Module 02 · Chapters 3

The Goods Market

Demand, the Keynesian cross, and the multiplier.

“Where output is set by demand, not by what firms can produce.”

~35 min· 4 sub-skills·7 exercisesExam frequency · high0% mastered

In the short run (a few quarters), prices and wages are sticky. Firms produce whatever is demanded. The level of output is set on the demand side. This module derives that result and extracts the most important number it gives us: the multiplier.
Aggregate demand (closed economy)
$Z \;=\; C(Y - T) + I + G$
$Z$
demand for goods
$C(Y-T)$
consumption — depends on disposable income
$I$
investment (treated as exogenous in this chapter)
$G$
government purchases
Linear consumption function
$C \;=\; c_0 + c_1\,(Y - T)$
$c_0$
autonomous consumption — what households spend even at zero income
$c_1$
marginal propensity to consume (MPC), 0 < c_1 < 1
$Y - T$
disposable income
When disposable income rises by €1, consumption rises by c_1 < 1 cents — the rest is saved.
Definition·Goods-market equilibrium
Output is determined by demand: $Y = Z$ . Substituting:

$Y = c_0 + c_1(Y - T) + I + G$

Solve for Y:

$Y^* = \frac{1}{1 - c_1}\,[c_0 - c_1 T + I + G]$

The term in brackets is autonomous demand. The factor $1/(1 - c_1)$ is the multiplier.
Figure · Keynesian cross
Multiplier rounds
ΔG = 50 · MPC = 0.60 · k = 2.50
ΔG50
MPC c₁0.60
50.0
R0
30.0
R1
18.0
R2
10.8
R3
6.5
R4
3.9
R5
Cumulative after 6 rounds: 119.2·Theoretical limit (∞ rounds): 125.0
Demand Z and 45° line Y = Z. Equilibrium where Z crosses 45°.
Multiplier
$k \;=\; \frac{1}{1 - c_1}$
If c₁ = 0.6, k = 1/(0.4) = 2.5. A €1 increase in autonomous demand (G or c₀ or I) raises equilibrium Y by €2.5.
Derivation · Where does the multiplier come from? — round-by-round
1. 01
  Round 0: government spends ΔG = €100. That's €100 of new demand directly.
  $\Delta Y_0 = \Delta G = 100$
2. 02
  Round 1: the €100 becomes income to whoever supplied G. They consume c₁ × ΔG of it.
  $\Delta C_1 = c_1\,\Delta G$
  The recipients spend their MPC fraction.
3. 03
  Round 2: their spending becomes income to others, who consume c₁ of that.
  $\Delta C_2 = c_1^2\,\Delta G$
4. 04
  Sum the geometric series: ΔY = ΔG × (1 + c₁ + c₁² + …) = ΔG / (1 − c₁).
  $\Delta Y = \Delta G \sum_{k=0}^{\infty} c_1^k = \frac{\Delta G}{1 - c_1}$
  Because |c₁| < 1, the geometric sum converges to 1/(1 − c₁).
  Predict
  If c₁ rises from 0.6 to 0.75, the multiplier becomes:
Common misconception
If everyone saves more, total saving rises.
Actually**Paradox of saving**: a higher c₀ reduction (more autonomous saving) lowers Y, which lowers actual saving back to where it started. In equilibrium S = I, and I is exogenous here — so saving cannot rise.
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If c₀ falls by 50, ΔY = −50/(1 − c₁). The fall in income offsets the rise in saving propensity exactly.
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Tip·Balanced-budget multiplier = 1
If the government raises G and T by the same amount, ΔG > 0 raises Y by ΔG/(1−c₁), but ΔT > 0 lowers Y by c₁·ΔT/(1−c₁). Net: $\Delta Y = \Delta G \,\frac{1 - c_1}{1 - c_1} = \Delta G$ . The balanced-budget multiplier is exactly 1.
Worked example · Worked example — solve for Y*
c₀ = 100, c₁ = 0.6, T = 100, I = 200, G = 200. Find equilibrium output.
1. 1
  Compute autonomous demand A = c₀ − c₁T + I + G.
  $A = 100 - 0.6 \times 100 + 200 + 200 = 440$
2. 2
  Apply the multiplier k = 1/(1 − c₁) = 1/0.4 = 2.5.
  $k = 2.5$
3. 3
  Y* = k × A = 2.5 × 440 = 1100.
  $Y^* = 1100$
✓ Equilibrium output Y* = 1100.
Exercise · multiple choice · +10 XP
Identify demand components
Which is **not** part of aggregate demand Z in a closed economy?
Exercise · numerical · +12 XP
Solve for equilibrium Y
c₀ = 80, c₁ = 0.5, T = 100, I = 120, G = 100. Compute equilibrium output Y*.
Your answer
Exercise · predict shift · +12 XP
Predict the shift — fiscal expansion
The government increases G by 100 (no change in T or I). Predict the equilibrium response.
Scenario: ΔG = +100, c₁ = 0.6, all else equal.
Exercise · numerical · +8 XP
Multiplier numerics
If MPC = 0.8, what is the multiplier k?
Your answer
Exercise · numerical · +14 XP
Tax change — sign and size of multiplier
c₁ = 0.6. The government raises taxes by ΔT = 50. Compute ΔY (use the tax multiplier).
Your answer
Exercise · multi step · +18 XP
Balanced-budget multi-step
ΔG = +100, ΔT = +100, c₁ = 0.6.
Context: A balanced fiscal expansion: spending and taxes both rise by 100. Compute the components of ΔY.
(a)ΔY from ΔG alone:
(b)ΔY from ΔT alone:
(c)Net ΔY (sum):
Exercise · multiple choice · +10 XP
Paradox of saving — what happens to S?
c₀ falls by 50 (households decide to save more autonomously). With G, T, I unchanged, what happens to equilibrium saving S?

Mastery check

5 questions · pass with 80%

Answer all five to confirm you've internalised the module. A passing run unlocks the next module.

Q1
MPC = 0.75. The multiplier is:
Q2
"In short-run equilibrium, output equals demand: Y = Z."
Q3
Why is the tax multiplier smaller (in absolute value) than the spending multiplier?
Q4
"An autonomous increase in saving raises equilibrium saving in this model."
Q5
c₀=50, c₁=0.5, T=100, I=100, G=200. Y* = ?

0 / 5 answered

Exam pitfalls

Computing the multiplier as 1/(1+c₁) instead of 1/(1−c₁).
Using the spending multiplier for a tax change. The tax multiplier is −c₁/(1−c₁) — strictly smaller.
Forgetting the balanced-budget multiplier equals 1 (not 0).
Treating ΔY = c₁ × ΔG as the full multiplier — that's only round 1.
Saying 'higher saving rate → higher saving'. In this model, S = I and I is exogenous; raising the saving rate only contracts output (paradox of saving).