The Marginal Rate of Substitution
Following the explanation in the text, you might expect that if two goods each exhibit diminishing marginal utility, then the marginal rate of substitution between them will also be diminishing. As we will show below, the marginal rate of substitution is equal to the ratio of the marginal utilities of the two goods, so this hypothesis seems reasonable. Unfortunately, this hypothesis is not generally true if the amount consumed of one good influences the marginal utility of the other good. For example, an increase in your butter consumption may decrease your marginal utility of margarine, or an increase in your coffee consumption may increase your marginal utility of cream. This note investigates the conditions under which the marginal rate of substitution is decreasing, as hypothesized in the text.
Consider the general utility function for two goods X and Y: U = U(X,Y) with = UX
> 0 and = UY > 0. That is, we assume that marginal utility of both goods is positive, but make no assumptions regarding the slope of marginal utility. (We have adopted the conventional notation of UX and UY as the partial derivatives of U with respect to X and Y for convenience.) By definition, the marginal rate of substitution (MRS) is equal to the slope of an indifference curve: MRS = . To find this derivative, we begin by taking the total differential of the utility function. dU = UXdX + UYdY and then set dU = zero to hold utility constant: 0 = UXdX + UYdY. A bit of rearranging gives the desired result: = . That is, the marginal rate of substitution is equal to the ratio of the marginal utilities. For convenience, let us define MRS = = R(X, Y). That is, UX and UY are each functions of X and Y, so their ratio is also a function of X and Y. Call this ratio R(X, Y).
Convexity of the indifference curves requires that their slopes be diminishing in absolute value; since the slope is negative, this means that the slope must be increasing in X. In symbols, we require that > 0. As before, we begin our task by taking the total differential of R(X, Y): dR = RXdX + RYdY where RX and RY are the partial derivatives of R with respect to X and Y. Dividing through by dX, we obtain the result, . Here we make use of the fact that = R(X, Y) = and substitute to find .
We are not finished, however, for we must find RX and RY. First, RX = so we will need to use the division rule: = - . Likewise, RY = = . The terms in parentheses are actually second derivatives of the utility function: = UXX (to continue our previous notational shorthand.) Likewise, = UYY and = UXY. Making these substitutions into , we find that = + . As messy as this looks, it can be simplified somewhat by multiplying and dividing the first term by Uy to get a common denominator and then rearranging: = . Finally!
Recall that the condition we seek is that > 0. The denominator is positive by our assumption of positive marginal utility for each good, so we only require that the numerator be positive. Now diminishing marginal utility of each good is assured if UXX and UYY are negative, in which case the last two terms of are positive. But what of the first term? It is certainly possible that UXY is sufficiently negative (an increase in consumption of X reduces the marginal utility of good Y) so that the first term swamps the positive effect of the last two. In other words, it is possible that the indifference curve is not convex despite the fact that each good has diminishing marginal utility.
Likewise, may be positive even if the last two terms are negative. This possibility requires that the first term be sufficiently positive (UXY > 0, so that increasing consumption of X increases the marginal utility of Y) so that it swamps the effect of the last two terms. That is, the indifference curve can be convex even if both goods exhibit increasing marginal utility! We are left with the uneasy conclusion that diminishing marginal utility is neither a sufficient nor a necessary condition for a diminishing marginal rate of substitution (convex indifference curves) and we must therefore simply state this as an assumption, albeit a reasonable one: The MRS between any pair of goods X and Y is diminishing.