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7. IS-LM and Aggregate Demand
A fundamental idea embodied in the income-expenditure accounting identity is that
one person’s spending becomes another person’s income. The simple circular flow
diagram presented in Figure 3.1 clearly illustrates this idea for a closed economy without
government. Re-examining that figure, notice that expenditure not only becomes income,
but the reverse is also true: income becomes expenditure. This interaction confounds
cause and effect. Do expenditures generate income? Or, does income generate
expenditures?
To examine this issue, macroeconomists have developed the concept of aggregate
demand, which is the total willingness of buyers to buy goods and services over a given
period of time. Aggregate demand is comparable to aggregate expenditure. However,
whereas aggregate expenditure is an accounting concept measuring the actual expenditure
on goods and services, aggregate demand is a modeling concept representing the buying
plans of buyers. One would expect buying plans to be contingent upon a variety of
factors. The central question of interest here is, “What determines buyers’ willingness to
buy?” Or, using economic jargon, “What determines aggregate demand?” This is an
important question since producers cannot sell all of their output if the willingness to buy
is insufficient.
7.1 Origin of the IS-LM Model
In his classic 1936 book, The General Theory of Employment, Interest, and Money,
John Maynard Keynes provided a critique of what he called the classical theory and
offered an alternative. Writing in the Great Depression, Keynes’ focus was on
understanding what determines the economy’s employment level.
The concept of
aggregate demand is central to Keynes’ theory.
Keynes presented few of his ideas using mathematical models. Preferring verbal
discourse, he specifically criticized mathematical formalization, saying “Too large a
portion of recent ‘mathematical’ economics are mere concoctions, as imprecise as the
initial assumptions they rest on, which allow the author to lose sight of the complexities
and interdependencies of the real world in a maze of pretentious and unhelpful symbols.”
Ironically, professional economists today use mathematical modeling much more than
economists did in Keynes’ day, though many economists today would whole-heartedly
agree with this quote by Keynes. Mathematical modeling has become popular because

it allows the assumptions of the analysis to be precisely stated. While words
allow for subtlety and complexity that is difficult to capture with mathematics,
they also allow for more varied interpretations, which can lead to confusion.
For example, although Keynes was an organized thinker and writer,
economists today still hotly debate the meaning of his words.
89
Aggregate
Demand
Chapter 7: IS-LM and Aggregate Demand
IS Curve
LM Curve

it allows the model builder to more carefully identify and study cause and
effect relationships. The interdependencies and complexities that Keynes
spoke of can make non-mathematical analysis exceedingly difficult.
Mathematical methods exist which can simplify complicated relationships.

