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Problem 4 - Optimal production The Acme Corp produces widgets using labor (L) and capital (K) according to the production, Q = F(L, K) = LI/4K1/4. A widget sells at price P = 16. A unit of labor costs W = 1 and a unit of capital costs R = 4. In the short run, the firm's capital is fixed at K = 1/2 and only labor can be set freely. In the long run, both capital and labor can be chosen freely. There are no fixed costs. 1. In the short run, if Acme Corp wants to produce an output level. Q, solve for its labor demand, LSR(Q). 2. Using your expression for LSR(Q), solve for the firm's short run cost function, CS#(Q). 3. Solve for the short run marginal cost function, ACS?(Q). With this, find Aeme's optimal short run output level OSH 1. In the long run. Acme can change both capital and labor as it pleases. Find the optimal long run capital and labor input choices for a given output level, Q. Denote the optimal long Fun input choices by L(Q) and K (Q). State the long run cost function C(Q) 5. Determine Acme's optimal output choice in the long run, Q. How much labor does Acme demand in the long run. L"? Compare with your answer to question 3. Explain the differencethat the following production function is given: ( - min(-, L.) a-) (6 points) For the above production function, find the elasticity of substitution? Find the Returns to Scale. b.) (8 points) Using the above production function: find the labor demand and capital demand as functions of output (Q). price of labor (w) and price of capital (r). Does the Law of Demand hold for each input? Are these inputs normal or inferior inputs in the production process? Are inputs complements or substitutes? Why? e-) (10 points) Find the cost function for the above production function. Verify the properties of the cost function d-) (8 points) Suppose that a firm wants to produce 36 units of output and w=1, 1=1. Find long run total cost. Suppose now that wage goes up to 2. Find the new long run total cost. Does firm substitute capital for labor? What is the percentage of cost saving relative to the case where firm is not able to substitute? e-) (4 points) Suppose that w-1, 1-1 and a firm has fixed amount of capital K - 108 in short nim (SR) Find the short run total cost, average total cost and marginal cost. What would be the short run total cost of producing 36 units, Compare your answer with long run total cost in part d. How and why are they different?c) Suppose that there is a 40% chance the wage will rise to $15 What are the new short-run profit maximizing inputs, output, and the associated profit?Problem 2 (20 Points) The production of Florida oranges uses two inputs: labor (L) and capital (K). The following production function describes how these inputs are combined to produce bushels of oranges. f(L, K) = 421/2 K a) Determine what kind of returns to scale this production function exhibits and the technical rate of substitution (HINT: labor is the "x" variable - the one that goes on the horizontal axis). b) Suppose the price of a bushel of oranges is $20, the wage is $5, and the rental rate is $8. In the short-run, capital is fixed at 10 units. Find the short-run profit maximizing inputs, output, and the associated profit