IS–LM to AD–AS · Lesson 2 of 10
The IS curve: the goods market
The Keynesian cross takes investment as given. But investment falls when borrowing gets dearer, so every interest rate gives a different equilibrium income. Plot those pairs and you have the IS curve, half of the model that the rest of this module is built on.
A question to hold on to
A closed economy has C = 2 + 0.5Y, government spending of 3 and investment I = 4 − 0.5r, where r is the interest rate. What is the slope of the IS curve?
IIT JAM 2024Solved at the end ↓
The idea
The piece the cross left out
In the Keynesian cross, output is autonomous spending times the multiplier. Investment sat inside autonomous spending as a fixed number. But the investment lesson of the foundations module showed that a firm invests in a project only if its return beats the cost of borrowing. When the interest rate rises, fewer projects pass, and investment falls. What matters is the real interest rate, the cost of borrowing after inflation (UGC NET 2023).
The simplest way to write it is a straight line:
The investment function
I₀ is the investment firms would do at a zero interest rate; b > 0 is how much investment falls per point of interest.
Output depends on investment, and investment depends on the interest rate, so output now depends on the interest rate. The IS curve is that relationship. Its name says what holds along it: planned Investment equals Saving, which is the same as saying that planned spending equals output. It is the set of combinations of income and the interest rate at which the goods market is in equilibrium (CUET 2021; Hyderabad 2012).
Interactive
Deriving IS, one interest rate at a time
Pick an interest rate. Investment is I₀ − br, so planned spending is a line in the Keynesian cross; where it crosses the 45° line is equilibrium output. Raise the interest rate and investment falls, the spending line shifts down, and output falls by b times the multiplier for every point. Mark each (Y, r) pair on a second diagram: the points line up into a downward-sloping curve. That is IS.
1. The Keynesian cross at r = 6%
2. The IS curve: each r and its output
The chain
r = 6% → investment is lower by b·r = 120 → autonomous spending 280 → output 280 ÷ (1 − 0.75) = 1,120.
The equation and its slope
IS: Y = 4 × (400 − 20r). One point off r moves output by b/(1 − c) = 80, so the slope is dr/dY = −(1 − c)/b = −0.0125.
Pause and predict
On the IS curve, as the interest rate falls, equilibrium income
The algebra
The equation of the IS curve
Put the investment function into the goods-market equilibrium and solve for Y exactly as in the cross. With a proportional tax, consumption is C₀ + c(1 − t)Y:
The IS curve
A = C₀ + I₀ + G (less cT₀ if there are lump-sum taxes) is autonomous spending at a zero interest rate. α = 1/(1 − c(1 − t)) is the Keynesian-cross multiplier. Written the other way up, r = A/b − Y/(αb).
CUET 2023 · standard
C = 300 + 0.8(1 − t)Y, I = 900 − 50i, G = 800 and t = 0.25. What is the IS equation?
- Collect autonomous spending.300 + 900 + 800 = 2,000.
- The slope of spending.c(1 − t) = 0.8 × 0.75 = 0.6, so α = 1/0.4 = 2.5.
- Solve.Y = 2.5(2,000 − 50i) = 5,000 − 125i.
Answer
Y = 5,000 − 125i. The options with 4,250 forget the autonomous consumption of 300; the ones with +125i have the sign of investment backwards.
JNU 2021 · standard
C = 15 + 0.5Yd, I = 150 − 10r and G = T = 20. Find the IS equation.
- Disposable income.Yd = Y − 20, so C = 15 + 0.5Y − 10 = 5 + 0.5Y.
- Equilibrium.Y = 5 + 0.5Y + 150 − 10r + 20, so 0.5Y = 175 − 10r.
Answer
Y = 350 − 20r. The companion question, with C = 5 + 0.6Y, I = 200 − 50r and no government, gives Y = 512.5 − 125r.
Induced investment
If investment also rises with income, I = I₀ + dY − br, the spending line gets steeper and the multiplier becomes 1/(1 − c − d). DSE 2006 had C = 2,000 + 0.4Yd, T = 1,000, G = 400 and I = 500 − 10r + 0.4Y: then 0.2Y = 2,500 − 10r and IS is Y = 12,500 − 50r.
The shape
What makes IS steep or flat
A fall of one point in the interest rate raises investment by b, and output by α·b. The bigger that response, the flatter IS, because a small change in r moves Y a long way. So IS is flatter when:
- investment is more sensitive to the interest rate (b larger);
- the multiplier is larger: a higher MPC (Hyderabad 2019), a lower tax rate, or, in an open economy, a lower propensity to import.
In the extreme, if investment ignores the interest rate (b = 0), output doesn’t depend on r at all and IS is vertical (JNU 2018, Hyderabad 2018). Lesson 5 shows why monetary policy then has no effect.
The slope of IS
In the (Y, r) diagram. GATE 2024 asked the direction: steeper with low interest elasticity of investment and with a high marginal propensity to save.
Pause and predict
Which does NOT affect the slope of the IS curve?
JNU 2014 · standard
With no government, C = 388 + 0.75Y − 15r and I = 1,863 + 0.05Y − 25r, where r is in percent. What is the slope of IS? What income clears the goods market at r = 15, and how much does it change if r falls to 6?
- Solve for Y.Y = 2,251 + 0.8Y − 40r, so 0.2Y = 2,251 − 40r and Y = 11,255 − 200r.
