# ECON 110 Open Economy Multiplier Derivation

St. Charles Community College
ECON 110 Principles of Macroeconomics

There are two expenditure multipliers that we will deal with in this class, and they are the expenditure multipliers for a closed economy (an economy without net exports) and the expenditure multipliers for an open economy (an economy which includes net exports).  Since one is simply an extension of the other, we shall start with the closed economy expenditure multiplier and then expand it for the other.

A Derivation of the Expenditure Multiplier (Closed Economy)

Starting With A Consumption Function:

John Maynard Keynes believed that consumption depended primarily on current income, and he stated this proposition as follows:

“The fundamental psychological law, upon which we are entitled to depend with great confidence both a priori from our knowledge of human nature and from the detailed facts of experience, is that men are disposed, as a rule and on the average, to increase their consumption as their income increases, but not by as much as the increase in their income.”

(Note:  See Keynes, J. M. , The General Theory of Employment, Interest, and Money, Great Minds Series, Prometheus Books, 1997, Page 96.)

If we plot this concept, we develop a curve as follows:

We see two thoughts expressed in this function:

1. Consumption increases as income increases.
2. The consumption function is an increasing function at a decreasing rate.  In other words, as our income increases consumption becomes a smaller part of our income.  Or, as our income increases, so does our saving.

We will now make a simplifying assumption and draw the consumption function as a straight line.

We make this simplifying assumption for two reasons:

1. We have to have a knowledge of calculus in order to work with a curved line, and calculus is not a prerequisite for this course.  We need to have a function on which we can use our algebra.
2. It doesn’t make any difference in this example.

We need to take a look at the slope of the line which we have labeled as the consumption function.  Remember that the slope of a line is defined as rise over run.

Mr. Keynes gave the slope of the consumption function the name of  “the marginal propensity to consume”, and he assumed that the function was constant in our society.

In your algebra classes you learned that the general equation for the slope of a line is represented as:    y = mx + b, where “m” was the slope of the line and “b” was the           y-intercept.    If we use the variables shown on the graphs and substitute “Co” for “b”  and  the marginal propensity to consume (MPC) for “m”, we can represent the consumption function as C = (MPC) (Y) + Co.

Developing the Expenditure Multiplier:

Looking at the gross domestic product of a closed economy from the expenditure side, we know that:

GDP = Y = C + I + G, where

GDP = Gross Domestic Product
Y = National Income
C = Consumption (after-tax)
I = Investment
G = Government Spending

A Derivation of the Expenditure Multiplier (Closed Economy) – Continued:

Now we substitute the consumption function into the above equation:

Y = (MPC) (Y) +Co + I + G

And we collect our terms:

Y – (MPC) (Y) = c + I + G

Factoring the terms on the left side of the equation:

Y ( 1 – MPC) = Co + I + G

Solving for Y:

Y = [1 / (1 – MPC)] (Co + I + G )   where

[1/ (1 – MPC)] is the term known as the expenditure multiplier.

It then follows that any change in any of the individual expenditure variables  will increase national income (Y) by a factor that is the expenditure multiplier.  We can show this by the following equation:

?Y = [ 1 / (1 – MPC)] [ ?( Co + I + G )]

For example, if investment spending in a given period in our economy were to increase by \$10 billion, the economy would grow by \$10 billion times the expenditure multiplier.  And in like manner, if government spending were to increase by \$10 billion, our economy would increase by \$10 billion times the expenditure multiplier.

On the other hand, if government spending were to decrease by \$10 billion, the economy would decrease by \$10 billion times the expenditure multiplier.  Therefore, we can assume that the expenditure multiplier will always be positive, but the expenditure variable can be either positive or negative.

Applying the Expenditure Multiplier:

Now let’s apply this principle.  Let’s assume that government wants to increase spending by \$10 billion so that much-needed maintenance on our national parks can be accomplished.  Let’s also assume the MPC = .80.  (Comment:  Since the MPC is the slope of the consumption function, it can never be greater than one or less than zero.)

The question to be answered here is how much this expenditure will increase national income (Y) or gross domestic income (GDP).

We’ll use the equation  ?Y = [1 / ( 1 – MPC)] (?G).  Note that Investment (I) and Consumption (C) are not involved here since the problem specifically states that we’re talking about a change in government spending.

Substituting:

?Y = [ 1 / ( 1 - .8)] (10) = ( 1 / .2) (10) = (5) (10) = \$50

Therefore, if MPC were 0.8, an increase in government spending of \$10 billion would increase the GDP by \$50 billion by the time the initial expenditure worked its way through the economy.

Summary:

With this information about the expenditure multiplier we can make the following general conclusions:

•    When a dollar is spent, that dollar does not “die”.  It moves through the economy being re-spent and re-spent, though at a decreasing rate.

Look at the dollars in your wallet or purse.  Very few of them will be “brand new”.  That means that they have been in someone else’s pocket or purse.

•    When Mr. Keynes said that government should spur the economy through government spending, he meant that (to use our example) government would not have to spend \$50 billion to improve the economy by \$50 billion.

A Derivation of the Expenditure Multiplier (Open Economy)

Now let’s change the model and look at the multiplier issue in an open economy.

Looking at the gross domestic product of an open economy from the expenditure side, we know that:

GDP = Y = C + I + G + NX, where

GDP = Gross Domestic Product
Y = National Income
C = Consumption (after-tax)
I = Investment
G = Government Spending
NX = Net Exports (exports minus imports)

In order to view the model with net exports included, we need to introduce an import function.  We know that in an open economy a portion of consumption will be due to imports.  We can assume that since imports are just a portion of consumption, the slope of the import function will be less than the slope of the consumption.  We can plot such a function, in a simplified version as follows:

Again, using our equation of a line as y = mx + b and substituting, we arrive at an equation of IM = (MPM) (Y) + IMo where:

IM = Imports
MPM = The slope of the import function, the Marginal Propensity toI Import
IMo  = The intercept of the import function

A Derivation of the Expenditure Multiplier (Open Economy) – Continued:

Now we substitute the consumption function and the import function into the national expenditure equation:

Y = (MPC) (Y) +Co + I + G + EX – [(MPM) (Y) + IMo]

Cleaning up the equation, we get:

Y = (MPC) (Y) + Co + I + G + EX – (MPM) (Y) - IMo

And we collect our terms:

Y – (MPC) (Y)  + (MPM) (Y) = Co + I + G + EX - IMo

Factoring the terms on the left side of the equation:

Y ( 1 – MPC + MPM) = Co + I + G + EX – IMo

If we isolate the multiplier terms, (1 – MPC + MPM) = [ 1 – (MPC – MPM)]

Reinserting the multiplier function into the equation and solving for Y:

Y = {1 / [1 –(MPC - MPM)]} (Co + I + G + EX - IMo )   where

{1/ [1 – (MPC - MPM)]} is the term known as the open economy expenditure multiplier.

It then follows that any change in any of the individual expenditure variables will increase national income (Y) by a factor that is the open economy expenditure multiplier.