Memoria Constructiva ꞏ

In particular, since only the price of good 1 is changing, we have

+

-

where

When - p^O) is small, this procedure provides a better approximation to the true compensating variation than does the area variation measure. However, if is large, we cannot tell which is the better approximation. It is entirely possible for the area variation measure to be superior. After all, its use guarantees some sensitivity of the approximation to demand behavior away from p^O, whereas the use of does not.

The Strong Axiom of Revealed Preference

have seen that in the context of consumer demand theory, consumer .choice satisfy the weak but not be capable of being generated by-a rational preference relation (see Sections

2.F

and We could therefore ask: Can we find a necessary and sufficient consistency condition on consumer demand behavior that is in the same style as the WA but that doesimply that demand behavior can be rationalized by preferences? The answer is "yes", and it was provided by Houthakker (1950) in the form of the strong axiom of revealed (SA), a kind of recursive closure of the weak

efinition The market demand function the strong axiom of revealed preference (the SA) if .for any list

with # x(pⁿ, wⁿ) for all n N

-

we have w^N

whenever for all n N 1.

In words, if is directly or indirectly revealed to x(p^N, w^N), cannot be (directly) revealed preferred to w^l) [so cannot be at (p^N, w^N)]. For example, the :SA was violated in Example 2F.1. It is clear that the SA is satisfied if demand originates in rational preferences. The converse is a deeper result. It is stated in Proposition the which is advanced, is presented in small type.

position If the Walrasian demand function w ) satisfies the strong axiom of revealed preference then there is a rational preference relation that rationalizes w ) , that is, such that for all (p, for every

# with

For an informal account of revealed preference theory after see (1982).

92 C H A P T E R 3 : C L A S S I C A L D E M A N D T H E O R Y

Proof: We Richter (1966). His proof is based on set theory and differs markedly from differential equations techniques used originally by

Define a relation on commodity vectors by letting x y whenever x y and we have and w for some ( p , w ) . The relation can be read as "directly revealed preferred to." From define a new relation to be read as "directly or indirectly revealed preferred to," by letting whenever there is a chain x² . . . , x^N with = x and x^N Observe that, by construction, is transitive. According to the SA, is also irreflexive impossible). A certain axiom of set theory (known as Zorn's lemma)

tells us following: relation (called a order)

an and transitive relation such that, first, x y implies second, whenever y, have either y or y x . Finally, we can define by letting whenever x = y or It is not now to verify that is complete and transitive and that w)

>

y whenever w and y w).

The proof of uses only single-valuedness of w). Provided choice is single-valued, the applies to abstract theory of choice of Chapter 1. The fact that budgets competitive is immaterial.

In Exercise 3.5.1, y o u a r e a s k e d to s h o w t h a t t h e W A is equivalent t o t h e S A when L = 2. by P r o p o s i t i o n 3.5.1, w h e n L = 2 a n d d e m a n d satisfies t h e W A , wc a rationalizing preference relation, a result t h a t we h a v e a l r e a d y seen in W h e n 2, however, t h e S A is s t r o n g e r t h a n t h e WA. In fact, us t h a t a choice-based theory of d e m a n d f o u n d e d o n t h e s t r o n g a x i o m is equivalent t o t h e preference-based theory of d e m a n d presented in this c h a p t e r .

The strong is therefore essentially equivalent both to the rational preference hypothesis and to symmetry and negative semidefiniteness of the matrix. We have

that the weak axiom is essentially equivalent to the negative semidefiniteness of the I t is therefore natural to ask whether there is an assumption on preferences that is weaker than rationality and that leads to a theory of consumer demand equivalent to that based on the WA. Violations of the SA mean cycling choice, and violations of the symmetry of the matrix generate path dependence in attempts to "integrate back" t o preferences.

This suggests preferences that may violate the transitivity axiom. See the appendix with W.

Shafer in Kihlstrom, and Sonnenschein (1976) for further discussion of this point.

A P P E N D I X A: C O N T I N U I T Y A N D D I F F E R E N T I A B I L I T Y P R O P E R T I E S

O F D E M A N D

In this appendix, we investigate t h e continuity and differentiability properties of t h e Walrasian c o r r e s p o n d e n c e w). We a s s u m e t h a t x f o r all (p, w)

a n d w).

