The Total Cost Of Producing A Certain Good Is Given By${ TC = 300 \ln(q+30) + 150 }$Find The Marginal Cost { (MC)$}$ And The Average Cost { (AC)$}$ Functions.a. { MC = \frac{300}{q+30} $}$ And [$ AC =
Introduction
In economics, the total cost of production is a crucial concept that helps businesses and policymakers understand the relationship between output and costs. The total cost function, denoted by TC, represents the total expenses incurred by a firm to produce a certain quantity of a good. In this article, we will explore the marginal cost (MC) and average cost (AC) functions, which are derived from the total cost function.
The Total Cost Function
The total cost function is given by:
where q represents the quantity of the good produced.
Marginal Cost Function
The marginal cost function represents the change in the total cost resulting from a one-unit increase in the quantity produced. Mathematically, it is represented as:
To find the marginal cost function, we need to differentiate the total cost function with respect to q.
Using the chain rule and the fact that the derivative of ln(u) is 1/u, we get:
Average Cost Function
The average cost function represents the total cost per unit of output. It is calculated by dividing the total cost by the quantity produced.
Substituting the total cost function, we get:
Properties of the Marginal and Average Cost Functions
The marginal cost function has several important properties:
- Increasing MC: The marginal cost function is increasing, meaning that as the quantity produced increases, the marginal cost also increases.
- Positive MC: The marginal cost function is always positive, indicating that the total cost increases as the quantity produced increases.
- MC = 0 at q = -30: The marginal cost function is equal to zero when the quantity produced is -30. This is because the total cost function is a logarithmic function, and the derivative of a logarithmic function is zero at the point where the argument is equal to 1.
The average cost function also has several important properties:
- Decreasing AC: The average cost function is decreasing, meaning that as the quantity produced increases, the average cost decreases.
- Positive AC: The average cost function is always positive, indicating that the total cost per unit of output is always greater than zero.
- AC = 0 at q = ∞: The average cost function approaches zero as the quantity produced approaches infinity. This is because the total cost function is a logarithmic function, and the logarithmic function approaches zero as the argument approaches infinity.
Economic Interpretation
The marginal cost function and the average cost function have important economic implications:
- MC = AC: The marginal cost function is equal to the average cost function when the quantity produced is equal to 30. This is because the marginal cost function is increasing, and the average cost function is decreasing.
- MC > AC: The marginal cost function is greater than the average cost function when the quantity produced is less than 30. This is because the marginal cost function is increasing, and the average cost function is decreasing.
- MC < AC: The marginal cost function is less than the average cost function when the quantity produced is greater than 30. This is because the marginal cost function is increasing, and the average cost function is decreasing.
Conclusion
In conclusion, the marginal cost function and the average cost function are important concepts in economics that help businesses and policymakers understand the relationship between output and costs. The marginal cost function represents the change in the total cost resulting from a one-unit increase in the quantity produced, while the average cost function represents the total cost per unit of output. The properties of the marginal and average cost functions, such as increasing MC and decreasing AC, have important economic implications.
References
- Mankiw, N. G. (2017). Principles of Economics. Cengage Learning.
- Varian, H. R. (2014). Microeconomic Analysis. W.W. Norton & Company.
Appendix
Derivation of the Marginal Cost Function
To derive the marginal cost function, we need to differentiate the total cost function with respect to q.
Using the chain rule and the fact that the derivative of ln(u) is 1/u, we get:
Derivation of the Average Cost Function
To derive the average cost function, we need to divide the total cost function by the quantity produced.
Substituting the total cost function, we get:
Properties of the Marginal and Average Cost Functions
The marginal cost function has several important properties:
- Increasing MC: The marginal cost function is increasing, meaning that as the quantity produced increases, the marginal cost also increases.
- Positive MC: The marginal cost function is always positive, indicating that the total cost increases as the quantity produced increases.
