Substitutability

BIBLIOGRAPHY

Substitutability is a fundamental concept in economics and is encountered in both consumer theory and producer theory. To illustrate, assume a finite number of commodities (say n ) and consider the following (direct) utility function,

u = f (X₁,X₂,....X_n),

where (x ₁, x ₂, …, x_n) is the commodity vector. Commodities i and j are said to be substitutes if

that is, if increased consumption of x_j reduces the marginal utility of x_i, and vice versa. If the second-order partial derivative is positive, then x_i and x_j are complements, and if it is zero, then x_i and x_j are independent.

Although the sign of the above second-order partial derivative is invariant under linear transformations of the utility function, it is not invariant under monotonic increasing transformations of the utility function. The demand system, however, has the advantage of being invariant under monotonic increasing transformations of the utility function. The demand system is obtained by maximizing u = f ( x ₁, x ₂, …, x_n ) subject to the budget constraint,

p₁x₁ + p₂x₂ + ... + p_nx_n=y,

where ( p ₁, p ₂, …, p_n, is the vector of prices and y is income. The demand systems is given by

x_i = x_i (p₁, p₂, ..., p_n,y), i = 1, 2, ... ,n

and gives the quantity demanded (of each commodity) as a function of the prices of all commodities and income.

In terms of the demand system, we can find out how the demand for good x_i changes if the price of good x_j changes. In particular, if Δx_i / Δp_j > 0, then x_i is a gross substitute for x_j , meaning that when x_j becomes more expensive the consumer switches to consuming x_i ; in other words, the consumer substitutes away from the relatively more expensive good towards the relatively less expensive good. If Δx_i / Δp_j > 0, then x_i is a gross complement for x_j , meaning that when x_j becomes more expensive the consumer reduces the consumption of x_j and also of x_i ; complements are goods that are consumed together (as, for example, coffee and sugar).

The definitions given above are in gross terms because they ignore the income effect— that is, the change in demand of good x_i due to the change in purchasing power as a result of the change in the price of good x_j . The Slutsky equation—see Mas-Colell, Whinston, and Green (1995) for more details—decomposes the total effect of a price change on demand into a substitution effect and an income effect, as follows:

where ∂ x_i | ∂ p_j is the total effect of a price change on demand, k_ij is the substitution effect of a compensated price change on demand, and – ∂ x | ∂ y)x _j is the income effect, resulting from a change in price (not in income). Hicks (1956) suggested using the sign of the cross-substitution effect (that is, the change in compensated demand) to classify goods as substitutes whenever k_ij is positive. In fact, according to Hicks (1956), k_ij > 0 indicates substi-tutability, k_ij > 0 indicates complementarity, and k_ij = 0 indicates independence.

One important property of the Slutsky equation is that the cross-substitution effects are symmetric, that is

k _ij = k _ji

This symmetry restriction may also be written in elasticity terms, as follows

where η _ij is the elasticity of demand of good i with respect to the price of good j, E_i is the income elasticity of demand of good i, and s = p_j x_j /y is the proportion of total expenditure devoted to good j.

The symmetrical terms in the above equation are the Allen elasticities of substitution, so that the equation can be written as

where denotes the Allen elasticity of substitution between goods i and j —see Allen (1938) for more details. If , goods i and j are said to be Allen substitutes, in the sense that an increase in the price of good j causes an increased consumption of good i. If, however, σ _ij^a > 0, then the goods are said to be Allen complements, in the sense that an increase in the price of good j causes a decreased consumption of good i.

The Allen elasticity of substitution is the traditional measure and has been employed to measure substitution behavior and structural instability in a variety of contexts. When there are more than two goods, however, the Allen elasticity may be uninformative and the Morishima elasticity of substitution,

is the correct measure of substitution elasticity—see Morishima (1967) and Blackorby and Russell (1989). Notice that measures the net change in the compensated demand for good j when the price of good i changes. Goods will be Morishima complements (substitutes) if an increase in the price of i causes x_i/x_j to decrease (increase).

In the producer context, the producers problem can be formulated as

C(W ₁, W ₂,..., W _n, y ) = min_x W ₁X ₁ + W ₂X ₂+ ... + W _nX _n

subject to

y = f (X₁,X₂, ... ,X_n),

where w_i denotes the price for input i, y denotes output, and x_i denotes input i. Inputs i and j are said to be substitutes if

If the second-order partial derivative is negative, then x_i and x_j are complements.

As in the consumer context, the degree of substi-tutability between any pair of factors in the producer context is also measured using the elasticity of substitution. The Allen elasticity of substitution in the cost minimization framework can be obtained by

where C_i is the partial derivative of the cost function with respect to the price of the i th factor, and C_ij is the partial derivative of C_i with respect to the price of the j th factor.ε_ij =∂ln x_i /∂ ln w_j is the elasticity of the demand for the i the factor (x_i) with respect to the price of the j th factor is the cost share of input j.

Again, as in the consumer context, when there are more than two goods, the Allen elasticity may be uninformative and the Morishima elasticity of substitution, calculated as

is the correct measure of substitution elasticity—see Morishima (1967) and Blackorby and Russell (1989).

The conceptual foundations of the Allen and Morishima elasticities of substitution are different. The Allen elasticity of substitution classifies a pair of inputs as direct substitutes (complements) if an increase in the price of one causes an increase (decrease) in the quantity demanded of the other, whereas the Morishima elasticity of substitution classifies a pair of inputs as direct substitutes (complements) if an increase in the price of one causes the quantity of the other to increase (decrease) relative to the quantity of the input whose price has changed. For this reason, the Morishima elasticity of substitution leans more toward substitutability. To put it differently, if two inputs are direct substitutes according to the Allen elasticity of substitution, theoretically they must be direct substitutes according to the Morishima elasticity of substitution, but if two inputs are direct complements according to the Allen elasticity of substitution, they can be either direct complements or direct substitutes according to the Morishima elasticity of substitution.

BIBLIOGRAPHY

Allen, R. G. D. 1938. Mathematical Analysis for Economists . London: Macmillan.

Blackorby, Charles, and R. Robert Russell. 1989. Will the Real Elasticity of Substitution Please Stand Up? (A Comparison of the Allen/Uzawa and Morishima Elasticities). American Economic Review 79 (4): 882–888.

Hicks, John R. 1956. A Revision of Demand Theory . Oxford: Clarendon Press.

Mas-Colell, Andreu, Michael D. Whinston, and Jerry R. Green. 1995. Microeconomic Theory . Oxford and New York: Oxford University Press.

Morishima. M. 1967. A Few Suggestions on the Theory of Elasticity (in Japanese). Keizai Hyoron ( Economic Review ) 16: 144–150.

Apostolos Serletis