Duality in Consumer Theory

Transcription

1 Pages from Chipman, J., D. McFadden and M. Richter: PTe!er'ences, Uncertainty and Optimality, Westview Press, Duality in Consumer Theory Vijay Krishna and Hugo Sonnenschein 1. Introduction The dual approach to demand theory is based on the fact that preferences can be represented in two forms other than the utility function; these are the expenditure function and the indirect utility function. Let U: R~...; R + be an upper semi-continuous unbounded utility function. The expenditure function e: R~ x R+...; R+ generated by U is defined by e(p,u) =min p. x subject to U (x) ~ u. The indirect utility function V: R~ x R+...; R+ generated by U is defined by V(p, /) =max U(x) subject to p. x!'/. The usefulness of the dual approach results from two facts: First, the Marshallian demand function can be computed from the indirect utility function by differentiation. Second. the Hicksian demand function can be computed from the expenditure function by differentiation. Both the Marshallian and Hicksian demand functions are obtained only as implicit functions when one derives demand directly from the utility function by the conventional Lagrange method. The purpose of this essay is to prove some strong duality theorems. We begin by defining a class of admissible utility functions U and we characterize the set of expenditure functions 8 and indirect utility functions V that are generated by U. We show that the function <P, which associates each U E U with the expenditure function generated by U, is invertible. Similarly, we show that the function X, which associates each U E U with the indirect utility function generated by U. is invertible. Finally, we provide formulas for tp- I,X-I. With the results in hand, one is able to pass from (a) each utility function in U to a unique equivalent expenditure function from 8 or indirect utility function from

2 DUALITY IN CONSUMER TIJEORY 45 V. (b) each expenditure function in 8 to a unique equivalent utility function from U or indirect utility function from V, (c) each indirect utility function from V to a unique equivalent utility function from U or expenditure function from The Class U of Utility Functions Our starting point is the class of utility functions V: R ~ --" R + satisfying the following conditions: (U I) V is non-decreasing; that is, x - x' E R ~ implies that V (x) ~ V (x'), (U2) V is upper semi-continuous; that is, for each u e R+, the set C(u) [x I V (x) ~ u) is closed, (U3) V (0) =0, (U4) V is unbounded. and (US) V is quasi-concave. that is, for each u e R+, the set C(u):; [x I Vex) ~ u} is convex. The class of utility functions satisfying these conditions is denoted by U. The power of our strong duality theorems derives in part from the fact that U admits essentially all preference relations that are considered in the neoclassical demand theory. Observe that the Slutsky theory can be developed without requiring preferences to be lower semi-continuous (Hurwicz and Uzawa. 1971), and that (US) is necessary if demand is to be single-valued.! Furthermore, demand functions which satisfy the strong axiom of revealed preference and have a convex range can be generated by elements of U. (UI) could be weakened at the cost of admitting negative prices and substantial complication. Conditions (U3) and (U4) do not restrict preferences. since a suitable monotonic transformation of a utility function satisfying (UI), (U2), and (US) will satisfy (UI) - (US) and represent the same preferences as V. Let V: R~ --"R+ satisfy (UI) - (U4). For p E R~, I e R+, and u e R+. denote by max( p, /) the problem of choosing x e R~ to maximize V (x) subject to p. x ~ / and by min (p,u) the problem of choosing x e R~ to minimize p. x subject to V (x) ~ u. The expenditure function e generated by V is defined by: e(p,u) is the value of min (p,u). Similarly, the indirect utility function V generated by V is defined by: V ( p, /) is the value of max (p, I). Provided that (U2) is satisfied, u is in the range of V, p e R~ and / e R+, V(p,e(p,u») ~ u, and (1) IV is lower semi-continuous if for each u E R~. the lower contour set {.x I V (.x) 5> u1 is closed.

