Comparative Statics and the Gross Substitutes Property of Consumer Demand
Anne-Christine Barthel1 and Tarun Sabarwal
Abstract. The gross substitutes property of demand is very useful in trying to understand stability of the standard Arrow-Debreu competitive equilibrium. Most attempts to derive this property rely on aspects of the demand curve, and it has been hard to derive this property using assumptions on the primitive utility function. Using new results on the comparative statics of demand in Quah (Econometrica, 2007), we provide simple and easy conditions on utility functions that yield the gross substitutes property. Quah provides conditions on utility functions that yield normal demand. We add an assumption on elasticity of marginal rate of substitution, which combined with Quah's assumptions yields gross substitutes. We apply this assumption to the family of constant elasticity of substitution preferences. Our approach is grounded in the standard comparative statics decomposition of a change in demand due to a change in price into a substitution eect and an income eect. Quah's assumptions are helpful to sign the income eect. Combined with our elasticity assumption, we can sign the overall eect. As a by-product, we also present conditions which yield the gross complements property.
PRELIMINARY AND INCOMPLETE
1 Department of Economics, University of Kansas, Lawrence, KS 66045 Email: [email protected]
1 2
1. Introduction
In the standard Arrow-Debreu model, the gross substitutes property of demand provides insight into the question of a competitive equilibrium's stability. Its usefulness in this matter entails the question what assumptions will guarantee this property. Most attempts to derive the gross substitutes property focus on characteristics of the demand curve. However, it has turned out to be rather di cult to guarantee this property by conditions on the primitive, the utility function, instead of the resulting demand function. In this paper, we will present easy conditions on the utility function that will yield the desired gross substitutes property. The approach taken has its foundation in the standard decomposition of the total eect of a price change on demand into a substitution and an income eect. The sign of the substitution eect can easily be determined by basic economic theory in the case of two goods, but requires additional assumptions on the marginal rate of substitution in the general case. With regard to the income eect, there are various recent works on comparative statics of demand that have focused on conditions that yield normal demand. We will use the assumptions on the utility function provided in Quah (2007) to sign the income eect and then introduce a simple and intuitive assumption on the elasticity of marginal rate of substitution, which will allow us to determine the direction of the overall eect of a price change. The question of gross substitutability and complementarity has been addressed in the literature in various dierent ways. Fisher (1972) nds that two goods are gross substitutes if the Allen- Uzawa elasticity of substitution between two commodities exceeds the maximum of the respective income elasticities of demand. As de ned in Uzawa (1962), the partial elasticity of substitution between two commodities depends on the expenditure function.2 Since the expenditure function and demand functions are results of optimization problems, Fisher's condition is not on the primitives. The intuition behind his condition is that indierence curves, that only have little curvature will yield gross substitutes; in the limiting case of perfect substitutes the indierence curves are straight lines. For gross complements, the opposite has to be true; indierence curves need to be su ciently curved, with the right-angled indierence curves of perfect complements as the extreme case. Later papers that establish results in accordance with this intuition, but impose conditions directly on the utility function, are Mirman and Ruble (2003) and Maks (2006). For the two good case, Mirman and Ruble (2003) state that for good 1 to be a gross substitute (complement) of good 2, MRS12 x2 needs to be decreasing (increasing) in x2. When extending this to the case of many goods, they nd that if MRSi,i+1 is independent of all goods j 6= i, i + 1 and increasing in good i + 1, for i < n=1, and MRSn−1,n is independent of all goods j 6= n=1, n, decreasing in good n=1, and increasing xn (decreasing) in good n, then good 1 is a gross complement (substitute) of good n. Maks (2006)
2 e(p,u) ∂ e(p,u) 2 ∂xi∂xj , where denotes minimum expenditures to obtain a given utility level σij = ∂e(p,u) ∂e(p,u) e(p, u) u ∂xi ∂xj 3 derives a condition with the same intuition for utility functions based on Walras' utility concept that are strongly separable or additive in each commodity. His shows that in case of a price change of good j, good i will be a gross substitute if good j 's elasticity of marginal utility with respect to xj exceeds -1. Many of the recent works on comparative statics theorems for constrained optimization prob- lems take a lattice-theoretic approach, as rst developed by Vives (1990) based on Topkis (1978). Milgrom and Roberts (1992) also focuses on monotone comparative statics results under cardinal assumptions (supermodular games), which is extended by Milgrom and Shannon (1994) to the ordinal case (quasisupermodular games). These results however cannot be easily applied to opti- mization problems where the constraint sets are not sublattices and comparable in the strong set order, such as budget sets at two levels of income. Quah (2007) expands Milgrom and Shannon's work by making it applicable to these types of optimization problems by using a weaker order on the constraint sets (Ci- exible set order). His comparative statics theorem provides easily veri able conditions on the utility function that yield normal demand, namely supermodularity and a form of concavity. Antoniadou (2007), which is largely based on Antoniadou (2004a,b), and Mirman and Ruble (2008) also address the question of normality of demand, but their approach introduces a direct value order to construct the relevant order-lattice structure. While Quah (2007) and Mirman and Ruble (2008) solely focus on income changes, Antoniadou (2007) also provides the relevant order-lattice structure for price changes in the case of 2 goods. In this paper, we will establish the gross substitutes property under assumptions on the prim- itives. We approach the problem by decomposing the eect of a price change in substitution and income eect. Existing results based on lattice-theoretic approaches can be used to determine the direction of the latter, while basic economic theory provides the sign of the former. Lastly, we provide a condition on the utility function similar to, but more general than the existing ones, in order to guarantee gross substitutes or complements. The next section introduces the theoretical framework for the lattice-based approach and de nes the gross substitutes property. Section 3 gives results for the two goods case, Section 4 addresses the extension to many goods. 4
2. Preliminaries
Quah (2007) introduces the following concepts to extend the comparative statics results from Milgrom and Shannon (1994) to problems where one constraint set does not dominate the other in the strong set order. This case is easily encountered even in simple optimization problems like the consumers maximizing utility subject to the budget constraint. Let X be a partially ordered set and let ∆and ∇ be two operations on X. Consider a function f : X → R. Then we can de ne the following properties for f.
