Suppose the firm produces two commodities A and B, where the quantity of the output of commodity A is qA and the quantity of the output of commodity B is qB. Suppose the prices of the two commodities are fixed and the firm is a price taker where the price for commodity A is pA and pB for commodity B.
A. Write the total revenue function of the firm.
B. Suppose the cost of producing commodity A is cA=10qA^2. Please write the marginal cost of commodity A (mcA).
C. Now suppose the cost of producing commodity B is cB=15qB^2+5qAqB. Please write the marginal cost of commodity B (mcB) and find out whether mcB is increasing or diminishing while the firm increases output of commodity A (qA).
d. Write the firm's profit function as a function of qA and qB.
e. Use the first order conditions to solve the optimized outputs of commodity A and commodity B: qA* and qB*.
f. Write the Hessian Matrix H.
g. Find out the optimized outputs you get from question (e) are maximizing the firm's profit or minimizing.
Suppose now the prices of the two commodities are not fixed but given by the demand functions as follows:
pA=840-2qA
pB=900-qA-3qB
h. Write the firm's profit function when the prices for the two commodities are relaxed.
i. Write the marginal profit function of commodity B (mpB).
j. Use the first order conditions to solve the optimized outputs with relaxed prices.
k. Write the new Hessian matrix and the new second order conditions.
l. Calculate the optimized profit of the firm under the relaxed prices.