Verifying a Profit Maximum with the Second-Order Condition

Calculating Optimal Employment Level

A manufacturing firm's profit (π), in dollars, is determined by the number of workers (N) it employs. The relationship is described by the function: π(N) = 200√N - 5N. To achieve the highest possible profit, what is the optimal number of workers the firm should hire?

Profit Optimization for a Manufacturing Firm

A firm's profit (π) is a function of the number of workers (N) it employs. At the current employment level of N = 50, the firm's management calculates that the derivative of the profit function with respect to the number of workers is positive (dπ/dN > 0). Based on this information, what is the most logical next step for the firm to take to increase its profit?

Interpreting the First-Order Condition for Profit Maximization

A firm's profit function is given by π(N) = 360N - 2N², where π is the profit in dollars and N is the number of workers. A student attempts to find the number of workers that maximizes profit by following these steps:

Step 1: Calculate the derivative of the profit function: dπ/dN = 360 - 4N. Step 2: To find the maximum, set the derivative to be greater than zero: 360 - 4N > 0. Step 3: Solve the inequality: 360 > 4N, which simplifies to N < 90. Step 4: Conclude that hiring any number of workers less than 90 will maximize profit.

In which step did the student first make a critical error in their optimization procedure?

A company's profit (π) is known as a function of the number of workers (N) it employs. To find the exact number of workers that will maximize profit, a specific calculus-based procedure is used. Arrange the following steps in the correct logical order to complete this procedure.

A firm's management determines that at an employment level of 150 workers, the rate of change of profit with respect to the number of workers is zero. This discovery is sufficient evidence to conclude that employing 150 workers will result in the highest possible profit.

A firm is analyzing its profit function, π(N), where π is profit and N is the number of workers. To find the profit-maximizing number of workers, the firm uses calculus. Match each mathematical expression or condition with its correct economic interpretation in this context.

Economic Rationale for Profit Maximization