ λ ⎣u xx′ − ( p′u xx′ p) −1 u xx′ pp′u xx′ ⎦ u xx′u xx′ ⎣ I n − ( p′u xx′ p) −1 pp′u xx′ ⎦
⎤            ⎡             −1             −1
⎤

⎡ −1           −1          −1       −1
⎡ −1
≡ λ ⎣u xx′ − ( p′u xx′ p) −1 u xx′ pp′u xx′ ⎦ u xx′ ⎣ u xx′ − ( p′u xx′ p) −1 u xx′ pp′u xx′ ⎦ ,
⎤                       −1          −1       −1
⎤

where the second line has been obtained by the distributive law of matrix multiplication and ad-

dition, and we have exploited the associative law of scalar multiplication to move the scalar

−1
term, ( p′u xx′ p ) −1 , around at our convenience.

−1          −1          −1       −1
The matrix ⎡ u xx′ − ( p′u xx′ p) −1 u xx′ pp′u xx′ ⎤ is clearly symmetric. We will prove that it has rank
⎣                                        ⎦

−1          −1          −1       −1
n−1. It also satisfies ⎡ u xx′ − ( p′u xx′ p) −1 u xx′ pp′u xx′ ⎤ p ≡ 0 . Therefore, it spans n−1 dimensions,
⎣                                        ⎦

with the one dimension of singularity the line from the origin through the point p.

By the Pythagorean theorem, any z ∈                 n
can be written as the sum of two parts: one that is par-

allel to p, say z1 = α p , and one that is orthogonal to p, say z2 p = 0 , so that z = z1 + z2 . There-
′

fore, suppose that z = αp, for some nonzero real number, α. Clearly, then, we have

z′Sz = α 2 p′Sp = 0 .

Now suppose that z ⊥ p , that is, z′p = 0 , and let

−1          −1          −1       −1
y = ⎡u xx′ − ( p′u xx′ p) −1 u xx′ pp′u xx′ ⎤ z .
⎣                                       ⎦

We will show that y′p = 0 as well, so that by the first-order conditions for an interior constrained

maximum, we also have y′u x = 0 since u x = λ p .

By the strict quasi-concavity of u(x), it follows that z′Sz = y′u xx′ y < 0 ∀ z ≠ 0: z′p = 0 .

Hence, we complete the proof of the negative semi-definiteness and rank n−1 of the Slutsky ma-

trix S by appealing to the definition of y to obtain

24

−1          −1          −1       −1
y′p = z ′ ⎡u xx′ − ( p′u xx′ p) −1 u xx′ pp′u xx′ ⎤ p
⎣                                       ⎦
−1            −1             −1         −1
= z ′u xx′ p − ( p′u xx′ p) −1 z ′u xx′ p( p′u xx′ p)              Q.E.D.
−1           −1
= z ′u xx′ p − z ′u xx′ p = 0.

In other words, any strictly monotone and strictly quasi-concave utility function has a Slutsky

substitution matrix at any interior solution for the quantities demanded that is symmetric, nega-

tive semi-definite, and has rank n–1.

Remark: This important property extends to the case of, say, n1 < n goods consumed in positive

quantities and n2 = n – n1 ≥ 1 goods not consumed (i.e., at a Kuhn-Tucker corner solution) in the

obvious way. The demands for the goods consumed in positive quantity only depend on the

prices of those goods, and the Slutsky substitution matrix for that subset of goods is symmetric,

negative semi-definite, and has rank n1-1. This issue will be analyzed in detail later in the course.

******************************************************************************

The Expenditure Function

For the purposes of connecting consumer choice theory to the theory of production, developing

true cost of living indices and measures of the economic welfare effects on consumers of

changes in prices or income, and a more comprehensive understanding of the dual structure of

demand functions, we now derive and analyze the consumer expenditure function. The expendi-

ture function is akin to the cost function in production economics, and is obtained by interchang-

ing the roles of the utility function and budget constraint:

{
e( p x , p y , u ) ≡ min p x x + p y y : ( x, y ) ∈     2
+, u ( x ,        }
y) ≥ u .

Thus, the expenditure function is defined as the minimum cost necessary to obtain a fixed level

25
Jeffrey T. LaFrance                                                                                  26

of utility, u, with market prices (px, py).

