Macroeconomics Lesson 6 – Aggregate Expenditure Model

Keynesian Cross & Aggregate Expenditures (AE) Model

Focus: Very‐short-run macro model where all individual prices are assumed fixed.
Consequence of fixed prices
- Overall price level stays constant.
- \text{Aggregate Demand (AD)} therefore directly determines real GDP (output).
Synonyms
- "Keynesian Cross" diagram = 45° line diagram.
- Same analytical core adopted in IS–LM, AS–AD introductions.
Guiding questions addressed in the lesson
- What determines expenditure plans when the price level is fixed?
- How is real GDP determined under fixed prices?
- Definition, size, and intuition of the expenditure multiplier.
- Logical bridge between Aggregate Expenditure (micro-foundation) and Aggregate Demand (macro curve).

Total spending identity (National‐income accounting):
- Y = C + I + G + X - M
- Y : Real GDP (also aggregate income)
- C : Consumption expenditure
- I : Investment (gross private domestic)
- G : Government expenditure on goods & services
- X : Exports
- M : Imports
Two-way causality under fixed prices (ceteris paribus)
- Rising Y ➔ higher planned C and M ➔ higher total AE.
- Higher AE ➔ firms raise production ➔ higher Y.
Key endogenous vs. exogenous components
- Endogenous (induced) with respect to Y: C,\;M.
- Exogenous (autonomous): I,\;G,\;X (also autonomous pieces of C and M).

Disposable income (after taxes): YD = Y - T
- T = Net taxes = Taxes − Transfer payments.
Household budget identity: YD = C + S
Behavioural relationships (holding interest rate, wealth, expectations, etc. constant)
- Consumption function: Planned C as a function of YD.
- Saving function: Planned S as a function of YD.
Graphical interpretation (Figure 6.1)
- 45° line where C = YD (zero saving).
- Above 45°: C > YD (dissaving; borrowing/wealth draw-down).
- Below 45°: C < YD (positive saving).

Definitions
- MPC = \dfrac{\Delta C}{\Delta YD}
- MPS = \dfrac{\Delta S}{\Delta YD}
Empirical classroom example (Figure 6.2)
- \Delta YD = \$2\ \text{trn} ➔ \Delta C = \$1.5\ \text{trn} ➔ MPC = 0.75.
- Same \Delta YD ➔ \Delta S = \$0.5\ \text{trn} ➔ MPS = 0.25.
Accounting identity proof
- Starting from C + S = YD take small changes: \Delta C + \Delta S = \Delta YD.
- Divide by \Delta YD ⇒ MPC + MPS = 1.

Short-run assumption: U.S. imports primarily driven by domestic real GDP.
Definition: \text{Marginal Propensity to Import (MPM)} = \dfrac{\Delta M}{\Delta Y}.
- Illustration: \Delta Y = \$1\ \text{trn} raises M by \$0.25\ \text{trn} ⇒ MPM = 0.25.

Planned AE formula under fixed prices
- AE = C + I + G + X - M (all "planned" values; imports subtracted).
Components by nature
- Induced expenditure: C - M portions that vary with Y.
- Autonomous expenditure: I + G + X (plus autonomous slices of C, M) – does not vary with Y.
AE schedule
- Tabular listing of planned AE for each potential Y.
AE curve
- Graph plotting AE (vertical axis) vs. Y (horizontal axis), typically upward sloping with slope < 1.
- Slope equals MPC - MPM when only C and M are induced and no taxes.

Accounting fact: Actual expenditure ≡ Actual Y for any point in time.
Equilibrium condition: AE_{planned} = Y.
- Graphically: AE curve intersects 45° line.
Inventory link (Figure 6.4)
- If AE_{planned} > Y ➔ unplanned inventory depletion ➔ firms increase output ➔ Y rises.
- If Y > AE_{planned} ➔ unplanned inventory accumulation ➔ firms cut output ➔ Y falls.
- Convergence continues until inventories are stable (equilibrium).

