Mec Homework
MEC 290 Problem Set #5 –due 4/1/2013
1
Problems
1. Consider the following production function, f (L; K) = p L+ 1p K, 3
where L is the quantity of labor, K is the quantity of capital, the wage rate is w; and the capital rental rate is r. (a) Compute the M RT SLK : Assume K = 81 and L = 25: Display the M RT SLK on the implied isoquant. (b) Does this production function display, decreasing, constant or increasing returns to scale? (c) (Short run) Suppose that the …rm has a binding agreement to (provided the …rm operates at all) employ exactly 64 people. p i. If the …rm is to produce some arbitrary level of output, Q 64, how much capital must it use? ii. What are the …rm’ total costs (an expression involving Q; w and r), again assuming s p Q 64? iii. Assuming w = 5 and r = 3, use Excel to calculate the …rm’ total costs, and plot s p 64: costs as a function of Q; for Q A. Do part (iii) again assuming r increases by $1? Do the plot on the same graph. B. Do part (iii) again assuming w increases by $1? Do the plot on the same graph. (d) (Long run) Now assume the …rm can vary both labor and capital. i. If the …rm has decided to produce some arbitrary level of output Q, what conditions must the cost-minimizing inputs satisfy? ii. If the …rm has decided to produce some arbitrary level of output Q, what is the the cost-minimizing pair of inputs? (The cost minimizing input pair will involve Q; w and r:) 1
iii. What are the …rm’ total costs (an expression involving Q; w and r)? Assuming s w = 5 and r = 3, calculate the …rm’ total costs, and plot cost as a function of Q: s Plot the total costs from part 1(c) on the same graph. 2. Read the article “UN Report: Let’ turn foul water from mass killer into global treasure.”The s article describes how investment in large scale water treatment plants will be needed. You may assume the only cost of water treatment is the up front investment, i.e., operating costs are minimal and can be ignored. A treatment plant is
1
Problems
1. Consider the following production function, f (L; K) = p L+ 1p K, 3
where L is the quantity of labor, K is the quantity of capital, the wage rate is w; and the capital rental rate is r. (a) Compute the M RT SLK : Assume K = 81 and L = 25: Display the M RT SLK on the implied isoquant. (b) Does this production function display, decreasing, constant or increasing returns to scale? (c) (Short run) Suppose that the …rm has a binding agreement to (provided the …rm operates at all) employ exactly 64 people. p i. If the …rm is to produce some arbitrary level of output, Q 64, how much capital must it use? ii. What are the …rm’ total costs (an expression involving Q; w and r), again assuming s p Q 64? iii. Assuming w = 5 and r = 3, use Excel to calculate the …rm’ total costs, and plot s p 64: costs as a function of Q; for Q A. Do part (iii) again assuming r increases by $1? Do the plot on the same graph. B. Do part (iii) again assuming w increases by $1? Do the plot on the same graph. (d) (Long run) Now assume the …rm can vary both labor and capital. i. If the …rm has decided to produce some arbitrary level of output Q, what conditions must the cost-minimizing inputs satisfy? ii. If the …rm has decided to produce some arbitrary level of output Q, what is the the cost-minimizing pair of inputs? (The cost minimizing input pair will involve Q; w and r:) 1
iii. What are the …rm’ total costs (an expression involving Q; w and r)? Assuming s w = 5 and r = 3, calculate the …rm’ total costs, and plot cost as a function of Q: s Plot the total costs from part 1(c) on the same graph. 2. Read the article “UN Report: Let’ turn foul water from mass killer into global treasure.”The s article describes how investment in large scale water treatment plants will be needed. You may assume the only cost of water treatment is the up front investment, i.e., operating costs are minimal and can be ignored. A treatment plant is