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Market demand and comparative statics when goods are normal
Journal of Mathematical Economics 39 (2003) 317–333
Market demand and comparative statics when goods are normal John K.-H. Quah Department of Economics, Oxford University, St. Hugh’s College, Oxford OX2 6LE, UK Received 9 February 2002; received in revised form 12 March 2003; accepted 18 March 2003
Abstract This paper examines the impact of the normality assumption on the structure of market demand and on general equilibrium comparative statics. We define a new notion of comparative statics which is fundamentally related to normality and examine its incidence in exchange, production and financial economies. © 2003 Elsevier Science B.V. All rights reserved. JEL classification: D11; D50; D51; D52 Keywords: Normal goods; Demand aggregation; Weak axiom; Monotonicity; Comparative statics
1. Introduction We say that a consumer’s demand function is normal if, with prices held fixed, demand for all goods increase as income increases. It is well-known that this property is not implied by utility maximization, but it is also clear that it is a very mild assumption to impose, provided goods are thought of in sufficiently broad categories. This paper offers a thorough examination of the implications of this property on general equilibrium comparative statics. Consider an exchange economy with l goods that has a mean endowment of ω and an equilibrium price of p, with both ω and p in Rl++ . While preferences remain unchanged, an endowment perturbation causes mean endowment to change to ω and the equilibrium price to p . What conditions are needed to guarantee that the price of a good ‘falls’ as its endowment increases? Since a scalar multiple of an equilibrium price is also an equilibrium price, the question implicitly assumes some suitable choice of scales. A typical analysis of this or similar problems will identify some good as a numeraire and examine the movement of prices relative to that good (see, for example, Mas-Colell et al., 1995, Section 17.G). E-mail address: [email protected] (J.K.-H. Quah). 0304-4068/03/$ – see front matter © 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0304-4068(03)00047-8
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In this paper, we wish to study a weaker comparative statics property. We wish to examine the conditions under which, for a pair of equilibrium outcomes, there exists some bundle of goods which, if it acts as a numeraire, will cause prices and endowments to move in opposite directions. In other words, the numeraire is not pre-identified but can be freely chosen. Formally, we say that the pair (p, ω) and (p , ω ) satisfies N-monotonicity (where ‘N’ stands for numeraire) if there exists a bundle of goods b, understood as an element in Rl+ , so that with prices normalized by scalar multiplication to satisfy (A) p · b = p · b, we have (B) (p − p ) · (ω − ω ) < 0. Condition (A) equalizes the value of the bundle b at the two prices so it effectively designates b as the numeraire. Condition (B) says that prices and endowments move in opposite direction. In particular, if the price change has resulted from an increase in the endowment of good 1, with the endowment of other goods remaining the same, the price of good 1 before the perturbation, p1 , will be greater than its value after the perturbation (p )1 . Clearly, we also have p1 /p · b < (p )1 /p · b, i.e. the price of good 1 relative to the price of the bundle b, has fallen. We show that if the economy is a single agent economy, then every pair of equilibrium outcomes will have the N-monotonic property if and only if the agent has a demand function which obeys the weak axiom and is normal.1 The situation becomes more complicated when we consider multi-agent exchange economies. In this setting, given any initial economy, we show that there will always exist endowment perturbations which lead to violations of N-monotonicity. In cases like these, no good and no bundle of goods could serve as a numeraire and guarantee that prices and endowments move in opposite directions. The breakdown of N-monotonicity is related to the breakdown of a sort of ‘aggregate normality’ condition: the change in the income distribution caused by the endowment perturbation is so large that the market does not buy more of every good even if average income has gone up and prices are (hypothetically) held fixed. That such perturbations exist is not the same as saying that violations of N-monotonicity are an empirically common phenomenon. Indeed, just as significantly, it can be shown that there is a large class of endowment perturbations for which the associated change in equilibrium prices will obey N-monotonicity. So one has to conclude that theory alone cannot tell us anything about the incidence of N-monotonicity; this is an empirical matter which hinges on the types of endowment perturbations that are likely to occur. The claims we have just made remain true even if the initial economy has a well or badly behaved excess demand function. So in this paper we caution against the casual conflation of comparative statics issues with those relating to dynamic stability and uniqueness. The latter two rely undeniably on the properties of excess demand, but its relationship with comparative statics is considerably less straightforward. While for certain types of endowment perturbations a well-behaved excess demand function (in particular, one satisfying the weak axiom) helps to guarantee N-monotonic equilibrium comparisons, this cannot be true for all perturbations. Similarly, endowment perturbations leading to N-monotonic equilibrium comparisons exist even when the excess demand function is badly behaved. The paper also shows how the concept of N-monotonicity could be extended to financial economies (with a possibly incomplete asset structure) and production economies. In the 1 The sufficiency part of this claim was in essence established by Nachbar (2002), though he did not refer to N-monotonicity as such.
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former, N-monotonicity guarantees that the prices and endowment of securities move in opposite directions, when prices are compared against a numeraire securities portfolio. In production economies, we show that if aggregate demand in the household sector satisfies N-monotonicity for a pair of equilibrium outcomes, then the property must also hold when comparing both economies in its entirety. This result follows from the profit maximization hypothesis. N-monotonicity guarantees that the price of a factor, i.e. an endowed good which does not give utility but which contributes to production, will also move in a direction opposite to that of its endowment. This price is measured relative to some bundle of utility giving goods; in other words, we are referring to movements in the ‘real’ factor price. This result also has implications for the factor-augmenting technology changes which are commonly considered in macroeconomics. For example, if a technology change causes each unit of a particular factor to behave like it was two, then the real factor price will not increase by a multiple greater than two. The paper is organized as follows. Section 2 studies market demand and formulates necessary and sufficient conditions for N-monotonicity. Section 3 examines N-monotonic comparative statics in exchange economies. Sections 4 and 5 extends the discussion to production and financial economies, respectively.
2. N-monotonic market demand We begin this section with a few definitions and a lemma which is fundamental to the results in this paper. A pair (p, x) and (p , x ) in Rl++ × Rl satisfies i. the weak axiom if either p · x > p · x or p · x > p · x ; ii. monotonicity if (p − λp ) · (x − x ) < 0 for λ chosen to satisfy p · x = λp · x ; iii. N-monotonicity if there exists a scalar λ > 0 such that (p −λp)·(x −x) < 0, and when p and p are not collinear, there also exists a vector b in Rl+ such that p · b = λp · b. We refer to b as the normalizing vector. We interpret p as the price vector of l goods, and x as the demand (either of a market or a single agent) at that price. It is easy to check that if the pair (p, x) and (p , x ) satisfies monotonicity or N-monotonicity, then it also satisfies the weak axiom. If the pair satisfies the weak axiom, we can assume that p · (x − x) > 0, in which case, for a sufficiently large positive λ, we have (p − λp) · (x − x) < 0. Given any p and λp, we can always find b = 0 such that p · b = λp · b, so what makes N-monotonicity stronger than the weak axiom is the requirement that b > 0. As we had pointed out in the introduction, in a comparative statics setting, b can be interpreted as a numeraire. So the specification that b > 0 is a compromise between pre-specifying a particular good as numeraire (which will mean that positive results can only be obtained under strong assumptions) and allowing the numeraire to be just any vector. It seems natural, given that we will be assuming that preferences are monotone, not to use as a numeraire a bundle of goods which has negative amounts of some goods. Furthermore, as this paper will make clear, N-monotonicity arises naturally in general equilibrium comparative statics, so it becomes a focus of enquiry even if one is merely interested in establishing a weaker property like the weak axiom. Our first result gives the precise conditions under
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Fig. 1. N-monotonicity between (p, x) and (p , x ).
which a pair (p, x) and (p , x ) satisfies N-monotonicity. It is fundamental to virtually everything else in this paper. Lemma 2.1. Consider a pair (p, x) and (p , x ) in Rl++ × Rl , where p and p are not collinear and p · x = p · x . Then the pair satisfies N-monotonicity with the normalizing vector b > 0 if and only if there exists c in Rl such that (i) x − c > 0 or x − c < 0 and b is collinear with x − c, (ii) p · c = p · x, (iii) (p , c) and (p, x) satisfies the weak axiom. Perhaps the best way of understanding conditions (i)–(iii) is to look at Fig. 1. Condition (ii) says that c should be on the budget line α, i.e. with price p and bundle x just affordable. Condition (iii) says that c rests on the line α and to the left of the bundle x. Condition (i) says that c must lie between the points m and n. In short, (p, x) and (p , x ) is N-monotonic if and only if a point c like the one in Fig. 1 can be found—on the line α, between m and x. The normalizing vector will then be x − c. Now it should be intuitively clear why normality plays a central role in this concept. Suppose that x and x are the demand bundles at prices p and p , respectively, of an individual utility maximizing agent with normal demand. Then c exists because we can always choose c to be the agent’s demand at income p · x—condition (iii) is guaranteed by revealed preference while (i) is guaranteed by normality. With the help of Lemma 2.1, we will now consider how N-monotonicity may arise in a market. We assume that the commodity space has l goods and the agents in the market form a finite set A. Each agent a has an income ya > 0 and a demand function fa . Formally, fa : Rl++ × R+ → Rl++ is a demand function if it is zero-homogeneous, i.e. fa (p, ya ) = fa (kp, kya ) for all k > 0, and satisfies p · fa (p, ya ) = ya . These properties are standard. We refer to the vector y = (ya )a∈A 0 as the market’s income distribution and denote the mean income by y¯ . An ordered pair (p, y) will be called a market situation. The (mean) |A| l market demand function F : Rl++ × R ++ → R++ maps each market situation (p, y) to the mean market demand, i.e. F(p, y) = a∈A fa (p, ya )/|A|. If we assume that the fa ’s are demand functions, then F is also zero-homogeneous and will satisfy p · F(p, y) = y¯ . For the rest of this paper, we shall assume that F satisfies these properties.
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We say that the demand function fa satisfies the weak axiom if for every ya , p and p , with p = p , the pair (p, fa (p, ya )) and (p , fa (p , ya )) satisfies the weak axiom. We say that F satisfies the weak axiom if for every y, p and p , with p = p , the pair (p, F(p, y)) and (p , F(p , y)) satisfies the weak axiom. Analogous definitions hold for the monotonicity and N-monotonicity properties. It is important to remember that when we refer to a particular property as holding for F we are varying prices while keeping the income distribution fixed. An agent with a demand function fa generated from a monotone preference where the indifference surfaces have no kinks will obey the weak axiom.2 Generally, the fact that fa obeys the weak axiom for all a is no guarantee that F obeys the weak axiom. However, in these situations, where one is varying prices while keeping the income distribution fixed, there is a wealth of theory with conditions guaranteeing that the weak axiom holds (see Mas-Colell et al., 1995, Chapter 4). For example, F obeys the weak axiom if fa obeys monotonicity for all agents. There is also considerable empirical evidence to support the view that F obeys the weak axiom (see Hildenbrand, 1994). So it is fair to say that this property is not overly strong. In this section, we investigate when F will obey N-monotonicity; more generally, we will allow the income distribution to change, and give conditions guaranteeing that a pair (p, F(p, y)) and (p , F(p , y ) satisfies N-monotonicity. To do that, we need an appropriate notion of normality for market demand. The demand function fa is normal at (p, ya ) if f(p, ya ) > f(p, ya ) when ya > ya and f(p, ya ) < f(p, ya ) when ya < ya . We call fa normal if it is normal at all (p, ya ) in Rl++ × R+ . The market demand function F satisfies Aaggregate normality between (p, y) and (p, y ) when the following holds: if y¯ > y¯ , then F(p, y) > F(p, y ); if y¯ < y¯ , then F(p , y) < F(p , y ); and if y¯ = y¯ , then F(p , y) = F(p , y ). We say that F obeys proportional aggregate normality if at any price p and income distributions y and ky, where k is a scalar, the pair (p, y) and (p, ky) satisfies aggregate normality. The next result gives sufficient conditions for N-monotonicity. Proposition 2.2. Suppose that F satisfies the weak axiom. Let (p, y) and (p , y ) be two non-collinear market situations and by multiplying (p , y ) with a scalar if necessary, assume that p · F(p, y) = y¯ . Then the pair (p, F(p, y)) and (p , F(p , y )) is N-monotonic if F satisfies aggregate normality between (p , y ) and (p , y). As we had argued earlier, that F obeys the weak axiom is a fairly mild assumption; hence, the focus of attention when considering whether a pair (p, F(p, y)) and (p , F(p , y )) 2 Suppose that agent a’s preference is such that there is more than one supporting price at a commodity bundle x, i.e. there is a kink at x. Then there are distinct prices p and p such that x = fa (p, ya ) = fa (p , ya ), which means that the pair (p, fa (p, ya )) and (p , fa (p , ya )) will violate our definition of the weak axiom. Note that our definition is slightly stronger than the usual one (see, for example, Mas-Colell et al., 1995), which is identical with ours, except that it does permit demand to be the same at two different price vectors. We need the slightly stronger definition because we establish N-monotonicity by assuming the weak axiom (plus other assumptions) and N-monotonicity is defined in terms of a strong inequality. If we modify the definition of N-monotonicity so that the inequalities are weak, our definition of the weak axiom can be correspondingly weakened. Our slightly stronger version of the weak axiom will hold if the agent’s demand function is generated from a C1 utility function u : Rl++ → R. In this case, each bundle x in Rl++ has at most one supporting price corresponding to the gradient of u at x.
