ECONOMICS 1B Week 5 Lesson 1: Keynesian Model in Open Economy - Comprehensive Notes

Key Concepts and Definitions

• National income (Y) and aggregate spending (A)
• Components of aggregate spending: consumption (C), investment (I), government spending (G), and net exports (NX = X − Z)
• Taxes and disposable income:
  • Proportional tax rate t, total tax T = tY
  • Disposable income: Y_d = Y − T = (1 − t)Y
• Consumption function:
  • Without taxes: C = a + cY, where a is autonomous consumption and c is the marginal propensity to consume (MPC)
  • With proportional tax: C = a + cY_d = a + c(1 − t)Y
• Open economy extensions:
  • Net exports NX = X − Z, where X are autonomous exports and Z are imports
  • Imports can be autonomous (Z_bar) plus induced imports (mY) that rise with income
• Equilibrium condition: Y = A (in macroeconomic equilibrium)
• The 45-degree line represents points where aggregate spending equals national income (Y = A)

The Keynesian Model: Basic Setup (Closed Economy Focus)

• Keynesian model purpose:
  • Explain fluctuations in aggregate demand at a fixed price level by identifying determinants of planned expenditure
• Government in the model:
  • Government spending (G) is an autonomous component added to aggregate spending: A = C + I + G
  • Taxes introduce a leakage via disposable income: Y_d = (1 − t)Y
• Aggregate expenditure and the effect of G:
  • With C = a + cY (no taxes), A = a + cY + I + G
  • Equilibrium condition: Y = A
  • This shifts the AE curve upward when G increases, moving the equilibrium from Y1 to Y2
• In summary:
  • The intercept of the AE curve increases by the autonomous component G
  • The 45-degree line intersects the new AE curve at a higher income level

The Multiplier in a Closed Economy

• Basic closed-economy multiplier (no taxes):
  • Equilibrium: Y = a + cY + I + G
  • Solve for Y: $Y = \frac{a + I + G}{1 - c}$
  • Multiplier for government spending: $\text{Multiplier}_{G} = \frac{1}{1 - c}$
• With proportional taxes (tax rate t):
  • Disposable income: Y_d = (1 − t)Y
  • Consumption: C = a + cY_d = a + c(1 − t)Y
  • Equilibrium: Y = a + c(1 − t)Y + I + G
  • Solve for Y: $Y = \frac{a + I + G}{1 - c(1 − t)}$
  • Multiplier for government spending with taxes: $\text{Multiplier}_{G} = \frac{1}{1 - c(1 − t)}$
• Intuition:
  • The multiplier measures how responsive national income is to an autonomous spending change
  • As MPC (c) increases or tax rate (t) decreases, the multiplier grows

The Multiplier in an Open Economy (Exports and Imports)

• Open-economy aggregate expenditure: $A = C + I + G + (X - Z) = C + I + G + NX$
• Imports contain autonomous and induced components:
  • Z = Z_{\text{bar}} + mY, where m is the marginal propensity to import (induced imports)
• Consumption in open economy with taxes remains: C = a + cY_d = a + c(1 − t)Y
• Substitution into equilibrium Y = A gives:
  • Start: Y = a + c(1 - t)Y + I + G + X - (Z_{\bar} + mY)
  • Rearrange: Y = a + I + G + X - Z_{\bar} + [c(1 - t) - m]Y
  • Move Y-terms to one side:
    Y[1 - c(1 - t) + m] = a + I + G + X - Z_{\bar}
  • Alternatively written as:
    Y[1 + m - c(1 - t)] = a + I + G + X - Z_{\bar}
• Open-economy multiplier (change in Y per change in autonomous spending such as G):
  • $\Delta Y / \Delta G = \frac{1}{1 + m - c(1 − t)}$
• Key implications:
  • If there are no imports (m = 0) and no tax (t = 0), we recover the closed-economy multiplier: $\frac{1}{1 - c}$
  • If imports rise with income (m > 0) or taxes exist (t > 0), the denominator increases and the multiplier falls
  • Exports (X) enter as autonomous injections that boost Y, but their impact is through the numerator constant; they do not directly alter the multiplier in the same way as G or T
• Net exports as a concept:
  • NX = X − Z
  • If X > Z, NX is positive and acts as an injection; if imports rise with income, NX can shrink the effectiveness of fiscal stimulus