The Lagrangean for this problem is

L = p x x + p y y + μ (u − u( x, y )) ,

and again we look for a saddle point of L, but in this case a relative minimum with respect to

(x,y) and a relative maximum with respect to the Lagrange multiplier, μ. The Kuhn-Tucker first-

order conditions are as follows:

∂L / ∂x = p x − μ∂u / ∂x ≤ 0, x ≥ 0, x × ∂L / ∂x = 0;

∂L / ∂y = p y − μ∂u / ∂y ≤ 0, y ≥ 0, y × ∂L / ∂y = 0; ;

∂L / ∂μ = u( x, y ) − u ≥ 0, μ ≥ 0, μ × ∂L / ∂μ = 0.

Let us concentrate on interior solutions, where ( x , y , μ ) >> (0,0,0) . Then we can take the ratio of

the first two Kuhn-Tucker conditions to eliminate μ to obtain

p x ∂u( x, y ) / ∂x
=                .
p y ∂u( x, y ) / ∂y

This condition – that the marginal rate of substitution of x for y be equated to the ratio of market

prices – is the same as we obtain for the utility maximization problem. However, in this problem

the budget constraint is replaced by the utility constraint,

u ( x, y ) = u .

The parameters in this problem are the market prices, (px, py), and the level of utility, u. Hence,

the optimal choice functions, which are known as Hicksian compensated demands,

x* ≡ g x ( p x , p y , u ),

y* ≡ g y ( p x , p y , u ),

26

are functions of prices and the utility level rather than prices and income. The optimal value of

the Lagrange multiplier is known as the marginal cost of utility,

μ* ≡ μ ( px , p y , u ) .

The expenditure function is obtained by substituting the optimal Hicksian demands for x and y

in the objective function,

e( p x , p y , u ) ≡ p x g x ( p x , p y , u ) + p y g y ( p x , p y , u ) .

We can apply the envelope theorem to the expenditure minimization problem, just as we do for

the utility maximization problem, and the results obtained are:

∂e( px , p y , u )
≡ g x ( p x , p y , u ),
∂p x

∂e( px , p y , u )
≡ g y ( p x , p y , u ), ,
∂p y

∂e( p x , p y , u )
≡ μ ( p x , p y , u ).
∂u

The first two identities are known as Hotelling’s/Shephard’s Lemma. Note that if we combine

the results of this lemma due to the envelope theorem with the definition of the expenditure func-

tion, we have

∂ e( p x , p y , u )      ∂ e( p x , p y , u )
e( p x , p y , u ) ≡                        px +                      py .
∂ px                      ∂ py

Therefore, by Euler’s theorem and its converse, the expenditure function is homogeneous of

degree one in prices. Moreover, since quantities demanded must be positive valued, the expen-

diture function is increasing in prices. Similarly, since shadow prices (Lagrange multipliers) for

inequality constraints are positive valued, the expenditure function is increasing in the utility

27
Jeffrey T. LaFrance                                                                              28

index, u. In addition, by the derivative property for homogeneous functions (i.e., if a function

that is homogeneous of degree k in (some of) its arguments, then its partial derivatives with re-

spect to each of those arguments will be homogeneous of degree k-1), the Hicksian compen-

sated demands are homogeneous of degree zero in prices.

The sufficient second-order condition for a constrained minimum for the expenditure minimiza-

tion problem is that the determinant of the Hessian matrix of the Lagrangean function with re-

spect to ( x, y , μ ) is strictly negative,

∂ 2u          ∂ 2u       ∂u
−μ          −μ            −
∂x 2          ∂x∂y       ∂x
∂ 2u         ∂ 2u         ∂u
C = −μ            −μ           −
∂x∂y         ∂x 2         ∂y
∂u           ∂u
−            −              0
∂x           ∂y

⎡ ⎛ ∂ 2u ⎞ ⎛ ∂u ⎞ ⎛ ∂u ⎞ ⎛ ∂ 2u ⎞ ⎛ ∂u ⎞2 ⎛ ∂ 2u ⎞ ⎛ ∂u ⎞ 2 ⎤
= − μ ⎢2 ⎜     ⎟⎜ ⎟⎜ ⎟ − ⎜ 2 ⎟⎜ ⎟ − ⎜ 2 ⎟⎜ ⎟ ⎥
⎢ ⎝ ∂x∂y ⎠ ⎝ ∂x ⎠ ⎝ ∂y ⎠ ⎝ ∂x ⎠ ⎝ ∂y ⎠ ⎝ ∂y ⎠ ⎝ ∂x ⎠ ⎥
⎣                                                           ⎦
= − μ H < 0,

iff the utility function is strictly quasi-concave in (x,y), where H is the bordered Hessian for the

utility function.

