3. A cost-minimizing firm has a production function of the form: 1 2 1 3 3 3 q = F(A,L,K) = ALK Where L is labor, K is capital, and A is what macroeconomists call total factor productivity or, for our purposes, technology. The firm is a price taker in input markets (no monopsony power) and pays w for each unit of labor hired and r for each unit of capital hired. Assume A is a constant and that the only costs to the firm come from L and K. A and q. How do L*(.) and K*(.) respond to a doubling of all input prices? Explain. a. Derive the conditional input demands L*(.) and K*(.) assuming labor and capital are both variable. These demands will depend on w,r, i. The firm's conditional input demand for labor, L* ( - ) is: O A. 3 2 NI L* (w,r, A,9) = q * (1/A)* (r/w) B. NI 2 L* (w,r, A, 9) = q *(1/A)* (r/w) O c. NI W 2 L* (w,r, A,9) = q *(1/A)*(r/w) OD. NI W 1 L* (W,r, A, 9) = q *(17A)*(r/w) ii. The firm's conditional input demand for capital, K* (. ) is: OA. K* (w,r, A, 9) = q *(1/A)* (w/r) OB. 3 NIC NIC K* (w,r, A, 9) = q *(1/A)* (w/r) O c. 2 K* (w,r, A, 9) = q *(1/A)* (w/r) OD. K* (w,r, A, 9) = q *(1/A)* (w/r) iii. Doubling all (both) input prices (w & r) will L* (.) and K* (.). A. Decrease ; increase O B. Decrease ; decrease O C. Increase ; decrease OD. Not change; not change b. Calculate the wage elasticity of labor demand. Sketch the labor demand curve. i. The wage elasticity of labor demand is: O A. -2/3 OB. 1 2 C. -1/2 od OD. -3/2 ii. This labor demand curve is best represented by Diagram A. D B. A OOOO C. C D. B c. Derive an expression for TC*(w,r, A, 9). How does TC*(.) respond to a doubling of all input prices? Explain. i. This firm's total cost function, TC * ( . ) is: O A. 3 TC* (w,r, A, 9) = 2* q *(1/A)* (w*r) B. 2 3 TC* (w,r, A, 9) = 2 * 97 * (1/A)* (w*r) OC. 3 2 1 3 TC* (W,r, A, 9) = 2 * q *(1/A)* (w*r) OD. 1 2 3 TC* (w,r, A, 9) = 2 * 97 * (1/A)* (w*r) ii. Doubling all (both) input prices (w & r) will TC* (.) A. Less than double B. More than double C. Double D. Not change The firm's research and development department discovers a breakthrough technology which effectively increases A from 1 to 8. If w=r= $64 / unit both before and after the technological advance, what will happen to the cost minimizing amounts of capital and labor? What will happen to the TC*(.) function? Sketch your TC*(.) function both before and after the technological change. What kind of technological change [neutral, labor augmenting, capital augmenting] has occurred? Explain with reference to the MRTS. i. This increase in A and its effect on the TC* ( . ) function is best represented by Diagram A. C B. A C D D. B because the MRTS (marginal rate of technical substitution) has ii. This technological change is A. Capital- augmenting ; decreased B. Labor-augmenting ; increased C. Neutral ; not changed D. Neutral ; increased 3. A cost-minimizing firm has a production function of the form: 1 2 1 3 3 3 q = F(A,L,K) = ALK Where L is labor, K is capital, and A is what macroeconomists call total factor productivity or, for our purposes, technology. The firm is a price taker in input markets (no monopsony power) and pays w for each unit of labor hired and r for each unit of capital hired. Assume A is a constant and that the only costs to the firm come from L and K. A and q. How do L*(.) and K*(.) respond to a doubling of all input prices? Explain. a. Derive the conditional input demands L*(.) and K*(.) assuming labor and capital are both variable. These demands will depend on w,r, i. The firm's conditional input demand for labor, L* ( - ) is: O A. 3 2 NI L* (w,r, A,9) = q * (1/A)* (r/w) B. NI 2 L* (w,r, A, 9) = q *(1/A)* (r/w) O c. NI W 2 L* (w,r, A,9) = q *(1/A)*(r/w) OD. NI W 1 L* (W,r, A, 9) = q *(17A)*(r/w) ii. The firm's conditional input demand for capital, K* (. ) is: OA. K* (w,r, A, 9) = q *(1/A)* (w/r) OB. 3 NIC NIC K* (w,r, A, 9) = q *(1/A)* (w/r) O c. 2 K* (w,r, A, 9) = q *(1/A)* (w/r) OD. K* (w,r, A, 9) = q *(1/A)* (w/r) iii. Doubling all (both) input prices (w & r) will L* (.) and K* (.). A. Decrease ; increase O B. Decrease ; decrease O C. Increase ; decrease OD. Not change; not change b. Calculate the wage elasticity of labor demand. Sketch the labor demand curve. i. The wage elasticity of labor demand is: O A. -2/3 OB. 1 2 C. -1/2 od OD. -3/2 ii. This labor demand curve is best represented by Diagram A. D B. A OOOO C. C D. B c. Derive an expression for TC*(w,r, A, 9). How does TC*(.) respond to a doubling of all input prices? Explain. i. This firm's total cost function, TC * ( . ) is: O A. 3 TC* (w,r, A, 9) = 2* q *(1/A)* (w*r) B. 2 3 TC* (w,r, A, 9) = 2 * 97 * (1/A)* (w*r) OC. 3 2 1 3 TC* (W,r, A, 9) = 2 * q *(1/A)* (w*r) OD. 1 2 3 TC* (w,r, A, 9) = 2 * 97 * (1/A)* (w*r) ii. Doubling all (both) input prices (w & r) will TC* (.) A. Less than double B. More than double C. Double D. Not change The firm's research and development department discovers a breakthrough technology which effectively increases A from 1 to 8. If w=r= $64 / unit both before and after the technological advance, what will happen to the cost minimizing amounts of capital and labor? What will happen to the TC*(.) function? Sketch your TC*(.) function both before and after the technological change. What kind of technological change [neutral, labor augmenting, capital augmenting] has occurred? Explain with reference to the MRTS. i. This increase in A and its effect on the TC* ( . ) function is best represented by Diagram A. C B. A C D D. B because the MRTS (marginal rate of technical substitution) has ii. This technological change is A. Capital- augmenting ; decreased B. Labor-augmenting ; increased C. Neutral ; not changed D. Neutral ; increased