it allows econometric analysis. To do science, theories must be tested against
the facts. Econometric models are mathematical models that allow this
testing.
John Hicks presented his view of Keynes’ work relative to Classical thought in his
famous 1937 essay, “Mr. Keynes and the ‘Classics’.” There Hicks introduced what is
now know as the IS-LM model. (For Hicks it was IS-LL.) In its graphical form, this
model includes an IS curve---the real interest rate and output combinations which are
consistent with investment being equal to savings. It also includes an LM curve---the real
interest rate and output combinations which are consistent with liquidity preference being
equal to the money supply. These relationships will now be explained in more detail.
7.2 The IS Curve
As a starting point, reconsider equation (2.1), which we reproduce here without the
time subscripts:
(7.1)
Y  C  I  G  NX .
The left side of (7.1) is the economy’s output level, which national income accountants
now measure as gross domestic product. The right side of (7.1) includes the expenditure
components of output.
To go from accounting to modeling, one can re-interpret (7.1) as an equilibrium
condition where plans are actualized. That is, suppose the left side of (7.1) still represents
the economy’s actual output level. However, rather than thinking of the right side of (7.1)
as the actual expenditure components of output as measured in the national income and
product accounts, think of it as aggregate demand as defined above. That is, we represent
the total willingness to buy goods and services as the sum of the willingness households
to buy consumption goods---consumption demand, the willingness of firms to buy
investment goods---investment demand, the willingness of government to buy goods--government purchases, and the net willingness of foreigners to buy goods. Letting AD
denote aggregate demand, this implies
(7.2) AD  C  I  G  NX
In order for the plans of buyers and sellers to be simultaneously fulfilled, the equilibrium
condition
90
Chapter 7: IS-LM and Aggregate Demand
(7.3)
Y  AD
must hold. Note that conditions (7.2) and (7.3) imply (7.1) must hold in equilibrium.
Will the economy tend toward an equilibrium? Disequilibrium is certainly possible, if
not likely. Keynes, like the Classical economists before him, felt disequilibrium would
motivate change that would tend to move the economy toward an equilibrium. However,
since Keynes, there has been dispute over what will tend to adjust to bring about the
equilibrium.
To examine this Keynesian-Classical dispute, we must add more detail to our model.
Keynes described an economy where consumption depends upon income. Since we are
considering an economy with government, let consumption depend upon disposable
income according to the consumption function
(7.4) C  C(Y  T ),
0  C'  1 .
The derivative C’ of the consumption function, represents what Keynes called the
marginal propensity to consume. Restricting C’ between zero and one indicates that a
one dollar increase in disposable income will lead to an increase in consumption but not
by the full dollar. The remaining fraction is saved, as can be shown using the income
allocation condition from chapter 2, reproduced here without time subscripts as
(7.5) Y  C  S  T  TF .
Consumption
Function
Marginal
Propensity to
Consume
Eliminating C from (7.5) using (7.4) and solving for S, one obtains the savings function
(7.6) S  Y  T  C(Y  T )  TF .
Equation (7.6) indicates that saving is a function of disposable income Y-T and transfers
to foreigners TF. Taking the derivative of the right side of (7.6) with respect to Y-T
yields 1-C’, the marginal propensity to save. Combining this knowledge with that
obtained above for the marginal propensity to consume, our model indicates that C’ of an
additional dollar of disposable income is spent on consumption, while the remaining
fraction 1-C’ is saved.
Keynes postulated that investment depends upon what he called the marginal efficiency
of capital, a measure of the expected return on investment. The marginal efficiency of
capital is comparable to the internal rate of return concept used in finance. For a given
prospective investment, there are expected cost outlays and expected returns. For the
typical investment, costs are typically incurred in more toward the present, while the
benefits are mostly received in the future. The internal rate of return is that rate which
discounts the benefits to the point where they are equal to the discounted costs.
A
prospective investment project looks more attractive as its internal rate of return
increases.
91
Savings
Function
Marginal
Propensity to
Save
Marginal
Efficiency of
Capital
Marginal
Efficiency of
Capital
Chapter 7: IS-LM and Aggregate Demand
Keynes noted that investment would be induced when the marginal efficiency of
capital exceeds the real rate of interest.
As such investment occurs, the marginal
efficiency of capital decreases, either because of to diminishing returns to capital or
because of an increase in the prices of investment goods. Alternatively, investment
would not be forthcoming when the real interest rate exceeds the marginal efficiency of
capital. In such a situation the marginal efficiency of capital would tend to increase,
either because depreciation would make capital increasingly scarce or because the prices
of investment goods would decrease. Consequently, Keynes argued that the marginal
efficiency of capital will tend to find an equilibrium where it is equal to the real rate of
interest.
Starting in such an equilibrium, a certeris paribus decrease in the real interest rate
would tend to promote investment. Hicks expressed this conclusion in the form of an
investment function, such as
(7.7) I  I (r ) ,
Investment
Function
Endogenous
Variable
Exogenous
Variable
I ' 0 ,
where r denotes the real interest rate. Investment occurs in response to an interest rate
decrease until the marginal efficiency of capital falls to the new interest rate level.
We now have a model of the economy’s circular flow of income and expenditure. It
consists of six equations: (7.2), (7.3), (7.4), (7.5), and (7.7).
Starting with (7.3) and using substitution and the other conditions, we obtain the
following series of relationships
Implicit
Function
(7.8)
Y  AD
Given as (7.3)
Y  C  I  G  NX
Substitution; (7.2)
Y  C(Y  T )  I (r )  G  NX
Substitution; (7.4) and (7.7)
Equation (7.8) is a variant of Hicks’ IS equation.
IS equation
The capital market is the market for saving. Using the equilibrium condition (7.3)
and substitution as follows:
Capital Market
(7.9)
Y  AD
Given as (7.3)
Y  C  I  G  NX
Substitution; (7.2)
S  T  G  NX  I  TF
Substitution; (7.5)
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Chapter 7: IS-LM and Aggregate Demand
Condition (7.9) can be thought of as a capital market equilibrium condition. On the left
side of (7.9) is the total saving supplied to the domestic economy, consisting of private
saving S, public saving T-G, and foreign saving -NX. On the right side of (7.9) is the
domestic demand for saving in the form of domestic investment I and transfers from
foreigners -TF.
The last series of equations shows that either (7.3) or (7.9) could be used as an
equilibrium condition in constructing our model. Hicks used a capital market condition
like (7.9). To be precise, Hicks examined an economy without government and without a
foreign sector, meaning condition (7.9) reduces to I=S. It is from this condition that the
IS curve gets its name. By using condition (7.3), we follow more recent tradition.
7.3 Using the Total Differential to Examine the IS equation
Now consider the IS equation (7.8) in more detail. As a single equation, it can
determine only one variable, typically called an endogenous variable. The Classical
approach is to assume that the real interest rate r is endogenous, adjusting to equate
saving and investment whenever they are not equal.
The remaining variables are
exogenous, determined by outside influences.
By treating r as endogenous, we can think of (7.8) as an implicit function determining
r in terms of the exogenous variables Y, T, G, and NX. We now show how the total
differential can be used to examine how changes in Y, T, G, and NX affect r. Totally
differentiating (7.9) gives
(7.10) dY  C' dY  C' dT  I ' dr  dG  d[ NX ] ,
where the differentials, dY, dT, dG, d[NX], and dr respectively represent the change in
output, the change in net taxes, the change in government purchases, the change in net
exports, and the change in the real interest rate. Solving (7.10) for dr yields
(7.11) dr 
1  C'
C'
1
1
dY  dT  dG  d[ NX ] .
I'
I'
I'
I'
Equation (7.11) indicates that the total change in the real interest rate, given by dr,
depends upon the change in output dY, the change in net taxes dT, the change in
government purchases dG, and the change in net exports d[NX].
The coefficients on the exogenous differentials are called multipliers because a given
multiplier indicates how the endogenous variable r changes in response to a one unit
r
increase in the particular exogenous variable. For example, the multiplier
is found
Y
by setting dT  dG  d[ NX ]  0 , which yields
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Chapter 7: IS-LM and Aggregate Demand
Multipliers
(7.12)
r 1  C '