De nition 1. (∆, ∇)-Supermodularitiy f is (∆, ∇)-supermodular if for all x, y ∈ X
f(x∇y) − f(y) ≥ f(x) − f(x∆y)
De nition 2. (∆, ∇)-Quasisupermodularitiy f is (∆, ∇)-quasisupermodular if for all x, y ∈ X
(1) f(x) ≥ f(x∆y) ⇒ f(x∇y) ≥ f(y) (2) f(x) > f(x∆y) ⇒ f(x∇y) > f(y)
Moreover, consider two non-empty subsets of X, S and S0. The operations ∆ and ∇ induce the following set order on S and S0.
De nition 3. (∆, ∇)-induced strong set order 0 0 S dominates S by the (∆, ∇)-induced strong set order (S ≥∆,∇ S) if and only if for all x ∈ S and for all y ∈ S0 x∆y ∈ S and x∇y ∈ S0.
After the introduction of the general case, now let X ⊆ Rl be a convex set and de ne the operations λ and λ on as follows for in : ∇i ∆i X λ [0, 1]
y if x ≤ y λ i i x∇i y = λx + (1 − λ)(x ∨ y) if xi > yi x if x ≤ y λ i i x∆i y = λy + (1 − λ)(x ∧ y) if xi > yi
Graphically, the points , , λ and λ form a backward-bending parallelogram instead of x y x∆i y x∇i y the rectangle generated by x, y and their join and meet. 5
Analogously to the previous case, we can de ne properties of a function f : x → R and introduce a new set order with respect to the operations λ λ. ∇i , ∆i De nition 4. Parallelogram Inequality
A function is λ λ - supermodular for some if (∇i , ∆i ) λ ∈ [0, 1] λ λ for all , in . f(x∇i y) − f(y) ≥ f(x) − f(x∆i y) x y X
De nition 5. Ci (C)-Supermodularity
A function is -supermodular if it is λ λ - supermodular for all λ λ in f : X → R Ci (∇i , ∆i ) (∇i , ∆i ) λ λ . Ci = {(∇i , ∆i ): λ ∈ [0, 1]} If f is Ci-supermodular for all i, then it is C-supermodular.
De nition 6. Ci (C)-Quasisupermodularity A function is -quasisupermodular if it is λ λ - quasisupermodular for all f : X → R Ci (∇i , ∆i ) λ λ in λ λ , that is (∇i , ∆i ) Ci = {(∇i , ∆i ): λ ∈ [0, 1]}
(1) λ λ f(x) ≥ f(x∆i y) ⇒ f(x∇i y) ≥ f(y) (2) λ λ f(x) > f(x∆i y) ⇒ f(x∇i y) > f(y)
If f is Ci-quasisupermodular for all i, then it is C-quasisupermodular.
De nition 7. Ci(C)- exible Set Order 0 0 Let S and S be subsets of the convex sublattice X. Then S dominates S in the Ci- exible set order ( 0 ) if for any in and in 0, there exists λ λ in such that λ is in 0and S ≥i S x S y S (∇i , ∆i ) Ci x∇i y S λ is in . x∆i y S 0 0 0 S dominates S in the C- exible set order (S ≥ S) if S ≥i S for all i.
The main result in Quah (2007) is the following comparative statics theorem.
Theorem 1. (Quah)
The function f : X → R is Ci-quasisupermodular (C-quasisupermodular) if and only if argmaxx∈S0 f(x) ≥i 0 argmaxx∈S f(x) (for all i) whenever S ≥i (≥) S.
Since it usese the Ci- exible set order, this result can be applied to optimization problems such as the consumer's utility maximization problem to obtain comparative statics results with respect to income changes. This result is useful and applicable, since Ci-quasisupermodularity follows from 6 properties that can be expected to hold in this context. The rst one is supermodularity, the other a form of concavity.
De nition 8. i-Concavity
A function f is i-concave if it is concave in direction v for any v > 0 with vi = 0.
Proposition 1. (Quah)
The function f : X → R is Ci-quasisupermodular if it is supermodular and i-concave.
Providing assumptions on the primitives to establish the gross substitutes property is the main focus of this paper. Generally, two goods are considered gross substitutes if a price increase of one of the goods results in nondecreasing demand for the other good. Moreover, a demand function is said to satisfy the gross substitutes property if an increase of the price of good i leads to nondecreasing demand for all goods j 6= i.
