Demand

BIBLIOGRAPHY

In economics, demand theory examines the purchasing behavior of an individual, or of a group of individuals, in terms of its responses to changes in purchasing constraints or other institutional factors. In the most basic setting, an individual chooses a bundle of commodities, considering both their prices and a maximum income that he or she can spend. It is assumed that the individual compares bundles of commodities according to his or her own preferences, and that he or she chooses to purchase the best bundle that is affordable; in economics, this type of behavior is called “rational” or “preference maximizing.” The first systematic analysis of this problem was done by Leon Walras in 1874. In 1886 Giovanni Antonelli studied the problem of constructing preferences that can explain a given demand function, and in 1915 Eugene Slutsky obtained a full set of implications of preference maximization on demand behavior. In the mid-twentieth century, the economist Paul Samuelson called for a revision of the theory, whereby observable behavior, rather than unobservable preferences, would constitute the foundation of demand theory. This proposal led to the development of revealed-preference analysis, an exhaustive series of conditions that must be satisfied by demand behavior, if it is consistent with preference maximization.

The standard problem of demand theory is that of an individual who chooses a bundle of L commodities, ranking different bundles according to his or her preferences. The individual faces prices p = (p ₁, …, p _L) and can spend up to a nominal income of m. Consumption bundles are denoted by x = (x ₁, …, x_L). A rational individual chooses a bundle x̑ that is:

1. feasible: p x = p ₁x ₁ + … + p_Lx_L ≤ m ; and
2. optimal: there does not exist an alternative bundle that the consumer can afford and prefers to x̑.

Under well-known assumptions on the preferences of the individual, one such x̑ is guaranteed to exist, and is uniquely defined. Then, x (p, m ) = x̑ is known as the Marshallian demand of the individual, and constitutes the central element of demand theory: It says how much of each commodity the consumer demands, as a function of the prices he or she encounters in the market and of the income he or she has.

Function x is homogeneous of degree zero: multiplying all prices and income by the same positive number does not change demand. When a consumer prefers more to less consumption of at least one commodity, Marshallian demand satisfies Walras’s law: the consumer will spend all his or her income, so p • x (p, m ) = m. When Marshallian demand is differentiable, these properties impose restrictions on the derivatives of the demand with respect to prices and income, which can be used to restrict empirical estimations of demand. More importantly, the substitution matrix, S, defined by S (p, m ) = D_p x (p, m ) + D_m x (p, m ) x (p, m )′, and which isolates the effects of changes in relative prices, is a symmetric, negativesemi definite matrix of rank L – 1. Suppose that the preferences of the consumer are represented by a utility function u. Then, the indirect utility function, v, defined by v (p, m ) = u (x (p, m ), and which measures the utility that the consumer obtains when he or she faces income m and prices p, is homogeneous of degree zero and, remarkably, satisfies Roy’s identity: the negative of the ratio of the derivative of the indirect utility with respect to the price of a commodity to the marginal utility of income equals the Marshallian demand for that commodity; formally, –(D_m v (p, m )^–1 D_p v (p, m ) = x (p, m )′.

Representability of preferences—via utility functions that assigned higher utility levels to more preferred bundles than to less preferred bundles—allows for an auxiliary problem: fixing a benchmark utility level ū, and given prices p, determine a bundle x̂ that:

1. gives at least that level of utility: û (x̂ ) ≥ û ; and
2. minimizes expenditure: every other bundle that gives utility of at least ú costs at least p • x̂.

The solution to this problem defines the Hicksian demand function, h (p, ū ) = x̂, whose cost defines the expenditure function, e (p, ū ) = p • h (p, ū ). Functions h (p, ū ) and e (p, ū ) are, respectively, homogeneous of degrees zero and one in prices. When small changes of consumption only induce small changes of utility (a condition known as “continuity of preferences”), the utility level given by the expenditure-minimizing bundle is exactly the required level: u (h (p, ū )) = û. Function e is strictly increasing in û and nondecreasing and concave in p, while Hicksian demand satisfies the “compensated law of demand” (this is not true for Marshallian demand), in a sense that Hicksian demands are nonincreasing in prices: (p – p ′) • (h (p, ū ) – h (p’, û )). Under differentiability, Hicksian demand satisfies Shephard’s lemma: the derivative of the expenditure function with respect to the price of a commodity equals the Hicksian demand for that commodity, D_pe (p, ū ) = h (p, ū ).

Duality theory establishes, with minor qualifications, that Marshallian and Hicksian demands mirror each other: (1) the solution of the expenditure minimization problem also solves the preference maximization problem at income equal to the minimized expenditure: x (p, e (p, ū )) = h (p, ū ); (2) the solution of the utility maximization problem also solves the expenditure minimization problem at benchmark utility equal to the maximized utility: h (p, v (p, m )) = x̂ (p, m ); (3) the maximal utility achievable with income equal to the minimized expenditure is the benchmark utility, v (p, e (p, ū )) = û ; and (4) with lower income, it is impossible to obtain at least the maximized utility level, e (p, v (p, m ) = m. Shephard’s lemma shows that the first equality yields the Slutsky decomposition: D_ph h (p, ū ) = S (p, e (p, ū )), which allows for empirical estimations of Hicksian demand. A commodity is said to be inferior if its demand decreases when the consumer has more income, and “Giffen” if it increases when the price of the commodity increases. It follows from Slutsky decomposition and the compensated law of demand (for Hicksian demands) that Giffen commodities are inferior.