31. Yet a third approach, based on linear programming techniques, was provided by Afriat (1967).

A P P E N D I X A : C O N T I N U I T Y A N D D I F F E R E N T I A B I L I T Y O F D E M A N D

Because w ) is, in general, a correspondence, we begin by introducing a generalization of the more familiar continuity property for functions, called upper

Definition ^3.AA.1: The Walrasian demand correspondence w ) is upper continuous at ( p , w) if whenever ( p " , wⁿ) ( p , w), xⁿ for all n, and

x we have x

In words, a demand correspondence is upper hemicontinuous at (p, w) if for any sequence of price wealth pairs limit of any sequence of optimal demand bundles is optimal (although not necessarily uniquely so) at the limiting price-wealth pair. If single-valued at all (p, w ) this notion is equivalent to the usual continuity property for functions.

Figure I depicts an upper hemicontinuous demand correspondence: When w) exhibits a jump in demand behavior at the price vector p, being for but suddenly becoming the interval of consumption bundles at p. It is upper hcmicontinuous because (the limiting optimum for p" along the sequence) is an element of segment [x, (the set of optima at price vector p). See Section M.H of the Mathematical Appendix for further details on upper hemicontinuity.

Proposition 3.AA.1: Suppose that is a continuous utility function representing locally nonsatiated preferences on the consumption s e t X = Then the derived demand correspondence w ) is upper hemicontinuous at all ( p , w ) 0.

Moreover, i f is a function if w ) has a single element for all ( p , then it is continuous at all ( p , w ) 0 .

To verify hcmicontinuity, suppose that we had a sequence { ( p " ,

( p . w) and a with x" for all n, such that 1 and w).

for all n, taking limits as n we conclude that Thus, is a consumption bundle when the set is However, since it is not optimal in this set, it be that for some

Figure 3.AA.1 An upper hemicontinuous Walrasian demand correspondence.

32. We the notation as synonymous with = This definition of upper only to that are "locally bounded" (see Section M.H of the Appendix). Under our assumptions, the Walrasian demand correspondence satisfies

this at w) 0.

C H A P T E R 3 : C L A S S I C A L D E M A N D T H E O R Y

continuity there is a y arbitrarily close to such that w and This bundle is illustrated in Figure 3.AA.2.

that if is enough, we will have w" [since wⁿ) (p, w)]. Hence, y is a n clement o f budget set and we must have u(xⁿ) because w").

Taking limits the continuity of then implies that which gives us contradiction. Wc must therefore w), establishing upper hemicontinuity of

also establishes continuity if w) is in fact a function.

Suppose that the consumption set is an arbitrary closed set X Then the continuity (or hemicontinuity) property still follows at any (p, that passes the following

test: that x X is affordable W). Then there is a X arbitrarily close to x and that costs less than w y W)." For example, in Figure 3.AA.3, commodity 2 is available only in indivisible unit amounts. The locally cheaper test then at price wealth point w) = (I, w), where a unit of good 2 becomes just can easily verify by examining the figure [in which the dashed line indicates the points I ) and z] that demand will fail to be upper hemicontinuous

= w. In particular, for price- wealth points (p", such that = 1 and >

w) involves only the consumption of good I; whereas a t w) = (I, w), we have w ) = I ) . Note that the proof of Proposition 3.AA.1 fails when the locally cheaper consumption condition does not hold because we cannot find a consumption bundle y with

described there.

Proposition has established that if w ) is a function, then it is continuous.

Often it is convenient that it be differentiable as well. We now discuss when this is so. We assume for the remaining paragraphs that is strictly quasiconcave and twice continuously differentiable and that 0 for all x.

A s we have shown Section the first-order conditions for the U M P imply that w ) is, for some 0, the unique solution of the system of L

+

1 equations in

+

unknowns:

Figure 3.AA.2 Finding a bundle y such that w and

Flgure 3.AA.3 (rlght) The locally cheaper test fails a t price wealth pair

= w,

R E F E R E N C E S

Therefore, the implicit .function theorem (see Section M.E of the Mathematical Appendix) tells us that the differentiability of the solution w) as a function of the parameters (p, w) of the system depends on the Jacobian matrix of this system having a nonzero determinant. The Jacobian matrix the derivative matrix of

+

I component functions with respect to the

+

1 variables (x, is

Since = and 0, the determinant of this matrix is nonzero if and only if the determinant of the of at x is nonzero:

This condition has a straightforward geometric interpretation. means that the set through x has a curvature at x; it is not (even infinitesimally) fat. This condition is a slight technical strengthening of strict quasiconcavity [just as the strictly concave function ⁼ has ⁼ a strictly quasiconcave function could a bordered Hessian determinant that is zero at a point].