De nition 9. Gross Substitutes Property of Demand 0 A (Marshallian) demand function x(p, w) satis es the gross substitutes property if xj(p , w) ≥ whenever 0 with 0 and 0 for all . xj(p, w) p > p pi > pi pj = pj i 6= j
A stronger version of this property can be de ned by replacing the weak inequality by a strict one, hence requiring demand of good j to be strictly increasing given an increase of the price of good i.
3. Price effects with two goods
Quah's new appraoach, the Ci- exible set order (see De nition 7), works well in cases like budget sets before and after an income change, where both sets have the same slope. However, there are fairly simple constraint sets, like budget sets with dierent slopes, that cannot be ordered in this way. For example consider a price change of good 1. The boundary of the smaller budget set will be steeper than the one of the larger set, which allows the conclusion that the bigger set dominates the smaller one in the C1- exible set order, but not in the C2- exible set order as shown in Figure 3.1. For price changes of good 2, the result is the mirror image. Hence this framework only gives comparative statics results for changes of a good's own price, but cannot provide insight into whether goods are gross substitutes or complements. We take the approach to consider the substitution and income eect of a price change separately to obtain results for this class of comparative statics problems. Consider the consumer's utility maximization problem in the case when there are only two goods and assume that the utility 7
Figure 3.1. Violation of Ci- exible set order by budget sets before and after a price change function u is locally nonsatiated. Let x∗be a maximizer for the initial budget set B(p, w) = {x ∈ X : p · x ≤ w}. By local nonsatiation, the budget constraint is binding at optimum, thus p · x∗ = w. Now suppose the price of good ( increases to 0 . As we know, this price change will i i = 1, 2) pi aect the consumer's demand in two ways. On the one hand, there is a substitution eect due to the change of the relative prices when the consumer's income will be adjusted in such a way, that the old preferred bundle x∗ is still aordable at new prices. On the other hand, the income eect will capture the change in demand when the consumer's income adjustment is reversed.
Proposition 2. If u is locally nonsatiated, the substitution eect on demand for good j will be non-negative (non-positive) when price of good i increases (decreases).
Proof. The proof will focus on the unbracketed claim, the bracketed version can be shown in similar fashion. Suppose price for good increases from to 0 . Because the budget constraint was binding i pi pi at the old optimal bundle x∗, the consumer will no longer be able to aord this bundle at the new prices. Now suppose the consumer is given a higher income 0 0 ∗ ∗ to compensate w = pixi + pjxj > w him for the price increase. Let xˆ be the optimal bundle at (p0, w0). Then, by local nonsatiation, ∗ ∗. This inequality can be rewritten as pixˆi + pjxˆj ≥ pixi + pjxj
(3.1) ∗ pj ∗ xˆi − xi ≥ − (ˆxj − xj ) pi At the compensated level of income 0 both bundles are aordable and 0 ∗ ∗ 0 w pixi +pjxj = pixˆi+pjxˆj = w0. Rearranging terms yields 8
(3.2) ∗ pj ∗ xˆi − xi = − 0 (ˆxj − xj ) pi p p p (p0 −p ) It follows from 3.1 and 3.2 that j ∗ j ∗ or equivalently j i i ∗ . − 0 (ˆxj −xj ) ≥ − (ˆxj −xj ) 0 (ˆxj −xj ) ≥ 0 pi pi pipi Since the rst term is positive, this implies ∗. xˆj ≥ xj
The other eect of a price change is the income eect, which captures the impact on demand when the consumer's income adjustment for the price change is reversed. In other words, the income eect compares the optimal consumption bundles after the price change at income levels w and w0. Since this is a comparative statics problem concerning an income change and the budget sets are comparable in the Ci- exible set order, Quah's result for normality of demand can be applied here.
Proposition 3. If u is supermodular and j-concave, the income eect of a price increase (decrease) of good i is non-positive (non-negative) on the demand for good j.
Proof. This result follows immediately from Proposition 1 from Quah (2007).
Since the substitution eect and income eect are opposite in sign for normal goods, it is not possible to tell whether the goods are gross substitutes or gross complements. For the former, the substitution eect needs to dominate the income eect and vice versa for the latter. In order to compare the income and substitution eect additional assumption on the utility function are needed. The question of interest now is under what assumptions on the primitives two goods will be gross substitutes or gross complements. Intuitively, the needed condition has to ensure that the indierence curves only have little curvature for gross substitutes; in the limiting case of perfect substitutes the indierence curves are straight lines. For gross complements, the opposite has to be true; indierence curves need to be su ciently curved, with the right-angled indierence curves of perfect complements as the extreme case. In other words, if the indierence curves are relatively straight, the substitution eect of a price change will outweigh the income eect and hence the two goods will be gross substitutes. Figure 3.2 depicts the two cases. In the following, a property that will yield the desired shape of the indierence curves will be introduced. It is similar to, but more general than the conditions in Mirman and Ruble (2003) and Maks (2006). The idea here is, that indierence curves will only have little curvature if the elasticity of the marginal rate of substitution between goods i and j with respect to good i is small. In order for two goods to be gross substitutes (complements), the absolute value of the marginal rate of substitution elasticity has to be less (greater) than 1. This condition extends the one in Maks (2006) to a wider class of of utility functions, since there is no restriction that marginal utilities 9
(a) Gross substitutes (b) Gross complements
Figure 3.2. Eect of a price change for dierent curvatures of indierence curves only depend on one commodity. It is easy to show that this more general condition reduces to the one discussed in Maks (2006) when only considering strongly separable utility functions.3 In the two goods case, our condition is very similar to the one proposed in Mirman and Ruble (2003). However, the extension to n commodities will yield a more general result. For simplicity, rst consider the case with only two goods and assume that the utility function is dierentiable. To begin with, consider only cases where the utility maximization problem has an interior solution. To guarantee this, we need to impose regularity conditions on the utility function, i. e. u is dierentiable, ∂u > 0, lim ∂u = ∞, lim ∂u = 0. Given these assumptions, the utility ∂xi ∂xi ∂xi xi→0 xi→∞ function does not cross the axes and therefore corner solutions are ruled out.
