Recover Utility From Exp

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Lecture 5 Rationalizable Demand Functions

1. So far our approach is to begin with assumptions about consumer prefer-ences and rationality, and derive the restrictions these assumptions imposeon consumer choices. Alternatively, we can begin with consumer choices,and ask whether these choices could have been made by a rational con-sumer whose preferences satisfy our axioms. If so, we say that the choicesare rationalizable or consistent with rational behavior.

2. Question: How do we know whether a dierentiable demand functionx (p; y) is a solution to a utility maximization problem:

maxx

u (x)

s:t: px y;

x 0;

for some quasiconcave, increasing utility functionu?

3. The question can be answered in the two steps. First, any functionE : 0

: (1)

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Since for each p,

fxju (x) ug fxjpx e (p; u)g ;

it follows thatfxju (x) ug A (u) :

In addition, everyx0 not belonging to the better-than set is excluded fromA (u). The reason is that the better-than set is convex and closed whenu is quasiconcave. By the separating hyperplane theorem, there exists ahyperplane that separates x0 from the better than set. If a hyperplaneseparatesx0 from the better-than set, so does the supporting hyperplanewith the same slope.

3. Thus,A (u) = fxju (x) ugthe set bounded by supporting hyperplanesand the better-than set are equivalent. We have used the better-than setto derive the expenditure function. It is also possible to do it the other wayroundusing the the expenditure function to derive the utility function.

4. So we start with A (u) dened in (1). Recall that for any p (from thediscussion of the Shepards lemma)

pxh (p; u) e (p; u) 0:

Thus, A (u) is non-empty (as xh (p; u) 2 A (u)). For each p, the setfx 2


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(a) Step 0: A (u) is non-empty. For any p; p

e (p; u) e (p; u) + re (p; u) (p p)= pre (p; u) (p p) + re (p; u) (p p)

= re (p; u)p:

The inequality follows from the concavity ofe, and the equality fol-lows from the fact that e is homogeneous of degree one in p. Finally,re (p; u) 0 as e increases in p. It follows re (p; u) 2 A (u).

(b) Step 1: u is well-dened. Since e (p; u) is strictly increasing andunbounded in u, the set fu 0jx 2 A (u)g is bounded from above.Letu denotesup fu 0jx 2 A (u)g, andu1; u2;:::be a sequence thatconverges to u, with ui 2 fu 0jx 2 A (u)g for all i. Suppose u =2fu 0jx 2 A (u)g. Then there exists some p >>0 such that px 0: (3)

(Note that x is xed in this problem.) This problem is hard to solve atthe rst glance because it involves an innite number of constraints. Butwe can transform this problem into one that is solvable.

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9. Denev(p; y) = max

u

u s.t. e (p; u) y:

Solving this problem is easy as e is increasing in u. Just sete (p; u) =y .

For example, ife = u (pr1+pre)1=r, then

v= y(pr1+pre)1=r

:

10. Instead of solving (3), we rst x p >>0 and consider the problem:

maxu

u s.t. px e (p; u) : (4)

11. By construction, the solution to this problem isv(p; px). Since we considerin (4) only a subet of constraints in (3) (as a result, any feasible solutionto (3) is feasible in (4)) for any p >>0; v(p; px) u (x). Hence

minp>>0 v(p; px) u (x) :

12. On the other hand, since e is increasing in u; for all p >>0

px e (p; v(p; px)) e

p; min

pv(p; px)

:

Hence,minpv(p; px) is a feasible solution to (3); therefore,

u (x) minp

v(p; px) :

13. It follows thatu (x) = min

pv(p; px) : (5)

This problem can be solved readily. See Reny pp. 78 for an example.

14. Here we dene v from e. There is an alternative interpretation. (5) willhold ifu is a utility function satisfying the standard properties and v isthe corresponding indirect utility function. See Reny pp. 79.

0.2 Integrability

1. We have shown how to construct a utility function from an expenditurefunction. Now, we can show that a demand functionx (p; y)is the solutionto a utility maximization problem if we can show that x can be generatedby an expenditure function.

2. Specically, given x (p; y) we want to know whether there exists e (p; u)

such that fori = 1;:::;n;

@e (p; u)

@pi=x (p; e (p; u)) : (6)

This is called the integrability problem because we are trying to solve asystem of partial dierential equations.

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3. First of all, note that if such a function e exists, then by dierentiatingthe above equation with respect to pj we obtain

@2e (p; u)

@pi@pj=

@x (p; e (p; u))

@pj+

@ x (p; e (p; u))

@y x (p; e (p; u)) ;

which is simply the ij element of the substitution matrix. Since @2e(p;u)@pi@pj

=

@2e(p;u)@pj@pi

, the substitution matrix must be symmetry. Thus a necessary

condition for the existence of a solution to the integrability problem isthat the substitution matrix is symmetric.