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satisfies N-monotonicity must primarily fall on the aggregate normality condition. If y ≥ y or y > y , then it is clearly satisfied if all agents have normal demand functions, so in these cases we can reasonably expect N-monotonicity to hold. It is also clear that beyond these cases aggregate normality will hold for many pairs of income distributions. However, short of all agents having parallel income expansion paths, violations of aggregate normality are always possible. To see this, consider a re-distribution of y , an income distribution y , which must also have a mean of y¯ . Since income expansion paths are non-parallel, there will be some good i for which F i (p , y ) < F i (p , y ), so aggregate normality is violated. For a strict violation, add to every agent’s income, so y = y + (, , . . . , ). Provided F is continuous, there will be a sufficiently small such that F i (p , y) < F i (p , y ), but y¯ > y¯ . In the next proposition, we show that aggregate normality is necessary for N-monotonicity. The result requires that we add some standard properties to F . We say that the market demand function F is nice if it is continuous and satisfies the following boundary condition: whenever the sequence (pn , yn ) tends to (p, y), where p is on the boundary of Rl++ and y¯ = 0, |F(pn , yn )| tends to infinity. It is well-known that this property will hold if all agents in the market have continuous, strongly monotone, and strongly convex preferences. Proposition 2.3. Suppose that F violates aggregate normality between (p , y) and (p , y ) and that F is nice and satisfies the weak axiom. Then there is p such that p · F(p, y) = y¯ and (p, F(p, y)) and (p , F(p , y )) violates N-monotonicity. Furthermore, p can be chosen to be arbitrarily close to p if y is sufficiently close to y . The next result concerns N-monotonicity when the income distribution is fixed. We omit its proof, which follows immediately from Propositions 2.2 and 2.3. Corollary 2.4. Suppose that F is nice and obeys the weak axiom. Then F is N-monotonic if and only if it obeys proportional aggregate normality. In particular, the demand function of a single agent which is nice and obeys the weak axiom will be N-monotonic if and only if it is normal.
3. N-monotonicity in exchange economies In this section, we explore the implications of Section 2’s results for general equilibrium comparative statics. We now assume that each agent a in the market defined in Section 2 also has an endowment of ωa in Rl+ ; we write ω = (ωa )a∈A and the mean endowment as ω. ¯ We assume that ω¯ 0. We denote this exchange economy by E(ω) and the collection of such economies by M, so M = {E(ω) : ω ∈ (Rl+ )|A| , ω¯ 0}. Two economies in M have the same agents with the same demand functions (and therefore the same market demand function F ) but different endowments. A price p is the equilibrium price of E(ω) if F(p, (p · ωa )a∈A ) = ω. ¯ The equilibrium set, denoted by E, is the set of ordered pairs (p, ω) where p is an equilibrium price of E(ω) in M.
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Throughout this section, we will adopt the perspective that the ‘initial’ economy is E(ω), with an equilibrium price of p. Endowments are then changed to ω and the equilibrium price becomes p . Abusing our terminology a little, we say that (p , ω ) and (p, ω) in E satisfies N-monotonicity if (p , ω¯ ) and (p, ω) ¯ satisfies N-monotonicity. The first thing we should note is the close relationship between this issue and our work in the last section. The pair (p , ω ) and (p, ω) in E is N-monotonic if and only if (p , F(p , y )) and (p, F(p, y)) is N-monotonic, where y = (p ·ωa )a∈A and y = (p·ωa )a∈A . Furthermore, the next result shows that we can associate an exchange economy to every market situation (p , y ). Proposition 3.1. Let (p, ω) be in E, write y = (p · ωa )a∈A , and suppose that F is nice. Then for every market situation (p , y ) there is ω with (p , ω ) in E and (p · ωa )a∈A = y such that ω is arbitrarily close to ω if (p , y ) is sufficiently close to (p, y). The first issue we wish to address is the incidence of N-monotonic comparative statics. In other words, of all the possible endowment perturbations of the initial endowment ω, how often can we expect N-monotonicity to hold between (p , ω ) and (p, ω)? The answer is that we cannot say, because, as we shall see, the theory tells us that there is a proliferation of both endowment perturbations which obey N-monotonicity and those which violate it. The intuition behind our claim comes from the results of the last section. As we had pointed out, income changes which obey aggregate normality and those which violate aggregate normality both exist; the former leads to N-monotonicity while the latter leads to violations of N-monotonicity. The first of our next two result shows how we can easily construct endowment perturbations which obey N-monotonicity; the second says that endowment perturbations which violate N-monotonicity must always exist. Proposition 3.2. Assume that the demand functions fa obey the weak axiom and let (p, ω) be in E. Suppose that for some (p , y ), F satisfies aggregate normality between (p , y ) and (p , y ), with y = (p · fa (p, p · ωa ))a∈A and y¯ = y¯ . Then there is ω , with (p , ω ) in E, such that (p · ωa )a∈A = y and the pair (p , ω ) and (p, ω) is N-monotonic. Furthermore, if F is nice, we can choose ω to be arbitrarily close to ω if (p , y ) is sufficiently close to (p, y). Note that y (in Proposition 3.2) can be easily constructed. For example, if fa obeys normality for all a, then for any choice of p , F obeys aggregate normality between (p , y ) and (p , y ) provided we choose y so that y > y or y < y .3 Proposition 3.3. Suppose that F is nice and obeys the weak axiom and let (p, ω) be in E. Provided all agents do not have parallel income expansion paths, there is a sequence (pn , ωn ) in E tending to (p, ω), with ω¯ n = ω¯ such that (pn , ωn ) and (p, ω) violates N-monotonicity for all n. 3 A related observation is that any endowment perturbation which leads to greater (or lesser) utility for all agents must lead to N-monotonicity. To see this, suppose that all agents are worse off after the perturbation. By revealed preference, ya = p · fa (p, ya ) > p · fa (p , ya ) = ya for all a, which implies aggregate normality if all agents have normal demand functions.
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Note that Proposition 3.3 imposes no assumptions on the initial economy’s excess demand ¯ So the existence of the sequence is function, i.e. the function Zω (p) = F(p, (p·ωa )a∈A )− ω. guaranteed even if Zω satisfies gross substitutability, the weak axiom or other nice properties. In a similar vein, we note that Proposition 3.2 imposes no restrictions on Zω either, so the result holds even when it does not satisfy any nice structural properties.4 Most of the positive comparative statics results in the literature are not like Proposition 3.2. Typically, they impose restrictions on the excess demand function, which then guarantee nice comparative statics for certain classes of endowment perturbations (see, for example, Mas-Colell et al., 1995, Proposition 17.G.3 and the discussion following it). Our next result is of the same type. It shows that when the excess demand function satisfies a version of the weak axiom, one can explicitly identify a set of endowment perturbations which lead to N-monotonicity. This result is closely related to Nachbar (1999, Theorem 2); indeed, one could think of it as a discrete version of that result. Nachbar (2001, Theorem 2) has also independently obtained a closely related discrete result.5 Proposition 3.4. The pair (p , ω ) and (p, ω) in E is N-monotonic if (A) p · Zω (p) > 0 when p and p are non-collinear and (B) F satisfies aggregate normality between (p, y) and (p, y ), where y = (p · ωa )a∈A and y = (p · ωa )a∈A . If all agents have normal demand functions, (B) will be satisfied if p · (ωa − ωa ) are of the same sign for all a. This occurs, for example, when the endowment of a good k increases for some agents. (A) is a weak axiom type condition on the excess demand function. This condition is strong; in particular it is considerably stronger than the property that F satisfies the weak axiom, essentially because the income distribution of the economy changes as prices change. Nevertheless, reasonable conditions guaranteeing this property are known (see Jerison, 1999; Quah, 1999). The local differentiable analog to (A) is (A ): vT ∂q Zω (p)v < 0 for v in Rl , non-zero and non-collinear with p. In the final result of this section, we identify a set of perturbations for which (A ) is both necessary and sufficient for N-monotonicity. We assume that all the demand functions fa are C1 and that p is a regular equilibrium price of E(ω), i.e. ∂q Zω (p) is of rank l − 1. In this case, we know that there is an open neighbourhood U of ω and a C1 function P : U → Rl++ such that for all ω in U, P(ω ) is an equilibrium price of E(ω ) and P(ω) = p. We refer to P as an equilibrium map at p. Proposition 3.5. Suppose that fa is C1 and normal for all a, p is a regular equilibrium price of E(ω) and let P : U → Rl++ be an equilibrium map at p. 4 To see that Proposition 3.2 involves no restrictions on the excess demand function we need only note that the conditions in the proposition are satisfied if all agents have homothetic preferences. This is far stronger than what the proposition needs, yet even this assumption imposes no structure on the excess demand function (Mantel, 1976). 5 Nachbar states his results in terms of a relationship linking the initial and final mean endowments and equilibrium prices, and a vector of the marginal propensity to consume (averaged across all agents). Provided this vector is in Rl+ , the relationship is what we refer to as N-monotonicity, with the marginal propensity to consume serving as the normalizing vector.
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i. If (A ) holds, then so does the following property (∗): there is an open neighbourhood U˜ of ω, with U˜ ⊂ U, such that for all ω in U˜ with p · (ωa − ωa ) either non-negative for all a or non-positive for all a, the pair (P(ω ), ω ) and (p, ω) obeys N-monotonicity. ii. If property (∗) holds, vT ∂q Zω (p)v ≤ 0 for all v in Rl .6
4. N-monotonicity in production economies The objective of this section is to extend our comparative statics analysis to production economies. Consider a production economy with l + m goods. The first l goods give utility to all consumers, while the other m goods do not and are useful only as factors of production. We will refer to these m goods as factors but it important to note that we are not excluding the possibility that utility-giving goods are useful as inputs to the production process and indeed we allow for m to equal 0. Production is chosen from a production possibility set Y contained in Rl+m . We denote an element in Y by (u, v) where u is in Rl and v is in Rm . A price vector will be written as (p, q) where p is the price of the first l goods and q the price of the m factors. We assume that there is a price (p, q) in Rl+m ++ such that arg max(u,v)∈Y (p, q) · (u, v) is non-empty and denote the subset of such prices by P ⊂ Rl+m ++ . For each (p, q) in P, the supply correspondence SY is defined by SY (p, q) = arg max(u,v)∈Y (p, q) · (u, v). The profit function ΠY is defined by ΠY (p, q) = (p, q) · SY (p, q). The consumer/household sector consists of a set A of agents. Agent a has a demand function fa . We denote by F(p, y) the market demand of the household sector when p is the price and y the income distribution. The agent a has an endowment ωa of utility-giving goods and µa of factors. He also has a claim on profits of αa , so αa ≥ 0 and a∈A αa = 1. At price (p, q) in P, agent a’s income is ya = (p, q) · (ωa , µa ) + αa ΠY (p, q) and his demand is fa (p, ya ), which is in Rl++ . This completes our specification of the production economy. Since we shall be studying the behaviour of this economy when endowments and technology are varied, we will emphasize its dependence on these by denoting the economy by E(ω, µ, Y), where µ = (µa )a∈A and ω is similarly defined. A price (p, q) in P is an equilibrium price if there is s(p, q) in S(p, q) such that (F(p, y), 0) = s(p, q) + (ω, ¯ µ). ¯ The right hand side of this equation is just the economy’s aggregate supply, with ω¯ and µ ¯ being, respectively, the mean endowments of utility giving goods and factors. The left hand side is the market demand of the household sector at price p and at the income distribution generated by (p, q), which we denote by y. Since F(p, y) lives in Rl++ , to make the equation formally correct, we write aggregate demand as (F(p, y), 0), where 0 is the zero vector in Rm . The type of technology changes we will consider are those of the factor augmenting variety, which are standard in macroeconomic models. For θ in Rm ++ , we define the set Yθ by Yθ = {(u, θ ⊗ v : (u, v) ∈ Y }, where θ ⊗ v refers to the vector (θ 1 v1 ,θ 2 v2 , . . . , θ m vm ). Technology changes of this sort are easy to analyze since its equilibrium effects could be replicated by a suitable change in the economy’s factor endowment. 6 An examination of the proof will reveal that (ii) can be strengthened by weakening the assumption. Part (ii) is still true if ‘obeys N-monotonicity’ in property (∗) is replaced by ‘obeys the weak axiom.’
