THE MULTIPLIER AND A KEYNESIAN MODEL OF A
MACROECONOMIC SYSTEM
I.
The Keynesian Model
definitional equations .......................................................................................... 1
identity ................................................................................................................ 1
causal relationships ............................................................................................. 2
consumption equation ......................................................................................... 2
autonomous consumption ................................................................................... 2
marginal propensity to consume ......................................................................... 2
import function.................................................................................................... 2
autonomous imports ............................................................................................ 2
marginal propensity to import ............................................................................. 2
A. The Reduced Form of a System ................................................................................ 2
dependent variable .............................................................................................. 2
endogenous” variable.......................................................................................... 2
independent variables.......................................................................................... 2
exogenous ........................................................................................................... 2
B. Multipliers and Policy Making ................................................................................. 3
sensitivity analysis .............................................................................................. 3
scenario planning ................................................................................................ 3
multiplier ............................................................................................................. 3
C. Diagrams of the System ............................................................................................ 4
Figure 9-1. Aggregate Expenditure........................................................................ 5
Figure 9-2. Shifts in Aggregate Expenditure ......................................................... 6
Figure 9-3. Multiplying Effects ............................................................................. 7
Table 1. Mulitplying Effects ............................................................................... 8
using EXCEL .....................................................Error! Bookmark not defined.
Table 9-1. Mulitplying Effects using EXCEL ........................................................ 8
II. Full Employment……………………………………………………….9
The study of the economic effects of policy requires an economy to be modeled
so that all of the important interactions within the economy can be examined. The study
of government policy requires the key actions of government to be included in the
economic model. Inevitably such a task requires that a system of equations be defined
which specifies how the economy works.
The National Income and Product Accounts (NIPA) set out some of the major
definitional equations- equations that are true by definition- with which to build a
model of the macroeconomy. The most important definitional equation is the definitional
link between income and expenditure:
Y= C + I + G + net X
This equation is an identity. It must always be true by definition.
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However, some equations define causal relationships that describe the behavior
of sectors of the economy. The most important one specified by economists is for
consumption, C:
C=a+b*Y
This consumption equation shows that consumption is affected by income, Y. Certain
behavioral constants have been found to apply to consumption through time:
autonomous consumption (“a”) and the marginal propensity to consume (“b”).
Autonomous consumption can be thought of as the consumption we must do regardless
of what happens to our income. The marginal propensity to consume represents how
much of each extra dollar we use for consumption; it should be positive and it should be
below 100%.
When the foreign sector is important to an economy then an “open economy”
model must be defined. Typically an open economy model includes an equation which
specifies an import function. Like the consumption equation the import equation is a
behavioral equation, showing how imports (IM) respond to income (Y):
IM= IMo +d*Y
Certain behavioral constants have been found to apply to imports through time:
autonomous imports (“IMo”) and the marginal propensity to import (“d”).
Autonomous imports are quite similar to autonomous consumption; can be thought of as
the imports we will import regardless of what happens to our income. The marginal
propensity to import represents how much of each extra dollar we use for imports; as in
the marginal propensity to consume, it should be positive and it should be below 100%.
However, for the duration of this reading, we will use a model of a “Closed economy” in
which imports are simply treated as exogenous, which means they are completely
determined outside of the model.
A. The Reduced Form of a System
Just these two equations define a system which acts very differently than when
the two equations are treated separately. To see how the system works we must find the
“reduced form” of the system. The reduced form equation shows one unknown
dependent variable on the left hand side (also referred to as the “endogenous” variable
because it is determined within the equation) and independent variables (“exogenous”)
and parameters on the right hand side. Let’s make income (Y) the endogenous variable
that is to be explained and the rest of the variables are exogenous. We can find the
reduced form by substituting the consumption function into the first equation as follows:
Y= C + I + G + net X
= (a + b*Y) + I + G + net X
Then we can subtract –b*Y from both sides so that it effectively appears only on the left
hand side:
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Y - b*Y = a + I + G + net X
This simplifies to:
Y*(1-b) = { a + I + G + net X }
Which finally, when dividing through by (1-b), simplifies to the reduced form:
Y = (1/(1-b))* { a + I + G + net X }
We have one dependent variable and one equation. Only parameters and independent
variables are found on the right hand side in a reduced form equation.
