Source: Lecture notes / textbook Ch. 2 (Texas A&M University)
Tags: marginal analysis, profit maximisation, demand function, inverse demand, revenue function, cost function, profit function, optimal price, optimal quantity, table method, calculus method, price setter, price searcher, Panera Bread, microchip manufacturer
Firms choose price and quantity to maximise profits, which requires knowing demand, revenue, and costs. You can estimate a linear demand curve from two price-quantity observations, then derive a profit function and find its maximum either by brute-force table methods (laborious, inexact) or by taking a derivative and setting it to zero (precise, fast).
Demand function
The relationship between price and quantity demanded: Qd = f(P). Shows how many units consumers will buy at each price.
Inverse demand function
The same relationship rearranged to express price as a function of quantity: P = f(Q). This is the form plotted with P on the y-axis and Q on the x-axis.
Revenue (R)
Total money the firm takes in. Calculated as R = P x Q. When expressed as a function of Q using the inverse demand, it becomes a quadratic (curved) function that captures the trade-off between price and quantity.
Cost function (C)
Total cost of production. In the examples here, it takes a linear form: C = Fixed Costs + (Variable Cost per Unit) x Q.
Fixed costs (overheads)
Costs the firm pays regardless of how much it produces, e.g. rent, plant operation costs.
Variable costs
Costs that change with the quantity produced, e.g. raw materials, labour per unit.
Profit function (Pi)
Pi = R - C. The difference between total revenue and total cost. For profit maximisation, you want to find the Q that makes this as large as possible.
Price setter / price searcher
A firm that has some control over the price it charges, as opposed to a price taker in perfect competition.
Marginal profit
The change in profit from producing one additional unit of output. In discrete terms: Delta Pi / Delta Q. In continuous (calculus) terms: dPi/dQ.
Cannibalization
When a new product takes sales away from the firm's own existing products rather than from competitors. A key concern for multiproduct firms.
You need two (P, Q) observations to estimate a linear demand line. The Panera Bread example uses two store locations testing different prices for the Chipotle Bacon Melt:
Store A (Bee Caves, Austin): P = $8.50, Q = 50 sandwiches/day
Store B (College Station): P = $8.00, Q = 75 sandwiches/day
To find the inverse demand equation (P as a function of Q):
Calculate the slope: (change in P) / (change in Q) = (-0.50) / (25) = -0.02
Plug one point into P = -0.02Q + b to find the intercept: 8.5 = -0.02(50) + b, so b = 9.5
Inverse demand: P = -0.02Q + 9.5
The slope is negative because of the law of demand: as price falls, quantity demanded rises.
Where does the demand come from? Two sources: old customers (risk of cannibalising existing products) and new customers (poached from competitors or genuinely new to the market).
This is the textbook's main worked example. A price-setting firm produces microchips in lots of 100.
Units throughout: Q is in lots, P is in thousands of dollars.
Demand and inverse demand:
Demand: Qd = 8.5 - 0.05P
Inverse demand: P = 170 - 20Q
The firm can choose either P or Q, not both. Once one is set, the demand curve pins down the other.
Revenue = P x Q. Substituting the inverse demand:
R(Q) = (170 - 20Q) x Q = 170Q - 20Q²
This is a parabola (opens downward). At low Q, price is high so revenue grows. Eventually, the price drops so much that revenue falls despite higher volume. Maximum revenue sits in the middle, but maximum revenue is not the goal: maximum profit is.
Variable cost: $38,000 per lot (materials, labour)
Fixed cost: $100,000 to run the plant regardless of output
C(Q) = 100 + 38Q (in thousands)
Example check: producing 3 lots costs C = 100 + 38(3) = $214,000.
Pi = R - C = (170Q - 20Q²) - (100 + 38Q)
Pi(Q) = -20Q² + 132Q - 100
This is also a downward-opening parabola. Its peak is the maximum profit.
Plug in whole-number values of Q and calculate profit:
Q (lots) | Profit (thousands) |
|---|---|
1 | -100 + 132 - 20 = 12 |
2 | -100 + 132(2) - 20(4) = 84 |
3 | -100 + 132(3) - 20(9) = 116 |
4 | -100 + 132(4) - 20(16) = 108 |
Maximum lies somewhere between Q = 3 and Q = 4. The method is laborious and inexact, and choosing the wrong integer could mean a meaningful profit loss.
Add a column tracking the change in profit as Q increases by 1:
Q: 1 to 2, marginal profit = 84 - 12 = 72 (positive, keep going)
Q: 2 to 3, marginal profit = 116 - 84 = 32 (positive, keep going)
Q: 3 to 4, marginal profit = 108 - 116 = -8 (negative, stop)
Rule: keep increasing output as long as marginal profit is positive. Stop when it turns negative. Still limited to whole numbers.
Take the derivative of the profit function and set it equal to zero:
dPi/dQ = -40Q + 132 = 0
Solving: Q = 132 / 40 = 3.3 lots
Plug back into inverse demand: P = 170 - 20(3.3) = $104 thousand
Plug into profit function: Pi = -20(3.3²) + 132(3.3) - 100 = $117,800
Advantages of the calculus method: it is faster and gives exact (non-integer) answers. The table method would have picked Q = 3 with profit = $116,000, missing $1,800 in potential profit.
Inverse demand (microchip example): P = 170 - 20Q
Revenue: R(Q) = 170Q - 20Q²
Cost: C(Q) = 100 + 38Q
Profit: Pi(Q) = -20Q² + 132Q - 100
Optimal Q (calculus): set dPi/dQ = 0, solve for Q
Slope of a line from two points: (P₂ - P₁) / (Q₂ - Q₁)
⚠️ The table method gives approximate answers only. The calculus method gives the exact optimum. Know both, but understand why the calculus approach is superior.
⚠️ A firm can choose P or Q, not both. The demand curve links them. Do not treat P and Q as independent decisions.
⚠️ Marginal profit turning negative is the signal to stop increasing output. This logic underpins all of marginal analysis.
⚠️ Revenue is maximised at a different Q than profit. The exam may test whether you can distinguish the two.
⚠️ Know how to estimate a linear demand from two (P, Q) points. This is a practical skill tested early in the chapter.
Q: How do you estimate a linear inverse demand function from two price-quantity observations?
A: Calculate the slope as (change in P) / (change in Q), then plug one point into P = slope x Q + b to solve for the intercept b.
Q: Why is the revenue function a curve (parabola) even though the demand function is a straight line?
A: Because R = P x Q, and substituting the linear inverse demand for P gives a quadratic in Q. The trade-off between rising quantity and falling price creates the curvature.
Q: Using the microchip example, what is the profit-maximising quantity, price, and profit level?
A: Q = 3.3 lots, P = $104,000, Profit = $117,800. Found by setting dPi/dQ = 0.
Q: What is the main disadvantage of the table (schedule) method for finding maximum profit?
A: It only evaluates whole-number quantities, so it is both laborious and inexact. It can miss the true optimum that falls between integers.
Q: Why does the profit function peak at a different quantity than the revenue function?
A: Profit subtracts costs from revenue. Costs increase with Q, so the profit peak is pulled to a lower Q than the revenue peak.
marginal analysis, profit maximisation, optimal pricing, demand estimation, inverse demand function, revenue function, cost function, profit function, price setter, price searcher, table method, schedule method, brute force method, calculus optimisation, derivative set to zero, first-order condition, Panera Bread pricing, microchip manufacturer, cannibalization, multiproduct firm, law of demand, fixed costs, variable costs, marginal profit