Short-period equilibrium: the IS-LM model
Lecture 20 – academic year 2014/15
Introduction to Economics
Fabio Landini

• Concepts of future value, expected value and expected payment flows
• Relationship between interest rate and expected value
• Relationship among firms’ investments, actual value and interest rate

Future value of 1 euro
Today
1
1 year
1(1+i)
2 years
1(1+i) 2 n years
1(1+i) n
Actual value of 1 euro that is paid in the future
Today 1 year 2 years n years
1 1/(1+i) 1/(1+i) 2 1/(1+i) n

The actual value is the maximum amount that one is willing to pay today in order to receive a given amount of money in the future.

i
Actual value of 1 euro that is paid in the future
1 year 5 years 10 years
0.01
0.99
0.05 0.952
0.951
0.784
0.905
0.614
0.10
0.20
0.909
0.833
0.621
0.402
0.386
0.162
The actual value is negatively related to the interest rate and the length of the period.

Suppose a firm is to decide whether of not to make an investment (e.g. buy a machine); the actual value of the expected payment flow is
AV= M
0
+ M
1
/(1+i) + M
2
/(1+i) 2 +….. M n
/(1+i) n
Net actual value = Actual value – cost of the machine
NAV = VA - C

The firm will undertake the investment if NAV > 0
NAV is negatively related to the interest rate
In the graph (follow), the investment will not be undertaken if the interest rate is higher than 7,5%
Variations of the interest rate change the amount of investment undertaken by firms
The value of the interest rate is determined by the financial market but it affects also the good market

NAV
0
0.075
i

Short-period equilibrium: the IS-LM model

Equilibrium in the short period
In the preceding lectures we examined the equilibrium in the good market and the financial market separately
What happens if we consider the equilibrium in both markets simultaneously?
What determines the overall equilibrium of the economic system in the short period?

Equilibrium in the short period
• Introduction
• Firms’ investment choices
• Construction of the IS curve
• Construction of the LM curve
• Equilibrium

Introduction
So fare we have separately considered:
• Equilibrium in the good market
1 equilibrium condition -> Y = Z
1 variable -> Y
• Equilibrium in the financial market
1 equilibrium condition -> M D = M S
1 variable -> i

Introduction
Now we consider the two market simultaneously…
This implies that :
• 2 equilibrium conditions
• 2 Curves
IS – equilibrium in the good market
LM – equilibrium in financial market
• 2 Variables -> Y , i

Introduction
To study the equilibrium in the short period we need to reconsider the two equilibrium condition examined in classes 17 and 18.
Before that we need to review our assumptions about investment decisions.

The investment function
In Class 17 we assumed that investment were exogenous (I=I
0
).
Now we want to make investment endogenous
(i.e., determined within the model).
Important: Remember that investments reflect the firms’ demand of new capital.

The investment function
What do firms look at when they decide how much to invest?
Mainly two things:
1)Sales
If I sell more I need to produce more -> I need more capital -> I invest more… Result: Investment (I) depends positively on sales.
If we assume no inventories -> Sales = production (Y)
For the sake of simplicity we can thus assume I as an increasing function of Y.

The investment function
2) Cost of the investment -> interest rate (i)
A firm computes the NAV of her investment which is inversely related to the interest rate.
This is true in the case in which: a) The firm borrows the money to pay for the investment -> the cost of the borrowing depends positively on the interest rate.

The investment function b) The firm uses her own funds -> Investments in capital are more convenient the lower the return of alternative financial assets.
E.g.: Treasury bonds (BOT) -> if the return of BOT increases, it becomes more convenient to invest in
BOT rather than to invest in machines.
Opportunity cost -> Cost to forgo alternative investments.

The investment function
In both cases: i -> financial cost (or opportunity cost)of investment -> I
Therefore we can conclude that investment depends positively on production and negatively on interest rate:
I = I(Y, i)
+ -

Construction of the IS curve
Let’s consider again the good market equilibrium
Aggregate demand -> Z = C + I + G
Consumption depends on disposable income Y D
C = C (Y D ) where Y D =Y-T
+
Investments depend positively on production and negatively on the interest rate
I = I(Y, i)
+ -
Public expenditure is exogenous (Government)
G = G
0 and T = T
0

Construction of the IS curve
By substituting the components of Z we get:
Z = C(Y-T
0
) + I(Y,i) + G
0
The equilibrium condition of the good market is Z=Y
Taking the two equation together we get:
Y = C(Y-T
0
) + I(Y,i) + G
0
In the equation we have two variables (Y and i) -> We can represent the equilibrium condition of the good market in a Cartesian diagram -> “IS Curve”

To construct the curve let’s consider the graph of the good market equilibrium (Class 17)
Z,Y
ZZ
A
Y
A
45
°
Y

The ZZ curve is associated with Z = C(Y-T
0 and could (is likely) be non-linear
) + I(Y,i) + G
0
Z,Y
ZZ
A
Y
A
45
°
Y

Construction of the IS curve
The new ZZ curve is now upward sloping for two reasons: because consumption increases with income and investment increases with sales.
The slope of the new ZZ is lower than the di 45degrees line (empirical evidence).
The ZZ curve has been built for a given value of i.