Comparative Statics

We conduct comparative statics analysis for the expenditure minimization problem in the same

manner as for the utility maximization problem, by applying Cramèr’s rule to the following lin-

ear system of equations obtained by taking the total differential of the first-order Kuhn-Tucker

conditions:

28

⎡    ∂ 2u           ∂ 2u          ∂u ⎤
⎢ −μ 2         −μ                −   ⎥
⎢    ∂x             ∂x∂y          ∂x ⎥
⎡ dx ⎤ ⎡ − dp x ⎤
⎢    ∂ 2u         ∂ 2u            ∂u ⎥ ⎢ ⎥ ⎢
⎢ −μ           −μ                − ⎥ dy = − dp y ⎥ .
⎢ ∂x∂y            ∂x 2            ∂y ⎥ ⎢ ⎥ ⎢           ⎥
⎢ d μ ⎥ ⎢ − du ⎥
⎢    ∂u           ∂u                 ⎥⎣ ⎦ ⎣            ⎦
⎢ −             −                 0 ⎥
⎢    ∂x           ∂y                 ⎥
⎣                                    ⎦

The effects of price changes on compensated quantities demanded are as follows:

∂ 2u              ∂u
−1 − μ                 −
∂x∂y              ∂x
∂g x ⎛ −1 ⎞                               ∂u − ( ∂u / ∂y )   − p2
2
∂ 2u
=⎜     ⎟ 0         −μ 2             −    =             = 3 y ;
∂p x ⎝ μ H ⎠              ∂x              ∂y     μH         μ H
∂u
0      −                    0
∂y

∂ 2u            ∂u
0       −μ             −
∂x∂y            ∂x
∂g x ⎛ −1 ⎞               ∂ 2u            ∂u ( ∂u / ∂x ) ∂u / ∂y   p p
=⎜     ⎟ −1        −μ             −      =                   = 3x y ;
∂p y ⎝ μ H ⎠              ∂x 2            ∂y        μH            μ H
∂u
0        −                 0
∂y

∂ 2u                       ∂u
−             −1 −
∂x 2                       ∂x
∂g y ⎛ −1 ⎞     ∂ 2u                      ∂u ( ∂u / ∂x ) ∂u / ∂y   p p
=⎜    ⎟ −μ                0       −      =                   = 3x y ;
∂px ⎝ μ H ⎠    ∂x∂y                       ∂y        μH            μ H
∂u
−                0           0
∂x

∂ 2u                         ∂u
− 2               0        −
∂x                           ∂x
⎛ −1 ⎞                         ∂u − ( ∂u / ∂x )
2
∂g y
∂ 2u                                   − p2
=⎜    ⎟ −μ                 −1 −    =             = 3 x .
∂px ⎝ μ H ⎠     ∂x∂y                 ∂y     μH         μ H
∂u
−                 0            0
∂x

29
Jeffrey T. LaFrance                                                                                     30

Note that ∂g x / ∂p x ≤ 0, ∂g y / ∂p y ≤ 0, ∂g x / ∂p y = ∂g y / ∂p x , and

(∂g x / ∂p x )(∂g y / ∂p y ) − (∂g x / ∂p y )(∂g y / ∂p x ) ≡ 0;

equivalently,                  (∂ 2e / ∂p x )(∂ 2e / ∂p 2 ) − (∂ 2e / ∂p x ∂p y )2 ≡ 0,
2
y

so that the matrix of compensated price effects is symmetric, negative semidefinite, and singular,

equivalently, the Hessian matrix for the expenditure function with respect to prices is singular,

which implies that e( p x , p y , u ) is 1° homogeneous and concave in ( p x p y ) .

Also note that since ∂u / ∂x = λ p x and ∂u / ∂y = λ p y in the utility maximization problem, we

μ
have C = μ H =           B , where B is the Hessian matrix for the Lagrangean function for utility
λ2

maximization subject to the budget constraint. Therefore, if λ = 1 μ , equivalently, if μ = 1 λ ,

for an appropriate choice of m, equivalently, an appropriate choice for u, then we have

∂g x / ∂p x = ∂h x / ∂p x + (∂h x / ∂m )h x ,

∂g x / ∂p y = ∂h x / ∂p y + (∂h x / ∂m)h y = ∂h y / ∂px + (∂h y / ∂m)h x = ∂g y / ∂p x ,

∂g y / ∂p y = ∂h y / ∂p y + (∂h y / ∂m )h y .

That is, under conditions in which the marginal utility of money is equal to the reciprocal of the

marginal cost of utility, the matrix of price effects for the Hicksian compensated demands is

equal to the matrix of Slutsky substitution effects for the ordinary Marshallian market demands.

