IB Maths IA idea · Banking, markets and economics

Setting a price: how to maximise profit from real data

Raise the price and you sell less. Lower it and you earn less per sale. Somewhere between is the best price, and you're going to find it from real data.

Banking, markets and economics
SL ideal
HL ideal
Maths AA Maths AI Your own data
Syllabus: AA 2.1, AA 2.9, AA 4.4, AA 5.3, AA 5.7, AA 5.8

By Photon Academy · Updated 13 November 2026

These ideas are starting points, not an IA to copy. Never copy an idea blindly: make the question, method and analysis your own, and check your plan with your teacher.

The big idea

Demand data links price p to the quantity q people buy. Fit a model q = f(p); then revenue is pq and profit is (p − c)q for a unit cost c. Differentiate, and the best price is where the gradient is zero. The mathematical heart is your choice of demand model (a line? an exponential decay? a power law?) and what that choice does to the answer. Use real data, justify your model with residuals, and test how sensitive the best price is to uncertainty in the data. At HL, bring in price elasticity or a non-linear demand function, and prove the stationary point is a maximum.

The maths you will use

Regression for demand. Fit $q = a - bp$ , $q = Ae^{-kp}$ and $q = Ap^{-k}$ to price and sales data, and judge each by its residuals.
Revenue and profit functions. Build $R(p) = pq(p)$ and $\Pi(p) = (p-c)q(p)$ with unit cost $c$ , and state the domain where they make sense.
Differentiation and optimisation. Solve $\Pi'(p) = 0$ , prove it is a maximum with $\Pi''(p) < 0$ , and compare with the revenue maximum.
Price elasticity. Compute $E = \frac{dq}{dp}\cdot\frac{p}{q}$ and show that the profit maximum satisfies $\frac{p-c}{p} = -\frac{1}{E}$ .
Sensitivity analysis. Change each coefficient by 10%, or refit with points removed, and measure the percentage change in the best price.

Key equations

\Pi(p) = (p-c)(a - bp), \quad p^{*} = \frac{a + bc}{2b}

q = Ae^{-kp} \quad \to \quad p^{*} = c + \frac{1}{k}

E = \frac{dq}{dp}\cdot\frac{p}{q}, \qquad \frac{p^{*}-c}{p^{*}} = -\frac{1}{E}

The maths in action

Two panels. Left: eleven points showing sales falling as price rises, with a straight line, an exponential curve and a power curve through them. Right: profit curves for the three models peaking near 8.5, 8.1 and rising without a peak in range for the power model. — (a) Simulated sales data with three fitted demand models. (b) Profit against price (unit cost 2): the linear model peaks at a price of 8.52 and the exponential at 8.13; the power model's best price, 26, lies far outside the data, which stop at 13.

Example research question

To adapt, never to copy

For one product sold at 8 or more different prices, which of a linear, an exponential and a power demand model fits the sales data best, and how far does the profit-maximising price change from one model to another and when the data are perturbed?

Change the context, the data or the method so the question is yours. Our Maths IA guide shows how to check it.

Check your research question

Names the product, the market and where the price and quantity data come from.
Names the demand model(s) you will fit and how you will judge them.
Says what the objective is, revenue or profit, and which costs are included.
Says how you will test the answer: sensitivity to the data, a second model, or a real check.
Is narrow: one product, one market and enough price points (at least 6 to 8).

Which course it fits

AA SL ideal
AA HL ideal
AI SL ideal
AI HL ideal

Syllabus links in Maths AA:

AA 2.1 Straight lines
AA 2.9 Exponential and logarithmic functions
AA 4.4 Correlation and the regression line of y on x
AA 5.3 Differentiating powers of x
AA 5.7 Second derivative and graph behaviour
AA 5.8 Maxima, minima, inflexion and optimisation

Steps

Read and predict. Sketch the demand shape you expect and say which price you think wins. Write down why.
Find or collect the data. Choose a source you can explain. Check you have at least 6 to 8 different prices with matching quantities.
Trial run (1 lesson). Plot the data and fit a line in a spreadsheet. Is the scatter reasonable? Are there outliers to decide about?
Choose the demand model. Fit two or three models. Use residuals, not R² alone, and say which you pick and why.
Build the functions. Define revenue and profit for your chosen model. State unit cost, and anything fixed.
Differentiate. Find the best price by hand, and show it is a maximum. Check against the graph and the data's range.
Test the sensitivity. Change one input at a time and re-solve. Rank the effects.
Go further. SL: a second model or a cost change. HL: elasticity or a non-linear demand function, with a justified optimum.
Write, then cut. Write in layout order. One worked derivation, then a table of results.
Check before your teacher's draft. Check notation, units, rounding to a sensible accuracy for the data, and reflection all the way through.

Make it yours

Pick a real market. A tuck shop, a bake sale, a second-hand listing site, or published prices and sales for a product.
Compare demand models. Linear, exponential and power.
Use elasticity. Compute price elasticity of demand and look at where it equals −1.
Add costs. A fixed cost, a variable cost, a bulk discount.
Test the sensitivity. Re-fit with some points removed, or change each coefficient by 10%.
Test it for real. A small survey: would you buy at this price?

HL extension

Prove the elasticity condition for the profit maximum, show that for a power demand model the best price depends only on $k$ and $c$ and exists only if $k > 1$ , and bound the best price using the uncertainty in the fitted coefficients.

The data you need

Record price and quantity sold for one product at 8 or more prices, from a stall, a school shop, an online listing site or a small survey, and say how the data were collected.

Traps

Maximising revenue when you meant profit: they are different questions with different answers.
Choosing the demand model because the software offered it, with no look at residuals.
Extrapolating the fitted line far outside the data (negative sales, endless demand).
Skipping the check that the stationary point is a maximum.
Data with no source, or no explanation of how it was collected.
A best price quoted to the cent when the data supports the nearest dollar.

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