Question

A firm has the following Cobb-Douglas production function: Q = 2L^0.5K^0.5. The firms pays $50 for each unit of labor (w) and $100 for each unit of capital (r).

a. Consider a short-run problem where the firm's level of capital is fixed at 25 units. In this case, derive an exression that shows the optimal amount of labor (L) the firm would want to use in order to produce Q units of output.

b. Use your answer for part (a) to derive a short-run cost function for the firm. Also derive expressions for the firm's margical cost (MC) and average total cost (ATC).

c. If the firm produces 100 units of output, what are the numerical values for the cost measures derived in part (b)?

Answer 1

Profit Function=@(L,K) =Revenue-Cost=Price *Q-(wL+rK)=2P*(LK)^0.5-(50L+100*25)

Finding FOC of above expression to get optimum values in terms of L

d@/dL=P(K/L)^(0.5)-50=0

P(K/L)^(0.5)=50

P*(25/L)^(0.5)=50

P/L^0.5=10

L=P^2/100..Optimal amount of labour

C=wL+rK

ATC=dC/dQ=(wL+rK)/2(LK)^0.5=0.5*(50L+2500)/(LK)^0.5=0.5*(50L+2500)/(5)*(L)^0.5=(50L+2500)*0.1/L^(0.5)

ATC=10L^0.5+500/L^0.5

MC=w=$50

Answer for c)

If Q=100

100=2(L*K)^0.5

2500=L*K

L=100

C=wL+rK=50*100+100*25=$7500