We will now show that this follows naturally from the special relationship between the expendi-

ture function and the indirect utility function. To this end, we set e( px , p y , u ) = m and

v ( p x , p y , m) = u . Then we obtain the following pair of identities:

30

e( px , p y , v ( p x , p y , m )) ≡ m ;

v ( px , p y , e( px , p y , u )) ≡ u .

The maximum level of utility that can be achieved with prices (px,py) and income just equal to

the minimum expenditure necessary to obtain the utility level u must be exactly u.

The minimum expenditure level necessary to obtain the maximum level of utility that can be

achieved with prices (px,py) and income m must be exactly m.

Mathematically, these two identities mean simply that the expenditure function and the indirect

utility function are inverse functions of one another with respect to the arguments u and m, re-

spectively. This means, in turn, that we can apply the inverse function theorem to differentiate

both sides of the first identity with respect to m and the second identity with respect to u:

⎛ ∂e( px , p y , v ( p x , p y , m )) ⎞ ∂v ( p x , p y , m)
⎜                                     ⎟                     ≡ 1;
⎝                ∂u                   ⎠        ∂m

⎛ ∂v ( px , p y , e( p x , p y , u )) ⎞ ∂e( p x , p y , u )
⎜                                     ⎟                     ≡ 1.
⎝               ∂m                    ⎠       ∂u

Now, recall from the envelope theorem that

∂v ( p x , p y , m)
≡ λ ( px , p y , m) ,
∂m

∂e( px , p y , u )
≡ μ ( px , p y , u) .
∂u

Evaluating m at e( p x , p y , u ) and u at v ( p x , p y , m ) in these two identities and substituting the re-

sults into the two identities just above, which are due to the inverse function theorem implies

1
λ ( px , p y , m) ≡                                       ,
μ ( p x , p y , v ( p x , p y , m))

31
Jeffrey T. LaFrance                                                                                                                    32

1
μ ( px , p y , u) ≡                                        .
λ ( p x , p y , e( p x , p y , u ))

Thus, for any set of prices and income that leads to a well-defined solution to the utility maximi-

zation problem, there is a level of utility (specifically, u = v ( p x , p y , m) ) associated with a well-

defined solution to the expenditure minimization problem. Conversely, for any set of prices and

utility that leads to a well-defined solution to the expenditure minimization problem, there is a

level of income (specifically, m = e( px , p y , u ) ) associated with a well-defined solution to the

utility maximization problem. These paired solutions are characterized by the expenditure and

indirect utility functions being inverses to one another with respect to income and utility, and the

marginal utility of money being the reciprocal of the marginal cost of utility. Moreover, the op-

timal choices for the quantities demanded are the same for both problems, i.e.:

g x ( p x , p y , v ( p x , p y , m )) ≡ h x ( p x , p y , m ) ;

g y ( px , p y , v ( p x , p y , m )) ≡ h y ( p x , p y , m ) ;

h x ( p x , p y , e( p x , p y , u )) ≡ g x ( px , p y , u ) ;

h y ( p x , p y , e( p x , p y , u )) ≡ g y ( px , p y , u ) .

These identities are extremely useful and important relationships. For example, we can differen-

tiate the last two with respect to prices to generate the symmetry and negativity conditions for

the Slutsky substitution terms in one easy step, e.g., combine the derivative condition

∂h x ( p x , p y , e( p x , p y , u )) ⎛ ∂h x ( p x , p y , e( p x , p y , u )) ⎞ ∂e( px , p y , u ) ∂g x ( p x , p y , u )
+⎜                                        ⎟                   ≡                       .
∂p x                    ⎜                 ∂m                     ⎟      ∂p x                 ∂p x
⎝                                        ⎠

with the envelope theorem result

32

∂e( p x , p y , u )
≡ g x ( p x , p y , u ) ≡ h x ( p x , p y , e( p x , p y , u )) ,
∂p x

The Hicksian compensated demands are the partial derivatives of the expenditure function with

respect to prices. It follows that the partial derivatives of the compensated demands with respect

to prices are the elements of the Hessian matrix for the expenditure function:

∂ 2 e( p x , p y , u )       ∂g x ( p x , p y , u )
≡                            ;
∂p x
2
∂p x

∂ 2 e( p x , p y , u )       ∂g x ( p x , p y , u )         ∂g y ( p x , p y , u )       ∂ 2 e( p x , p y , u )
≡                            ≡                              ≡                            ;;
∂p x ∂p y                       ∂p y                           ∂p x                      ∂p y ∂p x

∂ 2 e( p x , p y , u )       ∂g y ( p x , p y , u )
≡                            .
∂p 2
y                         ∂p y

Therefore, we have proven all of the following.

