The theory of consumer behaviour postulates that consumers behave "rationally" and choose among the consumption alternatives available to them in such a way as to maximise their satisfaction. Consumers are aware of the alternatives facing them and are able to evaluate their worth or utility. Thus, consumers have stable preference systems and the satisfaction they derive from the consumption of the various quantities of commodities they purchase, with the available income, can be described by means of a utility function. Although earlier economists, like Jevons and W alras, considered utility to be measurable or cardinal, 1 this is not strictly necessary and a rational consumer needs only to be able to rank commodities in order of preference by an ordinal2 utility function. It follows that each individual consumer, within the budget constraint, selects the particular combination, or basket, of commodities for which the utility function takes the largest value.
If we consider n commodities and indicate by qi (for i= l , ... , n) the quantities of them bought at time t3 and assume that the consumer's satisfaction is measured by the utility function:
1 A cardinal utility function is such that every combination of commodities consumed has a number associated with it representing its utility. For a discussion of how to measure the cardinal dimension of the utility concept from sources other than demand behaviour, and how to relate it to ordinal utility see Van Praag ( 1 994). For an earlier discussion of "measurable utility" see Ellsberg, 1 954.
2 An ordinal utility function does not assign numbers to measure utility, but simply ranks combinations of commodities in order of preference. The ranking is expressed mathematically by associating certain numbers with the various quantities of commodities consumed, b1,1t these numbers do not measure amounts of satisfaction. They only provide a ranking or ordering of preferences.
3 The utility function is defined with reference to consumption during a specific time period t. We do not take into account what happens after t, consumers make their decisions for only one such period at a time. The possibility to transfer consumption from one period to another is not taken into account. For
(2. 1 . 1)
where q is a (n x 1) column vector of quantities consumed, then the consumer will
chose a combination q* which will make U(q) as large as possible subject to the constraint:
(2. 1 .2)
where p =
(p1
, ... , Pn) is a column vector of prices and y represents the consumer's total expenditure or income. Constraint (2. 1 .2) simply states that the sum of the expenditures on the n commodities must be equal to the given amount of total expenditure.The maximisation of the utility function (2. 1 . 1) subject to constraint (2. 1 .2) under appropriate conditions will produce a system of n equations
q*= q(p, y) (2. 1 .3)
where q(.) represents a set of functions, the so-called demand equations, describing the quantities of the various commodities consumed under specific price/income conditions which maximise the consumer's satisfaction.
The conditions for maximisation are that (2. 1 . 1) has continuous derivatives up to the third order with positive first derivatives, all variables in (2. 1 .2) are continuous, and finally that the q* solution is strictly positive and unique. The positive sign of the first derivatives insures that an increase in the quantities consumed will generate an increase in utility.
The condition for positive first derivatives of the utility function and the existence of the third derivatives implies that for i, j = 1 , ... , n the (n x n) matrix of second derivatives
a discussion of multi-period consumption see Henderson and Quandt ( 1980, p.326-333) and Deaton ( 1 992).
(2. 1 .4)
called the Hessian, is symmetric and negative definite, this ensures that q* corresponds to a constrained maximum rather than a minimum or a saddle point (See Theil, 1 975, p.3).
To maximise (2. 1 . 1 ) subject to the budget constraint (2. 1 .2) we form the Lagrangian function
(2. 1 .5)
where A is an as yet unknown Lagrangian multiplier. Differentiating with respect to qi, the quantities of the commodities entering the consumer budget, and equating to
zero we obtain the set of n equations
(8U I
Oq_)
= A pi (for i = 1 , ... ,n) (2. 1 .6) Solving (2. 1 .6) for the n quantities qi and A gives the optimal consumption vector q* subject to the budget constraint. A direct result of utility maximisation is that every demand equation must be homogeneous of degree zero: if all prices and income are multiplied by a positive constant, the quantities demanded must remain constant (see Theil, 1 975, p.34).To define the functional form of q(p, y) for empirical studies, we need to choose a specific utility function. Following Klein and Rubin (1947- 1 948) we assume:
(2. 1 .7)
where {3i and fi are the parameters of the function. The qi are the quantities consumed
of the n commodities in the consumer's basket which must be positive and qi > fi.
The fi· if positive, may be interpreted as the minimum quantities needed for
subsistence.
From (2. 1 . 7) we can find the marginal utility of the ith commodity as the partial derivative of the utility function in
qi :
(2. 1 .8)
From the condition that all marginal utilities must be positive, and with the constraints on the quantities
qi
imposed in (2. 1 .7), it follows {Ji > 0. If we normalise the {3 parameters by dividing each of them by their sum, then(2. 1 .9)
To derive the empirical demand functions, we equate the RHS' s of (2. 1 .6) and (2. 1 .8) and obtain
(2. 1 . 1 0)
which, summed over n and, considering (2. 1 .9) and the constraint (2. 1 .2), yields (2. 1 . 1 1 )
If we now solve (2. 1 . 1 0) in A. and fit the result in (2. 1 . 1 1) we obtain the well known Linear Expenditure System (LES), frrst used by Stone ( 1954), and more recently by Solari ( 1 97 1 ), Deaton ( 1 975), Carlevaro (1975) and many others. For the ith commodity the demand equation becomes:
Piqi
=Yt Pi
+f3i
y -f3i :!
r k pk (2. 1 . 1 2)k•l
Equation (2. 1 . 12) describes the expenditure on each of the n commodities in a
consumer's budget as functions of prices and the consumer's income or total expenditure.
Demand systems based on functional relationships like (2. 1 . 1 2) are very attractive for the simplicity of their mathematical formulations and have been extensively used in
empirical studies. However, they depend on a utility function like
(2.1.7)
which is very restrictive, and rather unrealistic, as it assumes that the marginal utility of every commodity depends exclusively on its own quantity (Theil,1975,
p.6).We will apply the LES to our data as a first approximation to the description of households' consumption behaviour in Italy and New Zealand and also as a bench mark against which to compare the more advanced demand systems which we will go on to study in the next few chapters.