The Keynesian Cross Model Comprehensive Study Notes

Introduction to the Keynesian Cross Model

• Context and Scope:
  • The model, developed by Koen Vermeylen at the University of Amsterdam (February 2026), focuses on a closed economy in the short run.
  • Short Run Assumptions: In the short run, the Price level ($P$) is considered exogenous (fixed).
  • Focus: The model examines the equilibrating forces specifically within the goods market.
  • Outcome: The primary variable that adjusts to restore equilibrium is Aggregate Production or Income ($Y$).

Core Model Assumptions and Variables

• Economy Specifics:

  • Closed Economy: Defined by the absence of international trade, where Net Exports ($NX$) equal zero ($NX = 0$).
  • Equilibrium Condition: For a closed economy, the goods market is in equilibrium when production ($Y$) equals the sum of consumption ($C$), investment ($I$), and government purchases ($G$): $Y = C + I + G$.
  • Exogenous Variables:
    • Interest rate ($i$) is exogenous because the focus remains on the goods market.
    • Price level ($P$) is exogenous.
    • Inflation ($\pi$) is assumed to be zero ($\pi = 0$) to simplify calculations.
    • Real Interest Rate ($r$): Given $\pi = 0$, then $r = i$, making the real interest rate exogenous.

• Planned Aggregate Expenditures ($E$):

  • Definition: $E$ represents the total planned spending in the economy: $E = C + I + G$.

• Functional Components of the Model:

  • Consumption Function ($C$): $C = \bar{C} + c \cdot (Y - T)$.
    • $\bar{C}$ represents autonomous consumption.
    • $c$ is the marginal propensity to consume ($MPC$), where $0 < c < 1$.
    • $(Y - T)$ is disposable income.
  • Investment Function ($I$): $I = \bar{I} - b \cdot r$.
    • $\bar{I}$ represents autonomous investment.
    • $b$ is the sensitivity of investment to interest rates ($b > 0$).
  • Government Purchases ($G$): Exogenous constant ($G = \bar{G}$).
  • Taxes ($T$): Exogenous constant ($T = \bar{T}$).

The Goods Market Equilibrium

• Mathematical System:

  • The model consists of two equations with two unknowns ($Y$ and $E$):
    1. $E = \bar{C} + c \cdot (Y - T) + \bar{I} - b \cdot i + G$
    2. $Y = E$

• The E-Curve Characteristics:

  • As income ($Y$) increases, planned expenditures ($E$) increase.
  • However, $E$ increases by less than $Y$ because the slope of the curve is defined by the marginal propensity to consume ($c$), which is less than 1 ($0 < c < 1$).

Convergence and Market Dynamics

• Equilibrating Mechanisms:
  • Excess Demand: When $Y < \bar{C} + c \cdot (Y - T) + \bar{I} - b \cdot i + G$, there is a shortage of goods. This causes $Y$ to increase until equilibrium is reached.
  • Excess Supply: When $Y > \bar{C} + c \cdot (Y - T) + \bar{I} - b \cdot i + G$, there is a surplus. This causes $Y$ to decrease until equilibrium is reached.

Economic Shocks: Government Purchases and the Multiplier Effect

• Shock in Government Purchases ($G$):

  • An increase in $G$ shifts the $E$-curve upward.
  • This creates immediate excess demand, leading to an increase in $Y$.
  • As $Y$ increases, $E$ also increases (induced consumption effect), which continues until the goods market reaches a new equilibrium.

• The Multiplier Effect:

  • Observation: The change in production is greater than the change in government spending ($|\Delta Y| > |\Delta G|$).
  • Mathematical Derivation:
    1. Start with equilibrium: $Y = \bar{C} + c \cdot (Y - T) + \bar{I} - b \cdot i + G$
    2. Rearrange: $Y \cdot (1 - c) = \bar{C} - c \cdot T + \bar{I} - b \cdot i + G$
    3. Solve for $Y$: $Y = \frac{1}{1 - c} \cdot [\bar{C} - c \cdot T + \bar{I} - b \cdot i + G]$
    4. Calculate change: $\Delta Y = \frac{1}{1 - c} \cdot \Delta G$
  • The Multiplier: The term $\frac{1}{1 - c}$ is known as the multiplier of Government Purchases.

