The monopolist’s problem: existence, relaxation, and approximation

• Published in: Calculus of variations and partial differential equations 2005-09, Vol.24 (1), p.111-129
• Main Authors: Ghisi, Marina, Gobbino, Massimo
• Format: Article
• Language: English
• Subjects: Calculus; Calculus of variations; Economic models; Monopolies
• ISSN: 0944-2669; 1432-0835
• DOI: 10.1007/s00526-004-0317-2

We study a variational problem arising from a generalization of an economic model introduced by Rochet and Chone in [5]. In this model a monopolist proposes a set Y of products withprice list p : Y R. 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 prot. 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 dened through two nested variational problems. We prove that the minimum of G exists and coincides with the inmum of F. We also provide a variational approximation of G112 M. Ghisi, M. Gobbino. Finally, for every p P one denes F(p):= X Mp(x)d, G(p):= X mp(x)d. [PUBLICATION ABSTRACT]