B. Multipliers and Policy Making
Why should we go through the exercise of finding the reduced form equation?
Because it allows us to find the impact of policies and major events on our economy.
The reduced form equation provides information with which we can do sensitivity
analysis and scenario planning just as you can do in making a business plan for a firm.
Except that this kind of planning can also involve the entire economy.
Let’s see how we can use the above reduced form equation to examine the impact
of government expenditure on the economy. Suppose we increase government spending
by just $1.00. That means the new income for the economy (Ynew) would be:
Ynew =
(1/(1-b))* { a + I + (G+$1.00) + net X }
If we subtract the previous equation (without the $1.00) from this new equation, we can
find the change in income, Y:
Y = Ynew – Y
= (1/(1-b))* { a + I + (G+$1.00) + net X } - (1/(1-b))* { a + I + G + net X }
= (1/(1-b)) * $1.00
You should see that income (Y) would rise by the amount of $1*(1/(1-b)). The term,
(1/(1-b)), is called the multiplier. It shows how much income rises for every $1 of
government expenditure (or other exogenous expenditure).
To find the effect of government policy, the reduced form equation completely
eliminates the difficulty of finding out autonomous consumption (“a”), Investment (I),
the level of government expenditure (G), Consumption (C), income (Y), or net exports
(net X), even though all of those variables are in the equation. In this case the reduced
form equation has allowed us to focus only on the value of one parameter, the marginal
propensity to consume when we are trying to examine the effects of government policy.
In other words, finding the reduced form of a system enormously simplifies the problem
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of determining the effects of government policy or, for that matter, any other policy or
event.
What is the numerical value of the multiplier? Generally the Marginal propensity
to consume (“b”) on which the multiplier depends is close to 1.0, but never exceeds 1.0.
Let’s say it is 0.9. In other words, for every $1 rise in income people spend $.90
consumption. Then the multiplier becomes:
Multiplier = (1/(1-b)) = (1/(1-.10)) =10
That means every $1 rise in government expenditure causes a $10 income in the income
of the economy.
C. Diagrams of the System
What’s happening here? Is the government pulling a rabbit out of a hat? Is it
creating money out of thin air? No. The additional income comes from the multiplying
effects of expenditures turning into income which generates more expenditure which
raises income, etc., etc. We can see how this multiplying process works by using a graph
of the income and expenditure process. The fundamental identity, Y=C + I + G + net X,
can be represented as a 45 degree line where income is on the X-axis and expenditure on
the Y-axis. The 45 degree line consists of the points where income equals expenditures.
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Figure 1. Aggregate Expenditure
Expenditure
C+I+G+net X
equilibrium
C+I
C
=b= marginal propensity to consume
a
45o means Y=C+I+G+net X
income
In the same diagram the consumption function and the other expenditures can also
be represented. The consumption function, C = a + b*Y, is an upward sloping line which
starts at autonomous consumption, “a”, and has the slope, b, which is the marginal
propensity to consume (the consumption function is represented by the lowest upward
sloping line in the diagram). On top of the consumption function investment is added.
Since investment is not affected by income, it appears as a parallel upward shifting line.
On top of investment, government expenditures are added, which leads to the third line.
If government spends less with higher income, the third line may even be flatter than the
second line. In the diagram, net exports are presumed to be zero so that the third
represents total expenditures at every income level. This highest expenditure curve is
often referred to as aggregate expenditure.
Where aggregate expenditure intersects the 45 degree line, a macroeconomic
equilibrium occurs. Below the macroeconomic equilibrium the economy is spending too
much, inventories fall, and the attempt to produce more goods and services forces the
economy toward equilibrium and a higher income level. Above the equilibrium the
economy is spending too little, inventories start piling up, and production is curbed which
forces the economy downward to equilbrium. Equilibrium represents the point where
inventory levels are at a sustainable optimum.
