A competitive tech firm has a production function f(x1, x2) = 12 where x2 is the...

 

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A competitive tech firm has a production function f(x1, x2) = 12 where x2 is the number of machines and x1 is number of workers employed. The workers' wage is w = 4, the machine price is w216 and the output price is p = 16. Suppose that in the short run the firm can only vary the amount of workers it employs but not the number of machines and that the latter is fixed at 24 in the short run. (a) Find the firm's short run conditional factor demand for workers if the firm wants to produce y units of output. What is the firm's short run cost function for producing output y? (b) What are the firm's fixed costs, average variable costs, average costs and marginal costs of producing output y? Sketch the SRMC, AVC and SRAC curves on a graph. (c) What is the firm's short run supply curve? What is the profit-maximizing amount of output ySR that the firm will produce in the short run? At this output level how much profit or loss does the firm make? Suppose now that the firm is in the long run and can vary both its factors of production. (d) What are the firm's long run conditional factor demands for producing y units of output? What is its long run cost function? What is its long run supply curve? (e) Use your results from parts (a) and (d) to find and the short-run and long-run average costs for any y. Show that SRAC(y) > LRAC (y) for any y. Explain why. (f) Prove that the short-run and long-run average costs are equal and tangent to each other at the output level y which solves (y) = 4 where x (y) is the long-run conditional demand for machines from part (d). Explain why this is the case (a graph may help). (g) Assuming that the input and output prices stay at their given values, how much would the firm produce in the long run? A competitive tech firm has a production function f(x1, x2) = 12 where x2 is the number of machines and x1 is number of workers employed. The workers' wage is w = 4, the machine price is w216 and the output price is p = 16. Suppose that in the short run the firm can only vary the amount of workers it employs but not the number of machines and that the latter is fixed at 24 in the short run. (a) Find the firm's short run conditional factor demand for workers if the firm wants to produce y units of output. What is the firm's short run cost function for producing output y? (b) What are the firm's fixed costs, average variable costs, average costs and marginal costs of producing output y? Sketch the SRMC, AVC and SRAC curves on a graph. (c) What is the firm's short run supply curve? What is the profit-maximizing amount of output ySR that the firm will produce in the short run? At this output level how much profit or loss does the firm make? Suppose now that the firm is in the long run and can vary both its factors of production. (d) What are the firm's long run conditional factor demands for producing y units of output? What is its long run cost function? What is its long run supply curve? (e) Use your results from parts (a) and (d) to find and the short-run and long-run average costs for any y. Show that SRAC(y) > LRAC (y) for any y. Explain why. (f) Prove that the short-run and long-run average costs are equal and tangent to each other at the output level y which solves (y) = 4 where x (y) is the long-run conditional demand for machines from part (d). Explain why this is the case (a graph may help). (g) Assuming that the input and output prices stay at their given values, how much would the firm produce in the long run?