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General equilibrium comparative statics: discrete shocks in production economies
Journal of Mathematical Economics 40 (2004) 153–163
General equilibrium comparative statics: discrete shocks in production economies John H. Nachbar Department of Economics, Washington University, Box 1208, One Brookings Drive, St. Louis, MO 63130, USA
Abstract Nachbar [Econometrica 79 (5) (2002) 2065] established minimal conditions under which, following an infinitesimal shock to endowments in an exchange economy, changes in equilibrium prices are negatively related to changes in aggregate consumption. The present paper extends Nachbar (2002) to cover discrete shocks to technologies, ownership shares, and endowments in production economies. As in Nachbar (2002), the analyst’s choice of price normalization plays a key role. The required normalization is nonstandard but has a sensible interpretation. © 2003 Elsevier B.V. All rights reserved. Keywords: General equilibrium; Comparative statics; Weak Axiom
1. Introduction Consider a shock to technologies, ownership shares, or endowments in a production economy. A plausible conjecture is that (p䉫 − p∗ ) · (¯x䉫 − x¯ ∗ ) ≤ 0, where p䉫 − p∗ is the change in equilibrium prices and x¯ 䉫 − x¯ ∗ is the change in aggregate equilibrium consumption. Informally, changes in equilibrium prices are negatively related to changes in aggregate equilibrium consumption. It has been known at least since Hicks (1939) that this conjecture is false, in general, even in single consumer exchange economies. This paper provides minimal conditions under which the conjecture is, in fact, true. This paper is an extension of Nachbar (2002), which provided analogous results for infinitesimal shocks to endowments in exchange economies. Since the discrete results are easiest to interpret in relation to the infinitesimal results of Nachbar (2002), I begin by reviewing the earlier paper. E-mail address: [email protected] (J.H. Nachbar). 0304-4068/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0304-4068(03)00093-4
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The analysis in Nachbar (2002) requires that, regardless of how prices are actually normalized in the economy, prices be renormalized to satisfy p · µ ¯ = 1, where µ ¯ is the vector giving the economy’s aggregate marginal propensity to consume (MPC) each of the L goods. In a single consumer economy, the MPC is simply the derivative with respect to nominal wealth of that consumer’s demand. In multiconsumer economies, µ ¯ is a weighted sum of individual MPCs. The aggregate MPC price normalization is nonstandard. It has the unattractive property that it depends both on the reference equilibrium and, in multiconsumer economies, on the distribution of the endowment shocks. And estimating the aggregate MPC may be difficult. In defense of the aggregate MPC price normalization, Nachbar (2002) makes two observations. First, the aggregate MPC price normalization is necessary in the following sense. In single consumer economies, the MPC price normalization is the unique linear price normalization (up to scalar multiplication) for which (p䉫 − p∗ ) · (¯x䉫 − x¯ ∗ ) ≤ 0 holds for all endowment shocks. A similar but weaker uniqueness claim holds in multiconsumer economies. Second, if µ ¯ > 0 (all goods are weakly normal in the aggregate) then the µ ¯ normalization yields a sensible interpretation of the vector of normalized price changes.1 A decrease in the µ-normalized ¯ price of good 1 implies that the price of good 1 has fallen relative to the value of the composite commodity (µ ¯ 2, . . . , µ ¯ L ) > 0, which is the economy’s marginal consumption of all other goods. Since the comparative statics result in Nachbar (2002)considers a marginal change in endowment, interpreting relative price changes in terms of marginal consumption is, I think, natural. If, on the other hand, µ ¯ > / 0 (some goods are inferior in the aggregate) then the vector of normalized price changes may be difficult to interpret; see Nachbar (2002). I have glossed over the fact that, in multiconsumer economies, it is unclear what weighting scheme one should use for computing µ. ¯ Nachbar (2002) actually considers two different weighting schemes, yielding two different aggregate MPCs, µ ¯ γ and µ ¯ δ . For µ ¯ γ , the weight on i’s individual MPC is based on the value, at the original equilibrium prices, of the change in i’s equilibrium consumption. For µ ¯ δ , the weight on i’s individual MPC is based on the value, at the original equilibrium prices, of the change in i’s endowment. The µ ¯ γ and µ ¯ δ normalizations yield price change vectors corresponding to somewhat different interpretations of what one means by a relative price change. I find both interpretations reasonable. Relative to Nachbar (2002), the contribution of the present paper is three-fold. 1. This paper provides comparative statics results for discrete, as opposed to infinitesimal, shocks to endowments in exchange economies. This extension is straightforward but still worth recording.2 Corresponding to the aggregate MPC vectors µ ¯ γ and µ ¯ δ in Nachbar ¯ (2002), this paper uses the incremental propensity to consume (IPC) vectors µ ¯ x and µ ω. The interpretation of an IPC normalized price change is analogous to the interpretation of an MPC normalized price change. L L L L N RL + = {x ∈ R : x ≥ 0, ∀}, R+\0 = {x ∈ R+ : x = 0}, and R++ = {x ∈ R : x > 0, ∀}. Consider any L L L x ∈ R . Then x ≥ 0 means x ∈ R+ , x > 0 means x ∈ R+\0 , and x 0 means x ∈ RL ++ . 2 A discrete extension of the basic inequalities in Nachbar (2002) has been independently reported in Quah (2003). 1
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The µ ¯ ¯ γ result of Nachbar (2002), assumes that individual demand x result, like the µ satisfies a Weak Axiom of revealed preference (WA) condition. In contrast, the µ ¯ ω result, like the µ ¯ δ result of Nachbar (2002), assumes that aggregate demand satisfies a WA condition. The latter assumption is strong but it is necessary; see Remark 3 in Section 3.2.3 2. The µ ¯ ¯γ x result holds, trivially, in general production economies. The analogous µ result, for infinitesimal shocks, also holds in production economies, but its expression is somewhat cumbersome; this extension was therefore omitted from Nachbar (2002). Section 3.1 also contains an additional comparative statics result for the µ ¯ x normalization that links price changes directly to changes in the underlying production technology. Extending the µ ¯ ω result to production is problematic. I discuss this further in Remark 4 in Section 3.2. 3. This paper proves a uniqueness theorem for µ ¯ ¯ ωx that is the analog of the uniquex and µ ¯ δ discussed in Nachbar (2002) but not proved there in detail. ness result for µ ¯ γ and µ In particular, Lemma 1 of the present paper fills in an important step in the uniqueness argument for both µ ¯ γ and µ ¯ x. I focus here on supply shocks; the analysis of demand shocks, shocks to preferences, is comparatively straightforward. For demand shocks, it follows immediately from profit maximization that (p䉫 −p∗ )·(¯x䉫 − x¯ ∗ ) ≥ 0. In the case of an exchange economy, a demand shock results in no change in aggregate consumption, so that the result holds trivially. The remainder of this paper is organized as follows. Section 2 covers basic definitions, Section 3 contains the comparative statics results, and Section 4 discusses uniqueness. Section 5 concludes with a brief discussion of the literature.
2. Definitions 2.1. Basic notation There are I consumers and L consumption goods. xi ∈ RL + denotes a consumption vector for consumer i. x = (x1 , . . . , xI ) is a consumption allocation while x¯ = i xi is an aggregate consumption vector. L i i Let φi : RL+1 ++ ⇒ R+ be consumer i’s demand function: φ (p, m ) is consumer i’s i demand when prices are p 0 and nominal wealth is m > 0. The analysis extends easily to demand correspondences rather than demand functions. Assume that each φi satisfies Walras’s Law: p · φi (p, mi ) = mi . For any vector βi ∈ RL +\0 define f Di (p, βi ) = φi (p, p · βi ). f Di is individual demand in an exchange economy in which consumer i is given endowment βi . Note that f Di is defined even if the actual economy is, in fact, a production economy. 3 For recent work on sufficient conditions for aggregate demand to satisfy WA, see Jerison (1999) and Quah (1999).
