CB2 · explained

National income and the multiplier: where the rest of the spending comes from

Spend a thousand crore and national income rises by more than a thousand crore. That is not accounting magic, and the size of the "more" is not a free parameter: it is fixed by how much of each round of income leaks out before it is spent again.

CB2 · Business Economics · 10 min read

After this page you can

Trace a spending injection round by round, calculate the multiplier from the marginal propensities, and check that the leakages add back to the injection

The identity everything rests on

National income accounting starts from one fact: every rupee of output is simultaneously somebody's income and somebody's expenditure. A loaf is sold for twenty rupees; that twenty is the value of what was produced, it is the twenty the buyer spent, and it is distributed as wages, rent, interest and profit to the people who made it.

So the same total can be measured three ways and must come out the same: add up what was produced, add up what was earned, or add up what was spent. They are not three estimates of the same thing; they are three readings of one circulation. In practice the three differ because of measurement error, and the gap is the error, not a finding.

The expenditure reading is the one the rest of macroeconomics uses:

Y = C + I + G + (X − M)

consumption, investment, government spending, and exports less imports. Imports are subtracted because they are spending on output produced somewhere else, and Y is meant to measure output produced here.

Two distinctions are worth fixing because they are the ones that get confused. GDP measures output produced inside the borders, whoever owns the business. GNP measures output belonging to residents, wherever it was produced — so it adds income residents earn abroad and subtracts income foreigners earn here. A foreign-owned factory's profits count in India's GDP but not in its GNP; money sent home by an Indian working overseas counts in GNP but not GDP. And nominal figures use this year's prices while real figures use a fixed year's prices, so only real figures can tell you whether more was actually produced.

The mechanism: one person's spending is another's income

Now the part that makes macroeconomics different from arithmetic.

Government builds a road and pays out a thousand crore. That is one thousand crore of extra demand — but it does not stop there. The contractors, labourers and suppliers who received it are now a thousand crore richer, and they will spend part of it. What they spend is somebody else's income, who spends part of that. And so on.

The fraction of extra income that gets spent is the marginal propensity to consume, written c. It is marginal and it is out of extra income, which is why it is not the same as the average share of income consumed. The rounds are then

1,000 + 1,000c + 1,000c² + 1,000c³ + …

a geometric series with ratio c. Because c is between 0 and 1, it converges, and the sum is

total effect = 1,000 × 1/(1 − c)  ·  multiplier = 1/(1 − c)

The multiplier is not an extra assumption bolted on. It is the sum of a series each of whose terms is a real transaction between real people.

Leakages are what actually set the size

Here is the more useful way to read that formula, and the one that survives when the model gets more realistic. 1 − c is the fraction of each round's income that does not get passed on — the part that leaks out of the circulation. So

multiplier = 1 ÷ (the fraction of extra income that leaks out)

In the simplest model the only leakage is saving. A real economy has three.

Tax takes a share t of income before the household sees it, so only (1 − t) reaches disposable income.

Saving takes a share (1 − c) of what is left.

Imports take a share m of income and send the demand abroad, where it becomes somebody else's output.

Induced domestic spending per rupee of extra income is therefore c(1 − t) − m, and the multiplier is one over one minus that. Equivalently — and this is the version worth carrying — it is one over the total leakage rate:

leakage rate = t + (1 − c)(1 − t) + m
multiplier = 1 ÷ leakage rate

The two expressions are the same number, and checking that they agree is a good habit. The worked example does it and then checks something better: at the new equilibrium, the extra tax plus the extra saving plus the extra imports add back to exactly the original injection. They have to. The economy comes to rest precisely when the leakages have grown to match what was being injected, and that is why the multiplier is one over the leakage rate rather than some other function of it.

What the multiplier does not promise

Four conditions, and each one is a real limit rather than a caveat.

Spare capacity. The multiplier delivers extra output only if there are idle workers and idle machines to produce it. At full employment the extra demand meets a supply that cannot grow, and the result is higher prices instead. This is the single biggest qualification and the one that decides whether a stimulus is a good idea.

Time. The rounds are not instantaneous. Each takes as long as it takes for income to be received and spent again, so the full effect arrives over quarters, not weeks. The example below shows how much of it has landed after five rounds.

No offsetting reaction. If the extra government borrowing pushes interest rates up, investment falls and part of the gain is cancelled. That is crowding out, and it is the reason the multiplier measured in a fuller model is smaller than the one calculated here.

It works in reverse. A cut in spending contracts income by the same multiple. Nothing in the mechanism cares about the direction, and the rounds run just as reliably downwards.

One more thing the formula quietly tells you: a more open economy and a more heavily taxed one both have smaller multipliers, because both leak more per round. Fiscal policy is simply a weaker instrument in an open economy, and no amount of determination changes the arithmetic.

Questions students ask

Where does the extra output physically come from?

From resources that were idle. That is the whole claim, and it is why the condition matters so much: the multiplier says demand was the binding constraint, so putting demand into the system puts unemployed people and unused machines back to work. If nothing was idle, there is nothing for the extra demand to activate and it shows up in prices. The multiplier is a statement about an economy below capacity, not a general law.

Is the multiplier the same for every kind of spending?

No, and the reason is the first round. Money spent directly on domestic output starts with the full amount in circulation. A tax cut or a transfer payment starts smaller, because the household saves part of it immediately and imports part of it — the first round is already reduced before the series begins. So a given sum spent on construction moves demand more than the same sum given away, even though both then run through identical later rounds.

