1. Note that G is autonomous
Government and the income-expenditure model
 Consider an economy with a public sector:
Y  C + I + G + X - M [1]
 Assumptions:
G is autonomous--that is, determined independent of income.
Thus, G = Ga
Note that G is autonomous G Ga
Y = GDP

2. Substitute [2], [3], [4], [5], and [6] into [1]
Equilibrium in the open model with government
Y  C + I + G + X - M [1]
C = Ca + cY [2]
I = Ia [3]
G = G a [4]
X = Xa [5]
M = mY [6]
Substitute [2], [3], [4], [5], and [6] into [1]
Y = Ca + cY + Ia + Ga + Xa - mY [7]
Rearrange [5] to obtain:
[1 - c + m]Y = Ca + Ia + Ga + Xa [6]

3. To solve for equilibrium GDP, use the following “reduced form” equation.
Ca + Ia + Ga + Xa
Y* =
1 - c + m

4.  For an open economy with a public sector:
Numerical example
 For an open economy with a public sector:
Y = C + I + G + X - M
C = Y I = 60 G = 90 X = 40 M = .10Y
Thus:
Y* = =
220/.4 = 550

5. Numerical example Y  C + I + G + (X - M)
C = Y( ) I = 50 G = 90 X = 25 M = .10Y
To solve for equilibrium GDP (Y*):
Y* =
= 220/.4 = 550
( ) + .10

6. Graphical illustration
AE = Y AE
Slope = c - m
AE = Y Y
220
550 Y