The Open-Economy Income-Expenditure Model

2.3 The Open-Economy Income-Expenditure Model

A. Equilibrium
The equilibrium condition around which the model is built is: Y = C + I_d + G + X - M
Remember this means that total demand for national output equals national output.  But national absorption (C + I_d + G) does not have to equal national output, even in equilibrium, if the economy is open. Equilibrium still means what it did with a closed economy, which is to say that there is no change in inventories.  Equilibrium in no way implies trade balance.
We solve for a situation in which domestic investment is exactly at the level of planned purchases of plant and equipment; change in inventories is zero.)
B. Example
C = 10 + .8(Y-T) (Just like the consumption function from the closed economy)
S = -10 + .2(Y - T)
I_d = 23
G = 10
T = 10
M = .3Y
X = 15
 To find equilibrium Y:

Y =        C         + Id +  G +  X -   M
Y = 10 + .8(Y - 10)  + 23 + 10 + 15 - .3Y
Y = 50 + .5Y
.5Y = 50
Y = 100

Note that at this Y=100, C = 82, + I_d = 23, and G = 10 so C + Id + G (or E) = 115.
Check this against trade: if Y=100 then M = 30, while X = 15, so the country is indeed getting more stuff than it makes; in more technical language total absorption (E) exceeds total output by 15, which is the amount of the trade deficit. This can be seen in the diagram.
C. Thinking about the Example's Solution
At this point we have solved the problem by focusing on goods.  The equation
Y = C + I_d + G + X - M
draws our attention to the fact that in equilibrium, national income equals aggregate demand.
But we know from the diagrams that the financial flows must match as well.
Note that if Y = 100, S = 8. Note further that I_d = 23. Who is financing this gap of 15?  Foreigners.  And indeed 15 is precisely the observed gap between imports and exports, which has to be made up by such financing.  The relevant formula is:

 S = I_d + I_f

Remembering that I_f (a net concept) is negative in this case. The lower part of the diagram for this section graphs net exports and the gap between savings and domestic investment. Savings is also graphed by itself.  Here equilibrium is the point where the amount of financing forthcoming from foreigners is enough to fill the domestic savings-investment gap.
D. Further notes
In this model the level of output (which is also income) adjusts in response to changes in various exogenously-determined components of demand.  By assuming that X is exogenous, while M is a function of Y, we get a model that tells us what the resulting I_f is.  In other words the model produces a domestic equilibrium; if the external finance is forthcoming then it's an external equilibrium too.  As we just saw, you can solve this model for equilibrium Y either in terms of demand (Y = C + I_d + G + X - M) or in terms of the balance of payments (S - I_d = If = X - M).
Apart from some practice with the balances, this model provides a useful insight for countries with relatively open economies: any policy that raises income will worsen the trade balance.  (Although the model does not show it, higher domestic income may also reduce exports, as some goods that could be exported are sold locally instead.)  Note that by assuming that X is exogenous, we are considering a small country case. For a large country, an increase in it imports should raise foreign incomes, and lead to higher exports.
This model assumes that enough foreign finance is always available to cover a trade deficit.  Other  models do not make that assumption.  For example in the open-economy version of the IS-LM model, a model which includes interest rates, a higher domestic interest rate may be required to tempt foreign lenders.  Alternatively, a country may find itself limited, or rationed, in the amount of foreign finance it can obtain.  That might limit growth.