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Figure 2. Shifts in Aggregate Expenditure
Expenditure
New equilibrium
New
C+I+G+net X
$10 billion Old equilibrium
More
Govt
Expenditure
C+I+G+net X
45o
income
Suppose the government increases expenditures by $10 billion. Then aggregate
expenditure rises by the full $10 billion to a new aggregate expenditure curve (labeled
“New C+I+G+netX”). The equilibrium rises from the old equilibrium to the new
equilibrium. However, income rises by much more than $10 billion.
To see why it rises by more than $10 billion, we must follow the money trail. The
government spends the $10 billion, but the sellers from whom the government buys, see
the $10 billion as new income. With their new income they will save some and then
consume the rest. The marginal propensity to consume (MPC) tells us just how much of
the $10 billion will be consumed. Assuming an MPC of .9, the sellers will consume $9
billion (which equals $10 billion * 0.9). Of course, when the sellers consume, they are
buyers, not sellers. Their $9 billion of consumption adds to the total expenditure as
shown in the following diagram:
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Figure 3. Multiplying Effects
Expenditure
$10 b.
More
Expenditure
45o
Etc.
$10 b.
Etc.
Means $9 b.
More income
More income
Means $9
b. more Expend
Means $10 b.
More income
New
C+I+G+net X
C+I+G+net X
INCOME RISES BY
$100 BILLION.
income
But the diagram also shows that the money trail keeps multiplying onward.. The
consumers spend the $9 billion, but the new sellers from whom the consumers buy, see
the $9 billion as new income. The MPC tells us just how much of the $9 billion will be
consumed: the new sellers will consume $8.1 billion (which equals $9 billion * 0.9). Of
course, when the new sellers consume, they are buyers, not sellers. Their $8.1 billion of
consumption adds to the total expenditure as shown in the above diagram. The logic of
this multiplying process can be taken an infinite number rounds. Here’s how an EXCEL
spreadsheet can be programmed to carry out the calculation:
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Table 1. Mulitplying Effects using EXCEL
How the numbers appear
How the formulas appear
A
B
A
B
1 Expenditure Income Expenditure
Income
2
10
10
10
=A2
3
9
9
=0.9*B2
=A3
4
8.1
8.1
=0.9*B3
=A4
5
7.29
7.29
=0.9*B4
=A5
6
6.561
6.561
=0.9*B5
=A6
7
5.9049 5.9049
=0.9*B6
=A7
8
5.31441 5.31441
=0.9*B7
=A8
9
4.782969 4.782969
=0.9*B8
=A9
10
4.304672 4.304672
=0.9*B9
=A10
11
3.874205 3.874205
=0.9*B10
=A11
12
3.486784 3.486784
=0.9*B11
=A12
13
3.138106 3.138106
=0.9*B12
=A13
14
2.824295 2.824295
=0.9*B13
=A14
…
…
…
…
…
sum
74.58134 74.58134
=SUM(A2:A14)
=SUM(B2:B14)
Note: The formulas are on the right while the resulting numbers are shown at the
left. After labeling the columns, cell A2 shows the initial change in government
expenditure ($10 billion). Then cell B2 is set equal to A2 (in other words enter
“=A2”as shown at the right) because what is spent turns into someone else’s income
by definition (Expenditure=Income). But that income gets spent, although not all of
it. The marginal propensity to consume (we’re assuming it is .9) says that 90% of
the income is spent on consumption. So in cell A3, the marginal propensity to
consume is multiplied by the income in cell B2 (i.e. enter “=.9*B2” as shown in
cell A3 at the right). Then drag the formulas in both cells (cell B2 down first and
then cell A3) as far down as you want and sum the columns as shown at the bottom
of the middle column. Each additional row that you add represents another round of
spending based on the extra income created by the previous round of spending.
Of course all of this work is unnecessary with an understanding of the multiplier.
