Proof of the invisible hand: the optimal consumer allocation of time under price dispersion

The analysis of the labor-search-leisure model discovers some special optimality properties of the consumer choice under price dispersion. If the consumer is realistic about what he can buy with his efforts, the unit elasticity of his labor and search efforts with respect to consumption reproduces his initial consumption-leisure trade-off; it results in the equality of the marginal loss on search with its marginal benefit and the optimal consumption-leisure choice for any quantity purchased. The inequality of the marginal values of search, described by the satisficing approach, doesn’t take place, because this inequality represents the reproduction of the corner prior expectations by the same unit elasticity rule. If the consumer challenges the initial corner solution and start to work and to search, the unit elasticity rule reproduces high prior expectations and the following outcome produces the disappointment. As a result, the consumer either buys satisfactorily as well as optimally under the equality of the marginal values of search, or he quits the market without purchase. In this way the unit elasticity rule provides a powerful illustration of the consistency of consumer’s preferences. The consumer avoids the computational complexity of the marginal analysis because the unit elasticity rule automatically reproduces his realistic prior expectations how much leisure should be given up for consumption. However, the unit elasticity rule provides the optimality of the total efforts and tells nothing about their distribution between labor and search. But this distribution also is optimal, because it equalizes the marginal values of search with respect to the individual propensity to search. While the propensity to search gets the strong mathematical description ex-post with regard to the given price dispersion, it doesn’t exist ex-ante, on the level of prior expectations, when the consumer decides to give up some leisure time and starts to work and to search, but he hasn’t the reliable information on the price dispersion. It looks like the consumer makes the intuitive decision, or there is some hidden market mechanism, which provides the optimal allocation of his time between labor, search, and leisure, and confirms in this way the hypothesis of the invisible hand.