The Circular Flow Model for a Closed Economy: Classical Long-Run Equilibrium

Introduction to the Circular Flow Model for a Closed Economy

• Classical Theory Fundamentals:
  • Time Frame: This model operates under the assumptions of the long run.
  • Exogenous Variables: In the long run, aggregate production/output ($Y$) is considered exogenous ($Y = \bar{Y}$).
  • Market Focus: The primary focus is on the equilibrating forces within the goods market.
  • Equilibrating Mechanism: In a closed economy, the real interest rate ($r$) serves as the primary mechanism that brings the goods market into equilibrium.
  • Scope: A closed economy is defined by the absence of international trade, meaning net exports ($NX$) are equal to zero ($NX = 0$).

The Interest Rate

• The Nominal Interest Rate ($i$):

  • Definition: The nominal interest rate represents the nominal return on savings or the nominal cost of debt, expressed as a percentage ($\%$).
  • Temporal Representation: If an individual holds $e\,1$ in Period $t$, they will have $e\,1 + i_{t}$ in Period $t + 1$.\n

• The Real Interest Rate ($r$):

  • Definition: The real interest rate represents the real return on savings or the real cost of debt in terms of purchasing power, expressed as a percentage ($\%$).
  • Temporal Representation: If an individual has $1$ goods basket in Period $t$, it earns the equivalent of $1 + r_{t}$ goods baskets in Period $t + 1$.

• Mathematical Relation Between $i$ and $r$:

  • Step-by-Step Derivation:
    1. In Period $t$, a single goods basket is worth a price level of $P_{t}$. Thus, $e\,P_{t}$ is required to buy one basket.
    2. By Period $t+1$, the investment grows from $e\,P_{t}$ to $e\,P_{t} \times (1 + i_{t})$.
    3. The amount of goods baskets obtainable in Period $t+1$ is the total money divided by the new price level: $\frac{P_{t} \times (1 + i_{t})}{P_{t+1}}$.
    4. This quantity of future goods baskets must equal $1 + r_{t}$.
    5. Therefore: $1 + r_{t} = \frac{P_{t} \times (1 + i_{t})}{P_{t+1}}$.
    6. Using the definition of inflation ($\pi_{t+1}$), where $P_{t+1} = P_{t} \times (1 + \pi_{t+1})$, the equation becomes: $1 + r_{t} = \frac{1 + i_{t}}{1 + \pi_{t+1}}$.
  • The Fisher Equation Approximation:
    • The relationship is approximately: $1 + r_{t} \approx 1 + i_{t} - \pi_{t+1}$.
    • This simplifies to: $r_{t} \approx i_{t} - \pi_{t+1}$.

The Components of the Model

• Planned Aggregate Expenditures:

  • In a closed economy, total planned expenditures are the sum of consumption ($C$), investment ($I$), and government purchases ($G$).
  • The equilibrium condition is: $Y = C + I + G$.

• Private Consumption ($C$):

  • Consumption Function: $C = \bar{C} + c \times (Y - T)$.
  • Disposable Income ($Y - T$): The aggregate income ($Y$) minus taxes ($T$).
  • Marginal Propensity to Consume ($c$): A parameter where $0 < c < 1$. It measures how consumption reacts to a change in disposable income.
  • Autonomous Consumption ($\bar{C}$): An exogenous parameter representing consumption independent of income, often used to capture factors like consumer confidence.

• Planned Private Investment ($I$):

  • Investment Function: $I = \bar{I} - b \times r$.
  • Sensitivity to Interest Rates ($b$): A parameter where $b > 0$. It represents how sensitive investment spending is to changes in the real interest rate.
  • Autonomous Investment ($\bar{I}$): An exogenous parameter capturing investment independent of the interest rate, such as business confidence.

• Fiscal Policy Variables ($G$ and $T$):

  • Government purchases ($G$) and taxes ($T$) are treated as exogenous variables: $G = \bar{G}$ and $T = \bar{T}$.
  • Fiscal Expansion: A policy change in $G$ or $T$ designed to increase planned aggregate expenditures (e.g., increasing $G$ or decreasing $T$).
  • Fiscal Contraction: A policy change in $G$ or $T$ designed to decrease planned aggregate expenditures.
  • Budget Status:
    • Balanced Budget: $G = T$.
    • Government Surplus: $G < T$.
    • Government Deficit: $G > T$.