- Slope.dr/dY = −1/200 = −0.005. The options were −0.0175, −0.02 and −0.025: the answer is “none of the above”.
- Income at r = 15.11,255 − 3,000 = 8,255.
- A fall of 9 points.ΔY = 200 × 9 = 1,800.
Answer
−0.005, 8,255 and 1,800. Note that the interest rate affects consumption here as well as investment; both go into the slope.
A general version (CUET 2023): if demand is A(Y, r) and IS is the set of points with Y = A(Y, r), differentiating gives dY = AYdY + Ardr, so along IS dY/dr = Ar/(1 − AY).
The movers
What shifts IS
A change in the interest rate is a movement along IS (Hyderabad 2010). Anything else that changes spending at a given interest rate shifts the whole curve, by the change in autonomous spending times α:
- higher government purchases, lower taxes or higher transfers shift IS right (CUET 2025, 2026);
- more optimistic firms or households, for example a higher expected future marginal product of capital (JAM 2021), shift it right;
- self-fulfilling waves of optimism and pessimism, and changes in the demand for consumer goods, are IS shocks; a change in the demand for money is not (JAM 2025).
The size of the shift
Horizontal distance at any given r.
Pause and predict
Government spending rises by 40 with c = 0.75 and no income tax. By how much does IS shift to the right?
Looking ahead
Off the curve
Take a point to the right of IS. Income there is higher than the goods market can support at that interest rate: output exceeds planned spending, and there is an excess supply of goods. To the left of IS, spending exceeds output: an excess demand for goods, stocks run down and firms raise output. Lesson 4 pairs this with the same idea for the money market to see how the economy finds its way back.
Solving the opening question
The opening question, solved
IIT JAM 2024 · standard
C = 2 + 0.5Y, G = 3 and I = 4 − 0.5r. What is the slope of IS?
- Goods-market equilibrium.Y = 2 + 0.5Y + 4 − 0.5r + 3, so 0.5Y = 9 − 0.5r.
- Write r in terms of Y.r = 18 − Y.
- Read off the slope.dr/dY = −1, or by the formula, −(1 − c)/b = −0.5/0.5 = −1.
Answer
−1. Option −0.5 is the MPC or the investment coefficient on its own; 1 has the wrong sign.
Watch out for
The mistakes examiners are fishing for
- Axes. The IS diagram has income on the horizontal axis and the interest rate on the vertical. “Slope” means dr/dY.
- Flat against steep. A bigger multiplier or a more interest-sensitive investment makes IS flatter, not steeper.
- Moving along against shifting. The interest rate moves the economy along IS; everything else in autonomous spending shifts it.
- Dropping a term. Autonomous consumption, taxes times c and induced investment all have to be carried through. Most wrong options in IS questions are one dropped term.
Revise
The lesson on one screen
The lesson on one screen
- Investment
- I = I₀ − br: falls as the real interest rate rises.
- IS
- Pairs (Y, r) where planned spending = output (planned I = S).
- Equation
- Y = α(A − br), α = 1/(1 − c(1 − t)), A = autonomous spending at r = 0.
- Slope
- dr/dY = −1/(αb). Flatter with a bigger multiplier or a larger b; vertical if b = 0.
- Shifts
- G, T, confidence, expected returns: shift by α·ΔA. The interest rate moves along the curve.
- Off the curve
- Right of IS: excess supply of goods. Left: excess demand.
Practice
Check yourself
Compute
IS equation
C = 100 + 0.8Y and I = 300 − 20r, with no government. What is equilibrium income when r = 5?
Slope
c = 0.75, t = 0 and b = 50. What is the slope dr/dY of IS?
Shift
c = 0.8 and t = 0.25. Government spending rises by 60. How far does IS shift to the right?
Quick questions
1If investment does not respond to the interest rate, the IS curve is
2A rise in the MPC makes the IS curve
3The IS curve shifts when all of these change except
Real questions
Past papers, by level
Foundation
Definitions, directions of shifts and one-step problems. Most CUET, UGC NET and Hyderabad questions sit here.
7 questionsPractise →
Standard
Solving models, comparing slopes and multi-step reasoning, up to the hardest JNU, JAM and GATE questions.
12 questionsPractise →
Delhi School level
This topic’s hardest questions, from the DSE entrance, are worked through in Lesson 10: IS–LM in letters.
31 questionsLesson 10 →
All 19 past-paper questions for this lesson, easiest first: practise the full set →
These are questions from the TestStop bank. A free account answers 10 a day, each with its worked solution; All-Access opens every set with no daily limit.
Read next
All resources →MacroeconomicsIS–LM 3: the LM curveDeriving LM from money demand and a fixed money supply: its equation and slope, the vertical classical and horizontal liquidity-trap cases, an interest-rate peg, and shifts.MacroeconomicsMacroeconomics 4: the Keynesian cross and the multiplierPlanned spending and the 45° line, unplanned inventories, saving equal to investment, the multiplier as rounds of spending, deflationary and inflationary gaps, and the paradox of thrift.MacroeconomicsIS–LM to AD–AS, from zero to exam levelA module that builds the short-run macro model one curve at a time: the demand for money, the IS and LM curves, policy, aggregate demand and supply, the Phillips curve, expectations and the open economy, each with interactives and graded past-paper sets.