The Keynesian Cross
Write PAE as autonomous spending + cY, set it equal to Y, and solve. Then never report ΔY = ΔPAE: they differ by the multiplier 1/(1 − c). The past final gave this a 24-point question.
Why demand determines output in the short run
Prices do not adjust quickly or completely — limited information, long-term contracts, menu costs, consumers who punish "unfair" price changes, workers who resist wage cuts. Edwards's restaurant: a mom-and-pop place does not reprice by the hour, so on a busy night it produces more and on a quiet night less. Output is demand-driven. In the long run prices adjust and output returns to Y*.
The three equations
The multiplier, derived
Y(1 − c) = autonomous
Y = autonomous / (1 − c)The rounds: spending rises 1, income rises 1; consumption rises c; then c²; and 1 + c + c² + … = 1/(1 − c). My spending is your income. c = 1/3 gives 1.5; c = 0.8 gives 5.
The macro mpc is an average across households, and poorer households have mpc near 1 while rich ones are closer to 0 — so the composition of a stimulus matters, not just its size.
The worked example — learn this one cold
C = 10 + (1/3)Y, IP = 6, G = 5, NX = −1, in trillions.
Now the shock: C′ = 8 + (1/3)Y, I′ = 5, G = 5, NX′ = 0. Autonomous consumption fell 2, investment fell 1, net exports rose 1.
What shifts the PAE line
| Component | Moved by |
|---|---|
| C | Real rate r (↑r ⇒ ↓C); expectations about future income (consumer confidence); wealth (stocks, house prices); disposable income |
| IP | Real rate r (↑r ⇒ ↓I, via present value from Unit 14); expectations about future demand — "animal spirits" |
| G | Politics, wars, natural disasters. Taken as given. |
| NX | Taken as given for now; Unit 24 shows ↑r ⇒ ↓NX |
All three of C, I and NX fall when r rises. That single fact is the entire transmission mechanism of monetary policy.
The Great Depression, on this diagram
The October 1929 crash reduced wealth, crushed consumer confidence and created "wait and see" uncertainty. Christina Romer's data show spending collapsed immediately and specifically in durable goods — autos, furniture — not groceries. A fall in C̄: PAE shifts down in parallel, and the multiplier turns it into a much larger fall in Y.
Rules that earn points
- Collect everything into PAE = (autonomous) + cY. Get the intercept by setting Y = 0.
- Set PAE = Y and show the algebra line: "20 = (2/3)Y ⟹ Y = 30."
- For a shock, recompute the intercept, re-solve, and state both ΔPAE and ΔY.
- Explain the gap between them with 1/(1 − c). That sentence scores on its own.
- To offset, match the shift: ΔG = −ΔPAE, or a tax cut of −ΔPAE/c.
- Reporting ΔY = ΔPAE. They differ by the multiplier — that is the whole point.
- Forgetting that a tax change enters as −cT, so its effect is scaled by the mpc.
- Drawing PAE with slope 1. Its slope is the mpc, which is less than 1 — otherwise it never crosses the 45° line.
C = 12 + 0.75(Y − T), T = 8, IP = 10, G = 14, NX = −2 (all in $ trillions). Short-run equilibrium output is:
- A112
- B136
- C28
- D37.3
Answer
A — Y = 112.
Autonomous = 12 − 0.75(8) + 10 + 14 − 2 = 12 − 6 + 22 = 28. Multiplier = 1/(1 − 0.75) = 4. So Y = 28 × 4 = 112.
B is the trap: dropping the −cT term gives autonomous = 34 and Y = 136. Taxes always enter scaled by the mpc. (C reports the intercept; D divides by c instead of 1 − c.)
Autonomous spending falls by $3 billion and the mpc is 0.8. Equilibrium GDP changes by:
- A−$3 billion
- B−$3.75 billion
- C−$15 billion
- D−$2.4 billion
Answer
C — −$15 billion.
The multiplier is 1/(1 − 0.8) = 5, so ΔY = 5 × (−3) = −15.
A is the trap and the single most common error in the course: reporting ΔY = ΔPAE. The gap between them is the multiplier. (B uses 1/c; D multiplies by c.)
With mpc = 0.5, a $200 billion tax cut raises equilibrium GDP by:
- A$400 billion
- B$200 billion
- C$100 billion
- D$50 billion
Answer
B — $200 billion.