3 46 KRISHNA AND SONNENSCHEIN e(p,v(p,l) 5,1. (2) The following theorem lists properties of expenditure functions that are generated by a utility function. THEOREM 1: Let U: R+ ---? R~ satisfy (Ul) to (U4). The expenditure function e : R~ x R+ ---? R+ generated by U satisfies the following properties: (el) e(p, ) is non-decreasing, (e2) e(p, ) is lower semi-continuous. (e3) e(,u) is non-decreasing. (e4) e( p,u) =0 ifand only if u =0, (e5) e(p, ) is unbounded. (00) e(.u) is concave. and (e7) e(,u) is positive homogenous of degree one. PROOF: (el). (e3), (e4), and (e7) are easy to verify. (e2): To show that e(p,.) is lower semi-continous, we must show Jhat for each 1the set (u I e (p,u) 5,1) is closed. Let u" e {u I e(p,u) 5,1] and u" ---? uo. Let x" solve min (p,u") for each n, so p. x" = e(p,u"). The sequence {x"} lies in the compact set {x e R I I p. x 5, 1]. By taking a subsequence, there exists Xo such that x" ---? Xo and p. Xo 5, I. We verify that U (xo) :2: uo. Since U (x") :2: u" for all nand u" ---? uo, we have that for arbitrary r > 0, U (X"):2: u" :2: Uo r, for n sufficiently large. The upper semi-continuity of U, (U2), implies that U(xo):2: uo, since r was arbitrary, Hence, Xo is feasible for min (p,u 0). Thus, e( p,u0) 5, p. xo 5, I. (e5): We show that e(p,.) is unbounded as follows. Suppose there exists an 1 such that e(p,u) 5, 1 for all u. Consider the sequence e(p,n) for n = 1,2,3,... By (U4), for all n, there exists an x" such that e (p,u") = p. x". For all n, U (x") :2: n by definition and by hypothesis p. x" 5, I. Let Xo solve max(p,l). This implies that U(xo) :2: U(x") :2: n for all n. But this is impossible. Hence, e(p,. ) is unbounded. (e6): Let pi, p2 e R~ and t e [0,1]. Define p =tpl + (1 - t)p2 and e= te(pl,u)+(1-t)e(p2,u). We have to show that e(p,u):2: e. Let xsolve min (p,u). By definition, e(pl,u) 5, pi. x and e(p2,u) 5, p2. x. The desired conclusion follows immediately. Q.E.D. The next theorem lists properties of indirect utility functions that are generated by a utility function.

4 DUAUTY IN CONSUMER TIIEORY 47 THEOREM 2: Let U: R~ ~ R + satisfy (Ul) to (U4). The indirect utility function V generated by U satisfies the following properties: (VI) V (p,') is non-decreasing, (V2) V(p,') is upper semi-continuous. (V3) V (.,1) is non-increasing, (V4) V(p,O) == 0, (V5) V(p. ) is unbounded. (V6) V is quasi-convex. and (V7) V is positive homogenous of degree zero. PROOF: (VI), (V3), (V4), (V5), and (V7) are trivial to verify. (V2): To show that V is upper semi-continuous in I, we need to show that the set (I I V (p, /) ~ u) is closed. We will use the fact that under the hypotheses of the theorem, (1) and (2) hold; also e is non-decreasing in u. Let I" ~ 1 and I" e {I I V (p, I) ~ u}. By the definition of I", V (p, 1") ~ u. Since e is non-decreasing in u, I" ~ e (p,v(p, 1"») ~ e(p,u). Hence, 1 ~ e(p,u) and since V is non-decreasing in I, V(p, 1 ) ;;:: V(p,e(p,u») ;;:: u. Thus, 1 e {II V(p,/) ;;:: u). (V6): Let pi, p2 e R~,/I, 12 e R+ and t e [0,1]. Define Notice that {x I p x 5: I)c(x I pl x S; II)U(X I p2 x S; I2}. Therefore, V(p, I) 5: max(v(pl, Il),V(p2,/2)}. Q.E.D. 3. The Class 8 of Expenditure Functions Let 8 denote the class of functions e: R~ x R + ~ R+ that satisfy (el) to (e7), and let tp denote the function that associates each utility function U e U with the expenditure function generated by U: L {>(U)]( p,u) =min {p. x I U (x) ;;:: u}. Since U is a subset of the utility functions that satisfy (UI) to (U4), by Theorem 1,4> (U) c 8. The next theorem proves that 4> (U) =8 and that tp is invertible. 2 THEOREM 3: The function tp: U~ 8 is a bijection; furthermore. 4>.-1 is defined by lmcfadden (1978) and others (see Diewert. 1982) have established a bijection between cost/expenditure function and the level sets C (u).