Theorem 2. If u is locally nonsatiated, Ci- and Cj-quasisupermodular, satis es regularity condi- ∂ MRS tions and | ij xi |< (>)1, then the goods are gross substitutes (complements). ∂ xi MRSij Proof. The following proof will focus on the gross substitutes. The bracketed version for comple- ments can be shown accordingly. Now consider the case when pi increases. From Prop. 2 and Prop. 3 we know that ∗ and ∗∗, where ∗ is the original optimal bundle before the xˆj ≥ xj xˆj ≥ xj x price change, xˆ is the optimal bundle at new prices and compensated income and x∗∗ is the new maximizer after the price change. Since the budget constraint is binding at the old and new optimal bundle, we know that ∗ pixi + ∗ 0 ∗∗ ∗∗, which can be rewritten as pjxj = w = pixi + pjxj
3 ∂MRS ∂MU ij xi = ∂ ( MUi ) xi = ∂MUi xi − j xi ∂xi MRSij ∂xi MUj MRSij ∂xi MUi ∂xi MUi ∂MUj For strongly separable utility functions MUj only depends on xj , therefore = 0. As a result, marginal rate of ∂xi substitution elasticity and marginal utility elasticity are equivalent. 10
∗∗ ∗ ∗ 0 ∗∗. The total change of expenditures on each of the goods can be pj(xj − xj ) = pixi − pixi decomposed in the change resulting from the substitution and the income eect, resulting in pj(ˆxj − ∗ ∗∗ ∗ 0 0 0 ∗∗. First consider only the change in expenditures xj ) + pj(xj − xˆj) = pixi − pixˆi + pixˆi − pixi attributed to the income eect. Since ∗∗and ∗∗, 0 ∗∗ and ∗∗ . xˆi ≥ xi xˆj ≥ xj pi(xi − xˆi) < 0 pj(xj − xˆj) < 0 Hence expenditures on both goods decrease as a result of the income eect. Since ∗, the expenditures on good will increase when considering the substitution eect xˆj ≥ xj j only. Therefore we cannot draw any conclusion on the total change of expenditures on good j, because the income and substitution eect will cause them to go in opposite directions. The impact of the substitution eect on good i on the other hand is not as obvious. Using the Slutsky compensation, it can be written as
(3.3) 0 ∗ 0 ∗ 0 ∗ ∗ pixˆi − pixi = w − w − pj(ˆxj − xj ) = (pi − pi)xi − pj(ˆxj − xj )
∂ MRS ∂ MRS To use the assumption that | ij xi |< 1, we rst approximate the derivative ij ≈ ∂ xi MRSij ∂ xi ∆MRSij using a linear Taylor approximation. xˆi is a continuous function of pi, so for su - ∆xi ∗ ∆MRSij MRSij (ˆx)−MRSij (x ) ciently small price changes = ∗ . Assuming that the solution is interior, ∆xi xˆi−xi 0 pi pi ∗ p − p ∆MRSij MRSij (ˆx)−MRSij (x ) j j ∂ MRSij xi = ∗ = ∗ . Since | |< 1, there exists some > 0, ∆xi xˆi−xi xˆi−xi ∂ xi MRSij ∂ MRS ∆MRS such that | ij xi | + ≤ 1. For su ciently small price changes, || ij xi | − | ∂ xi MRSij ∆xi MRSij 0 0 p p p p i − i ∗ i − i ∗ ∂ MRS ∆MRS p p x p p x ij xi . Therefore, ij xi j j i , which implies j j i . ||< | |= ∗ pi ≤ 1 ∗ p0 ≤ 1 ∂ xi MRSij ∆xi MRSij x −xˆi x −xˆi i i pj i pj Again using the Slutsky compensation, this can be rewritten as ∗ 0 ∗ 0 . pj(ˆxj − xj ) ≥ (pi − pi)xi = w − w This condition and 3.3 imply
0 ∗ 0 ∗ 0 ∗ ∗ . pixˆi − pixi = w − w − pj(ˆxj − xj ) = (pi − pi)xi − pj(ˆxj − xj ) ≤ 0
Therefore, both the income and the substitution eect lead to a decrease of expenditures on good , so 0 ∗∗ ∗. Consequently, ∗∗ ∗which is equivalent to ∗∗ ∗. i pixi ≤ pixi pjxj ≥ pjxj xj ≥ xj
Example 1. CES Utility ρ ρ 1 Consider the CES utility function ρ with . Since the marginal u(x1, x2) = (x1 + x2) ρ ≤ 1 utilities for both goods are non-negative,4 u is monotone and locally non-satiated. For ρ ≤ 1, the CES utility function is concave5 and therefore u is 1,2-concave as well. Moreover, the utility
4 1−ρ ∂u = (xρ + xρ) ρ xρ−1 ≥ 0 ∂x1 1 2 1 1−ρ ∂u = (xρ + xρ) ρ xρ−1 ≥ 0 ∂x2 1 2 2 5 2 1−2ρ ∂ u ρ ρ ρ ρ−2 ρ for 2 = (ρ − 1)(x1 + x2) x1 x2 ≤ 0 ρ ≤ 1 ∂x1 2 1−2ρ ∂ u ρ ρ ρ ρ ρ−2 for 2 = (ρ − 1)(x1 + x2) x1x2 ≤ 0 ρ ≤ 1 ∂x2 11 function is supermodular for ρ ≤ 1,6 which in combination with 1- and 2- concavity implies that u is C1- and C2-supermodular (see Proposition 2 in Quah (2007)) and therefore also C1- and C2- quasisupermodular. The regularity conditions that guarantee an interior solution are satis ed for ρ < 1, and for 0 ≤ ρ ≤ 1 (ρ ≤ 0) the absolute value of the elasticity of the marginal rate of 7 substitution with respect to x1 is less (greater) than or equal to 1. Thus suppose 0 ≤ ρ < 1 (ρ ≤ 0) and then the above comparative statics theorem applies and the goods are gross substitutes (complements).
