Recent galactic conflict has boosted demand for new spaceships. Consider the firm ACME which uses capital,...

 

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Recent galactic conflict has boosted demand for new spaceships. Consider the firm ACME which uses capital, K, and labour, L, to produce spaceships according to the following production function: f (K, L) = K³L³. Let r and w be the prices of capital and labour respectively. P is the price of new spaceships. The markets for new spaceships, capital and labour are all perfectly competitive. a) Does the production function exhibit increasing-, decreasing- or constant returns to scale? Motivate your answer mathematically. (5 points) In the short run, the level of capital is fixed at K = K. b) Set up the short run profit maximization problem. Calculate the short run optimal level of labour. (5 points) In the long run, ACME can vary both K and L. c) Set up ACME's long run cost minimization problem by using the Lagrange method. (Hint: fix output at the level y.) What are the cost minimizing levels of K and L in the long run? (10 points) d) Derive ACME's long run cost function (hint: use your answer form c). Recent galactic conflict has boosted demand for new spaceships. Consider the firm ACME which uses capital, K, and labour, L, to produce spaceships according to the following production function: f (K, L) = K³L³. Let r and w be the prices of capital and labour respectively. P is the price of new spaceships. The markets for new spaceships, capital and labour are all perfectly competitive. a) Does the production function exhibit increasing-, decreasing- or constant returns to scale? Motivate your answer mathematically. (5 points) In the short run, the level of capital is fixed at K = K. b) Set up the short run profit maximization problem. Calculate the short run optimal level of labour. (5 points) In the long run, ACME can vary both K and L. c) Set up ACME's long run cost minimization problem by using the Lagrange method. (Hint: fix output at the level y.) What are the cost minimizing levels of K and L in the long run? (10 points) d) Derive ACME's long run cost function (hint: use your answer form c).

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