 0.
Y
I'
Notice that a multiplier is a partial derivative found by hold all exogenous variables
constant, except the one of interest. The multiplier (7.12) gives the change in r that
results from a one unit increase in Y. It also gives the slope of the IS curve, which is
typically drawn in (Y,r) space as shown in Figure 7.1
Figure 7.1: The IS Curve
r
Slope 
1  C'
I'
IS
Y
Money Demand
Transactions
Motive
Precautionary
Motive
Speculative
Motive
The multiplier is negative, indicating that an increase in output leads to a decrease in
the real interest rate. The magnitude of the decrease depends upon the size of the
marginal propensity to save 1-C’ and the sensitivity of investment to changes in the
interest rate I’. The interest rate decreases when output increases because an increase in
output generates and increase in income; the increase in income then increases private
saving. The increase in saving creates a surplus of saving. To encourage borrowers to
borrow the surplus, financial intermediaries decrease the real interest rate level. If the
propensity to save 1-C’ is small, then the income increase will not generate much of an
increase in saving, meaning interest rates will not have to increase much to eliminate the
saving surplus. Alternatively, if I’ is small, meaning investment is insensitive to a
change in the interest rate, then a relatively large decrease in the interest rate would be
necessary to encourage borrowers to borrow the saving surplus. Given these
interpretations of this multiplier, it is apparent that we would like to know the values of
C’ and I’ for our economy.
r C '
r
1

 0,
   0 indicate that a reduction in the budget
T I '
G
I'
deficit, either through a tax increase or government purchases decrease, will reduce the
real interest rate level . This occurs because a reduction in the budget deficit is equivalent
to an increase in public saving. The increase in public saving generates a surplus of
saving. Again, to encourage borrowers to borrow the surplus, financial intermediaries
decrease the real interest rate level.
The multipliers
Money Market
94
Chapter 7: IS-LM and Aggregate Demand
r
1
 0 indicates that a reduction in the trade deficit---which is
 [ NX ]
I'
equivalent to an increase the trade surplus or net exports---will increase the real interest
rate level. This occurs because a reduction in the trade deficit is equivalent to an decrease
in foreign saving. The decrease in foreign saving generates a shortage of saving. To
discourage borrowers from borrowing, financial intermediaries increase the real interest
rate level.
The multiplier