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Finally, let β = (β1 , . . . , βI ) be a profile with each βi ∈ RL +\0 and define f Di (p, βi ). f¯ D (p, β) = i
2.2. The aggregate wealth effect and the aggregate IPC Define ¯ ∆ (p, x¯ , β) = f¯ D (p, β) − x¯ . ψ ¯ (p, x¯ , β) is the aggregate wealth If x¯ is aggregate consumption when prices are p then ψ effect when the wealth of consumer i is changed to p · βi while prices remain unchanged. If p · ψ(p, x¯ , β) = 0, define µ ¯ (p, x¯ , β) =
1 ¯ (p, x¯ , β). ψ ¯ p · ψ (p, x¯ , β)
µ ¯ is the aggregate incremental propensity to consume (IPC) vector when wealth is changed to p · βi while prices remain unchanged. Consider any price pair (p∗ , p䉫 ). Since p∗ · µ ¯ (p∗ , x¯ , β) = 1, if p䉫 · µ ¯ (p∗ , x¯ , β) > 0 then p䉫 p䉫 · µ ¯ (p∗ , x¯ , β)
− p∗ ,
is a vector of normalized price changes, for the linear price normalization p · µ ¯ (p∗ , x¯ , β) = 1. 3. Comparative statics 3.1. The µ ¯ x results Fix preferences and let ξ denote a production economy. ξ specifies endowments, production sets, and ownership shares. It is natural to assume that there are factors or intermediate goods that are not directly consumed. Any such good is not included among the L consumption goods. Say that f Di satisfies the Weak Axiom (WA) (strong form) at the price pair (p∗ , p䉫 ) and the vector βi iff the inequalities p∗ · f Di (p䉫 , βi ) ≤ p∗ · f Di (p∗ , βi ) and p䉫 · f Di (p∗ , βi ) ≤ p䉫 · f Di (p䉫 , βi ) both hold iff f Di (p∗ , βi ) = f Di (p䉫 , βi ) iff p∗ and p䉫 are collinear.4 Theorem 1. Let p∗ be an equilibrium price vector of the ξ ∗ economy and let x∗ be the associated equilibrium consumption allocation. Let p䉫 be an equilibrium price vector of 4 The standard WA drops “iff p∗ and p䉫 are collinear.” This strengthening of WA is used in Theorem 1 and Theorem 2 to give a sharper statement of the conditions under which equilibrium prices do not, in fact, change. A similar comment applies to WA for aggregate demand and the µ ¯ ω result, Theorem 3.
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the ξ 䉫 economy and let x䉫 be the associated consumption allocation. Assume that x䉫i > 0 for all i. Assume that, for all i, f Di satisfies WA (strong form) at (p∗ , p䉫 ) and x䉫i . Let ¯ x = ψ ¯ (p∗ , x¯ ∗ , x䉫 ). ψ ¯ x = 0. Let µ ¯ ¯ (p∗ , x¯ ∗ , x䉫 ). Suppose further that p䉫 · µ ¯ 1. Suppose that p∗ · ψ x =µ x > 0. Then p䉫 ∗ − p · (¯x䉫 − x¯ ∗ ) ≤ 0, p䉫 · µ ¯ x ¯ x . with equality iff p䉫 is collinear with p∗ iff x¯ 䉫 − x¯ ∗ = ψ ¯ x = 0. Then 2. Suppose that ψ (p䉫 − p∗ ) · (¯x䉫 − x¯ ∗ ) ≤ 0, with equality iff p䉫 is collinear with p∗ iff x¯ 䉫 = x¯ ∗ . Proof. Since x¯ 䉫 − x¯ ∗ = x¯ 䉫 − f¯ D (p∗ , x䉫 ) + f¯ D (p∗ , x䉫 ) − x¯ ∗ , ¯ x . x¯ 䉫 − x¯ ∗ = x¯ 䉫 − f¯ D (p∗ , x䉫 ) + ψ
(1)
By Walras’s Law, for each i, p∗ · x䉫i = p∗ · f Di (p∗ , x䉫i ).
(2)
Since x䉫i = f Di (p䉫 , x䉫i ), WA implies p䉫 · f Di (p∗ , x䉫i ) ≥ p䉫 · x䉫i ,
(3)
with equality iff f Di (p∗ , x䉫i ) = x䉫i iff p䉫 and p∗ are collinear. Adding across consumers, (1), (2), and (3) imply that, for any scalar c > 0, ¯ x , (cp䉫 − p∗ ) · (¯x䉫 − x¯ ∗ ) ≤ (cp䉫 − p∗ ) · ψ
(4)
with equality iff f¯ D (p∗ , x䉫 ) = x¯ 䉫 iff p䉫 and p∗ are collinear. Finally, note that ¯ x . f¯ D (p∗ , x䉫 ) = x¯ 䉫 iff x¯ 䉫 − x¯ ∗ = ψ The proof of the two parts of Theorem 1 is then as follows. 1. Set c = 1/(p䉫 · µ ¯ x ). By construction, p䉫 ∗ ¯ x = 0. −p ·ψ p䉫 · µ ¯ x The proof then follows from (4). 2. Setting c = 1, this follows immediately from (4).
Remark 1. If µ ¯ x > 0 (all goods are normal in the aggregate) then the condition in ¯ Theorem 1 that p䉫 · µ x > 0 is satisfied automatically. Note that, depending on the distribution of wealth changes, one could have µ ¯ / 0 (some goods are inferior in the aggregate) x > even if every good is normal for every consumer. Similar comments apply to Theorem 3.
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Theorem 1 holds for production economies in the trivial sense that production plays no role in the result. But one can easily derive corollaries of Theorem 1 that link price changes more directly to production changes. The following provides one example. Given an economy ξ and a vector p of consumption good prices, let g(p, ξ) be a revenue maximizing vector of consumption goods. Thus, g(p, ξ) is a point on the frontier of the Production Possibility Set. I assume that there is at least one such point. If there is more than one, make a selection. I assume that for any economy ξ and price vector p under consideration, and for any scalar c > 0, the selection satisfies g(cp, ξ) = g(p, ξ). I further assume that for any economy ξ, equilibrium price vector p, and equilibrium allocation x under consideration, the selection satisfies g(p, ξ) = x¯ . Given economies ξ ∗ and ξ 䉫 , one way to measure the change in production possibilities is to consider the vector g(p, ξ 䉫 ) − g(p, ξ ∗ ) for some fixed price vector p. If the economy is an exchange economy then g(p, ξ 䉫 ) − g(p, ξ ∗ ) is simply the change in endowment. Theorem 2. Let p∗ be an equilibrium price vector of the ξ ∗ economy and let x∗ be the associated equilibrium consumption allocation. Let p䉫 be an equilibrium price vector of the ξ 䉫 economy and let x䉫 be the associated consumption allocation. Assume that x䉫i > 0, for all i. Assume that, for all i, f Di satisfies WA at (p∗ , p䉫 ) and x䉫i . Let ¯ x = ψ ¯ (p∗ , x¯ ∗ , x䉫 ). ψ ¯ x = 0. Let µ ¯ ¯ (p∗ , x¯ ∗ , x䉫 ). Suppose further that p䉫 · µ ¯ 1. Suppose that p∗ · ψ x =µ x > 0. Then p䉫 ∗ − p · (g(p∗ , ξ 䉫 ) − g(p∗ , ξ ∗ )) ≤ 0, (5) 䉫 p ·µ ¯x ¯ x . with equality iff p䉫 is collinear with p∗ iff g(p∗ , ξ 䉫 ) − g(p∗ , ξ ∗ ) = ψ ¯ 2. Suppose that ψx = 0. Then (p䉫 − p∗ ) · (g(p∗ , ξ 䉫 ) − g(p∗ , ξ ∗ )) ≤ 0.
