The Law of Demand. Werner Hildenbrand. Department of Economics, University of Bonn Lennéstraße 37, Bonn, Germany
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1 The Law of Demand Werner Hildenbrand Department of Economics, University of Bonn Lennéstraße 37, 533 Bonn, Germany September 5, 26 Abstract (to be written) The law of demand for a population of households asserts that the vector of price changes p R l and the resulting vector of mean demand changes F R l point in opposite directions, provided the price changes do not affect households incomes (total expenditure) and demand functions (preferences). Thus, the law asserts that the mean demand function F (p) is strictly monotone, i.e. (p q) (F (p) F (q)) < for all p, q R l ++, p q The law of mean demand implies, in particular, that for every commodity i the partial mean demand function F i is strictly decreasing in its own price p i and that the mean demand function F ( ) is invertible (existence of an inverse demand function). The goal of aggregation theory is to establish the law without assuming that households demand functions f h (p, x) are strictly monotone in p, since otherwise the law were a trivial consequence of a behavioral assumption on the micro-level, which is not implied by the hypothesis of Section 2 of the entry Demand Aggregation: Theory,... of The New Palgrave Dictionary of Economics, 2 nd edition.
2 utility maximization. (Cross reference to entry individual behavior, Theorem of Mitjuschin and Polterovich). Demand functions f h F are assumed to be continuous in p and x and satisfy the budget-identity p f(p, x) = x. The function f F satisfies the Weak Axiom of revealed preferences if for every price-income pair (p, x) and (p, x ), p f(p, x ) x implies p f(p, x) x, and satisfies the Axiom of revealed preferences, if f(p, x) f(p, x ) and p f(p, x ) x implies p f(p, x) > x. Every demand function which is derived from a continuous, strictly convex and non-saturated preference relation satisfies the Axiom yet not necessarily the law of demand. Theorem (Hildenbrand, 983). The function F (p) := f(p, x)ρ(x)dx is monotone, i.e., (p q) (F (p) F (q)) for all p, q in R l ++, if f F satisfies the Weak Axiom of revealed preferences and ρ is a density which is non-increasing on R + with ρ(x)dx < 2. The function F is strictly monotone, if, in addition, f satisfies the Axiom of revealed preferences and the expansion paths f(p,.) and f(q,.) have only in common for any p,q that are not colinear. Interpretation: The underlying micro-model is a population H of households which is indefinitely large ; mathematically, an atomless measure space, e.g. the unit interval [, ] with Lebesgue measure. Every household h [, ] is modeled by its income x(h) and the common demand function f. The income assignment x( ) is an integrable function whose distribution admits a density ρ. Thus, mean demand F (p) = f(p, x(h))dh = f(p, x)ρ(x)dx. Three questions are relevant:. Why a continuum of households? Does the result still hold approximately for a large but finite population? 2
3 2. Why a non-increasing income density? Does monotonicity of F fail if the density is first increasing and then decreasing? 3. Why a common demand function? Does the result extend to heterogenous populations in income and demand behavior? The discussion of these questions is simplified by assuming that f is continuously differentiable in p and x. Then monotonicity of F is equivalent with negative semi-definiteness (n.s.d.) of the Jacobian matrix p F (p) for all p, i.e., l i,j= v iv j pi F j (p) for all v R l, and the Weak Axiom for f is equivalent with n.s.d. of the Slutzky substitution matrix. Consequently, monotonicity of F follows from the positive semi-definiteness (p.s.d.) of the mean income effect matrix I(f, ρ) = I(f, x)ρ(x)dx, where I(f, x) = (f i (p, x) x f j (p, x)) i,j=,...,l. Question : The mean income effect matrix for a finite population H, i.e., #H H (f i(p, x h ) x f j (p, x h )) i,j = I H is p.s.d. if and only if for every v R l, v I H v = #H H g (x h ) where g(x) := (v f(p, 2 x))2. Assume that income x h is measured in multiples of (euro). Let π n := #{h H #H xh = n =: x n }, n =,,... Then () #H H g (x h ) = n= π ng (x n ) = n= (π n π n )g(x n ) + o( ) using the approximation (2) g (x n ) = (g(x n+) g(x n )) + o( ). Consequently, one needs π n π n, n =,..., to obtain a non-negative first term on the right hand side of (); this is the finite analogue of a non-increasing density. Thus, for a finite population with a small (which requires by π n π n a large population) one obtains the desired result up to the small term o( ). For a population H = [, ] one does not need the approximation (2) and hence o( ), since () becomes g (x)ρ(x)dx = g(x)ρ (x)dx (by partial integration), which is non-negative for a nonincreasing differentiable density ρ. Question 2: The mean income effect matrix I(f, ρ) is p.s.d. in each of the two extreme cases: either, ρ is non-increasing and no assumption on 3