Wc therefore, that w) is

if

and only determinant

of Hessian of is nonzero at w). It is worth noting the following interesting (which we shall not prove here): If w) is differentiable at (p, w), the has maximal possible rank; that is, the rank of w)

L

R E F E R E N C E S

S. (1907). The from expenditure data. Internutional Economic Review 67 77.

tipogrofia del translation: O n theory of political economy.] In Preferences, edited by J. L. and H. Sonnenschein. New York: Harcourt 1971

J., and Moore. Compensating variation, consumer's surplus, and welfare. American 70: 933

and J. Economics Consumer Cambridge, U.K.: Cambridge

(1960). Topological methods in cardinal utility. In Mathematical Methods in the Social Studies, by K. Arrow, S. Karlin, and P. Suppes. Stanford, Calif.: Stanford University Press.

W. Duality approaches to theory. Chap. in

2, edited by K. Arrow and M. Intriligator. Amsterdam: North- Green, J . W. Mathematical analysis and convexity with applications to economics.

Chap. I in Economics, edited by K. Arrow and M. Intriligator.

Amsterdam. North-Holland.

33. This applies only t o d e m a n d generated from a twice continuously differentiable utility function. It not be true when this condition is not met. F o r example, the d e m a n d function

W) =

+ +

^is a n d it is generated by the utility function

= Min which is not twice continuously a t all x. T h e substitution matrix for this function has all its entries equal t o zero a n d therefore h a s rank equal t o zero.

96 C H A P T E R 3 : C L A S S I C A L D E M A N D T H E O R Y

Exact consumer surplus and deadweight loss. American Economic Review 71: 662-76.

Hicks, J. (1939). Value Capital. Oxford: Clarendon Press.

Houthakker. H. (1950). Revealed preference and the utility function. 17:

Hurwicz, L., and (1971). O n the integrability of demand functions. Chap. 6 in Preferences, Utility edited by J . L. Hurwicz, and H. Sonnenschein. New York: Harcourt Brace, R.. A. and H. Sonnenschein. (1976). The demand theory of the weak axiom of

Econometricu 44: 971 78.

L. (1956--57). theory without a utility index. Review Economic Studies 24:

A. London: Macmillan.

A. Revealed preference after Samuelson, in Economics,

by Boston:

Richter, M. Revealed preference theory. Econometricu 34:

P. (1947). Cambridge, Mass.: Harvard

(1915). del consumatore. deyli Economisti 51: 1-26. [English translation: On the theory of the budget of the consumer, in in edited by

and K. Chicago: Richard Irwin, 1952.1

Stone, J. (1954). expenditure systems and demand analysis: An application to the pattern of

British 64: 51 1-27.

Vivcs, X. income A theory of consumer surplus downward sloping

54: 87 103.

E X E R C I S E S

3.R.I^A In text

preference relation defined o n t h e c o n s u m p t i o n set X = is s a i d t o be if a n d o n l y if x implies t h a t x y. S h o w t h a t if is transitive, locally

m o n o t o n e , t h e n it is m o n o t o n e .

c o n v e x preference relation t h a t is locally n o n s a t i a t e d b u t is n o t m o n o t o n e . 3.C.I^H Verify t h a t t h e lexicographic o r d e r i n g is c o m p l e t e , transitive, s t r o n g l y m o n o t o n e , a n d strictly

S h o w t h a t if is a c o n t i n u o u s utility f u n c t i o n r e p r e s e n t i n g t h e n is c o n t i n u o u s . S h o w t h a t if for every x t h e u p p e r a n d l o w e r c o n t o u r sets y x ) a n d

x a r c closed, t h e n is c o n t i n u o u s a c c o r d i n g t o Definition 3.C.1.

Exhibit a n e x a m p l e of a preference relation t h a t is n o t c o n t i n u o u s b u t is r e p r e s e n t a b l e by a utility f u n c t i o n .

Establish t h e following t w o results:

(a) A c o n t i n u o u s is h o m o t h e t i c if a n d o n l y if it a d m i t s a utility f u n c t i o n t h a t is h o m o g e n e o u s of d e g r e e o n e ; = for all a 0.

(b) A c o n t i n u o u s o n x q u a s i l i n e a r with respect t o t h e first c o m m o d i t y if a n d o n l y if it a d m i t s a utility f u n c t i o n of t h e f o r m = x,

+

^{. . .}

.

[ H i n t :

cxistcncc s o m e c o n t i n u o u s utility r e p r e s e n t a t i o n is g u a r a n t e e d b y P r o p o s i t i o n After (a) a n d a r g u e t h a t these p r o p e r t i e s o f a r e

E X E R C I S E S

Suppose that a two-commodity world, the consumer's utility function takes the form

=

+

This utility function is known as the constant elasticity substitution (or CES) utility function.