Equilibrium in the Goods and Loanable Funds Markets

• The Set-Up System:

  • $Y = \bar{Y}$
  • $C = \bar{C} + c \times (Y - T)$
  • $I = \bar{I} - b \times r$
  • $G = \bar{G}$
  • $T = \bar{T}$
  • Market Equilibrium Condition: $Y = \bar{C} + c \times (Y - T) + \bar{I} - b \times r + G$.

• Definitions of Saving:

  • Private Saving ($S_{pr}$): $S_{pr} = Y - T - C$.
  • Public Saving ($S_{pub}$): $S_{pub} = T - G$.
  • Total (National) Saving ($S$): $S = S_{pr} + S_{pub} = Y - C - G$.

• Equivalence of Markets:

  • The goods market is in equilibrium when aggregate production equals planned expenditures ($Y = C + I + G$).
  • Algebraic rearrangement: $Y - C - G = I$.
  • This implies $S = I$, meaning the supply of loanable funds equals the demand for loanable funds.
  • Conclusion: The real interest rate ($r$) reaches its equilibrium value in the loanable funds market.

• The Market for Loanable Funds Equations:

  1. Supply of Loanable Funds ($S$):
     • $S = Y - (\bar{C} + c \times (Y - T)) - G$
     • $S = (1 - c) \times Y + c \times T - \bar{C} - G$
     • The supply is a constant value ($S = \bar{S}$) because all variables in the expression are exogenous.
  2. Demand for Loanable Funds ($I$):
     • $I = \bar{I} - b \times r$
  3. Equilibrium:
     • $\bar{S} = \bar{I} - b \times r$

Convergence to Equilibrium

• Excess Demand Case:

  • If $Y < C + I + G$, there is excess demand in the goods market.
  • This is equivalent to $S < I$, which is excess demand in the loanable funds market.
  • Result: The interest rate $r$ increases, which reduces investment until equilibrium is restored.

• Excess Supply Case:

  • If $Y > C + I + G$, there is excess supply in the goods market.
  • This is equivalent to $S > I$, which is excess supply in the loanable funds market.
  • Result: The interest rate $r$ decreases, which stimulates investment until equilibrium is restored.

Applications: Economic Shocks

• Scenario 1: A Shock in the Supply of Loanable Funds (Increased $G$):

  • Suppose government purchases ($G$) increase.
  • Effect on Saving: Domestic saving ($S$) decreases because $S = Y - C - G$.
  • Graphical Shift: The vertical supply curve ($S-curve$) shifts to the left.
  • Market Implication: This creates excess demand for loanable funds at the initial interest rate.
  • Adjustment: The real interest rate $r$ increases.
  • Crowding Out: As $r$ rises, investment ($I$) decreases (moving along the $I-curve$). This phenomenon where fiscal expansion reduces private investment is called "crowding out."
  • Note: Similar results occur if taxes ($T$) decrease or autonomous consumption ($\bar{C}$) increases.

• Scenario 2: A Shock in the Demand for Loanable Funds (Increased Investment Demand):

  • Suppose autonomous investment ($\bar{I}$) increases due to business confidence.
  • Effect on Investment: The investment demand curve ($I-curve$) shifts to the right.
  • Adjustment: This creates excess demand for loanable funds. The real interest rate $r$ increases.
  • Result: Because the supply of saving ($S$) is fixed and vertical, the equilibrium quantity of investment does not change ($I_{1} = I_{2}$). The only change is the higher equilibrium interest rate ($r_{2} > r_{1}$).

Case Study: Military Spending and Interest Rates

• Context: Examining the relationship between government spending ($G$) and interest rates during wartime.
• Prediction: The model predicts that as military spending ($G$) increases, the real interest rate ($r$) should increase.
• Evidence: Historical data from the United Kingdom (1730–1919) comparing military spending as a percentage of GDP to the real interest rate on long-term government bonds supports this prediction.
• Critique and Nuance: While the data aligns with the model, it does not strictly prove it. Higher real interest rates during war could also be attributed to a high risk premium (the increased risk of government default during conflict), rather than just the crowding-out mechanism of the circular flow model.
• Source Citation: This case study is referenced from N. Gregory Mankiw and Mark P. Taylor: Macroeconomics, Second European Edition (2014).