## Basic Lecture Notes for Hall and Taylor, Chapter 7: The IS-LM Model

Maintained by: Leigh Tesfatsion
Email: tesfatsi@iastate.edu
Last Updated: 12 October 1996

PLEASE NOTE: These basic lecture notes on HT Chapter 7 are required for Econ 302. If possible, students should read these notes prior to attending class lectures on HT chapter 7. A more advanced detailed version of these notes is also available on this web site; the advanced notes are recommended but not required.

Basic References:
HT Chapter 7;
Study Guide, Chapter 7.

## A. Introduction

In HT5 we developed money market relationships, and in HT6 we developed product market (spending balance) relationships. In the current chapter these money and product market relationships are brought together within a single framework -- the IS-LM model. This model will allow us to begin to analyze the impact of monetary and fiscal policies on key macroeconomic variables.

As a first step in the construction of the IS-LM model, the previous HT6 modellings of investment and net exports will be generalized to permit dependence on the real interest rate.

## B. Investment and the Real Interest Rate

The key behavioral assumption for investment, based on empirical observation, is as follows. The demand for investment goods (i.e., physical capital goods) depends negatively on the real interest rate R paid on bonds (more generally, on financial assets). That is, investment demand is low when R is high and investment demand is high when R is low.

Intuitive rationale:

```
High R   -->  high expected cost of   -->  fewer planned investment
borrowing to finance         good purchases
investment

High R   -->  high return to lending  -->  more planned bond purchases

```

A Simple Algebraic Representation for Investment Demand:

```

I^D  =  e  -  dR    ,

where

R     =   the real interest rate on bonds;

e, d  =   exogenously given positive coefficients .

```

## C. Net Exports and the Real Interest Rate

The key assumption for net export demand in relation to the real interest rate R, based on empirical observation, is that an increase in the HC real interest rate R tends to cause a decline in net export demand and vice versa.

As depicted below, the reason for this inverse relationship between net export demand and the real interest rate is due to the intermediate effect of the change in R on the HC exchange rate E.

```             ------
|   more attractive for ROW
|   nationals to buy HC bonds,
|   that is, to lend to the HC
|
|
|                                  ------
|                                 |  relative demand for
|                                 |  HC currency increases
R higher -->|                                 |  as ROW nationals buy
|                              -->|  HC bonds and HC
|                                 |  nationals sell ROW
|                                 |  bonds
|                                  -------
| less attractive for HC
| nationals to buy ROW bonds,
| that is to lend to ROW
|
-------

-->  the price of HC currency   -->  increase in the nominal
relative to ROW currency        exchange rate E (ROW
increases                       currency per unit of
HC currency)

-->  HC goods more expensive to ROW        -->  EX^D lower and
and ROW goods less expensive               IM^D higher
to HC

-->  net export demand  NE^D = [EX^D - IM^D]  declines.

```

The relationship between net exports and the exchange rate E is more carefully developed by HT in Chapter 12. For now, we only use the implication that R is negatively related to net export demand. Specifically, HT make the behavioral assumption that net export demand is a simple linear function of both R and Y, as follows:

Simple Algebraic Representation for Net Export Demand:

```

(1)       NE^D  =  g  -  mY  -  nR ,

where

g, m, n = positive exogenously given coefficients.

```
Note that (1) implies that net export demand NE^D is a decreasing function of both real income Y and the real interest rate R. The reason for the negative dependence on Y was discussed in HT6.

## D. The IS-LM Model: Algebraic Formulation

The objective of this section is to combine the various behavioral assumptions for consumption demand, investment demand, net export demand, and money demand into an "IS-LM model" explaining how real income might be determined in the short run as the result of product market and money market equilibrium, taking as given the general price level P.

