Demand Analysis
2.2. Theory of Consumer Behavior and Demand Analysis 1. Basic Model
The basic objective of the theory of consumer behavior is to explain how a rational consumer chooses what to consume when confronted with various prices and a limited income. At this level of generality, the main usefulness of the theory for empirical purposes is that it establishes a set of constraints which demand parameters must satisfy, thus limiting the number of independent parameters to be estimated and ensuring consistency in the results obtained.
Consider an individual consumer whose utility function is u(q, z), where q is the vector of quantities of n commodities on which a consumption decision must be made and z are individual characteristics. The amount of income which can be spent is y, imposing a budget constraint pq = y, where pis an n-dimensional row vector of prices. The consumer’s objective function is to maximize utility with respect to q, subject to the budget constraint pq = y. This can be rewritten as:
Maxq, u(q,z)(y p q),
where is a Lagrange multiplier (see the Appendix).
The solution to this maximization problem is a set of n demand equations:
qi qi(p,y,z), i1,,n.
These n equations contain:
n income slopes qi
y or income elasticities i qi
y y qi, and n2price slopes pqi
j
or price elasticities Eij qi
pj pj
qi .
Goods can be categorized according to the signs and magnitudes of these elasticities as follows:
Categorization with respect to the income elasticity:
Normal good: i > 0 (i> 1 luxury; 0 < i< 1 necessity) Neutral good: i = 0
Inferior good: i < 0.
Categorization with respect to the own-price elasticity:
Non-Giffen good: Eii < 0 (Eii < –1 elastic; Eii > –1 inelastic) Giffen good: Eii > 0.
Categorization with respect to the cross-price elasticity:
Gross substitutes: Eij > 0 Gross complements: Eij < 0.
There are a number of constraints that these parameters must satisfy. They are:
a. The Engel equation that derives from the budget constraint:
(1) pi
i qyi 1 or wi
i i 1, where wi piqi
y are the budget shares.
b. The n Cournot equations that also derive from the budget constraint:
(2)
pi
i qpi j qj or wi
i Eij wj, for j= 1,,n.c. These two sets of equations together give the n Euler equations (that are consequently not additional restrictions) that state there is no money illusion; that is, that if all prices and income increase in the same proportion, demand remains unchanged:
(2)
Eij
j i 0, i1,,n.d. The n(n – 1)/2 Slutsky equations that express symmetry in substitution effects:
(3)
Eij wj
wi Eji wj(j i), fori j1,,n.
Using equations (1), (2) or (2), and (3), the n + n2 parameters of the system of demand equations are thus reduced to a smaller number of independent parameters, namely,
(nn2)1nn(n1)
2 1
2(n2n2 ).
Theory thus allows substantial reduction in the number of parameters to be estimated and, hence, in the amount of data needed for that purpose. If, for example, n is 10, the demand equations contain 110 parameters. The 56 constraints on these, however, imply that only 54 of them are independent.
For empirical work, time series data are usually needed to observe price changes and estimate price elasticities. However, these time series are generally short; and the number of independent parameters left by theory, even after imposing all constraints, remains excessive.
Additional constraints, consequently, need to be added by making the general model of consumer choice more restrictive. This can be accomplished with use of the concept of separability.
2.2.2. Separability and Stepwise Budgeting
The basic idea of separability is intuitively appealing. It postulates that commodities which interact closely in yielding utility can be grouped together while goods which interact only in a general way through the budget constraint are kept in separate groups. Items used for food, clothing, housing, transportation, and entertainment could thus constitute separable groups. While carrots and tomatoes compete closely in satisfying food utility, and movies and plays entertainment utility, carrots and movies compete for overall utility in a way similar to tomatoes and plays. Another way of understanding the idea of separability is through the concept of stepwise budgeting in the making of consumer choices. Due to the complexity for consumers in making choices among a very large number of alternatives, income is first allocated to budget categories, also called wants, such as food, clothing, and housing. In a second stage, the food budget is allocated to specific items such as carrots and tomatoes. It can be shown that, if separability in wants exists, the exact same final choices are made in two stages as in one single decision. Empirically, existence of separability reduces further the number of independent parameters to be estimated.
Several types of separability have been postulated, some more restrictive than others (see Brown and Deaton, 1972). The most restrictive, and also the most empirically useful, is one introduced by Frisch (1959), where each commodity belongs to a separate group (pointwise separability). The utility function is thus written as:
uul(ql)...un(qn).
This implies “want independence” in the sense that the marginal utility (MU) of a good i is independent of the quantity consumed of any other good j:
MUi u
qi dui
dqi, MUi
qj 0.
Because goods are not likely to be inferior, it implies that all goods are gross complements to each other. It is, consequently, an approach that works fairly well when dealing with commodities that are themselves broad aggregates, such as food, clothing, and housing. In this case, the cross- and own-price elasticities take the following forms, respectively:
Eij wj
ijwji, ij,
Eii 1
i(1wii)wii, where:
y y
is the “flexibility of money,”
= ∂u/∂y is the marginal utility of income, and is the indirect utility.
u(p,y)u(q(p,y))
Note that price elasticities are obtained as a function of the budget shares, the income elasticities, and the parameter . If is known, the price elasticities can be derived from cross-sectional household survey data which give measures of the budget shares and estimations of the income elasticities. In a sense, this is an extreme case where restrictive theoretical specifications allow for an estimation of behavior in response to price changes without even requiring observation of prices.
The flexibility of money can be measured from knowledge of the own-price elasticity, the income elasticity, and the budget share for one separable group i:
i(1wii) Eiiwii .
This is commonly done using the food group since it is the one for which prior knowledge of these elasticities tends to be most reliable.
The flexibility of money has been estimated for a large number of countries and time periods. Estimates range from –3 at low levels of per capita income (e.g., in Argentina and Chile) to –1.1 in the United States. In order to predict likely levels of , an empirical relation can be estimated between the measured and the corresponding level of real income. One such relation is:
ln ()1.87 ,
( 6.6)
0.60
(4.9) ln
y/P
, R2 0.46where y/P is real per capita income in 1957–59 dollars, with P = 100 in the base year, and the numbers in parentheses are t-statistics (see Bieri and de Janvry, 1972).