Market equilibrium under piecewise Leontief concave utilities

Leontief function is one of the most widely used function in economic modeling, for both production and preferences. However it lacks the desirable property of diminishing returns. In this paper, we consider piecewise Leontief concave (p-Leontief) utility function which consists of a set of Leontief-type segments with decreasing returns and upper limits on the utility. Leontief is a special case when there is exactly one segment with no upper limit. We show that computing an equilibrium in a Fisher market with p-Leontief utilities, even with two segments, is PPAD-hard via a reduction from Arrow-Debreu market with Leontief utilities. However, under a special case when coefficients on segments are uniformly scaled versions of each other, we show that all equilibria can be computed in polynomial time. This also gives a non-trivial class of Arrow-Debreu Leontief markets solvable in polynomial time. Further, we extend the results of [22,5] for Leontief to p-Leontief utilities. We show that equilibria in case of pairing economy with pLeontief utilities are rational and we give an algorithm to find one using the Lemke-Howson scheme.