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Lemma 4.1. The price (p, q) is an equilibrium price of E(ω, µ, Yθ ) if and only if (p, θ −1 ⊗q) is an equilibrium price of E(ω, θ ⊗ µ, Y). Note that θ −1 refers to the vector whose ith entry is 1/θ i , and θ ⊗ µ denotes (θ ⊗ µa )a∈A . Obtaining N-monotonic equilibrium comparisons between production economies is facilitated by the profit maximization hypothesis, which guarantees that the supply correspondence has a nice monotonic property. In particular, profit maximization guarantees that for (u, v) in S(p, q) and (u∗ , v∗ ) in S(p∗ , q∗ ), we have (p, q) · ((u, v) − (u∗ , v∗ )) ≥ 0. We say that S has the strong supply property if this condition holds and in addition, the inequality is strict when there exists t such that p = tp∗ and q = tq∗ .7 With this in place we can state the main result of this section. Proposition 4.2. Let (p, q) and (p , q ) be equilibrium prices of E(ω, µ, Y) and E(ω , µ , Yθ ) respectively and denote by y and y the income distributions generated at the two equilibria. Suppose that the pair (p, F(p, y)) and (p , F(p , y )) is N-monotonic and that SY has the strong supply property. Then the pair ((p, q), (ω, ¯ µ)) ¯ and ((p , θ −1 ⊗ q ), (ω¯ , θ ⊗ µ ¯ )) −1 satisfies N-monotonicity; when (p, q) and (p , θ ⊗ q ) are not collinear, the normalizing vector can be chosen in the form (b, 0), where b ∈ Rl+ \ {0} and 0 ∈ Rm . Note that Proposition 4.2 does not give conditions guaranteeing N-monotonicity for the household sector; instead it assumes that condition and says what it implies for the whole economy. N-monotonicity for household demand will be guaranteed if, for example, the conditions of Proposition 2.2 are satisfied. It is also not hard to see that, along the lines of Proposition 3.2, one can readily construct endowment perturbations for which N-monotonicity is satisfied by household demand. To appreciate the significance of Proposition 4.2, imagine that there is an increase in the endowment of the first factor, i.e. good m + 1, causing prices to move from (p, q) to (p , q ). The result says that provided N-monotonicity is satisfied in the household sector, the price q1 will be greater than (q )1 , using the normalization, p · b = p · b. This normalization means that factor prices are measured in ‘real’ terms, i.e. in terms of the number of units of some representative bundle of consumer goods. In short, when a factor’s endowment is increased, its real factor price will fall. What if endowments remain the same, but there is a technology change which augments the first factor, i.e. θ = (k, 1, 1, . . . , 1), with k > 1? This change is equivalent to an increase in the endowment of factor 1 by the multiple k. If the conditions of Proposition 4.2 are satisfied, we may conclude that q1 is greater than (q )1 /k. In other words, we obtain the intuitive conclusion that the price of factor 1 will not rise by a multiple of more than k. We may view Proposition 4.2 as a statement on the ‘observable restrictions’ of Walrasian equilibria in the spirit of Brown and Matzkin (1996). Assume that an observer can observe all prices, mean household demand and mean endowment. He will then be in a position to determine if N-monotonicity is satisfied in the household sector; if it is, then the model 7 The slight strengthening is a mild curvature condition on the production set. The reason for this condition is in essence the same as the one given in footnote 2 for disallowing kinks in agents’ preferences. The condition is needed to establish N-monotonicity and the latter is defined in the form of a strong rather than weak inequality.
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predicts that the same property will be extended to the whole economy. If he observes that it is not, the model has been refuted.
5. N-monotonicity when markets are incomplete In this section, we show that the results on N-monotonicity we have developed for complete markets can be extended to a certain class of financial markets with an incomplete asset structure. We assume that there are two dates, 0 and 1. At date 1 there are l states of the world, with one good in each state. The economy has m securities, with m ≤ l. The m × l matrix D gives the payoffs of these securities, with the ijth entry being the payoff of the ith security in state j. We assume that the rank of D is m. An agent a in this economy has an endowment of securities ωa in Rm satisfying DT ωa > 0. A weaker alternative assumption is to assume that the agent has an endowment of consumption in different states of nature, i.e. in Rl+ , which may or may not be achievable by some security portfolio. It is not completely clear the extent to which our results in this section will carry over to this more general case; our treatment will in any case have to be more complicated, so we will stick with our stronger assumption. We assume that there is no consumption at date 0 and every agent consumes only at date 1. Agent a’s consumption is governed by the demand function, fa which, following the definition in Section 2, is zero-homogeneous and satisfies the budget identity. In this context we interpret fa (p, ya ) in Rl++ as agent a’s demand for consumption in different states of the world at date 1, when he faces the state price vector p and has income ya at date 0. We also assume that his demand has the following properties: (a) fa (p, ya ) = DT θ for some θ in Rm and (b) fa (p, ˜ ya ) = fa (p, ya ) whenever Dp = Dp. ˜ These requirements will hold if the agent is maximizing his utility at date 0 subject to the budget set B∗ (p, ya ) = {x ∈ Rl++ : x = DT θ for some θ ∈ Rm , and p · x ≤ ya }. So the agent’s budget set now has the added restriction that the demand chosen has to be achievable via some portfolio of securities θ. Note that (b) is true because if Dp = Dp, ˜ then B∗ (p, ya ) = B∗ (p, ˜ ya ). We denote the set of arbitrage free security prices by Q, which we know from standard theory satisfies Q = {q : q = Dp for some p 0}. We write Ω = {ω ∈ Rm : DT ω > 0}. Given fa , we could define the security demand function, ga : Q × Ω → Rm given by ga (q, ωa ) = θ where θ satisfies fa (p, qT ωa ) = DT θ and p satisfies Dp = q. It is fairly straightforward to check that ga is well-defined. Furthermore, if the agent a has a utility function Ua , then ga (q, ωa ) solves the problem: maximize Ua (DT θ) subject to q · θ ≤ q · ωa . A financial (or security market) economy consists of the set A of agents, each of whom has an endowment of securities ωa in Ω and a demand function fa satisfying the properties identified above. We denote this economy by F(ω), where ω = (ωa )a∈A . A price q in Q is an equilibrium price if a∈A ga (q, ωa )/|A| = ω, ¯ where ω¯ is the economy’s mean endowment of securities. We denote the equilibrium set by EF —this consists of the ordered pairs (q, ω) where q is an equilibrium price of the economy F(ω). We are interested in conditions which guarantee N-monotonicity between the equilibrium outcomes of two financial economies. To this end we will need to consider the plausibility of normality and weak axiom type conditions in this context. Normality for fa is defined in the usual way, and while not guaranteed by utility maximization remains a reasonable
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assumption. The definition of the weak axiom, however, needs modification. We say that fa satisfies the weak axiom if whenever Dp = Dp, ˜ and p˜ · fa (p, ya ) ≤ ya then p · fa (p, ˜ ya ) > ya . The additional condition requiring Dp = Dp˜ is needed to guarantee that demand is different at the two prices. Finally, the notion of N-monotonicity itself must also be modified. A pair (q, θ) and (q , θ ), both in Q × Rm , is defined as N-monotonic if there is a positive scalar λ such that (q − λq) · (θ − θ) < 0 and when q and q are not collinear, there ˆ N-monotonicity says that the also exists θˆ in Rm such that DT θˆ > 0 and q · θˆ = λq · θ. equilibrium price and endowment move in opposite directions, provided a suitable portfolio (θˆ in our definition), which generates weakly positive consumption in all states of the world, is chosen as the numeraire securities portfolio. With these in place, we can state the next result. It is in the same spirit as Proposition 3.2. It shows how endowment perturbations which obey N-monotonicity can be readily constructed in this context. Proposition 5.1. Assume that the demand functions fa obey the weak axiom and let (q, ω) be in EF . Suppose that for some (p , y ), F satisfies aggregate normality between (p , y ) and (p , y ) where q = Dp and y = (q · ga (q, ωa ))a∈A . Then there is ω with (q , ω ) in EF such that (q · ωa )a∈A = y , and the pair (q , ω ) and (q, ω) is N-monotonic. It is also possible to write down a result for financial economies which is the analog of Proposition 3.4. The excess demand function for securities of the economy F(ω ) is defined in the usual way: for any price qˆ in Q, Zω (ˆq) = a∈A ga (ˆq, ωa )/|A| − ω¯ . Proposition 5.2. The pair (q , ω ) and (q, ω) in EF is N-monotonic if (A) q · Zω (q) > 0 and (B)fa is normal for all a and q·(ωa −ωa ) are either non-negative for all a or non-positive for all a. Unlike the situation for standard exchange economies, the primitive conditions guaranteeing that the excess demand function for securities obey the weak axiom (property (A)) are less well-explored. But one simple case is well-known and straightforward to verify: when ¯ a∈A and ω = (ta ω¯ )a∈A , the economies have collinear endowments, i.e. when ω = (ta ω) with ta > 0 and a∈A ta = 1, and all agents have monotonic security demand functions. Sufficient utility conditions for the latter can be found in Quah (2003). Note also that for such perturbations q · (ωa − ωa ) are clearly all of the same sign. Acknowledgements I am grateful to the Economic and Social Research Council for providing me with financial support under its Research Fellowship Scheme. I would like to thank John Nachbar for some stimulating discussions and without whose earlier work this paper would probably not have been conceived. Comments from the editors and referee of this journal, and from participants at the 2002 GE conferences in Athens and Minneapolis also helped to improve the presentation and sharpen the focus of this paper. A more leisurely account of the material
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in this paper, together with some other results can be found in Quah (2001). Finally, I would like to thank Felicity Nyan for help with preparing the manuscript and much else besides.
Appendix A Proof of Lemma 2.1. For sufficiency, assume that p · x < p · x (the proof for the other case is the same). Choose c such that it satisfies (i)–(iii) and set b = x − c. Choose λ such that p · b = λp · b. Then (p − λp) · (x − x) = (p − λp) · (x − c + c − x) = (p − λp) · (c − x) = −λp · (c − x) < 0. Note that the second equality follows from our choice of b and λ, the final equality follows from (ii), and the final inequality from (iii). To proof necessity, suppose that (p − λp) · (x − x) < 0 for b satisfying p · b = λp · b. Choose c such that p · c = p · x and x − c is collinear with b. Then c satisfies (i) and (ii). Since the pair is N-monotonic, the expression (p − λp) · (x − x) = −λp · (c − x) must be negative. This implies that p · c > p · x, which is condition (iii). 䊐 To prove Proposition 2.2, we will use the following lemma.8 Lemma A. Let (p, y) and (p , y ) be two market situations. Re-scale (p , y ) to satisfy p · F(p, y) = y¯ . Assuming p = p , the pair (p, F(p, y)) and (p , F(p , y )) is N-monotonic if there exists some distribution y such that (1) y¯ = y¯ and F satisfies aggregate normality between (p , y ) and (p , y ); and (2) F satisfies the weak axiom between (p , F(p , y )) and (p, F(p, y)). The normalizing vector can be chosen to be collinear to F(p , y ) − F(p , y ) if y¯ = y¯ ; if y¯ = y¯ , any vector in Rl+ can serve as a normalizing vector. Proof. If y¯ = y¯ , by (1), y¯ = y¯ so normality implies that F(p , y ) = F(p , y ). Since p · F(p, y) = y¯ , (2) implies that p · F(p , y ) > p · F(p, y). Therefore, for any λ > 0, (p − λp) · (F(p , y ) − F(p, y)) = −λp · (F(p , y ) − F(p, y)) = −λp · (F(p , y ) − 䊐 F(p, y)) < 0. If y¯ = y¯ , we may apply Lemma 2.1 with c = F(p , y ). Condition (ii) of Lemma 2.1 is satisfied since p · c = p · F(p , y ) = y¯ = y¯ = p · F(p, y); (iii) is satisfied because of condition (2) in this theorem; finally, (i) is satisfied because of aggregate normality between (p , y ) and (p , y ). Proof of Proposition 2.2. We omit the easy case when p and p . If they are not, we apply Lemma A, but to do that we need to specify an appropriate y . In this case, choosing y = y will satisfy conditions (1) and (2) of Lemma A. 䊐 Proof of Proposition 2.3. We divide the proof into a few claims. We omit the proof of the first claim which is quite standard. (It is also found in Quah, 2001.) 䊐 8 Lemma A is a sort of decomposition theorem. If there is one agent in the market, conditions (1) and (2) correspond to the income and substitution effects. In a multi-agent market, this decomposition can be done in many ways, each corresponding to a particular choice of y (see Lemma A. This lemma is used in the proofs of Propositions 2.2, 3.2 and 3.4; in each of them, a different y is chosen.