Suppose that is differentiable and strictly quasiconcave and that the Walrasian demand w) is differentiable. Show the following:

(a) If of degree one, then the Walrasian demand function w) and strictly convex and quasilinear. Normalize =

(a) Show that the Walrasian demand functions for goods 2,. . . , L are independent of wealth. What does this imply about the wealth effect (see Section 2.E) of demand for good

(b) that the indirect utility function can be written in the form w) = w

+

for

(c) for simplicity, that L = 2, and write the consumer's utility function as

= Now, however, let the consumption set be so that there is a nonnegativity constraint on consumption of the numeraire x , . Fix prices p, and examine how the consumer's Walrasian demand changes as wealth w varies. When is the nonnegativity constraint on the numeraire irrelevant'!

Consider again the CES utility function of Exercise 3.C.6, and assume that = = 1.

(a) Compute the Walrasian demand and indirect utility functions for this utility function.

(b) Verify that two functions satisfy all the properties of Propositions 3.D.2 and 3.D.3.

(c) the Walrasian demand correspondence and indirect utility function for the case of utility and the case of Leontief utility (see Exercise 3.C.6). Show that the CES Walrasian demand and indirect utility functions approach these as approaches 1 and -m, respectively.

(d) The substitution between goods 1 and 2 is defined as

Show for utility function, w) = thus justifying its name. What

is linear, Leontief, and utility functions?

the three-good setting in which the consumer has utility function

(a) Why you assume that a

+ +

⁼ 1 without loss of generality'? D o so for the rest of the problem.

(b) Write down the first-order conditions for the UMP, and derive the consumer's Walrasian dcmand and indirect utility functions. This system of demands is known as the

and is due to Stone (1954).

(c) Verify that these dcmand functions satisfy the properties listed in Propositions 3.D.2 and 3.D.3.

There are two commodities. We are given two budget sets and described, by ( I , I), w" = and = = 26. The observed choice a t w^O)

is At have a choice that =

(a) the region of permissible choices if the choices and are consistent with maximization of preferences.

(h) the region of permissible choices if the choices x^O and are consistent with of preferences that arc quasilinear with respect to the good.

of permissible choices if the choices and are consistent with of preferences that are quasilinear with respect to the second good.

(d) the region of permissible choices if the choices x^O and are consistent with maximization of preferences for which both goods are normal.

(e) Detcrminc the region of choices if the choices x^O and are consistent with of homothetic preferences.

The ideal to answer this exercise relies on (good) pictures as much as possible.]

Show all (p, =

Prove solution to the E M P exists if and there is some x satisfying

Show that if the consumer's preferences are convex, then u) is a convex set. Also show that if is strictly convex, then u) is single-valued.

Show if is homogeneous of degree one, then u) and u) are homo- geneous of one in they can be written as u) = and u) =

Consider the constant elasticity of substitution utility function studied in Exercises 3.C.h and with a , = a, = 1. Derive its Hicksian demand function and expenditure function. Verify the properties of Propositions 3.E.2 and 3.E.3.

Show that if is quasilinear with respect to good 1, the Hicksian demand functions for goods 2 , . . . , d o not depend on u. What is the form of the expenditure function in this case?

Douglas utility function, verify that the relationships in (3.E.1) and (3.E.4) hold. that the expenditure function can be derived by simply inverting the indirect utility function, and vice versa.

E X E R C I S E S

Use the relations in (3.E.I) to show that the properties of the indirect utility function identified in Proposition 3.D.3 imply Proposition 3.E.2. Likewise, use the relations in (3.E.1) to prove that Proposition 3.E.2 implies Proposition 3.D.3.

Use the relations in and (3.E.4) and the properties of the indirect utility and expenditure functions to show that Proposition 3.D.2 implies Proposition 3.E.4. Then use these

to prove that Proposition 3.E.3 implies Proposition 3.D.2.

Prove formally that a closed, convex set K equals the intersection of the half-spaces that contain it (use the separating hyperplane theorem).

Show by means of a graphic example that the separating hyperplane theorem does not hold for nonconvcx sets. Then argue that if is closed and not convex, there is always some

cannot separated from

that Proposition is implied by Roy's identity (Proposition 3.G.4).

Verify for the case of a Douglas utility function that all of the propositions in Section 3.G hold.

(linear system) utility function given in Exercise 3.D.6.

(a) Derive the Hicksian demand and expenditure functions. Check the properties listed in Propositions 3.E.2 3.E.3.

(b) Show that derivatives of expenditure function are the Hicksian demand function you derived in

that the equation holds.