7.4 The LM Curve
Hicks argued that Keynes’ innovation was to introduce the notion of “liquidity
preferences”---the idea that money demand depends upon the rate of interest. Money
demand is the willingness of people to hold money. The Classical approach is to assume
that people hold money to make transactions. Because people with higher incomes make
more transactions, the Classical approach is to assume money demand L depends upon
the income level Y :
(7.13) L  L(Y ) ,
L' 0 .
In addition to the transactions motive, Keynes spoke of a precautionary motive---hold
money for unexpected circumstances---and a speculative motive---hold money in
anticipation of a bargain purchase.
In any case, the interest rate represents the
opportunity cost of holding money. Thus, Keynes argued that an increase in the interest
rate would reduce the preference for liquidity; i.e. reduce money demand. Adding the
real interest rate i to the money demand function (7.13), one obtains
(7.14) L  L(Y ,r ) ,
LY  0 ,
Lr  0 .
While Keynes and Hicks thought of money demand in nominal terms, here we follow
the approach that has become more standard: Define L as real money demand and P as
the price of output so that PL represents nominal money demand. Letting M denote the
nominal money supply, the money market is in equilibrium when
(7.15) PL  M .
Substituting the money demand function (7.14) into (7.15), one obtains Hicks’ LM
equation:
(7.16) PL(Y ,r )  M .
As Hicks explained, Keynes distinguished himself from the Classical perspective by
assuming (7.16) determines r. Most significantly, the rate of interest depends upon the
quantity of money.
95
LM Equation
Chapter 7: IS-LM and Aggregate Demand
However, Hicks introduced the IS-LM diagram specifically to show that the
Keynesian perspective on the money market is not really much different from the
Classical perspective once the money market model---represented by the LM equation--is combined with the capital market model---represented by the IS equation. We will
consider that framework momentarily. However, we first examine the shape of the LM
curve, something Hicks described as “the most important thing in Keynes’ book.”
Totally differentiating (7.16), one obtains
(7.17) PLY dY  PLr dr  LdP  dM .
Solving for dr, one obtains
(7.18) dr 
1
L
L
dM  Y dY  dP .
PLr
Lr
Lr
r
L
  Y  0 gives the slope of the LM curve. The slope of the LM
Y
Lr
curve drawn in Figure 7.2 is not constant but rather increases at an increasing rate. This
is the shape Hicks credited to Keynes and deemed so important. Mathematically, this
shape arises if LY is constant as Y and r change, while Lr gets large as r gets small and
Lr gets small as r gets large. LY remaining constant is consistent with Classical monetary
theory, the idea being that people hold the fraction LY of any increase in income in the
form of money. Since r represents the opportunity cost of holding money, it also makes
sense that Lr decreases as r increases. At low interest rates, a one percent increase in the
interest rate would represent a large percentage increase in the cost of holding money.
Thus, a small increase in the interest rate would encourage people to significantly reduce
their money holdings. Alternatively, a small decrease would greatly increase people’s
willingness to hold money. This situation is known as a liquidity trap. At high interest
rates, people are already economizing on their cash balances. Thus, an increase in the
interest rate will not lead to much of a decrease in the willingness to hold money. In
contrast to the liquidity trap, here the economy is liquidity constrained.
Hicks
summarized these conditions by saying (1) there is some minimum below which the
interest rate is unlikely to go and (2) there is a maximum level of income that can be
financed with a given money supply. These assumptions generate the LM curve shown in
Figure 7.2.
The multiplier
Figure 7.2: The LM Curve
r
LM
96
Chapter 7: IS-LM and Aggregate Demand
L
Slope   Y
Lr
Y
r
1
r
L
   0 . The

 0 and
P
Lr
M PLr
first multiplier indicates that an increase in the money supply reduces the interest rate
level, while the second multiplier indicates that inflation increases the interest rate level.
Because Y is being held constant as these changes occur, these multipliers also indicate
that an increase in the money supply shifts the LM curve to the right---r increases with Y
fixed, while an increase in the price level shifts the LM curve to the left---r decreases
with Y fixed.
Equation (7.18) also generates the multipliers
Figure 7.3 illustrates the curve shift associated with a money supply increase. Notice
that the money supply increase allows for a higher real income level. However, the
minimum real interest rate level remains unchanged. This is a Keynesian notion which
remains widely held today: An increase in the money supply can be helpful in that it can
relax a liquidity constraint thereby allowing for an increase in real income, however the
ability of an increase in the money supply to decrease interest rates is limited.
Figure 7.3: Effect of a Money Supply Increase
r
LM1 LM 2
Y
97
Liquidity Trap
Liquidity
Constrained
Chapter 7: IS-LM and Aggregate Demand
7.5 Analyzing the IS-LM Model Using Comparative Static Analysis
7.5.1 Equilibrium and the Implicit Function Theorem
Hicks used his IS-LM model to explain why Keynes and Classical economists differed
as to expected effect of an increase in the inducement to invest. The Classical perspective
was that an increase in the inducement to invest would increase the interest rate level with
no change in output, while Keynes claimed that an increase in the inducement to invest
would lead to an increase in real income with no change in the interest rate level. To
formally examine a change in the inducement to invest, let  be an exogenous variable
which measures the willingness to invest at the going interest rate, and expand the
investment function (7.7) so that
(7.19) I  I (r , ) ,
Inducement to
Invest
Animal Spirits
Ir  0 ,
I  0 .
The variable  can be thought of as a measure of Keynes’ animal spirits. Keynes argued
that investment is volatile partly because the willingness to invest is partly driven by
unexplainable, spontaneous “animal spirits.” Using the investment function (7.19), the
IS-LM model can be formally presented as:
(7.20) Y  C (Y  T )  I (r , )  G  NX
IS equation
(7.21) PL(Y ,r )  M
LM equation
Variables (8): Y, T, r, , G, NX, P, M
Endogenous (2): r, Y
Exogenous (6): T, G, M, , NX, P
The IS-LM diagram introduced by Hicks is obtained by plotting the IS and LM curves
in the same space.
Figure 7.4: The IS-LM Diagram
r
LM
r1
IS
98
Chapter 7: IS-LM and Aggregate Demand
Y1
Y
Figure 7.4 illustrates how the two equations (7.20) and (7.21) determine the
economy’s real interest rate level and real income level. The IS curve shows the interest
rate-income combinations which satisfy the IS equation (7.20). The LM curve shows the
interest rate-income combinations which satisfy the LM equation (7.21). Thus, only the
interest rate income combination (Y1 ,r1) satisfies (7.20) and (7.21) simultaneously. It is
this combination which the IS-LM model predicts for the economy.
A more careful analysis first requires the use of the implicit function theorem. As it is
usually stated, the implicit function theorem applies to a system of equations presented in
the form:
F 1 ( y1 ,..., yn ; x1 ,..., xm )  0
F 2 ( y1 ,..., yn ; x1 ,..., xm )  0
(7.22)