NOTE ON TERMINOLOGY: The reason for the term "IS" is that product market equilibrium was traditionally expressed as an equality between investment and saving in planning terms. [Below it is equivalently expressed as an equality between aggregate demand Y^D and actual supply Y.] Also, the reason for the term "LM" is that the money market equilibrium condition was traditionally expressed as a condition of the form L(Y,R)=M, where L(Y,R) denoted the money demand function and M denoted money supply.

```EQUATIONS DESCRIBING THE ECONOMY IN PERIOD T

Equation                Verbal Description

Product Market (Spending Balance) Relations:

#Supply Side#:

(1) Realized income       Y  =  C + I + G + NE      Accounting Identity
For Realized Income

#Demand Side#:

(2) Consumption Function  C^D  =  a + bY_d             Behavior Assumption

(3) Disposable Income     Y_d   =  [1-t]Y              Definition of Y_d

(4) Investment Demand     I^D   =  e - dR              Behavior Assumption

(5) Net export Demand     NE^D  =  g - mY - nR         Behavior Assumption

(6) Aggregate Demand      Y^D = C^D + I^D + G + NE^D   Definition of Y^D

#Product Market Equilibrium#:

(7) Supply=Demand        I = I^D          Equilibrium Condition

(8) Supply=Demand        C = C^D          Equilibrium Condition

(9) Supply=Demand        NE = NE^D        Equilibrium Condition

```

REMARK: What about the equality of realized output supply Y and aggregate demand Y^D? Given equations (7) through (9), it immediately follows from equations (1) and (6) that Y=Y^D, hence the latter equilibrium condition does not have to be separately assumed.

```

Money Market Relations:

#Supply Side#:

(10)  Money Supply Function    M^S = M    Government Monetary Policy Rule

#Demand Side#:

(11)  Money Demand Function    M^D/P = kY - hR^N       Behavioral Assumption

(12)  Nominal Interest Rate    R^N  =  R + INF        Definition

#Money Market Equilibrium#:

(13)  Supply=Demand            M^S  =  M^D             Equilibrium Condition

CLASSIFICATION OF VARIABLES (CAUSAL STRUCTURE)

#Endogenous Variables#: (Variables appearing in the model equations
for period T whose solution values are determined by these equations,
hence determined #during# period T)

Y^D, Y, C^D, C, I^D, I, NE^D, NE, M^D, M^S, Y_d, R, R^N,

Note there are thirteen equations and thirteen endogenous variables to be
solved for.

#Exogenous Variables#: (Variables appearing in the model equations
whose values are fixed known quantities, hence determined #prior#
to period T)

Government policy variables:  t, G, M

Coefficients a, b, e, d, g, m, n, k, h (all positive)

Price level P  (#temporary# classification)

Inflation rate INF  (#temporary classification#)

NOTE: In HT8, price adjustment relations for P and INF will be
introduced into the IS-LM model, which will change the classification
of P and INF.

```

## E. The IS-LM Model: Graphical Representation

We will now reduce equations (1) through (13) to just two equations in Y and R -- the IS and LM equations. We will then graphically depict the IS and LM equations in the Y-R plane to obtain a representation of the economy useful for policy analysis purposes.

Key Definitions:

The IS Curve is a schedule showing all combinations of the real interest rate R and the level of real income Y that satisfy the product market (spending balance) relations (1) through (9). The LM Curve is a schedule showing all combinations of the real interest rate R and the level of real income Y that satisfy the money market relations (10) through (13).

The IS Curve:

By definition, the IS Curve is the graph of the product market relations (1) through (9) in the Y-R plane after all endogenous variables except Y and R have been substituted out. Taking equation (1) as the basic starting point for these substitutions, the result is one equation in Y and R, as follows:

```

Y  =  (a + b[1-t]Y)  +  (e - dR)  +  G  +  (g - mY - nR)

```
In order to graph this equation, it is useful to rearrange terms so that R is expressed as a linear function of Y in slope-intercept form, as follows:
```

#The IS Equation#:

a + e + G + g              1 - b[1-t] + m
(IS)   R  =  ---------------       -   -----------------   Y
d + n                       d + n

R-intercept            slope dR/dY = -[1-b(1-t)+m]/[d+n]
(Value of R when Y=0)      (note the minus sign is included)

