The money multiplier and the introduction of taxes

This article goes over the economics of the money multiplier, and goes over an example:

Imagine an economy that has no imports or taxes. An increase in autonomous expenditure of 2 billion increases equilibrium expenditure by 8 billion.

a) calculate the multiplier
b) calculate the marginal propensity to consume
c) what would happen to the multiplier if an income tax is introduced

An increase in autonomous expenditure means that there is an increase in the minimum amount of spending done (for example, sustenance only consumption which is necessary regardless of income).  This point is represented on the Keynesian Cross graph (the 45 degree line) by its positive intersection on the Y axis.


Since an increase of 2 billion, leads to an increase in equilibrium expenditure by 8 billion, it is necessary to figure out how exactly this happens and why.

When a dollar is spent in the economy, a certain percentage of that money is saved, and the rest of it is then spent again on other goods and services.  For example, if you spent $100 at the store, it is possible that $20 (20%) will be saved, and the other $80 will be used to purchase other goods and services by employees or owners of the store.  Of this $80, $16 (20%) will be saved, and $64 will be spent.  This process continues until the entirety of the $100 is eventually saved, and thus not entering the circular flow directly.  So this initial $100 being spent would result in a total of:

$100+$80+$64+$51.2+$40.96+$32.77+$26.21+.... = $500

How did I get $500?  There is a neat trick to finding out what the money multiplier is, or how to use the money savings rate or marginal propensity to consume to find initial or final money values.  This equation is:

Initial monetary change/savings rate = final monetary amount

For my example above:

100/.2 = 500

Another way to think about this using the terminology above is:

Change in autonomous expenditure / 1 - marginal propensity to consume = equilibrium expenditure

So we know that the change in autonomous expenditure was $2 billion, and that the final equilibirum expenditure was $8 billion.  So we can  write the following equation where X represents the marginal propensity to consume.

$2 / (1-X) = $8

Now multiply both sides by (1-X) to get:

$2 = $8 - 8X, and add 8X and subtract $2 from both sides to get:

8X = $6, now divide both sides by 8 to get:

X = .75, so the marginal propensity to consume is .75.  This is good, because we expect the MPC to be less than 1, if it was greater than 1 we would be worried.  This also tells us that the savings rate is 25%.

Now, if a tax was introduced, this would lower the MPC because people would have less money to spend.

Let's imagine a 5% income tax is imposed, this would lower the MPC to .70 (.75 - .05).  This would mean that the $2 billion change in autonomous expenditure would result in only:

2/.30 = $6.667 billion equilibrium expenditure, a change of about $1.33 billion by introducing this 5% income tax!