DANBY  Macro Flows Tutorial   |   1.3   |   General  Solution
 

Here is a general solution to for the income-expenditure model.

Y = C + Ip + G       equilibrium condition
C = a + b(Y-T)       consumption function
Ip = Ip               investmentis fixed
G = G                 government spending is fixed
T = T                 taxes are fixed
(We use red to indicate constants -- fixed values)

Substituting into the first equation, we get

Y = a + b(Y-T) + Ip + G

multiplying through,

Y = a + bY - bT + Ip + G

subracting bYfrom both sides,

Y - bY = a + - bT Ip + G

re-expressing the left side

Y(1 - b) = a + - bT Ip + G

and dividing through by (1 - b)

    a + - bT + Ip + G
Y = -----------------
         (1 - b)

gives us the final expression for equilibrium Y.  What this shows is that any change in government spending or in capital investment will have a multiplied effect on national income -- multiplied by

    1
 -------
 (1 - b)

which is called, not surprisingly, the "multiplier."