```
[Compare Hall and Taylor, equ.(7-5), p. 188.] The IS Curve is the graph of the IS equation in the Y-R plane. Since all of the coefficient terms b, t, m, d, and n are assumed to be positive, the intercept of the IS Curve is positive and the slope dR/dY of the IS Curve is negative. Also, it follows from the derivation of the IS Curve that the product market is in a demand=supply equilibrium at every point on the IS Curve.

```

R
|
|
a+e+G+g    *
---------   |  is
d+n      |     is                      IS Curve with slope
|        is
|           is                 dR        1-b[1-t]+m
|              is              --  =  - ------------
|                 is           dY           d+n
|                     is
|                        is
|                           is
|                              is
|                                 is
|
|
--------------------------------------------  Y
0

The IS Curve

```

Four Different Ways the IS Curve can be Affected By Variable Changes:

1. Movements Along the IS Curve:
Given fixed values for all variables apart from Y and R, any change in Y causes a movement along the IS Curve as R correspondingly adjusts to keep (IS) satisfied.
2. Parallel Shifts of the IS Curve:
A change in any variable entering into the R-intercept of the IS Curve but not the slope of the IS Curve causes the IS Curve to undergo a parallel shift.
1. EXAMPLE 1: Suppose there is an increase in the coefficient a appearing in the consumption function. It follows from (IS) that the coefficient a appears in the R-intercept of the IS Curve but not in the slope of the IS Curve. Consequently, the IS Curve undergoes a parallel upward shift -- for each Y, a larger R.
2. EXAMPLE 2: Suppose that real government expenditure G decreases. It follows from (IS) that G appears in the R-intercept of the IS Curve but not in the slope of the IS Curve. Consequently, the IS Curve undergoes a parallel downward shift -- for each Y, a smaller R.
3. Rotations of the IS Curve:
A change in any variable entering into the slope of the IS Curve but not into the R-intercept of the IS Curve rotates the IS Curve around the R-intercept.
1. EXAMPLE: What happens if t increases? First note from (IS) that t enters into the slope of the IS Curve but not into the R-intercept of the IS Curve. Also, if t increases, then [1-b(1-t)+m] increases. Thus, for any given Y, an increase in t results in a smaller right hand side of (IS) and hence in a smaller value for R. It follows that the IS Curve rotates downward around the unchanged R-intercept when t increases.
4. Simultaneous rotation and shift of the IS Curve: A change in any variable entering into both the slope of the IS Curve and the R-intercept of the IS Curve results in a simultaneous rotation and shift of the IS Curve. For the model at hand, the only such variables are d and n.

The LM Curve:

By definition, the LM Curve is the graph of the money market relations (10) through (13) in the Y-R plane after all endogenous variables other than Y and R have been substituted out. The result of these substitutions is one equation in Y and R, as follows:

```
M/P  =  kY - hR - hINF

where

M, P, k, h, INF  =  variables with given values.

```
Recall that, for now, we are retaining the dependence of money demand on the nominal interest rate, R+INF, which is the true opportunity cost of holding money. HT simplify their analysis by assuming money demand depends on the real interest rate R rather than on the nominal interest rate R+INF. [compare HT equation (7-3), p. 186]:

In order to graph this equation, it is useful to rearrange terms so that R is expressed as a linear function of Y in slope-intercept form, as follows:

```

#The LM Equation#:

(LM)   R   =   - [ (M/hP) + INF ]       +           [k/h]Y

R-intercept               slope dR/dY = k/h
(Value of R when Y=0)

```

The LM Curve is the graph of the LM equation in the Y-R plane. Since both k and h are assumed to be positive, the LM Curve slopes upward, in the sense that Y and R increase and decrease together along the LM Curve. The R-intercept of the LM Curve is negative if the inflation rate INF in nonnegative; otherwise it could be positive. Also, it follows from the derivation of (LM) that the money market is in equilibrium at each point on the LM Curve.

```
R
|
|                                           lm
|                                       lm
|                                   lm  -- LM Curve
|                               lm      with slope
|                           lm
|                       lm              dR      k
|                    lm                 --  =  ---
|                lm                     dY      h
0   -------------*--------------------------------- Y
|         lm
|     lm        [M/kP] + hINF/k
| lm
-[M/hP]-INF *
|

The LM Curve

```

How is the LM Curve affected by variable changes? In complete analogy to the IS Curve, the LM Curve shifts or rotates depending on whether changes occur to variables appearing in the R-intercept of the LM Curve, in the slope of the LM Curve, or in both.

