  Although production is an integral part of the Arrow-Debreu market model,
most of the work in theoretical computer science has so far concentrated on
markets without production, i.e., the exchange economy. This paper takes a
significant step towards understanding computational aspects of markets with
production.
  We first define the notion of separable, piecewise-linear concave (SPLC)
production by analogy with SPLC utility functions. We then obtain a linear
complementarity problem (LCP) formulation that captures exactly the set of
equilibria for Arrow-Debreu markets with SPLC utilities and SPLC production,
and we give a complementary pivot algorithm for finding an equilibrium. This
settles a question asked by Eaves in 1975 of extending his complementary pivot
algorithm to markets with production.
  Since this is a path-following algorithm, we obtain a proof of membership of
this problem in PPAD, using Todd, 1976. We also obtain an elementary proof of
existence of equilibrium (i.e., without using a fixed point theorem),
rationality, and oddness of the number of equilibria. We further give a proof
of PPAD-hardness for this problem and also for its restriction to markets with
linear utilities and SPLC production. Experiments show that our algorithm runs
fast on randomly chosen examples, and unlike previous approaches, it does not
suffer from issues of numerical instability. Additionally, it is strongly
polynomial when the number of goods or the number of agents and firms is
constant. This extends the result of Devanur and Kannan (2008) to markets with
production.
  Finally, we show that an LCP-based approach cannot be extended to PLC
(non-separable) production, by constructing an example which has only
irrational equilibria.