n
F ( y1 ,..., yn ; x1 ,..., xm )  0
where the n equations implicitly define a set of n functions which determine each of the n
endogenous variables (y1,...,yn) in terms of the m exogenous variables (x1,...,xn).
Rewriting the IS and LM equations in this form gives the system
A1 (Y , r ; G , T , M , , NX , P )  Y  C (Y  T )  I (r , )  G  NX  0
(7.23)
A2 (Y , r ; G , T , M , , NX , P )  PL (Y , r )  M  0
For the system (7.22), if some technical details are satisfied1, the implicit function
theorem tells us that the endogenous y variables are determined implicitly as functions of
the exogenous x variables if the Jacobian determinant is not equal to zero. For the
general system (7.22) the Jacobian determinant is given by
F 1
y1
F 2
(7.24) J  y
1
F 1
F 1

y2
yn
2
F
F 2

y2
yn
F n
y1
F n
F n

y2
yn




1
See Fundamental Methods of Mathematical Economics by Alpha Chiang for a careful, understandable
presentation of the implicit function theorem.
99
Implicit
Function
Theorem
Chapter 7: IS-LM and Aggregate Demand
For our system (7.23), the Jacobian determinant is
Jacobian
Determinant
Linear System
Cramer’s Rule
A1
(7.25) A  Y2
A
Y
A1
r  1  C'  I r  PL [1  C']  PI L  0 .
r
r Y
A2
PLY PLr
r
Because the determinant (7.25) for our system is negative, we know our system has a
solution.
7.5.2 Comparative Static Multipliers and Cramer’s Rule
When an exogenous variable changes, the equilibrium described by the IS and LM
equations is perturbed. However, under the assumption that the IS and LM equations
always hold, the economy must always be in equilibrium. Thus, the endogenous
variables Y and r must adjust to new equilibrium levels. Comparative static analysis
involves comparing equilibrium states. The analysis is static because time plays no
essential role. That is, we do not consider the disequilibrium path the economy might
follow as it moves from one equilibrium to another. This is unfortunate since many
interesting things happen can happen out of equilibrium.
However, ignoring
disequilibrium greatly simplifies the analysis while still providing a forecast of change
consistent with the assumption that the new equilibrium will be reached.
To obtain comparative statics multipliers for our system, the next step is to totally
differentiate the system (7.23). Doing so, one obtains
Comparative
Static
Analysis
Comparative
Statics
Multipliers
(7.26)
dY  C' dY  C' dT  Ir dr  I d  dG  dNX  0
PLY dY  PLr dr  LdP  dM  0
The system (7.26) contains two equations and two endogenous differentials---dY and dr.
It is a linear system because the differentials dY and dr only appear as coefficients. There
are two ways in which this linear system can be solved. One method is by using
substitution. Using substitution is easier for systems with fewer equations and fewer
variables. However, it can become a tedious art for more complicated systems. Here, we
demonstrate a second method called Cramer’s Rule---a general, systematic method for
solving a linear system.
To use Cramer’s rule, the linear system must first arranged in the matrix form Ax=d.
The matrix A has n rows and n columns, corresponding to the system’s n equations and n
endogenous variables. The vector x has n rows and consists of the system’s n endogenous
variables. The vector d has n rows and consists of all terms which do not contain an
endogenous variable. Immediately below, we show this matrix system in general for
two equation system and then show how our system (7.26) is written in this matrix form:
100
Chapter 7: IS-LM and Aggregate Demand
a11 a12   x1   d1 
a
    
