Final answer:
To maximize profit, Firm S should produce 6 units of steel, and Fishery F should produce an unknown quantity of fish, which depends on the value of variable x. The equations MRs = MCS and MRf = MCF are used to determine the optimal quantities of steel and fish produced, where MRs represents the marginal revenue from selling steel and MCS represents the marginal cost of producing steel, while MRf represents the marginal revenue from selling fish and MCF represents the marginal cost of producing fish.
Step-by-step explanation:
To maximize their profits, Firm S and Fishery F need to produce steel and fish in quantities where their marginal costs equal the prices of steel and fish, respectively. Firm S's cost function is given by cs(s; x) = s^2 + (x - 4)^2, while Fishery F's cost function is cf(f; x) = f^2 + xf. The price of steel is $12 and the price of fish is $10. To find the quantities of steel and fish that maximize each firm's profit, we need to equate their marginal costs with the respective prices: For Firm S, MRs = MCS, where MRs is the marginal revenue from selling steel and MCS is the marginal cost of producing steel. Since the price of steel is $12, MRs = 12. Differentiating the cost function of steel with respect to s gives us MCS = 2s. Equating MRs and MCS, we get 12 = 2s, which implies s = 6.
Similarly, for Fishery F, MRf = MCF, where MRf is the marginal revenue from selling fish and MCF is the marginal cost of producing fish. Since the price of fish is $10, MRf = 10. Differentiating the cost function of fish with respect to f gives us MCF = 2f + x. Equating MRf and MCF, we get 10 = 2f + x. To solve for the quantities of steel and fish produced, we need to solve the two equations we obtained. However, since we don't have a specific value for the variable x, we can't determine the exact quantities produced without further information.