Maths IA · Modelling · topic 83 of 97
How do banks create money? The money multiplier · IB Maths IA idea: guide, sample graph and step-by-step plan
a chain of deposits: that's your seriesSomeone deposits 1,000. The bank lends most of it out, and that loan is deposited at another bank, which lends again. You're going to find how far this chain can go, and whether banks really work that way.
- ✓ you wonder where money actually comes from
- ✓ you like a series that has a neat sum to infinity
- ✓ you want to test a textbook formula against real data
The big idea
Each time a bank receives a deposit it keeps a fraction as reserves and lends the rest, and the loan comes back as a new deposit. The deposits form a geometric sequence, and their total is a geometric series. For a reserve ratio R the series has a finite limit, so the whole banking system turns one deposit into a fixed multiple of it. Add the public's wish to hold some cash and the multiplier changes again. The algebra is the easy part. The marks come from testing the model: compare the formula with the real ratio of broad money to base money, explain the gap, and say what the model leaves out about how modern banks actually lend.
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 →
Write the deposit in round k and add the first n rounds, then let n grow.
→ A finite geometric sum and then an infinite one. Say when the infinite sum is allowed and how fast the chain nears its limit.Find the number of rounds needed to create 90% or 99% of the final total, for several reserve ratios.
→ An exponential inequality solved with logarithms. A lower R gives a bigger limit but needs many more rounds.Suppose the public keeps a fraction c of its money as notes and coins. Rebuild the series.
→ A new ratio in the series and a new multiplier. Compare it with the simple one and say which is closer to data.Take a central bank's broad money and base money series and compute their ratio over time.
→ Your formula against a published number. The gap, and how it moves in a crisis, becomes part of your evaluation.Treat the multiplier as a function of R and find how much it changes for a small change in R.
→ Differentiation of a reciprocal function. Show why a cut from 10% to 5% matters far more than 20% to 15%.Read how central banks now describe money creation, where loans create deposits first.
→ The model flips its direction of cause. Decide what still holds and what does not.Your Maths IA research question: run the check
which ratio? which data?Write your own question first. Then make sure it ticks every box:
- Names the quantity you will find: the multiplier, the number of rounds, or the gap between model and data.
- States the model's assumptions: the reserve ratio, whether cash is held, and that every loan is re-deposited.
- Says which real series you compare with, from which central bank and which dates.
- Goes beyond quoting 1/R: there is a series to derive and a test of the result.
- Says what you will not model: interest rates, loan demand, bank capital and regulation.
The model
A deposit comes in, the bank keeps a share and lends the rest, and the loan is deposited again. Draw the first rounds for your own reserve ratio.
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.
Why this question, what a reserve ratio is, and what you expect the total to be before you calculate.
The deposit in each round, the sequence spotted, every symbol defined, the first rounds checked in a table.
The sum of n rounds, the sum to infinity, and why the common ratio makes it converge.
The number of rounds needed to reach a chosen share of the limit, for several reserve ratios.
The rebuilt series, the new multiplier and a comparison with the simple one on one graph.
Your formula beside a published ratio of broad money to base money. A number for the gap and a reason for it.
Back to your aim: the multiplier for your assumptions, and how close real banking comes to it.
Loans that create deposits, banks that hold spare reserves, changing demand. Say which assumption matters most.
Where every series, rule and explanation came from.
What your Maths IA graph could look like
draw the limit as a line!A made-up banking system to show the form of a strong graph, not the answer: a 1,000 deposit and a 10% reserve ratio. Yours will look different. example only
- The sum rises quickly at first, then flattens against the limit. The graph shows that at a glance.
- Axis labels give units: rounds on x, currency units on y.
- Draw the limit as its own line, so the reader sees what the series is heading for.
- Plot your spreadsheet rows as points: agreement with the curve is your evidence the formula is right.
- Another graph to consider: the multiplier against the reserve ratio, one curve for each cash ratio.
The maths in action
worked through, not sketched!The graph above was a sketch with made-up numbers. This figure works the idea through from the model itself, so you can see what a finished figure looks like. The source line under it says where every number comes from.
