We will start with a very simple economy and work our way up to something a little more complicated.
First, a couple of definitions:
Leakage: a flow in the circular flow diagram that decreases aggregate demand (S, T, IM). In other words a leakage takes money out of the economy and puts is somewhere else such as the financial system , the government, or foreign markets.
Injection: a flow in the circular flow diagram that increases aggregate demand (I, G, EX).
Injections add money into the economy by way of investment spending, government spending, or spending by foreign consumers.
Autonomous Consumption: The amount of consumption that would occur if GDP = 0.
In equilibrium,
If leakages < injections, then AD>Y and inventories will rise.
1) Very Simple Economy
Leakages = Injections
S+T+IM = I+G+EX
If leakages > injections, then AD<Y and inventories will fall.
First, consider and economy with only consumption (C) and output (Y).
Y = output
AD = C
We know that in equilibrium,
AD = Y

Y = C in this simple economy
Assume that the consumption function takes the following form:
C = a + b*DI,
Where, by definition, a = autonomous consumption
b = MPC
Furthermore, assume that there are no taxes, so the consumption function simplifies as follows:
DI = Y-T = Y-0 = Y, so,
C = a + b*Y
We will solve for the equilibrium in this economy given the following values for our variables: a = autonomous consumption = 100 b = MPC = 0.90
Therefore,
C = 100 + .9*Y
Set Y = C
Y = 100 + .9*Y
0.1*Y = 100
Y = 1000 = output at equilibrium
AD = C = 100 + .9(1000) = 1000
Therefore, output is equal to demand.

2) Adding Investment
Assume I = 50
AD = C + I
= (100 +.9*Y) + 50
Again, set Y = AD
Y = (100 +.9*Y) + 50
0.1*Y = 150

Y = 1500
3) Now with G and T
Assume I = 50
G = 100
T = 75
This complicates our consumption function:
C = a + b*DI
= a + b*(Y-T)
= 100 + .9(Y-T)
In equilibrium, Y = AD
Y = C + I + G
= 100 + .9(Y-75) + 50 + 100
= 250 + .9*Y – 67.5
0.1*Y = 182.5
Y = 1825
Let’s take a look at savings in this example:
By definition, S = DI – C = (Y – T) – C
C = 100 + .9(1825-75) = 1675
S = (1825 – 75) – 1675 = 75
We can now look at leakages and injections:
Leakages = S + T + IM = 75 + 75 + 0 = 150
Injections = I + G + EX = 50 + 100 + 0 = 150
As discussed earlier, if imbalances exist between leakages and injections, changes in inventories will occur. Therefore, if leakages

injections, the economy is not at an equilibrium level of output. To illustrate this point, consider the following two deviations from the equilibrium level of output:
Case I: Suppose Y = 1800
C = 100 + .9 (1800 – 75) = 1652.5
AD = C + I + G
= 1652.5 + 50 + 100 = 1802.5
=> AD > Y = 1800
=> inventories fall by 2.5
S = (Y – T) – C
= 1800 – 75 – 1652.5 = 72.5
So,
Leakages = S + T + IM = 72.5 + 75 + 0 = 147.5
Injections = I + G + EX = 50 + 100 + 0 = 150

=> leakages < injections
=> inventories fall by 2.5
Case II: Suppose Y = 2000
C = 100 + .9 (2000 – 75) = 1832.5
AD = C + I + G
= 1832.5 + 50 + 100 = 1982.5
=> AD < Y = 2000
=> inventories rise by 17.5
S = (Y – T) – C
= 2000 – 75 – 1832.5 = 92.5
So,
Leakages = S + T + IM = 92.5 + 75 + 0 = 167.5
Injections = I + G + EX = 50 + 100 + 0 = 150
=> leakages > injections
=> inventories rise by 17.5
This should give you an idea of how to determine the demand side equilibrium mathematically. Notice, however, that we have not yet considered imports and exports. The import market, in particular, causes some problems that we will explore later.