Properties of the expenditure function:

1. e( p x , p y , u ) is increasing in ( p x , p y , u ) ;

2. e( p x , p y , u ) is 1º homogeneous in ( p x , p y ) ;

3. e( p x , p y , u ) is concave in ( p x , p y ) ;

4. e( p x , p y , u ) satisfies Hotelling’s/Shephard’s Lemma:

∂e( px , p y , u )
≡ g x ( px , p y , u) ,
∂p x

∂e( p x , p y , u )
≡ g y ( px , p y , u) .
∂p y

The Indirect Utility Function

33
Jeffrey T. LaFrance                                                                              34

The maximum level of utility that can be obtained given prices px and py and income m,

v (px , p y , m ) ≡ u(h x (px , p y , m), h y (p x , p y , m )) ,

is called the indirect utility function.

Properties of the indirect utility function:

1. v (px , p y , m) is continuous for all (p x , p y , m)         (0,0,0) .

2. v (px , p y , m) is decreasing in (px , p y ) (px, py), increasing in m.

3. v (px , p y , m) is quasi-convex in (px , p y ) .

4. v (px , p y , m) is 0° homogeneous in (px , p y , m) .

5. Roy's Identity:

⎛ ∂v ( p x , p y , m ) ∂p x ⎞
h x ( px , p y , m ) ≡ − ⎜                           ⎟,
⎜ ∂v ( px , p y , m ) ∂m ⎟
⎝                           ⎠

⎛ ∂v ( px , p y , m ) ∂p y ⎞
h y ( px , p y , m ) ≡ − ⎜                          ⎟.
⎜ ∂v ( p x , p y , m ) ∂m ⎟
⎝                          ⎠

We will prove these results in two separate ways. The first derivation uses calculus and the re-

sults that we have already obtained regarding the properties of the Marshallian demands and the

marginal utility of money. The second approach uses arguments from set theory and relies only

on the properties of the budget set, the maximization hypothesis, and the existence of an optimal

solution to the consumer’s choice problem. This second method is very important because it

shows us that the properties of the indirect utility function hold under much more general condi-

tions than the calculus dependent methods might suggest. On the other hand, when it comes to

the practical derivation of comparative statics for a particular utility function or demand model,

34

calculus methods are indispensable. Therefore, it is my belief that a sound understanding of both

methodologies are essential in microeconomic theory.

Calculus Arguments

For this approach, we will assume throughout that the utility function is twice continuously dif-

ferentiable and strictly quasiconcave in (x,y). All of the monotonicity results can then be ob-

tained by simply applying the envelope theorem to the utility maximization problem,

∂v ( p x p y , m) / ∂p x ≡ −λ ( p x , p y , m )h x ( p x , p y , m) ≤ 0,

∂v ( p x p y , m) / ∂p y ≡ −λ ( px , p y , m)h y ( px , p y , m) ≤ 0,

∂v ( p x p y , m) / ∂p x ≡ λ ( p x , p y , m) ≥ 0,

where we have made use of the fact that the optimal quantities demanded and the marginal utility

of money are positive valued. Note that Roy’s Identity follows from these envelope theorem re-

sults by simply taking the ratio of the first and third equations and the ratio of the second and

third equations above.

Euler’s theorem and its converse imply that v ( p x , p y , m ) is 0º homogeneous iff

⎛ ∂v ( p x p y , m) ⎞      ⎛ ∂v ( p x p y , m) ⎞      ⎛ ∂v ( p x p y , m) ⎞
⎜                   ⎟ px + ⎜
⎜                   ⎟ py + ⎜
⎟                          ⎟m ≡ 0
⎝       ∂p x        ⎠      ⎝      ∂p y         ⎠      ⎝       ∂m          ⎠

∀ ( px , p y , m) ∈   3
+.   From our application of the envelope theorem above, we can evaluate the

three required derivatives of v ( px , p y , m ) as follows:

35
Jeffrey T. LaFrance                                                                                                                    36

⎛ ∂v ( p x , p y , m) ⎞      ⎛ ∂v ( p x , p y , m ) ⎞      ⎛ ∂v ( p x , p y , m) ⎞
⎜                     ⎟ px + ⎜
⎜                      ⎟ py + ⎜
⎟                            ⎟m ≡
⎝       ∂p x          ⎠      ⎝       ∂p y           ⎠      ⎝        ∂m           ⎠

−λ ( p x , p y , m )h x ( p x , p y , m ) p x − λ ( p x , p y , m )h y ( p x , p y , m ) p y + λ ( p x , p y , m ) m ≡

λ ( px , p y , m) ⎡ m − p x h x ( px , p y , m) − p y h y ( p x , p y , m ) ⎤ ≡ 0,
⎣                                                         ⎦

where the last identity results from the Kuhn-Tucker complementary slackness condition for the

shadow price and the budget constraint. It is worthwhile to restate this with emphasize.