4 the shape of the income expansion path f i (p, ) or, no assumption on ρ yet linearity of f i (p, ). There must be results in between. Indeed, if the curvature of all income expansion paths f i (p, ) is limited and the unimodal density ρ is sufficiently skewed, then I(f, ρ) is p.s.d. Example: All income expansion paths restricted to the interval [, x] are polynomials of degree n (note that, no non-linear f i (p, ) can be a polynomial on R + ) and ρ is concentrated on [, x]. Then, I(f, g) is p.s.d. if and only if the matrix M(n, ρ) := ((i + j)m i+j ) i,j=,...,n is p.s.d. where m k := x k ρ(x)dx (Hildenbrand, 994, Appendix 6). Let the densities ρ m be as in Figure. 2 x ρ m m For every n there exists m(n) > such that I(f, ρ m ) is p.s.d. if m m(n); e.g. n = 2, m(2) =.38 x or n = 3, m(3) =.4 x. For a more general analysis see Chiappori (985) and Hildenbrand (994). Question 3: A population of households that is heterogeneous in income and demand functions is described by a joint distribution µ of income and demand functions, i.e., µ is a distribution on R + F. (A reader not familiar with distributions on function spaces might replace F by a finite set F ). As before, the marginal distribution of income admits a density ρ. The x 4
5 conditional distribution of demand functions given the income level x is denoted by ν(x). Then mean demand F (p) := f(p, x)dµ = f(p, x)ρ(x)dx R + F where f(p, x) := f(p, x)dν(x). Consequently, the Theorem or the extensions discussed under Question 2 imply that F (p) is monotone pro- F vided the function f satisfies the Weak Axiom. This approach to derive monotonicity for a heterogeneous population is the most direct, yet not the most general way (see Hildenbrand, 994). It is well-known (Hicks, 956, p.53) that f does not necessarily satisfy the Weak Axiom, even if individual demand functions are derived from utility maximization. The following two assumptions (which, again, are not the most general ones) imply that f satisfies the Weak Axiom (a) independence: ν(x) does not depend on x (b) increasing dispersion: the distribution D(x + ), >, is more dispersed than the distribution D(x), where D(ξ) denotes the distribution (in the commodity space R l ) of individual demand of all households with income ξ at the price p (i.e., D(ξ) is the image distribution of ν under the mapping f f(p, ξ)). Generalizing the one-dimensional case where the variance is a measure of dispersion one chooses the positive definiteness of the covariance matrix as a measure of dispersion for distributions on R l. Thus, increasing dispersion means that for >, covd(x + ) covd(x) is positive semidefinite. Assumptions (a) and (b) are quite restrictive, in particular, the independence assumption. Therefore one partitions the whole population H into subpopulations H(a) by stratifying with respect to a certain vector a of household attributes (household size, age,...) and than requires assumptions (a) and (b) for each subpopulation H(a). The role of stratifying is to reduce the heterogeneity in demand behavior. In the extreme case, where stratifying leads to a homogeneous subpopulation in demand behavior, assumption (a) and (b) are trivially satisfied. If the income density 5
6 of each subpopulation H(a) is non-increasing on R + or if the extension discussed in Question 2 apply, the mean demand of each subpopulation is monotone and hence also the mean demand of the whole population, since monotonicity is additive. A more general definition of increasing dispersion and a detailed discussion is given in Hildenbrand (994). For an empirical study of the Law of Demand see Härdle et al. (99). A broader discussion of the law of demand and related properties including cases where income is price dependent is contained in the entry law of demand by M. Jerison and J. Quah. Werner Hildenbrand References Chiappori, P Distribution of income and the law of demand. Econometrica 53, Härdle, W., Hildenbrand, W. and Jerison, M. 99. Empirical evidence on the law of demand. Econometrica 59, Hicks, J.R A Revision of Demand Theory. London: Oxford University Press. Hildenbrand, W On the law of Demand. Econometrica 5, Hildenbrand, W Market Demand. Princeton: Princeton University Press. 6
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