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Claim 1. Suppose that F : Rl++ ×R++ → Rl++ is nice and satisfies the weak axiom. Fixing the income distribution at y, for all x in Rl++ , there is a unique p such that F(p, y) = x. The map from x to p, which we denote by P, is continuous and satisfies the following boundary condition: if xn tends to x on the boundary of Rl+ , |P(xn )| must tend to infinity. Assume that y¯ < y¯ and that F violates normality between (p , y) and (p , y ). (The case of y > y can be treated in the same way.) Then F(p , y) is not contained in the set C = {x ∈ Rl++ : p · x = y¯ and x < F(p , y )}. By the separating hyperplane theorem, there is x¯ satisfying p · x¯ = y¯ , q in Rl , and a scalar M such that q · x¯ = M, q · F(p , y) > M and q · c < M for all c in C. Define S = {x ∈ Rl++ : p · x = y¯ and q · x = M} and the map E : S → Rl by E(x) = P(x) − p , where P follows the definition in Lemma A. Note that x · E(x) = 0 for all x in S. Claim 2. There is x∗ in S with t · E(x∗ ) = 0 for all t in T = {t ∈ Rl : p · t = q · t = 0}. To see this, we first choose some xˆ in S. By the boundary property on P (and therefore E), we know that there is a compact set J, such that xˆ · E(x) > 0 for x not in J. Given this, we can choose a compact and convex set in S, call it K, which contains J such that xˆ ·E(x) > 0 whenever x is on the boundary of K (with respect to the relative topology). By the Gale–Debreu–Nikaido Lemma (see Debreu, 1982), there is x∗ in K such that x · E(x∗ ) ≤ 0 for all x in K. Since xˆ is in K, we know that x∗ is not on the boundary of K. We claim that t · E(x∗ ) = 0 for all t in T . Without loss of generality, assume that t · E(x∗ ) > 0 for some t. For δ positive and sufficiently close to zero, x = x∗ + δt must be in K (since x∗ is not on the boundary) and so x · E(x∗ ) = δt · E(x∗ ) ≤ 0, which is a contradiction. Claim 3. Suppose x∗ satisfies the conditions in Claim 2. Then c · P(x∗ ) < y¯ for all c in C. By our choice of q, for any c in C, q · c < M while q · F(p , y) > M, so there is α in (0, 1) with q · [αF(p , y) + (1 − α)c] = M. Clearly then [αF(p , y) + (1 − α)c] − x∗ is in T (as defined in Claim 2) so that ([αF(p , y) + (1 − α)c] − x∗ ) · E(x∗ ) = 0. Since x∗ · E(x∗ ) = 0, and E(x∗ ) = P(x∗ )−p , we may re-write this equation as [αF(p , y)+(1−α)c]·P(x∗ ) = y¯ . By the weak axiom, P(x∗ ) · F(p , y) > y¯ , so we may conclude that c · P(x∗ ) < y¯ . We have shown that there is x∗ satisfying p · x∗ = y¯ such that a vector c satisfying the conditions of Lemma 2.1 does not exist. This implies that the pair (p , F(p , y )) and (P(x∗ ), x∗ ) = (P(x∗ ), F(P(x∗ ), y)) violates N-monotonicity. Suppose now that yn tends to y . For each n we know that there is pn such that p · F(pn , yn ) = y¯ n with the pair (p , F(p , y )) and (pn , F(pn , yn )) violating N-monotonicity for all n. We wish to show that pn tends to p . Define the scalar λn by specifying that zn = λn F(p , y ) satisfies p · zn = y¯ n . Note that λn tends to 1. Since N-monotonicity is violated, zn · pn ≤ y¯ n for all n (otherwise, F(p , y ) will be a normalizing vector). Without loss of generality we assume that F(pn , yn ) has a limit of x∗ . If this limit is on the boundary, then because the image of F is in Rl++ , |pn | will tend to infinity. This cannot happen since zn · pn is bounded and zn is bounded away from the boundary. So we assume that x∗ is in Rl++ , in which case there is p∗ such that F(p∗ , y ) = x∗ with pn tending to p∗ . Since zn tends to F(p , y ), taking limits, we obtain F(p , y ) · p∗ ≤ y¯ . On the other hand,
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p · F(pn , yn ) = y¯ n , so p · F(p∗ , y ) = y¯ . To preserve the weak axiom, we must have p∗ = p . Proof of Proposition 3.1. There are many ways of constructing the economy E(ω ). We will choose a construction that satisfies the continuity property we require. Define the scalar , y )−K ω) K by setting p ·F(p , y ) = y¯ = Kp · ω. ¯ Let ωa = ωa +λa ω+(F(p ¯ ¯ and choose 9 λa so that p · ωa = ya . Summing ya across all agents, we see that K = 1 + a∈A λa /|A|. This also means that ω¯ = F(p , y ). If F is nice (and so continuous), F(p , y ) will converge to ω¯ if (p , y ) converges to (p, y). It is clear that with our construction of the economy E(ω ), ω will converge to ω. 䊐 Proof of Proposition 3.2. We omit the simple case when p and p are collinear. When they are not, we apply Lemma A.10 By multiplying (p, y) with a scalar if necessary, we have p · F(p, y) = y¯ . Note that F(p, y) = ω, ¯ and because (p, ω) is in E, y¯ = p · ω¯ = y¯ . F also satisfies aggregate normality between (p , y ) and (p , y ) by assumption, so condition (1) is satisfied. Furthermore, p · fa (p, ya ) > p · fa (p, ya ) since a obeys the weak axiom. Summing across the agents, we have p · F(p, y ) > p · F(p, y), so condition (2) is satisfied. Therefore, Lemma A says that (p, F(p, y)) and (p , F(p , y )) obeys N-monotonicity. Applying Proposition 3.1 then gives us our result. 䊐 Proof of Proposition 3.3. Since agents’ income expansion paths are not parallel, there is yn tending to y such that y¯ n = y¯ and (p, y) and (p, yn ) violate aggregate normality for all n. By Proposition 2.3, there is pn tending to p such that (pn , F(pn , yn )) and (p, F(p, y)) 䊐 violate N-monotonicity for all n. Proposition 3.1 then gives us the desired result. Proof of Proposition 3.4. We omit the simple case when p and p are collinear. When they are not, we apply Lemma A, where conditions (A) and (B) in the proposition correspond to conditions (2) and (1) in the lemma. (Unfortunately, the notation here is confusing—the roles of p and p are reversed in the two results.) Let y = (p · ωa )a∈A . By re-scaling (p , y ) if necessary, we have y¯ = y¯ ; (B) guarantees that F satisfies aggregate normality between (p, y) and (p, y ), so (1) is satisfied. Condition (A) says that p · Zω (p) = p · (F(p, y ) − ω¯ ) = p · (F(p, y ) − F(p , y )) > 0. Therefore, (2) is satisfied. We conclude that the pair (p, F(p, y)) = (p, ω) ¯ and (p , F(p , y )) = (p , ω¯ ) is N-monotonic. 䊐 Proof of Proposition 3.5. Part (i) is an application of Proposition 3.4. (A ) implies that ˜ P(ω ) · Zω (p) > 0, so there is an open neighborhood of ω, U˜ ⊂ U, such that for ω in U, that (A) is satisfied. Clearly if fa s are normal and p · (ωa − ωa ) are of the same sign for all a, (B) is also satisfied. Applying Proposition 3.4 gives us the result. 䊐 For (ii), we define Θk = {θ ∈ Rl : p · θ = 0, θ = 0, and θ < k} and consider ω ˜ so the pair (p, ω) and such that ωa = ωa + θ for all a. For k sufficiently small, Θk is in U, (P(ω ), ω ) is N-monotonic; in particular, it satisfies the weak axiom, and since p· ω¯ = p· ω¯ , 9 10
Note that we have ignored the boundary constraints in our construction of ωa . The decomposition used in this result is also used by Nachbar (2001, Theorem 1) to prove a related theorem.
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we have P(ω ) · ω¯ > P(ω ) · ω¯ or 0 > P(ω ) · θ.11 The latter means that y y where y = (P(ω ) · ωa )a∈A and y = (P(ω ) · ωa )a∈A . Since fa s are normal, F(P(ω ), y ) > F(P(ω ), y ) = ω¯ . So then p · ω¯ = p · ω¯ < p · F(P(ω ), y ) or 0 < p · Zω (P(ω )). Now it is straightforward to show via the inverse function theorem that there is an open neighborhood V of p such that for any p in V non-collinear with p, there is θ in Θk such that with (ωa )a∈A = (ωa + θ)a∈A , P(ω ) is collinear with p . So we obtain p · Zω (p ) > 0 for all p in V and non-collinear with p. This implies that ∂q Zω (p) is negative semi-definite. Proof of Lemma 4.1. The lemma follows immediately from the the following observations: (a) the vector (u, v) is in SYθ (p, q) if and only if (u, θ ⊗ v) is in SY (p, θ −1 ⊗ q); (b) from (a), we know that ΠYθ (p, q) = ΠY (p, θ −1 ⊗ q) and, furthermore, the income of agent a at price (p, q) in the economy E(ω, µ, Y) is equal to the income of the agent a at price (p, θ −1 ⊗ q) in the economy E(ω, θ ⊗ µ, Y); (c) consequently, household demand for consumer goods in the two situations is identical, while supply of consumer goods is also the same in the two situations. 䊐 Proof of Proposition 4.2. At the equilibrium price (p , q ), the economy E(ω , µ , Yθ ) has aggregate household demand of F(p , y ); we denote the equilibrium supply by (u , v ), which must be in SYθ (p , q ). By Lemma 2.1 and its proof, the economy E(ω , θ ⊗ µ , Y) has an equilibrium price at (p , θ −1 ⊗ q ), with aggregate demand of F(p , y ) and equilibrium supply of (u , θ ⊗ v ). which is in SY (p , θ −1 ⊗ q ). By definition of N-monotonicity, there is b > 0 in Rl such that (p − λp) · (F(p , y ) − F(p, y)) ≤ 0, for λ satisfying p · b = λp · b, with equality only if p and p are collinear. (In the case where p and p are collinear, simply choose λ to satisfy p = λp and b to be any vector in Rl+ \ {0}.) By the definition of SY , (p , θ −1 ⊗ q ) · ((u , θ ⊗ v ) − (u, v)) ≥ 0 and (p, q) · ((u , θ ⊗ v ) − (u, v)) ≤ 0 where (u, v) in SY (p, q) is the equilibrium supply of E(ω, µ, Y) at the equilibrium price (p, q). If (p , θ −1 ⊗ q ) and (p, q) are not collinear but p and p are collinear, then by the strong supply property, both inequalities must be strict. Therefore, we obtain, ((p , θ −1 ⊗q )−λ(p, q))·((u , θ⊗v )−(u, v)) ≥ 0, with the inequality being strict if p and p are collinear. Since (ω, ¯ µ) ¯ = (F(p, y), 0)−(u, v) and (ω¯ , θ ⊗ µ ¯ ) = −1 (F(p , y ), 0) − (u , θ ⊗ v ) we find that ((p , θ ⊗ q ) − λ(p, q)) · ((ω¯ , θ ⊗ µ ¯ ) − (ω, ¯ µ)) ¯ equals ((p , θ −1 ⊗ q ) − λ(p, q)) · ((F(p , y ), 0) − (F(p, y), 0) − (u , θ ⊗ v ) + (u, v)) < 0. We now consider the case where (p, q) and (p , θ −1 ⊗ q ) are collinear. Since the pair (p, F(p, y)) and (p , F(p , y )) is N-monotonic, it also satisfies the weak axiom, and therefore we can find λ such that (p − λp) · (F(p, y) − F(p , y )) < 0. Since (p, q) and (p , θ −1 ⊗ q ) are collinear, SY (p, q) = SY (p , θ −1 ⊗ q ), which means that (p , θ −1 ⊗ q ) · ((u , θ ⊗ v ) − (u, v)) = (p, q) · ((u , θ ⊗ v ) − (u, v)) = 0. So ((p , θ −1 ⊗ q ) − λ(p, q)) · ((ω¯ , θ ⊗ µ ¯ ) − (ω, ¯ µ)) ¯ equals ((p , θ −1 ⊗ q ) − λ(p, q)) · [(F(p , y ), 0) − (F(p, y), 0)] = 䊐 (p − λp) · (F(p , y ) − F(p, y)), which must be strictly negative. Proof of Proposition 5.1. We omit the simple case where q and q are collinear. Choose p such that Dp = q, write y = (p · DT ωa )a∈A and scale (p, y) so that p · F(p, y) = y¯ . 11 N-monotonicity is generally stronger than the weak axiom, but when p · ω ¯ = p · ω¯ , the pair (p, ω) and (P(ω ), ω ) is N-monotonic if and only if it satisfies the weak axiom.
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Now we employ Lemma A in the same manner as in Proposition 3.2, which leads us to conclude that the pair (p, F(p, y)) and (p , F(p , y )) is N-monotonic. Lemma A says that if y¯ = y¯ any vector in Rl+ can be a normalizing vector; if y¯ = y¯ , without loss of generality we assume that y¯ > y¯ , in which case we may choose as the normalizing vector b = F(p , y ) − F(p , y ) > 0. Formally, there is λ > 0 such that (p − λp ) · (F(p, y) − F(p , y )) < 0 and λp · b = p · b. But F(p, y) = DT ω¯ and F(p , y ) = DT α for some α in Rm . Furthermore, since the range of F is contained in the range of DT , there is θ such that b = DT θ. Re-writing the inequality we obtain, (q − λq ) · (ω¯ − α) < 0 when λq · θ = q · θ. 䊐 It remains for us to construct a financial economy with a mean endowment of α. The method developed in Proposition 3.1 will work here as well. First define K to satisfy y¯ = Kq · ω. ¯ Then for each agent a, let ωa = ωa + λa ω¯ + (α − Kω) ¯ and choose λa so that ya = q · ωa . One can easily check that ω¯ = α. Proof of Proposition 5.2. Consider the exchange economy where agent a has demand function fa and endowment DT ωa . This exchange economy has an excess demand function, ˜ DT ω . Now it is easy to check that p with Dp = q is an equilibrium which we denote by Z price of this economy, just as p satisfying Dp = q is an equilibrium price of the economy with ˜ DT ω (p) > 0, endowment distribution (DT ωa )a∈A . Condition (A) can be re-written as p · Z while (B) just says that p · (DT ωa − DT ωa ) are all of the same sign. So the conditions of Proposition 3.4 are satisfied and we conclude that (p, DT ω) ¯ and (p , DT ω¯ ) is N-monotonic with the normalizing vector parallel to F(p, (q · ωa )a∈A ) − F(p, (q · ωa )a∈A ). Then by essentially repeating the argument in Proposition 5.1 we see that the pair (q, ω) ¯ and (q , ω¯ ) is N-monotonic. 䊐 References Brown, D.J., Matzkin, R.L., 1996. Testable restrictions on the equilibrium manifold. Econometrica 64 (6), 1249– 1262. Debreu, G., 1982. Existence of competitive equilibrium. In: Arrow, K.J., Intriligator, M.D. (Eds.), Handbook of Mathematical Economics, vol. II. North Holland, New York. Hildenbrand, W., 1994. Market Demand. Princeton University Press, Princeton. Jerison, M., 1999. Dispersed excess demands, the weak axiom and uniqueness of equilibrium. Journal of Mathematical Economics 31, 15–48. Mas-Colell, A., Whinston, M.D., Green, J.R., 1995. Microeconomic Theory. Oxford University Press, Oxford. Mantel, R., 1976. Homothetic preferences, and community excess demand functions. Journal of Economic Theory 12, 197–201. Nachbar, J.H., 2001. General equilibrium comparative statics: discrete shocks in production economies. Washington University, St Louis. Journal of Mathematical Economics, in press. Nachbar, J.H., 2002. General equilibrium comparative statics. Econometrica 70 (5), 2065–2074. Quah, J.K.-H., 1999. The Weak Axiom and Comparative Statics. Nuffield College Working Paper, W15, Oxford. Quah, J.K.-H., 2001. Comparative Statics and Welfare Theorems When Goods are Normal. Nuffield College Working Paper, W24, Oxford. Quah, J.K.-H., 2003. The law of demand and risk aversion. Econometrica 71 (2), 713–721.