Does government borrowing to pay for it cancel the effect?

Partly, by two routes. Borrowing can push interest rates up and reduce investment, which is crowding out. And households expecting future taxes to repay the debt may save more today, which weakens the consumption rounds — the strong form of that argument says they save the entire amount and the effect disappears, which requires households to be remarkably far-sighted and is not what is usually observed. Both effects shrink the multiplier. Neither is generally thought to remove it in an economy with spare capacity.

One thousand crore spent once, passed from hand to hand

02,5005,0001510Round of spending₹ crore so farit never passes ₹5,000 croreafter 5 rounds: ₹3,361.60 crore
  • what had already arrived
  • what this round added
Everybody spends four fifths of whatever they receive, so the first ₹1,000 crore keeps moving and the running total climbs. Each round adds less than the one before. Add every round together and the answer is exactly ₹5,000 crore — but no finite number of rounds ever gets there, which is what the bars show.

A worked example

An invented case, built for this page. A government spends an extra ₹1,000 crore on a road. The figures below are chosen so the arithmetic is exact; they do not describe any real economy or any real project.

  1. Part 1: the simplest case, no tax and no trade

    Marginal propensity to consume c = 0.8, so the only leakage is saving of 0.2 per rupee.

    RoundExtra spendingRunning total
    1₹1,000.00 cr₹1,000.00 cr
    2₹800.00 cr₹1,800.00 cr
    3₹640.00 cr₹2,440.00 cr
    4₹512.00 cr₹2,952.00 cr
    5₹409.60 cr₹3,361.60 cr

    Each round is 0.8 of the one before. The series does not stop at round 5; it sums to

    multiplier = 1 ÷ (1 − 0.8) = 1 ÷ 0.2 = 5
    total effect = 1,000 × 5 = ₹5,000 crore

    Check the running total a second way. After n rounds the series has delivered 1 − cn of its total. After five rounds 0.85 = 0.32768, so 1 − 0.32768 = 0.67232, and 5,000 × 0.67232 = ₹3,361.60 crore — the figure in the table. Two-thirds of the effect has arrived after five rounds and the last third takes considerably longer, which is the practical form of the lag.

  2. Part 2: put the leakages in

    Now a more honest economy. Income tax takes t = 0.20, households spend c = 0.8 of what is left, and the marginal propensity to import is m = 0.15 of income.

    Induced domestic spending per rupee of extra national income:

    c(1 − t) − m = 0.8 × 0.8 − 0.15 = 0.64 − 0.15 = 0.49
    multiplier = 1 ÷ (1 − 0.49) = 1 ÷ 0.51 = 1.9608
    total effect = 1,000 × 1.9608 = ₹1,960.78 crore

    Check it through the leakage rate instead:

    tax 0.20 + saving 0.2 × 0.8 = 0.16 + imports 0.15 = 0.51

    The same 0.51, reached without computing induced spending at all. The two routes have to agree, because what is not passed on is exactly what leaked.

    And note the size of the change: the multiplier has fallen from 5.00 to 1.96. Two leakages, neither of them enormous on its own, have cut the effect of the same ₹1,000 crore by more than half. The multiplier is a statement about leakages, not about the spending.

  3. Part 3: the check that proves the equilibrium

    At rest, national income is ₹1,960.78 crore higher. Work out how much extra leaks out of that higher income each year:

    extra tax = 0.20 × 1,960.78 = ₹392.16 cr
    extra saving = 0.16 × 1,960.78 = ₹313.73 cr
    extra imports = 0.15 × 1,960.78 = ₹294.12 cr
    total leakage = ₹1,000.00 cr

    Exactly the injection. Nothing was arranged to make that happen — it is forced, because (0.20 + 0.16 + 0.15) ÷ 0.51 = 0.51 ÷ 0.51 = 1. The economy stops expanding at precisely the income level where the leakages have grown to swallow the whole of the extra spending, and not a rupee before. That is what the multiplier formula is really saying, and it is the single best check on any multiplier calculation: if the leakages at the new income do not add back to the injection, something is wrong.

  4. Part 4: the same arithmetic in reverse

    Suppose instead the ₹1,000 crore of road spending is cancelled.

    effect = −1,000 × 1.9608 = −₹1,960.78 crore

    The rounds run downwards with the same ratio: the contractor's suppliers lose income, spend less, and so on. Nothing in the mechanism distinguishes the directions, which is why a consolidation large enough to matter is expected to reduce output by a multiple of itself in an economy with spare capacity.

  5. What this answer does not say

    It says nothing about whether output can actually rise by ₹1,960.78 crore. If the economy is already at full employment there are no idle workers or machines to produce the extra goods, the demand meets a fixed supply, and the result is higher prices rather than higher real output. It also assumes nothing else reacts — in particular that interest rates stay put. Let borrowing push rates up and investment will fall, cancelling part of the gain. The next page puts that reaction in and the multiplier shrinks again.

Where this fits in CB2

This page explains one idea from CB2 Business Economics. The full CB2 syllabus, with what each part is examined on, sits on the course page, and past-paper style mock exams with full solutions are where you find out whether the idea has stuck under time pressure.

Other CB2 topics explained

Stuck on a term rather than the idea? The actuarial glossary defines the notation used above. If something here is wrong or unclear, tell us — these pages get corrected.