Definition: Factor by which a change in autonomous expenditure (∆A) is magnified into a larger change in equilibrium Y (∆Y).
- Multiplier = \dfrac{\Delta Y}{\Delta A}.
Algebraic derivation with AE-curve slope ((z))
- Multiplier = \dfrac{1}{1 - z} where z = \dfrac{dAE}{dY}.
Special cases
- No taxes, no imports: z = MPC ⇒ Multiplier = \dfrac{1}{1 - MPC} = \dfrac{1}{MPS}.
- With taxes/imports, z = MPC\,(1 - t) - MPM ((t) = marginal tax rate), so multiplier shrinks.
Numerical illustrations (Figures 6.5 & 6.6)
- Slope z = 0.75 (no taxes/imports) ⇒ Multiplier = 4.
- Adding taxes & imports ⇒ slope z = 0.5 ⇒ Multiplier = 2.
Intuition
- Initial autonomous rise (e.g., investment) boosts Y.
- Higher income induces more consumption, further elevating Y.
- Successive rounds continue but diminish geometrically (determined by MPC).

Figure 6.7 reveals sequential rounds: Round 0 = ∆A, Round 1 = MPC·∆A, Round 2 = MPC²·∆A, …
Link to business‐cycle turning points
- Peaks & troughs often triggered by changes in autonomous components (investment, exports, fiscal shifts).
- ↑ Autonomous Expenditure ➔ unplanned inventory falls ➔ expansion.
- ↓ Autonomous Expenditure ➔ unplanned inventory rises ➔ recession.
Graphical business cycle (Figure 5.1)
- Trend growth vs. short-run fluctuations (expansion A → peak; peak B → recession → trough).

Real-world firms rarely keep prices frozen indefinitely.
- Persistent inventory gaps prompt both quantity and price adjustments.
Moving beyond Lesson 6
- AS-AD model (Lesson 7) incorporates price level flexibility: simultaneous determination of Y and price level.
- AE model provides the horizontal slice (fixed‐price case) underlying the downward-sloping AD curve.

AE curve: Relationship between planned spending and Y at a given price level.
AD curve: Relationship between price level (P) and equilibrium Y when AE interacts with price-dependent mechanisms (wealth effect, interest‐rate effect, exchange-rate effect).
Derivation mechanism (Figures 6.8a & 6.8b)
1. Start with AE₀ at price level 110 and equilibrium Y = \$13\ \text{trn} (Point B on AD).
2. Increase price level to 130 ➔ wealth ↓ & substitution effects ➔ AE curve shifts down to AE₁.
- New equilibrium Y = \$12\ \text{trn} (Point A) – movement up along AD.
1. Decrease price level to 90 ➔ AE shifts up to AE₂; equilibrium Y = \$14\ \text{trn} (Point C) – movement down along AD.
Key insight: Each possible price level maps to a distinct AE curve; connecting equilibrium points traces the AD curve.

Fiscal stimulus (∆G or tax cuts raising autonomous C) exerts multiplied impact, magnitude depending on MPC, tax/import leakages.
Automatic stabilisers (income taxes, import leakages) dampen multiplier ➔ smoother business cycles.
AE analysis underlies short-run Keynesian policy prescriptions; AS-AD required for price-level & inflation considerations.

National income identity: Y = C + I + G + X - M
Disposable income: YD = Y - T
Household allocation: YD = C + S
Marginal propensities: MPC = \dfrac{\Delta C}{\Delta YD},\; MPS = \dfrac{\Delta S}{\Delta YD},\; MPC + MPS = 1
Marginal propensity to import: MPM = \dfrac{\Delta M}{\Delta Y}
Equilibrium condition (Keynesian Cross): AE_{planned}(Y) = Y
Multiplier (general): k = \dfrac{1}{1 - z} where z is AE-slope.
No‐tax / no‐import case: k = \dfrac{1}{1 - MPC} = \dfrac{1}{MPS}.

Lesson 7 will combine AE insights with Aggregate Supply to build the full AS-AD framework, explaining simultaneous movements in output and the price level.