When the MPC is .90, the multiplier on government expenditure is given as we saw
above by:
Multiplier = (1/(1-b)) = (1/(1-.10)) =10
Multiplying the multiplier by the $10 billion change in government expenditure gives us
the total increase of $100 billion. And we haven’t had to go through all of the rounds to
get there.
However for complicated systems of equations the reduced form may be very
difficult to find. The process of using EXCEL to solve for the rounds of equations may
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prove a much easier task. Furthermore, the EXCEL approach helps us to visualize what
is happening in the economy, step-by-step, as policy works to stimulate or destimulate
the economy.
II. FULL EMPLOYMENT AND THE BUSINESS CYCLE
Multiplying effects can be stopped when an economy reaches its capacity. The
capacity of a economy can be defined by any constraining resource. Typically, there may
be many resources that are constrained before the multiplying effects of an expanding
economy are fully stopped. However, traditionally economists have viewed labor as the
principal constraint on the economy. Particularly during war time an economy may have
very low unemployment because everyone is needed. When capacity is constrained by
labor, the economy is said to reach "full employment." Historically, there have been
periods in American history where the economy has been constrained by other factors.
For example, in the seventies, there were significant commodity shortages such as oil,
wood, and construction materials that were limiting growth of the U.S. economy.China’s
economy in 2004 began to face commodity constraints. Many developing countries are
not constrained so much by labor as they are by capital constraints or "human capital"
constraints. In other words, they do not have the resources, infrastructure or the trained
people to break out of the constraints of poverty.
The capacity of the economy can be represented quite easily. It is simply a
vertical line in a diagram of the relationship between income and aggregate expenditure
as in the following diagrams.
Figure 9-4
Shifts of Aggregate Expenditures
Aggregate Expenditure Shifts UPWARD
(A)
Aggregate Expenditure Shifts Downward:
(B)
EXP
EXP
New equilibrium
(C)
(D)
EXP
EXP
FE
FE
New equilibrium
New equilibrium
New equilibrium
FE
FE
45o
45o
45o
45o
Y
Y
Y
Y
FE= Full Employment
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These four diagrams represent the four basic stages of a business cycle:
(a)
The economy functions below full employment (FE), but is growing toward full
capacity. Inflation, Interest rate, and wage changes are moderate and there are few
disequilibrium problems in the economy. Nevertheless unemployment rates, government
deficits, and trade surpluses are historically high.
(b)
The economy pushes beyond full employment, experiences disequilibria (rapidly
diminishing inventories, high capacity utilization) and fast rising prices/ wages/ interest
rates. Government deficits fall and may move toward surplus while the trade balance
deterirorates.
(c )
The growth rate of GDP declines due to disequilibria (eg. large unfilled back
orders and shortages of resources) and the higher prices/ wages/ interest rates.
Government deficits rise again with continued deterioration in the trade balance.
Unemployment starts to grow.
(d)
Through multiplying effects the downward momentum of the economy continues
throwing people out of work and causing bankruptcies. Prices/ wages/ interest rates
come down. Government deficits rise substantially, but the trade balance is restored.
With the above four diagrams in mind, you now have a model of how each of the
indicators should be related to each other. When you read through the media, you should
be able to choose which of the four situations depicted above would best fit the pattern of
macroeconomic indicators that you find.
INDEX of terms
“endogenous” variable 2
45 degree line .............. 3
aggregate expenditure . 5
autonomous
consumption ............ 2
causal relationships ..... 1
consumption equation . 2
definitional equations .. 1
dependent variable ...... 2
exogenous ................... 2
full employment……..9
identity ........................ 1
independent variables.. 2
macroeconomic
equilibrium .............. 5
marginal propensity to
consume .................. 2
multiplier ..................... 3
multiplying effects ...... 3
reduced form ............... 2
reduced form equation 2
scenario planning ........ 2
sensitivity analysis ...... 2
system of equations ..... 1
The Reduced Form of a
System ..................... 2
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