Exact Consumer's Surplus and Deadweight Loss

Consumer's surplus is a widely used tool in applied welfare economics. Both economic theorists and cost benefit analysis often use consumer's surplus despite its somewhat dubious reputation. The basic idea is to evaluate the value to a consumer or his willingness to for a change in price of a good from say pricep? to pricep'. Because price changes affect consumer welfare, an evaluation of this effect is often a key input to public policy decisions. Yet consumer's surplus is probably the most controversial of widely used economic concepts. Both Paul Samuelson and Ian Little conclude that the economics profession would be better off without it. It is my feeling of the situation that substantial agreement exists on the correct quantities to be measured: the amount the consumer would pay or would need to be paid to be just as well off after the price change as he was before the price change. The quantities correspond to John Hicks' compensating variation measures. An alternative measure which takes ex post price change utility as the basis of comparison is Hicks' equivalent variation.' The controversy arises in the measurement of these quantities. The usual measurement procedure is to use the area to the left of the Marshallian (market) demand curve between two price levels. Jules Dupuit originated this measure of welfare change, and Alfred Marshall and Hicks derived appropriate conditions for its use. The primary condition for the area to the left of the demand curve to correspond to the compensating variation is to have constant marginal utility of income. Marshall gave this condition, and if it holds, the same quantity will be derived as the area to the left of the compensated (Hicksian) demand curve. This area to the left of the compensated demand curve is exactly what the compensating variation and equivalent variation measure. Thus the constant marginal utility of income is a sufficient condition for Marshallian consumer's surplus to be equal to Hicks' consumer's surplus. In this case Arnold Harberger's plea to use the welfare triangle as one-half times the product of the price change times the quantity change to measure deadweight loss corresponds to the correct theoretical amount of welfare change. In a recent paper, Robert Willig derives bounds for the percentage difference between the correct measure of either the compensating or equivalent variation and the Marshallian measure derived form the market demand curve. His bounds, which depend on the income elasticity of demand for the single good in the region of price change being considered as well as the proportion of the consumer's income spent on the good, demonstrate that the Marshallian consumer's surplus is often a good approximation to Hicks' consumer's surplus. The fact that the proportion of the consumer's income spent matters as well as the income elasticity was first pointed out by Harold Hotelling. Willig contends that the approximation error will be less than the errors involved in estimating the demand curve. Thus he hopes to remove the need for apology that applied economists often need to give to theorists who remark on the inappropriateness of using Marshallian consumer's surplus to measure welfare change. However, in this paper I show that for the case primarily considered by Willig of a single price change, which is also the situation in which consumer's surplus is often used in applied work, no approximation is necessary. *Professor of economics, Massachusetts Institute of Technology, and research associate, National Bureau of Economic Research. I would like to thank Peter Diamond, Erwin Diewert, Daniel McFadden, Robert Merton, Robert Solow, Hal Varian, Joel Yellin, and the referees for help and comments. Research support from the National Science Foundation is acknowledged. 'The reason that we still have two, rather than one, of Samuelson's six measures of consumer's surplus arises from an index number problem of the correct basis for the welfare comparison. I will give both measures but plan to concentrate on the compensating variation.