Sometimes you can be asked to find the equilibrium value
from the Keynesian Cross or the 45 degree line.
This means that we have to find the point where planned expenditures are
equal to GDP by finding where the planned expenditure line (our C + I + G) is
actually equal to the GDP in the economy.
Sometimes these lines will not cross or be in equilibrium and this is
due to the fact that inventories are either being built up or used up which
will allow for discrepancies between C + I + G and actual GDP. Please review a previous post on the
construction of the Keynesian Cross (or 45 degree line) and how inventorieswork for more information.

This particular post is going over the process to find the
equilibrium value mathematically assuming that you are given the marginal
propensity to consume value, investment value, government spending value and
taxes. Here is the specific problem:

In an economy with no exports or imports, C = 0.94(DI) + 87;
I = 15; G = 25; and T = 25. Taxes do not vary with the level of income. In
addition, in this closed economy GDP = NI = PI. All figures are in billions.
Find the level of equilibrium in the economy, i.e., when actual expenditures
equal planned expenditures in the economy.

While notation may vary from textbook to textbook or from
class to class, generally C is consumption, I is investment, G is government
spending, T is taxes, DI is disposable income (GDP – T), GDP is gross domestic
product also commonly notated as Y, and NI/PI are national income and planned
income respectively. Other books will
call NI/PI aggregate expenditure and planned expenditure respectively.

Since we are solving for equilibrium in this economic model
we want GDP to be equal to C + I + G. We
are given values for I and G, so we need to find out how to incorporate GDP, C
and T. GDP is the number we are solving
for so we should keep it as our “unknown” variable. C is given as .94(GDP-T), and T is given. Setting this equation up gives us:

GDP = .94(GDP – 25) + 87 + 15 + 25

We can multiply .94 times GDP and 25 to get:

GDP = .94*GDP – 23.5 + 87 + 15 + 25

Subtract .94 GDP from both sides to get:

.06*GDP = 87 + 25 + 15 - 23.5

Now add up the terms on the right hand side, then divide
both sides by .06 to get:

GDP = (87 + 25 + 15 - 23.5)/.06 = 1,725

This means that if GDP is 1,725 we will be at the point
where the two curves cross and therefore be at the economic equilibrium. This means that there is no change in
business inventories (and investment).
These equations are shown below in a graph. Because the marginal propensity to consume is
so close to 1, the slope of the curves are pretty similar and that is why they
are so close to each other.