B. J. Brown
Professor of Economics
Pace University
THE SUBSTITUTION EFFECT AND INDIFFERENCE CURVE CURVATURE
Intuitively, one thinks of substitutes as goods which can serve the same purpose, and for which, therefore, changes in quantity demanded will be relatively large. Also, we have learned that substitutes= indifference curves have relatively little curvature, that indifference curve curvature is associated with the substitution effect, and that therefore the law of demand is likely to hold true for substitutes because of the large aubstitution effect. One link, then, in this sketch of our view of substitutes is the idea that indifference curve curvature is associated with the magnitude of the own-price substitution effect: the less curved the indifference curve, the greater the own-price substitution-effect change in quantity demanded. In this paper I show that the curvature of the indifference curve plays the same role in determining the size of the income effect as it does in determining the size of the substitution effect, i.e., that it is not true, as has been believed, that indifference curve curvature finds its unique expression in the size of the substitution effect. The implication is that the law of demand is not more likely to hold true for a good with a flatter indifference curve than for a good with greater indifference curve curvature.
The association between indifference curve curvature and the size of the substitution effect was established by Hicks. In his part of the "Reconsideration" (1934), his analysis of the effect of a change in own-price centers on the diagram (1934, p.66) in Figure 1, and he writes (p. 66)
...the extent to which R will be pushed to the right of Q ... will depend upon the curvature of the indifference curves, that is to say, upon the elasticity of substitution. The greater the elasticity of substitution -- the greater will be the divergence between the [income-consumption curve] and the [price-consumption curve].
The increase in demand for a commodity X, which results from a fall in its price, depends therefore partly upon the income-elasticity of demand for X, and partly upon the elasticity of substitution between X and Y. We can in fact look upon the increase in demand as consisting of two parts, one of which is due to the increase in real income which a fall in the price of X entails, the other to the opportunity of substituting X for other goods which results from the fall in the relative price of X.
The same analysis, with the same diagram, appears in Value and Capital (1946, p. 31). Here Hicks christens the movement along the income-consumption curve Athe Income Effect,@ and the movement along the indifference curve Athe Substitution Effect@ (p. 32). Although he no longer mentions the elasticity of substitution, the diagram itself shows the magnitude of the substitution effect depending on the degree of curvature of the indifference curve.
In demonstratinge that indifference curve curvature relates similarly to both income and substitution effects, the first step involves getting the own-price Slutsky equation in a new way. Analysis of this version of the Slutsky equation yields the result just stated. The result is then used to show that in determining whether a good is Giffen, neither indifference curve curvature nor subjective substitutability have any influence on the size of the substitution effect. I also point to the identical implications in equations in Allen=s part of the ARe consideration@ (1934). Following Hicks, initially I discuss the two-goods case. Later I discuss extension to the n-goods case.