Expected Consumer's Surplus as an Approximate Welfare Measure
Willig (1976) argues that the change in consumer's surplus is often a good approximation to the willingness to pay for a price change: if the income elasticity of demand is small, or the price change is small, then the percentage error from using consumer's surplus is small. If the price of a good is random, then the change in expected consumer's surplus (ECS) equals a consumer's willingness to pay for a change in its distribution if and only if its demand is independent of income and the consumer is risk neutral. We ask how well the change in ECS approximates the willingness to pay if these conditions fail. We show that the difference between the change in ECS and willingness to pay is of higher order than the L_1 distance between the price distributions if and only if the indirect utility function is additively separable in the price and income. If additively separability fails, then the percentage error from using ECS is unbounded for small distribution changes, and is always nonzero in the limit except for knife-edge cases. If, however, the distribution change is smooth on the space of random variables, and either the initial price is nonrandom or state-contingent payments are possible, then the change in ECS might approximate the willingness to pay well. Unfortunately, this smoothness condition necessarily fails in some important applications of ECS.
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