Lecture 10 · Slides L10 · Dowling-style problems 6.7 to 6.17 · about 50 min

Income Determination: Multipliers and Comparative Statics

The Keynesian model with income taxes and imports, solved with calculus: every multiplier is a partial derivative of equilibrium income. Worked through the lecture's two full examples, including full-employment targets, budget effects and the balance of payments.

By the end you can

Solve the three-sector model with T = T₀ + tY for equilibrium income
Derive the government, lump-sum tax and tax-rate multipliers as partial derivatives
Show why the balanced-budget multiplier is 1 with lump-sum taxes and below 1 with income taxes
Solve the four-sector model and its export, import and tax multipliers
Find the policy change needed to reach full employment and its effect on the budget

The idea: multipliers are derivatives

Lecture 8 found multipliers by reasoning through spending rounds. This lecture does it the fast way. Solve the model for equilibrium income $\bar Y$ as a function of every exogenous variable, then take partial derivatives. Each derivative is a multiplier. This is comparative statics: comparing equilibria before and after a change.

Closed and open multipliersFormula sheet →

\text{closed: } \frac{1}{1 - MPC} \qquad \text{open: } \frac{1}{1 - (MPC - MPM)}

Use it when a quick reminder from Lecture 8. The models below generalise both.

The three-sector model with an income tax

Three-sector modelModelFormula sheet →

\begin{aligned} Y &= C + I_0 + G_0 \\ C &= C_0 + bY_d, \quad Y_d = Y - T \\ T &= T_0 + tY \end{aligned}

$C_0$: autonomous consumption
$b$: MPC, 0 < b < 1
$T_0$: autonomous (lump-sum) taxes
$t$: marginal tax rate, 0 < t < 1
$I_0, G_0$: exogenous investment and government spending

Use it when taxes rise with income. This is the lecture's Example 1 setup.

Equilibrium income, three-sector modelKey resultFormula sheet →

\bar{Y} = \frac{1}{1 - b + bt}\,\big(C_0 - bT_0 + I_0 + G_0\big)

Use it when every three-sector numerical question. Plug in, then differentiate for multipliers.

Now differentiate.

Government spending multiplierFormula sheet →

\frac{\partial \bar Y}{\partial G_0} = \frac{1}{1 - b + bt} > 0

Use it when G changes. Positive because 0 < b < 1.

Autonomous tax multiplierFormula sheet →

\frac{\partial \bar Y}{\partial T_0} = \frac{-b}{1 - b + bt} < 0

Use it when lump-sum taxes change. Negative, and smaller in size than the G multiplier.

The tax rate appears in the denominator, so use the quotient rule:

Tax-rate multiplierFormula sheet →

\frac{\partial \bar Y}{\partial t} = \frac{-b\,(C_0 - bT_0 + I_0 + G_0)}{(1 - b + bt)^2} = \frac{-b\,\bar Y}{1 - b + bt} < 0

Use it when the marginal tax rate changes. Because a parameter inside the multiplier changes, this only approximates the effect of a finite change.

Balanced-budget multipliers

Example 2: lump-sum taxes only ( $T = T_0$ , so $t = 0$ ). The two multipliers are $\frac{1}{1-b}$ and $\frac{-b}{1-b}$ . Raise $G_0$ and $T_0$ by one unit each:

\frac{1}{1 - b} + \frac{-b}{1 - b} = \frac{1 - b}{1 - b} = 1

The balanced-budget multiplier is exactly 1.

Example 3: income taxes ( $T = T_0 + tY$ ). Raise $G_0$ and $T_0$ by one unit each:

Balanced-budget multiplier with an income taxFormula sheet →

\frac{1}{1 - b + bt} + \frac{-b}{1 - b + bt} = \frac{1 - b}{1 - b + bt} < 1

Use it when G and autonomous taxes rise together but taxes also depend on income. The effect is positive but smaller than 1.

Why below 1? Total tax revenue rises by more than $\Delta T_0$ : the extra income also generates $t\,\Delta Y$ of tax. So the budget is not really balanced after the change.