(6)
with equality iff p䉫 is collinear with p∗ iff g(p∗ , ξ 䉫 ) = g(p∗ , ξ ∗ ). Proof. By profit maximization, (p䉫 − p∗ ) · (g(p∗ , ξ 䉫 ) − g(p䉫 , ξ 䉫 )) ≤ 0. Inequalities (5) and (6) then follow as a corollary to Theorem 1 and the fact that one can write, ¯ x . g(p∗ , ξ 䉫 ) − g(p∗ , ξ ∗ ) = [g(p∗ , ξ 䉫 ) − g(p䉫 , ξ 䉫 )] + [¯x䉫 − f¯ D (p∗ , x䉫 )] + ψ It remains to prove the iff statements for when (5) and (6) hold with equality. To begin, if (5) or (6) hold with equality then, by the same argument as for Theorem 1, p䉫 and p∗ must be collinear.
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If p䉫 and p∗ are collinear then it follows, just as in the proof of Theorem 1, that x¯ 䉫 − x¯ ∗ = ¯ x . Since, by collinearity, g(p∗ , ξ 䉫 ) = g(p䉫 , ξ 䉫 ) = x¯ 䉫 , and since g(p∗ , ξ ∗ ) = x¯ ∗ , it ψ ¯ x . follows that g(p∗ , ξ 䉫 ) − g(p∗ , ξ ∗ ) = ψ ¯ x then the price normalization implies that (5) Finally, if g(p∗ , ξ 䉫 ) − g(p∗ , ξ ∗ ) = ψ ∗ 䉫 holds with equality, while if g(p , ξ ) = g(p∗ , ξ ∗ ) then (6) holds with equality trivially. Remark 2. Inequality (5) in Theorem 2 is “dual” to Theorem 4.2 in Quah (2003). In the latter, technology changes are modeled as, in effect, changes in the endowments of factors (non-consumption goods). Theorem 4.2 in Quah (2003) states that there is a negative inner product between properly normalized price changes, of both consumption goods and factors, and changes in endowments or effective endowments, again of all goods. The input-based perspective of Theorem 4.2 in Quah (2003) and the output-based perspective of Theorem 2 in the present paper coincide in exchange economies. 3.2. The µ ¯ ω result Restrict attention to exchange economies. Let ωi ∈ RL +\0 be consumer i’s endowment. i 1 I ω be the aggregate Let ω = (ω , . . . , ω ) be the endowment allocation and let ω¯ = endowment. Assume ω¯ 0. f¯ D satisfies the Weak Axiom (WA) (strong form) at the price pair (p∗ , p䉫 ) and the endowment profile ω iff the inequalities p∗ · f¯ D (p䉫 , ω) ≤ p∗ · f¯ D (p∗ , ω) and p䉫 · f¯ D (p∗ , ω) ≤ p䉫 · f¯ D (p䉫 , ω) both hold iff f¯ D (p∗ , ω) = f¯ D (p䉫 , ω) iff p∗ and p䉫 are collinear. Theorem 3. Let p∗ be an equilibrium price vector of the ω∗ economy and let x∗ be the associated equilibrium consumption allocation. Let p䉫 be an equilibrium price vector of the ω䉫 economy and let x䉫 be the associated equilibrium consumption allocation. Assume ¯ ω = ψ ¯ (p∗ , x¯ ∗ , ω䉫 ). that f¯ D satisfies WA (strong form) at (p∗ , p䉫 ) and ω䉫 . Let ψ ¯ ω䉫 = 0. Let µ ¯ ¯ (p∗ , x¯ ∗ , ω䉫 ). Suppose further that p䉫 ·µ ¯ 1. Suppose that p∗ ·ψ ω =µ ω > 0. Then p䉫 ∗ − p · (¯x䉫 − x¯ ∗ ) ≤ 0, p䉫 · µ ¯ x ¯ ω . with equality iff p䉫 is collinear with p∗ iff x¯ 䉫 − x¯ ∗ = ψ ¯ ω = 0. Then 2. Suppose that ψ p䉫 − p∗ · (¯x䉫 − x¯ ∗ ) ≤ 0, with equality iff p䉫 is collinear with p∗ iff x¯ 䉫 = x¯ ∗ . Proof. Since x¯ 䉫 − x¯ ∗ = x¯ 䉫 − f¯ D (p∗ , ω䉫 ) + f¯ D (p∗ , ω䉫 ) − x¯ ∗ , one gets, in place of ¯ ω . The proof is then the (1) in the proof of Theorem 1, x¯ 䉫 − x¯ ∗ = x¯ 䉫 − f¯ D (p∗ , ω䉫 ) + ψ
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same as that of Theorem 1, except that it applies WA to f¯ D rather than to the individual f Di . ¯ ω , the particular Remark 3. In view of the decomposition x¯ 䉫 − x¯ ∗ = x¯ 䉫 − f¯ D (p∗ , ω䉫 )+ ψ WA assumption made in Theorem 3 is necessary as well as sufficient. Remark 4. Extending Theorem 3 to production is problematic. There are unresolved questions regarding the force of assuming WA for aggregate demand in production economies. See Grodal and Hildenbrand (1989) and Hildenbrand (1989) but also Jerison (1999). These issues are in addition to the difficulties with aggregation that arise even in exchange economies. Further, the production version of µ ¯ ω requires that one compute the wealth profile generated by the new technology at the old equilibrium prices. This wealth profile may be undefined. Things are much simpler if one can assume collinearity: consumer i has a fixed share αi of the endowment of every good and of the profit of every firm. In this case, one can ignore factors and factor prices and compute consumer i’s wealth as simply her share of revenue from consumption goods, αi p · g(p, ξ). The potential problems with WA alluded to above stem from the interaction of factor prices and the wealth profile and hence do not arise in collinear economies. Given collinearity, one can easily derive the production analogs of Theorem 1 and Theorem 2 for µ ¯ ω ; I will not do so explicitly. 4. Uniqueness Throughout this section, assume that an exchange economy equilibrium exists for any endowment allocation under consideration. I begin by recording the fact that µ ¯ ¯ x and µ x are, in the following sense, independent of aggregate consumption. Recall that consumer i’s preferences are strongly monotone iff, for L i i i i any xi ∈ RL + and γ ∈ R+\0 , i strictly prefers x + γ to x . Lemma 1. The notation is the same as in Theorems 1 and 3. Consider any aggregate consumption vector x¯ 0 such that p∗ · x¯ = p∗ · x¯ 䉫 . ¯ x = 0, so that µ ¯ 1. Assume x䉫i > 0, for all i. Assume that p∗ · ψ ω is well defined. Assume that each consumer’s preference is strongly monotone and has a utility representation that is continuous and concave. Then there is an economy ξ # and an equilibrium consumption allocation x# for the ξ # economy such that x¯ # = x¯ and µ ¯ (p∗ , x¯ ∗ , x# ) = µ ¯ x. ∗ ¯ 2. Assume that p · ψω = 0, so thatµ ¯ x is well defined. Then there is an endowment ¯ (p∗ , x¯ ∗ , ω# ) = µ ¯ allocation ω# such that ω¯ # = x¯ and µ ω. Proof. The claim for µ ¯ ¯ , it suffices to give each consumer ω is trivial: by the definition of µ the endowment ω#i = x¯
p∗ · ω䉫i p∗ · ω¯ 䉫
0,
since this ensures that p∗ · ω#i = p∗ · ω䉫i .
