IS-LM Model

A diagram for the General Theory

Keynes published The General Theory of Employment, Interest, and Money in February 1936. It was long, contentious, and famously short on summarising equations. Within eighteen months, a young John Hicks at Cambridge had produced a four-equation, two-curve diagram that captured what he took to be the essential mechanical content of the book. The paper, Mr Keynes and the Classics: A Suggested Interpretation, appeared in Econometrica in April 1937 and quietly became the way almost everyone learned Keynesian macroeconomics for the next sixty years. Alvin Hansen's 1953 textbook gave it the Y-on-the-horizontal-axis, r-on-the-vertical form that every undergraduate now recognises; Paul Samuelson's Economics made it canonical for a generation of students.

What Hicks did was simple in retrospect. He observed that Keynes's argument could be split into two markets — a goods market in which output adjusts to equate aggregate demand with aggregate supply, and a money market in which the interest rate adjusts to equate money demand with money supply — and that each market separately defines a locus of (Y, r) pairs consistent with its own equilibrium. The intersection of those two loci picks out the unique (Y, r) at which both markets clear simultaneously. The whole short-run macroeconomy reduces to a single sketch on the back of an envelope.

The IS curve: the goods market

The goods market clears when planned aggregate demand equals output. With the four standard components,

Y = C(Y − T) + I(r) + G + NX

Consumption C depends on disposable income Y − T (with marginal propensity to consume MPC, typically about 0.7 in the U.S.). Investment I depends negatively on the real interest rate r — higher rates raise the cost of capital, fewer projects clear the hurdle. Government spending G and net exports NX are taken as exogenous in the basic closed-economy version (NX = 0 in the simplest form).

Trace what happens as r changes. Cut the interest rate. Investment I rises. The rise in I increases aggregate demand directly, which raises output, which raises disposable income, which raises consumption — and so on, around the Keynesian multiplier loop. In the simplest linearised version the new equilibrium Y rises by

ΔY = (1 / (1 − MPC)) × ΔI

where 1/(1 − MPC) is the multiplier, about 3.3 at MPC = 0.7. So every (Y, r) point at which the goods market is in equilibrium has the property that lower r pairs with higher Y. Plot those points, and you get a downward-sloping locus — the IS curve. Its slope's steepness depends on two things: the interest-sensitivity of investment (steep IS if I is interest-inelastic) and the size of the multiplier (flatter IS for a larger multiplier).

The LM curve: the money market

The money market clears when real money demand equals real money supply:

M / P = L(Y, r)

The real supply M/P is set by the central bank (in the textbook version with fixed P). Real money demand L(Y, r) has two arguments: it rises with Y because more income means more transactions and therefore more cash on hand, and it falls with r because higher interest rates make bonds more attractive than cash. Keynes called the negative interest-elasticity of money demand liquidity preference — the willingness to sacrifice yield for the convenience of holding cash.

Trace what happens as Y changes. Raise income. Real money demand rises (the L curve shifts up at any r). But the supply is fixed: M/P has not changed. The only way to bring demand back down to supply is to raise the interest rate, which makes bonds more attractive at the margin. So every (Y, r) point at which the money market clears has the property that higher Y pairs with higher r. Plot those points, and you get an upward-sloping locus — the LM curve.

Three regions of LM are worth distinguishing.

Region	Y range	r elasticity of L	LM slope	Interpretation
Liquidity trap	Low Y	Infinite	Horizontal	Zero lower bound — money and bonds are perfect substitutes
Intermediate range	Normal Y	Finite, negative	Upward	The textbook regime
Classical range	High Y	Near zero	Vertical	Money demand is interest-inelastic — pure quantity theory

In the intermediate range the curve has the standard upward slope. At very low rates LM goes flat — the liquidity trap, where money and bonds are indistinguishable. At very high rates LM goes vertical — the classical case, where the quantity theory of money holds and the price level alone adjusts.

Short-run equilibrium

The simultaneous equilibrium is the intersection of IS and LM. At that single (Y*, r*) pair both markets clear: investment equals saving, money demand equals money supply, there is no excess demand or supply anywhere in the system, and no household, firm, or bank has an unmet plan. It is short-run equilibrium because the price level P is held fixed; in the longer run, P adjusts and the LM curve shifts as M/P changes, eventually returning the economy to the vertical long-run aggregate supply curve.