0° homogeneity of v ( px p y , m) in ( p x , p y , m ) is due to the linear budget constraint.

The only remaining property that we need to develop is quasi-convexity. To accomplish this, we

calculate the second-order partial derivatives of v ( px , p y , m ) with respect to px and py from the

above identities for the first-order partial derivatives obtained from the envelope theorem:

∂ 2v ( p x , p y , m)        ∂λ ( p x , p y , m)                                              ∂h x ( p x , p y , m )
≡−                          h ( px , p y , m) − λ ( px , p y , m)
x
;
∂p x
2
∂p x                                                              ∂p x

∂ 2v ( p x , p y , m)        ∂λ ( p x , p y , m)                                              ∂h x ( p x , p y , m )
≡−                          h ( px , p y , m) − λ ( px , p y , m)
x
;
∂p x ∂p y                      ∂p y                                                              ∂p y

∂ 2v ( px , p y , m)          ∂λ ( px , p y , m )                                              ∂h y ( p x , p y , m )
≡−                          h y ( px , p y , m) − λ ( px , p y , m)                            ;
∂p 2
y                         ∂p y                                                              ∂p y

∂ 2v ( px , p y , m)          ∂λ ( p x , p y , m )                                             ∂h y ( p x , p y , m )
≡−                          h ( px , p y , m) − λ ( px , p y , m)
y
.
∂p y ∂p x                      ∂p x                                                              ∂p x

From these terms, we obtain the bordered Hessian matrix for v ( px , p y , m ) in prices as

⎡ −(∂λ / ∂p x )h x − λ (∂h x / ∂p x ) −(∂λ / ∂p y )h x − λ (∂h x / ∂p y ) −λ h x ⎤
⎢                                                                                ⎥
B = ⎢ −(∂λ / ∂px )h y − λ (∂h y / ∂p x ) −(∂λ / ∂p y )h y − λ (∂h y / ∂p y ) −λ h y ⎥ .
⎢                                                                                ⎥
⎢
⎣              −λ h x                              −λ h y                  0 ⎥   ⎦

36

Quasi-convexity requires that B ≤ 0 . From the above expression for B, this determinant is

B = ⎡ −(∂λ / ∂p x )h y − λ (∂h y / ∂px ) ⎤ ( −λ h x )( −λ h y )
⎣                                    ⎦

+ ⎡ −(∂λ / ∂p y )h x − λ (∂h x / ∂p y ) ⎤ ( −λ h x )( −λ h y ) − ⎡ −(∂λ / ∂p y )h y − λ (∂h y / ∂p y ) ⎤ ( −λ h x )2
⎣                                     ⎦                        ⎣                                     ⎦

− ⎡ −(∂λ / ∂p x )h x − λ (∂h x / ∂p x ) ⎤ ( −λ h y )2
⎣                                     ⎦

⎡⎛ ∂λ ⎞ y     ⎛ ∂h y ⎞ ⎛ ∂λ ⎞ x    ⎛ ∂h x ⎞ ⎤       ⎡ ⎛ ∂λ ⎞ x 2 y        ⎛ ∂h y ⎞ x 2 ⎤
= −λ 2 h x h y ⎢ ⎜       h +λ⎜       +⎜       h +λ⎜          + λ 2 ⎢⎜         (h ) h + λ ⎜
⎟⎥
⎟                                                     (h ) ⎥
⎟
⎢⎝ ∂p x ⎠
⎟ ⎜     ⎟
∂px ⎠ ⎝ ∂p y ⎠     ⎜ ∂p y ⎟ ⎥                ⎟
⎢ ⎜ ∂p y ⎟            ⎜ ∂p y ⎟
⎟      ⎥
⎣             ⎝                    ⎝      ⎠⎦        ⎣⎝       ⎠            ⎝      ⎠      ⎦

⎡⎛ ∂λ ⎞ x y 2        ⎛ ∂h x ⎞ y 2 ⎤
+λ ⎢⎜
2
⎟ h (h ) + λ ⎜      ⎟ (h ) ⎥
⎢⎝ ∂p x ⎠
⎣                    ⎝ ∂p x ⎠      ⎥
⎦