 21 a22   x2  d2 
(7.27)
 I r  dY  C ' dT  I d  dG  dNX 


PLr   dr  
 LdP  dM

1  C'
 PL
 Y
Here, the system’s endogenous variables are the endogenous differentials; i.e., x1=dY and
x2=dr.
The second step in using Cramer’s rule is to calculate the determinant A . Notice that
this determinant is the Jacobian (7.25). Thus, one nice thing about using Cramer’s rule is
that the main requirement of the implicit function theorem can be checked along the way.
From (7.25) above, we know that A  [1  C'] PLr  I r PLY  0 .
The third step involves calculating the determinants A1 and A2 , where the matrix Ai
is created by replacing column i in the matrix A by the vector d. For our system the
determinants A1 and A2 are given by
d1
d2
(7.28) A1 
a12
 C ' dT  I d  dG  d [ NX ]  I r

a22
 LdP  dM
PLr
  PLr C' dT  PLr I d  PLr dG  PLr d[ NX ]  Ir LdP  Ir dM
and
(7.29) A2 
a11 d1 1  C '  C ' dT  I d  dG  d [ NX ]

a12 d 2
PLY
 LdP  dM
 [1  C']dM  [1  C']LdP  PLY C' dT  PLY I d  PLY dG  PLY d[ NX ]
The final step in using Cramer’s rule is to construct the solutions for the endogenous
variables. In general, and for our example system, this is done as follows:
(7.30) x1 
A1
A
 dY 

 PLr C '
PL I
PL
PL
I L
dT  r  d  r dG  r d [ NX ]  r dP
A
A
A
A
A
Ir
dM
A
101
Chapter 7: IS-LM and Aggregate Demand
and
(7.31)
x2 
A2
A
 dr 

PLY C '
PL I
PL
PL
dT  Y  d  Y dG  Y d [ NX ]
A
A
A
A
[1  C'] L
[1  C']
dP 
dM
A
A
As before, the multipliers are the coefficients on the exogenous differentials. Because
A  0 , the sign of the particular multiplier is the opposite sign of the coefficient’s
numerator. What can be learned from these multipliers? This question is explored in the
next few sections.
7.5.3 Changes in Fiscal Policy
The term fiscal policy is used to describe government policy regarding government
purchases and net taxes. To begin interpreting the multipliers, note that the multipliers
for government purchases and net exports are the same. Presenting these multipliers in
detail, we have
(7.32)
Y
Y
1


 0 and
G  [ NX ] [1  C ']  I LY
r
Lr
(7.33)
LY
Lr
r
r


 0.
LY
G  [ NX ]
[1  C']  I r
Lr
Fiscal Policy
An increase in G or NX shifts the IS curve up or to the right, which increases Y and r.
The multipliers for G and NX are identical because the exogenous shock to the economy
is precisely the same. A one dollar increase in G and a one dollar increase in NX each
increase aggregate expenditure by one dollar. Such an increase in spending is sometimes
referred to as “autonomous” because G and NX are exogenous variables, meaning any
change in their levels is independent of changes in any other variables in the model. For
this reason, the multiplier (7.32) is often called the autonomous spending multiplier.
The increase in spending caused by an increase in G or NX is a direct increase. This
direct increase can generate indirect increases in spending because an increase in one
person’s spending becomes an increase in another’s income, which increases
consumption spending, which further increases income. For any direct one dollar increase
in spending, the autonomous spending multiplier gives the total direct and indirect
102
Chapter 7: IS-LM and Aggregate Demand
increase in spending generated as direct increase ripples through the economy.
Examining (7.32) notice that the size of this total increase depends upon the slope of the
LM curve -LY/Lr . For a given marginal propensity to consume C’, the autonomous
spending multiplier is as large as it can be---1/[1-C’]>1---when the slope of the LM
curve is equal to zero. To the contrary, as the slope of the LM curve -LY / Lr increases,
the autonomous spending multiplier eventually approaches zero.
Examining (7.33) note that the extent to which the interest rate level increases also
depends upon LY / Lr. The ratio -LY / Lr gives the increase in the interest rate which
occurs when the real income level increases by one unit, assuming the monetary authority
does not change the money supply. This interest rate increase “crowds out” investment
spending equal to IrLY / Lr . This crowding out is minimal when LY / Lr is near zero.
However, when LY/Lr is infinitely large the crowding out is complete---any increase in
government purchases G or net exports NX is exactly completely offset by a decrease in
investment I so that consumption spending C and total spending Y remain unchanged.
Expansionary
Policy
Contractionary
Policy
Balanced
Autonomous
Budget
Spending
Multiplier
Multiplier
Next, consider an increase in net taxes T. A one dollar increase in net taxes reduces
disposable income Y-T by one dollar, meaning consumption decreases by C’ dollars. This
direct decrease in aggregate demand then ripples through the economy. Consequently,
the tax multiplier for income is simply -C’ times the autonomous spending multiplier
(7.32). In detail the tax multipliers are
C'
(7.34)
LY
Lr
Y
C'
r

