ECON30005 · Money and Banking
A Simple OLG Model with Money and Money Neutrality
Weeks 2-3 build the subject's central structural model: an overlapping-generations (OLG) economy where the young save in capital and real money balances and consume when old. Solving the saver's problem delivers a money-demand equation, price determination P = M/[N(w − k)], and the result that steady-state capital does not depend on the money stock — so a permanent money increase is neutral in the long run. This is the hardest optimisation in the subject and a prime Section 3 model; expect true/false on neutrality and a worked derivation.
What this chapter covers
- 01The OLG environment: two-period-lived generations, work when young, consume when old, save in capital and real balances
- 02Utility from real balances and old-age consumption: u = θ·log(m/P) + (1−θ)·log(c_o)
- 03Budget constraints and the first-order condition giving money demand m/P = θ(1 + 1/i)w
- 04Money demand rises with income w and falls with the nominal interest rate i — the Keynesian result, derived analytically
- 05The exact nominal rate (1 + i) = (1 + r)(1 + π) versus the Fisher approximation i ≈ r + π
- 06Price determination: P = M/[N(w − k)], nominal money over the real goods the young sell
- 07Adding production y = A·k^α, factor prices r = αA·k^(α−1) and w = (1−α)A·k^α, the transition equation and the steady state
- 08Long-run neutrality: steady-state capital is independent of M, so a permanent money increase raises P equiproportionately with no real effect
Long-run neutrality of a permanent money increase
- +1Identify what pins down the real side. Steady-state capital solves k = {A(1−θ)α(1−α)/[θ + α(1−θ)]}^(1/(1−α)), which contains no M — capital per worker does not depend on the money supply. Hence the real wage w = (1−α)A·k^α and the real rate r = αA·k^(α−1) are also unchanged.
- +1So (i): capital per worker k, output y = A·k^α, the real wage and the real interest rate are all unchanged by the money increase. Real allocation is untouched.
- +1Price level (ii): since w and k are unchanged, the real quantity of goods the young sell, N(w − k), is unchanged, so P = M/[N(w − k)] is proportional to M. With M′ = 1.25M, the new price level is P′ = 1.25P — the price level rises by exactly 25%.
- +1Real balances (iii): nominal money and the price level both rise 25%, so real money balances m/P are unchanged, and so is every other real variable. A permanent, anticipated money increase changes only nominal magnitudes — money is neutral in the long run.
Key terms
- Overlapping-generations (OLG) model
- A model where two-period-lived generations coexist: the young work and save (in capital and real money balances), the old consume. It is the subject's core structural model of money demand, price determination and neutrality.
- OLG money-demand equation
- The solution m/P = θ(1 + 1/i)w to the young saver's problem: real money demand rises with income w and falls with the nominal interest rate i, reproducing Keynesian liquidity preference from micro-foundations.
- Price determination (OLG)
- P = M/[N(w − k)]: the price level equals the nominal money held by the old divided by the real goods the young sell them. Holding the real side fixed, P is proportional to M.
- Exact nominal interest rate
- (1 + i) = (1 + r)(1 + π), linking the nominal rate to the real rate r and gross inflation. The Fisher approximation i ≈ r + π drops the r·π cross term and is valid only when r and π are small.
- Steady state
- A situation where capital per worker is constant across periods, so output, the real wage, the real rate and (with constant M) the price level are constant and inflation is zero. Steady-state capital is independent of the money supply.
- Long-run money neutrality
- The result that a permanent, one-off change in the money supply leaves all real variables unchanged and raises the price level in the same proportion — a consequence of steady-state capital not depending on M.
A Simple OLG Model with Money and Money Neutrality FAQ
Why does a permanent money increase leave the real economy unchanged?
Because steady-state capital per worker is determined by preferences and technology (θ, α, A) and contains no money term. Since capital pins down the real wage and the real interest rate, those are unchanged too, and with the real goods the young sell fixed, the price level P = M/[N(w − k)] simply scales with M. Nominal money and prices rise together, so real balances and every real variable are unchanged. That is long-run neutrality.
When should I use the exact nominal rate rather than the Fisher approximation?
Use the exact relation (1 + i) = (1 + r)(1 + π) whenever r or π is large, because the approximation i ≈ r + π drops the r·π cross term. For example a real return of 12% with 25% inflation gives an exact nominal rate of (1.12)(1.25) − 1 = 40%, but the approximation gives only 37% — the 3% gap is r·π. In the exam, if the numbers are small the approximation is fine; if they are large, use the product form.
What is the intuition for the OLG money-demand equation?
The young split their wage between capital, which earns the real return, and real money balances, which pay no nominal return but yield liquidity services in utility. The first-order condition trades these off, and the nominal interest rate i is the opportunity cost of holding money. That is why m/P = θ(1 + 1/i)w falls with i and rises with income w — the same comparative statics Keynes assumed, here derived analytically.
Can AI help me with the OLG model in ECON30005?
Yes. This is the model students most often ask Sia about. It can set up the young and old budget constraints, take the first-order condition, and solve for money demand or the steady state one line at a time, and it can check your neutrality reasoning. Use it to rehearse the derivation rather than to answer a graded quiz, and confirm assessment details on Canvas.
Exam move
This is the derivation to over-practise. Write out, from a blank page, the young and old budget constraints, the substitution to a single-variable objective, the first-order condition, and the money-demand solution m/P = θ(1 + 1/i)w — then state the two comparative-statics signs. Separately, be fluent with price determination P = M/[N(w − k)] and the neutrality argument: steady-state k has no M, so real variables are fixed and P scales with M. Keep the exact nominal-rate relation (1 + i) = (1 + r)(1 + π) alongside the Fisher approximation and know the worked counterexample where they diverge. Because this model recurs in Weeks 3, 5 and 8, mastering it early pays off repeatedly, and it is a prime candidate for the 10-mark Section 3 worked model. Confirm the exam structure on Canvas.
Working through A Simple OLG Model with Money and Money Neutrality in ECON30005? Sia is AskSia’s AI Economics tutor — ask any ECON30005 A Simple OLG Model with Money and Money Neutrality question and get a clear, step-by-step explanation grounded in how ECON30005 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