Moreover, notice that for the CES utility function Mirman and Ruble's condition coincides with the one proposed here. Using either condition, goods will be gross substitutes (complements) if 0 ≤ ρ ≤ 1 (ρ ≤ 0).8
Now consider cases where the solution to the utility maximization problem is not interior, that is one the consumer only consumes one of the two goods and nothing of the other one. First, there is the trivial case of perfect substitutes. For these preferences, the marginal rate of substitution is
∗ ∗ p1 ∗ ∗ constant along the utility curve. The consumer chooses x = (x , 0) if MRS12 > and x = (0, x ) 1 p2 2 p1 if MRS12 < . So as long as the relationship between the price ratio and the MRS stays the same p2 after a price change of either good, the optimal consumption bundle will remain unchanged. If the price change of good i changes the inequality between the marginal rate of substitution and the price ratio, then the optimal quantity of good j will increase, i.e for a price increase of good 1 and 0 p1 p1 > MRS12 > , the consumer will move from consuming no good 2 at all to only consuming p2 p2 good 2. Therefore, in the case of perfect substitutes the demand for good j is nondecreasing in the price of good i. The other class of preferences where corner solutions often occur is quasilinear utility. In this case, interior and corner solutions are both possible. Assuming the optimal bundle before the price
p1 change was a corner solution on the x1-axis with MRS12 > , the optimal consumption bundle p2 will remain the same if the price ratio decreases, e.g. p1 goes down or p2 goes up. If the price change causes the price ratio to increase such that it exceeds the marginal rate of substitution at the original optimal bundle, the consumer will move to an interior solution where xj will move in p1 the same direction as pi. Similarly, if the initial corner solution lies on the x2-axis and MRS12 < , p2 a price change that increases the price ratio will not change the optimal choice; a change that will
6 2 2 1−2ρ ∂ u ∂ u ρ ρ ρ ρ−1 ρ−1 for ∂x ∂x = ∂x ∂x = (1 − ρ)(x1 + x2) x1 x2 ≥ 0 ρ ≤ 1 1 2 2 1 ( 7 1 − ρ for 0 ≤ ρ ≤ 1 | ∂MRS12 x1 |=| ρ − 1 |= ∂x1 MRS12 ρ − 1 for ρ ≤0 8 x1 ρ−1 MRS12 = ( ) x2 ρ−1 MRS12 x1 x = ρ 2 x2 MRS12 ∂( ) xρ−1 x2 =-ρ 1 ≤(≥)0 ⇔ ρ ≥ (≤)0 ∂x2 ρ+1 x2 12 decrease the relative price below the MRS at x∗will shift the consumer's utility maximizing bundle to an interior solution and the change in xj will have the same sign as the price change. Thus, also in this case the demand of good j is non-decreasing in the price of good i.
Example 2. Substitutes and Complements in Input Another application of Theorem 2 can be found in the rm's production decision. Suppose the rm's production function is given by 2 . Standard assumptions assure the unique F : R++ → R existence of a cost minimizing input bundle x(p, q) = argmin p · x given strictly positive 2 {x∈R++:F (x)≥q} input prices p and output q. Alternatively, x(p, q) can be found by maximizing output while limiting expenditures to p · x(p, q), hence x(p, q) = argmax F (x). x∈B(p,p·x(p,q)) Quah's comparative statics theorem allows us to make statements about how xi(p, q) changes with respect to output or its own price. Theorem 2 takes it a step further, because it yields the answer to the question how optimal inputs react to price changes of other inputs. In other words, it allows us to examine the eect of a change of pi on xj(p, q). Under the assumptions that F is monotone, Ci- and Cj-quasisupermodular (which is the case if it is supermodular and i-, j-concave) ∂MRT S and | ij xi |< (>)1, the optimal amount of input j will increase (decrease) as the price ∂xi MRT Sij for input i goes up.
