Market Demand: Theory and Empirical Evidence

by Werner Hildenbrand

Overview

In a major work that is the culmination of over a decade of intensive research, Werner Hildenbrand presents a new theory of market demand, the principal aim of which is to identify the conditions under which the Law of Demand holds true. Hildenbrand argues that the Law of Demand is due mainly to the "heterogeneity" of the population of households. In his view, "rationality" of individual behavior plays only a minor role. While the traditional approach to the theory of market demand is to analyze the question, To what extent are the postulated properties of individual behavior preserved by going from individual to market demand?, this book asks the question, Which properties of the market demand function are created by the aggregation process?.

Two hypotheses on the population of households play a key role in Hilden-brand's thinking. The first is the "increasing dispersion" and the second the "increasing spread" of households' demand. These hypotheses can easily be interpreted and are a priori plausible. For a positive theory of market demand, according to Hildenbrand, it is more important that the hypotheses are well supported by empirical evidence. His claims in this important new book are based on a nonparametric statistical data analysis of the U.K. Family Expenditure Survey and the French Enquête Budget de Famille.

Originally published in 1994.

The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Product Details

ISBN-13:	9781400863716
Publisher:	Princeton University Press
Publication date:	07/14/2014
Series:	Frontiers of Economic Research , #215
Sold by:	Barnes & Noble
Format:	eBook
Pages:	216
File size:	9 MB

Read an Excerpt

Market Demand: Theory and Empirical Evidence

---

By Werner Hildenbrand

PRINCETON UNIVERSITY PRESS

Copyright © 1994 Princeton University Press
All rights reserved.
ISBN: 978-0-691-03428-7

---

CHAPTER 1

Introduction

Pure economics has a remarkable way of producing rabbits out of a hat—apparently a priori propositions which apparently refer to reality. It is fascinating to try to discover how the rabbits got in; for those of us who do not believe in magic must be convinced that they got in somehow. —J. R. Hicks, Value and Capital, (1946), p. 23

1.1 The Law of Market Demand

In this book I shall develop a theory of market demand. The principal aim of this theory is to identify the conditions under which the Law of Demand will hold. I shall defend the thesis that the Law of Demand is mainly due to the heterogeneity of the population of households; the "rationality" of individual households plays only a minor role. After explaining what I understand by the Law of Demand, it will become clear that my goal is quite modest. However, I believe, it is fundamental.

What does the Law of Demand assert? First of all the Law of Demand does not refer to the demand of an individual household, but to market demand, that is to say, to the mean demand of a large population of households, for example, to all private households in Germany or the United Kingdom. To emphasize this, I shall often refer to the Law of Market Demand.

Second, the Law of Market Demand does not assert that for any particular commodity, say h, and any two time periods t and τ, for example the years 1991 and 1992, the actual mean demand per period of that commodity, Qht and Qhτ and the actual average prices, pht and phτ, during these periods are related by

(pht - phτ)(Qht - Qhτ) < 0. (1.1)

That is to say, an increase (decrease) in the price of commodity h from period t to period τ is followed by a decrease (increase) in mean demand per period of that commodity. However, the first empirical studies on demand in the nineteenth century analyzed exactly this type of relation.

One might expect that relation (1.1) holds if the two time periods are not far apart from each other and if the prices of all relevant commodities other than commodity h do not change. However, the actual evolution of prices over time would hardly respect such a ceteris paribus clause. If all prices change simultaneously there is, of course, no reason why relation (1.1) should hold.