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#i #i For µ ¯ x , the analogous approach is problematic because x , unlike ω , is endogenous. Instead, use the following argument, in the style of Negishi (1960). Fix a continuous, concave utility representation for each i. Let K = {k ∈ RI+ : i ki = 1}; K is a set of utility weights. Let Xx¯ be the set of consumption allocations for which aggregate consumption equals x¯ . I construct a correspondence from K × Xx¯ to itself as follows. Let gA : K ⇒ Xx¯ be the correspondence that maps each k ∈ K to the set of consumption allocations in Xx¯ that maximize the k-weighted sum of utilities. Let gB : Xx¯ ⇒ K be the correspondence that maps each x ∈ Xx¯ to the set of k ∈ K that minimize ki p∗ · xi − x䉫i . i
Let g : K ×Xx¯ ⇒ K ×Xx¯ be the correspondence defined by g(x, k) = (gB (x), gA (k)). One can verify that, given the assumptions on preferences, g is nonempty and convex valued and upper hemicontinuous. Therefore, by the Kakutani fixed point theorem, g has a fixed point, say (k# , x# ). One can verify that x# is efficient and that p∗ · x#i = p∗ · x䉫i for each i. Let ξ # be the exchange economy (which is just a particular form of production economy) with endowments ω#i = x#i > 0. This economy has an equilibrium in which each i’s consumption is, in fact, x#i (no trade). The proof follows. The next result records the observation that if the desired inner product inequality holds for one linear price normalization then it fails for any other linear price normalization whose normalization vector lies on the “wrong side” of the change in consumption vector. Define ∗ Λp∗ + = {λ ∈ RL + : p · λ = 1}. Lemma 2. Consider any two vectors λ1 , λ2 ∈ Λp∗ + . Suppose that x¯ # − x¯ ∗ lies in the open cone spanned by λ1 and λ2 . If
p# ∗ · (¯x# − x¯ ∗ ) < 0, − p p# · λ 1 then
p# ∗ · (¯x# − x¯ ∗ ) > 0. − p p# · λ 2
I omit the proof, which is a trivial calculation. The uniqueness result is then as follows. Theorem 4. The notation is the same as in Theorems 1–3. 1. Assume x䉫I > 0, for all i. Assume that µ ¯ x > 0. Assume that each consumer’s preference is strongly monotone and has a utility representation that is continuous and concave. Then µ ¯ x is the unique λ ∈ Λp∗ + for which #
p ∗ − p · (¯x# − x¯ ∗ ) ≤ 0, p# · λ
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for every economy ξ # , equilibrium price vector p# , and equilibrium consumption allo¯ (p∗ , x¯ ∗ , x# ) = µ ¯ cation x¯ # such that µ x. 2. Assume that µ ¯ x > 0. Then µ ¯ x is the unique λ ∈ Λp∗ for which #
p ∗ − p · (¯x# − x¯ ∗ ) ≤ 0, p# · λ for every economy ω# , equilibrium price vector p# , and equilibrium consumption allo¯ (p∗ , x¯ ∗ , ω# ) = µ ¯ cation x¯ # such that µ x. Proof. The arguments for either case are nearly identical. For concreteness, consider the ¯ µ ¯ x claim. By Theorem 3, µ x does indeed have the desired property. To see that it is the ∗ only element of Λp + with this property, consider any λ ∈ Λp∗ + , λ = µ ¯ ω . By Lemma 1, # # there is an ω such that ω¯ lies in the open cone spanned by λ and µ ¯ ω and such that # # ¯ ) − p∗ ] · (¯ µ ¯ (p∗ , ω¯ ∗ , ω# ) = µ ¯ x# − x¯ ∗ ) < 0. The proof x . By Theorem 3, [p /(p · µ ω then follows from Lemma 2. One implication of these results is negative: even for a single consumer exchange economy, there is no λ ∈ Λp∗ + for which p䉫 ∗ − p · (¯x䉫 − x¯ ∗ ) ≤ 0, p䉫 · λ holds for every endowment allocation ω䉫 , equilibrium price vector p䉫 , and equilibrium consumption allocation x䉫 . See also the discussion in Nachbar (2002). Remark 5. The results of this section differ from their differential analogs in Nachbar (2002) in the following respects. First, for infinitesimal shocks, the analog of the condition p∗ · x¯ = p∗ · x¯ 䉫 in Lemma 1 can be dropped. Second, the concave utility representation assumption in Lemma 1 and Theorem 4 is satisfied almost automatically in the setting of Nachbar (2002) because the standard assumptions to ensure differentiability of demand are almost sufficient to ensure that preferences have a concave utility representation, at least in a neighborhood of equilibrium consumption in the reference economy; see Proposition 2.6.4 in Mas-Colell (1985). Third, the restriction to λ > 0 is to ensure that p · λ > 0 for any p 0. In the differential case, because all price changes are local, this restriction can be dropped; it suffices to restrict λ to Λp∗ = {λ ∈ RL : p∗ · λ = 1}. In particular, the differential analog of the uniqueness claim holds even if µ ¯ > / 0, meaning that some goods are inferior in the aggregate.
5. The Literature Nachbar (2002) contains a detailed literature review, so I will be brief. Other than Nachbar (2002) itself, the closest predecessor to this paper is the following result, which appears in Chapter 4 of Dixit and Norman (1980). Consider an exchange economy with a single consumer with homothetic preferences. Normalize prices so that
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e(p, 1) = 1, where e is the consumer’s expenditure function. Then, (p䉫 − p∗ ) · (¯x䉫 − x¯ ∗ ) ≤ 0. Finding a fixed price normalization that works for all reference endowments and all endowment shocks is possible in Dixit and Norman (1980) because, with a single homothetic consumer, the expenditure function integrates the different MPC vectors that obtain at different reference equilibria. In view of the necessity of the MPC price normalization for infinitesimal shocks in single consumer economies, this sort of integration is required if (p䉫 − p∗ ) · (¯x䉫 − x¯ ∗ ) ≤ 0 is to hold for all reference endowments and all endowment shocks. Such integration is possible with a single homothetic consumer but not, even with a single consumer, in general. The e(p, 1) = 1 price normalization is not linear. As noted in Section 4, there is no fixed linear price normalization that works for all reference endowments and all endowment shocks. But, in the case of a single homothetic consumer, the IPC vector µ ¯ (= µ ¯ ¯ x =µ ω) ∗ ¯ is collinear with the reference endowment (assuming p · ψ = 0). Hence, if there is a single homothetic consumer then, for any fixed reference endowment ω¯ ∗ and (almost) any endowment shock, the inequality (p䉫 − p∗ ) · (¯x䉫 − x¯ ∗ ) ≤ 0 holds for the linear price normalization p · ω¯ ∗ = 1. This is the Laspeyres price normalization.
Acknowledgements I would like to thank John Quah for helpful discussions. And I would like to thank Washington University’s Center for Political Economy for its support. References Dixit, A., Norman, V., 1980. Theory of International Trade. Cambridge University Press, Cambridge, UK. Grodal, B., Hildenbrand, W., 1989. The Weak Axiom of revealed preference in a productive economy. Review of Economic Studies 56 (4), 635–639. Hicks, J.R., 1939. Value and Capital. Clarendon Press, Oxford. Hildenbrand, W., 1989. The Weak Axiom of revealed preference for market demand is strong. Econometrica 57, 979–985. Jerison, M., 1999. Dispersed excess demands, the Weak Axiom and the uniqueness of equilibrium. Journal of Mathematical Economics 31, 15–48. Mas-Colell, A., 1985. The Theory of General Equilibrium. A Differentiable Approach. Cambridge University Press, Cambridge, UK. Nachbar, J.H., 2002. General equilibrium comparative statics. Econometrica 79 (5), 2065–2074. Negishi, T., 1960. Welfare economics and existence of equilibrium for a competitive economy. Metroeconomica 12, 92–97. Quah, J.K.H., 1999. The Weak Axiom and Comparative Statics. Nuffield College Working Paper, 1999, W15, Oxford University. Quah, J.K.H., 2003. Market demand and comparative statics when goods are normal. Journal of Mathematical Economics 39, 317–333.