The diagram's analytical power is that it converts an entire macroeconomic policy question into a question about which curve moves and by how much.

Fiscal policy in IS-LM

An increase in government spending G (a fiscal expansion) raises aggregate demand at every interest rate. The IS curve shifts to the right by ΔG/(1 − MPC) — the simple Keynesian multiplier applied to the spending change. With the LM curve unchanged, the new intersection lies up and to the right of the old one. Output rises. The interest rate also rises.

The rise in r is not a side effect — it is the channel through which IS-LM modifies the simple multiplier story. Higher r reduces investment I, so the actual rise in Y is smaller than the textbook ΔG/(1 − MPC) prediction. This reduction is called crowding out: government spending displaces some private investment by driving up the cost of borrowing.

Pure-IS multiplier:  ΔY/ΔG = 1/(1 − MPC)     (~3.3 at MPC = 0.7)
IS-LM multiplier:    ΔY/ΔG = smaller        (~1 to 2 in practice)
Crowding-out gap:    depends on LM slope    (steep LM → more crowding)

How much crowding out depends on the slope of LM. A flat LM (highly interest-elastic money demand) means r barely rises and almost no investment is crowded out — fiscal policy is highly effective. A steep LM means r rises sharply and crowding out is strong — fiscal policy is weak. In the limiting case of vertical LM (the classical case), crowding out is complete: ΔY = 0. In the limiting case of horizontal LM (the liquidity trap), crowding out is zero and fiscal multipliers are at their textbook maximum.

Monetary policy in IS-LM

An increase in the real money supply M/P (a monetary expansion) shifts the LM curve to the right. At the old interest rate the money market would be in excess supply, so r must fall to make households willing to hold the extra balances. With IS unchanged, the new intersection lies down and to the right of the old one. The interest rate falls. Output rises.

The transmission mechanism is entirely through investment. The drop in r raises I along the IS curve; the rise in I raises Y via the multiplier; the rise in Y raises money demand back toward the new larger supply. Monetary policy works because it changes r — and r only matters because investment depends on it. If investment were interest-inelastic (a vertical IS), monetary expansion would lower r without raising Y at all. If money demand were interest-inelastic (a vertical LM), the shift in LM would be powerful but again the mechanism would still run through r and I.

Notice the symmetry. Both fiscal and monetary expansions raise Y, but they move r in opposite directions: fiscal raises r, monetary lowers r. The policy mix matters. A government wanting to expand output without bidding up rates can use monetary policy alone; a government wanting to keep rates from collapsing during a fiscal stimulus can run the two policies together.

The liquidity trap

The textbook case of policy paralysis. At the zero lower bound, nominal interest rates cannot fall further — holding cash returns 0%, so no rational saver will accept a negative-yielding bond. In the IS-LM diagram, LM becomes horizontal at r ≈ 0 over the relevant range of Y. Any further increase in M is absorbed entirely as idle balances. The LM curve, in effect, cannot shift downward beyond the floor.

The consequences for the two policy levers are dramatic and opposite. Monetary policy is impotent: shifting LM rightward does nothing because the relevant section is already at the floor; r does not fall, so I does not rise, so Y does not rise. Fiscal policy, by contrast, becomes maximally effective: an IS shift along a horizontal LM produces the full Keynesian multiplier with no crowding-out at all.

This is the textbook case for fiscal stimulus during a deep recession and was the explicit framework behind both the Japanese policy debate of the 1990s and the post-2008 Western policy debate. It is also the rationale for unconventional monetary policy — quantitative easing, forward guidance, negative deposit rates — which are attempts to make monetary policy effective even when conventional rate-cutting is at the floor.

A worked numerical example

Take a stylised economy with

C = 100 + 0.7 (Y − T)         T = 100
I = 200 − 1,000 r              G = 100
M/P = 500                      L(Y, r) = 0.5 Y − 2,000 r

Solve for the IS and LM equations.

