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 = Io
G = Go
T = To
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.

Equate Y with AE, solve for Y*:

AE = Y = [m(Y - To) + b] + Io + Go + (0 - 0)
Y* = (-mTo + b + Io + Go)/(1 - m)

Step 2 - Consider a change in T
Suppose we allow for a change in taxes. If so, then:

T = To + DT

Step 3 - Recalculate the new equilbrium:
Equate Y with the new AE, solve for Y**:

AE = Y = [m(Y - [To + DT]) + b] + Io + Go + (0 - 0)
Y** = (-m[To +
DT] + b + Io + Go)/(1 - m)

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

Y** - Y* = [(-m[To + DT] + b + Io + Go)/(1 - m)] - [(-mTo + b + Io + Go)/(1 - m)]
Y** - Y* = -m
DT/(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.