IS-LM model: finding the equilibrium
The IS-LM model links the goods market and the money market. The IS curve shows which interest rates and incomes clear the goods market, the LM curve does the same for the money market. Where they cross you find the economy’s income and interest rate.
Formulas and rules
- IS (goods market): Y = C(Y − T) + I(i) + G, investment falls as the interest rate rises
- LM (money market): M/P = L(Y, i) = k·Y − l·i
- Equilibrium: solve LM for i and put it into IS
- Fiscal policy: IS shifts, the interest rate rises, investment falls (crowding out)
- Fiscal multiplier = (1/(1−c)) / (1 + h·k/(l·(1−c))), smaller than 1/(1 − c)
Your turn
Three problems on this topic. Type numbers like 12.5 or 1,250 and elasticities with their sign.
Task
An economy has C = 180 + 0.6·(Y − T), T = 50, I = 1,120 − 40·i, G = 250; money demand L = 0.25·Y − 80·i, real money supply M/P = 530 (amounts in € bn, i in percentage points). Find income and the interest rate in IS-LM equilibrium. By how much do income and the interest rate rise if G rises by € 20 bn?
Income Y* in € bn
Interest rate i* in %
Rise in income ΔY in € bn
Rise in the interest rate in percentage points
Numbers like 12.5 or 1,250; elasticities with their sign (−0.75); percentages as a number (12.5). Round to 2 decimal places.
Worked example
An economy has C = 190 + 0.9·(Y − T), T = 150, I = 345 − 40·i, G = 100; money demand L = 0.25·Y − 60·i, real money supply M/P = 770 (amounts in € bn, i in percentage points). Find income and the interest rate in IS-LM equilibrium. By how much do income and the interest rate rise if G rises by € 60 bn?
Worked solution
- 1.IS: Y = 190 + 0.9(Y − 150) + 345 − 40i + 100 → Y = 10 × (500 − 40i)
- 2.LM: 770 = 0.25Y − 60i → i = (0.25Y − 770) / 60
- 3.Put LM into IS and solve: Y* = 3,800, i* = (0.25 × 3,800 − 770) / 60 = 3%
- 4.Fiscal multiplier with the interest rate response: 10 / (1 + 10 × 40 × 0.25 / 60) = 3.75; ΔY = 3.75 × 60 = 225
- 5.Δi = 0.25 × ΔY / 60 = 0.94 percentage points. Without the interest rate response it would be 600: the rest is crowded out by lower investment.
Typical mistake: Using the simple multiplier 1/(1 − c). In IS-LM, higher income raises money demand and so the interest rate, which lowers investment: the effect is smaller.
Common questions
What is crowding out?
Higher government spending raises income and so money demand. With the same money supply the interest rate rises and private investment falls, so part of the government’s demand pushes out private demand.
When is fiscal policy especially strong?
When money demand reacts strongly to the interest rate (flat LM curve) or investment hardly reacts to it (steep IS curve). Then the interest rate rises little, or crowds out little investment.