Chapter 7: Adding Government and Trade to the Simple Macro Model — Study Notes

Government and the Aggregate Expenditure (AE) Model

Government Purchases and Net Taxes

• Government purchases (G) are considered autonomous in the macroeconomic model, meaning they do not depend on the current level of national income.

• Net taxes (T) are defined as total tax revenue minus total transfer payments:

$T = \text{total tax revenue} - \text{total transfer payments}$

• Government purchases do not include transfer payments (e.g., pensions, unemployment benefits).

Net Tax Rate

• The net tax rate (t) measures how much net tax revenue increases when national income increases by $1.

• Mathematically, if the net tax rate is constant:

$T = tY$

• Where t is the net tax rate and Y is national income.

Taxes in the AE Model

• Taxes affect aggregate expenditure (AE) indirectly by reducing disposable income (Yd):

$Y_d = Y - T$

• Consumption depends on disposable income, so taxes reduce consumption and thus AE.

Government Budget Balance

• The government budget can be in one of three states:

  • Budget deficit: $G > T$

  • Budget surplus: $G < T$

  • Balanced budget: $G = T$

Adding Government and Taxes to the Simple Macro Model

Aggregate Expenditure with Government

• The aggregate expenditure function with government is:

$AE = C + I + G$

• But now, consumption (C) depends on disposable income:

$Y_d = Y - T$

• Government affects equilibrium GDP both directly (through G) and indirectly (through T).

Example: Balanced Budget Calculation

• Given: $G = 520$, net tax rate $t = 0.14$

• Balanced budget condition: $G = T$

• Since $T = tY$, set $520 = 0.14Y$

• Solve for $Y$:

$Y = \frac{520}{0.14}$ $Y \approx 3714$

• So, equilibrium income for a balanced budget is approximately 3,714.

Open Economy: Net Exports and Imports

Exports and Imports

• Exports (X) are autonomous with respect to domestic income but depend on foreign income and relative prices.

• Imports (IM) rise as national income increases.

Marginal Propensity to Import (m)

• The marginal propensity to import (m) is the increase in desired imports when national income rises by $1$:

$m = \frac{\Delta \text{imports}}{\Delta Y}$

Net Exports Function

• The net exports (NX) function is:

$NX = X - mY$

• As income rises, imports rise, so net exports decrease with income.

Relative Prices and Net Exports

• If domestic prices rise relative to foreign prices:

  • Imports rise, net exports decrease (NX shifts downward).

• If domestic prices fall:

  • Imports fall, net exports increase (NX shifts upward).

Graphing the Net Exports Function

• Given: $X = 60$, $m = 0.15$

• Net exports function: $NX = 60 - 0.15Y$

• Vertical intercept (Y = 0): $NX = 60$

• X-intercept (NX = 0): $0 = 60 - 0.15Y \Rightarrow Y = 400$

• The NX line is downward sloping from (0, 60) to (400, 0).

The Multiplier with Government and Foreign Trade

Leakages: Taxes and Imports

• Both taxes and imports act as leakages from the spending stream, reducing the size of the multiplier.

Marginal Propensity to Spend on Domestic Output (z)

• Defined as:

$z = MPC(1 - t) - m$

• Where:

  • MPC = marginal propensity to consume

  • t = net tax rate

  • m = marginal propensity to import

Multiplier Formula

• The multiplier in this model is:

$Multiplier = \frac{1}{1 - z}$

• As leakages (taxes and imports) increase, z decreases, and so does the multiplier.

Example: Calculating the Multiplier and Change in Income

• Given: $z = 0.45$, $\Delta G = -1.0$

• Multiplier: $Multiplier = \frac{1}{1 - 0.45} = \frac{1}{0.55} \approx 1.82$

• Change in equilibrium income:

$\Delta Y = Multiplier \times \Delta G$ $\Delta Y = 1.82 \times (-1) = -1.82$

• So, equilibrium income decreases by about 1.82 units.

Aggregate Expenditure Function and Equilibrium

AE Function and Equilibrium Output

• General form: $AE = A + zY$

• Where A is autonomous expenditure and z is the marginal propensity to spend on domestic output.

• Equilibrium occurs where $Y = AE$.

Example: AE and Equilibrium Calculation

• Given: $AE = 750 + 0.60Y$

• Equilibrium: $Y = 750 + 0.60Y$

• $0.40Y = 750 \Rightarrow Y = 1875$

• If autonomous consumption drops by $250$:

  • Multiplier: $\frac{1}{1 - 0.60} = 2.5$

  • Change in equilibrium: $\Delta Y = 2.5 \times (-250) = -625$

Demand-Determined Output and the Fixed-Price Model

• In the AE model (Chapters 6–7), the price level is assumed constant.

• Firms supply whatever output is demanded at that price level.

• National income is determined by aggregate expenditure (demand-determined output).

• This assumption changes in Chapter 8, where price level and aggregate supply are introduced.

Summary Table: Key Equations in the Extended AE Model

Concept	Equation	Description
Disposable Income	$Y_d = Y - T$	Income after taxes
Net Taxes	$T = tY$	Net tax revenue as a function of income
Imports	$IM = mY$	Imports as a function of income
Net Exports	$NX = X - mY$	Exports minus imports
Marginal Propensity to Spend	$z = MPC(1 - t) - m$	Fraction of income spent on domestic output
Multiplier	$Multiplier = \frac{1}{1 - z}$	Effect of autonomous spending on equilibrium income

Key Takeaways

• Government spending and taxes are crucial additions to the macro model, affecting both aggregate expenditure and equilibrium income.

• Net exports introduce the open economy dimension, with imports acting as a leakage from the spending stream.

• The multiplier is reduced by taxes and imports, reflecting leakages from the circular flow of income.

• In the fixed-price model, output is demand-determined; this assumption is relaxed in later chapters.

Additional info: Some explanations and examples have been expanded for clarity and completeness, following standard macroeconomics textbook treatments.