General equilibrium comparative statics: discrete shocks in production economies John H. Nachbar Department of Economics, Washington University, Box 1208, One Brookings Drive, St. Louis, MO 63130, USA
Abstract Nachbar [Econometrica 79 (5) (2002) 2065] established minimal conditions under which, following an infinitesimal shock to endowments in an exchange economy, changes in equilibrium prices are negatively related to changes in aggregate consumption. The present paper extends Nachbar (2002) to cover discrete shocks to technologies, ownership shares, and endowments in production economies. As in Nachbar (2002), the analyst’s choice of price normalization plays a key role. The required normalization is nonstandard but has a sensible interpretation. © 2003 Elsevier B.V. All rights reserved. Keywords: General equilibrium; Comparative statics; Weak Axiom
1. Introduction Consider a shock to technologies, ownership shares, or endowments in a production economy. A plausible conjecture is that (p䉫 − p∗ ) · (¯x䉫 − x¯ ∗ ) ≤ 0, where p䉫 − p∗ is the change in equilibrium prices and x¯ 䉫 − x¯ ∗ is the change in aggregate equilibrium consumption. Informally, changes in equilibrium prices are negatively related to changes in aggregate equilibrium consumption. It has been known at least since Hicks (1939) that this conjecture is false, in general, even in single consumer exchange economies. This paper provides minimal conditions under which the conjecture is, in fact, true. This paper is an extension of Nachbar (2002), which provided analogous results for infinitesimal shocks to endowments in exchange economies. Since the discrete results are easiest to interpret in relation to the infinitesimal results of Nachbar (2002), I begin by reviewing the earlier paper. E-mail address: [email protected] (J.H. Nachbar). 0304-4068/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0304-4068(03)00093-4
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The analysis in Nachbar (2002) requires that, regardless of how prices are actually normalized in the economy, prices be renormalized to satisfy p · µ ¯ = 1, where µ ¯ is the vector giving the economy’s aggregate marginal propensity to consume (MPC) each of the L goods. In a single consumer economy, the MPC is simply the derivative with respect to nominal wealth of that consumer’s demand. In multiconsumer economies, µ ¯ is a weighted sum of individual MPCs. The aggregate MPC price normalization is nonstandard. It has the unattractive property that it depends both on the reference equilibrium and, in multiconsumer economies, on the distribution of the endowment shocks. And estimating the aggregate MPC may be difficult. In defense of the aggregate MPC price normalization, Nachbar (2002) makes two observations. First, the aggregate MPC price normalization is necessary in the following sense. In single consumer economies, the MPC price normalization is the unique linear price normalization (up to scalar multiplication) for which (p䉫 − p∗ ) · (¯x䉫 − x¯ ∗ ) ≤ 0 holds for all endowment shocks. A similar but weaker uniqueness claim holds in multiconsumer economies. Second, if µ ¯ > 0 (all goods are weakly normal in the aggregate) then the µ ¯ normalization yields a sensible interpretation of the vector of normalized price changes.1 A decrease in the µ-normalized ¯ price of good 1 implies that the price of good 1 has fallen relative to the value of the composite commodity (µ ¯ 2, . . . , µ ¯ L ) > 0, which is the economy’s marginal consumption of all other goods. Since the comparative statics result in Nachbar (2002)considers a marginal change in endowment, interpreting relative price changes in terms of marginal consumption is, I think, natural. If, on the other hand, µ ¯ > / 0 (some goods are inferior in the aggregate) then the vector of normalized price changes may be difficult to interpret; see Nachbar (2002). I have glossed over the fact that, in multiconsumer economies, it is unclear what weighting scheme one should use for computing µ. ¯ Nachbar (2002) actually considers two different weighting schemes, yielding two different aggregate MPCs, µ ¯ γ and µ ¯ δ . For µ ¯ γ , the weight on i’s individual MPC is based on the value, at the original equilibrium prices, of the change in i’s equilibrium consumption. For µ ¯ δ , the weight on i’s individual MPC is based on the value, at the original equilibrium prices, of the change in i’s endowment. The µ ¯ γ and µ ¯ δ normalizations yield price change vectors corresponding to somewhat different interpretations of what one means by a relative price change. I find both interpretations reasonable. Relative to Nachbar (2002), the contribution of the present paper is three-fold. 1. This paper provides comparative statics results for discrete, as opposed to infinitesimal, shocks to endowments in exchange economies. This extension is straightforward but still worth recording.2 Corresponding to the aggregate MPC vectors µ ¯ γ and µ ¯ δ in Nachbar ¯ (2002), this paper uses the incremental propensity to consume (IPC) vectors µ ¯ x and µ ω. The interpretation of an IPC normalized price change is analogous to the interpretation of an MPC normalized price change. L L L L N RL + = {x ∈ R : x ≥ 0, ∀}, R+\0 = {x ∈ R+ : x = 0}, and R++ = {x ∈ R : x > 0, ∀}. Consider any L L L x ∈ R . Then x ≥ 0 means x ∈ R+ , x > 0 means x ∈ R+\0 , and x 0 means x ∈ RL ++ . 2 A discrete extension of the basic inequalities in Nachbar (2002) has been independently reported in Quah (2003). 1
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The µ ¯ ¯ γ result of Nachbar (2002), assumes that individual demand x result, like the µ satisfies a Weak Axiom of revealed preference (WA) condition. In contrast, the µ ¯ ω result, like the µ ¯ δ result of Nachbar (2002), assumes that aggregate demand satisfies a WA condition. The latter assumption is strong but it is necessary; see Remark 3 in Section 3.2.3 2. The µ ¯ ¯γ x result holds, trivially, in general production economies. The analogous µ result, for infinitesimal shocks, also holds in production economies, but its expression is somewhat cumbersome; this extension was therefore omitted from Nachbar (2002). Section 3.1 also contains an additional comparative statics result for the µ ¯ x normalization that links price changes directly to changes in the underlying production technology. Extending the µ ¯ ω result to production is problematic. I discuss this further in Remark 4 in Section 3.2. 3. This paper proves a uniqueness theorem for µ ¯ ¯ ωx that is the analog of the uniquex and µ ¯ δ discussed in Nachbar (2002) but not proved there in detail. ness result for µ ¯ γ and µ In particular, Lemma 1 of the present paper fills in an important step in the uniqueness argument for both µ ¯ γ and µ ¯ x. I focus here on supply shocks; the analysis of demand shocks, shocks to preferences, is comparatively straightforward. For demand shocks, it follows immediately from profit maximization that (p䉫 −p∗ )·(¯x䉫 − x¯ ∗ ) ≥ 0. In the case of an exchange economy, a demand shock results in no change in aggregate consumption, so that the result holds trivially. The remainder of this paper is organized as follows. Section 2 covers basic definitions, Section 3 contains the comparative statics results, and Section 4 discusses uniqueness. Section 5 concludes with a brief discussion of the literature.
2. Definitions 2.1. Basic notation There are I consumers and L consumption goods. xi ∈ RL + denotes a consumption vector for consumer i. x = (x1 , . . . , xI ) is a consumption allocation while x¯ = i xi is an aggregate consumption vector. L i i Let φi : RL+1 ++ ⇒ R+ be consumer i’s demand function: φ (p, m ) is consumer i’s i demand when prices are p 0 and nominal wealth is m > 0. The analysis extends easily to demand correspondences rather than demand functions. Assume that each φi satisfies Walras’s Law: p · φi (p, mi ) = mi . For any vector βi ∈ RL +\0 define f Di (p, βi ) = φi (p, p · βi ). f Di is individual demand in an exchange economy in which consumer i is given endowment βi . Note that f Di is defined even if the actual economy is, in fact, a production economy. 3 For recent work on sufficient conditions for aggregate demand to satisfy WA, see Jerison (1999) and Quah (1999).
