Question

Suppose that a firm's production function is given by \(Q=12 L-L^{2},\) for \(L=0\) to \(6,\) where \(L\) is labor input per day and \(Q\) is output per day. Derive and draw the firm's demand for labor curve if the firm's output sells for \(\$ 10\) in a competitive market. How many workers will the firm hire when the wage rate is \(\$ 30\) per day? \(\$ 60\) per day? (Hint: The marginal product of labor is \(12-2 L\).)

Short answer

The firm will hire 4.5 workers if the wage rate is $30 per day and it will hire 3 workers if the wage rate is $60 per day.

Step-by-step solution

Compute the Value of the Marginal Product of Labor (VMPL)

The VMPL is calculated as the multiplication of the price of the output and the marginal product of labor. So, for the given problem, since the price of the output is $10 and the marginal product of labor is \(12 - 2L\), the VMPL would be calculated as \(VMPL = 10 * (12 - 2L) = 120 - 20L\).

Create the Demand for Labor Curve

The demand for labor curve is the graphical representation of the VMPL with respect to labor (L). It is a downward sloping line that starts from (0, 120) and its slope is -20.

Determine Labor Demand at Wage Rate = $30

A firm will hire workers until the wage rate equals the VMPL. So, when the wage rate is $30, equate the VMPL to the wage rate such that \(30 = 120 - 20L\). Solving this will give \(L = 4.5\). Therefore, the firm will hire 4.5 workers at $30 per day.

Determine Labor Demand at Wage Rate = $60

Again, a firm will hire workers until the wage rate equals the VMPL. So, when the wage rate is $60, equate the VMPL to the wage rate such that \(60 = 120 - 20L\). Solving this will give \(L = 3\). Therefore, the firm will hire 3 workers at $60 per day.