We study the problem of computing equilibrium prices in a Fisher market with linear utilities and linear single-constraint production units. This setting naturally appears in ad pricing where the sum of the lengths of the displayed ads is constrained not to exceed the available ad space. There are three approaches to solve market equilibrium problems: convex programming, auction-based algorithms, and primal-dual. Jain, Vazirani, and Ye recently proposed a solution using convex programming for the problem with an arbitrary number of production constraints. A recent paper by Kapoor, Mehta, and Vazirani proposes an auction-based solution. No primal-dual algorithm is proposed for this problem.
In this paper we propose a simple reduction from this problem to the classical Fisher setting with linear utilities and without any production units. Our reduction not only imports the primal-dual algorithm of Devanur et al. to the single-constraint production setting, but also: i) imports other simple algorithms, like the auction-based algorithm of Garg and Kapoor, thereby providing a simple insight behind the recent sophisticated algorithm of Kapoor, Mehta, and Vazirani, and ii) imports all the nice properties of the Fisher setting, for example, the existence of an equilibrium in rational numbers, and the uniqueness of the utilities of the agents at the equilibrium.