Understanding the Relationship Between Marginal Cost and Average Total Cost
In economics, the relationship between marginal cost and average total cost is a fundamental concept. This article aims to clarify the nuances of this relationship, particularly proving mathematically that as the average total cost (ATC) increases, the marginal cost (MC) is always greater than ATC.
Defining the Concepts
Let's start by defining the key terms involved in this analysis:
**Marginal Cost (MC):** The cost of producing one additional unit of a good or service. This is the total change in cost brought about by increasing the level of production by one unit. **Average Total Cost (ATC):** The total cost of production divided by the number of units produced. It includes both fixed costs (FC) and variable costs (VC).Mathematically, we can express the total cost (TC) as:
$$TC FC VC$$
Where:
FC is the fixed cost, which remains constant regardless of the level of output. VC is the variable cost, which changes with the level of output.Mathematical Relationship
Given the above definitions, we can write the average total cost (ATC) as:
$$ATC frac{TC}{Q}$$
Where (Q) is the quantity of units produced. Let's break this down step by step.
Step-by-Step Derivation
**Step 1: Express Total Cost in Terms of Fixed and Marginal Costs:**We know that:
$$TC Q cdot MC$$
The total cost is a linear function of the quantity produced, adjusting only the variable costs while fixed costs remain constant.
**Step 2: Substitute and Simplify the ATC Equation:**Substitute (TC Q cdot MC) into the ATC equation:
$$ATC frac{Q cdot MC}{Q}$$
This simplifies to:
$$ATC MC frac{FC}{Q}$$
**Step 3: Analyze the Components of ATC:****Fixed Costs (FC) and Quantity (Q):**The term (frac{FC}{Q}) represents the fixed cost per unit, which decreases as (Q) increases. This is because the fixed cost is spread over more units.
**Marginal Cost (MC) and Average Marginal Cost (AMC):**The term (MC) represents the cost of producing the next unit, while (AMC frac{Q cdot MC}{Q} MC), the average of these marginal costs.
**Step 4: Prove that Marginal Cost is Greater than ATC:**Since (frac{FC}{Q}) is decreasing as (Q) increases, we have:$$ATC MC frac{FC}{Q}$$
Given that (frac{FC}{Q}) is always positive, it follows that:
$$ATC
Therefore, (MC) must be greater than ATC to ensure the equality holds.
Examples
To illustrate, let's consider an example. Suppose the fixed costs (FC) are $100, the quantity (Q) is 10 units, and the marginal cost (MC) for each unit is $10.
At 10 units: **Total Cost (TC):**$$TC Q cdot MC 10 cdot 10 100 100 200$$
**Average Total Cost (ATC):**$$ATC frac{TC}{Q} frac{200}{10} 20$$
At 20 units (assuming the marginal cost remains $10): **Total Cost (TC):**$$TC Q cdot MC 20 cdot 10 200 200 400$$
**Average Total Cost (ATC):**$$ATC frac{TC}{Q} frac{400}{20} 20$$
In this scenario, even though ATC remains constant, the marginal cost (MC) is still $10, which is greater than ATC.
Conclusion
The mathematical relationship between marginal cost and average total cost shows that as the average total cost increases, the marginal cost must also be greater than the average total cost. This is due to the decreasing nature of the fixed cost per unit as production increases, which ensures that the increase in marginal cost always exceeds the increase in average total cost.