Tangency as a Profit Minimum with a Highly Concave No-Shirking Wage Curve

Sketching the Profit Minimum Scenario with a Concave No-Shirking Wage Curve

Further Reading on Curve-Sketching, Convexity, and Finding Maxima/Minima

A firm identifies an employment level, N*, where the first derivative of its profit function with respect to employment equals zero. To determine the nature of this point, the firm calculates the second derivative of the profit function at N* and finds that its value is positive. What is the correct interpretation of this finding?

Evaluating a Hiring Recommendation

Interpreting Stationary Points in Profit Functions

A firm finds an employment level where the slope of its revenue function is exactly equal to the slope of its total wage cost function. This employment level is always the point of maximum profit for the firm.

The Role of the Second-Order Condition in Profit Maximization

A firm has identified a level of employment where the marginal revenue from an additional worker equals the marginal cost of that worker. For this employment level to represent a true profit maximum, and not a minimum, the second derivative of the profit function with respect to employment must be ________.

A firm is analyzing its profit function, which depends on the level of employment (N). Match each mathematical condition with its correct economic interpretation in the context of finding the optimal employment level.

A firm wants to find and confirm the level of employment (N) that truly maximizes its profit. Arrange the following steps in the correct logical sequence that the firm should follow.

Diagnosing an Unexpected Profit Calculation

A firm's profit is the difference between its revenue and its total wage cost, both of which are functions of the number of employees (N). The firm identifies a potential profit-maximizing employment level, N*, where the slope of the revenue function equals the slope of the total wage cost function. The firm also observes that its total wage cost function is unusually concave at this level of employment. What is the most likely implication of this observation for verifying if N* is a true profit maximum?

Further Reading on Curve-Sketching, Convexity, and Finding Maxima/Minima

Productive Efficiency Analysis

A firm's total cost of production is described by the function TC(q) = 50 + 30q - 6q² + q³, where q is the quantity of output. At what level of output does the firm's average variable cost reach its minimum?

For a firm with the total cost function TC(q) = 100 + 40q + 2.5q², the marginal cost is always greater than the average variable cost for all positive levels of output (q > 0).

A firm's production process is characterized by the total cost function TC(q) = a + bq + cq², where q is the quantity of output and a, b, and c are positive constants. Match each economic cost concept below with its correct mathematical representation derived from this function.

Comprehensive Cost Analysis for a Firm

Marginal Cost Interpretation

A firm's total cost of production is given by the function TC(q) = 100 + 20q - 5q² + q³. To determine the firm's short-run supply curve, you must follow a specific analytical procedure. Arrange the following steps in the correct logical order to complete this analysis.

A firm's total cost of production is represented by the function TC(q) = 200 + 10q + 2q², where q is the quantity of output. The marginal cost of producing the 5th unit of output is ____.

Evaluating a Production Decision

A firm's total cost of production is described by the function TC(q) = 50 + 12q - 3q² + q³, where q is the quantity of output. At the specific level of output where the firm's average variable cost (AVC) reaches its minimum value, what is the corresponding value of the firm's marginal cost (MC)?