Question

Suppose that we have perfectly competitive input markets (for both capital and labour) and output markets. Firm Tomato Harvesting produces canned tomatoes which it sells at $50 (it is a big can of tomatoes!). Suppose that initally the firm’s production technology is given by: f(k,l)= √l

A technological innovation has occured however! A new tomato harvester has been invented by a professor at UC Davis. If the firm employs the tomato harvester, the new production technology is given by: f(k,l) = k^(0.25) √l The rental rate of capital is $2.
(a): Suppose the market wage is $3. How much more or less labour will the firm use once it switches to the new production technology with the tomato harvester in the long run?

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