The Role of Exports and Imports in Determining A and Y

• Exports (X):
  • Autonomous with respect to domestic income (Y)
  • An increase in X raises A and thereby Y, through the multiplier mechanism
• Imports (Z):
  • Can be autonomous (Z_{\bar}) or induced (mY)
  • Induced imports (mY) create leakage from the domestic circular flow, reducing the multiplier effect
• Net exports effect:
  • Net exports enter the AE function as an additional injection: (X - Z)
  • If exports rise or imports fall, A and Y rise; if imports rise with income, the multiplier effect weakens

Fiscal Policy in an Open Economy: Practical Implications

• Expansionary policy (↑ G or ↓ T) in an open economy:
  • Increases equilibrium income via the multiplier, but the effect is dampened by induced imports (mY)
  • Net exports may worsen if imports rise with income or if there is a current account link to the balance of payments
• Key conclusions:
  • With autonomous G, aggregate spending and income rise via the multiplier, but the multiplier itself is reduced by open-economy leakages (m and Z_{\bar})
  • Proportional taxes reduce the size of the multiplier by reducing disposable income and consumption (via c(1 − t) term)
  • Exports act as autonomous injections, increasing equilibrium income similarly to other autonomous spending
  • Imports act as leakages; when they respond to income, they reduce the effectiveness of fiscal stimulus via the multiplier

Diagrams and Graphical Interpretation (What Happens in the AE Diagram)

• Closed economy (no G or tax changes mentioned):
  • The initial aggregate spending curve is determined by C + I (and sometimes initial G)
  • An increase in autonomous spending shifts the AE curve upward from A1 to A2, raising equilibrium income from Y1 to Y2
• With government spending and proportional taxes:
  • The autonomous component of spending includes G, shifting A upward; the tax reduces disposable income and flattens the AE response
  • The new equilibrium is at a higher income, but the slope of the AE curve is reduced by the tax (and further reduced in the open-economy case by m)
• In open economy diagrams, the inclusion of X and Z shifts the intercept and slope of the AE curve depending on m and t, producing a smaller overall increase in Y for a given ΔG

Worked Example: The Multiplier in Practice

• Case: Multiplier calculation from an example (Figure 28.5 context)
  • Initial equilibrium expenditure: ΔE = 0? Approximately R16 trillion (example uses currency R)
  • Autonomous expenditure increases by ΔA = R0.5 trillion
  • Resulting change in equilibrium expenditure: ΔY = R2 trillion
  • Multiplier: $\text{Multiplier} = \frac{\Delta Y}{\Delta A} = \frac{2}{0.5} = 4$
• Insight:
  • The multiplier magnitude depends on the slope of the AE curve; a steeper AE slope (higher MPC) yields a larger multiplier

The Multiplier Derivation (Alternative View via MPS/Tax/Imports)

• Relationship among MPC, MPS, and the multiplier:
  • Since MPC + MPS = 1, a common closed-economy version is: $\text{Multiplier} = \frac{1}{\text{MPS}}$
  • Example: If MPS = 0.25, then Multiplier = 4
• In open economies with taxes and imports, the practical multiplier can be expressed as:
  • $\text{Multiplier} = \frac{1}{1 + m - c(1 - t)}$
  • This reduces to $\frac{1}{1 - c}$ when m = 0 and t = 0