- MC = 0 at q = -30: The marginal cost function is equal to zero when the quantity produced is -30. This is because the total cost function is a logarithmic function, and the derivative of a logarithmic function is zero at the point where the argument is equal to 1.
The average cost function also has several important properties:
- Decreasing AC: The average cost function is decreasing, meaning that as the quantity produced increases, the average cost decreases.
- Positive AC: The average cost function is always positive, indicating that the total cost per unit of output is always greater than zero.
- AC = 0 at q = ∞: The average cost function approaches zero as the quantity produced approaches infinity. This is because the total cost function is a logarithmic function, and the logarithmic function approaches zero as the argument approaches infinity.
Q&A: Marginal and Average Cost Functions =============================================
Q: What is the marginal cost function?
A: The marginal cost function represents the change in the total cost resulting from a one-unit increase in the quantity produced. It is calculated by differentiating the total cost function with respect to q.
Q: How is the marginal cost function related to the average cost function?
A: The marginal cost function is related to the average cost function in that they are both derived from the total cost function. However, the marginal cost function represents the change in the total cost resulting from a one-unit increase in the quantity produced, while the average cost function represents the total cost per unit of output.
Q: What is the economic interpretation of the marginal cost function?
A: The marginal cost function has several important economic implications. For example, when the marginal cost function is greater than the average cost function, it indicates that the total cost per unit of output is increasing. Conversely, when the marginal cost function is less than the average cost function, it indicates that the total cost per unit of output is decreasing.
Q: What is the average cost function?
A: The average cost function represents the total cost per unit of output. It is calculated by dividing the total cost function by the quantity produced.
Q: How is the average cost function related to the marginal cost function?
A: The average cost function is related to the marginal cost function in that they are both derived from the total cost function. However, the average cost function represents the total cost per unit of output, while the marginal cost function represents the change in the total cost resulting from a one-unit increase in the quantity produced.
Q: What is the economic interpretation of the average cost function?
A: The average cost function has several important economic implications. For example, when the average cost function is decreasing, it indicates that the total cost per unit of output is decreasing. Conversely, when the average cost function is increasing, it indicates that the total cost per unit of output is increasing.
Q: What is the relationship between the marginal cost function and the average cost function?
A: The marginal cost function and the average cost function are related in that they are both derived from the total cost function. However, the marginal cost function represents the change in the total cost resulting from a one-unit increase in the quantity produced, while the average cost function represents the total cost per unit of output.
Q: How do the marginal cost function and the average cost function change as the quantity produced increases?
A: The marginal cost function increases as the quantity produced increases, while the average cost function decreases as the quantity produced increases.
Q: What is the economic implication of the marginal cost function being greater than the average cost function?
A: The economic implication of the marginal cost function being greater than the average cost function is that the total cost per unit of output is increasing.
Q: What is the economic implication of the marginal cost function being less than the average cost function?
A: The economic implication of the marginal cost function being less than the average cost function is that the total cost per unit of output is decreasing.
Q: How do the marginal cost function and the average cost function relate to the law of diminishing marginal returns?
A: The marginal cost function and the average cost function are related to the law of diminishing marginal returns in that they both indicate that the total cost per unit of output is increasing as the quantity produced increases.
Q: What is the relationship between the marginal cost function and the law of diminishing marginal returns?
A: The marginal cost function is related to the law of diminishing marginal returns in that it indicates that the total cost per unit of output is increasing as the quantity produced increases.
Q: How do the marginal cost function and the average cost function change as the quantity produced approaches infinity?
A: The marginal cost function approaches zero as the quantity produced approaches infinity, while the average cost function approaches zero as the quantity produced approaches infinity.
Q: What is the economic implication of the marginal cost function approaching zero as the quantity produced approaches infinity?
A: The economic implication of the marginal cost function approaching zero as the quantity produced approaches infinity is that the total cost per unit of output is approaching zero.
Q: What is the economic implication of the average cost function approaching zero as the quantity produced approaches infinity?
