Profit Maximization and the Average Cost Curve

Definition of Average Cost (AC): Average cost is defined as the cost per unit of output. It is calculated by dividing the total cost (TC) by the quantity of output ( $Q$ ). $AC = \frac{TC}{Q}$
Total Cost Components: To understand average cost more deeply, one must recognize that total cost is the sum of fixed costs (FC) and variable costs (VC). $TC = FC + VC$
Long Format of Average Cost: By substituting the components of total cost into the average cost equation, it can be written as: $AC = \frac{FC}{Q} + \frac{VC}{Q}$

The Impact of Fixed Costs (FC): Fixed costs do not change as quantity ( $Q$ ) changes. Consequently, the term $\frac{FC}{Q}$ decreases as production increases.
- Numerical Example: If fixed costs are $100$ and $Q = 1$ , the fixed cost per unit is $100$ . If $Q = 10$ , the fixed cost per unit drops to $10$ . As $Q$ gets larger, this number continues to fall.
The Impact of Variable Costs (VC): Variable costs increase with the quantity produced. Observations of marginal cost curves indicate that at a certain point, variable costs will begin to increase faster than the quantity of output.
- This causes the term $\frac{VC}{Q}$ to eventually increase.
The Typical U-Shape: The average cost curve is driven by two opposing forces:
1. At low levels of production, the falling fixed cost per unit drives the total average cost down.
2. As production continues to increase, the rising variable cost per unit eventually dominates, driving the average cost back up.
- This results in a characteristic shape that falls, reaches a minimum point, and then rises.

Intersection at the Minimum: Mathematically, the marginal cost (MC) curve always passes through the minimum point of the average cost (AC) curve.
The Grade Metaphor (Marginal vs. Average): To understand the movement of the average cost curve relative to marginal cost, consider a student's grades:
- Falling Average: If your current average grade is $80\%$ and your next (marginal) test score is $60\%$ , your average will decrease. Whenever the marginal value is below the average, the average must be falling.
- Rising Average: If your average is $80\%$ and your next (marginal) test score is $90\%$ , your average will increase. Whenever the marginal value is above the average, the average must be rising.
- Flat Average (Minimum Point): If your average is $80\%$ and your next (marginal) test score is exactly $80\%$ , the average remains flat. This is equivalent to the minimum point of the AC curve where $MC = AC$ .

Rearranging the Profit Equation: Profit is traditionally defined as total revenue (TR) minus total cost (TC). $\text{Profit} = TR - TC$
Substitution Using Price and Average Cost:
1. Total Revenue is price times quantity: $TR = P \times Q$ .
2. Since $AC = \frac{TC}{Q}$ , total cost can be expressed as: $TC = AC \times Q$ .
The Final Profit Formula: By substituting these into the profit equation and factoring out $Q$ , we derive: $\text{Profit} = (P - AC) \times Q$
Diagrammatic Representation:
- Profit is shown as the area of a rectangle on the graph.
- Height: The difference between the price ( $P$ ) and the average cost ( $AC$ ) at the profit-maximizing level of output.
- Width: The total quantity ( $Q$ ) produced.

Loss Scenarios: Profit maximization does not always result in a positive profit; sometimes the goal is to minimize losses. This occurs when the market price is below the minimum of the average cost curve.
Example Case Study ( $17$ Breakeven Price):
- If the price is below $17$ (the minimum average cost), any quantity chosen where $P = MC$ will result in a price that is lower than the average cost at that quantity.
- In this scenario, $\text{Profit} = (P - AC) \times Q$ yields a negative number, representing a loss.
The Breakeven Point: The price of $17$ serves as the breakeven point. It is the minimum value on the average cost curve. To generate a profit, a firm must receive a price that exceeds this minimum.

Entry Incentive: Firms will seek to enter an industry in the long run if the price is above the average cost (P > AC), as this indicates positive economic profits.
Exit Incentive: Firms will exit an industry in the long run if the price falls below the average cost curve (P < AC), as they are incurring losses.
Equilibrium: When price equals the minimum average cost ( $P = AC$ ), profits are zero, and there is no incentive to enter or exit.
Normal Profits vs. Zero Profits: In economics, "zero profits" means the firm is covering all costs, including the opportunity costs of labor and capital. This is what is colloquially referred to as "normal profits."

Definition of Sunk Costs: A sunk cost is a cost that, once incurred, cannot be recovered.
Entry Barriers (The Oil Well Example): Drilling an oil well is a sunk cost. A firm should not enter an industry just because the price is slightly above average cost. They must expect the price to remain above average cost long enough to recover the initial entry (sunk) costs.
Exit Barriers: Exit costs can also exist. In the United States, exiting the oil industry might require filling a well with cement or shuttering it.
- Because of these costs, firms might "weather the storm" during periods where P < AC rather than exiting immediately.
- Firms only exit if they expect the price to stay below the average cost for an extended period of time.
Lifetime Profits: Entry and exit are calculated based on expected lifetime profits rather than just immediate, short-term data.