Market demand and comparative statics when goods are normal John K.-H. Quah Department of Economics, Oxford University, St. Hugh’s College, Oxford OX2 6LE, UK Received 9 February 2002; received in revised form 12 March 2003; accepted 18 March 2003
Abstract This paper examines the impact of the normality assumption on the structure of market demand and on general equilibrium comparative statics. We define a new notion of comparative statics which is fundamentally related to normality and examine its incidence in exchange, production and financial economies. © 2003 Elsevier Science B.V. All rights reserved. JEL classification: D11; D50; D51; D52 Keywords: Normal goods; Demand aggregation; Weak axiom; Monotonicity; Comparative statics
1. Introduction We say that a consumer’s demand function is normal if, with prices held fixed, demand for all goods increase as income increases. It is well-known that this property is not implied by utility maximization, but it is also clear that it is a very mild assumption to impose, provided goods are thought of in sufficiently broad categories. This paper offers a thorough examination of the implications of this property on general equilibrium comparative statics. Consider an exchange economy with l goods that has a mean endowment of ω and an equilibrium price of p, with both ω and p in Rl++ . While preferences remain unchanged, an endowment perturbation causes mean endowment to change to ω and the equilibrium price to p . What conditions are needed to guarantee that the price of a good ‘falls’ as its endowment increases? Since a scalar multiple of an equilibrium price is also an equilibrium price, the question implicitly assumes some suitable choice of scales. A typical analysis of this or similar problems will identify some good as a numeraire and examine the movement of prices relative to that good (see, for example, Mas-Colell et al., 1995, Section 17.G). E-mail address: [email protected] (J.K.-H. Quah). 0304-4068/03/$ – see front matter © 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0304-4068(03)00047-8
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In this paper, we wish to study a weaker comparative statics property. We wish to examine the conditions under which, for a pair of equilibrium outcomes, there exists some bundle of goods which, if it acts as a numeraire, will cause prices and endowments to move in opposite directions. In other words, the numeraire is not pre-identified but can be freely chosen. Formally, we say that the pair (p, ω) and (p , ω ) satisfies N-monotonicity (where ‘N’ stands for numeraire) if there exists a bundle of goods b, understood as an element in Rl+ , so that with prices normalized by scalar multiplication to satisfy (A) p · b = p · b, we have (B) (p − p ) · (ω − ω ) < 0. Condition (A) equalizes the value of the bundle b at the two prices so it effectively designates b as the numeraire. Condition (B) says that prices and endowments move in opposite direction. In particular, if the price change has resulted from an increase in the endowment of good 1, with the endowment of other goods remaining the same, the price of good 1 before the perturbation, p1 , will be greater than its value after the perturbation (p )1 . Clearly, we also have p1 /p · b < (p )1 /p · b, i.e. the price of good 1 relative to the price of the bundle b, has fallen. We show that if the economy is a single agent economy, then every pair of equilibrium outcomes will have the N-monotonic property if and only if the agent has a demand function which obeys the weak axiom and is normal.1 The situation becomes more complicated when we consider multi-agent exchange economies. In this setting, given any initial economy, we show that there will always exist endowment perturbations which lead to violations of N-monotonicity. In cases like these, no good and no bundle of goods could serve as a numeraire and guarantee that prices and endowments move in opposite directions. The breakdown of N-monotonicity is related to the breakdown of a sort of ‘aggregate normality’ condition: the change in the income distribution caused by the endowment perturbation is so large that the market does not buy more of every good even if average income has gone up and prices are (hypothetically) held fixed. That such perturbations exist is not the same as saying that violations of N-monotonicity are an empirically common phenomenon. Indeed, just as significantly, it can be shown that there is a large class of endowment perturbations for which the associated change in equilibrium prices will obey N-monotonicity. So one has to conclude that theory alone cannot tell us anything about the incidence of N-monotonicity; this is an empirical matter which hinges on the types of endowment perturbations that are likely to occur. The claims we have just made remain true even if the initial economy has a well or badly behaved excess demand function. So in this paper we caution against the casual conflation of comparative statics issues with those relating to dynamic stability and uniqueness. The latter two rely undeniably on the properties of excess demand, but its relationship with comparative statics is considerably less straightforward. While for certain types of endowment perturbations a well-behaved excess demand function (in particular, one satisfying the weak axiom) helps to guarantee N-monotonic equilibrium comparisons, this cannot be true for all perturbations. Similarly, endowment perturbations leading to N-monotonic equilibrium comparisons exist even when the excess demand function is badly behaved. The paper also shows how the concept of N-monotonicity could be extended to financial economies (with a possibly incomplete asset structure) and production economies. In the 1 The sufficiency part of this claim was in essence established by Nachbar (2002), though he did not refer to N-monotonicity as such.
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former, N-monotonicity guarantees that the prices and endowment of securities move in opposite directions, when prices are compared against a numeraire securities portfolio. In production economies, we show that if aggregate demand in the household sector satisfies N-monotonicity for a pair of equilibrium outcomes, then the property must also hold when comparing both economies in its entirety. This result follows from the profit maximization hypothesis. N-monotonicity guarantees that the price of a factor, i.e. an endowed good which does not give utility but which contributes to production, will also move in a direction opposite to that of its endowment. This price is measured relative to some bundle of utility giving goods; in other words, we are referring to movements in the ‘real’ factor price. This result also has implications for the factor-augmenting technology changes which are commonly considered in macroeconomics. For example, if a technology change causes each unit of a particular factor to behave like it was two, then the real factor price will not increase by a multiple greater than two. The paper is organized as follows. Section 2 studies market demand and formulates necessary and sufficient conditions for N-monotonicity. Section 3 examines N-monotonic comparative statics in exchange economies. Sections 4 and 5 extends the discussion to production and financial economies, respectively.
2. N-monotonic market demand We begin this section with a few definitions and a lemma which is fundamental to the results in this paper. A pair (p, x) and (p , x ) in Rl++ × Rl satisfies i. the weak axiom if either p · x > p · x or p · x > p · x ; ii. monotonicity if (p − λp ) · (x − x ) < 0 for λ chosen to satisfy p · x = λp · x ; iii. N-monotonicity if there exists a scalar λ > 0 such that (p −λp)·(x −x) < 0, and when p and p are not collinear, there also exists a vector b in Rl+ such that p · b = λp · b. We refer to b as the normalizing vector. We interpret p as the price vector of l goods, and x as the demand (either of a market or a single agent) at that price. It is easy to check that if the pair (p, x) and (p , x ) satisfies monotonicity or N-monotonicity, then it also satisfies the weak axiom. If the pair satisfies the weak axiom, we can assume that p · (x − x) > 0, in which case, for a sufficiently large positive λ, we have (p − λp) · (x − x) < 0. Given any p and λp, we can always find b = 0 such that p · b = λp · b, so what makes N-monotonicity stronger than the weak axiom is the requirement that b > 0. As we had pointed out in the introduction, in a comparative statics setting, b can be interpreted as a numeraire. So the specification that b > 0 is a compromise between pre-specifying a particular good as numeraire (which will mean that positive results can only be obtained under strong assumptions) and allowing the numeraire to be just any vector. It seems natural, given that we will be assuming that preferences are monotone, not to use as a numeraire a bundle of goods which has negative amounts of some goods. Furthermore, as this paper will make clear, N-monotonicity arises naturally in general equilibrium comparative statics, so it becomes a focus of enquiry even if one is merely interested in establishing a weaker property like the weak axiom. Our first result gives the precise conditions under
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Fig. 1. N-monotonicity between (p, x) and (p , x ).
which a pair (p, x) and (p , x ) satisfies N-monotonicity. It is fundamental to virtually everything else in this paper. Lemma 2.1. Consider a pair (p, x) and (p , x ) in Rl++ × Rl , where p and p are not collinear and p · x = p · x . Then the pair satisfies N-monotonicity with the normalizing vector b > 0 if and only if there exists c in Rl such that (i) x − c > 0 or x − c < 0 and b is collinear with x − c, (ii) p · c = p · x, (iii) (p , c) and (p, x) satisfies the weak axiom. Perhaps the best way of understanding conditions (i)–(iii) is to look at Fig. 1. Condition (ii) says that c should be on the budget line α, i.e. with price p and bundle x just affordable. Condition (iii) says that c rests on the line α and to the left of the bundle x. Condition (i) says that c must lie between the points m and n. In short, (p, x) and (p , x ) is N-monotonic if and only if a point c like the one in Fig. 1 can be found—on the line α, between m and x. The normalizing vector will then be x − c. Now it should be intuitively clear why normality plays a central role in this concept. Suppose that x and x are the demand bundles at prices p and p , respectively, of an individual utility maximizing agent with normal demand. Then c exists because we can always choose c to be the agent’s demand at income p · x—condition (iii) is guaranteed by revealed preference while (i) is guaranteed by normality. With the help of Lemma 2.1, we will now consider how N-monotonicity may arise in a market. We assume that the commodity space has l goods and the agents in the market form a finite set A. Each agent a has an income ya > 0 and a demand function fa . Formally, fa : Rl++ × R+ → Rl++ is a demand function if it is zero-homogeneous, i.e. fa (p, ya ) = fa (kp, kya ) for all k > 0, and satisfies p · fa (p, ya ) = ya . These properties are standard. We refer to the vector y = (ya )a∈A 0 as the market’s income distribution and denote the mean income by y¯ . An ordered pair (p, y) will be called a market situation. The (mean) |A| l market demand function F : Rl++ × R ++ → R++ maps each market situation (p, y) to the mean market demand, i.e. F(p, y) = a∈A fa (p, ya )/|A|. If we assume that the fa ’s are demand functions, then F is also zero-homogeneous and will satisfy p · F(p, y) = y¯ . For the rest of this paper, we shall assume that F satisfies these properties.
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We say that the demand function fa satisfies the weak axiom if for every ya , p and p , with p = p , the pair (p, fa (p, ya )) and (p , fa (p , ya )) satisfies the weak axiom. We say that F satisfies the weak axiom if for every y, p and p , with p = p , the pair (p, F(p, y)) and (p , F(p , y)) satisfies the weak axiom. Analogous definitions hold for the monotonicity and N-monotonicity properties. It is important to remember that when we refer to a particular property as holding for F we are varying prices while keeping the income distribution fixed. An agent with a demand function fa generated from a monotone preference where the indifference surfaces have no kinks will obey the weak axiom.2 Generally, the fact that fa obeys the weak axiom for all a is no guarantee that F obeys the weak axiom. However, in these situations, where one is varying prices while keeping the income distribution fixed, there is a wealth of theory with conditions guaranteeing that the weak axiom holds (see Mas-Colell et al., 1995, Chapter 4). For example, F obeys the weak axiom if fa obeys monotonicity for all agents. There is also considerable empirical evidence to support the view that F obeys the weak axiom (see Hildenbrand, 1994). So it is fair to say that this property is not overly strong. In this section, we investigate when F will obey N-monotonicity; more generally, we will allow the income distribution to change, and give conditions guaranteeing that a pair (p, F(p, y)) and (p , F(p , y ) satisfies N-monotonicity. To do that, we need an appropriate notion of normality for market demand. The demand function fa is normal at (p, ya ) if f(p, ya ) > f(p, ya ) when ya > ya and f(p, ya ) < f(p, ya ) when ya < ya . We call fa normal if it is normal at all (p, ya ) in Rl++ × R+ . The market demand function F satisfies Aaggregate normality between (p, y) and (p, y ) when the following holds: if y¯ > y¯ , then F(p, y) > F(p, y ); if y¯ < y¯ , then F(p , y) < F(p , y ); and if y¯ = y¯ , then F(p , y) = F(p , y ). We say that F obeys proportional aggregate normality if at any price p and income distributions y and ky, where k is a scalar, the pair (p, y) and (p, ky) satisfies aggregate normality. The next result gives sufficient conditions for N-monotonicity. Proposition 2.2. Suppose that F satisfies the weak axiom. Let (p, y) and (p , y ) be two non-collinear market situations and by multiplying (p , y ) with a scalar if necessary, assume that p · F(p, y) = y¯ . Then the pair (p, F(p, y)) and (p , F(p , y )) is N-monotonic if F satisfies aggregate normality between (p , y ) and (p , y). As we had argued earlier, that F obeys the weak axiom is a fairly mild assumption; hence, the focus of attention when considering whether a pair (p, F(p, y)) and (p , F(p , y )) 2 Suppose that agent a’s preference is such that there is more than one supporting price at a commodity bundle x, i.e. there is a kink at x. Then there are distinct prices p and p such that x = fa (p, ya ) = fa (p , ya ), which means that the pair (p, fa (p, ya )) and (p , fa (p , ya )) will violate our definition of the weak axiom. Note that our definition is slightly stronger than the usual one (see, for example, Mas-Colell et al., 1995), which is identical with ours, except that it does permit demand to be the same at two different price vectors. We need the slightly stronger definition because we establish N-monotonicity by assuming the weak axiom (plus other assumptions) and N-monotonicity is defined in terms of a strong inequality. If we modify the definition of N-monotonicity so that the inequalities are weak, our definition of the weak axiom can be correspondingly weakened. Our slightly stronger version of the weak axiom will hold if the agent’s demand function is generated from a C1 utility function u : Rl++ → R. In this case, each bundle x in Rl++ has at most one supporting price corresponding to the gradient of u at x.
