Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


d. A particular marginal cost function has the property that it is positive and decreasing. The cost of increasing production from A units to 2A units is greater than the cost of increasing production from 2A units to 3A units.

Verified step by step guidance

1. Recall that the marginal cost function, denoted as \(MC(x)\), represents the rate of change of the total cost function \(C(x)\) with respect to the quantity \(x\). Mathematically, \(MC(x) = C'(x)\).
2. Given that \(MC(x)\) is positive and decreasing, this means \(MC(x) > 0\) for all \(x\) and \(MC'(x) < 0\), so the marginal cost is getting smaller as production increases.
3. The cost of increasing production from \(A\) units to \$2A$ units is given by the integral of the marginal cost over that interval: \(\int_{A}^{2A} MC(x) \, dx\).
4. Similarly, the cost of increasing production from \$2A\( units to \)3A$ units is \(\int_{2A}^{3A} MC(x) \, dx\).
5. Since \(MC(x)\) is positive and decreasing, the values of \(MC(x)\) on the interval \([2A, 3A]\) are less than those on \([A, 2A]\). Therefore, the integral over \([A, 2A]\) is greater than the integral over \([2A, 3A]\), meaning the cost of increasing production from \(A\) to \$2A\( units is greater than from \)2A\( to \)3A$ units.

Key Concepts

Marginal Cost Function

Marginal cost represents the additional cost incurred by producing one more unit of a good. It is mathematically the derivative of the total cost function with respect to quantity. Understanding how marginal cost behaves helps analyze changes in total cost over different production levels.

Relationship Between Marginal Cost and Total Cost

The total cost of increasing production over an interval is the integral of the marginal cost over that interval. If marginal cost is positive and decreasing, the total cost increases but at a diminishing rate. Comparing costs over different intervals requires evaluating the area under the marginal cost curve.

Properties of Decreasing Functions and Inequalities

A decreasing marginal cost means the cost per unit falls as production increases. This affects the total cost increments over intervals of equal length differently. Understanding how decreasing functions influence integrals and sums is essential to determine which production interval has a higher total cost increase.