(d) Verify own-substitution terms are negative and that compensated cross-price arc symmetric. whatever attach to the remaining ones. It turns out that this ordinal property is not only necessary but also sufficient for the existence of a n additive separable representation.

should attempt a proof. This is very hard. See Debreu

(c) Show that the Walrasian and Hicksian demand functions generated by an additively separable utility function admit no inferior goods if the functions are strictly concave.

(You assume and interiority to answer this question.)

(d) Suppose that all are identical and twice differentiable. Let =

Show that if 1 for all then the Walrasian demand w) has the so-called for all and k

commodities.) Suppose there are two groups of desirable moditics, and with corresponding prices and The consumer's utility function is y),

her is Suppose that prices for goods y always vary in proportion t o one another, so write = For any number 0, define the function

= Max

Y

z .

C H A P T E R 3 : C L A S S I C A L D E M A N D T H E O R Y

(a) Show that if imagine that the goods in the economy are x and a single composite commodity that z) is the consumer's utility function, and that a is the price of the composite commodity, then the solution to Max,,, z )

+

w will give the consurncr's actual of x and =

(b) Show that properties of Walrasian demand functions identified in Propositions 3.D.2 and 3.G.4 hold for w) and a, w).

( c ) properties in Propositions 3.E.3, and 3.G.1 to 3.G.3 hold for the Hicksian demand functions derived using z).

(F. M . Fisher) A consumer in a three-good economy (goods denoted x , , x,, and prices with wealth level w 0 has demand functions for commodities 1 and

2 by

where Cireek letters are nonzero constants.

(a) Indicate how to calculate the demand for good 3 (but d o not actually d o it).

(b) Are the functions for and appropriately homogeneous?

(c) the restrictions on the numerical values of a, and 6 implied by utility (d) your results in part (c), for a fixed level of draw the consumer's indifference

in the plane.

(e) does your to (d) imply about the form of the consumer's utility function

A striking duality is obtained by using the concept of indirect demand function. Fix w

some w = from now on, we write = = The indirect

is the inverse of that is, it is the rule that assigns to every commodity bundle price vector such that = 1). Show that

Deduce from Proposition 3.G.4 that

Note that this a completely symmetric expression. Thus, direct (Walrasian) demand is the of indirect utility, and indirect demand is the normalized derivative of direct utility.

The indirect utility function w) is logarithmically homogeneous if =

w) for [in other words, w) = w)), where w) is homo-

geneous of degree one]. Show that if .) is logarithmically homogeneous, then

=

matrix from the indirect utility function

For a function of the Gorman form w) = which properties will the functions and have to satisfy for w) to qualify as an indirect utility function?

E X E R C I S E S

Verify that an indirect utility function in Gorman form exhibits linear expansion curves.

What restrictions on the Gorman form correspond to the cases of homothetic and quasilinear preferences'?

that the indirect utility function w ) is a polynomial of degree n on w (with cocfficicnts that may depend on p ) . Show that any individual wealth-expansion path is contained in a linear of at most dimension n 1. Interpret.

matrix records the (Walrasian) demand substitution effects for a consumer endowed with preferences and consuming three goods at the prices = 1, p , = 2,

and =

Supply missing numbers. Does the resulting matrix possess all the properties of a matrix'!

Consider the utility function

=

+

( a ) Find demand functions for goods I and 2 as they depend on prices and wealth.

( b ) Find compensated demand function

(c) Find the expenditure function, and verify that u) = u).

( d ) indirect utility function, and verify Roy's identity.

the expenditure function

= exp

+

( a ) What restrictions on a , , . . . . . . are necessary for this to be derivable from utility

( b ) indirect utility that corresponds to it.

(c) Verify Roy's identity and the equation

[From Suppose L = 2. Consider a "local" indirect utility function in some neighborhood of price wealth pair by

( a ) Verify local demand function for the first good is

( b ) Verify that the local expenditure function is

C H A P T E R 3 : C L A S S I C A L D E M A N D T H E O R Y

(c) Verify that the local Hicksian demand function for the first commodity is

Show that good is related to every other good by a chain of (weak) substitutes;

that is, for any goods and k, either 0, or there exists a good r such that and 0, or there i s . .

.

^, and so on. [ H i n t : Argue first the case of two commodities. next the insights on composite commodities gained in Exercise 3.G.5 to handle the case of three, and then L, commodities.]

Show that if u ) is continuous, increasing in homogeneous of degree one, and concave in p, then the utility function = Sup {u: where

Show that if u ) is continuous, increasing in homogeneous of degree one, and concave in p, then the utility function = Sup {u: where

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