Example 4: the full three-sector problem

The lecture’s main worked example. Use the model above with $C_0 = 100$ , $b = 0.75$ , $t = 0.20$ , $T_0 = 240$ , $I_0 = 90$ , $G_0 = 330$ .

(d, e) Reaching full employment. Full-employment income is 1,000. The gap is $1000 - 850 = 150$ .

(f, g) What does each policy do to the budget? Start at $\bar Y = 850$ : $T = 240 + 0.2(850) = 410$ and $G = 330$ , a surplus of 80.

(h, i) Changing the tax rate. Raising the 20% tax rate by 10 per cent means $\Delta t = 0.10 \times 0.20 = 0.02$ .

The four-sector model

Add exports and imports. Imports depend on disposable income:

Four-sector modelModelFormula sheet →

\begin{aligned} Y &= C + I_0 + G_0 + X_0 - Z \\ C &= C_0 + bY_d, \quad Z = Z_0 + zY_d \\ Y_d &= Y - T, \quad T = T_0 + tY \end{aligned}

$X_0$: exports
$Z$: imports
$Z_0$: autonomous imports
$z$: marginal propensity to import (MPM)

Use it when the problem has trade. The slides write imports as M or Z; the algebra is the same.

Substituting and solving as before:

Equilibrium income, four-sector modelKey resultFormula sheet →

\bar Y = \frac{C_0 - bT_0 + I_0 + G_0 + X_0 - Z_0 + zT_0}{1 - b + bt + z - zt}

Use it when every four-sector numerical question.

Change	Multiplier	Sign
Exports $X_0$	$\frac{1}{1 - b + bt + z - zt}$	$> 0$
Autonomous imports $Z_0$	$\frac{-1}{1 - b + bt + z - zt}$	$\lt 0$
Autonomous taxes $T_0$	$\frac{-b + z}{1 - b + bt + z - zt}$	$\lt 0$ since $z \lt b$
MPM $z$	$\frac{-\bar Y_d}{1 - b + bt + z - zt}$	$\lt 0$

Example 5: the full four-sector problem

$b = 0.9$ , $t = 0.2$ , $C_0 = 125$ , $T_0 = 150$ , $I_0 = 92.5$ , $G_0 = 600$ , $X_0 = 150$ , $Z_0 = 55$ , $z = 0.15$ .

(d, e) Full employment is 2,075, so the gap is 75. Government spending must rise by $75/2.5 = 30$ . Alternatively, the autonomous tax multiplier is $\frac{-0.9 + 0.15}{0.4} = -1.875$ , so taxes must fall by $75/1.875 = 40$ .

(f, g) Budget. Income rises by 75 either way, raising revenue by $0.2 \times 75 = 15$ . The spending route costs $30 - 15 = 15$ ; the tax-cut route costs $40 - 15 = 25$ .

(h, i) Balance of payments. $B/P = X_0 - Z = X_0 - Z_0 - z(Y - T_0 - tY)$ .

(j) A lower MPM. If $z$ falls by one percentage point: $\bar Y_d = 2000 - 150 - 0.2(2000) = 1450$ , so $\Delta \bar Y \approx \frac{-1450}{0.4}(-0.01) = +36.25$ .

Switch the diagram to Open to see Example 5 and try these changes yourself.

Summary and review

Three-sector: $\bar Y = \frac{C_0 - bT_0 + I_0 + G_0}{1 - b + bt}$ .
Multipliers: $\frac{1}{1-b+bt}$ for $G_0$ ; $\frac{-b}{1-b+bt}$ for $T_0$ ; $\frac{-b\bar Y}{1-b+bt}$ for $t$ .
Balanced budget: 1 with lump-sum taxes; $\frac{1-b}{1-b+bt} \lt 1$ with income taxes.
Four-sector: denominator $1 - b + bt + z - zt$ ; exports $+$ , autonomous imports $-$ , taxes $\frac{-b+z}{\cdot}$ .
Spending reaches full employment more cheaply for the budget than a tax cut; a tax cut worsens the external balance more.