5 48 KRISHNA AND SONNENSCHEIN cp-i (e)](x) =max B (x), where B(x)= (u E R+ I p'x :?: e(p,u), for all p E R~). PROOF: The proof proceeds in three steps. Step 1: For e ES let qrl (e):; U. We first show that U: R~ ~ R + is well defined. Let x E R~. By (ea), 0 E B (x) and so B (x) is not empty. To assure that U is well defined we verify that B (x) is bounded above and closed. The former follows from the fact that e(p,.) is unbounded, (e5), and the latter follows from the lower semi-continuity of e (p,. ) and the fact that the inner. section of closed sets is closed. Since (el) implies that if u E B (x) and u' ~ u, then u' E B (x), it follows that B (x) is an interval. We now verify properties of U. (UI): Let x :?: x'. By definition of B, B(x):JB(x'), and so U(x) ~ U(x'). Hence U is non-decreasing. (U2): We must show that for all u, C (u):; (x I U (x) ~ u) = {x I for all p, p. x ~ e(p,u)} is closed. Let x" E C (u) for all n and x" ~ xo. For all p, and for all n, p. x" ~ e(p,u). Thus p. Xo :?: e(p,u) for all p, and so Xo E C (u). Hence, U is upper semi-continuous. (U3) is a direct consequence of (ea). (U4): We argue by contradition. Suppose there exists a usuch that for all x E R~, U (x) < U. Using the definition of U, this means that for all x E RL there exists ape R~ such that e(p, u) > p. x. Since e (.,u) is concave, it is bounded on P ={p E R~ I II p II = I]. Let ebe an upper bound of e(., u) on P, and let x= (e, e,...e) E R~. Now, for all p E R~, ( ) < (- - ':'\_--2-. e lip II ' u - e - lip II e, e,..., e, - lip II x. But this contradicts our hypothesis. (U5): Since C(u)= (x I U(x) ~ u)::: {x I p x ~ e(p,u), for all pe R~) is convex for each u, U is quasi-concave. Step 2: We next show that cp(q,-i (e» = e for all e E 8. Let e E 8 and U = cp-i(e). We will prove that for each (po,uo),e(po,uo)=min{po.x I U(x) ~ u). Using (e7), this implies that for all p E R~, e(p, u) ~ p. x.

