Consider a production economy with three commodities: the first two are not produced while the third is produced using commodities one and two as inputs. Production is performed by an arbitrary number of identical firms each with cost function
$$
c\left(p_1, p_2, y_3\right)=\sqrt{p_1 \overline{p_2}}\left(1+y_3^2\right)
$$
where $p_1$ and $p_2$ are the prices of commodities one and two and $y_3$ the output of commodity three. Assume that consumers have strictly positive initial endowments of commodities one and two and that utility functions $u_i\left(x_i\right)$ are strictly monotonic and continuous.
(a) Compute the supply correspondence for a representative firm and demonstrate that for some prices it fails to be convex-valued.
(b) Describe the effect of "convexifying" this economy on the cost function of each of the firms.
(c) Assuming that an equilibrium exists for the convexified economy, apply the Shapley-Folkman Theorem to prove the existence of an approximate equilibrium for this economy: i.e., a price vector $p$ such that all markets clear, every consumer receives a commodity bundle which maximizes utility subject to his or her budget constraint, and the production of all but three firms is both feasible and profit maximizing.
(d) Use the Shapley-Folkman Theorem to prove that the approximate equilibrium can allow all but one firm a profit maximizing, feasible production vector.