Question

Suppose that we have perfectly competitive input markets and output markets. Firm "Tomato Harvesting" produces canned tomatoes which it sells at $50. Suppose that initially the firm's production technology is given by:

f(k,l) = l^(1/2)

Technological innovation has occurred, however. A new harvester has been invented by a professor. If the firm employs the tomato harvester, the new production technology is given by:

f(k.l) = k^0.25l^(1/2)

The tomato harvester represents 1300 units of capital. At what hourly market wage will the firm switch from employing their labor only technology to adopting the tomato harvester (and employing their new production technology)?

Answer 1

We see that initially the firm’s production technology is given by q = l^0.5. This implies that MPL is 0.5/l^0.5 and

Value of MPL is price x 0.5/l^0.5 which becomes 50*0.5/l^0.5 or 25/l^0.5. At the optimum level VMPL should be

equal to the wage rate so we have 25/l^0.5 = 3. This gives l* = (25/3)^2 = 69.44 units and output = 69.44^0.5 =

8.33

With new production function q = k^0.25 l^0.5, we have MRTS = MPL/MPK

= 0.5*(l^-0.5)*(k^0.25) / 0.25*(k^-0.75)*(l^0.5)

= 2k/l

Wage rental ratio is 3/2 or 1.5. At the optimal input mix firm has MRTS = wage rental ratio or 2k/l = 1.5 of k =

0.75l. To produce the same output we have 8.33 = (0.75l)^0.25 l^0.5 or l = 18.58

Hence the firm has reduced the use of labor from 69.44 units to 18.58 units