6 DUAUfY IN CONSUMER 11IEORY 49 The concavity of e(., uo), (e6), implies that there exists an XO IS R' such that for all p IS R~ (3) First, note that x o IS R~. This is verified by choosing p =po + e i (where e i is the jth component of the unit basis) in (3) for j = 1,2,..., I, and using the fact e (., uo) is non-decreasing, (e3). Second, we show e (po, u 0) == p O xo. Again, from (3) we have that for all t > 0, e (tpo,uo) S; e (po, uo) + (tpo - po). xo. Using (e7), the homogeneity of e, this can be rewritten as, (t -1)e(po, uo) S; (t -1) po. xo. For t> 1 this yields e(po, uo) S; po xo. For t < 1, the opposite inequality is obtained. Thus e(po, u O ) = po. x O. Now, using e(po, uo) po xo in (3) yields e(p,uo) S; p xo for all p IS R~. By the definition of B (xo), UO IS B(xo), and U (xo) == max B (xo) ~ uo. Thus, X O is feasible for mine po, uo). Furthermore, if x IS R~ satisfies U (x) ~ uo, then UO IS B (x), or equivalently, p. x ~ e(p,uo) for all p. In O O O particular p. x ~ e(p,u 0) == p. xo. Thus X O minimizes p. x subject to U(x) ~ uo and e(po, uo)=minpo x subject to U(x) ~ uo. Therefore, U generates the expenditure function e. Step 3: Finally, we show that for all U e U, 0/-1 [ q>(u)] = U. For U IS U, let J(U) =e and -l(e) == U*. We claim that U* = U. Let x* IS R~. First, notice that u* = U (x*) e B (x*), since for any p IS R~, e( p,u*) S; p. x*. Next, if u' > u*, then we claim that u'., B(x*). Now, u' > u* implies that x*., C(u')== (x I U(x) 2: u'}. C(u') is closed, convex, and satisfies the condition: x IS C ( u') and (x' - x) E Ri implies that X'IS C ( u'). Hence, there exists a p* E R~, such that p*. x* <p*. x for all x e C (u,).3 Thus, p*. x* < e(p*, u')andu'., B(x*). We have shown that U(x*)=maxB(x*). Hence U* ==U. Q.E.D. 'nus uses the following fact, which is a modification of a standard separating hyperplane problem: Suppose C c R~ is a closed, convex set that satisfies the condition that %. C and %' - % R~ imply that %' C. II %0 * C. then there exists a po. R~+ such that for all %. C, po. %0 < po. %. (To prove this fact. observe that if p R~ - ( 0 I separates %0 from C, then we can construct a po R~, very dose to p. that still separates %0 from C.)

7 50 KRISHNA AND SONNENSCHEIN 4. The Class V of Indirect Utility Functions Let V denote the class of functions V: R ~ x R + -+ R + that satisfy (VI) - (V7). and let X denote the function that associates each utility function U E U with the indirect utility function generated by U: [ X (U)]( p. I) == max {U (x) : p. x ~ I]. Since U is a subset of the utility functions that satisfy (U1) - (U4). by Theorem 2. X (U) E V. Theorem 4 proves that X (U) = V and that Xis invertible. THEOREM 4: The funclion X: U -+ V is a bijection; furlhermore, X-I is defined by [X-I(V)](x)==max (u I V(P. p x) ~ u for all p E R~}. The proof follows from a sequence of Lemmas. Lemma 1 introduces a function "': V-+ fj and shows that it is well defined. Lemma 2 shows that", is a bijection and defines ",-I. Lemma 3 establishes that X=",-I 0", and hence that X is a bijection. Finally. Lemma 4 shows that X-I is defined by lx-i (V)](x) =max{u I V(P.p x):?! u.forallper~}. LEMMA 1: Suppose Ve V. Define ",(V): R~ xr+ -+R+ by [",(V)](P.u) == sup [I I V(P. /) < use. Then. "': V-+ fj is well defined. Before we begin the proof. we draw attention to Figure 1. which indicates that elements of fj and V are not necessarily continuous in u and I respectively and shows the relation between V and ",(V). V(p.' ) 14 / I 13 u\ 12 I} "'vj i I} I u\ II. i i II Figure 1