## F. Solving the IS-LM Model for Y and R

DEFINITION: An economy is said to be in an IS-LM equilibrium if both the product market and the money market are in a demand=supply equilibrium. Graphically, then, an economy is in an IS-LM equilibrium whenever it is at an intersection point of the IS and LM Curves in the Y-R plane.

Note that involuntary unemployment may exist in an IS-LM equilibrium, for there is no guarantee that Y is equal to potential GDP Y*. Consequently, although the product and money markets are in equilibrium in an IS-LM equilibrium, by definition, there is no guarantee that the labor market is in equilibrium. This situation is depicted in the following graph.

```
R
|
is
|
|    is                         lm
|
|            is           lm
|
R^o |..................E^o
|                   .
|             lm    .       is
|                   .
|       lm          .               is
|                   .
--------------------------------------------- Y
0                   Y^o         Y*

IS-LM Equilibrium at E^o with Y^o less than Y*

```

How can the IS-LM equilibrium values for Y and R be determined algebraically?

To determine these solution values, it is useful to first restate the IS-LM model (1) through (13) in its reduced two-equation form, as follows:

```
The IS Equation (Product Market Equilibrium):

a + e + G + g              1 - b[1-t] + m
(IS)   R  =  ---------------       -   -----------------   Y
d + n                       d + n

The LM Equation (Money Market Equilibrium):

M                  k
(LM)   R   =   - [ ---- + INF ]   +   ---  Y
hP                 h

```

By construction, the values for Y and R that solve the two relations (IS) and (LM) constitute an intersection point of the IS and LM Curves in the Y-R plane. That is, they are the values for Y and R that simultaneously satisfy equations (IS) and (LM) and hence constitute the IS-LM equilibrium values for Y and R.

Determining explicit analytical expressions for these equilibrium values is straightforward if a bit messy. They are found by using one of the equations to solve for R as a function of Y, and then substituting this value for R into the second equation. For example, using (LM) to solve for R, and substituting this value into (IS), one obtains one equation in the one unknown Y:

```

- [ (M/hP) + INF ]  +  [k/h]Y

=  [a+e+G+g]/[d+n]  - ([1-b(1-t)+m]/[d + n])Y .

```
Collecting terms in Y on the left, putting all other terms on the right, and abbreviating [1-b(1-t)+m] by A for expositional simplicity, one obtains:
```

[k/h]Y  +  (A/[d+n])Y  =  [ (M/hP) + INF ] + [a+e+G+g]/[d+n]

or

( [k/h] + A/[d+n] )Y   =  [ (M/hP) + INF ] + [a+e+G+g]/[d+n] .

```
Now divide through by the term in parantheses multiplying Y to obtain the IS-LM equilibrium solution value for Y:
```
[M/hP]  +  INF  + [a+e+G+g]/[d+n]
Y^o     =  ------------------------------------
[k/h]   +    A/[d+n]

```
The IS-LM equilibrium solution value R^o for R can now be found by substituting Y^o for Y either in the IS equation or the LM equation and then solving for R.

Once the solution values Y^o and R^o are obtained for Y and R, solution values for all remaining endogenous variables in the equations (1) through (13) can be obtained by substituting the solution values for Y and R back into these equations.

For example, given Y^o and R^o, the solution value for consumption C is given by

```

C^o  =   a   +   b[1-t]Y^o  .

```

## G. The Aggregate Demand Curve

DEFINITION: The aggregate demand curve, or AD Curve for short, is a schedule showing the aggregate spending Y of the economy at each price level P, assuming the economy is in an IS-LM equilibrium at each price level P.

As one would expect from microeconomics, the AD Curve is downward sloping. That is, the higher the price level P, the lower the demand. There is, however, a major difference between the AD Curve and the concept of a demand curve in microeconomics: the AD Curve incorporates market equilibrium conditions whereas the micro demand curve does not.