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satisfies N-monotonicity must primarily fall on the aggregate normality condition. If y ≥ y or y > y , then it is clearly satisfied if all agents have normal demand functions, so in these cases we can reasonably expect N-monotonicity to hold. It is also clear that beyond these cases aggregate normality will hold for many pairs of income distributions. However, short of all agents having parallel income expansion paths, violations of aggregate normality are always possible. To see this, consider a re-distribution of y , an income distribution y , which must also have a mean of y¯ . Since income expansion paths are non-parallel, there will be some good i for which F i (p , y ) < F i (p , y ), so aggregate normality is violated. For a strict violation, add to every agent’s income, so y = y + (, , . . . , ). Provided F is continuous, there will be a sufficiently small such that F i (p , y) < F i (p , y ), but y¯ > y¯ . In the next proposition, we show that aggregate normality is necessary for N-monotonicity. The result requires that we add some standard properties to F . We say that the market demand function F is nice if it is continuous and satisfies the following boundary condition: whenever the sequence (pn , yn ) tends to (p, y), where p is on the boundary of Rl++ and y¯ = 0, |F(pn , yn )| tends to infinity. It is well-known that this property will hold if all agents in the market have continuous, strongly monotone, and strongly convex preferences. Proposition 2.3. Suppose that F violates aggregate normality between (p , y) and (p , y ) and that F is nice and satisfies the weak axiom. Then there is p such that p · F(p, y) = y¯ and (p, F(p, y)) and (p , F(p , y )) violates N-monotonicity. Furthermore, p can be chosen to be arbitrarily close to p if y is sufficiently close to y . The next result concerns N-monotonicity when the income distribution is fixed. We omit its proof, which follows immediately from Propositions 2.2 and 2.3. Corollary 2.4. Suppose that F is nice and obeys the weak axiom. Then F is N-monotonic if and only if it obeys proportional aggregate normality. In particular, the demand function of a single agent which is nice and obeys the weak axiom will be N-monotonic if and only if it is normal.
3. N-monotonicity in exchange economies In this section, we explore the implications of Section 2’s results for general equilibrium comparative statics. We now assume that each agent a in the market defined in Section 2 also has an endowment of ωa in Rl+ ; we write ω = (ωa )a∈A and the mean endowment as ω. ¯ We assume that ω¯ 0. We denote this exchange economy by E(ω) and the collection of such economies by M, so M = {E(ω) : ω ∈ (Rl+ )|A| , ω¯ 0}. Two economies in M have the same agents with the same demand functions (and therefore the same market demand function F ) but different endowments. A price p is the equilibrium price of E(ω) if F(p, (p · ωa )a∈A ) = ω. ¯ The equilibrium set, denoted by E, is the set of ordered pairs (p, ω) where p is an equilibrium price of E(ω) in M.
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Throughout this section, we will adopt the perspective that the ‘initial’ economy is E(ω), with an equilibrium price of p. Endowments are then changed to ω and the equilibrium price becomes p . Abusing our terminology a little, we say that (p , ω ) and (p, ω) in E satisfies N-monotonicity if (p , ω¯ ) and (p, ω) ¯ satisfies N-monotonicity. The first thing we should note is the close relationship between this issue and our work in the last section. The pair (p , ω ) and (p, ω) in E is N-monotonic if and only if (p , F(p , y )) and (p, F(p, y)) is N-monotonic, where y = (p ·ωa )a∈A and y = (p·ωa )a∈A . Furthermore, the next result shows that we can associate an exchange economy to every market situation (p , y ). Proposition 3.1. Let (p, ω) be in E, write y = (p · ωa )a∈A , and suppose that F is nice. Then for every market situation (p , y ) there is ω with (p , ω ) in E and (p · ωa )a∈A = y such that ω is arbitrarily close to ω if (p , y ) is sufficiently close to (p, y). The first issue we wish to address is the incidence of N-monotonic comparative statics. In other words, of all the possible endowment perturbations of the initial endowment ω, how often can we expect N-monotonicity to hold between (p , ω ) and (p, ω)? The answer is that we cannot say, because, as we shall see, the theory tells us that there is a proliferation of both endowment perturbations which obey N-monotonicity and those which violate it. The intuition behind our claim comes from the results of the last section. As we had pointed out, income changes which obey aggregate normality and those which violate aggregate normality both exist; the former leads to N-monotonicity while the latter leads to violations of N-monotonicity. The first of our next two result shows how we can easily construct endowment perturbations which obey N-monotonicity; the second says that endowment perturbations which violate N-monotonicity must always exist. Proposition 3.2. Assume that the demand functions fa obey the weak axiom and let (p, ω) be in E. Suppose that for some (p , y ), F satisfies aggregate normality between (p , y ) and (p , y ), with y = (p · fa (p, p · ωa ))a∈A and y¯ = y¯ . Then there is ω , with (p , ω ) in E, such that (p · ωa )a∈A = y and the pair (p , ω ) and (p, ω) is N-monotonic. Furthermore, if F is nice, we can choose ω to be arbitrarily close to ω if (p , y ) is sufficiently close to (p, y). Note that y (in Proposition 3.2) can be easily constructed. For example, if fa obeys normality for all a, then for any choice of p , F obeys aggregate normality between (p , y ) and (p , y ) provided we choose y so that y > y or y < y .3 Proposition 3.3. Suppose that F is nice and obeys the weak axiom and let (p, ω) be in E. Provided all agents do not have parallel income expansion paths, there is a sequence (pn , ωn ) in E tending to (p, ω), with ω¯ n = ω¯ such that (pn , ωn ) and (p, ω) violates N-monotonicity for all n. 3 A related observation is that any endowment perturbation which leads to greater (or lesser) utility for all agents must lead to N-monotonicity. To see this, suppose that all agents are worse off after the perturbation. By revealed preference, ya = p · fa (p, ya ) > p · fa (p , ya ) = ya for all a, which implies aggregate normality if all agents have normal demand functions.
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Note that Proposition 3.3 imposes no assumptions on the initial economy’s excess demand ¯ So the existence of the sequence is function, i.e. the function Zω (p) = F(p, (p·ωa )a∈A )− ω. guaranteed even if Zω satisfies gross substitutability, the weak axiom or other nice properties. In a similar vein, we note that Proposition 3.2 imposes no restrictions on Zω either, so the result holds even when it does not satisfy any nice structural properties.4 Most of the positive comparative statics results in the literature are not like Proposition 3.2. Typically, they impose restrictions on the excess demand function, which then guarantee nice comparative statics for certain classes of endowment perturbations (see, for example, Mas-Colell et al., 1995, Proposition 17.G.3 and the discussion following it). Our next result is of the same type. It shows that when the excess demand function satisfies a version of the weak axiom, one can explicitly identify a set of endowment perturbations which lead to N-monotonicity. This result is closely related to Nachbar (1999, Theorem 2); indeed, one could think of it as a discrete version of that result. Nachbar (2001, Theorem 2) has also independently obtained a closely related discrete result.5 Proposition 3.4. The pair (p , ω ) and (p, ω) in E is N-monotonic if (A) p · Zω (p) > 0 when p and p are non-collinear and (B) F satisfies aggregate normality between (p, y) and (p, y ), where y = (p · ωa )a∈A and y = (p · ωa )a∈A . If all agents have normal demand functions, (B) will be satisfied if p · (ωa − ωa ) are of the same sign for all a. This occurs, for example, when the endowment of a good k increases for some agents. (A) is a weak axiom type condition on the excess demand function. This condition is strong; in particular it is considerably stronger than the property that F satisfies the weak axiom, essentially because the income distribution of the economy changes as prices change. Nevertheless, reasonable conditions guaranteeing this property are known (see Jerison, 1999; Quah, 1999). The local differentiable analog to (A) is (A ): vT ∂q Zω (p)v < 0 for v in Rl , non-zero and non-collinear with p. In the final result of this section, we identify a set of perturbations for which (A ) is both necessary and sufficient for N-monotonicity. We assume that all the demand functions fa are C1 and that p is a regular equilibrium price of E(ω), i.e. ∂q Zω (p) is of rank l − 1. In this case, we know that there is an open neighbourhood U of ω and a C1 function P : U → Rl++ such that for all ω in U, P(ω ) is an equilibrium price of E(ω ) and P(ω) = p. We refer to P as an equilibrium map at p. Proposition 3.5. Suppose that fa is C1 and normal for all a, p is a regular equilibrium price of E(ω) and let P : U → Rl++ be an equilibrium map at p. 4 To see that Proposition 3.2 involves no restrictions on the excess demand function we need only note that the conditions in the proposition are satisfied if all agents have homothetic preferences. This is far stronger than what the proposition needs, yet even this assumption imposes no structure on the excess demand function (Mantel, 1976). 5 Nachbar states his results in terms of a relationship linking the initial and final mean endowments and equilibrium prices, and a vector of the marginal propensity to consume (averaged across all agents). Provided this vector is in Rl+ , the relationship is what we refer to as N-monotonicity, with the marginal propensity to consume serving as the normalizing vector.
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i. If (A ) holds, then so does the following property (∗): there is an open neighbourhood U˜ of ω, with U˜ ⊂ U, such that for all ω in U˜ with p · (ωa − ωa ) either non-negative for all a or non-positive for all a, the pair (P(ω ), ω ) and (p, ω) obeys N-monotonicity. ii. If property (∗) holds, vT ∂q Zω (p)v ≤ 0 for all v in Rl .6
4. N-monotonicity in production economies The objective of this section is to extend our comparative statics analysis to production economies. Consider a production economy with l + m goods. The first l goods give utility to all consumers, while the other m goods do not and are useful only as factors of production. We will refer to these m goods as factors but it important to note that we are not excluding the possibility that utility-giving goods are useful as inputs to the production process and indeed we allow for m to equal 0. Production is chosen from a production possibility set Y contained in Rl+m . We denote an element in Y by (u, v) where u is in Rl and v is in Rm . A price vector will be written as (p, q) where p is the price of the first l goods and q the price of the m factors. We assume that there is a price (p, q) in Rl+m ++ such that arg max(u,v)∈Y (p, q) · (u, v) is non-empty and denote the subset of such prices by P ⊂ Rl+m ++ . For each (p, q) in P, the supply correspondence SY is defined by SY (p, q) = arg max(u,v)∈Y (p, q) · (u, v). The profit function ΠY is defined by ΠY (p, q) = (p, q) · SY (p, q). The consumer/household sector consists of a set A of agents. Agent a has a demand function fa . We denote by F(p, y) the market demand of the household sector when p is the price and y the income distribution. The agent a has an endowment ωa of utility-giving goods and µa of factors. He also has a claim on profits of αa , so αa ≥ 0 and a∈A αa = 1. At price (p, q) in P, agent a’s income is ya = (p, q) · (ωa , µa ) + αa ΠY (p, q) and his demand is fa (p, ya ), which is in Rl++ . This completes our specification of the production economy. Since we shall be studying the behaviour of this economy when endowments and technology are varied, we will emphasize its dependence on these by denoting the economy by E(ω, µ, Y), where µ = (µa )a∈A and ω is similarly defined. A price (p, q) in P is an equilibrium price if there is s(p, q) in S(p, q) such that (F(p, y), 0) = s(p, q) + (ω, ¯ µ). ¯ The right hand side of this equation is just the economy’s aggregate supply, with ω¯ and µ ¯ being, respectively, the mean endowments of utility giving goods and factors. The left hand side is the market demand of the household sector at price p and at the income distribution generated by (p, q), which we denote by y. Since F(p, y) lives in Rl++ , to make the equation formally correct, we write aggregate demand as (F(p, y), 0), where 0 is the zero vector in Rm . The type of technology changes we will consider are those of the factor augmenting variety, which are standard in macroeconomic models. For θ in Rm ++ , we define the set Yθ by Yθ = {(u, θ ⊗ v : (u, v) ∈ Y }, where θ ⊗ v refers to the vector (θ 1 v1 ,θ 2 v2 , . . . , θ m vm ). Technology changes of this sort are easy to analyze since its equilibrium effects could be replicated by a suitable change in the economy’s factor endowment. 6 An examination of the proof will reveal that (ii) can be strengthened by weakening the assumption. Part (ii) is still true if ‘obeys N-monotonicity’ in property (∗) is replaced by ‘obeys the weak axiom.’
