Abstract

In this paper, we study the functional \(\Phi\) that arises in numerous economic applications, in particular, in the monopolist problem. A special feature of these problems is that the domains of such functionals are nonclassical (in our case, increasing convex functions). We use an appropriate minimax theorem to prove the duality relation for \(\Phi\). In particular, an important corollary is obtained stating that the dual functional (defined on a space of measures and known as the “Beckmann functional) attains its minimum. The present approach also provides simpler proofs of some previously known results.

中文翻译：

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论垄断者问题及其双重性

摘要

在本文中，我们研究了在许多经济应用中出现的函数\(\Phi\)，特别是在垄断问题中。这些问题的一个特点是，此类函数的域是非经典的（在我们的例子中，是递增凸函数）。我们使用适当的极小极大定理来证明\(\Phi\)的对偶关系。特别是，获得了一个重要的推论，表明对偶泛函（在测度空间上定义并称为“贝克曼泛函”）达到了最小值。本方法还提供了一些先前已知结果的更简单的证明。