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?

(b): Explain intuitively why labour demanded by the firm went up or down in part (a). In doing so, describe the two forces that operate when capital is introduced into the technology.

(c): Suppose that there can only be one tomato harvester per farm (i.e., once adopted, capital can be considered fixed). The tomato harvester represents 1300 units of capital (note: 1300^(0.25) = 6). At what hourly market wage will the firm switch from employing their labour only technology to adopting the tomato harvester (and employing their new production technology)? [hint: find labour demand functions for each case, then set the profits in each case equal to solve for w]

Answer 1

A)

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