The IS-LM model and crowding out, solved with numbers

CB2 · Business Economics · 11 min read

The problem with the multiplier answer

The multiplier calculation holds the interest rate fixed. That is a strong assumption, because a fiscal expansion raises income, higher income makes people want to hold more money for transactions, and if the quantity of money has not changed then something has to give — and what gives is the interest rate. Once it rises, investment falls, and part of the expansion is cancelled by its own side effect.

IS-LM is the smallest model that captures that feedback. It has two markets, two curves and one job: find the output and the interest rate at which both markets clear at the same time.

The IS curve: the goods market, and why it slopes down

IS is the set of output-and-interest-rate pairs at which the goods market clears — planned spending equals output.

The mechanism is a two-step chain. A higher interest rate makes borrowing dearer, so some investment projects that cleared their hurdle no longer do, and investment falls. Less investment means less demand, which the multiplier then amplifies, so equilibrium output falls. Higher rate, lower output: the curve slopes down.

Two features are worth reading off it before any algebra.

Its steepness depends on how sensitive investment is to the interest rate. If investment barely responds, a big rate change is needed to shift output and IS is steep. If investment is very sensitive, IS is flat.

Its position depends on everything else that affects demand at a given interest rate — government spending, taxes, exports, confidence. A rise in government spending shifts IS to the right, and by how much is given by the multiplier of the earlier page.

The LM curve: the money market, and why it slopes up

LM is the set of pairs at which the money market clears — the demand for real money balances equals the supply.

Money is demanded for two reasons that pull on different variables. People need it to transact, and transactions rise with income, so money demand rises with Y. And holding money means forgoing interest, so money demand falls as the interest rate rises.

Now hold the money supply fixed and raise income. Money demand has gone up; supply has not. The only thing that can restore balance is a higher interest rate, making people economise on the money they hold. Higher output needs a higher rate: the curve slopes up.

Its steepness depends on how sensitive money demand is to the interest rate: insensitive money demand means a steep LM, because the rate has to move a long way to free up any balances. Its position depends on the real money supply — the nominal quantity divided by the price level — so the central bank shifts LM to the right by expanding the money supply, and inflation shifts it left by reducing the real value of a given nominal quantity.

Crowding out, which is the whole point of the model

Put the two together. Government spending rises, IS shifts right. If the interest rate could stay put, output would rise by the full multiplier amount — the horizontal distance the IS curve moved. But the economy does not travel horizontally; it travels up the LM curve, because the higher income is pulling the interest rate with it.

So the new equilibrium has higher output and a higher interest rate. And the higher rate reduces investment, which through the same multiplier reduces output. The output gained is the horizontal shift minus the loss from fallen investment, and that loss is crowding out.

The arithmetic below measures it exactly, and the measurement is the thing to take away: crowding out equals the fall in investment multiplied by the multiplier, and that is precisely the gap between the simple answer and the IS-LM answer. The two routes to the number agree, which is what tells you the mechanism has been understood rather than asserted.

How much gets crowded out depends on the slopes, and the two extreme cases are worth knowing because they are the limiting arguments people reach for in real debates.

A vertical LM — money demand completely insensitive to the interest rate — means output cannot change at all. The rate rises until investment has fallen by exactly as much as government spending rose. Fiscal policy moves the composition of demand and nothing else: complete crowding out.

A horizontal LM — money demand so sensitive that the rate does not move, which is what a central bank holding rates fixed produces, and what a liquidity trap at the lower bound looks like — means no crowding out at all, and the simple multiplier is right.

Real economies are between the two, which is why fiscal policy works but works less well than the multiplier alone suggests.

Where the model stops: it is all demand

IS-LM determines the demand for output. Whether that demand turns into real output or into higher prices depends on the supply side, and that is where aggregate demand and aggregate supply take over.

Below capacity, with idle workers and machines, short-run aggregate supply is relatively flat and extra demand mostly becomes extra output. The gap between actual output and what the economy could sustainably produce is the output gap, and a negative one is what makes a demand stimulus worth considering. At or above capacity, supply cannot grow, aggregate supply is steep or vertical, and extra demand mostly becomes inflation. In the long run, when wages and prices have adjusted, aggregate supply is taken to be vertical at the economy's productive capacity, and demand policy can move prices but not sustained output.

So the honest reading of an IS-LM answer is: this is how much demand moved. Whether the economy can deliver it is a separate question with a separate diagram, and the two have to be asked in that order.

Where marks are lost

Shifting the wrong curve. Government spending, taxes and confidence move IS. The money supply and the price level move LM. A change in the interest rate moves neither — it is a movement along both.

Quoting the simple multiplier as the answer. If a question gives an LM curve, it wants the IS-LM answer. The simple multiplier is the right answer only to the different question of what would happen at a fixed interest rate, and it is worth computing both and naming the difference as crowding out.

Solving one market and stopping. Both equations have to hold. Substituting into one and reporting the result is the commonest way to lose most of a long question, and it is caught by checking the answer in the equation you did not use.

Treating higher output as proof of a good policy. The model says nothing about capacity. An expansion that fills an output gap and one that arrives at full employment look identical in IS-LM and are completely different in their results.

