Question

A firm in a perfectly competitive market uses Labor ( L, number of workers) and Capital ( K, number of machines) as input to produce hat. Assume the firm has purchased K=25 machines, with the price Pk=$100 dollars each. This firm now needs to make a short-run decision: choosing the optimal number of workers L to maximize its profit. Assume - Labor market is competitive so each worker receives an hourly wage w=25. - The market equilibrium price for the output, hat, is P=$120. - The firm's has the following Cobb-Douglass production function, Q=K1/2(AL)1/2 where Q is the level of outputs and A is a parameter that captures the productivity of labor. The greater A is the more productive the workers are. Throughout the question, assume A=4.

1. Calculate the level of outputs, Q, when the firm hires 100 workers (L=100).
2. Based on the production function in (5), write L as a function of Q.
3. Assuming that the total cost consists of fixed cost (ie, cost of capital: =PkK ) and variable cost (ie, cost of labor: =wL ), write the firm's profit as a function of Q.
4. Find the profit-maximizing quantity; that is, the level of output Q that maximizes the firm profit.

Answer 1

1. Q = 100. 2. L = (Q^2 / (A*K))^(1/2). 3. Profit = P * Q - (Pk * K + w * L). 4. Find profit-maximizing quantity by finding Q that maximizes profit.

Step-by-step explanation:

1. To calculate the level of outputs, Q, when the firm hires 100 workers (L=100), we can use the Cobb-Douglas production function: Q = K^(1/2) * (A*L)^(1/2). Substituting the given values, we have Q = 25^(1/2) * (4*100)^(1/2) = 5 * 20 = 100.

2. Based on the production function, we can write L as a function of Q as follows: Q = K^(1/2) * (A*L)^(1/2). Rearranging the equation, we get L = (Q^2 / (A*K))^(1/2).

3. The firm's profit can be written as the difference between total revenue and total cost. Total revenue is given by TR = P * Q, where P is the market price. Total cost is the sum of fixed cost and variable cost, which is given by TC = Pk * K + w * L. Therefore, the firm's profit is given by the equation Profit = TR - TC = P * Q - (Pk * K + w * L).

4. To find the profit-maximizing quantity, we need to find the level of output Q that maximizes the firm's profit. We can do this by taking the derivative of the profit function with respect to Q and setting it equal to zero. By solving for Q, we can find the profit-maximizing quantity.