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Chapter 4: Problem 15

Show analytically that if marginal cost is greater than average cost, then the derivative of average cost with respect to quantity satisfies \(a^{\prime}(q)>0\)

Short Answer

Expert verified
If MC > AC, then MC influences AC's growth, thus \( a'(q) > 0 \).

Step by step solution

01

Understanding the Relationship

To solve this problem, we need to understand the relationship between average cost (AC) and marginal cost (MC). The average cost function is the total cost (C) divided by the quantity (q), i.e., \( AC = rac{C}{q} \). Marginal cost is the derivative of total cost with respect to quantity, i.e., \( MC = C'(q) \).
02

Differentiating Average Cost

We are required to find the derivative of the average cost with respect to quantity \( q \), which involves differentiating the function \( a(q) = \frac{C}{q} \). Using the quotient rule, we have: \[ a'(q) = \frac{C'(q) \cdot q - C}{q^2} \].
03

Using Given Condition

We know from the problem statement that \( MC = C'(q) > a(q) \). Replace \( C'(q) \) with \( MC \) in the expression for \( a'(q) \): \[ a'(q) = \frac{MC \cdot q - C}{q^2} \].
04

Analyzing the Expression

To show that \( a'(q) > 0 \), we need \( MC \cdot q - C > 0 \). This implies \( MC \cdot q > C \). Since \( a(q) = \frac{C}{q} \), then \( C = a(q) \cdot q \). Substitute back: \[ MC \cdot q > a(q) \cdot q \].
05

Conclusion

Since \( MC > a(q) \), multiplying both sides by \( q \) gives \( MC \cdot q > a(q) \cdot q \), ensuring that \( a'(q) > 0 \). Therefore, the derivative of the average cost with respect to quantity \( q \) is positive, confirming that when marginal cost is greater than average cost, the slope of the average cost curve is positive.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Average Cost
Average cost is a fundamental concept in economics and helps businesses understand how costs evolve with production. It is calculated by dividing the total cost (C) by the total quantity (q) produced. This gives the cost per unit of output, providing insights into how efficiently resources are being utilized.

In mathematical terms, the average cost (AC) can be expressed as:
  • \( AC = \frac{C}{q} \)
Understanding average cost is crucial for businesses to determine their pricing strategies and profit margins. Lower average costs often grant competitive advantages, allowing businesses to lower prices or increase profit margins.

The average cost may change with scale. As more units are produced, fixed costs are spread over more units, often lowering the average cost. However, variable costs may also rise, impacting the average cost accordingly. This fluctuating nature makes understanding average cost vital for scaling production efficiently.
Derivative of Average Cost
The derivative of average cost provides information about how the average cost per unit changes as the quantity produced varies. By examining this derivative, businesses can understand whether increasing production will increase or decrease their average cost per unit.

To compute the derivative of the average cost function \( a(q) = \frac{C}{q} \), we use the quotient rule. The quotient rule for derivatives states that for a function \( f(x) = \frac{g(x)}{h(x)} \), its derivative is:
  • \( f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{h(x)^2} \)
Applying this to our average cost function, we find:
  • \( a'(q) = \frac{MC \cdot q - C}{q^2} \)
Here, \( C'(q) \), the derivative of the total cost, is equivalent to the marginal cost (MC). This derivative helps in analyzing how different levels of production influence average costs, supporting businesses in making decisions about optimal production quantities.
Marginal Cost Analysis
Marginal cost analysis is an essential tool for businesses to evaluate how the cost to produce one additional unit of output affects overall profitability. It is defined as the change in total cost resulting from a one-unit change in output quantity. In formula terms, the marginal cost (MC) is:
  • \( MC = C'(q) \)
This metric is valuable for decision-making processes regarding production levels. When marginal cost is greater than average cost, it means each additional unit is more expensive than the average, leading to an increase in the average cost per unit.

Analyzing the relationship between marginal cost and average cost can reveal whether a company is operating efficiently. If the marginal cost is less than the average cost, producing additional units will lower the average cost, indicating potentially beneficial economies of scale. However, if the marginal cost exceeds average cost, focusing on cost-management strategies might be advisable to avoid inefficient production and increased average costs.

Marginal cost analysis guides companies in pricing and production strategies, helping balance between cost control and production efficiency.

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Most popular questions from this chapter

A company manufactures only one product. The quantity, \(q\), of this product produced per month depends on the amount of capital, \(K\), invested (i.e., the number of machines the company owns, the size of its building, and so on) and the amount of labor, \(L\), available each month. We assume that \(q\) can be expressed as a Cobb-Douglas production function: $$ q=c K^{\alpha} L^{\beta} $$ where \(c, \alpha, \beta\) are positive constants, with \(0<\alpha<1\) and \(0<\beta<1 .\) In this problem we will see how the Russian government could use a Cobb-Douglas function to estimate how many people a newly privatized industry might employ. A company in such an industry has only a small amount of capital available to it and needs to use all of it, so \(K\) is fixed. Suppose \(L\) is measured in man-hours per month, and that each man-hour costs the company \(w\) rubles (a ruble is the unit of Russian currency). Suppose the company has no other costs besides labor, and that each unit of the good can be sold for a fixed price of \(p\) rubles. How many man-hours of labor per month should the company use in order to maximize its profit?

A reasonably realistic model of a firm's costs is given by the short-run Cobb- Douglas cost curve $$ C(q)=K q^{1 / a}+F $$ where \(a\) is a positive constant, \(F\) is the fixed cost, and \(K\) measures the technology available to the firm. (a) Show that \(C\) is concave down if \(a>1\). (b) Assuming that \(a<1\), find what value of \(q\) minimizes the average cost.

Dwell time, \(t\), is the time in minutes that shoppers spend in a store. Sales, \(s\), is the number of dollars they spend in the store. The elasticity of sales with respect to dwell time is 1.3. Explain what this means in simple language.

Use the first derivative to find all critical points and use the second derivative to find all inflection points. Use a graph to identify each critical point as a local maximum, a local minimum, or neither. \(f(x)=\frac{x^{3}}{6}+\frac{x^{2}}{4}-x+2\)

The Tojolobal Mayan Indian community in Southern Mexico has available a fixed amount of land. \({ }^{16}\) The proportion, \(P\), of land in use for farming \(t\) years after 1935 is modeled with the logistic function $$ P=\frac{1}{1+3 e^{-0.0275 t}} . $$ (a) What proportion of the land was in use for farming in \(1935 ?\) (b) What is the long-run prediction of this model? (c) When was half the land in use for farming? (d) When is the proportion of land used for farming increasing most rapidly?
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