Consider for example a translog production function, F (x1, x2) = ln(x1 +1)+ln(x2 +1)+ln(x1 + 9 10 1)·ln(x2 +1). It is easy to check that F is monotone, concave and therefore 1- and 2-concave and 12 supermodular.11 Moreover, | ∂MRT S12 x1 |< 1, therefore the 2 inputs are gross substitutes ∂x1 MRT S12 and hence the optimal amount of input of good 2 will increase if the price for input 1 goes up. Other functional forms that will yield gross substitutes are CES production functions for 0 ≤ ρ ≤ 1 and production functions with a root polynomial or logarithmic speci cation.
Example 3. Linear PPF in General Equilibrium Model Consider an open economy that produces 2 goods using the inputs capital and labor. Assume that the production functions for both goods 2 , are linear. As a result, the fi : R+ → R+ i = 1, 2 PPF of these two goods will also be linear. On the consumption side, suppose the representative consumer has a locally non-satiated, 1- and 2-concave and supermodular utility function U. The
p1 markets for both goods are perfectly competitive and in equilibrium MRS12 = = MRTS12. p2
9 ∂F 1 1 = + ln(xj + 1) · for i 6= j, i, j = 1, 2 ∂xi xi+1 xi+1 2 ln((x +1) 10 ∂ F = − 1 − j < 0 for i 6= j, i, j = 1, 2 2 (x +1)2 (x +1)2 ∂xi i i 2 11 ∂ F = 1 > 0 for i 6= j, i, j = 1, 2 ∂xi∂xj (xi+1)(xj +1) 12 x2+1 1+ln(x2+1) MRTS12 = · x1+1 1+ln(x1+1) ∂MRT S12 x1 x1 | |=| − 2 2 |< 1 ∂x1 MRT S12 (x1+1) [1+ln(x1+1)] 13
Now suppose there is technological progress in the production process of one of the goods, i.e. good 2. The new PPF will be steeper than the original one, thus the relative price between the goods and the marginal rate of substitution need to increase for the economy to return to equilibrium. To determine the eect of this change of technology on the optimal quantity of the other good, Theorem 2 can be applied.
−x1 −x2 Consider for example the negative exponential utility function u(x1, x2) = 1 − e − e . u is monotone,13 1- and 2-concave14 as well as supermodular.15 The elasticity of the marginal rate of 16 substitution with respect to good 1 is equal to x1, therefore the goods will be gross substitutes if x1 < 1 and gross complements for x1 > 1. In this context, the optimal choice of good 1 will increase if x1 > 1 and decrease if x1 < 1 if technological progress occurs in the production process of good 2.
4. Price effects with n goods
The next question to be addressed is how this result can be generalized to the case of n goods. As previously, at rst the utility function is assumed to satisfy the regularity conditions that guarantee an interior solution and the substitution eect will be identi ed using the Slutsky compensation. The identi cation of the substitution eects sign is no longer as straightforward as in the previous case with two goods. It is a well-established result in microeconomic theory that the substitution eect of a price change of good i is negative for that good in the general case with n goods as well. However, the sign of the substitution eect for the other n − 1 goods is no longer clear. Obviously, if the substitution eect leads to an increase in xi, there has to be at least one good among the remaining n − 1 goods for which the substitution eect goes in the opposite direction of good i because of the Slutsky compensation.17 Therefore, additional properties need to be imposed on the objective function to guarantee a non-negative substitution eect for all goods other than i.
Proposition 4. If u is locally nonsatiated and MRSik is independent of all goods l 6= i, k, the substitution eect on demand for good j will be positive (negative) when price of good i increases (decreases).
Proof. The proof will focus on the unbracketed claim, the bracketed version can be shown in similar fashion. Suppose price for good increases from to 0 . Because the budget constraint was i pi pi
13 ∂u = e−xi > 0 for i = 1, 2 ∂xi 14 2 ∂ u −xi for 2 = −e < 0 i = 1, 2 ∂xi 2 15 ∂ u = 0 for i 6= j, i = 1, 2 ∂xi∂xj −x −x 16 e 1 ∂MRS12 x1 e 1 x1 MRS12 = , | |=| − · |= x1 e−x2 ∂x1 MRS12 e−x2 e−x1 e−x2 17 0 ∗ P ∗ 0 0 P pixi + pj xj = w = pixˆi + pj xˆj i6=j i6=j 14 binding at the old optimal bundle x∗, the consumer will no longer be able to aord this bundle at the new prices. Now suppose the consumer is given a higher income 0 0 ∗ P ∗ w = pixi + pjxj > w j6=i to compensate him for the price increase. Let be the optimal bundle at 0 0 . Then, xˆ (pi, p−i, w ) P 0 ∗ P ∗. This inequality can be rewritten as pixˆi + pjxˆj = w > w = pixi + pjxj j6=i j6=i
(4.1) ∗ X pj ∗ xˆi − xi > − (ˆxj − xj ) pi j6=i At the compensated level of income 0 both bundles are aordable and 0 ∗ P ∗ 0 w pixi + pjxj = pixˆi + j6=i P 0 pjxˆj = w . Rearranging terms yields j6=i
X pj (4.2) xˆ − x∗ = − (ˆx − x∗) i i p0 j j j6=i i
0 P pj ∗ P pj ∗ P pj (pi−pi) It follows from 4.1 and 4.2 that − 0 (ˆxj−x ) > − (ˆxj−x ) or equivalently 0 (ˆxj− p j pi j pip j6=i i j6=i j6=i i ∗ . Since the rst term is always positive, this implies that there exists such that ∗. xj ) > 0 j xˆj > xj Moreover, given that pi increases and that the solution is interior, MRSij increases since MRSij = pi pj for i 6= j. The marginal rate between any two goods other than i, MRSjk = for j 6= k, pj pk j, k 6= i, however remains unchanged. In other words, ∂MRSjk ∂MRSjk d MRSjk = dxj + dxk = 0 , ∂xj ∂xk given that MRSij is independent of all goods other than i and j for all j. MRSjk is decreasing in good j and increasing in good k , hence it is easy to see that dxj and dxk need to have the same sign for the above equation to hold. Consequently, xj and xk need to move in the same direction . Furthermore, the changes in xi and xj need to be of opposite signs because of the Slutsky compensation ( 0 ∗ P ∗ 0 0 P ). Therefore, decreases and all pixi + pjxj = w = pixˆi + pjxˆj xi xj i6=j i6=j increase.
