This is a question from my intermediate micro economics text book. Any help is very appreciated!
Given Info:
Company ST (a company which offers custom travel-planning services) is a profit-maximizing firm whose technology is described by the production function Q = F(L,K) = [Min(L,K)]^0.5. L is labor and K is capital. ST is a price-taker in the input markets, paying w for each unit of labor and r for each unit of capital. ST faces no costs other than those generated by labor and capital; ST can sell each unit of its output at the market price P.
Part A:
Assuming that ST can freely choose both labor (L) and capital (K), derive expressions for its profit-maximizing input demands, K*(r,w,P) and L*(r,w,P), its output supply function Q*(r,w,P), and its (maximized) profit function, Π*(r,w,P).
Illustrate ST’s profit maximizing choice in an isoquant – isocost diagram.
Part B:
If increased competition in the travel services industry causes P to fall, what will happen to the quantity of travel services sold by ST? To the number of workers hired by ST?
Part C:
If there is an increase in the market wage rate (w) but r and P remain the same, what will happen to the number of workers ST will hire? The number of units of capital? The number of units of output it will supply? Its profits? Explain why these results are different from the cost-minimization results.
My attempt at a solution:
Profit = Total Revenue - Total Cost
Total Revenue = PQ Total Cost = wL + rK
Maximize Profit (L,K)
dΠ/dL ==> 0 = P* [dF(L,K)]/dL -w
Marginal Product of Labor = w/p
dΠ/dK ==> 0 = P* [dF(L,K)]/dK -r
Marginal Product of Capital = r/p
Marginal Product of Labor = [dF(L,K)]/dL = w/p
MPl = 0 if L >= K or 0.5(L)^-0.5 if L
MPk = 0 if L=K
MPl/MPk = MRTS (the marginal rate of technical substitution) = (w/r) = 0 if L>K or infinity if L
This is as far as I was able to get. Thanks for your help!