The IS-LM Model

**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.

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.

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 .

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.

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.

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*.- 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.
- 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*.- 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.

- 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
*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

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.

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 .

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 YThis 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]

Note that, by construction, changes in P lead to movements along the AD Curve as Y adjusts to keep (AD) satisfied.

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 . | . | . P_1 |..........AD | . . | . . | . . P_2 |.......................AD | . . . | . . . | . . . P_3 |........................................AD | . . . . ------------------------------------------------- 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.