Hence, the substitution eect can be generalized for n goods under additional assumptions on the marginal rate of substitution between good i and any other good. The result for the income eect obtained earlier carries over to this case. As already observed in the above discussion with two goods, the substitution and income eect of a good other than i go in opposite directions and therefore the net eect is unclear. Once again the idea is to nd assumptions on the utility function that will yield gross substitutes (complements). The condition on the utility function that will allow us to do this again focuses on the elasticity of the marginal rate of substitution between the good i and another good k with respect to xi. If the absolute value of this elasticity is less than 1, the marginal rate of subsitution increases relatively 15 slow as xi decreases, therefore the indierence curves are characterized by relatively small curvature. On the other hand, if the absolute value of the marginal rate of substitution elasticity exceeds 1, the marginal rate of substitution reacts relatively strong to changes in xi and hence the indierence curves display more curvature. This condition applies to a wider class of utility functions than the one proposed in Maks (2006), which only considers strongly separable utility functions. It will also be used to establish the gross substitutes property of demand in a more straightforward manner than Mirman and Ruble (2003), who use more complex value orders. The rst step towards the gross substitutes property of demand is the existence of a gross substitute of good i among the other n − 1 goods.
Proposition 5. If u is locally nonsatiated, Cj -quasisupermodular for all j=1,...,n , satis es regu- larity conditions and | ∂ MRSik xi |< (>)1 for some k 6= i, then there exists a gross substitute ∂ xi MRSik (complement) among the other goods for su ciently small price changes of good i.
Proof. The following proof will focus on the gross substitutes. The bracketed version for comple- ments can be shown accordingly. Now consider the case when pi increases. From above we know that ∗ and ∗∗ for all , where ∗ is the original optimal bundle before the price xˆj ≥ xj xˆj ≥ xj j 6= i x change, xˆ is the optimal bundle at new prices and compensated income and x∗∗is the new maximizer after the price change. Since the budget constraint is binding at the old and new optimal bundle, we know that ∗ pixi + P ∗ 0 ∗∗ P ∗∗ , which can be rewritten as P ∗∗ ∗ ∗ 0 ∗∗ . The pjxj = w = pixi + pjxj pj(xj − xj ) = pixi − pixi j6=i j6=i j6=i total change of expenditures on each of the goods can be decomposed in the change resulting from the substitution and the income eect, resulting in P ∗ ∗∗ ∗ 0 [pj(ˆxj − xj ) + pj(xj − xˆj)] = pixi − pixˆi + j6=i 0 0 ∗∗ . First consider only the change in expenditures attributed to the income eect. Since pixˆi − pixi ∗∗ and ∗∗ for each , 0 ∗∗ and P ∗∗ . Hence expenditures xˆi ≥ xi xˆj ≥ xj j pi(xi − xˆi) < 0 pj(xj − xˆj) < 0 j6=i on both good decrease as a result of the income eect. The impact of the substitution eect on good i on the other hand is not as obvious. Using the Slutsky compensation, it can be written as
(4.3) 0 ∗ 0 X ∗ 0 ∗ X ∗ pixˆi − pixi = w − w − pj(ˆxj − xj ) = (pi − pi)xi − pj(ˆxj − xj ) j6=i j6=i . ∂ MRS ∂ MRS To use the assumption that | ij xi |< 1, we rst approximate the derivative ij ≈ ∂ xi MRSij ∂ xi ∆MRSij using a linear Taylor approximation. xˆi is a continuous function of pi, so for su - ∆xi ∗ ∆MRSij MRSij (ˆx)−MRSij (x ) ciently small price changes = ∗ . Assuming that the solution is interior, ∆xi xˆi−xi 0 pi pi ∗ p − p ∆MRSij MRSij (ˆx)−MRSij (x ) j j ∂ MRSij xi = ∗ = ∗ . Since | |< 1, there exists some > 0, ∆xi xˆi−xi xˆi−xi ∂ xi MRSij 16
∂ MRS ∆MRS such that | ij xi | + ≤ 1. For su ciently small price changes, || ij xi | − | ∂ xi MRSij ∆xi MRSij 0 0 p p p p i − i ∗ i − i ∗ ∂ MRS ∆MRS p p x p p x ij xi . Therefore, ij xi j j i , which implies j j i . ||< | |= ∗ pi ≤ 1 ∗ p0 ≤ 1 ∂ xi MRSij ∆xi MRSij x −xˆi x −xˆi i i pj i pj Again using the Slutsky compensation, this can be rewritten as P ∗ 0 ∗ 0 pj(ˆxj −xj ) ≥ (pi −pi)xi = w −w j6=i . This condition and 4.3 imply 0 ∗ . pixˆi − pixi ≤ 0 Therefore both the income and the substitution eect lead to a decrease of expenditures on good i , so Pp (x∗∗ − x∗) ≥ 0 . Consequently, x∗∗ ≥ x∗ for some good j . j j j j0 j0 0 j6=i
Proposition 5 only guarantees the existence of a gross substitute (complement) among the n − 1 goods for good i. However, it fails to provide insight in the eect of a price change on each of these n − 1 goods individually. As for the positive substitution eect for all goods other than i, the additional assumption, that the marginal rate of substitution between any two goods only depends on the quantities of these two goods, will allow us to draw wider-ranging conclusions in comparative statics problems with price changes.
