Welcome to your study space
Understand economics at your own pace: step-by-step guided courses, and exercises to practice.
The fiscal multiplier
Why €1 spent by the state can return more than €1 of GDP — but rarely as much as people say.
ΔY = k × ΔG, with k = 1 / (1 − c)Problem / motivation
The government launches €2 billion of public works. Will the country's activity rise by less than €2 billion, by exactly €2 billion… or by more?
Keynes answers: by more — and here is why. The worker paid on the building site does not put all their wage aside: they spend part of it, which becomes a shopkeeper's income, part of which is spent in turn… The initial spending RICOCHETS, wave after wave. This amplification has a name: the fiscal MULTIPLIER. Two pieces of notation first: Δ (the Greek letter “delta”) means “change in”, and Y is the country's total output — GDP (course “GDP”). So ΔY = “how much GDP moves”, ΔG = “how much public spending moves” (G for Government).
That leaves the question of HOW MUCH it ricochets. It all depends on the share of each extra euro received that people spend again — economists call it the marginal propensity to consume, written c: “marginal” means “on each ADDITIONAL euro”, not on the whole income. This course first builds the formula wave by wave; then it makes the return journey, from the paper model to the figures economists ACTUALLY measure — two worlds further apart than you might imagine.
Assumptions
In your view, which assumptions are needed for this model to hold? Jot down your ideas — no lead is wrong, this is your worksheet.
Your worksheet is still empty. Go for it: propose at least one idea.
0 idea(s) proposedFormalization
Let's follow €2bn injected (ΔG), with c = 0.75 — households spend 75 cents of every extra euro received. Wave 0: the €2bn paid to the firms on the site. Wave 1: their employees spend 75% of it again, that is 2 × 0.75 = €1.5bn. Wave 2: those who banked that €1.5bn spend 75% of it, that is €1.125bn. Wave 3: €0.84bn… Each wave is the previous one MULTIPLIED by c. The total GDP added is the sum of them all: ΔY = ΔG × (1 + c + c² + c³ + …).
This constant multiplier from one wave to the next is called the RATIO of the sequence — it is constant thanks to assumption 2, and it equals c thanks to assumption 3. Since c is below 1, the waves SHRINK: 2; 1.5; 1.125; 0.84… An endless addition can therefore give a FINITE total — exactly like 1 + ½ + ¼ + ⅛ + … which comes to 2, not infinity. Mathematicians call this a GEOMETRIC SERIES.
How do you sum an infinity of terms? By a balance-sheet trick. Call S the total of the ricochets for €1 injected: S = 1 + c + c² + c³ + … Multiply it by c: cS = c + c² + c³ + … — which is exactly S, minus its first term: cS = S − 1. So S = 1 + cS, hence S − cS = 1, that is S(1 − c) = 1 and finally S = 1 / (1 − c). That number S is the MULTIPLIER k. Three lines, and it is born.
All that remains is to apply k to the injection: ΔY = k × ΔG, with k = 1 / (1 − c). Read it both ways: the more people spend again (a large c), the smaller the denominator 1 − c, and the more k explodes — c = 0.75 gives k = 4, c = 0.9 gives k = 10. And if you meet the notation ΔY = ΔG / (1 − c) in the simulator, do not look for a second formula: it is the same one, with k replaced by its value. Click each term:
Solving / calculation
Let's take the €2bn of public works again with c = 0.75. First count the waves by hand, then check with the formula — and above all, make the journey from the paper model to the real figures.
- Waves 0 to 3, one by one (each = the previous × 0.75)
2 + 1.5 + 1.125 + 0.84≈ €5.47bn — already 68% of the total - The total of ALL the waves, through the formula
k = 1 / (1 − 0.75) = 4; ΔY = 4 × 2ΔY = €8bn - From toy to reality — each wave also loses taxes (t ≈ 30%) and imports (m ≈ 25%)
ratio = 0.75 × (1 − 0.30) − 0.25 = 0.275k = 1 / (1 − 0.275) ≈ 1.4 - The same public works, in an open and taxed economy
ΔY ≈ 1.4 × 2≈ €2.8bn — and not 8
Same mechanism, two worlds: the simple model promises €8bn, the version with leaks returns €2.8bn. And the empirical studies (interpretation, point 2) put the real multiplier between about 0.5 and 1.5 depending on conditions — less still. The ricochet is quite REAL; it is its SCALE that the paper model exaggerates. Keep both: the mechanism to understand, the leaks so as not to have illusions.
k = 1 / (1 − ratio), with ratio = c × (1 − t) − mBoth leaks start at zero: you are in the simple model. Raise taxes and imports to watch k collapse towards real-world values — and look at the number of waves needed for the effect to work through.
- Wave 0 (injection) 2
- Wave 1 1,5
- Wave 2 1,13
- Wave 3 0,84
- All the following ones 2,53
- Total effect (ΔY) 8 %
Economic interpretation
Three readings so as not to draw the wrong conclusion: which amplifies most, what it is really worth, and when it arrives.
Public spending enters the circuit IN FULL: it is wave 0. A tax cut, by contrast, first lands in households' pockets, and they spend only the share c of it — wave 0 is already trimmed. Hence a smaller tax multiplier: c/(1 − c) instead of 1/(1 − c). With c = 0.75: 3 against 4. Two readings of that gap, true for any c: the tax multiplier is exactly c TIMES the spending one (3 = 0.75 × 4), and spending always keeps exactly ONE point of lead (4 − 3 = 1; at c = 0.8, it would be 5 − 4). (In the MODEL. Empirically the ranking is disputed: some targeted tax cuts do better than some spending. Keep the mechanics, not a league table.)
The k = 4 of the simple model exists nowhere. Empirical work puts the public spending multiplier at around 0.5 to 1.5 — and above all, it DEPENDS on conditions: close to 0 to 0.5 when the economy runs at full tilt (capacity is missing: assumption 1 violated, and the stimulus feeds prices), but clearly above 1 in a recession, when unemployed people and idle machines are waiting (Auerbach & Gorodnichenko; Ramey's survey). The more open an economy, the more the leak through imports trims it further.
ΔY = k × ΔG says NOTHING about the calendar — yet each wave takes months: to pay, to bank, to spend again. In the simulator, at c = 0.75, it takes 9 waves to reach 90% of the effect, that is more than two years if a wave lasts a quarter. Like monetary policy (course “The policy rate and transmission”), a stimulus arrives with a LAG — at the risk of producing its effects once the recession is already over.
Limits / critiques
Exercises
The marginal propensity to consume is c = 0.8. What is the value of the multiplier k (simple model)?
The government injects €4bn and c = 0.6. How much is wave 2 worth — the second ricochet AFTER the injection (in €bn)?
Simple model, c = 0.75. The government cuts taxes by €2bn instead of spending €2bn. By how much does GDP rise (in €bn)?
True or false: the fiscal multiplier measured by economists is commonly 4.
With k = 3, the government increases its spending by €2bn. By how much does GDP rise (in €bn)?
A realistic economy: c = 0.75, tax leak t = 30%, imports m = 25%. What is k (one decimal)?
When is the multiplier at its LARGEST?
True or false: the total effect of a stimulus works through in a few weeks.