Skip to content

Lesson M13.L02: Keynesian Cross Model: Formal Treatment

Module: National Accounting, Keynesian Income-Expenditure Model and Fiscal Policy Level: intermediate Duration: 30 minutes Learning Objective: Solve for equilibrium income in the Keynesian cross model with all four sectors. Data as of: 2024 Provenance: OpenStax Macro 3e | MIT OCW 14.02

Explanation

The Keynesian cross model determines equilibrium output (Y*) as the level at which aggregate expenditure (AE) equals output (Y). With all four sectors, expenditure is:

AE = C + I + G + NX

Consumption function: C = a + b(Y − T̄), where a > 0 is autonomous consumption, b ∈ (0,1) is the marginal propensity to consume (MPC), Y is national income, and T̄ is lump-sum net taxes (autonomous).

Investment: I = Ī (autonomous, fixed in this model — we relax this in M13.L03).

Government spending: G = Ḡ (autonomous/exogenous).

Net exports: NX = X̄ − mY, where X̄ is autonomous exports and m > 0 is the marginal propensity to import. As Y rises, imports rise proportionally; exports are exogenous.

Equilibrium condition: Y = AE. Substituting:

Y = a + b(Y − T̄) + Ī + Ḡ + X̄ − mY Y = a − bT̄ + bY + Ī + Ḡ + X̄ − mY Y − bY + mY = a − bT̄ + Ī + Ḡ + X̄ Y(1 − b + m) = a − bT̄ + Ī + Ḡ + X̄

\[Y^* = \frac{a - b\bar{T} + \bar{I} + \bar{G} + \bar{X}}{1 - b + m}\]

The denominator (1 − b + m) is the leakage rate: saving (1 − b) and imports (m) are both withdrawals from the expenditure circular flow. The numerator is total autonomous spending net of tax-induced consumption reduction.

45-degree diagram logic: The AE line has slope (b − m) < 1 (since b < 1 and m > 0). Equilibrium is where the AE line crosses the 45° line (Y = AE). An upward shift in any autonomous component (G, I, X, a) shifts AE up, raising Y* by the multiplied amount.

Worked Example

Parameters: a = 200, b = 0.8, T̄ = 100, Ī = 150, Ḡ = 250, X̄ = 180, m = 0.1

Step 1 — Compute the numerator (autonomous spending net of taxes):

a − bT̄ + Ī + Ḡ + X̄ = 200 − (0.8)(100) + 150 + 250 + 180 = 200 − 80 + 150 + 250 + 180 = 700

Step 2 — Compute the denominator (leakage rate):

1 − b + m = 1 − 0.8 + 0.1 = 0.3

Step 3 — Solve for Y*:

Y* = 700 / 0.3 = $2,333.3bn (rounded to $2,333bn)

Step 4 — Verify by computing AE at Y = 2,333:

C = 200 + 0.8(2,333 − 100) = 200 + 0.8(2,233) = 200 + 1,786.4 = 1,986.4 I = 150 G = 250 NX = 180 − 0.1(2,333) = 180 − 233.3 = −53.3 AE = 1,986.4 + 150 + 250 − 53.3 = 2,333.1 ✓ (rounding difference)

Common Misconception

Misconception: In the open-economy Keynesian cross, the equilibrium formula is Y* = (a + bT̄ + Ī + Ḡ + X̄)/(1 − b + m) — i.e., taxes enter with a positive sign in the numerator.

Correction: Taxes enter with a negative sign because higher taxes reduce disposable income (Y − T̄), which reduces consumption. The correct numerator term is −bT̄ (not +bT̄). A tax increase of ΔT̄ reduces Y* by bΔT̄/(1 − b + m), which is negative — an intuitive result.

Practice Prompts

  1. In the Keynesian cross model with four sectors, what are the two sources of leakage from the expenditure stream, and how do they appear in the multiplier denominator? → Answer: The two leakages are (1) saving, captured by (1 − b) — the fraction of each additional dollar of income not consumed — and (2) imports, captured by m — the fraction of each additional dollar of income spent on foreign goods. Together they form the denominator (1 − b + m), which reduces the multiplier effect.

  2. Using b = 0.75, m = 0.15, T̄ = 200, a = 100, Ī = 120, Ḡ = 180, X̄ = 140, calculate equilibrium income Y*. → Answer:

  3. Numerator: 100 − (0.75)(200) + 120 + 180 + 140 = 100 − 150 + 120 + 180 + 140 = 390
  4. Denominator: 1 − 0.75 + 0.15 = 0.40

  5. Y* = 390 / 0.40 = $975bn

  6. In the model above, if autonomous exports X̄ increase by \(20bn (e.g., due to stronger Chinese demand for Australian iron ore), by how much does Y\* increase? Interpret this in the context of Australia's resource export dependence. → **Answer:** ΔY\* = ΔX̄ × 1/(1 − b + m) = 20 × (1/0.40) = 20 × 2.5 = **\)50bn**. The $20bn export boost is multiplied 2.5 times. This illustrates Australia's structural sensitivity to commodity export demand: a sharp rise in iron ore exports to China amplifies through domestic consumption, creating a sizeable income effect well beyond the direct export revenue.

Visual — Keynesian Cross and the Multiplier

The Keynesian cross Output / income, Y Aggregate expenditure, AE 45° line: Y = AE AE₀ AE₁ Intercept = a − bT̄ + Ī + Ḡ + X̄ Autonomous-demand increase shifts AE upward E₀ E₁ Y₀ Y₁ slope = b − m Multiplier gap ΔA

Figure: The AE line crosses the 45° line where planned expenditure equals output. A rise in autonomous spending shifts AE upward and increases equilibrium income by more than the initial shift — the multiplier effect.

Further Resources