The idea: two markets, one rate, one output

The interest rate depends on output, and output depends on the interest rate. The IS–LM model (John Hicks, 1937, from Keynes's General Theory) solves both at once. The IS curve lists the pairs of rate r and output Y where the goods market clears: what people plan to spend equals what is produced. The LM curve lists the pairs where the money market clears: the money people want to hold equals the money there is. Only where the curves cross are both markets in balance.

IS-LM diagram: interest rate r up, output Y across. The IS curve falls from upper left to lower right, the LM curve is flat at 0 percent up to Y = 800 and then rises. They cross at Y = 1000, r = 5 percent.
IS (goods market) and LM (money market) cross at Y = 1000, r = 5%: the only point where both markets clear.

The price level is fixed, so this is a short-run model. The page uses small linear equations you can check by hand:

PieceEquationAt the start
consumptionC = 100 + 0.75·(Y − T)775
investmentI = I₀ − 10·r, I₀ = 175125
government, taxesG = 100, T = 100
money demandL = 0.25·Y − 10·r200
money supplyM = 200

The IS curve: the goods market

Firms produce what people buy, so output settles where Y = C + I + G. Spending feeds on itself. One extra unit of spending is someone's income, and they spend 0.75 of it, which is the next person's income, and so on. The total is the multiplier 1 / (1 − 0.75) = 4. Solving gives Y = 4·(A − 10·r), where A = 100 + I₀ + G − 0.75·T = 300 is the spending that does not depend on income.

A higher rate makes fewer investment projects pay, so I falls and, through the multiplier, Y falls 40 per point. That is why IS slopes down. A change in G, T or confidence (I₀) shifts the whole curve sideways by 4 times the change in A. The "Goods market" panel shows C, I and G as one bar next to output Y.

The LM curve: the money market

People hold money to pay for things, so money demand rises with income: 0.25 per unit of Y. Holding money instead of bonds costs the interest the bonds would pay, so money demand falls as the rate rises: 10 per point. This is Keynes's liquidity preference. The central bank fixes M. Setting L = M gives r = (0.25·Y − M) / 10: more output needs a higher rate to make people part with money, so LM slopes up.

The rate cannot go below zero, because cash always pays 0%. Below Y = M / 0.25 = 800 the curve lies flat on the axis. More money moves LM down and to the right: 5 points lower for each 50 of M.

Fiscal policy and crowding out

The government buys 50 more (Demo: fiscal expansion, crowding out). At the old rate of 5% the multiplier would take output from 1000 to 1200. But the extra income raises money demand, and with M fixed people sell bonds to get money. Bond prices fall and the rate rises. The higher rate cuts investment from 125 to 100. The new point is Y = 1100, r = 7.5%. Half of the multiplier was crowded out by the higher rate.

Two IS-LM panels. Left, fiscal expansion G +50: IS shifts right, the point moves from (1000, 5%) up to (1100, 7.5%), and a ring at (1200, 5%) marks the simple multiplier. Right, monetary expansion M +50: LM shifts down, the point moves from (1000, 5%) down to (1100, 2.5%).
Both policies add 100 to output, but fiscal policy raises the rate (and cuts investment) while monetary policy lowers it (and raises investment).

Monetary policy

The central bank buys bonds with 50 of new money (Demo: monetary expansion). At the old output, people would hold the extra money only if bonds paid less, so the rate would drop to 0%. The lower rate lifts investment, output rises, and rising output raises money demand again. The rate settles at 2.5% with Y = 1100 and investment at 150.

ChangeCurveYrI
start10005%125
G +50IS right11007.5%100
M +50LM down11002.5%150
G +50 and M +50both12005%125

Doing both (Demo: policy mix) keeps the rate at 5%, so nothing is crowded out and output gets the full multiplier: 1200.

The liquidity trap

Firms lose confidence twice (I₀ −50, −50): output falls to 900 and then to 800, and the rate falls to 0%. Now the central bank adds 50 of money (Demo: liquidity trap). Nothing happens. The rate is already zero, no bond pays less than cash, and the extra money sits idle. This is the liquidity trap: Japan after the late 1990s and the US and Europe after 2008.

Two IS-LM panels where IS crosses the flat part of LM at r = 0. Left: M +50 moves the kink of LM from 800 to 1000 but the point stays at (800, 0%). Right: G +50 shifts IS right and the point moves along the flat LM from (800, 0%) to (1000, 0%).
In the liquidity trap more money changes nothing, while government spending gets the full multiplier 4 because the rate does not rise.

Fiscal policy works best here. G +50 lifts output from 800 to 1000, the full 4 × 50. The rate stays at 0%, because the extra money demand is met from idle money, so nothing is crowded out. A smaller slump (Demo: investment slump, rate cut to zero) shows the limit from the other side: one rate cut brings output back to 1000, but only by using up all the room down to 0%.

The model keeps prices fixed. If the price level rises, the same M buys less, so real money falls and LM shifts up: a higher rate and lower output. Plotting output against the price level gives the downward-sloping AD curve of the AD–AS model, which then lets prices move. The circular flow page shows the multiplier as money passing between households and firms.

What the page leaves out

The curves are straight lines and the price level never moves, so there is no inflation and no difference between real and nominal rates. There is no foreign trade or exchange rate (that is the Mundell–Fleming model), no expectations, and one interest rate stands for every rate in the economy. Modern central banks set a rate rather than a quantity of money, so many textbooks now draw LM as a flat line at the policy rate (the IS–MP model). Wealth, debt and the time it takes for policy to work are left out.