Question;Problem #1Consider the following production functionQ = [L(2? L)]2for L? [0, 2].i) Find the average product of laborii) Find the marginal product of laboriii) For what value of L is the marginal product of labor the same as the average product?Problem #2Which of the following production functions exhibits constant returns to scale?i) Q = F (K, L) = 2 + K? 3L+ii) Q = F (K, L) = K + KL1iii) Q = F (x1, x2,..., xN) = (x? + x? +... + x?)?12Niv) Q = F (K, L) = K 2 L2Problem #3Consider the following cost functionC(Q) = 200? 20Q + Q2.i) Find the average costsii) Find the marginal costsiii) Plot both the marginal and the average costs. Be precise in the neighborhood of the lowest value of the average costsProblem #4Let the production function be given by12Q = F (K, L) = AK 3 L 3.Assume that the rental price of capital is r and the prevailing wage is w.1i) Find the optimal amounts of labor and capital hired by a cost-minimizingrm that needs to produce Q0 units of output.ii) How does the optimal ratio of K change with A? Would you interpret L increases in A as capital saving technological progress?iii) Find the total costs of production as a function of r, w, and Q0.iv) Imagine that A doubles and at the same time the factor prices r and wdouble what is the impact on the total cost of production of Q0 units of output?Problem #5Consider the following production function11Q = AK 2 L 2.?i) Imagine that capital is xed at K. Assume that factor prices are r and w.What is the optimal amount of labor hired by a cost minimizing rm interestedin producing Q0 units of output?ii) How many units of capital would this rm hire if it was free to select itscapital optimally??iii) For what level of output, Q0, is the original value of K optimal?Problem #6Consider the following production function1Q = F (K, L) = (K? + L?)?.Assume that the price of output is equal to p.i) Assume that the rm employs K units of capital and L units of labor.Assume that the factor rental prices are r and w respectively. What is the levelof prot earned?ii) Find the marginal product of laboriii) Find the marginal product of capitaliv) State the conditions that describe the optimal amounts of K and Lv) What is the maximum amount of prot earned?
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