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Homework answers / question archive / A firm has Cobb-Douglas production function y = KL
A firm has Cobb-Douglas production function y = KL. Input prices are as follows: rental rate on capital r = 4, wage is w = 1. a) Suppose in SR capital is fixed at 5 units, find short run TC function. b) Use Lagrangean to derive firm's LR conditional factor demands for capital and labor (You can substitute input prices into the total cost to make calculations easier.) From FOC you should obtain that in cost-minimizing combination of K and L TRS= input price ratio, MPx = , or = 5; After substituting into the constraint get K* = 19, L = 2/9. Reality check: K and L enter production function symmet- rically, they are relatively equally productive, capital is more expensive, given the inpit prices firm optimally demands less capital. TC function: TC=wL+rk, substitute input prices and factor demands: TC =240 + 4+ 5 = 4/9 c) What combination of inputs minimizes total cost of producing 100 units of output? What is the total cost of producing 100 units of output? (Find how much labor and capital the firm needs: L=20 and K=5, then find how much it will cost to buy it. You can calculate TC = 1.20+4.5 = 40, or plug 100 into the TC function to get the same number.) d) Use diagram to show the results from part (b): plot an isoquant for y = 100, show the cost-minimizing combination of inputs and respective isocost (be careful with the slopes). d) Use your diagram to show the impact of wage going up to 2: what is the slope of the new isocost? Will firm demand more or less labor? capital?
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