What happens if i changes? Let’s assume i -> I -> Z
Equilibrium A -> A’
Z,Y
Y
A
> Y
A
’ , it follows that in equilibrium i -> Y
Y
A
A’
A
ZZ (i)
ZZ’(i’) i’>i
Y
A
’
45
°
Y

Construction of the IS curve
There exist different combination of i and Y that ensures the equilibrium
To maintain the equilibrium when i -> Y
Explanation:
• i -> cost of funds -> I -> Z -> (in equil.) Y
• + multiplier effects -> Y -> C and I -> Z ->(in equil.) Y

Construction of the IS curve
In the equilibrium of the good market the relationship between Y and i is decreasing
The IS curve is decreasing in the diagram (Y,i) because if i then Z and to maintain the equilibrium also Y
The slope of the IS curve depends on the sensitivity of investment to the interest rate and the value of the multiplier.

i
IS
Y
The curve shows all the pairs (Y,i) such that the good market is in equilibrium.

i
A
IS
Y
• Point A -> good market is in equilibrium
Y
• The level of Y and the interest rate generate a demand that equalize the supply.

Points outside the IS curve
The points above the IS curve are points characterized by excess supply.
The points below the IS curve are points characterized by excess demand.

i i
2 B
ES i
1
A
IS
Y
1
Y
Above the IS curve -> excess supply.
Y = Y
1 and i = i
2
> i
1
(B) generates a demand lower than the level of supply.

i i
1 i
2
A
C
ED
IS
Y
1
Below the IS -> excess demand.
Y= Y
1 and i = i
2
< i
1
(C) generates a demand higher than the level of supply.
Y

The IS curve is built for given value of T, G and C
0
.
Changes in the value of T, G and C
0 shift the IS curve.
If G and C increases, the demand for given i increases; if follows that production must increase as well to maintain the equilibrium between demand and supply.
If T increases, the demand for given i decreases; it follows that the level of production must decrease as well to maintain the equilibrium between demand and supply.

i
What happens to the IS curve if T
0 changes?
T
0
-> Y D -> C -> Z -> (in equilibrium) Y for given i
Is curve shifts leftward i
Y’ Y
Per T’>T
Per T
IS
IS’
Y

Construction of the LM curve
Let’s take a look again at the equilibrium in financial market.
The demand of money depends is positively related to income and negatively related to the interest rate:
M D = $YL(i)
The supply of money is fixed by the Central Bank
M S exogenous

Construction of the LM curve
Equilibrium condition in financial markets
M S = $YL(i)
Equilibrium in financial markets -> nominal income ($Y).
Equilibrium in good market -> real income (Y).
To homogenize the variables we divide both sides of equation M S = $YL(i) for the price level P (measured through the GDP deflator).
We obtain M S /P = YL(i), where M S /P is the real money supply.

Construction of the LM curve
Equilibrium condition in financial markets:
M S /P = YL(i)
In this equation we have two variables (Y e i) -> We can represent the above equilibrium condition in a
Cartesian diagram -> LM curve
To build the LM curve let’s examine the effect of Y in financial markets.

Construction of the LM curve
In Class 18 we saw that if Y then M D . If the money supply is given, then the equilibrium is maintained through i -> M D
In equilibrium there is a positive relations between Y and i -> the LM cure is increasing

i
E
’ i
E i
Y -> M D shifts rightward
Y -> E -> E’ -> i in equilibrium
M D M D ’
M S
E
E’ Per Y
2
>Y
1
Per un dato Y
1
M

i
LM
Y
The LM curve shows all the pairs (Y,i) for which the financial markets are in equilibrium.

i
LM i
1
A
Y
1
Y
The combination i
1 and Y
1 ensures that A is an equilibrium of the financial markets.

Points outside the LM curve
Points on the right of the LM curve are points characterized by an excess demand of money.
Points on the left of the LM curve are points characterized by and excess supply of money.

i
LM
ED i
1
A
B
Y
1
Y
2
Y
In point B there is the same interest rate as in A but an higher income. Therefore, for given M S -> M D >M S

i
LM
ES i
1
C
A
Y
3
Y
1
Y
In point C, there is the same interest rate as in A but lower income. Therefore, for given M S -> M D <M S

The LM curve is built for given M S .
If M S increases, an excess supply of money is created.
Individuals purchase financial assets to get rid of the money in excess.
The interest rate reduces and the demand for money increases.
The new equilibrium will be characterized by the same income and lower interest rate.

i
1 i
2 i
What happens to the LM curve if M S increases?
M S -> given Y i -> M D -> LM shifts rightward.
LM LM’
Y
1
Y

Short-period equilibrium of the economy ->
The equilibrium of the good market and of the financial market are simultaneously satisfied ->
IS and LM in the same diagram.

i
LM
E
IS
E – short-period equilibrium of the economy.
In E we are on the IS -> eq. in good markets
In E we are on the LM -> eq. in financial markets.
Y

i
LM
E i
E
IS
Y
E i
E
– equilibrium interest rate and Y
E
Y
– equilibrium income.
The pair (Y
E
, i
E
) is the only one for which the good market and the financial market are simultaneously in equilibrium.