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Finally, let β = (β1 , . . . , βI ) be a profile with each βi ∈ RL +\0 and define f Di (p, βi ). f¯ D (p, β) = i
2.2. The aggregate wealth effect and the aggregate IPC Define ¯ ∆ (p, x¯ , β) = f¯ D (p, β) − x¯ . ψ ¯ (p, x¯ , β) is the aggregate wealth If x¯ is aggregate consumption when prices are p then ψ effect when the wealth of consumer i is changed to p · βi while prices remain unchanged. If p · ψ(p, x¯ , β) = 0, define µ ¯ (p, x¯ , β) =
1 ¯ (p, x¯ , β). ψ ¯ p · ψ (p, x¯ , β)
µ ¯ is the aggregate incremental propensity to consume (IPC) vector when wealth is changed to p · βi while prices remain unchanged. Consider any price pair (p∗ , p䉫 ). Since p∗ · µ ¯ (p∗ , x¯ , β) = 1, if p䉫 · µ ¯ (p∗ , x¯ , β) > 0 then p䉫 p䉫 · µ ¯ (p∗ , x¯ , β)
− p∗ ,
is a vector of normalized price changes, for the linear price normalization p · µ ¯ (p∗ , x¯ , β) = 1. 3. Comparative statics 3.1. The µ ¯ x results Fix preferences and let ξ denote a production economy. ξ specifies endowments, production sets, and ownership shares. It is natural to assume that there are factors or intermediate goods that are not directly consumed. Any such good is not included among the L consumption goods. Say that f Di satisfies the Weak Axiom (WA) (strong form) at the price pair (p∗ , p䉫 ) and the vector βi iff the inequalities p∗ · f Di (p䉫 , βi ) ≤ p∗ · f Di (p∗ , βi ) and p䉫 · f Di (p∗ , βi ) ≤ p䉫 · f Di (p䉫 , βi ) both hold iff f Di (p∗ , βi ) = f Di (p䉫 , βi ) iff p∗ and p䉫 are collinear.4 Theorem 1. Let p∗ be an equilibrium price vector of the ξ ∗ economy and let x∗ be the associated equilibrium consumption allocation. Let p䉫 be an equilibrium price vector of 4 The standard WA drops “iff p∗ and p䉫 are collinear.” This strengthening of WA is used in Theorem 1 and Theorem 2 to give a sharper statement of the conditions under which equilibrium prices do not, in fact, change. A similar comment applies to WA for aggregate demand and the µ ¯ ω result, Theorem 3.
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the ξ 䉫 economy and let x䉫 be the associated consumption allocation. Assume that x䉫i > 0 for all i. Assume that, for all i, f Di satisfies WA (strong form) at (p∗ , p䉫 ) and x䉫i . Let ¯ x = ψ ¯ (p∗ , x¯ ∗ , x䉫 ). ψ ¯ x = 0. Let µ ¯ ¯ (p∗ , x¯ ∗ , x䉫 ). Suppose further that p䉫 · µ ¯ 1. Suppose that p∗ · ψ x =µ x > 0. Then p䉫 ∗ − p · (¯x䉫 − x¯ ∗ ) ≤ 0, p䉫 · µ ¯ x ¯ x . with equality iff p䉫 is collinear with p∗ iff x¯ 䉫 − x¯ ∗ = ψ ¯ x = 0. Then 2. Suppose that ψ (p䉫 − p∗ ) · (¯x䉫 − x¯ ∗ ) ≤ 0, with equality iff p䉫 is collinear with p∗ iff x¯ 䉫 = x¯ ∗ . Proof. Since x¯ 䉫 − x¯ ∗ = x¯ 䉫 − f¯ D (p∗ , x䉫 ) + f¯ D (p∗ , x䉫 ) − x¯ ∗ , ¯ x . x¯ 䉫 − x¯ ∗ = x¯ 䉫 − f¯ D (p∗ , x䉫 ) + ψ
(1)
By Walras’s Law, for each i, p∗ · x䉫i = p∗ · f Di (p∗ , x䉫i ).
(2)
Since x䉫i = f Di (p䉫 , x䉫i ), WA implies p䉫 · f Di (p∗ , x䉫i ) ≥ p䉫 · x䉫i ,
(3)
with equality iff f Di (p∗ , x䉫i ) = x䉫i iff p䉫 and p∗ are collinear. Adding across consumers, (1), (2), and (3) imply that, for any scalar c > 0, ¯ x , (cp䉫 − p∗ ) · (¯x䉫 − x¯ ∗ ) ≤ (cp䉫 − p∗ ) · ψ
(4)
with equality iff f¯ D (p∗ , x䉫 ) = x¯ 䉫 iff p䉫 and p∗ are collinear. Finally, note that ¯ x . f¯ D (p∗ , x䉫 ) = x¯ 䉫 iff x¯ 䉫 − x¯ ∗ = ψ The proof of the two parts of Theorem 1 is then as follows. 1. Set c = 1/(p䉫 · µ ¯ x ). By construction, p䉫 ∗ ¯ x = 0. −p ·ψ p䉫 · µ ¯ x The proof then follows from (4). 2. Setting c = 1, this follows immediately from (4).
Remark 1. If µ ¯ x > 0 (all goods are normal in the aggregate) then the condition in ¯ Theorem 1 that p䉫 · µ x > 0 is satisfied automatically. Note that, depending on the distribution of wealth changes, one could have µ ¯ / 0 (some goods are inferior in the aggregate) x > even if every good is normal for every consumer. Similar comments apply to Theorem 3.
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Theorem 1 holds for production economies in the trivial sense that production plays no role in the result. But one can easily derive corollaries of Theorem 1 that link price changes more directly to production changes. The following provides one example. Given an economy ξ and a vector p of consumption good prices, let g(p, ξ) be a revenue maximizing vector of consumption goods. Thus, g(p, ξ) is a point on the frontier of the Production Possibility Set. I assume that there is at least one such point. If there is more than one, make a selection. I assume that for any economy ξ and price vector p under consideration, and for any scalar c > 0, the selection satisfies g(cp, ξ) = g(p, ξ). I further assume that for any economy ξ, equilibrium price vector p, and equilibrium allocation x under consideration, the selection satisfies g(p, ξ) = x¯ . Given economies ξ ∗ and ξ 䉫 , one way to measure the change in production possibilities is to consider the vector g(p, ξ 䉫 ) − g(p, ξ ∗ ) for some fixed price vector p. If the economy is an exchange economy then g(p, ξ 䉫 ) − g(p, ξ ∗ ) is simply the change in endowment. Theorem 2. Let p∗ be an equilibrium price vector of the ξ ∗ economy and let x∗ be the associated equilibrium consumption allocation. Let p䉫 be an equilibrium price vector of the ξ 䉫 economy and let x䉫 be the associated consumption allocation. Assume that x䉫i > 0, for all i. Assume that, for all i, f Di satisfies WA at (p∗ , p䉫 ) and x䉫i . Let ¯ x = ψ ¯ (p∗ , x¯ ∗ , x䉫 ). ψ ¯ x = 0. Let µ ¯ ¯ (p∗ , x¯ ∗ , x䉫 ). Suppose further that p䉫 · µ ¯ 1. Suppose that p∗ · ψ x =µ x > 0. Then p䉫 ∗ − p · (g(p∗ , ξ 䉫 ) − g(p∗ , ξ ∗ )) ≤ 0, (5) 䉫 p ·µ ¯x ¯ x . with equality iff p䉫 is collinear with p∗ iff g(p∗ , ξ 䉫 ) − g(p∗ , ξ ∗ ) = ψ ¯ 2. Suppose that ψx = 0. Then (p䉫 − p∗ ) · (g(p∗ , ξ 䉫 ) − g(p∗ , ξ ∗ )) ≤ 0.
