The Froman Sausage Company uses capital (K) and labor (L) to produce output (y) ac- cording...

 

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The Froman Sausage Company uses capital (K) and labor (L) to produce output (y) ac- cording to the production function y(L, K) = L²K. Let r and w be the price of capital and labor, respectively, and let p be the price of the output good. Assume that the markets for capital, labor, and output are perfectly competitive. (a) Formulate an expression for the Froman Sausage Company's marginal rate of technical substitution between labor and capital. Evaluate the rate when L = 2 and K = 4, and give a one-sentence interpretation of this number. (b) Plot the firm's iso-cost line and iso-quant curve with L on the horizontal axis and K on the vertical axis. Label axes, lines/curves, and the slope of the iso-cost line. (c) Does the firm's production exhibit increasing, decreasing, or constant returns to scale? Justify your answer. (d) In the short run, the level of capital is fixed at K = K. Formulate an expression for the short-run optimal level of labor to produce y units of output. (e) In the long run, the Froman Sausage Company can vary both K and L. Set up the company's long-run cost-minimization problem using the Lagrangian and list the first- order conditions that characterize the optimal combination of inputs. (f) Formulate expressions for the firm's long-run cost-minimizing levels of labor and capital as a function of the output level y. Derive the Froman Sausage Company's long-run cost curve, c(y). (g) Is there a profit-maximizing level of output, y? Explain. (Hint: think about returns to scale) The Froman Sausage Company uses capital (K) and labor (L) to produce output (y) ac- cording to the production function y(L, K) = L²K. Let r and w be the price of capital and labor, respectively, and let p be the price of the output good. Assume that the markets for capital, labor, and output are perfectly competitive. (a) Formulate an expression for the Froman Sausage Company's marginal rate of technical substitution between labor and capital. Evaluate the rate when L = 2 and K = 4, and give a one-sentence interpretation of this number. (b) Plot the firm's iso-cost line and iso-quant curve with L on the horizontal axis and K on the vertical axis. Label axes, lines/curves, and the slope of the iso-cost line. (c) Does the firm's production exhibit increasing, decreasing, or constant returns to scale? Justify your answer. (d) In the short run, the level of capital is fixed at K = K. Formulate an expression for the short-run optimal level of labor to produce y units of output. (e) In the long run, the Froman Sausage Company can vary both K and L. Set up the company's long-run cost-minimization problem using the Lagrangian and list the first- order conditions that characterize the optimal combination of inputs. (f) Formulate expressions for the firm's long-run cost-minimizing levels of labor and capital as a function of the output level y. Derive the Froman Sausage Company's long-run cost curve, c(y). (g) Is there a profit-maximizing level of output, y? Explain. (Hint: think about returns to scale)

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a Marginal Rate of Technical Substitution MRTS The MRTS is the rate at which the firm can substitute capital for labor while maintaining the same outp... View the full answer