Businesses use cost, revenue, and profit functions to describe how money changes as the quantity produced and sold changes. Calculus makes these functions more useful by measuring their instantaneous rates of change. These rates are called marginal cost, marginal revenue, and marginal profit.
They help estimate what happens when production increases by one more unit.
If C(q) is cost, R(q) is revenue, and P(q) is profit, then their derivatives tell how fast each quantity changes at a specific output q. Marginal cost C'(q) estimates the extra cost of producing the next unit, while marginal revenue R'(q) estimates the extra income from selling the next unit. Marginal profit P'(q) = R'(q) - C'(q) shows whether producing another unit is expected to increase or decrease profit.
A key decision rule is that profit is often maximized near the quantity where marginal revenue equals marginal cost, with attention to whether profit changes from increasing to decreasing.
Key Facts
- Profit function: P(q) = R(q) - C(q).
- Marginal cost: MC = C'(q), the instantaneous rate of change of cost with respect to quantity.
- Marginal revenue: MR = R'(q), the instantaneous rate of change of revenue with respect to quantity.
- Marginal profit: MP = P'(q) = R'(q) - C'(q).
- Profit has a critical point when P'(q) = 0, which means MR = MC.
- A maximum profit point usually occurs where P'(q) changes from positive to negative.
Vocabulary
- Cost function
- A function C(q) that gives the total cost of producing q units.
- Revenue function
- A function R(q) that gives the total income from selling q units.
- Profit function
- A function P(q) that gives total profit and is found by subtracting cost from revenue.
- Marginal value
- The derivative of a business function, interpreted as the approximate change caused by producing or selling one more unit.
- Tangent line
- A line that touches a curve at one point and has the same slope as the curve at that point.
Common Mistakes to Avoid
- Confusing total cost with marginal cost. Total cost C(q) is the full cost at q units, while marginal cost C'(q) is the rate at which cost is changing at q units.
- Assuming marginal revenue is always the selling price. This is only true for a constant price model, and in many demand models the price changes as quantity changes.
- Maximizing revenue instead of profit. A business can have high revenue but low or negative profit if costs are also high.
- Stopping when MR = MC without checking the behavior nearby. MR = MC gives a critical point for profit, but you must confirm that profit changes from increasing to decreasing.
Practice Questions
- 1 A company has cost C(q) = 500 + 12q + 0.04q^2 dollars. Find the marginal cost C'(q), then estimate the cost of producing the 101st unit using q = 100.
- 2 A product has revenue R(q) = 80q - 0.2q^2 and cost C(q) = 300 + 20q. Find P(q), P'(q), and the production quantity that maximizes profit.
- 3 At q = 250 units, a company has MR = 18 dollars per unit and MC = 24 dollars per unit. Explain whether producing slightly more than 250 units is expected to increase or decrease profit, and why.