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

Open Economy Multiplier with Proportional Taxes

When taxes are proportional to income rather than lump-sum, the consumption function changes. With $T = tY$ (where $t$ is the tax rate), disposable income becomes $Y_d = Y - tY = (1-t)Y$. Consumption is then $C = \bar{C} + c(1-t)Y$, where $c$ is the marginal propensity to consume.

The equilibrium condition in an open economy is:

Substituting $C = \bar{C} + c(1-t)Y$ and $M = \bar{M} + mY$ (where $m$ is the marginal propensity to import):

Collecting terms with $Y$ on the left:

Therefore, the open economy multiplier with proportional taxes is:

$\alpha = \frac{1}{1 - c(1-t) + m}$ …

The multiplier tells us how much equilibrium income changes when autonomous spending (government expenditure, investment, or exports) changes by one rupee. In a closed economy with lump-sum taxes, the multiplier is $\frac{1}{1-c}$, where $c$ is the marginal propensity to consume. But in an open economy with proportional taxes, two additional leakages shrink the multiplier: a fraction $t$ of each rupee earned goes to taxes, and a fraction $m$ (the marginal propensity to import) leaks out to foreign goods.

The intuition is straightforward. Suppose the government spends an extra rupee; it becomes someone's income. With lump-sum taxes an extra rupee of income is fully disposable at the margin, so consumption rises by $c$ rupees. With proportional taxes, however, a fraction $t$ of that rupee is first taken as tax, leaving disposable income of $(1-t)$ rupees, so consumption rises by only $c(1-t)$. On top of this, imports rise with income: because $M = \bar{M} + mY$, a fraction $m$ of the extra rupee of income is spent on foreign rather than domestic goods and so leaks out of the domestic circular flow. The demand for domestic output therefore rises by only $c(1-t) - m$ rupees in the next round, and this smaller injection keeps circulating, leaking to saving, taxes and imports each time. The cumulative effect is weaker than in the simple closed-economy case, which is exactly why the denominator carries both extra leakages and the multiplier is $\frac{1}{1 - c(1-t) + m}$.

Derivation

Start with the equilibrium condition in an open economy:

where $Y$ is national income, $C$ is consumption, $I$ is investment, $G$ is government expenditure, $X$ is exports, and $M$ is imports.

Here $\bar{C}$ is autonomous consumption, $c$ is the marginal propensity to consume, and $T = tY$ is the proportional tax.

Imports depend on income:

where $\bar{M}$ is autonomous imports and $m$ is the marginal propensity to import.

Substitute the consumption and import functions into the equilibrium condition:

Collect all terms involving $Y$ on the left:

Denote the autonomous spending (everything on the right) as $\bar{A} = \bar{C} + I + G + X - \bar{M}$. Then:

The coefficient $\frac{1}{1 - c(1-t) + m}$ is the open economy multiplier with proportional taxes.

Interpretation

The denominator $1 - c(1-t) + m$ captures three forces:

• The term $c(1-t)$ is the marginal propensity to consume out of national income after taxes. It measures how much of each additional rupee of income stays in the circular flow as domestic consumption.
• The term $m$ is the marginal propensity to import, a direct leakage. …