Practical applications of the Scarf algorithm usually involve production. Assuming constant returns and the absence of intermediate production, it is possible to solve for equilibrium in such models by iterating through a low dimensional simplex even though the number of produced commodities is very large. This exercise gives a flavor of such applications.
Consider an economy with four commodities and a single consumer. Commodities 1 and 2 are produced subject to constant returns to scale with a Cobb-Douglas production function using commodities 3 and 4 as inputs. Average cost of production for commodities 1 and 2 is given by $p_3^\beta p_4^{1-\beta}$ and $p_3^{1-\beta} p_4^\beta$ respectively where $0<\beta<1$. The consumer has initial endowment $w=(0,0, b, b)$, $b>0$, of commodities 3 and 4 , which she is willing to supply perfectly inelastically. Adopting the normalization $p_3+p_4=1$, we conclude that consumer wealth equals $b$. The consumer's preferences are described by a Cobb-Douglas utility function $x_1^\alpha x_2^{1-\alpha}, 0<\alpha<1$, for commodities 1 and 2 . (Commodities 3 and 4 are not desired by the consumer, which explains why her supply of these input commodities is perfectly inelastic.)
(a) Show that, in solving for equilibrium, it suffices to clear only one of the markets for the "primary factors" (i.e., either commodity 3 or commodity 4) rather than three out of four markets as would be the case in pure exchange. Find the Walrasian equilibrium prices for commodities 3 and 4 as functions of the parameters $\alpha$ and $\beta$.
(b) Letting $\alpha=\beta=1 / 4$, solve for the equilibrium prices for all four commodities.
(c) Focusing solely on the prices of the primary factors, illustrate the Merrill algorithm for this economy using a grid denominator $D=8$ and initial estimate $\left(\pi_3, \pi_4\right)=(7,1)$. Show how the path through the simplex can be represented by a series of matrices with the movement from matrix to matrix governed by the Kuhn replacement method.
(d) If the consumer's utility is an arbitrary quasi-concave function of all four commodities, does it still suffice to clear the markets for commodities 3 and 4 ? If not, what more needs to be assumed for this procedure to suffice?