IS: Y = 100 + 0.7(Y − 100) + (200 − 1,000 r) + 100
   Y = 100 + 0.7 Y − 70 + 200 − 1,000 r + 100
   0.3 Y = 330 − 1,000 r
   Y = 1,100 − 3,333 r        (IS)

LM: 500 = 0.5 Y − 2,000 r
   Y = 1,000 + 4,000 r        (LM)

Set IS = LM:

1,100 − 3,333 r = 1,000 + 4,000 r
100 = 7,333 r
r* = 0.0136  (≈ 1.36%)
Y* = 1,000 + 4,000 × 0.0136 = 1,054.5

Now suppose G rises from 100 to 150 (fiscal expansion). The IS intercept rises by 50/0.3 = 166.7, giving

IS': Y = 1,266.7 − 3,333 r
1,266.7 − 3,333 r = 1,000 + 4,000 r
266.7 = 7,333 r
r* = 0.0364  (≈ 3.64%)
Y* = 1,000 + 4,000 × 0.0364 = 1,145.5

ΔY = 91.0    Δr = +2.28 percentage points

The naive Keynesian multiplier (1/(1 − MPC) = 3.33) applied to ΔG = 50 would predict ΔY = 166.7. The IS-LM model gives ΔY = 91.0 — about 55% of the naive prediction. The remaining 75.7 of output is crowded out by the rising interest rate. This is the IS-LM correction to pure Keynesian arithmetic.

Or instead, suppose M/P rises from 500 to 600 (monetary expansion, G held at 100). The LM intercept changes:

LM': 600 = 0.5 Y − 2,000 r  →  Y = 1,200 + 4,000 r
1,100 − 3,333 r = 1,200 + 4,000 r
−100 = 7,333 r
r* = −0.0136  (impossible — would violate ZLB)

The arithmetic produces a negative r*. In practice the economy hits the zero lower bound and slips into the liquidity-trap regime. This is the IS-LM diagnostic that monetary expansion at a low starting rate runs out of room.

A catalogue of shifts

Shock	Curve that shifts	Direction	Δ Y	Δ r
G rises (government spending)	IS	Right	+	+
T rises (taxes)	IS	Left	−	−
Autonomous C rises (consumer confidence)	IS	Right	+	+
Autonomous I rises (business expectations)	IS	Right	+	+
NX rises (export demand)	IS	Right	+	+
M rises (monetary expansion)	LM	Right	+	−
P falls (real money supply rises)	LM	Right	+	−
Money demand rises (financial panic)	LM	Left	−	+

Two general patterns. (1) Shifts of IS move Y and r in the same direction. (2) Shifts of LM move Y and r in opposite directions. The pattern lets you read off the cause of a recession from the comovement of output and interest rates: if Y and r both fall together, the shock is on the IS side (collapse of investment demand, fiscal contraction, drop in consumer confidence); if Y falls while r rises, the shock is on the LM side (a rise in money demand, a contraction of money supply).

Mundell-Fleming: open-economy extension

The closed-economy IS-LM model assumes no international capital flows and no exchange rate. Robert Mundell (1962) and Marcus Fleming (1962) extended it to a small open economy with perfect capital mobility. The extension adds two ingredients: net exports NX depending on the real exchange rate e, and an interest-parity condition tying the domestic rate r to the world rate r* plus expected currency depreciation.

Two limiting cases dominate the textbook treatment.

Regime	Fiscal expansion	Monetary expansion	Mechanism
Fixed exchange rate	Fully effective	Impotent	Central bank loses reserves defending peg, so M endogenously adjusts back
Floating exchange rate	Impotent	Doubly effective	Fiscal: currency appreciates, NX falls, crowding-out via trade. Monetary: currency depreciates, NX rises, on top of domestic effect

From this comes the famous impossible trinity (or trilemma): a country can have at most two of the following three — a fixed exchange rate, free capital mobility, and an independent monetary policy. The Bretton Woods system chose fixed rates and independent policy, suppressed capital mobility, and held until capital controls leaked. The post-1973 floating-rate system chose floating and independent policy. The Eurozone chose fixed rates (the euro) and capital mobility, surrendering independent national monetary policy. Every actual policy regime is a corner of this trilemma.

From IS-LM to AD-AS

IS-LM holds the price level P fixed and so produces a strictly short-run equilibrium. To handle inflation, one builds AD (aggregate demand) by varying P and tracing how the IS-LM intersection moves in (Y, P) space. A lower P raises M/P, which shifts LM right, which raises Y at the new intersection. The locus of P-Y pairs is the aggregate demand curve. Combine it with an aggregate supply curve — short-run upward-sloping, long-run vertical at potential output — and the standard AD-AS apparatus emerges as a price-level wrapper around an IS-LM core.

