Question 11 of 17advanced🔧 ApplyNumerical4 marks

Calculate the open economy multiplier with proportional taxes, T = tY, instead of lump-sum taxes as assumed in the text.

Correct Answer

The open economy multiplier with proportional taxes (T = tY) is:

Multiplier = 1/(1 - c + ct + m)

or equivalently,

Multiplier = 1/[1 - c(1-t) + m]

where:

  • c = marginal propensity to consume
  • t = tax rate
  • m = marginal propensity to import
Exercise: EXERCISES | Q: 12 | (Page Page 12)
For More Understanding

Explanation

The textbook shows the derivation for lump-sum taxes where the multiplier equals 1/(1-c+m). For proportional taxes T = tY, the student must modify the consumption function. Since disposable income YD = Y - T = Y - tY = Y(1-t), consumption becomes C = c(Y-tY) = cY(1-t). Substituting this into the equilibrium equation Y = Ā + cY(1-t) - mY and solving yields Y(1-c+ct+m) = Ā, giving the multiplier 1/(1-c+ct+m). This shows that proportional taxes reduce the multiplier value compared to lump-sum taxes because higher income leads to higher taxes, which reduces disposable income and consumption.

Solution Steps

  1. Step 1: Start with the equilibrium condition for an open economy: Y = C + I + G + X - M

  2. Step 2: With proportional taxes T = tY, disposable income YD = Y - T = Y - tY = Y(1-t)

  3. Step 3: Substitute consumption function: C = c(Y-tY) = cY(1-t)

  4. Step 4: Write equilibrium: Y = Ā + cY(1-t) - mY, where Ā represents autonomous expenditure

  5. Step 5: Rearrange: Y = Ā + cY - ctY - mY

  6. Step 6: Collect Y terms: Y - cY + ctY + mY = Ā

  7. Step 7: Factor out Y: Y(1 - c + ct + m) = Ā

  8. Step 8: Solve for Y: Y = Ā/(1 - c + ct + m)

  9. Step 9: Therefore, the open economy multiplier with proportional taxes = 1/(1 - c + ct + m)

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