 0 and

 0.
T [1  C ']  I LY
T [1  C']  I LY
r
r
Lr
Lr
An change in fiscal policy affects the budget deficit G-T. An expansionary policy is
one the generates an increase in aggregate demand. Alternatively, a contractionary policy
decreases aggregate demand. Note that an increase in government purchases is
expansionary, whereas an increase in net taxes is contractionary. What happens to total
spending if government purchases and net taxes each increase by a dollar so that the
budget deficit remains unchanged? We can consider this question by examining the so
called balanced budget multiplier,
(7.35)
Y Y
1  C'


0
G T [1  C ' ]  I LY
r
Lr
The balanced budget multiplier is positive because a dollar increase in government
purchases directly increases aggregate demand by one dollar, whereas a one dollar
increase in net taxes generates a direct decrease in aggregate demand of C’ dollars as
explained above. Thus, as long as there is a propensity to save---i.e., C’<1, a balanced
103
Crowding Out
Chapter 7: IS-LM and Aggregate Demand
budget policy is expansionary while G and T are increasing and contractionary while G
and T are decreasing.
7.5.4 Change in the Inducement to Invest
Next, consider an increase in the inducement to invest, something of particular interest
to Hicks. The coefficients on d in (7.30) and (7.31) respectively tells us how a one unit
increase in the inducement to invest affects the real income level and real interest rate
level. Presenting these multipliers in detail, we have
(7.32)
Y
I

0
 [1  C ']  I LY
r
and
Lr
LY I
Lr
r

 0.
L

[1  C ']  I r Y
Lr
An increase in the inducement to invest shifts the IS curve up or to the right, increasing Y
and r. More precisely, a one unit increase in the inducement to invest  leads to a direct
increase in investment equal to I. The spending multiplier (7.31) then multiplies this
direct investment spending increase into the total given by the income multiplier in
(7.32).
Again, the extent to which interest rates increase depends the slope of the LM curve LY/Lr. Recognizing this, Hicks argued that the main difference between Keynes’ “general
theory” and Classical theory was a different assumption about the slope of the LM curve.
As noted above, the extreme version of Classical monetary theory, which Hicks called
“crude quantity theory,” contains only a transactions demand for money, and any effects
which the interest rate level might have on money demand are ignored. This implies Lr =
0 and -LY/Lr = +. With the economy liquidity constrained, the “crude” Classical theory
predicts that an increase in the inducement to invest will increase the interest rate level
without much change in real income. To the contrary, Keynes predicted an increase in
real income, and consequently an increase in employment, with little change in the
interest rate level. Hicks showed that Keynes’ prediction is consistent with an economy
in a liquidity trap, where Lr is near - and -LY/Lr is near zero---an assumption crude to
the same degree as that made in the Classical’s crude quantity theory but at the opposite
extreme. Thus, the brilliance of Hicks’ IS-LM invention was that it offered a more
general theory than Keynes’ general theory---one that includes the Keynesian and
Classical extremes as special cases.
7.5.5 Changes in Monetary Policy
To examine the impact of money, consider the model’s money multipliers
104
Chapter 7: IS-LM and Aggregate Demand
(7.33)
Ir
PLr
Y

0
M [1  C']  I LY
r
and
Lr
1  C'
PLr
r

 0.
M [1  C']  I LY
r
Lr
Keynes stressed that an increase in the money supply will have little positive impact on a
depressed economy because people will tend to hold the additional money rather than
lend it or spend it due to the high levels of risk and uncertainty. As mentioned, this
liquidity trap idea can be modeled by making Lr large. As Lr increases, both of the money
multipliers in (7.33) approach zero. That is, as Keynes suggested, expansionary monetary
policy is useless in a liquidity trap because it affects neither the income level nor the
interest rate level. Because a liquidity trap would likely occur only in a severe recession,
Hicks referred to Keynes general theory as the “Economics of Depression.”
Hicks also used the IS-LM apparatus to show that a money supply increase can be
useful if the economy is operating in the liquidity constrained condition where the LM
curve is near vertical. Adopting the Classical extreme Lr = 0, dY/dM becomes 1/PLY >0.
Notice that there is no interest rate parameter in this multiplier. Here, real income
increases directly in response to the money supply increase rather than indirectly through
an increase in investment caused by lower interest rates. In contrast to his
characterization of “crude” Keynesian economics as the Economics of Depression, this
“crude” Classical view could be called the “Economics of Prosperity,” a state where
people have so economized on their cash balances that the opportunity cost of holding
money is ignored.
Demand-Side
Supply-Side
7.5.6 IS-LM and Aggregate Demand
An increase in the price level P shifts the LM curve up or to the left, generating a
decrease in Y and an increase in r. In detail, the price multiplier for income is
(7.34)
Ir L
PLr
Y