Adding the Phillips curve closes the system. The Phillips curve relates inflation to the output gap (Y − Y_potential): output above potential produces inflation, output below potential produces disinflation. Plugging this into AD-AS gives an inflation dynamics that converges to long-run potential output regardless of monetary policy — the Friedman-Phelps natural-rate proposition that monetary expansion cannot permanently raise output. The vertical long-run AS curve in this composite story is the long-run neutrality of money, derived from the IS-LM core plus inflation dynamics.

Limitations and the modern alternative

• No microfoundations. The consumption function, investment function, and money demand function are aggregate behavioural relations not derived from optimising households and firms. Modern macro insists on starting from agents maximising utility and profits subject to budget constraints, then aggregating.
• Lucas critique (1976). The slopes and intercepts of IS and LM depend on agents' expectations about policy. When the policy regime changes — for instance, when the central bank switches from money-supply targeting to interest-rate targeting — the curves themselves shift, so the model's predictions about a new policy can be wrong precisely when the policy actually changes.
• The wrong policy variable. Most modern central banks set an interest rate (the federal funds rate, ECB main refinancing rate, BOE Bank Rate) and let the money supply adjust endogenously. IS-LM treats M as the policy variable and r as endogenous, which inverts the actual decision structure.
• Static. The model is comparative-static: it tells you the new equilibrium after a shock but not the dynamics of adjustment. Real economies have inertia, expectations, and gradual price adjustment that the IS-LM diagram suppresses.
• No expectations. Rational forward-looking expectations — the dominant assumption in modern macro — do not appear. IS-LM treats consumption as a function of current income and investment as a function of the current interest rate, when both should depend on expected lifetime income and the expected future path of rates.

The mainstream research alternative since the 1990s is the dynamic stochastic general equilibrium (DSGE) model. The simplest three-equation New Keynesian core consists of

(NK-IS)   y_t = E_t[y_{t+1}] − (1/σ)(i_t − E_t[π_{t+1}] − r*)
(NK-PC)   π_t = β E_t[π_{t+1}] + κ y_t
(Taylor)  i_t = r* + π_t + φ_π (π_t − π*) + φ_y y_t

where y_t is the output gap, π_t is inflation, i_t is the nominal rate, and the parameters σ, β, κ, φ are derived from household and firm optimisation. The reduced-form comparative statics still look like IS-LM — a Taylor-rule shock raises r and lowers Y, a fiscal shock raises both — but the curves are derived rather than postulated, expectations are forward-looking, and the central bank sets r directly. IS-LM has become a teaching tool whose mechanical intuition is preserved inside a more rigorous framework.

Common pitfalls

• Confusing the curves with supply and demand for output. IS is not a supply curve for output and LM is not a demand curve for output. Both are equilibrium loci — the set of (Y, r) pairs at which a particular market clears. They look superficially like supply-demand but the logic is different.
• Forgetting that P is fixed. The whole diagram is short-run. Any conclusion about long-run output must come from embedding IS-LM in AD-AS and letting prices adjust.
• Treating M as if the central bank actually controls it. Modern central banks set interest rates and let the money supply float to whatever quantity the system demands at that rate. The textbook IS-LM mechanics still work, but the language must be inverted: the central bank picks a point on LM (an r) and the money supply is implied.
• Reading off too much detail from slopes. The actual interest-elasticity of investment and the actual interest-elasticity of money demand are empirical magnitudes, both heavily contested. The diagram is a qualitative organising device, not a quantitative forecasting model.
• Ignoring the liquidity trap as a real-world regime. Pre-2008, the liquidity trap was a textbook curiosity. Post-2008 (and post-1995 in Japan) it is the operating regime of large advanced economies for most of a decade at a time. The flat segment of LM is not an edge case but a recurrent state.
• Equating "crowding out" with complete crowding out. Crowding out is partial in the standard intermediate range and zero in the liquidity trap. Saying "fiscal stimulus is always neutralised by crowding out" is a vertical-LM claim that the data do not support.