Barry Haworth

University of Louisville

Department of Economics

Economics 202

### Derivation of the Tax Multiplier

A good first step in deriving any multiplier within the AE model
is to determine equilibrium income (i.e. Y*). We start with the
following (algebraic) set of equations:

C = m(DI) + b

I = I_{o}

G = G_{o}

T = T_{o}

X = M = 0

(C = consumption spending, m = marginal propensity to consume,
b = autonomous consumption, DI = disposable income,
I = investment expenditure, G = government spending,
X = exports, M = imports, and T = tax revenue)

The "o" subscripts are provided to distinguish these
variables as autonomous expenditures.

**Step 1 - Find equilibrium:** (not sure how? click
here)

Equate Y with AE, solve for Y*:

AE = Y = [m(Y - T_{o}) + b] + I_{o} +
G_{o} + (0 - 0)

Y* = (-mT_{o} + b + I_{o} + G_{o})/(1 - m)

**Step 2 - Consider a change in T**

Suppose we allow for a change in taxes.
If so, then:

**Step 3 - Recalculate the new equilbrium:**

Equate Y with the new AE, solve for Y**:

AE = Y = [m(Y - [T_{o} + DT]) + b] + I_{o} + G_{o} + (0 - 0)

Y** = (-m[T_{o} + DT] + b + I_{o} + G_{o})/(1 - m)

**Step 4 - Determine the change in Y:**

Subtract Y* from Y**:

Y** - Y* = [(-m[T_{o} + DT] + b + I_{o} + G_{o})/(1 - m)] -
[(-mT_{o} + b + I_{o} + G_{o})/(1 - m)]

Y** - Y* = -mDT/(1 - m)

DY =
-mDT/(1 - m)

DY/DT
= -m/(1 - m)

This is the tax multiplier we worked with on the Multiplier handout. Note that this
multiplier makes one particularly unrealistic (simplifying) assumption. That is, that
tax revenues don't change with changes in consumer income. This assumption doesn't change
any of the basic results we get when working with this tax multiplier, but it does put a restriction on how detailed our conclusions might otherwise be.