Question

1. A firm that produces shirts has a production function q=f(K,L)=KL/10, that has a cost price of labor= $10 and cost price of capital=$100. Find the firm’s long run average cost function, and marginal cost function. Graph AC(q) and MC(q) and identify the firm’s long-run supply curve.

Answer 1

Here the production function is, => Q=K*L/10, => MPL = K/10 and MPK = L/10. So, the MRTS is the given by.

=> MRTS = MPL/MPK = (K/10)/(L/10) = K/L, => MRTS = K/L. Now, the labor cost and the capital cost are “W=$10” and “R=$100” respectively. At the equilibrium the MRTS must be equal to input price ratio.

=> MRTS = W/R, => K/L = W/R = 10/100 = 1/10, => K = L/10.

The production function is, => Q = K*L/10 = K*K, => K = Q^0.5. From the above condition we got “L = 10*K”.

=> L = 10*Q^0.5. So, the cost function id given by.

=> C = W*L + R*K = W*(10*Q^0.5) + R*Q^0.5 = 10*(10*Q^0.5) + 100*Q^0.5 = 100*Q^0.5 + 100*Q^0.5.

=> C = 200*Q^0.5. So, the average cost function is “AC = C/Q = 200*Q^0.5/Q = 200*Q^(-0.5).

=> AC = 200*Q^(-0.5).

The marginal cost function is, => dC/dQ = 200*0.5*Q^(-0.5) = 100*Q^(-0.5), => MC = 100*Q^(-0.5).

Here both the LRAC and LRMC are downward sloping. The above fig shows the LRAC and LRMC. Here LRMC is completely below of LRAC, => any point on the MC implied the producer will incur loss. So, the LR supply curve is the AC curve here.