Specifically, the AD Curve is derived from the succession of intersection points (Y^o,R^o) of the IS and LM Curves as the price level P is varied. More precisely, it is the graph of the resulting solution values Y^o against the corresponding P values. Note that P enters into the R-intercept of the LM Curve but that P does not enter at all into the IS Curve. Thus, changes in P shift the LM Curve but do not affect the position of of the IS Curve.

For example, consider what happens when P increases:

```

P increases --->  M/P decreases      --> For each possible Y, R goes
[excess demand for money]     #up# to dampen money demand

--> LM Curve shifts #up#

--> New intersection point of the
IS and LM Curves occurs at
a #lower# level of Y

```
This sequence of events illustrates why the AD Curve is downward sloping in the Y-P plane: higher P values correspond to lower levels of Y (and vice versa). However, it also illustrates how the AD Curve is based on market equilibrium conditions. In particular, by construction, the economy is in an IS-LM equilibrium at every point on the AD Curve.

Algebraic Derivation of the AD Curve:

```STEP 1:  Consider once again the IS and LM Curves, where A
abbreviates the expression [1-b(1-t)+m]:

(IS Curve)    R  =  [a+e+G+g]/[d+n]   -  (A/[d+n])Y

(LM Curve)    R  =  - [ (M/hP) + INF ]  +   [k/h]Y

STEP 2:  Substitute out for R, leaving one equation in Y and P:

[a+e+G+g]/[d+n] - (A/[d+n])Y =  - [ (M/hP) + INF ] +  [k/h]Y

STEP 3:  Collect terms in Y and in P:

( [a+e+G+g]/[d+n] + INF) +  (M/hP)  =  ([k/h] + A/[d+n])Y .

STEP 4:  Divide through by the coefficient multiplying Y.  This
yields the IS-LM equilibrium value Y^o for Y as a function of P,
given by

(AD)        Y^o    =    V  +  B/P

where

( [a+e+G+g]/[d+n]  +  INF)
V     =     -----------------------------
[k/h]  +  A/[d+n]

[M/h]
B     =         -------------------     .
[k/h]  +  A/[d+n]

```

When will the AD Curve shift and/or rotate in response to a change in some exogenous variable other than P? Answer: Whenever the variable enters into either V or B. Or equivalently, whenever the variable appears somewhere in the IS and LM equations that are used to derive the AD Curve.

Graphical Derivation of the AD Curve:

```
R                              lm
|                          lm
|                      lm -- LM Curve for P = P_1
|     is           lm
|       is     lm                                lm
|          E1                                lm
|      lm     is                         lm
|  lm            is                  lm   --LM Curve for P=P_2
|                  is            lm
|                     is     lm
|                        E2                              lm
|                    lm      is                      lm
|              lm                is              lm  --LM Curve for P=P_3
|         lm                         is      lm
|                                        E3
|                                    lm      is
------------------------------------------------------  Y
0        Y^o(P_1)     Y^o(P_2)        Y^o(P_3)

.
.
P    .
|    .
|    .
|    .     .
|    .     .
|    .     .
|    .     .             .
|    .     .             .
|    .     .             .
|    .     .             .               .
-------------------------------------------------  Y
0    V   Y^o(P_1)     Y^o(P_2)        Y^o(P_3)

The Aggregate Demand Curve in the Y-P Plane

```

Policy in the IS-LM Model: Introductory Comments

BASIC QUESTION: What can government do in the short run to affect Y and R through changes in its monetary policy variable M and its fiscal policy variables G and t?

Monetary Policy

What happens to the IS Curve, the LM Curve, and the AD Curve if the Fed decreases the money supply M?

ALGEBRAIC ANSWER: The LM Curve shifts up, that is, R is larger for each given Y. The IS Curve is not affected. The new IS-LM intersection point is at higher R and lower Y. Since the given level of P is unaffected by this change in M, this implies a lower value of Y for the given P, i.e., the AD Curve shifts to the left.

Such a monetary policy change is called a contractionary monetary policy change, since it contracts Y.