⎡ ⎛ ∂h x ⎞ y 2 ⎛ ∂h x ∂h y ⎞ x y ⎛ ∂h y ⎞ x 2 ⎤
= λ 3 ⎢⎜       ⎟ (h ) − ⎜     +       h h +⎜        (h ) ⎥ .
⎢ ⎝ ∂p x ⎠        ⎜ ∂p y ∂p x ⎟
⎟      ⎜ ∂p y ⎟
⎟      ⎥
⎣                 ⎝           ⎠      ⎝      ⎠      ⎦

Now, recall the following comparative statics results for the Marshallian demands that we de-

rived earlier:

∂h x    ⎛ λ p 2 ⎞ ⎛ ∂h x ⎞ x
= −⎜ y ⎟ − ⎜
⎜ H ⎟ ⎝ ∂m ⎟
h ;
∂p x    ⎝       ⎠        ⎠

∂h x    ⎛ λ p p ⎞ ⎛ ∂h x ⎞ y
= −⎜ x y ⎟ − ⎜      ⎟h ;
∂p y    ⎝   H ⎠ ⎝ ∂m ⎠

∂h y    ⎛ λ p x p y ⎞ ⎛ ∂h y ⎞ x
= −⎜           ⎟−⎜      ⎟h ;
∂p x    ⎝ H ⎠ ⎝ ∂m ⎠

∂h y    ⎛ λ p 2 ⎞ ⎛ ∂h y ⎞ y
= −⎜ x ⎟ − ⎜        ⎟h ,
∂p y    ⎝  H ⎠ ⎝ ∂m ⎠

where |H| > 0 is the determinant of the bordered Hessian matrix for u(x,y). Substituting these re-

lationships into the bracketed term in the last expression above for |B| gives

37
Jeffrey T. LaFrance                                                                                       38

⎡ ⎛ ⎛ λ p 2 ⎞ ⎛ ∂h x ⎞ ⎞               ⎛ ⎛ λ p 2 ⎞ ⎛ ∂h y ⎞ y ⎞ x 2 ⎤
B = λ ⎢− ⎜ ⎜ y ⎟ + ⎜
3
⎟ h x ⎟ (h y )2 − ⎜ ⎜ x ⎟ + ⎜        ⎟ h ⎟ (h ) ⎥
⎢ ⎜ ⎜ H ⎟ ⎝ ∂m ⎠ ⎟                     ⎜ H                    ⎟
⎣ ⎝⎝        ⎠              ⎠           ⎝ ⎝       ⎠ ⎝ ∂m ⎠ ⎠          ⎥
⎦

⎡ ⎛ λ p x p y ⎞ ⎛ ∂h x ⎞ y ⎛ ∂h y ⎞ x ⎤
+λ h h ⎢ 2 ⎜
3 x y
⎟+⎜      ⎟h + ⎜     ⎟ h ⎥.
⎢ ⎝ H ⎠ ⎝ ∂m ⎠
⎣                           ⎝ ∂m ⎠ ⎥  ⎦

Finally, grouping and canceling all common terms eliminates all of the expressions involving

income effects, so that we are left with

⎛ λ4         ⎞                                             ⎛ λ 4m 2 ⎞
B = −⎜            ⎟ ⎡( px h ) + ( p y h ) + 2 p x p y h h ⎤ = − ⎜        ⎟ ≤ 0,
x 2         y 2             x y
⎣                                      ⎦
⎝ H          ⎠                                             ⎝ H ⎠

where the last inequality follows because, again, the marginal utility of money is positive and the

determinant of the bordered Hessian for the utility function is strictly positive.

Set Theoretic Arguments

We will now prove the monotonicity, homogeneity, and quasiconvexity properties of the indirect

utility function using a much more general approach. This approach relies on the properties of

the budget set and the underlying maximization hypothesis. As a result, it demonstrates the prop-

erties of v(px, py, m) for very general conditions on u(x, y).