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Lemma 4.1. The price (p, q) is an equilibrium price of E(ω, µ, Yθ ) if and only if (p, θ −1 ⊗q) is an equilibrium price of E(ω, θ ⊗ µ, Y). Note that θ −1 refers to the vector whose ith entry is 1/θ i , and θ ⊗ µ denotes (θ ⊗ µa )a∈A . Obtaining N-monotonic equilibrium comparisons between production economies is facilitated by the profit maximization hypothesis, which guarantees that the supply correspondence has a nice monotonic property. In particular, profit maximization guarantees that for (u, v) in S(p, q) and (u∗ , v∗ ) in S(p∗ , q∗ ), we have (p, q) · ((u, v) − (u∗ , v∗ )) ≥ 0. We say that S has the strong supply property if this condition holds and in addition, the inequality is strict when there exists t such that p = tp∗ and q = tq∗ .7 With this in place we can state the main result of this section. Proposition 4.2. Let (p, q) and (p , q ) be equilibrium prices of E(ω, µ, Y) and E(ω , µ , Yθ ) respectively and denote by y and y the income distributions generated at the two equilibria. Suppose that the pair (p, F(p, y)) and (p , F(p , y )) is N-monotonic and that SY has the strong supply property. Then the pair ((p, q), (ω, ¯ µ)) ¯ and ((p , θ −1 ⊗ q ), (ω¯ , θ ⊗ µ ¯ )) −1 satisfies N-monotonicity; when (p, q) and (p , θ ⊗ q ) are not collinear, the normalizing vector can be chosen in the form (b, 0), where b ∈ Rl+ \ {0} and 0 ∈ Rm . Note that Proposition 4.2 does not give conditions guaranteeing N-monotonicity for the household sector; instead it assumes that condition and says what it implies for the whole economy. N-monotonicity for household demand will be guaranteed if, for example, the conditions of Proposition 2.2 are satisfied. It is also not hard to see that, along the lines of Proposition 3.2, one can readily construct endowment perturbations for which N-monotonicity is satisfied by household demand. To appreciate the significance of Proposition 4.2, imagine that there is an increase in the endowment of the first factor, i.e. good m + 1, causing prices to move from (p, q) to (p , q ). The result says that provided N-monotonicity is satisfied in the household sector, the price q1 will be greater than (q )1 , using the normalization, p · b = p · b. This normalization means that factor prices are measured in ‘real’ terms, i.e. in terms of the number of units of some representative bundle of consumer goods. In short, when a factor’s endowment is increased, its real factor price will fall. What if endowments remain the same, but there is a technology change which augments the first factor, i.e. θ = (k, 1, 1, . . . , 1), with k > 1? This change is equivalent to an increase in the endowment of factor 1 by the multiple k. If the conditions of Proposition 4.2 are satisfied, we may conclude that q1 is greater than (q )1 /k. In other words, we obtain the intuitive conclusion that the price of factor 1 will not rise by a multiple of more than k. We may view Proposition 4.2 as a statement on the ‘observable restrictions’ of Walrasian equilibria in the spirit of Brown and Matzkin (1996). Assume that an observer can observe all prices, mean household demand and mean endowment. He will then be in a position to determine if N-monotonicity is satisfied in the household sector; if it is, then the model 7 The slight strengthening is a mild curvature condition on the production set. The reason for this condition is in essence the same as the one given in footnote 2 for disallowing kinks in agents’ preferences. The condition is needed to establish N-monotonicity and the latter is defined in the form of a strong rather than weak inequality.
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predicts that the same property will be extended to the whole economy. If he observes that it is not, the model has been refuted.
5. N-monotonicity when markets are incomplete In this section, we show that the results on N-monotonicity we have developed for complete markets can be extended to a certain class of financial markets with an incomplete asset structure. We assume that there are two dates, 0 and 1. At date 1 there are l states of the world, with one good in each state. The economy has m securities, with m ≤ l. The m × l matrix D gives the payoffs of these securities, with the ijth entry being the payoff of the ith security in state j. We assume that the rank of D is m. An agent a in this economy has an endowment of securities ωa in Rm satisfying DT ωa > 0. A weaker alternative assumption is to assume that the agent has an endowment of consumption in different states of nature, i.e. in Rl+ , which may or may not be achievable by some security portfolio. It is not completely clear the extent to which our results in this section will carry over to this more general case; our treatment will in any case have to be more complicated, so we will stick with our stronger assumption. We assume that there is no consumption at date 0 and every agent consumes only at date 1. Agent a’s consumption is governed by the demand function, fa which, following the definition in Section 2, is zero-homogeneous and satisfies the budget identity. In this context we interpret fa (p, ya ) in Rl++ as agent a’s demand for consumption in different states of the world at date 1, when he faces the state price vector p and has income ya at date 0. We also assume that his demand has the following properties: (a) fa (p, ya ) = DT θ for some θ in Rm and (b) fa (p, ˜ ya ) = fa (p, ya ) whenever Dp = Dp. ˜ These requirements will hold if the agent is maximizing his utility at date 0 subject to the budget set B∗ (p, ya ) = {x ∈ Rl++ : x = DT θ for some θ ∈ Rm , and p · x ≤ ya }. So the agent’s budget set now has the added restriction that the demand chosen has to be achievable via some portfolio of securities θ. Note that (b) is true because if Dp = Dp, ˜ then B∗ (p, ya ) = B∗ (p, ˜ ya ). We denote the set of arbitrage free security prices by Q, which we know from standard theory satisfies Q = {q : q = Dp for some p 0}. We write Ω = {ω ∈ Rm : DT ω > 0}. Given fa , we could define the security demand function, ga : Q × Ω → Rm given by ga (q, ωa ) = θ where θ satisfies fa (p, qT ωa ) = DT θ and p satisfies Dp = q. It is fairly straightforward to check that ga is well-defined. Furthermore, if the agent a has a utility function Ua , then ga (q, ωa ) solves the problem: maximize Ua (DT θ) subject to q · θ ≤ q · ωa . A financial (or security market) economy consists of the set A of agents, each of whom has an endowment of securities ωa in Ω and a demand function fa satisfying the properties identified above. We denote this economy by F(ω), where ω = (ωa )a∈A . A price q in Q is an equilibrium price if a∈A ga (q, ωa )/|A| = ω, ¯ where ω¯ is the economy’s mean endowment of securities. We denote the equilibrium set by EF —this consists of the ordered pairs (q, ω) where q is an equilibrium price of the economy F(ω). We are interested in conditions which guarantee N-monotonicity between the equilibrium outcomes of two financial economies. To this end we will need to consider the plausibility of normality and weak axiom type conditions in this context. Normality for fa is defined in the usual way, and while not guaranteed by utility maximization remains a reasonable
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assumption. The definition of the weak axiom, however, needs modification. We say that fa satisfies the weak axiom if whenever Dp = Dp, ˜ and p˜ · fa (p, ya ) ≤ ya then p · fa (p, ˜ ya ) > ya . The additional condition requiring Dp = Dp˜ is needed to guarantee that demand is different at the two prices. Finally, the notion of N-monotonicity itself must also be modified. A pair (q, θ) and (q , θ ), both in Q × Rm , is defined as N-monotonic if there is a positive scalar λ such that (q − λq) · (θ − θ) < 0 and when q and q are not collinear, there ˆ N-monotonicity says that the also exists θˆ in Rm such that DT θˆ > 0 and q · θˆ = λq · θ. equilibrium price and endowment move in opposite directions, provided a suitable portfolio (θˆ in our definition), which generates weakly positive consumption in all states of the world, is chosen as the numeraire securities portfolio. With these in place, we can state the next result. It is in the same spirit as Proposition 3.2. It shows how endowment perturbations which obey N-monotonicity can be readily constructed in this context. Proposition 5.1. Assume that the demand functions fa obey the weak axiom and let (q, ω) be in EF . Suppose that for some (p , y ), F satisfies aggregate normality between (p , y ) and (p , y ) where q = Dp and y = (q · ga (q, ωa ))a∈A . Then there is ω with (q , ω ) in EF such that (q · ωa )a∈A = y , and the pair (q , ω ) and (q, ω) is N-monotonic. It is also possible to write down a result for financial economies which is the analog of Proposition 3.4. The excess demand function for securities of the economy F(ω ) is defined in the usual way: for any price qˆ in Q, Zω (ˆq) = a∈A ga (ˆq, ωa )/|A| − ω¯ . Proposition 5.2. The pair (q , ω ) and (q, ω) in EF is N-monotonic if (A) q · Zω (q) > 0 and (B)fa is normal for all a and q·(ωa −ωa ) are either non-negative for all a or non-positive for all a. Unlike the situation for standard exchange economies, the primitive conditions guaranteeing that the excess demand function for securities obey the weak axiom (property (A)) are less well-explored. But one simple case is well-known and straightforward to verify: when ¯ a∈A and ω = (ta ω¯ )a∈A , the economies have collinear endowments, i.e. when ω = (ta ω) with ta > 0 and a∈A ta = 1, and all agents have monotonic security demand functions. Sufficient utility conditions for the latter can be found in Quah (2003). Note also that for such perturbations q · (ωa − ωa ) are clearly all of the same sign. Acknowledgements I am grateful to the Economic and Social Research Council for providing me with financial support under its Research Fellowship Scheme. I would like to thank John Nachbar for some stimulating discussions and without whose earlier work this paper would probably not have been conceived. Comments from the editors and referee of this journal, and from participants at the 2002 GE conferences in Athens and Minneapolis also helped to improve the presentation and sharpen the focus of this paper. A more leisurely account of the material
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in this paper, together with some other results can be found in Quah (2001). Finally, I would like to thank Felicity Nyan for help with preparing the manuscript and much else besides.
Appendix A Proof of Lemma 2.1. For sufficiency, assume that p · x < p · x (the proof for the other case is the same). Choose c such that it satisfies (i)–(iii) and set b = x − c. Choose λ such that p · b = λp · b. Then (p − λp) · (x − x) = (p − λp) · (x − c + c − x) = (p − λp) · (c − x) = −λp · (c − x) < 0. Note that the second equality follows from our choice of b and λ, the final equality follows from (ii), and the final inequality from (iii). To proof necessity, suppose that (p − λp) · (x − x) < 0 for b satisfying p · b = λp · b. Choose c such that p · c = p · x and x − c is collinear with b. Then c satisfies (i) and (ii). Since the pair is N-monotonic, the expression (p − λp) · (x − x) = −λp · (c − x) must be negative. This implies that p · c > p · x, which is condition (iii). 䊐 To prove Proposition 2.2, we will use the following lemma.8 Lemma A. Let (p, y) and (p , y ) be two market situations. Re-scale (p , y ) to satisfy p · F(p, y) = y¯ . Assuming p = p , the pair (p, F(p, y)) and (p , F(p , y )) is N-monotonic if there exists some distribution y such that (1) y¯ = y¯ and F satisfies aggregate normality between (p , y ) and (p , y ); and (2) F satisfies the weak axiom between (p , F(p , y )) and (p, F(p, y)). The normalizing vector can be chosen to be collinear to F(p , y ) − F(p , y ) if y¯ = y¯ ; if y¯ = y¯ , any vector in Rl+ can serve as a normalizing vector. Proof. If y¯ = y¯ , by (1), y¯ = y¯ so normality implies that F(p , y ) = F(p , y ). Since p · F(p, y) = y¯ , (2) implies that p · F(p , y ) > p · F(p, y). Therefore, for any λ > 0, (p − λp) · (F(p , y ) − F(p, y)) = −λp · (F(p , y ) − F(p, y)) = −λp · (F(p , y ) − 䊐 F(p, y)) < 0. If y¯ = y¯ , we may apply Lemma 2.1 with c = F(p , y ). Condition (ii) of Lemma 2.1 is satisfied since p · c = p · F(p , y ) = y¯ = y¯ = p · F(p, y); (iii) is satisfied because of condition (2) in this theorem; finally, (i) is satisfied because of aggregate normality between (p , y ) and (p , y ). Proof of Proposition 2.2. We omit the easy case when p and p . If they are not, we apply Lemma A, but to do that we need to specify an appropriate y . In this case, choosing y = y will satisfy conditions (1) and (2) of Lemma A. 䊐 Proof of Proposition 2.3. We divide the proof into a few claims. We omit the proof of the first claim which is quite standard. (It is also found in Quah, 2001.) 䊐 8 Lemma A is a sort of decomposition theorem. If there is one agent in the market, conditions (1) and (2) correspond to the income and substitution effects. In a multi-agent market, this decomposition can be done in many ways, each corresponding to a particular choice of y (see Lemma A. This lemma is used in the proofs of Propositions 2.2, 3.2 and 3.4; in each of them, a different y is chosen.
