Question

For production function $f(L,K) = \sqrt{min(L,K)}$ , find the long-run supply curve (L and K are variable). How does a change in w shift the graph of the function?

Answer 1

From the production function we have Q^2 = min(K, L) or K = Q^2 and L = Q^2. The cost function is C = wL + rK

or C = w*Q^2 + r*Q^2. This is simplified to C = Q^2(w + r). Now marginal cost is MC = 2Q(w + r) and so the supply

function is P = MC. This implies we have the supply curve expressed as P = 2Q(w + r) or Q = 0.5P/(w + r). An

increase in the wage rate is likely to increase denominator and so the expression 0.5P/(w + r) is reduced. This

implies that quantity supplied is reduced at every price so supply curve shifts to the left.