4. It turns out that by the Frobenius theorem that this condition is sucientas well. Ife is a solution to (6), we still need to show that e processes allthe properties of an expenditure function.

5. Theorem: e processes all the properties of an expenditure function if andonly ifx (p; y) satises budget balancedness, and its substitution matrixis symmetry and negative semidenite.

6. We wont go through the proof. Only note that we have the Sheppardslemma by construction. Sincex 0,e must be increasing inp= p:Also bydenition,e is concave if and only if the substitution matrix is negativesemidenite. You are asked to show in the problem set that x satisesbudget balancedness if and only ife is homogeneous to degree one.

7. To conclude, we have showed that any demand function x that satisesbudget balancedness and possesses a symmetric and negative semide-nite substitution matrix is rationalizable by a strictly increasing quasicon-

cave utility function. Since we have already shown thatx generated by astrictly increasing quasiconcave utility function must possess these proper-ties. Thus, budget balancedness, symmetry and negative semidenitenessis the only conclusions we can draw if all we know is that the consumerhas a quasiconcave, strictly increasing utility function.

8. This is an extremely important result because it allows economists todirectly write down demand functions that possess additional propertiesthat are desirable for empirical estimation. As long as the function satisesbudget balancedness, symmetry and negative semideniteness, we knowthat it is consistent with rational behavior. There is thus no need to startexplicitly with a utility function. The problem with the latter approachis that it is dicult to make sure that the demand function generated by

a utility function possesses the properties we want.

9. Here we present the duality theorem as an intermediate step of recoveringthe underlying utility function. But the result is important in its ownright. Later on we will learn that the exact same argument allows us torecover the production function from the cost function.

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1 Revealed Preference

1. Thus far, we have developed consumer theory on the basis of axioms im-posed on consumer preferences. One (minor) drawback of this approach isthat preferences are not directly observablewhat we observe are choicesand not the underlying preferences. In this section, we can actually obtainthe same consumer theories on the basis of axioms imposed on consumerchoices.

2. Denition: A consumers choice behavior satises the Weak Axiom ofRevealed Preference (WARP) if for every distinct pair of bundles x0, x1

withx0 chosen at prices p0 andx1 chosen at prices p1, then

p0:x1 p0:x0 )p1:x0 > p1:x1:

The Weak Axiom says that if a consumer chooses x0

at somep0

whenx1

isalso feasible, then he will never choose x1 over x0 when both are feasible.If the consumer chooses x1 atp1, then x0 must be infeasible at p1.

3. Note that the Weak Axiom requires only pairwise consistency; it does notincorporate any notion of transitivity. Thus, a consumer who obey theWeak Axiom may not the rational. On the other hand, a consumer whosepreferences are complete, transitive, and strictly convex will always obeythe Weak Axiom. (Note that convexity is needed to ensure the uniquenessof the optimal consumption bundle.)

4. Examples: Do the following consumer behavior satises the Weak Axiom?

(a)

p1 p2 q1 q2Year 1 1 1 1 3Year 2 2 1 2 1

p1 p2 q1 q2Year 1 1 1 1 3Year 2 2 1 0.5 4

1.1 Implications of the Weak Axiom

The Weak Axiom, together with budget balancedness, imply 1) that consumerchoice must be homogenous of degree 0 in p and y, and 2) the compensated lawof demand.

(Reny pp. 88) Theorem 1. Let x (p; y) denote the consume choice when theprice vector is p and the income is y . WA and budget balancedness imply thatx (p; y) is h.t.d. 0.

This is straightforward. A proportional change in price and income leaves thebudget set unchanged. And the weak axiom essentially requires that the optimalconsumer choice be unique for each budget set. (Note: Budget balancedness is

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needed because the current form of WA implies the consistency of pairwisechoice only when the consumer always use up his income. I suppose one may

in principle state a more general form of the WA would imply h.t.d. 0 withoutbudget balancedness, but I havent done it myself.)

Proof: Suppose x0 = xp0; y0

and x1 = x

tp0; ty0

for some t > 0. By

budget balancedness, we have the expenditure in

tp0x1 = ty0

= tp0x0:

Hence, by WA, x1 must equal to x0.Denition: We call (p0; y0) a Slutsky compensated price change from (p; y)

ifp0x (p; y) = y0. That is if the new incomey0 is adjusted in such a way thatx (p; y) is still feasible under p0.