A: The economic implication of the average cost function approaching zero as the quantity produced approaches infinity is that the total cost per unit of output is approaching zero.
Q: How do the marginal cost function and the average cost function relate to the concept of economies of scale?
A: The marginal cost function and the average cost function are related to the concept of economies of scale in that they both indicate that the total cost per unit of output is decreasing as the quantity produced increases.
Q: What is the relationship between the marginal cost function and the concept of economies of scale?
A: The marginal cost function is related to the concept of economies of scale in that it indicates that the total cost per unit of output is decreasing as the quantity produced increases.
Q: How do the marginal cost function and the average cost function change as the quantity produced increases in the short run?
A: The marginal cost function increases as the quantity produced increases in the short run, while the average cost function decreases as the quantity produced increases in the short run.
Q: What is the economic implication of the marginal cost function increasing as the quantity produced increases in the short run?
A: The economic implication of the marginal cost function increasing as the quantity produced increases in the short run is that the total cost per unit of output is increasing.
Q: What is the economic implication of the average cost function decreasing as the quantity produced increases in the short run?
A: The economic implication of the average cost function decreasing as the quantity produced increases in the short run is that the total cost per unit of output is decreasing.
Q: How do the marginal cost function and the average cost function relate to the concept of diseconomies of scale?
A: The marginal cost function and the average cost function are related to the concept of diseconomies of scale in that they both indicate that the total cost per unit of output is increasing as the quantity produced increases.
Q: What is the relationship between the marginal cost function and the concept of diseconomies of scale?
A: The marginal cost function is related to the concept of diseconomies of scale in that it indicates that the total cost per unit of output is increasing as the quantity produced increases.
Q: How do the marginal cost function and the average cost function change as the quantity produced increases in the long run?
A: The marginal cost function increases as the quantity produced increases in the long run, while the average cost function decreases as the quantity produced increases in the long run.
Q: What is the economic implication of the marginal cost function increasing as the quantity produced increases in the long run?
A: The economic implication of the marginal cost function increasing as the quantity produced increases in the long run is that the total cost per unit of output is increasing.
Q: What is the economic implication of the average cost function decreasing as the quantity produced increases in the long run?
A: The economic implication of the average cost function decreasing as the quantity produced increases in the long run is that the total cost per unit of output is decreasing.
Q: How do the marginal cost function and the average cost function relate to the concept of increasing returns to scale?
A: The marginal cost function and the average cost function are related to the concept of increasing returns to scale in that they both indicate that the total cost per unit of output is decreasing as the quantity produced increases.
Q: What is the relationship between the marginal cost function and the concept of increasing returns to scale?
A: The marginal cost function is related to the concept of increasing returns to scale in that it indicates that the total cost per unit of output is decreasing as the quantity produced increases.
Q: How do the marginal cost function and the average cost function change as the quantity produced increases in the short run and the long run?
A: The marginal cost function increases as the quantity produced increases in both the short run and the long run, while the average cost function decreases as the quantity produced increases in both the short run and the long run.
Q: What is the economic implication of the marginal cost function increasing as the quantity produced increases in both the short run and the long run?
A: The economic implication of the marginal cost function increasing as the quantity produced increases in both the short run and the long run is that the total cost per unit of output is increasing.
Q: What is the economic implication of the average cost function decreasing as the quantity produced increases in both the short run and the long run?
A: The economic implication of the average cost function decreasing as the quantity produced increases in both the short run and the long run is that the total cost per unit of output is decreasing.
Q: How do the marginal cost function and the average cost function relate to the concept of decreasing returns to scale?
A: The marginal cost function and the average cost function are related to the concept of decreasing returns to scale in that they both indicate that the total cost per unit of output is increasing as the quantity produced increases.
Q: What is the relationship between the marginal cost function and the concept of decreasing returns to scale?
A: The marginal cost function is related to the concept of decreasing returns to scale in that it indicates that the total cost per unit of output is increasing as the quantity