Samuelson’s program looked for conditions on demand data that were equivalent to the existence of the unobservable preferences: Given a series of observations of prices, incomes, and demanded bundles, do there exist preferences such that, at every observation, the observed demand is rational according to those preferences (x (p_t, m_t) = x_t)? A necessary condition for the existence of such preferences is the Weak Axiom of Revealed Preference, or WARP: If p_t • x_t ≤ m_t and x_t ≠ x_t’, then p_t′ • x_t > m_t; that is, if the bundle purchased at observation t′, x_t, could have been purchased at t (since it was affordable) but bundle x_t was purchased instead, then it must be that the consumer prefers bundle x_t to bundle x_t′. Then, at observation t′, when the consumer actually purchased bundle x_t′, it must be that the more preferred bundle, x_t, was not affordable. WARP is not, however, a sufficient condition. It compares only pairs of bundles, whereas rationality requires comparisons of sequences of bundles, since it implies that preference is a transitive relation. A transitive chain of reasonings—like the ones posed by WARP, known as the Strong Axiom of Revealed Preference —is a necessary and sufficient condition for rationality.

Antonelli’s “integrability” result shows that homogeneity of degree zero, symmetry and negative semi-definiteness of the substitution matrix, and Walras’s law exhaust the necessary conditions of a rational Marshallian demand. Using the solution to the system of differential equations implied by S, one can construct an expenditure function whose variation with respect to prices can be used to preferences that, when maximized, would yield the observed demand behavior.

The separability problem, first considered by John Hicks, studies how a group of commodities whose prices are kept in fixed proportions can be treated as a single, composite commodity whose price is constructed through the aggregation of the individual prices. This analysis provides the basis for the construction of price indices. A related problem assumes that preferences satisfy the following separability property: a group of commodities exists whose variation is ranked by the individual independently of the level of consumption of all other commodities. In this case, there is a “subutility” function, defined for the group in question, and the overall utility depends on the consumption of the group only via the subutility level. When preferences satisfy this separability condition, demand can be determined in a two-stage process: first, the individual decides how much to spend in the group, and then the individual decides how to allocate expenditures across commodities in the group.

Suppose that there is a set of consumers {1, …, i, …, I }, with individual variables indexed by i. The aggregate demand function is x (p, m ¹, …, m ^I ) = x ¹(p, m ¹) + … + x ^I (p, m ^I ). The aggregation (or representative consumer) problem asks whether one can find an individual demand function x̄ such that x̂ (p, m ¹, …, m ^I ) = x̂ (p, m ¹ + … + m ^I ). The answer to this question is, in general, no, for it requires that all individuals have parallel wealth-expansion paths (the trajectories defined by changing individual incomes at given prices). Also, even under aggregation, individual satisfaction of WARP does not imply its satisfaction in the aggregate. WARP, in the aggregate, is obtained whenever individual Marshallian demands satisfy the law of demand, a condition that does not generally occur. (The definition of aggregate demand given here should not be confused with Hicks’s aggregate demand, which is used in macroeconomic models and in national accounts.)

In the standard setting, commodities can be interpreted to accommodate intertemporal problems. If there is only one commodity in each time period, relative prices represent interest rates, and an impatient consumer will anticipate consumption unless interest rates compensate for his or her impatience. In general, if intertemporal preferences are convex, an individual prefers a consumption plan that smooths consumption, in the sense that it avoids high levels of consumption in some periods and low levels in other periods. A canonical case for consumption smoothing is the life-cycle model originally proposed by Franco Modigliani, who divided life into three periods: (1) youth, when income is low; (2) adulthood, when income is high; and (3) retirement, when income is once again low. A consumer without restrictions will choose a smooth consumption plan that includes accruing debt during youth, repaying this debt and saving during adulthood, and spending savings during retirement. Consumption is determined by lifetime (discounted) income, while temporary deviations from this “permanent” income are accommodated by savings. A more general version of this model, involving different individuals at different stages in life, is known as the overlapping generations model. This model constitutes the basic tool for the economics of Social Security.

When the interpretation of commodities includes different states of nature, the model permits the study of uncertainty. In this case, convexity of preferences means that the individual dislikes risk and will avoid large consumption in some states if it implies low consumption in other states, unless prices compensate for this effect. This effect means that, if available, the individual will use insurance opportunities, or, alternatively, that groups of individuals will prefer risk-sharing schemes that disseminate risks.

Applied work usually imposes particular functional forms useful for estimation. An important, flexible functional form is the Almost Ideal Demand System (AIDS), which imposes an expenditure function of the form e(p,û) = α(p) + β(p)û where ,and some conditions on the parameters of these two functions are imposed. Applied work also uses “hedonic” models to impose additional structure on demand systems. It is assumed that individuals do not directly care about the commodities, but only about their attributes (physical features), which are, normally, objectively observable.

SEE ALSO Aggregate Demand; Aggregate Supply; Equilibrium in Economics; Excess Demand; Hedonic Prices; Markets; Rationality; Samuelson, Paul A.; Supply; Utility Function

BIBLIOGRAPHY

Ando, Albert, and Franco Modigliani. 1963. The “Life-Cycle” Hypothesis of Savings: Aggregate Implications and Tests. American Economic Review 53 (1): 55-84.

Deaton, Angus, and John Muelbauer. 1980. Economics and Consumer Behavior. Cambridge, U.K.: Cambridge University Press.

Hildenbrand, Werner. 1994. Market Demand. Princeton, NJ: Princeton University Press.

Samuelson, Paul. 1938. A Note on the Pure Theory of Consumer’s Behaviour. Economica 5: 61-71.

Andrés Carvajal