Barry Haworth
University of Louisville
Department of Economics
Economics 202

### Derivation of the Government Expenditure 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.

Step 1 - Find equilibrium: (not sure how? click here)
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 G
Suppose we allow for a change in govt. spending. If so, then:

G = Go + DG

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

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

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

Y** - Y* = [(-mTo + b + Io + [Go + DG])/(1 - m)] - [(-mTo + b + Io + Go)/(1 - m)]
Y** - Y* = (
DG)/(1 - m)
DY = (DG)/(1 - m)
DY/DG = 1/(1 - m)

This is the expenditure multiplier we worked with on the Multiplier handout. Note that this multiplier makes a few unrealistic (simplifying) assumptions. For example, one assumption is that government expenditures don't automatically change when GDP changes. Another is that changes in interest rates don't affect GDP. These assumptions don't change any of the basic results we get when working with this multiplier, but it does put a restriction on how detailed our conclusions might otherwise be.