Question

Assume that output is given by Q(L,K)=50 K^0.5 L^0.5 with price of labour L = w and price of capital K = r

1.If capital in the short run is fixed at K what is the short-run total cost?

2.Write the values for the derivatives of the Total cost with respect to w and r. Does Shephard’s lemma hold in this case?

Answer 1

1)Q=50*K^0.5*L^0.5

Q/(50*K^0.5)=L^0.5

L=(Q^2)/(2500*K)

Total cost= L*w +K*r=(Q^2)/(2500*K)]*w+k*r

Total cost=(w*Q^2)/(2500*K) +k*r

2)∆Total cost/∆w=(Q^2)/(2500*K)

∆Total cost/∆r=k

Shepherd lemma holds as partial derivative of total cost with respect to input price is equal to conditional demand of input.