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Claim 1. Suppose that F : Rl++ ×R++ → Rl++ is nice and satisfies the weak axiom. Fixing the income distribution at y, for all x in Rl++ , there is a unique p such that F(p, y) = x. The map from x to p, which we denote by P, is continuous and satisfies the following boundary condition: if xn tends to x on the boundary of Rl+ , |P(xn )| must tend to infinity. Assume that y¯ < y¯ and that F violates normality between (p , y) and (p , y ). (The case of y > y can be treated in the same way.) Then F(p , y) is not contained in the set C = {x ∈ Rl++ : p · x = y¯ and x < F(p , y )}. By the separating hyperplane theorem, there is x¯ satisfying p · x¯ = y¯ , q in Rl , and a scalar M such that q · x¯ = M, q · F(p , y) > M and q · c < M for all c in C. Define S = {x ∈ Rl++ : p · x = y¯ and q · x = M} and the map E : S → Rl by E(x) = P(x) − p , where P follows the definition in Lemma A. Note that x · E(x) = 0 for all x in S. Claim 2. There is x∗ in S with t · E(x∗ ) = 0 for all t in T = {t ∈ Rl : p · t = q · t = 0}. To see this, we first choose some xˆ in S. By the boundary property on P (and therefore E), we know that there is a compact set J, such that xˆ · E(x) > 0 for x not in J. Given this, we can choose a compact and convex set in S, call it K, which contains J such that xˆ ·E(x) > 0 whenever x is on the boundary of K (with respect to the relative topology). By the Gale–Debreu–Nikaido Lemma (see Debreu, 1982), there is x∗ in K such that x · E(x∗ ) ≤ 0 for all x in K. Since xˆ is in K, we know that x∗ is not on the boundary of K. We claim that t · E(x∗ ) = 0 for all t in T . Without loss of generality, assume that t · E(x∗ ) > 0 for some t. For δ positive and sufficiently close to zero, x = x∗ + δt must be in K (since x∗ is not on the boundary) and so x · E(x∗ ) = δt · E(x∗ ) ≤ 0, which is a contradiction. Claim 3. Suppose x∗ satisfies the conditions in Claim 2. Then c · P(x∗ ) < y¯ for all c in C. By our choice of q, for any c in C, q · c < M while q · F(p , y) > M, so there is α in (0, 1) with q · [αF(p , y) + (1 − α)c] = M. Clearly then [αF(p , y) + (1 − α)c] − x∗ is in T (as defined in Claim 2) so that ([αF(p , y) + (1 − α)c] − x∗ ) · E(x∗ ) = 0. Since x∗ · E(x∗ ) = 0, and E(x∗ ) = P(x∗ )−p , we may re-write this equation as [αF(p , y)+(1−α)c]·P(x∗ ) = y¯ . By the weak axiom, P(x∗ ) · F(p , y) > y¯ , so we may conclude that c · P(x∗ ) < y¯ . We have shown that there is x∗ satisfying p · x∗ = y¯ such that a vector c satisfying the conditions of Lemma 2.1 does not exist. This implies that the pair (p , F(p , y )) and (P(x∗ ), x∗ ) = (P(x∗ ), F(P(x∗ ), y)) violates N-monotonicity. Suppose now that yn tends to y . For each n we know that there is pn such that p · F(pn , yn ) = y¯ n with the pair (p , F(p , y )) and (pn , F(pn , yn )) violating N-monotonicity for all n. We wish to show that pn tends to p . Define the scalar λn by specifying that zn = λn F(p , y ) satisfies p · zn = y¯ n . Note that λn tends to 1. Since N-monotonicity is violated, zn · pn ≤ y¯ n for all n (otherwise, F(p , y ) will be a normalizing vector). Without loss of generality we assume that F(pn , yn ) has a limit of x∗ . If this limit is on the boundary, then because the image of F is in Rl++ , |pn | will tend to infinity. This cannot happen since zn · pn is bounded and zn is bounded away from the boundary. So we assume that x∗ is in Rl++ , in which case there is p∗ such that F(p∗ , y ) = x∗ with pn tending to p∗ . Since zn tends to F(p , y ), taking limits, we obtain F(p , y ) · p∗ ≤ y¯ . On the other hand,
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p · F(pn , yn ) = y¯ n , so p · F(p∗ , y ) = y¯ . To preserve the weak axiom, we must have p∗ = p . Proof of Proposition 3.1. There are many ways of constructing the economy E(ω ). We will choose a construction that satisfies the continuity property we require. Define the scalar , y )−K ω) K by setting p ·F(p , y ) = y¯ = Kp · ω. ¯ Let ωa = ωa +λa ω+(F(p ¯ ¯ and choose 9 λa so that p · ωa = ya . Summing ya across all agents, we see that K = 1 + a∈A λa /|A|. This also means that ω¯ = F(p , y ). If F is nice (and so continuous), F(p , y ) will converge to ω¯ if (p , y ) converges to (p, y). It is clear that with our construction of the economy E(ω ), ω will converge to ω. 䊐 Proof of Proposition 3.2. We omit the simple case when p and p are collinear. When they are not, we apply Lemma A.10 By multiplying (p, y) with a scalar if necessary, we have p · F(p, y) = y¯ . Note that F(p, y) = ω, ¯ and because (p, ω) is in E, y¯ = p · ω¯ = y¯ . F also satisfies aggregate normality between (p , y ) and (p , y ) by assumption, so condition (1) is satisfied. Furthermore, p · fa (p, ya ) > p · fa (p, ya ) since a obeys the weak axiom. Summing across the agents, we have p · F(p, y ) > p · F(p, y), so condition (2) is satisfied. Therefore, Lemma A says that (p, F(p, y)) and (p , F(p , y )) obeys N-monotonicity. Applying Proposition 3.1 then gives us our result. 䊐 Proof of Proposition 3.3. Since agents’ income expansion paths are not parallel, there is yn tending to y such that y¯ n = y¯ and (p, y) and (p, yn ) violate aggregate normality for all n. By Proposition 2.3, there is pn tending to p such that (pn , F(pn , yn )) and (p, F(p, y)) 䊐 violate N-monotonicity for all n. Proposition 3.1 then gives us the desired result. Proof of Proposition 3.4. We omit the simple case when p and p are collinear. When they are not, we apply Lemma A, where conditions (A) and (B) in the proposition correspond to conditions (2) and (1) in the lemma. (Unfortunately, the notation here is confusing—the roles of p and p are reversed in the two results.) Let y = (p · ωa )a∈A . By re-scaling (p , y ) if necessary, we have y¯ = y¯ ; (B) guarantees that F satisfies aggregate normality between (p, y) and (p, y ), so (1) is satisfied. Condition (A) says that p · Zω (p) = p · (F(p, y ) − ω¯ ) = p · (F(p, y ) − F(p , y )) > 0. Therefore, (2) is satisfied. We conclude that the pair (p, F(p, y)) = (p, ω) ¯ and (p , F(p , y )) = (p , ω¯ ) is N-monotonic. 䊐 Proof of Proposition 3.5. Part (i) is an application of Proposition 3.4. (A ) implies that ˜ P(ω ) · Zω (p) > 0, so there is an open neighborhood of ω, U˜ ⊂ U, such that for ω in U, that (A) is satisfied. Clearly if fa s are normal and p · (ωa − ωa ) are of the same sign for all a, (B) is also satisfied. Applying Proposition 3.4 gives us the result. 䊐 For (ii), we define Θk = {θ ∈ Rl : p · θ = 0, θ = 0, and θ < k} and consider ω ˜ so the pair (p, ω) and such that ωa = ωa + θ for all a. For k sufficiently small, Θk is in U, (P(ω ), ω ) is N-monotonic; in particular, it satisfies the weak axiom, and since p· ω¯ = p· ω¯ , 9 10
Note that we have ignored the boundary constraints in our construction of ωa . The decomposition used in this result is also used by Nachbar (2001, Theorem 1) to prove a related theorem.
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we have P(ω ) · ω¯ > P(ω ) · ω¯ or 0 > P(ω ) · θ.11 The latter means that y y where y = (P(ω ) · ωa )a∈A and y = (P(ω ) · ωa )a∈A . Since fa s are normal, F(P(ω ), y ) > F(P(ω ), y ) = ω¯ . So then p · ω¯ = p · ω¯ < p · F(P(ω ), y ) or 0 < p · Zω (P(ω )). Now it is straightforward to show via the inverse function theorem that there is an open neighborhood V of p such that for any p in V non-collinear with p, there is θ in Θk such that with (ωa )a∈A = (ωa + θ)a∈A , P(ω ) is collinear with p . So we obtain p · Zω (p ) > 0 for all p in V and non-collinear with p. This implies that ∂q Zω (p) is negative semi-definite. Proof of Lemma 4.1. The lemma follows immediately from the the following observations: (a) the vector (u, v) is in SYθ (p, q) if and only if (u, θ ⊗ v) is in SY (p, θ −1 ⊗ q); (b) from (a), we know that ΠYθ (p, q) = ΠY (p, θ −1 ⊗ q) and, furthermore, the income of agent a at price (p, q) in the economy E(ω, µ, Y) is equal to the income of the agent a at price (p, θ −1 ⊗ q) in the economy E(ω, θ ⊗ µ, Y); (c) consequently, household demand for consumer goods in the two situations is identical, while supply of consumer goods is also the same in the two situations. 䊐 Proof of Proposition 4.2. At the equilibrium price (p , q ), the economy E(ω , µ , Yθ ) has aggregate household demand of F(p , y ); we denote the equilibrium supply by (u , v ), which must be in SYθ (p , q ). By Lemma 2.1 and its proof, the economy E(ω , θ ⊗ µ , Y) has an equilibrium price at (p , θ −1 ⊗ q ), with aggregate demand of F(p , y ) and equilibrium supply of (u , θ ⊗ v ). which is in SY (p , θ −1 ⊗ q ). By definition of N-monotonicity, there is b > 0 in Rl such that (p − λp) · (F(p , y ) − F(p, y)) ≤ 0, for λ satisfying p · b = λp · b, with equality only if p and p are collinear. (In the case where p and p are collinear, simply choose λ to satisfy p = λp and b to be any vector in Rl+ \ {0}.) By the definition of SY , (p , θ −1 ⊗ q ) · ((u , θ ⊗ v ) − (u, v)) ≥ 0 and (p, q) · ((u , θ ⊗ v ) − (u, v)) ≤ 0 where (u, v) in SY (p, q) is the equilibrium supply of E(ω, µ, Y) at the equilibrium price (p, q). If (p , θ −1 ⊗ q ) and (p, q) are not collinear but p and p are collinear, then by the strong supply property, both inequalities must be strict. Therefore, we obtain, ((p , θ −1 ⊗q )−λ(p, q))·((u , θ⊗v )−(u, v)) ≥ 0, with the inequality being strict if p and p are collinear. Since (ω, ¯ µ) ¯ = (F(p, y), 0)−(u, v) and (ω¯ , θ ⊗ µ ¯ ) = −1 (F(p , y ), 0) − (u , θ ⊗ v ) we find that ((p , θ ⊗ q ) − λ(p, q)) · ((ω¯ , θ ⊗ µ ¯ ) − (ω, ¯ µ)) ¯ equals ((p , θ −1 ⊗ q ) − λ(p, q)) · ((F(p , y ), 0) − (F(p, y), 0) − (u , θ ⊗ v ) + (u, v)) < 0. We now consider the case where (p, q) and (p , θ −1 ⊗ q ) are collinear. Since the pair (p, F(p, y)) and (p , F(p , y )) is N-monotonic, it also satisfies the weak axiom, and therefore we can find λ such that (p − λp) · (F(p, y) − F(p , y )) < 0. Since (p, q) and (p , θ −1 ⊗ q ) are collinear, SY (p, q) = SY (p , θ −1 ⊗ q ), which means that (p , θ −1 ⊗ q ) · ((u , θ ⊗ v ) − (u, v)) = (p, q) · ((u , θ ⊗ v ) − (u, v)) = 0. So ((p , θ −1 ⊗ q ) − λ(p, q)) · ((ω¯ , θ ⊗ µ ¯ ) − (ω, ¯ µ)) ¯ equals ((p , θ −1 ⊗ q ) − λ(p, q)) · [(F(p , y ), 0) − (F(p, y), 0)] = 䊐 (p − λp) · (F(p , y ) − F(p, y)), which must be strictly negative. Proof of Proposition 5.1. We omit the simple case where q and q are collinear. Choose p such that Dp = q, write y = (p · DT ωa )a∈A and scale (p, y) so that p · F(p, y) = y¯ . 11 N-monotonicity is generally stronger than the weak axiom, but when p · ω ¯ = p · ω¯ , the pair (p, ω) and (P(ω ), ω ) is N-monotonic if and only if it satisfies the weak axiom.
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Now we employ Lemma A in the same manner as in Proposition 3.2, which leads us to conclude that the pair (p, F(p, y)) and (p , F(p , y )) is N-monotonic. Lemma A says that if y¯ = y¯ any vector in Rl+ can be a normalizing vector; if y¯ = y¯ , without loss of generality we assume that y¯ > y¯ , in which case we may choose as the normalizing vector b = F(p , y ) − F(p , y ) > 0. Formally, there is λ > 0 such that (p − λp ) · (F(p, y) − F(p , y )) < 0 and λp · b = p · b. But F(p, y) = DT ω¯ and F(p , y ) = DT α for some α in Rm . Furthermore, since the range of F is contained in the range of DT , there is θ such that b = DT θ. Re-writing the inequality we obtain, (q − λq ) · (ω¯ − α) < 0 when λq · θ = q · θ. 䊐 It remains for us to construct a financial economy with a mean endowment of α. The method developed in Proposition 3.1 will work here as well. First define K to satisfy y¯ = Kq · ω. ¯ Then for each agent a, let ωa = ωa + λa ω¯ + (α − Kω) ¯ and choose λa so that ya = q · ωa . One can easily check that ω¯ = α. Proof of Proposition 5.2. Consider the exchange economy where agent a has demand function fa and endowment DT ωa . This exchange economy has an excess demand function, ˜ DT ω . Now it is easy to check that p with Dp = q is an equilibrium which we denote by Z price of this economy, just as p satisfying Dp = q is an equilibrium price of the economy with ˜ DT ω (p) > 0, endowment distribution (DT ωa )a∈A . Condition (A) can be re-written as p · Z while (B) just says that p · (DT ωa − DT ωa ) are all of the same sign. So the conditions of Proposition 3.4 are satisfied and we conclude that (p, DT ω) ¯ and (p , DT ω¯ ) is N-monotonic with the normalizing vector parallel to F(p, (q · ωa )a∈A ) − F(p, (q · ωa )a∈A ). Then by essentially repeating the argument in Proposition 5.1 we see that the pair (q, ω) ¯ and (q , ω¯ ) is N-monotonic. 䊐 References Brown, D.J., Matzkin, R.L., 1996. Testable restrictions on the equilibrium manifold. Econometrica 64 (6), 1249– 1262. Debreu, G., 1982. Existence of competitive equilibrium. In: Arrow, K.J., Intriligator, M.D. (Eds.), Handbook of Mathematical Economics, vol. II. North Holland, New York. Hildenbrand, W., 1994. Market Demand. Princeton University Press, Princeton. Jerison, M., 1999. Dispersed excess demands, the weak axiom and uniqueness of equilibrium. Journal of Mathematical Economics 31, 15–48. Mas-Colell, A., Whinston, M.D., Green, J.R., 1995. Microeconomic Theory. Oxford University Press, Oxford. Mantel, R., 1976. Homothetic preferences, and community excess demand functions. Journal of Economic Theory 12, 197–201. Nachbar, J.H., 2001. General equilibrium comparative statics: discrete shocks in production economies. Washington University, St Louis. Journal of Mathematical Economics, in press. Nachbar, J.H., 2002. General equilibrium comparative statics. Econometrica 70 (5), 2065–2074. Quah, J.K.-H., 1999. The Weak Axiom and Comparative Statics. Nuffield College Working Paper, W15, Oxford. Quah, J.K.-H., 2001. Comparative Statics and Welfare Theorems When Goods are Normal. Nuffield College Working Paper, W24, Oxford. Quah, J.K.-H., 2003. The law of demand and risk aversion. Econometrica 71 (2), 713–721.