Fish and Coconuts,
Let's decentralize the model from exercise 5.2. Now, let's think of "Chuck" as two firms and one consumer. Each firm (i.e., the "fish firm" that produces good 1 and the "coconut firm" that produces good 2) has the production function S(L) = VL
Both firms take the price of their good as given. The labor supply is L = 100, supplied inelastically. For simplicity, let's fix p2 = 1, and let pı = p; so throughout this problem, we'll be interested in the equilibrium price ratio p = pı/P2. Assuming the two firms take the price of their good as given, find their equilibrium profit-maximizing combinations of outputs, Y1 and Y2, as a function of p. You may do this in one of two ways:
Easy way: Use the fact that the firms will produce at the point along the PPF that sets MRT = p. Remember that you found the equation of the PPF and an expression for the MRT in exercise 5.2!
Hard/thorough way: Find the firms' individual supply functions, Si(p, w) and S2(w) (since P2 = 1). Find their labor demands, and set labor demand equal to the total labor supply.
Solve for the equilibrium wage rate as a function of p — that is, w* (p). Finally, plug w* (p) back into the supply functions to get Y* (P) = S(P, w*(p)) and Y; (p) = S2(w*(p)).