Quasiconvexity:

Let ( p x , p 0 ), ( p1 , p1 ) ∈
0
y       x    y
2
+   denote any two price vectors and define their convex combination

by p x = (1 − t ) p x + tp1 and p 2 = (1 − t ) p 0 + tp1 for any t ∈ [0,1] . Let (x0,y0) maximize u(x,y) sub-
2              0
x       y              y     y

ject to p x x + p 0 y ≤ m , while (x1,y1) maximizes u(x,y) subject to p1 x + p1 y ≤ m . The convex
0
y                                                    x      y

combination of (x0,y0) and (x1,y1) is x 2 = (1 − t ) x 0 + tx1 and y 2 = (1 − t ) y 0 + ty1 . The budget sets

for each of these three price vectors are

38

{
B 0 = ( x, y ) ∈       2
+                     }
: px x + p0 y ≤ m ,
0
y

{
B1 = ( x, y ) ∈        2
+                     }
: p1 x + p1 y ≤ m ,
x      y

{
B 2 = ( x, y ) ∈       2
+                     }
: px x + p 2 y ≤ m .
2
y

Claim: B 2 ⊆ B0 ∪ B1 , i.e., if p x x + p 2 y ≤ m , then either p x x + p 0 y ≤ m , or p1 x + p1 y ≤ m , or
2
y
0
y             x      y

both.

Suppose that this is not the case. Then

p x x + p 2 y = ⎡ (1 − t ) p x + tp1 ⎤ x + ⎡ (1 − t ) p 0 + tp1 ⎤ y ≤ m
2
y     ⎣
0
x⎦      ⎣            y     y⎦

while both p x x + p 0 y > m and p1 x + p1 y > m . But if we multiply the first inequality by (1-t),
0
y            x      y

the second by t, and add the two results together, we obtain

(
(1 − t ) p x x + p 0 y + t p1 x + p1 y = ⎡(1 − t ) px + tp1 ⎤ x + ⎡ (1 − t ) p 0 + tp1 ⎤ y > (1 − t )m + tm = m
0
y) (     x      y     )
⎣
0
x⎦      ⎣            y     y⎦

by the associative and distributive laws of elementary algebra. Clearly, this is a contradiction,

and we are led to the conclusion that B 2 ⊆ B0 ∪ B1 .

Denote the values of the indirect utility function for each of the three specified price vectors by

{                         }
v ( pix , piy , m ) ≡ max u ( x, y ) : ( x, y ) ∈ Bi , i = 0, 1, 2.

Then

y              {
v ( p x , p 2 , m) ≤ max v ( p x , p 0 , m ), v ( p1 , p1 , m) ,
2                        0
y             x    y    }
This is precisely the requirement for quasiconvexity in prices,

0
x             y     y               {
v ((1 − t ) p x + tp1 ,(1 − t ) p 0 + tp1 , m ) ≤ max v ( p x , p 0 , m), v ( p1 , p1 , m ) .
0
y            x    y     }

39
Jeffrey T. LaFrance                                                                                                      40

Monotonicity:

Now suppose that ( p x , p 0 ) ≤ ( p1 , p1 ) , i.e., p x ≤ p1 and p 0 ≤ p1 . Then clearly B1 ⊆ B0 , since
0
y        x    y
0
x       y    y

both the x and y intercepts are shifted towards the origin. Clearly, then, we also must have

v ( p1 , p1 , m) ≤ v ( p x , p 0 , m ) since we can not reach a higher level on an objective function with a
x    y
0
y

smaller feasible set of choices. On the other hand, suppose that m 0 ≤ m1 . Then B0 ⊆ B1 since

the    budget         line   has      shifted        outwards           in         a      parallel    fashion.   Therefore,

v ( p x , p y , m 0 ) ≤ v ( px , p y , m1 ) and the indirect utility function is decreasing in prices and increas-

ing in income.

Homogeneity:

Rewriting the budget constraint, p x x + p y y ≤ m , equivalently as ( p x / m ) x + ( p y / m ) y ≤ 1 , the

budget set B is unchanged for all proportional changes in ( p x , p y , m ) :

{
B ≡ ( x, y ) ∈        2
+   : px x + p y y ≤ m     }
≡ {( x, y ) ∈    2
+   : ( px / m) x + ( p y / m) y ≤ 1      }
≡ {( x, y ) ∈    2
+   : (tp x / tm) x + (tp y / tm) y ≤ 1       }
≡ {( x, y ) ∈    2
+   : tp x x + tp y y ≤ tm .   }
Therefore, scaling all prices and income by the same factor does not change the feasible choice

set or the optimal choices for (x,y), and thus can not have any impact on the maximal level of

utility attained:

v (tp x , tp y , tm ) ≡ v ( p x , p y , m ) .

This is precisely the condition for zero degree homogeneity.

40

```
DOCUMENT INFO
Shared By:
Categories:
Stats:
 views: 168 posted: 1/9/2010 language: English pages: 40
How are you planning on using Docstoc?