Questions for Practice (Activity 1) and Solutions

• Question 1: In the Keynesian model of an open economy, aggregate expenditure consists of:
  • Options: a) C + I + G, b) C + I + G + NX, c) C + I, d) G + Exports only
  • Correct answer: b) C + I + G + Net Exports (NX)
• Question 2: Which of the following reduces the size of the multiplier in an open economy?
  • Options: a) Higher MPC, b) Lower taxes, c) Higher marginal propensity to import (MPM), d) Increase in government spending
  • Correct answer: c) Higher marginal propensity to import (MPM)
• Question 3: Which of the following is an example of a "leakage" in the Keynesian model?
  • Options: a) Government spending, b) Investment, c) Savings, d) Consumption
  • Correct answer: c) Savings
• Question 4: An increase in exports will result in:
  • Options: a) A decrease in aggregate expenditure, b) An increase in aggregate expenditure and national income, c) No effect on national income, d) A decrease in the multiplier
  • Correct answer: b) An increase in aggregate expenditure and national income
• Question 5: Which policy would increase equilibrium national income in an open economy?
  • Options: a) Cutting government spending, b) Increasing imports, c) Raising taxes, d) Increasing government spending
  • Correct answer: d) Increasing government spending

Connections to Theory and Real-World Relevance

• Links to foundational principles:
  • Keynesian emphasis on aggregate demand and its determinants (C, I, G, X, Z)
  • The role of fiscal policy as a stabilization tool in the short run
  • The open-economy considerations that link domestic policies to foreign sector outcomes
• Practical implications:
  • In small open economies, fiscal stimulus can be less effective due to leakage via imports
  • Policymakers must consider exchange rates, balance of payments, and current account when using G or T to influence Y
  • Exports provide a buffer against domestic downturns, but depend on external conditions (growth abroad, exchange rates)

Summary of Key Equations (Reference Toolbox)

• Aggregate spending in open economy:
  • $A = C + I + G + (X - Z)$
• Consumption with taxes:
  • $C = a + c(1 - t)Y$
• Imports (induced):
  • Z = Z_{\bar} + mY
• Equilibrium in open economy:
  • Y = a + I + G + X - Z_{\bar} + [c(1 - t) - m]Y
  • Equivalent rearranged form: Y[1 + m - c(1 - t)] = a + I + G + X - Z_{\bar}
• Open-economy multiplier (change in Y for a change in autonomous spending like G):
  • $\Delta Y / \Delta G = \frac{1}{1 + m - c(1 - t)}$
• In the closed economy benchmark (no taxes, no imports):
  • $\Delta Y / \Delta G = \frac{1}{1 - c}$
• Net exports:
  • $NX = X - Z$
• Disposable income under proportional tax:
  • $Y_d = (1 - t)Y$
• 45-degree rule and equilibrium concept:
  • Equilibrium occurs where $Y = A$, and the intersection with the 45-degree line marks the output level

Ethical, Philosophical, and Practical Implications

• Policy trade-offs:
  • Fiscal expansion improves output but can worsen current account if imports rise; balance with exchange-rate policies and monetary conditions
• Distributional effects:
  • Tax changes affect disposable income and consumption differently across income groups; redistribution effects are not captured in the basic model but are important in practice
• Temporal considerations:
  • The model captures short-run dynamics; long-run impacts involve other channels (potential output, inflation, debt sustainability)

What Happens Next? (Eduvos Flipped Classroom Context)

• Structure:
  • Pre-class self-study via myLMS, with practice activities and questions
  • In-class active learning with peers and lecturer, focusing on higher-order thinking and application
• Objective for today:
  • Understand how government spending affects production and income in both closed and open economy settings
  • Describe the impact of a proportional income tax on the multiplier
  • Apply the simple Keynesian model to fiscal policy scenarios
  • Analyze how exports and imports alter income in the domestic economy

Quick Reference: Key Formulas to Memorize

• Closed economy (no taxes, no imports):
  • $Y = \frac{a + I + G}{1 - c}$
  • $\text{Multiplier} = \frac{1}{1 - c}$
• Closed economy with proportional tax:
  • $Y = \frac{a + I + G}{1 - c(1 - t)}$
  • $\text{Multiplier} = \frac{1}{1 - c(1 - t)}$
• Open economy with induced imports:
  • $\Delta Y / \Delta G = \frac{1}{1 + m - c(1 - t)}$
• Net exports and imports in the AE identity:
  • $A = C + I + G + (X - Z)$
• Imports specification:
  • Z = Z_{\bar} + mY
• Disposable income and consumption with tax:
  • $Y<em>d = (1 - t)Y, \quad C = a + cY</em>d = a + c(1 - t)Y$