Further evidence of positively sloping marginal revenue.

I. Introduction

The importance of a non-monotonic marginal revenue function has been well documented by Beckman and Smith [1], who, in a recent article in this journal, investigate the behavior of marginal revenue derived from a CES utility function. For the standard CES utility function, the authors prove that the marginal revenue function is strictly positive, downward sloping, and asymptotic to the axes when the elasticity of substitution is greater than unity. Introduction of a subsistence requirement into the standard CES utility function, however, results in a marginal revenue function that is non-monotonic. These results are obtained in the two good case, for which the authors derive the marginal revenue function from the consumer's inverse demand function for good 1. The inverse demand function for good 1 has the general form [p.sub.1]/y = g([x.sub.1], [x.sub.2]). The price of good 1 divided by income is a function of the quantities of the two goods. This requires us to hold the quantity of good 2 constant when moving along the demand curve for good 1.

However, in many applications it may be more plausible to consider the marginal revenue function derived from the Marshallian or ordinary demand function. The Marshallian demand function for good 1 has the general form [x.sub.1] = f([p.sub.1], [p.sub.2], y). Movements along the Marshallian demand curve for good 1 require us to hold the price of good 2 and income constant. This is a more reasonable case when prices are exogenous to the consumer. In this paper we examine the behavior of the marginal revenue function derived from the Marshallian demand function which corresponds to a CES utility function with minimum subsistence requirements. We are able to demonstrate that the resulting marginal revenue function is also non-monotonic.

Before proceeding to the next section we should acknowledge that there are special cases when the marginal revenue function derived from the Marshallian demand function is the same as that derived from the inverse demand function. In the two good case, if the Marshallian (inverse) demand for good 1 does not depend on the price (quantity) of good 2 then one will get the same marginal revenue function from either type of demand. The Cobb-Douglas utility function is one such example. The Marshallian demand for good 1 has the form [x.sub.1] = [[Alpha].sub.1] [center dot] y/[p.sub.1] and the inverse demand has the form [p.sub.1] = [[Alpha].sub.1] [center dot] y/[x.sub.1]. Both give the same revenue function, [R.sub.1] = [[Alpha].sub.1] [center dot] y, and the same marginal revenue function, M[R.sub.1] = 0. However, for the CES demand functions considered here, the Marshallian (inverse) demand for good 1 does depend on the price (quantity) of good 2. The resulting marginal revenue function will depend on which type of demand is used in its derivation.