Consider a monopolist with market inverse demand P(Q) = a-Q. The firm's production function is Q...

  

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Consider a monopolist with market inverse demand P(Q) = a-Q. The firm's production function is Q = F(L, K) = (L + K). The total cost of inputs is thus wL + rK. To maximize its profit, the firm must solve max(a-Q)Q-wL-rK L.K subject to 2 (L+K)=Q. Assume that all parameters are positive. While the optimal choice for the firm corresponds to the labor and capital, the "choice" of interest in the market is the quantity. As such, we would like to convert this into a problem of choosing quantity. To do so, you will solve this problem in two steps. To begin, consider the following intermediate problem min wL+rK L.K subject to (L+ K) = Q. a Verfiry that a solution to the KT conditions for the cost minimization problem is a global minimizer. b Compute the solutions to the KT conditions for the cost minimization problem. (Note that since and K are not defined at negative values, you do not need to consider the implicit constraints of L 0 and K > 0). C Given the value function from the cost minimization problem C(Q, w, r) = wL* (Q, w, r)+ rK*(Q, w,r), the monopolists problem can be converted to max(a - b)Q-C(Q, w,r) subject to Q0. Verify that a solution to the KT conditions is a global maximizer. Consider a monopolist with market inverse demand P(Q) = a-Q. The firm's production function is Q = F(L, K) = (L + K). The total cost of inputs is thus wL + rK. To maximize its profit, the firm must solve max(a-Q)Q-wL-rK L.K subject to 2 (L+K)=Q. Assume that all parameters are positive. While the optimal choice for the firm corresponds to the labor and capital, the "choice" of interest in the market is the quantity. As such, we would like to convert this into a problem of choosing quantity. To do so, you will solve this problem in two steps. To begin, consider the following intermediate problem min wL+rK L.K subject to (L+ K) = Q. a Verfiry that a solution to the KT conditions for the cost minimization problem is a global minimizer. b Compute the solutions to the KT conditions for the cost minimization problem. (Note that since and K are not defined at negative values, you do not need to consider the implicit constraints of L 0 and K > 0). C Given the value function from the cost minimization problem C(Q, w, r) = wL* (Q, w, r)+ rK*(Q, w,r), the monopolists problem can be converted to max(a - b)Q-C(Q, w,r) subject to Q0. Verify that a solution to the KT conditions is a global maximizer.