Maths IA · Modelling · topic 37 of 40

Setting a price: how to maximise profit from real data · IB Maths IA idea: guide, sample graph and step-by-step plan

real data makes this yours

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.

Difficulty●●○ Medium
KitPrice and sales data, a spreadsheet or GDC
TimeAbout 12 to 15 hours
FitsAA and AI, SL and HL
  • ✓ you like calculus with a real purpose
  • ✓ you can find or collect price and sales data
  • ✓ you want to defend a model with evidence

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.

Make it yours: pick one twist

Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.

Swipe for more twists →

Pick a real market

A tuck shop, a bake sale, a second-hand listing site, or published prices and sales for a product.

→ Your own data and a context to explain. Messy data forces decisions about outliers.
Compare demand models

Linear, exponential and power.

→ The same data can give different best prices; residuals decide, and you discuss why.
Use elasticity

Compute price elasticity of demand and look at where it equals −1.

→ Links calculus to economics, and shows why the revenue-maximising price is not the profit-maximising one.
Add costs

A fixed cost, a variable cost, a bulk discount.

→ Your profit function gains terms; a piecewise function needs care.
Test the sensitivity

Re-fit with some points removed, or change each coefficient by 10%.

→ A clear picture of how far the best price moves, and honesty about uncertainty.
Test it for real

A small survey: would you buy at this price?

→ A real check on the model, plus a discussion of survey bias.

Your Maths IA research question: run the check

write it FIRST, then check it

Write your own question first. Then make sure it ticks every box:

  • 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).

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The picture to draw

A demand line fitted to your price and quantity data. The shaded rectangle is the revenue p × q at one price. Draw yours with your own axes and units.

qprevenue = p x qdemand line fitted to your databest price: profit, not revenue!

How your Maths IA could be laid out

The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.

Introduction and aim~1.5 pages
Your question; why this product; what you expect the best price to depend on.
The data~2 pages
Where it came from, how it was collected, units, and anything removed and why.
Choosing a demand model~2.5 pages
Plots, candidate models, regression output explained in your own words, and residuals.
Revenue and profit functions~2 pages
Each function defined with units; where each term comes from; the costs you include.
Finding the maximum~2.5 pages
Derivative, solving, and a check that it is a maximum (second derivative or sign change); sense check against the data.
Sensitivity~2 pages
How the best price moves if the data, the model or the cost changes.
Conclusion~1 page
Your best price, how sure you are, and what a seller should do with it.
Reflectionbuilt in, plus ~0.5 page
Not one paragraph at the end: a comment after each result or choice (a poor fit, a surprise, a decision you changed), then a specific evaluation of the model's limits.
Bibliographynot counted
Every source for data, theory and any code or software used.

What your Maths IA graph could look like

profit AND revenue: nice ✓

Made-up numbers to show the form of a strong graph, not the answer: a stall selling one item at a unit cost of $2. Yours will look different. example only

2468101214160100200300400500600Price p / $ per unitDaily profit / $profit model (unit cost $2)revenue modelprofit from your sales dataprofit peaks at a higherprice than revenue: why?ILLUSTRATIVE · MADE-UP DATA
  • Price on the x-axis and profit on the y-axis, both with units.
  • Profit and revenue on the same axes show why the best price changes once costs are included.
  • Real data will scatter more than this: say how you dealt with the scatter.
  • The peak is where the gradient is zero: calculate it, then check it against the graph.
  • Another graph to consider: demand (q against p) with two fitted models, and residuals underneath each.

Which input moves the best price most?

rank them: don't just list

Re-solve for the best price each time one input changes, then rank the changes by their effect on your result. Here the demand line's two coefficients matter more than a one-dollar error in cost.

Demand intercept (± 10%)8.8%Demand slope (± 10%)8%Unit cost (± $1)5.9%check this input first% change in the profit-maximising priceILLUSTRATIVE · MADE-UP NUMBERS

Your game plan

trial fit before the real data

Tick steps off as you go. We'll remember on this device.

0 of 10 done

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

Maths you'll need

Traps to avoid in this Maths IA

  • 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.

More Maths IA ideas like this

Make sure it's yours

This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.

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