(6)
with equality iff p䉫 is collinear with p∗ iff g(p∗ , ξ 䉫 ) = g(p∗ , ξ ∗ ). Proof. By profit maximization, (p䉫 − p∗ ) · (g(p∗ , ξ 䉫 ) − g(p䉫 , ξ 䉫 )) ≤ 0. Inequalities (5) and (6) then follow as a corollary to Theorem 1 and the fact that one can write, ¯ x . g(p∗ , ξ 䉫 ) − g(p∗ , ξ ∗ ) = [g(p∗ , ξ 䉫 ) − g(p䉫 , ξ 䉫 )] + [¯x䉫 − f¯ D (p∗ , x䉫 )] + ψ It remains to prove the iff statements for when (5) and (6) hold with equality. To begin, if (5) or (6) hold with equality then, by the same argument as for Theorem 1, p䉫 and p∗ must be collinear.
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If p䉫 and p∗ are collinear then it follows, just as in the proof of Theorem 1, that x¯ 䉫 − x¯ ∗ = ¯ x . Since, by collinearity, g(p∗ , ξ 䉫 ) = g(p䉫 , ξ 䉫 ) = x¯ 䉫 , and since g(p∗ , ξ ∗ ) = x¯ ∗ , it ψ ¯ x . follows that g(p∗ , ξ 䉫 ) − g(p∗ , ξ ∗ ) = ψ ¯ x then the price normalization implies that (5) Finally, if g(p∗ , ξ 䉫 ) − g(p∗ , ξ ∗ ) = ψ ∗ 䉫 holds with equality, while if g(p , ξ ) = g(p∗ , ξ ∗ ) then (6) holds with equality trivially. Remark 2. Inequality (5) in Theorem 2 is “dual” to Theorem 4.2 in Quah (2003). In the latter, technology changes are modeled as, in effect, changes in the endowments of factors (non-consumption goods). Theorem 4.2 in Quah (2003) states that there is a negative inner product between properly normalized price changes, of both consumption goods and factors, and changes in endowments or effective endowments, again of all goods. The input-based perspective of Theorem 4.2 in Quah (2003) and the output-based perspective of Theorem 2 in the present paper coincide in exchange economies. 3.2. The µ ¯ ω result Restrict attention to exchange economies. Let ωi ∈ RL +\0 be consumer i’s endowment. i 1 I ω be the aggregate Let ω = (ω , . . . , ω ) be the endowment allocation and let ω¯ = endowment. Assume ω¯ 0. f¯ D satisfies the Weak Axiom (WA) (strong form) at the price pair (p∗ , p䉫 ) and the endowment profile ω iff the inequalities p∗ · f¯ D (p䉫 , ω) ≤ p∗ · f¯ D (p∗ , ω) and p䉫 · f¯ D (p∗ , ω) ≤ p䉫 · f¯ D (p䉫 , ω) both hold iff f¯ D (p∗ , ω) = f¯ D (p䉫 , ω) iff p∗ and p䉫 are collinear. Theorem 3. Let p∗ be an equilibrium price vector of the ω∗ economy and let x∗ be the associated equilibrium consumption allocation. Let p䉫 be an equilibrium price vector of the ω䉫 economy and let x䉫 be the associated equilibrium consumption allocation. Assume ¯ ω = ψ ¯ (p∗ , x¯ ∗ , ω䉫 ). that f¯ D satisfies WA (strong form) at (p∗ , p䉫 ) and ω䉫 . Let ψ ¯ ω䉫 = 0. Let µ ¯ ¯ (p∗ , x¯ ∗ , ω䉫 ). Suppose further that p䉫 ·µ ¯ 1. Suppose that p∗ ·ψ ω =µ ω > 0. Then p䉫 ∗ − p · (¯x䉫 − x¯ ∗ ) ≤ 0, p䉫 · µ ¯ x ¯ ω . with equality iff p䉫 is collinear with p∗ iff x¯ 䉫 − x¯ ∗ = ψ ¯ ω = 0. Then 2. Suppose that ψ p䉫 − p∗ · (¯x䉫 − x¯ ∗ ) ≤ 0, with equality iff p䉫 is collinear with p∗ iff x¯ 䉫 = x¯ ∗ . Proof. Since x¯ 䉫 − x¯ ∗ = x¯ 䉫 − f¯ D (p∗ , ω䉫 ) + f¯ D (p∗ , ω䉫 ) − x¯ ∗ , one gets, in place of ¯ ω . The proof is then the (1) in the proof of Theorem 1, x¯ 䉫 − x¯ ∗ = x¯ 䉫 − f¯ D (p∗ , ω䉫 ) + ψ
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same as that of Theorem 1, except that it applies WA to f¯ D rather than to the individual f Di . ¯ ω , the particular Remark 3. In view of the decomposition x¯ 䉫 − x¯ ∗ = x¯ 䉫 − f¯ D (p∗ , ω䉫 )+ ψ WA assumption made in Theorem 3 is necessary as well as sufficient. Remark 4. Extending Theorem 3 to production is problematic. There are unresolved questions regarding the force of assuming WA for aggregate demand in production economies. See Grodal and Hildenbrand (1989) and Hildenbrand (1989) but also Jerison (1999). These issues are in addition to the difficulties with aggregation that arise even in exchange economies. Further, the production version of µ ¯ ω requires that one compute the wealth profile generated by the new technology at the old equilibrium prices. This wealth profile may be undefined. Things are much simpler if one can assume collinearity: consumer i has a fixed share αi of the endowment of every good and of the profit of every firm. In this case, one can ignore factors and factor prices and compute consumer i’s wealth as simply her share of revenue from consumption goods, αi p · g(p, ξ). The potential problems with WA alluded to above stem from the interaction of factor prices and the wealth profile and hence do not arise in collinear economies. Given collinearity, one can easily derive the production analogs of Theorem 1 and Theorem 2 for µ ¯ ω ; I will not do so explicitly. 4. Uniqueness Throughout this section, assume that an exchange economy equilibrium exists for any endowment allocation under consideration. I begin by recording the fact that µ ¯ ¯ x and µ x are, in the following sense, independent of aggregate consumption. Recall that consumer i’s preferences are strongly monotone iff, for L i i i i any xi ∈ RL + and γ ∈ R+\0 , i strictly prefers x + γ to x . Lemma 1. The notation is the same as in Theorems 1 and 3. Consider any aggregate consumption vector x¯ 0 such that p∗ · x¯ = p∗ · x¯ 䉫 . ¯ x = 0, so that µ ¯ 1. Assume x䉫i > 0, for all i. Assume that p∗ · ψ ω is well defined. Assume that each consumer’s preference is strongly monotone and has a utility representation that is continuous and concave. Then there is an economy ξ # and an equilibrium consumption allocation x# for the ξ # economy such that x¯ # = x¯ and µ ¯ (p∗ , x¯ ∗ , x# ) = µ ¯ x. ∗ ¯ 2. Assume that p · ψω = 0, so thatµ ¯ x is well defined. Then there is an endowment ¯ (p∗ , x¯ ∗ , ω# ) = µ ¯ allocation ω# such that ω¯ # = x¯ and µ ω. Proof. The claim for µ ¯ ¯ , it suffices to give each consumer ω is trivial: by the definition of µ the endowment ω#i = x¯
p∗ · ω䉫i p∗ · ω¯ 䉫
0,
since this ensures that p∗ · ω#i = p∗ · ω䉫i .
