Question

The Ajax Coal Company is the only hirer of labor in its area. It can hire any number of female workers or male workers it wishes. The supply curve for women is given by \\[ l_{f}=100 w_{f} \\] and for men by \\[ l_{m}=9 w_{m}^{2} \\] where \(w_{f}\) and \(w_{m}\) are the hourly wage rates paid to female and male workers, respectively. Assume that Ajax sells its coal in a perfectly competitive market at \(\$ 5\) per ton and that each worker hired (both men and women) can mine 2 tons per hour. If the firm wishes to maximize profits, how many female and male workers should be hired, and what will the wage rates be for these two groups? How much will Ajax earn in profits per hour on its mine machinery? How will that result compare to one in which Ajax was constrained (say, by market forces) to pay all workers the same wage based on the value of their marginal products?

Short answer

#Answer# The labor supply curve for female workers is l_f = 100w_f. To maximize profits, Ajax Coal Company should hire 1000 female workers.

Step-by-step solution

Express the labor supply curves for female and male workers

We are given the labor supply curves for female and male workers as follows: For female workers: \\[ l_{f} = 100w_{f} \\] For male workers: \\[ l_{m} = 9w_{m}^2 \\]

Find the value of the marginal product of labor

The coal price is \(\$5\) per ton, and each worker can mine 2 tons per hour, so the value of the marginal product of labor (VML) is: \\[ VML = 2 \times 5 = \$10 \\]

Calculate the optimal number of female and male workers hired

To maximize profits, the company needs to hire workers until the value of their marginal products equals their wage rates. So, this equality holds for both female and male workers: For female workers: \\[ VML = w_{f} \\] For male workers: \\[ VML = w_{m} \\] Using the given VML value, we get: \\[ w_{f} = 10 \\ w_{m} = 10 \\] Now, plug the wage rates into the labor supply equations: For female workers: \\[ l_{f} = 100w_{f} = 100(10) = 1000 \\] For male workers: \\[ l_{m} = 9w_{m}^2 = 9(10)^2 = 900 \\] So, the company should hire 1000 female workers and 900 male workers to maximize profits.

Calculate Ajax's profits per hour

Ajax's total hourly revenue is: \\[ Revenue = (\text{number of workers}) \times (\text{tons/hour/worker}) \times (\text{price/ton}) \\ = (1000 + 900) \times 2 \times 5 = 1900 \times 10 = 19000 \\] Ajax's total hourly wages for both groups are: \\[ Wages = (1000w_{f} + 900w_{m}) = 1000(10) + 900(10) = 19000 \\] Thus, Ajax's profits per hour on its mine machinery are: \\[ Profits = Revenue - Wages = 19000 - 19000 = \$0 \\]

Compare the results with the constrained wage scenario

If Ajax was constrained to pay equal wages based on the value of the workers' marginal product, the situation would be different. As the value of the marginal product of labor is \(\$10\), the wages would be equal for both genders, and the equilibrium in the labor market would happen faster than in the initial scenario. However, the profits would still be zero since the wages equal the value of the workers' marginal product. In both cases, Ajax Coal Company earns no profits per hour on mine machinery because all the revenues are paid to the workers as wages.