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

Malakhov, Sergey (2020): Proof of the invisible hand: the optimal consumer allocation of time under price dispersion.

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The analysis of the static 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 of 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 rule cannot work without the optimal consumer labor-search trade-off. It means that there is some inner market mechanism, which leads the producer along the production possibility frontier to make an offer that unintendently optimizes the consumer’s allocation of time. So to the consumer, it feels under the unit elasticity rule like the producer is more about being at the right place and the right time with a price, which summurizes optimally consumer’s efforts on purchase and in this way maximizes his consumption-leisure utility function. Generally, the phenomenon of the Invisible hand describes the unintended social benefits of individual self-interested actions. But used more narrowly, it refers to the quote, where Adam Smith speaks about the phenomenon, which leads self-interested producers to promote the public interest. And the unit elasticity rule provides the proof of the Invisible hand in this narrow meaning.

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