Theorem 2. (Reny pp. 89, Mas-Colell Prop. 2.F.1) Suppose the demand

functionx (p; y) satises budget balancedness, then x (p; y) satises WA if andonly if for any compensated price change from an initial situation (p; y) to anew price-income pair (p0; y0) = (p0; p0x (p; y)), we have

(p0 p) [x (p0; y0) x (p; y)] 0: (7)

The above equation is another version of the compensated law of demand.We saw before another version where income is adjusted to keep utility constant.Here the income is changed to keep the optimal bundle under the initial pricesand income feasible. The Sheppards lemma means that for small changes inprices the two versions is identical. A consumer must consumer less of good iwhen it price increases. Since a rational consumer obeys the Weak Axiom, it isobvious that the compensated law of demand based on constant utility implies

the compensated law of demand based on Slutsky income compensation.Proof:1. WA and BB)(7).The result is immediate ifx (p0; y0) = x (p; y). Supposex (p0; y0) 6= x (p; y).

By budget balancedness and the denition of Slutsky income compensation, wehave

p0x (p; y) = p0x (p0; y0) :

By WA, x (p0; y0)is not feasible under (p; y). Thus,

px (p0; y0)> px (p; y) :

Adding the two equation yields the desired result.2. (7) and BB)WA

The compensated law of demand would be violated if the Weak Axiom doesnot hold for some compensated price changes. Part 2 then follows from thefact if the Weak Axiom fails at some price-income pairs, then it fails at somecompensated price changes. Formally, the Weak Axiom holds if and only if, forany two price-income pairs (p; y)and (p0; y0), we havepx (p0; y0)> w0 whenever

px (p0; w0) = w amdx (p; y) 6=x (p0; y0).

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Proof: (See Mas-Colell pp.31) Suppose we have (p0; y0) and (p00; y00) suchthat

x (p0; y0) 6=x (p00; y00) ;

p0x (p00; y00)< y0;

p00x (p0; y0)< y00:

. Draw gure. Now consider the budget line that passes through x (p0; w0) andx (p00; w00). Denote this budget line bypx = y. Now, it is clear from the gurethat this new budget line is in the interior of the union of the two initial budgetsets. That is, for allx such that px = y, either p0x < y0 or p00x < y00. Thus,eitherp0x (p; y)< y0 orp00x (p; y)< y00. In the rst case, we have p0x (p; y)< y0

andpx (p0; y0) = y which violates the compensated law of demand. In the secondcase, we have p00x (p; y)< y00 andpx (p00; y00) = y.

The above theorem tells us given BB that WA is equivalent to the compen-

sated law of demand.Suppose, in addition, that the demand function is dierentiable. We can

write (7) as

(p+tz p) [x (p+tz; (p+tz) x (p; y)) x (p; y)] 0;

wherez 2


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The nal property of the Marshallian demand, namely, the symmetry of thesubstitution matrix, is not a consequence of the Weak Axiom when there are

more than two goods. (See Mas-Colell Prob 2.F.15 for a counter example.) Toobtain symmetry we need a stronger axiom.

Defn: (Mas-Colell pp. 91) The market demand function x (p; y)satises theStrong Axiom of Revealed Preference if for any list

p1; y1

;:::; (pn; yn)

with xpi+1; yi+1

6= x

pi; yi

for all i n 1, we have pnx

p1; y1

> yn

whenever pixpi+1; yi+1

yn for all i n 1.

In words, the Strong Axiom says that ifxi+1 is revealed to be preferred toxi for all i = 1;:::;n 1, then x1 cannot be revealed to be better than x1.

Thm: (Mas-Colell pp.91) Ifx (p; y) satises the Strong Axiom of RevealedPreference then there is a rational preference relation that rationalizesx (p; w).

Proof: See Mas-Colell pp.92.The theorem means that the substitution matrix of x (p; y) that satises

the Strong Axiom must be symmetric as the substitution matrix of a rationalconsumer is symmetric.

Note that 1) Any demand function generated a rational preference relationmust by denition obeys the Strong Axiom. 2) While the theorem says thatchoice behavior that obeys the Strong Axiom is consistent with rational behav-ior, it does not say that the consumer must be rational, for SA says nothingabout choice behavior when the choice set is not a budget set.

We wont go over the proof. But the basic idea is the following: the SAstipulates any observed consumer choices under linear budget constraints mustbe transitive. However, some consumption bundles cannot directly or indirectly

compared under linear budget constraints. (For example, a bundle in the non-convex part of a preference relation is never chosen under any price.) Hence,the ordering implied by the strong axiom is incomplete. The proof shows thatthis partial ordering can be extended to the whole commodity space in a waythat doesnt violate transitivity.

When n = 2, the Weak Axiom and the Strong Axiom are equivalent. Wecan show that when n = 2 budget balancedness and homogeneity of degree 0implies that the substitution matrix is symmetric. (You are asked to prove thisin the problem set.) This means that any demand function that obeys the weakaxiom is consistent with utility maximization and, hence, must also obey thestrong axiom.

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