To avoid this ceteris paribus clause on prices, one can consider the following weakening of relation (1.1). Consider a comprehensive collection of commodities, say h = 1, ..., l. Let pt = (p1t, ..., plt) denote the vector of actual average prices in period t and let Qt = (Qlt, ..., Qlt) denote the vector of actual demands per period t. One can now ask whether the vector of prices and the vector of demands for two time periods t and τ are related by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.2)

Relation (1.2) says that the vector pt - pτ of price changes and the vector Qt - Qτ of demand changes point in opposite directions. This clearly does not imply relation (1.1) for every commodity: For example, in Figure 1.1,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Whether the actual evolution of prices and demand over time satisfies relation (1.2) is an interesting empirical question, which to my knowledge has never been analyzed. However, the Law of Demand, as understood in this book, does not refer to the actual evolution of prices and demand, but to hypothetical price changes within the same period. This requires the market demand to be modelled as a schedule: If the average price vector in period t were p, then the market demand per period t would be

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This notation emphasizes the dependence of market demand on the price system. Naturally, there are other determinants of demand, like income, tastes, or other household characteristics. If they are not mentioned explicitly in the notation, then this means that these determinants of demand are considered fixed during the period for which the market demand function is modelled. I shall say more on these determinants of demand later.

The Law of Demand asserts that the market demand function Ft(·) is strictly monotone, i.e.,

(p - q) · (Ft(p) - Ft(q)) < 0 (1.3)

for any two different price vectors p and q (see Figure 1.1).

This clearly implies a downward sloping partial market demand curve for every commodity. Indeed, if only the price of commodity h is changing (all other prices remain fixed), then the inequality (1.3) reduces to

(ph - qh)(Fht(p) - Fht (q)) < 0.

Thus, the partial demand curve

ph [??] Fht(p1, ..., ph, ..., pl)

is a decreasing function. This partial Law of Demand, however, is much weaker than the Law of Demand, because, in the definition of monotonicity, no restriction is made on the vector of price changes.

I have not yet commented on the nature of the commodities being consumed. It was, however, assumed that the "quantity" and the "price" of a commodity are well-defined. One cannot rule out a priori that the Law of Demand may hold for some commodities or some degree of commodity aggregation and not for others. More details on the nature of the commodities being consumed will be given in Chapter 2, where I distinguish "elementary" (microeconomic) commodities and "commodity aggregates". The composite commodity theorem of Hicks–Leontief, which is explained in Section 2.3, will play an important role, in particular, when I discuss empirical demand data made available by Family Expenditure Surveys.

If the market demand function Ft (·) is differentiable, then the partial demand functions are downward sloping if the partial derivatives [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are negative, that is to say, the diagonal of the Jacobian matrix

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is negative. The Law of Demand requires, however, that this Jacobian matrix be negative definite for every price vector p (see Appendix 1), which clearly is more demanding than a negative diagonal.

The results (predictions) of partial equilibrium analysis—as everybody knows—depend crucially on the assumption of downward sloping market demand curves, which implies, in particular, a unique and stable partial equilibrium. The Law of Demand plays the same role for a multimarket demand-supply analysis.

A competitive price equilibrium of a multicommodity demandsupply system, [F(·), S(·)], is defined as a solution of the system of l equations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This equilibrium should be well-determined if one wants to carry out a comparative static analysis. Strict monotonicity of the excess demand system F(p) - S(p) implies that the equilibrium, if it exists, is unique and globally stable with respect to the Walrasian tâtonnement process. Downward sloping partial excess demand functions are not a sufficient condition to derive these results. The Law of Demand is also useful in other applications to price theory, for example, in monopolistic competition, monopoly, or oligopoly price theory. I have chosen competitive price theory as a particularly simple illustration.

If one wants to model the actual evolution over time of an economy by a sequence of "temporary equilibria", then it is crucial that the temporary equilibrium in every period be well-determined. Again, properties of the short-run market demand function F, such as being subject to the Law of Demand, are important.

1.2 Wald's Axiom

The equilibrium of a demand–supply system can be made determinate by a weaker restriction on the excess demand function. This property was first formulated by Wald in 1936, I shall, therefore, refer to it as Wald's Axiom.

For the market demand function F this axiom asserts that for any two price vectors p and q,

p · F(q) ≤ q · F(q) implies q · F(p) ≥ p · F(p)

or, equivalently,

(p - q) · F(q) ≤ 0 implies (p - q) · F(p) ≤ 0.