0
L
P
[1  C ']  I r Y
Lr
The increase in the price level decreases the real money supply creating an excess demand
for money. The decrease in the real interest rate necessary to eliminate this initial excess
money demand is L/PLr. This generates a direct decrease in investment spending equal to
IrL/PLr . The income multiplier (7.34) captures the total increase in spending which
results as the direct decrease in investment spending ripples through the economy.
Note that all of the multipliers in the IS-LM model are obtained without any reference
to the production process. This is because the IS-LM model only represents the demand-
105
Aggregate
Demand Curve
Chapter 7: IS-LM and Aggregate Demand
side of the economy. The demand-side of the economy determines aggregate demand by
recognizing the constraints on the willingness and ability to buy.
Alternatively, the
supply side of the economy determines aggregate supply by recognizing the constraints on
the willingness and ability to produce and sell. Because it does not recognize the real
constraints on the ability to produce in the form of limited labor, capital, raw materials,
and technology, the IS-LM model does not contain a supply-side.
The equilibrium Y value is the aggregate demand level determined by the IS-LM
model. The willingness and ability to buy is constrained by both current income and
accumulated assets. The product market, represented by the IS equation, captures the
effect of current income on the willingness and ability to buy. The money market,
represented by the LM equation, captures the effect of assets. If either the product market
or money market is in disequilibrium, the buying power present in the economy changes
as the real income and real asset levels change. Only when both markets are in
equilibrium is total willingness/ability to buy---i.e., aggregate demand---in a state of
equilibrium.
The aggregate demand curve is the set of (P,Y) combinations associated with a
simultaneous product market-money market equilibrium. The multiplier (7.34) tells us
that the slope of the aggregate demand curve is negative as shown in Figure 7.5. By
presenting Y on the vertical axis and P on the horizontal axis, we go against economic
tradition. Nearly all economics texts do the opposite. However, we are considering Y as
being dependent upon P, meaning Figure 7.5 is mathematically proper. As we have
drawn it, the slope of the aggregate demand curve becomes steeper---i.e., output is more
sensitive to a change in the price level---when the autonomous spending multiplier is
larger, when investment is more sensitive to a change in the interest rate level, and when
money demand is less sensitive to a change in the interest rate level.
Figure 7.5: The Aggregate Demand Curve
Y
AD
P
The IS-LM model provides an answer to the question posed at the beginning of this
chapter. Where does aggregate demand come from? The IS-LM model tells us that it
depends upon
106
Chapter 7: IS-LM and Aggregate Demand




the (exogenous) variables T,G, M, , NX, P;
the particular form of the consumption function as characterized by the marginal
propensity to consume C’;
the particular form of the investment function as characterized by the sensitivity of
investment to interest rates Ir and the sensitivity of investment to animal spirits I;
the particular form of the money function as characterized by the sensitivity of money
demand to interest rates Lr and the sensitivity of money demand to income LY.
To construct a more compete macro model, we must construct a model of the supply side
and then combine it with this demand-side model. A supply-side model is developed in
the next chapter.
Key Words: Chapter 7
Aggregate Demand
LM-Curve
Consumption Function
Savings Function
Marginal Efficiency of Capital
IS Equation
Capital Market
Exogenous Variable
Multipliers
Transactions motive
Speculative motive
LM Equation
Liquidity Constrained
Animal Spirits
Jacobian determinant
Comparative Statics Multipliers
Cramer’s Rule
Crowding out
Contractionary Policy
Demand-Side
Aggregate Demand Curve
Key Concepts: Chapter 7
Review Questions: Chapter 7
107
IS-Curve
Aggregate Supply
Marginal Propensity to Consume
Marginal Propensity to Save
Investment Function
Product Market
Endogenous Variable
Implicit Function
Money Demand
Precautionary motive
Money market
Liquidity Trap
Inducement to Invest
Implicit Function Theorem
Comparative Static Analysis
Linear System
Autonomous spending multiplier
Expansionary Policy
Balanced Budget Multiplier
Supply-Side