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#i #i For µ ¯ x , the analogous approach is problematic because x , unlike ω , is endogenous. Instead, use the following argument, in the style of Negishi (1960). Fix a continuous, concave utility representation for each i. Let K = {k ∈ RI+ : i ki = 1}; K is a set of utility weights. Let Xx¯ be the set of consumption allocations for which aggregate consumption equals x¯ . I construct a correspondence from K × Xx¯ to itself as follows. Let gA : K ⇒ Xx¯ be the correspondence that maps each k ∈ K to the set of consumption allocations in Xx¯ that maximize the k-weighted sum of utilities. Let gB : Xx¯ ⇒ K be the correspondence that maps each x ∈ Xx¯ to the set of k ∈ K that minimize ki p∗ · xi − x䉫i . i
Let g : K ×Xx¯ ⇒ K ×Xx¯ be the correspondence defined by g(x, k) = (gB (x), gA (k)). One can verify that, given the assumptions on preferences, g is nonempty and convex valued and upper hemicontinuous. Therefore, by the Kakutani fixed point theorem, g has a fixed point, say (k# , x# ). One can verify that x# is efficient and that p∗ · x#i = p∗ · x䉫i for each i. Let ξ # be the exchange economy (which is just a particular form of production economy) with endowments ω#i = x#i > 0. This economy has an equilibrium in which each i’s consumption is, in fact, x#i (no trade). The proof follows. The next result records the observation that if the desired inner product inequality holds for one linear price normalization then it fails for any other linear price normalization whose normalization vector lies on the “wrong side” of the change in consumption vector. Define ∗ Λp∗ + = {λ ∈ RL + : p · λ = 1}. Lemma 2. Consider any two vectors λ1 , λ2 ∈ Λp∗ + . Suppose that x¯ # − x¯ ∗ lies in the open cone spanned by λ1 and λ2 . If
p# ∗ · (¯x# − x¯ ∗ ) < 0, − p p# · λ 1 then
p# ∗ · (¯x# − x¯ ∗ ) > 0. − p p# · λ 2
I omit the proof, which is a trivial calculation. The uniqueness result is then as follows. Theorem 4. The notation is the same as in Theorems 1–3. 1. Assume x䉫I > 0, for all i. Assume that µ ¯ x > 0. Assume that each consumer’s preference is strongly monotone and has a utility representation that is continuous and concave. Then µ ¯ x is the unique λ ∈ Λp∗ + for which #
p ∗ − p · (¯x# − x¯ ∗ ) ≤ 0, p# · λ
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for every economy ξ # , equilibrium price vector p# , and equilibrium consumption allo¯ (p∗ , x¯ ∗ , x# ) = µ ¯ cation x¯ # such that µ x. 2. Assume that µ ¯ x > 0. Then µ ¯ x is the unique λ ∈ Λp∗ for which #
p ∗ − p · (¯x# − x¯ ∗ ) ≤ 0, p# · λ for every economy ω# , equilibrium price vector p# , and equilibrium consumption allo¯ (p∗ , x¯ ∗ , ω# ) = µ ¯ cation x¯ # such that µ x. Proof. The arguments for either case are nearly identical. For concreteness, consider the ¯ µ ¯ x claim. By Theorem 3, µ x does indeed have the desired property. To see that it is the ∗ only element of Λp + with this property, consider any λ ∈ Λp∗ + , λ = µ ¯ ω . By Lemma 1, # # there is an ω such that ω¯ lies in the open cone spanned by λ and µ ¯ ω and such that # # ¯ ) − p∗ ] · (¯ µ ¯ (p∗ , ω¯ ∗ , ω# ) = µ ¯ x# − x¯ ∗ ) < 0. The proof x . By Theorem 3, [p /(p · µ ω then follows from Lemma 2. One implication of these results is negative: even for a single consumer exchange economy, there is no λ ∈ Λp∗ + for which p䉫 ∗ − p · (¯x䉫 − x¯ ∗ ) ≤ 0, p䉫 · λ holds for every endowment allocation ω䉫 , equilibrium price vector p䉫 , and equilibrium consumption allocation x䉫 . See also the discussion in Nachbar (2002). Remark 5. The results of this section differ from their differential analogs in Nachbar (2002) in the following respects. First, for infinitesimal shocks, the analog of the condition p∗ · x¯ = p∗ · x¯ 䉫 in Lemma 1 can be dropped. Second, the concave utility representation assumption in Lemma 1 and Theorem 4 is satisfied almost automatically in the setting of Nachbar (2002) because the standard assumptions to ensure differentiability of demand are almost sufficient to ensure that preferences have a concave utility representation, at least in a neighborhood of equilibrium consumption in the reference economy; see Proposition 2.6.4 in Mas-Colell (1985). Third, the restriction to λ > 0 is to ensure that p · λ > 0 for any p 0. In the differential case, because all price changes are local, this restriction can be dropped; it suffices to restrict λ to Λp∗ = {λ ∈ RL : p∗ · λ = 1}. In particular, the differential analog of the uniqueness claim holds even if µ ¯ > / 0, meaning that some goods are inferior in the aggregate.
5. The Literature Nachbar (2002) contains a detailed literature review, so I will be brief. Other than Nachbar (2002) itself, the closest predecessor to this paper is the following result, which appears in Chapter 4 of Dixit and Norman (1980). Consider an exchange economy with a single consumer with homothetic preferences. Normalize prices so that
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e(p, 1) = 1, where e is the consumer’s expenditure function. Then, (p䉫 − p∗ ) · (¯x䉫 − x¯ ∗ ) ≤ 0. Finding a fixed price normalization that works for all reference endowments and all endowment shocks is possible in Dixit and Norman (1980) because, with a single homothetic consumer, the expenditure function integrates the different MPC vectors that obtain at different reference equilibria. In view of the necessity of the MPC price normalization for infinitesimal shocks in single consumer economies, this sort of integration is required if (p䉫 − p∗ ) · (¯x䉫 − x¯ ∗ ) ≤ 0 is to hold for all reference endowments and all endowment shocks. Such integration is possible with a single homothetic consumer but not, even with a single consumer, in general. The e(p, 1) = 1 price normalization is not linear. As noted in Section 4, there is no fixed linear price normalization that works for all reference endowments and all endowment shocks. But, in the case of a single homothetic consumer, the IPC vector µ ¯ (= µ ¯ ¯ x =µ ω) ∗ ¯ is collinear with the reference endowment (assuming p · ψ = 0). Hence, if there is a single homothetic consumer then, for any fixed reference endowment ω¯ ∗ and (almost) any endowment shock, the inequality (p䉫 − p∗ ) · (¯x䉫 − x¯ ∗ ) ≤ 0 holds for the linear price normalization p · ω¯ ∗ = 1. This is the Laspeyres price normalization.
Acknowledgements I would like to thank John Quah for helpful discussions. And I would like to thank Washington University’s Center for Political Economy for its support. References Dixit, A., Norman, V., 1980. Theory of International Trade. Cambridge University Press, Cambridge, UK. Grodal, B., Hildenbrand, W., 1989. The Weak Axiom of revealed preference in a productive economy. Review of Economic Studies 56 (4), 635–639. Hicks, J.R., 1939. Value and Capital. Clarendon Press, Oxford. Hildenbrand, W., 1989. The Weak Axiom of revealed preference for market demand is strong. Econometrica 57, 979–985. Jerison, M., 1999. Dispersed excess demands, the Weak Axiom and the uniqueness of equilibrium. Journal of Mathematical Economics 31, 15–48. Mas-Colell, A., 1985. The Theory of General Equilibrium. A Differentiable Approach. Cambridge University Press, Cambridge, UK. Nachbar, J.H., 2002. General equilibrium comparative statics. Econometrica 79 (5), 2065–2074. Negishi, T., 1960. Welfare economics and existence of equilibrium for a competitive economy. Metroeconomica 12, 92–97. Quah, J.K.H., 1999. The Weak Axiom and Comparative Statics. Nuffield College Working Paper, 1999, W15, Oxford University. Quah, J.K.H., 2003. Market demand and comparative statics when goods are normal. Journal of Mathematical Economics 39, 317–333.