The IS-LM Model Explained: IS Curve, LM Curve and a Worked Example

The IS-LM model finds the level of output (Y) and the interest rate (r) at which the goods market and the money market are both in equilibrium, holding the price level fixed. The IS curve shows every combination of r and Y where planned spending equals output, the LM curve shows every combination where money demand equals the real money supply, and the economy sits where the two curves cross.

IS-LM is a short-run model usually taught in college intermediate macroeconomics. AP Macroeconomics uses three separate graphs instead (the money market, loanable funds and AD-AS), and IS-LM is the machine that ties the first and last of those together. This guide explains each curve, shows how policy shifts them, and solves a complete numeric example.

The IS curve: goods market equilibrium

IS stands for investment and saving. A point on the IS curve is a pair (r, Y) at which total planned spending equals total output:

Y = C + I + G

Consumption depends on disposable income, C = a + b(Y − T), where b is the marginal propensity to consume. Investment depends negatively on the interest rate, I = e − f·r, because a higher rate raises the cost of borrowing for new equipment and buildings. Government purchases G and taxes T are set by policy.

Why the IS curve slopes down

Start at a point on IS and lower the interest rate. Cheaper borrowing raises investment. Higher investment raises income, which raises consumption, which raises income again: the multiplier effect. So a lower r goes with a higher Y, and the curve slopes downward. The IS curve is flatter when investment is very sensitive to r (a large f) or when the multiplier is large (a high MPC), because then a small change in r produces a big change in Y.

What shifts the IS curve

Anything that changes spending at a given interest rate shifts IS. Higher government purchases, lower taxes, stronger business confidence or higher consumer wealth shift it right. The horizontal size of the shift equals the change in spending times the multiplier. With an MPC of 0.75, the spending multiplier is 1 ÷ (1 − 0.75) = 4, so a 50 rise in G shifts IS right by 200. A tax cut shifts IS by less, because households spend only the MPC share of it: the shift is MPC ÷ (1 − MPC) × ΔT, so a 50 tax cut shifts IS right by 3 × 50 = 150.

The LM curve: money market equilibrium

LM stands for liquidity preference and money. A point on the LM curve is a pair (r, Y) at which the demand for real money balances equals the supply:

M / P = L(r, Y)

The real money supply M / P is fixed by the central bank (M) and the price level (P), which is why the real money balances idea matters here. Money demand rises with income, because people need more money for more transactions, and falls with the interest rate, because holding cash means giving up the interest a bond would pay. This is the same money demand curve you draw in the money market.

Why the LM curve slopes up

Raise income. People want to hold more money, so money demand shifts right. The money supply has not changed, so the interest rate must rise until the higher opportunity cost of holding money brings quantity demanded back down to the fixed supply. A higher Y therefore goes with a higher r, and the LM curve slopes upward.

What shifts the LM curve

An increase in the money supply, or a fall in the price level, raises M / P and shifts LM right (downward): at every income level the interest rate that clears the money market is lower. A cut in M, a rise in P, or a jump in money demand at every income level shifts LM left.

Worked example: solving for Y and r

Use these equations, with r measured in percent:

Step 1. Derive the IS curve. Substitute into Y = C + I + G:

Y = 100 + 0.75(Y − 100) + 200 − 20r + 150

Expand: 0.75 × (Y − 100) = 0.75Y − 75. Add the constants: 100 − 75 + 200 + 150 = 375. So Y = 375 + 0.75Y − 20r.

Subtract 0.75Y from both sides: 0.25Y = 375 − 20r. Divide by 0.25: Y = 1,500 − 80r.

Step 2. Derive the LM curve. The real money supply is 1,000 ÷ 2 = 500. Set money demand equal to it: 0.5Y − 10r = 500. Add 10r and divide by 0.5: Y = 1,000 + 20r.

Step 3. Set IS equal to LM. 1,500 − 80r = 1,000 + 20r, so 500 = 100r and r = 5 (percent).

Step 4. Find Y. From IS: Y = 1,500 − 80 × 5 = 1,500 − 400 = 1,100. From LM: Y = 1,000 + 20 × 5 = 1,100. Both agree.

Check one: the goods market. C = 100 + 0.75 × (1,100 − 100) = 100 + 750 = 850. I = 200 − 20 × 5 = 100. C + I + G = 850 + 100 + 150 = 1,100, which equals Y.

