We study a variational problem arising from a generalization of an economic model introduced by Rochet and Chone ́. In this model a monopolist proposes a set Y of products with pricelist p.Each rational consumer chooses which product to buy by solving a personal minimum problem, taking into account his/her tastes and economic possibilities. The monopolist looks for the optimal price list which minimizes costs, hence maximizes the profit. This leads to a minimum problem for functionals F(p) (the “pessimistic cost expectation”) and G(p) (the “optimistic cost expectation”), which are in turn defined through two nested variational problems. We prove that the minimum of G exists and coincides with the infimum of F. We also provide a variational approximation of G by smooth functionals defined in finite dimensional Euclidean spaces.
The monopolist's problem: existence, relaxation, and approximation
GHISI, MARINA;GOBBINO, MASSIMO
2005-01-01
Abstract
We study a variational problem arising from a generalization of an economic model introduced by Rochet and Chone ́. In this model a monopolist proposes a set Y of products with pricelist p.Each rational consumer chooses which product to buy by solving a personal minimum problem, taking into account his/her tastes and economic possibilities. The monopolist looks for the optimal price list which minimizes costs, hence maximizes the profit. This leads to a minimum problem for functionals F(p) (the “pessimistic cost expectation”) and G(p) (the “optimistic cost expectation”), which are in turn defined through two nested variational problems. We prove that the minimum of G exists and coincides with the infimum of F. We also provide a variational approximation of G by smooth functionals defined in finite dimensional Euclidean spaces.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
