How do you show that the minimum average cost occurs when the average cost equals the marginal cost?
To show that the minimum average cost occurs when the average cost equals the marginal cost, we need to understand the relationships between average cost, marginal cost, and how they change as production levels vary.
1. To begin, let's define average cost and marginal cost:
- Average Cost (AC): It represents the cost per unit of output, calculated by dividing the total cost (TC) by the number of units produced (Q). Mathematically, AC = TC/Q.
- Marginal Cost (MC): It represents the additional cost incurred by producing one more unit of output. It can be calculated by taking the derivative of the total cost function with respect to quantity (MC = dTC/dQ).
2. The relationship between average cost and marginal cost is crucial in determining the behavior of costs at different production levels:
- When marginal cost is less than the average cost (MC < AC), producing one more unit of output decreases the average cost. This occurs when the marginal cost is lower than the average cost, pulling the average cost down.
- When marginal cost is greater than the average cost (MC > AC), producing one more unit of output increases the average cost. This occurs when the marginal cost is higher than the average cost, pushing the average cost up.
- When marginal cost is equal to average cost (MC = AC), producing one more unit does not impact the average cost. At this point, the average cost is neither decreasing nor increasing.
3. Now, let's examine the relationship between average cost, marginal cost, and the minimum average cost:
- If the marginal cost is below the average cost (MC < AC), the average cost will continue to decline as more units are produced. In this case, the minimum average cost does not occur when the average cost equals the marginal cost.
- If the marginal cost is above the average cost (MC > AC), the average cost will increase with each additional unit of output. Again, here the minimum average cost does not occur when the average cost equals the marginal cost.
- The only possibility for minimum average cost occurs when the marginal cost is equal to the average cost (MC = AC). At this point, the average cost neither decreases nor increases, indicating the lowest possible average cost.
4. To summarize, when the average cost equals the marginal cost (AC = MC), it indicates the point of minimum average cost. This relationship signifies that producing an additional unit of output does not affect the average cost.
Therefore, by understanding the relationships between average cost, marginal cost, and their behavior at different production levels, we can see that the minimum average cost occurs precisely when the average cost equals the marginal cost.