CHAPTER IV. THEORETICAL FRAMEWORK
4.3 Properties of Marshallian Demand Function
By manipulating the first order conditions presented earlier in (4.2) and (4.3) one can derives important properties pertaining the parameters of demand functions. The knowledge on this properties helps researcher in resolving the problem of estimation.
The results of changes in prices of commodities and income level of the consumer are described by the partial derivatives of first order conditions. There are in general, four basic properties of demand functions: namely, adding up, homogeneity, negativity, and symmetry that are important in providing a testable hypothesis to test the rationality of consumer behavior. The properties of demand functions guide the empirical analysis in testing consumer behavior from real world data. The properties are always effective irrespective of the form of utility function and take the form of mathematical restrictions on the derivatives of the demand functions.
4.3.1 Adding Up
Adding up condition comes from the budget restriction and the monotonic property of the preference. According to this property, because the representative consumer is assumed to spend exhaustively all of their income, the income or total expenditure should be the addition of the values of the Marshallian demand function. Formally, it is expressed as
Pq Pnqn q
P1 1 2 2 ... y
which is the linear budget constraint given in (4.1). Substituting (P, y) for qi we get
qi(P,y)
i pi y (4.4) By writing in the elasticity notation, one obtains the following equation:
1 1
1eywneny
w ; (4.5) where w1 is the budget share of good 1, and e1x is the income elasticity. This condition, which says the sum of weighted share of income elasticities, is equal to one; the weights being the budget shares of the commodities. This condition is known as an Engel aggregation.
39 By writing in the elasticity form, the following expression is obtained
0 commodity and
e
jj is the own price elasticity. This condition, which says that the sum of the own price and cross price elasticities weighted by their budget shares due to change in price ofj
th commodity is equal to the negative of the change in price ofj
th commodity, is known as Cournout aggregation.4.3.2 Homogeneity
The Marshallian demand functions are homogeneous of degree zero in price and income, meaning that if we multiply all the prices and income by a constant k, the optimal quality demanded of commodities is unchanged.
According to Euler‘s theorem, if a function f(y) is homogeneous of degree, then derivatives of this function satisfy the following properties:
one says that the sum of all own and cross price elasticities is equal to the negative of the income elasticity. This condition given by the homogeneity property of demand function is also referred to as the row constraint. If there are n demand equations then there will be n restriction of the utility maximization problem with a budget constraint.
The homogeneity restriction is not particularly useful for single commodities studies since such studies seldom, if ever, include all prices and income (Currie, 1972).
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The negativity property places the following inequality restriction on
s
ij; the diagonal elements must be non-positive for all i, ) 0 assumption of quasi concavity of the utility function by which the second derivative with respect to any price is negative.Symmetry condition implies that if budget shares and one set of cross prices elasticities are known along with income and own price elasticities, another set of cross price elasticities could be calculated.
In applied demand analysis, the properties of demand functions discussed above have important implications in terms of testing the hypothesis of consumer theory, in imposing certain restrictions on the parameters of estimations and the expected signs of elasticities.
By Engel aggregation in adding up property,
1
So only n-1 of the income elasticities are independent.
By homogeneity property, calculated, which reduces the number of independent elasticities by12 2
(n n). In practice
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to derive all price and income elasticities we need to estimates n2n parameters ( n2 price and n income elasticities). Using the properties of demand functions, namely homogeneity, Engel aggregation and symmetry, the number of independent parameters to be estimated can be reduced to
This is very useful when the applied researcher is faced with the problem of a small number of goods to be analyzed is 10, and then the total number of elasticities to be estimated is 110. However, using the above restrictions, only 54 parameters need to be estimated. Also, expected signs of elasticities can be derived from these restrictions. For example, using homogeneity, if all cross elasticities and the income elasticity for a good are positive then own price elasticity should be negative.
An alternative to derived demand functions is made available by duality principle. This is achieved by introducing the indirect utility function. This is done by inserting demand function q = q (y, p) into a utility function U = U (q) to give maximum attainable utility as a function of y and p, noted as demand function could be retrieved from indirect utility functions by applying Roy‘s Identity (Roy, 1942) according to which maximization subject to an expenditure constraint, one can derive demand functions by minimization of expenditures subject to a utility constraint.
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The duality theorem implies that the solution to the maximization problem is identical to the solution of its minimization dual when the constraint to the maximization problem is appropriately defined. By applying this principle, the behavior functions for the xi's are solved simultaneously for the primal and the dual. Simply put, this means that the levels of the solution values are the same. That is, xiM
= xiU
as can be seen in the graph in the two goods case. But it also means that equations (3) and (4) are equal at that point.
Marshallian demand function is observable but not predictive. The Hicksian is predictive but not observable. Combining both is the advantage of using the duality principle.
By using of duality principle, the demand functions may be established from derivation of cost function, the minimum cost of obtaining a fixed level of U at given price. Deriving the cost function with respect of price (Shepard‘s Lemma) leads to Hicksian or compensated demand functions. Hicksian demand function is the relationship between the goods purchased, prices and utility.
This result has important implications in applied demand analysis. If a functional form is assumed for V (P, Y) then the estimable form of Marsallian demand equations could be derived using Roy‘s identity and will have the same structure as the ones derived from direct utility function (Barten and Bohm, 1982). The approach to derive demand functions using indirect utility function is also amenable for applications in welfare economics and index number analysis since it represents the allocations to achieve the maximum utility levels under different prices and income (Jorgenson et. al., 1982).