8 DUAllTY IN CONSUMER TIlEORY 51 PROOF OF LEMMA 1: Observe that since V(p,. ) is upper semi-continuous, [\ji(v)](p,u) =min (II V(p, I) 2! u). (4) To prove the lemma we must show that for all V ev, \ji(v) e 8. Let e ",(V). (ei), (e3), (e4), (e5), and (e7) follow directly from the definition of "'(V). (e2): To prove the lower semi-continuity of e(p,.) we use (4). Let u" e (u I e(p,u) ~ 1 ) and u"--') uo. For all n, e(p,u") ~ 1. By (4), V(p, 1 ) 2! u",foralln. Thus, V(p, 1 ) 2! uo,andsoe(p,uo) ~ 1. Hence, e(p,.) is lower semi-continuous. (e6): To prove the concavity of e(,u), let pl,p2er~ and Ie [0,1]. Define p::: tp 1+ (1- I) p2 and e== le(pl,u) + (1- t) e(p2,u). We must show thate(p,u) 2! e. For each r > 0, there exists an II such that II> e(pl,u) - rand V(pl,/l) < u. Similarly, there exists an 12 such that 12> e(p2,u) rand V(p2,/2) < u. Define I=t/ l +(I-t)/ 2 By the quasi-convexity of V, V(p,l) ~ max {V (pi'll), V(p2,/2») < u. Furthermore, I> e- r. But by the definition of e, e(p,u) 2! I> e- r. Thus, for all r > 0, e(p,u) > e- r. Hence, e(p,u) 2! e. Q.E.D. LEMMA 2: The function", : V--') 8 is a bijection; furthermore, \ji-1 is defined by, [",-1 (e)](p,l) =max (u I e(p,u) ~ I}. PROOF: Observe that [",-I(e)](p,/)=inf[u I e(p,u»i}. (5) The proof preceeds in three steps. Step 1.' For e e 8, let ",-I(e) ::: V. We first verify that Ve V. (VI), (V3). (V4), (V5), and (V7) follow directly from the definition of ",-1. (V2): We must show that the set [I I V(p, /) 2! u} is closed for all u e R+. Let V (p, 1") 2! u for all n, and I" --') 1. By definition, V (p, 1") 2! u implies that/" 2! e(p,u). Thus,/o 2! e(p,u) and hence, V(p, I) 2! u. (V6): Let (pl,/l) and (p2,/2) be such that V(pl,/l) ~ UO and V (p2,/2) Suo. Define (,p,l):::t(pl,/i)+{l t)(p2,/2) for 0 S t S 1.

9 52 KRISHNA AND SONNENSCHEIN 1 Using (5), we have that for all n, UO + - e (u I e(pl,u) > II} n (u I n e(p2,u) >/2}. Hence, by the concavity of e(,u), (00), e(p,u o + 1.) n ~ t. e(pl,uo + 1.) + (1- t)e(p2,uo + 1.) > 'i. Thus, for all n, UO + 1. e n n n {u I e( p,u) > I}. Using (5) again, yields that V (p,/) s; u + 1., for all n, and _ n thus V(p, I) S; uo. Step 2: We now show that", [",-1 (e)] = e for each e e 6. For e e 6, let ",-1 (e) = V and ",(V):::: e"'. We show that e:::: e"'. Let (p,u) be arbitrary. The definition of ",-l(e) yields V(p, e(p,u» ~ u. Using the fact and applying the definition of ",(V) implies that e(p,u) ~ e'" (p,u). On the other hand (4) implies that V(p,e"'(p,u» ~ u. Using this fact in the definition of ",-1 (e) immediately implies that e(p,u) S; e'" ( p,u). Hence, e = e"', Step 3: Finally, we show that for each V e V, ",-1 ["'(V)] = V. For V e V, let "'(V) = e and ",-1 (e) = V*. We show that V = V*. Let (p, I) be arbitrary, The definition of ",-l(e) yields e(p,v*(p, I) S; I. Using this fact in (4) then implies that V( p,l) C! V'" (p, I). On the other hand, (4) implies that e(p,v(p,/) s;/. Applying the definition of ",-1 (e) yields V (p, I) S; V* (p, I). Hence, V = V*. LEMMA 3: X = ",-1 0 <P PROOF: For U e U, let (U)= e. By Theorem I, e e 6. We must show that ",-1 (e) is the indirect utility function generated by U. Let Xo solve max( p,/), It is sufficient to show that U (xo):::: [",-1 (e )]( p, /). Since (U)=e, e(p,u(x» S; p xo S; I. Thus, U(xo)e (u I e(p,u) S; I}. It remains to show that it is the largest element of this set; that is, if u > U(x ), then e(p,u) > I. Suppose u' > U{xo) and e(p,u') S; I. Let x' be the solution to min(p,u'). Thus, U (x') ~ u' > U (xo) and p. x' = e{p,u') S; I. This contradicts the fact that XO solves max(p, I). Hence, u' > U(x ) implies that e(p,u') > I. Thus, U(xo) = max (u I e(p,u) S; I} :::: [",-1 (e)](p,/). Q.E.D. LEMMA 4: [X- 1 (V)](x) = max (u I V(p,p. x) ~ u, for all p e R~). PROOF: By Lemma 3, X = ",-1 o, and since both 1> and", are bijections, so are Xand X-I = -1 0 ",. For V e V, let "'(V) =e. We claim that (u I V(p, p. x) ~ u, for all p e R~)= {u Ip x ~ e(p,u), for allper~}. This is an immediate

