Mathematical and Graphical Explanation of the Multiplier Model

Aggregate Income and Expenditure

• Aggregate income (Y) equals aggregate expenditure.
• Components of aggregate expenditure:
  • Consumption (C) by households
  • Investment (I) by businesses
  • Government expenditure (G)
  • Exports (EX) - Expenditure by foreigners
  • Imports (IM) - Expenditure on goods from outside the country; net export is the difference between exports and imports.

Marginal Propensity to Consume (MPC)

• Focus on deriving MPC, assuming other factors are constant.
• Equation: $Y = C + I + G + EX - IM$
• Assumptions:
  • Investment (I) is constant.
  • Government expenditure (G) is constant.
  • Exports (EX) are constant.
  • Imports (IM) have a constant part and a part that varies with consumption.

Consumption Function

• Consumption has two elements:
  • Autonomous consumption (constant part)
  • Part that varies with income (disposable income)

Disposable Income

• Disposable income is income after tax plus transfers:
  • $Disposable Income = Income - Tax + Transfer$

Tax Disaggregation

• Tax has two parts:
  • Constant tax (autonomous)
  • Tax that increases with income

Import Disaggregation

• Import has two elements:
  • Constant part
  • Part dependent on new income

Combining Equations

• Replace disposable income with its value in the aggregate income equation:
  • $Y = C<em>0 + c(Y - T + R) + I + G + EX - (IM</em>0 + mY)$

Factoring and Rearranging

• Multiply through and rearrange terms:
  • $Y = C<em>0 + cY - cT + cR + I + G + EX - IM</em>0 - mY$

Further Simplification

• Disaggregate tax (T) into its components:
  • $Y = C<em>0 + cY - c(T</em>0 + tY) + cR + I + G + EX - IM_0 - mY$

Grouping Terms

• Bring all terms with Y to the left side:
  • $Y - cY + ctY + mY = C<em>0 - cT</em>0 + cR + I + G + EX - IM_0$

Factoring Out Y

• Factor out Y on the left side:
  • $Y(1 - c + ct + m) = C<em>0 - cT</em>0 + cR + I + G + EX - IM_0$

Final Equation

• $Y = \frac{Autonomous Expenditure}{1 - c + ct + m}$
• Autonomous expenditure includes $C<em>0 - cT</em>0 + cR + I + G + EX - IM_0$
• The denominator involves marginal propensity to consume.

Analyzing the Multiplier Equation

• Three levels to consider:
  • Individual elements (c, t, m)
  • The pink level: $1 - c + ct + m$
  • The blue level: the entire equation

Impact of Changes

• If c (marginal propensity to consume) increases:
  • The pink level decreases.
  • The blue level (overall multiplier) increases.
  • Delta Y (change in income) increases (direct relationship).
• If c decreases:
  • The pink level increases.
  • The blue level decreases.
  • Delta Y decreases.
• If t (tax rate) or m (import rate) increases:
  • The pink level increases.
  • The blue level decreases.
  • Delta Y decreases (inverse relationship).

Real-World Implications

• Y represents increased income and expenditure.
• Higher MPC leads to greater income.
• Higher tax rates reduce income.
• More spending on imports reduces domestic income.

Multiplier vs. Marginal Propensity to Consume

• Multiplier: $\frac{1}{1 - MPC}$
• If MPC < 1, the condition is met.

Marginal Propensity to Expand

• Broader measure including all spending on local products and services.
• Consumption expenditure in the United States is over 70% of the total, making it very important.

Potential Output

• Similar to the production possibility curve.
• Divides the quadrant into three areas:
  • Efficient (on the line)
  • Inefficient (inside the line)
  • Unattainable (outside the line)

Equilibrium Output

• Equilibrium output isn't always the same as potential output.
• Equilibrium can fall outside the efficient production level.

Inflationary Scenario

• Current equilibrium is beyond potential output (inflationary).
• Need to bring the economy back to the potential output level.
• Full employment level of production is the same as potential output.

Recessionary Scenario

• Current equilibrium is below potential output (recession).
• Need to increase expenditure.

Government Policy

• Fiscal policy can adjust government spending.
• Reducing expenditure by 30 generates an income decrease of $90 (multiplier effect).
• Increasing government spending by 30 generates an income increase of $120.

Solving Example Problems (Whiteboard Examples):

First Problem:

• Equation: $\Delta Y = \frac{\Delta AE}{1 - 0.5}$
• Target: Reduce $ 200 billion of income.
• Calculations: $\Delta AE = \frac{-200}{0.5} = -100$
• Solution: Government should reduce expenditure by $ 100 billion.

Second Problem:

• Equation: $\Delta Y = \frac{\Delta AE}{1 - 0.75}$
• Target: Increase $120 billion income.
• Calculations: $\Delta AE = 4 \times \Delta AE$
  $\Delta AE = \frac{120}{4} = 30$
• Solution: Government should spend $30 billion.

Third Problem:

• Equation: $\Delta Y = \frac{\Delta AE}{1 - 0.66}$
• Target: Reduce $90 billion of income.
• Calculations: $\Delta AE = 3 \times \Delta AE$
  $\Delta AE = \frac{-90}{3} = -30$
• Solution: Government has to reduce $30 billion.