What does the reserve ratio do to the limit?
Rank the reserve ratios on one number: the limit of total deposits. Work it out for your own ratios, and explain why halving R doubles it.
Your game plan
Tick steps off as you go. We'll remember on this device.
0 of 10 done
- Read and predictFind out what a reserve ratio is. Before any algebra, guess how many times a 1,000 deposit can be lent out and re-deposited.
- Fix the assumptionsChoose a reserve ratio you can justify and decide whether the public holds cash. Write each assumption down.
- Write the sequenceFind the deposit in round 1, 2, 3 and spot the common ratio. Define every symbol.
- Trial run (1 lesson)Build a 10-round spreadsheet and compare its running total with a formula for the sum. Fix any mismatch now.
- Sum to infinityDerive the limit, and say why the common ratio lets the series converge.
- Count the roundsUse logarithms to find how many rounds reach 90% of the limit, for three or four reserve ratios.
- Add cash holdingsRebuild the series with a cash ratio and find the new multiplier. Compare the two models on one graph.
- Go further, with a reasonCompare with a real ratio of broad money to base money, or differentiate the multiplier with respect to R. HL: this is where the rigour for criterion E comes from.
- Reflect as you goAfter every result, write one line on what it means in the real setting and what it assumes. These lines become your evaluation.
- Cut, check, then sendCheck every graph has labelled axes with units, every symbol is defined once and the conclusion answers your aim. Then send your teacher one draft.
The maths you'll use
- Geometric sequence – Write the deposit in round as and identify the first term and common ratio.
- Finite and infinite geometric series – Sum the first rounds, then take the limit to get and state the condition for convergence.
- Logarithms – Solve for to find how many rounds create 90% of the limit.
- Rational functions and differentiation – Treat and as functions; differentiate to measure sensitivity to .
- Comparing model and data – Compute a real ratio from two published series, plot it against the model and quantify the gap.
Key equations
Syllabus links
- 1.3Geometric sequences and series
- 1.7Rational exponents and laws of logarithms
- 1.8Infinite geometric series
- 2.9Exponential and logarithmic functions
- 5.6Standard derivatives; chain, product and quotient rules
- 1.3Geometric sequences and series
- 1.9Laws of logarithmsHL
- 1.11Infinite geometric seriesHL
- 2.5Linear, quadratic, exponential, cubic and sinusoidal models
- 2.9Logarithmic, logistic, piecewise and half-life modelsHL
Read up first
Data and sources to start from
Download a broad money series and a monetary base series from one central bank for the same dates, and divide one by the other.
- FRED: M2 money stock (M2SL), United States
- FRED: Monetary base, total (BOGMBASE), United States
- Bank of England: Statistics
References to cite
- McLeay, M., Radia, A., & Thomas, R. (2014). Money creation in the modern economy. Bank of England Quarterly Bulletin, 54(1), 14-27.
- Mankiw, N. G. (2021). Macroeconomics (10th ed.). Worth Publishers. (Chapter on the monetary system, money multiplier.)
- Federal Reserve Bank of St. Louis. (n.d.). Federal Reserve Economic Data (FRED). https://fred.stlouisfed.org/
Cite what you actually use, in your own format, and say where each number came from.
Traps to avoid in this Maths IA
check the ratio is below 1- Quoting 1/R as the multiplier without deriving it from the series.
- Using the infinite sum without checking that the common ratio is between −1 and 1.
- Forgetting that the first deposit is counted in the total, and then being out by one term.
- Assuming every loan is deposited again, with no cash held and no spare reserves.
- Comparing the formula with a real ratio but not saying which series (M1, M2, M3) you used.
- Treating the textbook story as how banks really work without checking it against a central bank source.
More Maths IA ideas like this
How much should you bet or invest? The Kelly criterion●●○
Credit cards: how long does the minimum payment take?●●○
Retirement savings: how long will the money last?●●○
Bitcoin's 21 million cap: a geometric series in code●●○
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.