Taxes enter as −cT, so a 200 cut raises autonomous spending by only 0.5 × 200 = 100. The multiplier is 1/(1 − 0.5) = 2, so ΔY = 100 × 2 = 200.
A is the trap: treating the tax cut like government purchases, which enter one-for-one, and multiplying the full 200 by 2. (C stops at ΔPAE.)
Output is determined by demand in the short run but not the long run because:
- AFirms in the short run do not maximize profit
- BPrices and wages adjust slowly and incompletely, so firms respond with quantity
- CPotential output Y* is itself set by planned spending
- DThe multiplier only operates in the short run
Answer
B — nominal rigidity.
Limited information, long-term contracts, menu costs, customers who punish "unfair" price changes and workers who resist wage cuts all mean prices are sticky, so firms meet a change in demand by changing quantity. In the long run prices adjust and output returns to Y*, set by labor, capital and technology.
C is the trap: it inverts the whole framework. Y* is a supply-side object; demand determines Y around it, not Y* itself.
An economy has C = 8 + 0.6(Y − T), T = 10, IP = 9, G = 12, NX = 1, all in $ trillions.
(a) Write PAE as a function of Y and give its vertical intercept.
(b) Find short-run equilibrium output.
(c) Investment falls from 9 to 6. Find the new equilibrium, and state ΔPAE, ΔY and the multiplier.
(d) Potential output is 60. Find the increase in G that restores it, and the tax cut that would do the same job.
Model answer
(a) Autonomous = C̄ − cT + IP + G + NX = 8 − 0.6(10) + 9 + 12 + 1 = 8 − 6 + 22 = 24.
(b) Set PAE = Y:
(c) New autonomous = 24 − 3 = 21, so PAE′ = 21 + 0.6Y.
ΔPAE = −3. ΔY = −7.5. The multiplier is 1/(1 − 0.6) = 2.5, and 3 × 2.5 = 7.5. Output falls by more than the shift because the PAE line slopes up: lower income cuts consumption, which cuts income again, round after round.
(d) Raise G by 3. Government purchases enter autonomous spending one-for-one, so the intercept returns to 24 and Y to 60. A tax cut of 5 does the same job: taxes enter as −cT, so ΔPAE = −c·ΔT = −0.6(−5) = +3. Check: 8 − 0.6(5) + 6 + 12 + 1 = 24. ✓
On the diagram, the shock shifts PAE down in parallel by 3 and the equilibrium slides down the 45° line from 60 to 52.5; the policy shifts it back up by 3.
- Subtracting cT, not T, when building autonomous spending.
- The explicit line "24 = 0.4Y ⟹ Y = 60" — setting PAE equal to Y.
- Reporting ΔPAE and ΔY separately and naming the multiplier that links them.
- Sizing the offset to the PAE shift (3), not the output gap (7.5).
- Scaling the tax cut by 1/c to get 5, and checking it.
- A parallel downward shift on the diagram, with the 45° line and both equilibria labelled.
Explain why the October 1929 stock market crash produced a depression rather than just a bad year for investors. Identify the component of PAE that moved, the evidence, and what happened on the Keynesian Cross.
Model answer
The crash shifted autonomous consumption C̄ down hard, through three channels: it destroyed wealth, it crushed consumer confidence, and the volatility produced "wait and see" postponement of purchases.
The evidence is Christina Romer's: consumer spending collapsed immediately, and specifically in durable goods — autos and furniture, the postponable purchases — while groceries held up. That pattern is what a confidence-and-uncertainty shock looks like.
On the diagram, the PAE line shifts down in parallel by the size of the shock, and the equilibrium slides down the 45° line. Output falls by considerably more than the shift, because the multiplier 1/(1 − c) turns lost income into lost consumption into lost income again. A fall in demand became a fall in production.
Unit 22 adds the second blow: monetary contraction drove the real interest rate up, shifting PAE down a second time.
- Naming C̄ as the component, not "demand" in general.
- All three channels: wealth, confidence, wait-and-see.
- Romer's durable-goods evidence, with the contrast against non-durables.
- A parallel downward shift of PAE, with ΔY > ΔPAE and the multiplier named.
- Mentioning the monetary contraction as a second, separate downward shift.