ECONOMIC INTUITION: With a decrease in the money supply, there is more demand for money in the economy than supply -- that is, there is an excess demand for money in the money market. In order to convince people to demand less money (i.e., to hold more bonds) so that the money market can regain a state of equilibrium, the interest rate (i.e., the opportunity cost of holding money) must increase for any given level of Y. The higher interest rate R also discourages investment and net exports, which lowers real income Y through the multiplier process.

Conversely, an increase in M would result in a new intersection point at a lower R and a higher Y, implying a move to the right of the AD Curve.

Note that the IS-LM model (1) through (13) assumes that the Fed cannot directly control R. Rather, the Fed has an indirect control over R through its control over M. That is, the model assumes that the Fed can "peg" R at any level in the short run (i.e., in period T) by setting the money supply M at an appropriate level.

This mimics what actually goes on in the U.S. when the Fed decides to increase or decrease the Federal funds rate, the rate which banks charge each other for loans. The Fed does not directly control the Federal funds rate. However, it exerts a strong indirect control over the Federal funds rate through its control of the money supply, and also through its control over the Federal discount rate (the rate the Fed charges banks for loans). For simplicity, the IS-LM model (1) through (13) does not incorporate the Federal discount rate option for the Fed.

Fiscal Policy: Government Expenditure

What happens to the IS Curve, the LM Curve, and the AD Curve if the government increases its expenditure level G?

ALGEBRAIC ANSWER: The expenditure level G enters into the numerator of the R-intercept of the IS Curve but it does not enter into the slope of the IS Curve. Thus, an increase in G shifts up the IS curve. That is, R is higher for each given Y. Since G does not enter into the LM Curve, the LM Curve is not affected by a change in G. It follows that the new intersection point with the LM curve is at a higher value for R and a higher value for Y. Since P is unaffected by the change in G, this implies a higher Y for the given P; that is, the AD Curve shifts to the right.

Such a policy change is called an expansionary fiscal policy change, since it results in a higher level of Y.

ECONOMIC INTUITION: The first-round effect of an increase in G causes Y to increase at the given level of R. This leads to an increase in M^D, hence an excess demand for money. To encourage people to hold less money, the opportunity cost of holding money must increase, i.e., R must increase; but this leads to decreases in investment I and net exports NE which #partially# offset the initial increase in Y.

REMARK: The decrease in private sector investment I in response to the initial increase in G is referred to as a "crowding out effect" -- the increase in government expenditures is said to have "crowded out" private investment I due to the resulting rise in the real interest rate R.

By similar arguments, a decrease in G would lead to a lower value for R and a lower value for Y, and hence a shift to the left in the AD Curve.

Fiscal Policy: Tax Rate

What happens to the IS Curve, the LM Curve, and the AD Curve if the government decreases the tax rate t?

ALGEBRAIC ANSWER: The tax rate t enters into the slope and the Y-intercept of the IS Curve, but not into the R-intercept of the IS Curve. Specifically, a decrease in t causes the negative slope of the IS Curve to decrease in magnitude (i.e., to become less negative) and the positive Y-intercept to increase in magnitude. The R-intercept stays fixed. The IS Curve thus rotates outward around the R-intercept, implying that R is higher for each given Y. Since t does not enter into the LM Curve, the LM Curve is not affected by a change in t. It follows that the new intersection point with the LM Curve is at a higher value for R and a higher value for Y. Since the predetermined price level P is unaffected by the change in t in the current period, this implies a higher IS-LM equilibrium value for Y for any given P; that is, the AD Curve shifts to the right.

Such a policy change is called an expansionary fiscal policy change, for obvious reasons.

ECONOMIC INTUITION: The first-round effect of a decrease in t causes C and hence Y to increase. This leads to an increase in M^D (hence an excess demand for money), and also to a further increase in C in response to the first-round increase in Y which tends to further increase Y (although this increase in Y is partially offset by a corresponding decrease in net exports). To encourage people to hold less money, the opportunity cost of holding money, R, must increase; but this leads to decreases in investment I and net exports NE which partially offset the initial increase in Y. 