Wald's Axiom clearly is a weaker condition on the market demand function F than the Law of Demand, which can be expressed as

(p - q) · F(p) < (p - q) · F(q).

Consequently, if the expression on the right side of this inequality is nonpositive, then the expression on the left side must be negative.

Figure 1.2 shows a situation where Wald's Axiom is satisfied but not monotonicity. Wald's Axiom can be interpreted as a restricted monotonicity property of F. Indeed, it is equivalent to (p - q) · (F(p) - F(q)) ≤ 0 provided the vector p - q of price changes is orthogonal to F(q), i.e., (p - q) · F(q) = 0.

Wald's Axiom for a continuous excess demand system W(p) = F(p) - S(p) implies only that the set {p [member of] Rl++|W(p) = 0} of equilibrium price vectors is a convex set; it does not imply that there is at most one solution (see Appendix 2). However, under some additional regularity assumptions one can show that the set of solutions of W(p) = 0 is typically (generically) a discrete or even a finite set. In this case, of course, Wald's Axiom implies the uniqueness of the equilibrium price vector.

1.3 Validation of Hypotheses on Market Demand

Does the Law of Market Demand need a justification, or is the Law of Demand and Wald's Axiom prima facie plausible or a priori evident?

The prima facie plausibility cannot be based on direct experience or empirical observations because the Law of Demand and Wald's Axiom refer to hypothetical situations. Strictly speaking, one can, at best, observe only one point (pt, Ft(pt)) of the postulated market demand function Ft(·). If one believes, on whatever grounds, that the market demand function changes only slowly over time, or does not change at all, even in this case, the empirical data from time series will not exhibit sufficient price variations to test the Law of Demand or Wald's Axiom. The lack of time-invariance of economic relationships poses a fundamental difficulty for an empirical inductive validation of these relationships. Neither can the a priori evidence be derived from introspection since both properties refer to market demand, and introspection can only refer to individual demand, which I shall discuss later.

One can, of course, argue that the foregoing questions are altogether irrelevant; only the conclusions (predictions) that are derived from an assumption have to be tested. In Friedman's words "... the only relevant test of the validity of a hypothesis is comparison of its predictions with experience. The hypothesis is rejected if its predictions are contradicted ("frequently" or more often than predictions from an alternative hypothesis); it is accepted if its predictions are not contradicted; great confidence is attached to it if it has survived many opportunities for contradiction." Alas, Friedman's methodological prescription—theory as a tool or an instrument—is hard to apply. What are the precise predictions that are derivable from the Law of Demand? Without additional hypotheses, obviously, no prediction is possible. If the Law of Demand, for example, is embedded in a competitive demand-supply analysis (this seems to be Friedman's framework), then one can think of comparative static results as predictions. It is well-known, however, that comparative static results that are derived solely from the Law of Demand are not specific enough. Testing a theory by its predictions is indeed a difficult task in economics.

Undoubtedly, one has to insist on a justification if one is to model market demand as a monotone function or as a function satisfying Wald's Axiom. Such a justification requires a more explicit definition of market demand that clearly should involve the demand of individual households; after all, market demand is defined as mean demand of households' demand. However, what do we know about the demand behavior of individual households?

1.4 On Individual Behavior: Introspection and Plausibility

The relationship in a given period t between the current average price vector p, the current disposable income xi, and the current demand vector yl of a household i is described by a "black box" or a function yi = fil(p, xl) (see Figure 1.3).

Microeconomics fills this black box by postulating certain models of household behavior. In doing so it is claimed that one derives the relationship fl from more basic concepts, like preferences and expectations. In this introduction, as a short cut, I take the short-run demand function as a primitive concept. Clearly, the function fl is typically not homogeneous in (p, x), and does not satisfy the identity p · fi(p, x) [equivalent to] x, since current total expenditure might well be different from current disposable income. More will be said on this topic in Section 2.2.