• Intuitive Breakdown (The Rounds of Spending):

  • Round 1: $G \uparrow$ by $\Delta G \Rightarrow E \uparrow$ by $\Delta G \Rightarrow Y \uparrow$ by $\Delta G$
  • Round 2: $Y \uparrow$ by $\Delta G \Rightarrow E \uparrow$ by $c \cdot \Delta G \Rightarrow Y \uparrow$ by $c \cdot \Delta G$
  • Round 3: $Y \uparrow$ by $c \cdot \Delta G \Rightarrow E \uparrow$ by $c^2 \cdot \Delta G \Rightarrow Y \uparrow$ by $c^2 \cdot \Delta G$
  • Summation: $\Delta Y = [1 + c + c^2 + ...] \cdot \Delta G = \frac{1}{1 - c} \cdot \Delta G$

Economic Shocks: Taxation Policies

• Shock in Taxes ($T$):

  • An increase in $T$ decreases disposable income, causing the $E$-curve to shift downward.
  • This leads to excess supply, resulting in a decrease in $Y$.
  • The process continues until a new equilibrium is established ($Y$ decreases, which further decreases $E$, but by a smaller magnitude).

• Tax Multiplier Formula:

  • $\Delta Y = -\frac{c}{1 - c} \cdot \Delta T$

Economic Shocks: Private Sector Volatility and Animal Spirits

• **Shock in Autonomous Consumption ($\bar{C}$) or Investment ($\bar{I}$):

  • An increase in $\bar{C}$ or $\bar{I}$ shifts the $E$-curve upward, leading to an increase in $Y$ via the multiplier effect.

• Concept of Animal Spirits:

  • Proposed by Keynes, "animal spirits" (consumer or business confidence) can lead to self-fulfilling prophecies.
  • Increased confidence leads to higher aggregate production and income ($Y$).
  • This increase in $Y$ further boosts planned aggregate expenditures ($E$), which may retrospectively justify the initial increase in confidence.

Economic Shocks: Interest Rates and the IS Curve

• Shock in the Interest Rate ($i$):

  • An increase in $i$ increases the cost of borrowing, decreasing planned investment ($I$).
  • This shifts the $E$-curve downward, creating excess supply and leading to a decrease in equilibrium production ($Y$).
  • Conclusion: There is an inverse relationship between the interest rate and equilibrium production ($i \uparrow \Rightarrow Y \downarrow$).

• The IS-Curve Defined:

  • The IS-curve describes the relationship between interest rates and equilibrium production in the goods market.
  • Movement Along the Curve: Occurs when $i$ changes ($i \uparrow \Rightarrow I \downarrow \Rightarrow \text{Excess Supply} \Rightarrow Y \downarrow$).
  • Shift of the Curve: Occurs when variables other than interest rates change:
    • Fiscal Expansion: (e.g., increase in $G$ or decrease in $T$) causes excess demand at a given $i$, shifting the IS-curve to the right.
    • Fiscal Contraction: Causes excess supply at a given $i$, shifting the IS-curve to the left.

Case Study: Fiscal Stabilization and Historical Applications

• Stabilization and Fine-tuning:

  • According to the model, fiscal policy (adjusting $G$ or $T$) can be used to stabilize the economy to reach potential output ($Y = Y_n$).
  • Fiscal expansion is used if $Y < Y_n$.
  • Fiscal contraction is used if $Y > Y_n$.

• Macroeconomic Skepticism regarding Fine-tuning:

  • Timing Problems:
    1. Deciding on policy changes for $G$ or $T$ takes time.
    2. Implementing these changes takes time.
    3. The multiplier effect takes time to propagate through the economy before affecting $Y$ fully.
  • Import Leakage: In open economies, some of the stimulus spending may "leak" out to purchase foreign goods, reducing the effectiveness of the multiplier.

• Large-Scale Applications:

  • Fiscal policy is considered potentially effective for shocking an economy out of severe recessions, such as the Great Depression (Keynes' original context).
  • The Great Recession (2007-2009):
    • In 2009, the U.S. implemented a stimulus package roughly equivalent to 5% of GDP.
    • Analysis: The economy recovered more slowly than forecasted. Proponents of the model argue the economy was more damaged than initially understood, while skeptics suggest the fiscal stimulus was ineffective.