Dr. Barry Haworth
University of Louisville
Department of Economics
Economics 202

### Summary Sheet on Multipliers

1. What is a multiplier? It is a way of seeing how sensitive the economy is to an expenditure injection/leakage. That is, it tells us the effect of a change in expenditures. Multipliers also reveal two basic ideas related to changes in expenditure:

a. the necessary DG, DI, DT is always going to be less than the DY
(i.e. you don't have to change expenditure by as much as the gap you're trying to close)

b. both DG and DI are positively related to DY, while DT is negatively related to DY
(i.e. if G or I increase, Y increases; and if T increases, Y decreases)

[Note: Y = real GDP, I = Investment, G = Government spending, T = Taxes, m = MPC]

2. Why do we need a multiplier? A multiplier tells us how much expenditure needs to change in order to close a GDP gap. For example, if equilibrium GDP is \$2 trillion and Potential GDP is \$3 trillion, there is a recessionary gap of \$1 trillion. The multiplier can tell us exactly how much we need to increase expenditure, in order to close the gap. Therefore, the multiplier is just a tool which allows us to properly employ fiscal policy.

3. How do I use a multiplier? There are two basic types of multipliers: the expenditure multipliers and the tax multiplier.

(i) The expenditure multiplier: DY/DG or DY/DI = 1/(1-m)
(ii) The tax multiplier: DY/DT = -m/(1-m)
(to see a simple derivation of the government expenditure multiplier, click here)

Before moving on, let's think about what the equations for these two multipliers are saying:

a. The expenditure multiplier is greater than the tax multiplier.
(i.e. for a certain DY, the DT is greater than the corresponding DG or DI)
b. There is a positive relationship between the MPC and each multiplier.
(i.e. as the MPC gets larger, each multiplier gets larger)

4. Can you give me a simple example of how a multiplier might be used? Here are two different, but related examples of how multipliers could be used.

Example 1: The MPC is 0.8 and the government wants to know how much GDP is affected after increasing government spending by \$20 million versus a decrease in taxes of \$20 million.

Plug this information into equations (i) and (ii):

c. If DG = 20, what is the DY?

DY/20 = 1/(1-0.8)
DY/20 = 5
DY = 5 x 20 = 100
Increasing G by \$20 million increases Y by \$100 million.

d. If DT = -20, what is the DY?

DY/-20 = -0.8/(1-0.8)
DY/-20 = -4
DY = -4 x -20 = 80
Decreasing T by \$20 million increases Y by \$80 million.

Example 2: The MPC is 0.8 and there is a recessionary gap of \$20 billion. How much should the government either increase spending or decrease taxes to close this gap?

e. What DG makes DY = 20:

20/DG = 5
DG = 20/5
DG = 4
Increasing G by \$4 billion increases Y by \$20 billion.

f. What DT makes DY = 20:

20/DT = -4
DT = 20/-4
DT = -5
Decreasing T by \$5 billion increases Y by \$20 billion.

[Note: DG, DI, DT and DY represent changes in the variables G, I, T and Y respectively]