10 DUAUTYINCONSUMER TIlEORY 53 consequenceof(4),sincev(p,p x):!: uifandonlyife(p.u) s; p x. Now, ['V-l (e)](x) = max (u I p. x :!: e(p,u), for all p e R~} = max {u I V (p, p. x) :!: u, for all p e R~ 1, and writing e ='V( V) yields the required expression. Q.E.D. 5. Strong Duality Theorems The previous results are summarized in the following theorem. THEOREM 5: The following diagram commutes and 1>, X, and 'V are bijections. U v \ Figure 2 Starting with an arbitrary expenditure function e from 8, U =1>-1 (e) is the unique utility function in U that generates e. U can alternatively be obtained by applying 'V-I and the X-I to e. Similarly, starting with an arbitrary indirect utility function V from V, U =X-I (V) is the unique utility function that generates V. and the unique utility function that generates 'V(V), etc. A Comment The key to the strong duality theorems is the choice of the class of utility functions U. Since we do not require lower semi-continuity for each U e U, we allow for the possibility that the indirect utility function generated by an element of U is not continuous. Furthermore, under our assumptions, each e e 1> (U) is not strictly increasing in u, and so e = (U) and V =X(U) are in general not inverse functions for each fixed p. This requires that we relate ' (U) and X(U) by a method that is not standard; this is the role of 'V. As we said in the introduction, we prefer to work with U, rather than the subset U* of U obtained

11 54 KRISHNA AND SONNENSCHEIN by restricting utility functions to be continuous because U is sufficient for the classical theorems of demand analysis. In addition, previous attempts to identify 'II (U*) have involved awkward conditions. (See Diewert 1982, Theorems 1 and 2 and the comments which follow.) 2 Figure 3 This fact is illustrated by the example of Figure 3: U is upper semicontinuous and generates an expenditure function e which belongs to e. The function e is strictly increasing, it inverts (for each p) to give the continuous indirect utility function V = X(U), and in short it satisfies just about every natural condition that an expenditure function can be expected to satisfy. However. it is clear that e cannot be generated by any continuous utility function. Since e t!i t:p (U*), the example proves that : u* ---+ e is not a bijection; the example also indicates why it is difficult to describe'll (U*). An alternative route for obtaining a strong duality theorem is to choose U** c U* so that'll (U**) c I; can be naturally described and'll: U** ---+ 'II (U**) is a bijection. A tidy theory would require, in addition, that X (U**) can be naturally described and that X: U** ---+ X(U**) c X(U) is a bijection.

12 DUALITY IN CONSUMER 11lEORY 55 References Diewert. E. "Duality Approaches to Microeconomic Theory." In HaNibook of MalhemtJlical Ecol'lOmics, vol. 2, edited by K. J. Arrow and M. D. Intriligator, Amsterdam: Nonh-HoUand Hurwicz, Land H. Uzawa. "On the Integl1lbility of Demand Functions." In Pr~f~r~lIus, Utility and D~mtJnd, edited by J. S. Chipman, M. N. Richter, and H. Sonnenschein New York: Harcourt, Bl1Ite, Jovanovich, McFadden. D. "Cost, Revenue. and Profit Functions." In ProducliOll ECOllOmics: A Duo.l Approach to Theory and ApplicaliollS. vol. 1. edited by M. Fuss and D. McFadden Amsterdam: Nor!h-HoUand