Chapter 6: The Simplest Short-Run Macro Model – Consumption, Aggregate Expenditure, and Equilibrium

Consumption Function and Its Components

Definition and Structure

The consumption function describes the relationship between household consumption and disposable income. It is a foundational concept in short-run macroeconomic models, especially in analyzing aggregate expenditure (AE).

• General Form: $C = a + bYD$

• a = Autonomous consumption (the intercept): Consumption when disposable income is zero.

• b = Marginal Propensity to Consume (MPC) (the slope): The fraction of additional income that is spent on consumption.

• YD = Disposable income: Income after taxes and transfers.

Example: If the consumption function is $C = 600 + 0.5YD$, then autonomous consumption is 600, and the MPC is 0.5.

Finding the Consumption Function from a Graph

1. Intercept: Identify where the consumption line crosses the vertical axis (when $YD = 0$). This gives autonomous consumption ($a$).

2. Slope (MPC): Select two points on the line, calculate the change in consumption ($\Delta C$) over the change in disposable income ($\Delta YD$): $MPC = \frac{\Delta C}{\Delta YD}$

3. Equation: Substitute the values into $C = a + bYD$.

Example Calculation:

• At $YD = 0$, $C = 600$ ($a = 600$)

• At $YD = 1200$, $C = 1200$; at $YD = 2400$, $C = 1800$

• $MPC = \frac{1800 - 1200}{2400 - 1200} = \frac{600}{1200} = 0.5$

• So, $C = 600 + 0.5YD$

Aggregate Expenditure (AE) and Equilibrium

Components of Aggregate Expenditure

Aggregate Expenditure (AE) is the total planned spending in the economy. In the simplest model (no government, no foreign sector):

• $AE = C + I$

• C = Consumption

• I = Planned Investment (assumed autonomous)

Example: If $C = 200 + 0.8Y$ and $I = 300$, then $AE = 200 + 0.8Y + 300 = 500 + 0.8Y$

Autonomous Expenditure

• Autonomous AE is the sum of all components of AE that do not depend on income (e.g., autonomous consumption and investment).

• For $C = 200 + 0.8Y$ and $I = 300$, autonomous AE is $200 + 300 = 500$.

Equilibrium National Income

Equilibrium occurs where planned aggregate expenditure equals actual output/income:

• $Y = AE$

• To solve for equilibrium income, set $Y = a + bY + I$ and solve for $Y$.

Example:

• $Y = 200 + 0.8Y + 300$

• $Y = 500 + 0.8Y$

• $0.2Y = 500$

• $Y = 2500$

Table Example: Calculating C and AE

Given $C = 200 + 0.8Y$ and $I = 300$, for $Y = 1250$:

• $C = 200 + 0.8 \times 1250 = 1200$

• $AE = 1200 + 300 = 1500$

Disposable Income vs. National Income

• In the absence of government and taxes, disposable income ($Y_d$) equals national income ($Y$).

• When government and taxes are introduced, $Y_d = Y - T$ (where $T$ is taxes).

Marginal Propensity to Consume (MPC) and Save (MPS)

Definitions

• MPC: The fraction of an additional dollar of income that is consumed. $MPC = \frac{\Delta C}{\Delta Y}$

• MPS: The fraction of an additional dollar of income that is saved. $MPS = 1 - MPC$

Table Example: Calculating MPC and MPS

Disposable Income	Consumption
0	15
30	36

• Change in income: $\Delta Y = 30$

• Change in consumption: $\Delta C = 36 - 15 = 21$

• $MPC = \frac{21}{30} = 0.70$

• $MPS = 1 - 0.70 = 0.30$

Key Relationship: $MPC + MPS = 1$

Aggregate Expenditure Model and the 45° Line

Understanding the AE Diagram

• The 45° line on the AE diagram shows all points where $AE = Y$ (actual output equals planned expenditure).

• If the AE curve is above the 45° line at a given $Y$, then $AE > Y$:

  • Firms' inventories fall

  • Firms increase output

  • Income/output rises toward equilibrium

• If the AE curve is below the 45° line at a given $Y$, then $AE < Y$:

  • Inventories accumulate

  • Firms decrease output

  • Income/output falls toward equilibrium

• Equilibrium is where the AE curve intersects the 45° line.

Example: At $Y_1$, if $AE > Y_1$, output will rise. At $Y_4$, if $AE < Y_4$, output will fall. Equilibrium is at $Y_3$ where $AE = Y$.

Summary Table: Key Formulas and Concepts

Concept	Formula	Description
Consumption Function	$C = a + bYD$	Relationship between consumption and disposable income
Aggregate Expenditure	$AE = C + I$	Total planned spending (simple model)
Equilibrium Income	$Y = AE$	Where output equals planned expenditure
MPC	$MPC = \frac{\Delta C}{\Delta Y}$	Marginal propensity to consume
MPS	$MPS = 1 - MPC$	Marginal propensity to save

Quick Exam Memory Tricks

• Intercept of consumption function = autonomous consumption

• Slope of consumption function = MPC

• For equilibrium: Write AE equation, set $Y = AE$, solve for $Y$

• For tables: Compute MPC first, then MPS

• On AE diagram: Above 45° line → output rises; below → output falls

Additional info: The above notes synthesize and expand on the provided examples, connecting them to the broader context of Chapter 6 in a typical macroeconomics textbook. The multiplier concept, government, and trade are not covered here but are foundational for subsequent chapters.