Check two: the money market. Money demand is 0.5 × 1,100 − 10 × 5 = 550 − 50 = 500, which equals M / P.

Fiscal policy: G rises by 50

The IS equation becomes 0.25Y = 425 − 20r, or Y = 1,700 − 80r. IS shifts right by 200, exactly the multiplier times the change in G. Set it against the unchanged LM: 1,700 − 80r = 1,000 + 20r, so 700 = 100r, r = 7 and Y = 1,000 + 20 × 7 = 1,140.

Output rose by only 40, not 200. The reason is the interest rate. Higher income raised money demand, the rate climbed from 5 to 7, and investment fell from 100 to 200 − 20 × 7 = 60. A drop of 40 in investment times the multiplier of 4 removes 160 of output, and 200 − 160 = 40. That is crowding out worked out to the number. Check: C = 100 + 0.75 × 1,040 = 880, and 880 + 60 + 200 = 1,140.

Monetary policy: M rises from 1,000 to 1,040

Real balances rise to 1,040 ÷ 2 = 520. LM becomes 0.5Y − 10r = 520, or Y = 1,040 + 20r. Against the original IS: 1,500 − 80r = 1,040 + 20r, so 460 = 100r, r = 4.6 and Y = 1,040 + 92 = 1,132. The rate falls, investment rises to 200 − 92 = 108, and output rises by 32. Check: C = 100 + 0.75 × 1,032 = 874, and 874 + 108 + 150 = 1,132.

Fiscal and monetary expansion both raise output, but they push the interest rate in opposite directions. That is why fiscal expansion crowds out investment and monetary expansion crowds it in.

How IS-LM relates to the AP Macroeconomics graphs

Three of the AP graphs connect directly to IS-LM: the money market, loanable funds and aggregate demand.

The money market graph is one slice of LM

The AP money market graph has a vertical money supply and a downward-sloping money demand, and it gives one equilibrium interest rate. That graph assumes a particular level of income. If income rises, money demand shifts right and the rate rises. Plot each income level against the rate it produces and you have traced out the LM curve. The familiar AP chain (the Fed buys bonds, the money supply rises, the interest rate falls, investment rises, AD shifts right) is an LM shift followed by a move down the IS curve, as in the monetary policy transmission mechanism.

Loanable funds tells the same crowding out story another way

In the loanable funds market, a government deficit raises the demand for loanable funds (or cuts national saving), which raises the real interest rate and lowers private investment. IS-LM reaches the same conclusion through the money market: higher spending raises income, income raises money demand, and the rate rises. The two models disagree only in where they place the pressure on interest rates, not in the direction of the result.

IS-LM produces the aggregate demand curve

IS-LM holds the price level fixed. Now change it. If P rises from 2 to 2.5, real balances fall from 500 to 1,000 ÷ 2.5 = 400, and LM becomes Y = 800 + 20r. Against the original IS: 1,500 − 80r = 800 + 20r, so r = 7 and Y = 1,500 − 560 = 940. A higher price level gives lower output, which is a downward-sloping aggregate demand curve. This is the interest rate effect from the AP course, made explicit. Anything that shifts IS or LM at a given P shifts AD, and the AD-AS model then decides how much of that shift shows up as output and how much as inflation.

The liquidity trap case

If the interest rate is stuck at its lower bound, the LM curve is flat over that range. Shifting LM right does nothing to r, so monetary policy loses its grip, while fiscal policy moves output by the full multiplier because there is no rise in r to crowd out investment. The liquidity trap post covers this case in detail.

Common mistakes

• Mixing up which curve is which market. IS is the goods market (spending equals output). LM is the money market (money demand equals money supply).
• Shifting AD when the price level changes. Inside IS-LM a change in P shifts LM, but in the AD-AS graph that same change is a movement along AD. Only changes in spending, taxes, the money supply or money demand at a given P shift AD.
• Using the full multiplier for fiscal policy. The multiplier gives the shift of IS. Actual output rises by less whenever LM slopes upward, because the interest rate rises.
• Moving along the curve instead of shifting it. A change in Y caused by a change in r is a movement along IS. A change in G or T is a shift of IS.
• Treating IS-LM as a long-run model. It holds prices fixed, so it describes the short run, before wages and prices adjust.