Consider a firm with the production function The wage rate

Consider a firm with the production function . The wage rate and rental rate on capital are w and r, respectively.

1.Use the Lagrangian to derive the long-run cost function for this firm.

2.Suppose the government provides a subsidy of $10 per unit of capital to the fir. Rewrite the long-run cost function.

Solution

a.

The Lagrangian form is given below-

L = wL + rK + (q – LK)

K = wL/r

L = (qr/W)^1/2 and K = (qw/r)^1/2

The cost function is C(w,r,q) = w(qr/W)^1/2 + r(qw/r)^1/2 = 2(qwr)^1/2

b.

Replace w with w+t in the cost function above

C(w+t,r,q) = 2(q(w+t)r)^1/2