Questions students ask

Why does a fixed money supply force the interest rate up?

Because the two markets are linked by the demand for money. Higher income means more transactions and so a desire to hold more cash. With the supply of money unchanged, that extra desire can only be satisfied by people selling other assets to get cash — which pushes those assets' prices down and their yields up. The interest rate rising is the mechanism by which the desire to hold more money is choked off until it matches the money that exists.

Does it matter whether the deficit is financed by borrowing or by printing money?

Very much, and the model shows why. Borrowing shifts IS alone, and the whole crowding-out story follows. Financing the same spending by expanding the money supply shifts IS and LM to the right together, so the interest rate need not rise and little or nothing is crowded out — at the cost of a larger money supply, with whatever that implies for prices. Same spending, different financing, different result.

Is IS-LM still worth learning if it leaves out expectations and prices?

It is worth learning for what it does: it is the simplest apparatus in which fiscal and monetary policy interact rather than being studied one at a time, and it produces crowding out, the policy mix and the liquidity trap from two lines of algebra. Its silence on expectations, inflation and the supply side is exactly why the aggregate demand and supply framework exists, and the sensible order is to get the demand side solidly first.

A worked example

An invented economy, built for this page. All quantities are in thousands of crore of rupees a year and r is the interest rate in per cent. The coefficients were chosen to make every answer exact; nothing here describes a real country.

Step 1: build the IS curve

Set output equal to planned spending, with T = 0.25Y so that Y − T = 0.75Y:

Downward sloping, as the mechanism requires. Note where the 0.4 came from: 1 − 0.6, and 0.6 is c(1 − t) = 0.8 × 0.75. So the simple multiplier in this economy is 1/0.4 = 2.5, and we shall need that number shortly.

Step 2: build the LM curve

Set money demand equal to the real money supply:

Upward sloping, as it must be.

Step 3: solve both at once

Check in the equation not used to find Y: LM gives 1,500 + 50(5) = 1,750. Agreed.

Now check the goods market from first principles, which catches any slip in the algebra:

And the money market: 0.2(1,750) − 10(5) = 350 − 50 = 300, which is the money supply. Agreed.

Step 4: a fiscal expansion, G from 400 to 500

Rebuild IS with the new G. Only the constant changes:

LM is untouched, because neither the money supply nor the price level moved. Solve again:

Full check at the new point: T = 468.75, Y − T = 1,406.25, C = 100 + 1,125 = 1,225, I = 300 − 150 = 150, G = 500, and 1,225 + 150 + 500 = 1,875. Agreed. Money: 0.2(1,875) − 10(7.5) = 375 − 75 = 300. Agreed.

Step 5: measure the crowding out

Output rose from 1,750 to 1,875, so ΔY = 125 from ΔG = 100. The realised multiplier is 1.25.

What would the simple multiplier have predicted? Hold r at 5% and read the new IS curve: Y = 2,250 − 250 = 2,000, so ΔY = 250 and the multiplier is 2.5 — the figure computed in step 1.

Now confirm that 125 from the mechanism rather than by subtraction. The interest rate rose from 5% to 7.5%, so investment fell:

The same 125, reached through the fall in investment and the multiplier. That agreement is the point of the exercise: crowding out is not a correction factor, it is the fall in investment amplified by exactly the same multiplier that amplified the government spending.

Step 6: the alternative — monetary expansion instead

Go back to the original economy and leave G at 400. Instead let the central bank raise the real money supply from 300 to 350.

Check: T = 468.75, C = 1,225, I = 300 − 50 = 250, G = 400, total 1,225 + 250 + 400 = 1,875. Agreed. Money: 375 − 25 = 350. Agreed.

The two policies compared

Both policies reach exactly the same output. They get there with opposite interest rates and opposite investment. The fiscal route buys 125 of output by adding 100 of government spending and destroying 50 of private investment; the monetary route buys the same 125 by adding 50 of investment through a cheaper interest rate. One expands the state's share of demand, the other the private sector's.

That is the policy mix, and it is the most practically useful thing IS-LM produces: the choice between the two instruments is not only about how much demand you want, but about what you want the economy to be doing with it. An economy that needs more productive capacity in ten years' time would rather have the 250 of investment than the 150.

The caution that has to go with the answer

Every figure above is a statement about demand. Whether output can actually be 1,875 rather than 1,750 depends on whether the economy has 125 of spare capacity. If it does not, the extra demand meets a supply that cannot grow and shows up as higher prices — and a higher price level reduces the real money supply, which shifts LM back to the left and undoes part of the output gain by itself. That feedback is where this model hands over to aggregate demand and aggregate supply.

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

• Price elasticity of demand, and what it decides about a price rise
• Income and substitution effects: splitting one price change in two
• Profit maximisation and the shutdown decision, from one cost table
• Cournot and Bertrand: why the same two firms give two different answers
• Adverse selection and moral hazard, and what an insurer does about them
• National income and the multiplier: where the rest of the spending comes from
• How monetary policy reaches the economy: the transmission mechanism

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.