Theorem 3. If u is locally nonsatiated, Cj -quasisupermodular for all j=1,...,n , satis es regularity
∂ MRSik xi conditions, MRSij is independent of all goods l 6= i, j and | |< (>)1 for some k 6= i, ∂ xi MRSik then all goods j 6= i are gross substitutes (complements) of good i.
Proof. As previously, the proof will focus on the case of gross substitutes. From Proposition 5 we know that there exists a substitute good of good i among the other goods. At an interior solution,
pi pj MRSij = for all j and MRSjk = for all j 6= k, j, k 6= i. As pi increases, so does MRSij. pj pk pj Since MRSjk is independent of all goods other than j and k, increasing in good k and remains pk unchanged at the new optimal solution, the marginal utilities of all goods other than i need to move in the same direction. Because both the substitution eect and the income eect have led to a decrease in the optimal quantity of good i, MUi has increased. Moreover, the existence of a substitute yields that needs to decrease and ∗∗ ∗ for all . MUj xj ≥ xj j 6= i
Example 4. CES utility (3 goods case) ρ ρ ρ 1 Consider the CES utiliy function ρ . Like in the previously mentioned u(x1, x2, x3) = (x1 +x2 +x3) 2 goods case, u is monotone and locally nonsatiated. Furthermore, it is quasiconcave and C1-, C2- and C3-quasisupermodular if ρ ≤ 1. All marginal rates of substitution MRSij only depend only 18 ∂MRS goods i and j and are decreasing in good j. Moreover, | ij xi |=| ρ − 1 |< 1 for all i 6= j ∂xi MRSij and 0 < ρ ≤ 1.
xρ−1 xρ−1 xρ−1 18 1 , 1 , 2 MRS12 = ρ−1 MRS13 = ρ−1 MRS23 = ρ−1 x2 x3 x3 17
Therefore, if we consider a price increase of good 1, by Theorem 5 optimal consumption of either good 2 or good 3 has to increase. Theorem 3 goes even further and says that more of both goods will be consumed after the price change. It can easily be seen in Footnote 18that if either x2 or x3 goes up, the other one has to increase equally for MRS23 to remain unchanged. This needs to be p2 the case at an interior solution where one of the rst order condition requires MRS23 = , since p3 the price ratio between these two goods has not changed.
The following example is an application of Theorem 3. Milgrom and Shannon (1994) show how results for games with strategic complements can be applied to a general equilibrium model with gross substitutes.
Example 5. General Equilibrium Model with Gross Substitutes Let there be L + 1 goods in an economy with good 0 as the numéraire. For each other good n, a market maker announces a price pn and his objective is to minimize excess demand dn(pn, p−n) = P for that good, where denotes the demand of consumer for good and i∈I xin(pn, p−n) − en xin i n en is the total endowment of it. So his payo can be written as πn(pn, p−n) = − | dn(pn, p−n) |.
In Milgrom and Shannon, a game with strategic complementarities is de ned as follows. N players each have a strategy set Sn, partially ordered by ≥n, and a payo function πn(xn, x−n). Moreover, the following conditions need to be satis ed for all n :
(1) Sn is a compact lattice;
(2) πn is upper semi-continuous in xn for x−n xed and continuous in x−n for xed xn;
(3) πn is quasisupermodular in xn and satis es the single crossing property in (xn; x−n). The previously described ctional game with a market maker for each good n satis es the conditions for a game with strategic complementarities (GSC) if the economy exhibits gross substitutes. This is the case if dn is continuous and decreasing in pn and continuous and monotone nondecreasing in p−n for all n. The above results can be used to get these properties making assumptions directly on the utility functions instead of the excess demand function. Hence this game is a GSC if the con- sumers' utility functions are monotone, locally nonsatiated, quasiconcave, Cj -quasisupermodular for all j=1,...,L+1 , satis es regularity conditions, MRSij is independent of all goods l 6= i, j and | ∂ MRSik xi |< 1 for some k 6= i. ∂ xi MRSik Examples of utility functions that satisfy these conditions include as previously mentioned CES utlity for ρ ≤ 1 and monotone logarithmic or translog utility functions.
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