This unknown functional relationship that is represented by the black box can be viewed either as stochastic or as deterministic. In the former case, a household's demand function [??]l is a random function, that is to say, [??]l(p, xi) is a random vector. One can always write the random demand [??]i as the sum of the expected demand and a random variable with expectation zero:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In modelling the behavior of an individual household, the random term [??]l (p, xi) might be essential. However, if one is interested in modelling mean demand of a large population under the hypothesis of weak stochastic interaction among the households, then the individual randomness can be eliminated. Indeed, the mean demand of the population is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the law of large numbers implies that the random variable

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is negligible provided the population is sufficiently large. Thus, for the purpose of modelling market demand of a large population it suffices—if one accepts the assumption of a weak stochastic interaction—to model the demand behavior of an individual household by its expected demand function.

How are we to model household demand behavior? Do empirical studies of household behavior establish or at least suggest some general properties of the individual expected demand functions? By properties I naturally think of qualitative predictions about how a household reacts to changes in prices or changes in income. Given the hypothetical nature of the demand function fi, I am afraid the answer is negative. The empirical studies typically refer to groups of households that are stratified by suitable demographic characteristics like household size, sex, or profession of household head. Empirical analysis typically shows the relevance of these demographic characteristics, but they do not suggest general properties of the individual demand functions. Also, to my knowledge, laboratory experiments have not led, at least up to now, to well established regularities. I am afraid that all properties that have been formulated so far for individual demand functions, for example, the hypothesis of utility maximization or the Weak Axiom of Revealed Preference (which I shall discuss later), are entirely grounded on a priori reasonings. In the face of this fundamental difficulty in justifying properties of individual demand functions, economists often make reference to "introspection." Some economists see here an essential difference between the natural sciences and economics. It is argued that a physicist can hardly imagine how he would react if he were a molecule. However, every economist makes daily demand decisions. To rely on introspection in justifying a hypothesis has been forcefully criticized by Hutchison (1938).

(Continues...)

---

Excerpted from Market Demand: Theory and Empirical Evidence by Werner Hildenbrand. Copyright © 1994 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

---

Table of Contents

Preface

1 Introduction 3

1.1 The Law of Market Demand 3

1.2 Wald's Axiom 7

1.3 Validation of Hypotheses on Market Demand 9

1.4 On Individual Behavior: Introspection and Plausibility 10

1.5 Substitution and Income Effects 15

1.6 Increasing Spread of Households' Demand 19

1.7 Family Expenditure Data: Increasing Spread of Conditional Demand 22

1.8 Increasing Dispersion 26

2 Market Demand 30

2.1 The Distribution of Households' Demand 30

2.2 A Microeconomic Model: The Distribution of Households' Characteristics 35

2.3 Hicks-Leontief Composite Commodity Theorem 49

2.4 Family Expenditure Surveys: The Data 52

3 Increasing Dispersion 72

3.1 The Hypothesis of Increasing Dispersion of Households' Demand 74

3.2 Examples 88

3.3 Empirical Evidence of Increasing Dispersion of Households' Demand 93

Conclusion to Chapter 3 119

Notes on Chapter 3 120

4 The Law of Demand 122

4.1 The Hypothesis of Increasing Spread of Households' Demand 123

4.2 Empirical Evidence of Increasing Spread 133

4.3 Deductive Validation of Hypothesis 3 156

Notes on Chapter 4 166

App. 1. Monotone Functions 168

App. 2. Wald's Axiom 171

App. 3. The Weak Axiom of Revealed Preference and the Slutsky Decomposition 175

App. 4. Monotonicity of Individual Demand Functions 180

App. 5. Spread and Dispersion 185

App. 6. The Structure of the Matrix B [actual symbol not reproducible] 187

References 195

Author Index 199

Subject Index 201

Index of Frequently Used Symbols 205