Chapter 5: IS-LM Model
Calculation exercises
Summary of the IS relation
Previously in the simple Keynesian cross model
The goods market assumptions
1. Closed economy, therefore Z is equal to C(Y-T) +I +G
2. The price level is fixed P fixed
• Equilibrium in the goods market is when production (output) Y, is equal to the demand
for goods, Z. This equilibrium condition is called the Investment- savings (IS )relation.
• In the simple Keynesian cross model, the interest rate did not affect the demand for
goods. The equilibrium condition was given by:
Summary of the IS relation
When we allow the interest rate to affect the goods market:
• Investments become endogenous and depend on the level of sales (+) and the
interest rate (-) such that in the investments behave like consumption:
• The investment function becomes:
Summary of the IS relation
• The new equilibrium condition in the goods market becomes :
• The equation shows the equilibrium condition in the goods market,
and it relates production Y to the interest rate. Mainly that:
• For a given value of the interest rate, demand is an increasing function of
output, because:
• An increase in production/output leads to an increase in income increases in disposable
income and therefore consumption.
• An increase in output also leads to an increase in investment.
Summary of the IS relation
• We can use the equilibrium conditions to derive the IS relation by tracing how
equilibrium in the goods market changes as the interest rate changes i.e., how does the
equilibrium level of demand and output change when we vary the interest rate.
• We can trace where the 45-degree line and demand (Z) curve will intersect at the
different levels of interest rates.
• We then can plot the different equilibrium points associated with the given level of the
interest rate (i) to derive the IS curve.
Summary of IS relation
• Properties of the IS curve:
• The derived IS curve shows all values of Y and i where the goods market is in
equilibrium.
• Equilibrium in the goods market implies that an increase in the interest rate
leads to a decrease in output, which is why the IS curve is downward facing.
• An increase in the interest rate decreases the demand for goods at any level
of output, leading to a decrease in the equilibrium level of output.
• Changes in factors that decrease the demand for goods, given the interest
rate, shift the IS curve to the left, decreasing output
• Changes in factors that increase the demand for goods, given the interest
rate, shift the IS curve to the right.
Summary of the LM relation
Liquidity-Money LM relation assumptions:
1. Price level is fixed
2. Central bank controls money supply:
• The CB controls the money supply by buying or selling bonds in the bonds market
i.e, in Open Market Operations.
• The assets of the central bank are the bonds it holds. The liabilities are the stock of
money in the economy.
• An open market operation in which the central bank buys bonds and issues money
increases both assets and liabilities by the same amount.
• In an expansionary open market operation, if the central bank buys R 10 million worth of
bonds, it will increase the money supply by R10 million.
• In a contractionary open market operation, if the central bank sells R 10 million worth of
bonds, it will decrease the money supply by R 10 million.
Summary of the LM relation
• Equilibrium condition in the financial markets is that money supply
equal money demand:
• The equilibrium condition relates the interest rate to output.
• The interest rate is determined by the equality of the supply of and the
demand for money,
• The demand for money is dependent on income (+) and the interest rate on
bonds (-)
Summary of the LM relation
Summary of the LM relation
Properties of the LM curve:
• LM curve slopes upwards because the equilibrium in financial markets implies
that, for a given real money supply, an increase in the level of income, which
increases the demand for money, leads to an increase in the interest rate.
• An increase in the money supply shifts the LM curve down.
• A decrease in the money supply shifts the LM curve up.
Summary of IS-LM Model
Why do we think about shifts of the IS curve to the left or to the right,
but about shifts of the LM curve up or
down? The reason:
● We think of the goods market as determining Y given i, so we want to
know what happens to Y when an exogenous variable changes. Y is on
the horizontal axis and moves right or left.
● We think of financial markets as determining i given Y, so we want to
know what happens to i when an exogenous variable changes. i is on
the vertical axis and moves up or down.
Summary of IS-LM Model
IS relation: Y = C(Y − T ) + I(Y, i) + G
LM relation: M/P = YL(i)
IS-LM model is a system of two equations
in two unknowns, Y and i.
Summary of IS-LM Model
• Factors that shift the IS curve:
• Changes in autonomous consumer expenditure
• Changes in government spending
• Changes in taxes
• Changes in interest rates don’t cause shifts in the IS curve, they cause
movement along the IS curve.
• Factors that shift the LM curve
• Changes in money supply (Ms)
• Autonomous changes in money demand (we have assumed this is zero)
Mathematical derivation of IS and LM
relations
Remember:
IS relation: Y = C(Y − T ) + I(Y, i) + G
LM relation: M/P = YL(i)
Two equations and two unknowns, Y and i.
Deriving the IS curve
Question 1 : The following equations describe an economy: C = 100 +
0.8Y; I = 50-25i; G = 50; T=50. Derive the equations for IS curve and
represent it graphically.
Deriving the IS curve
• Answer 1
• Equilibrium in the goods market is when output (Y) is equal to demand
(Z):
Y=Z
Y= C(Y-T) + I+G
Y= 100 + 0.8 (Y-50)+ 50-25i+ 50
Y=200+0,8Y-40-25i
Y-0.8Y= 160-25i
Y(1-0.8)=160-25i
0.2Y=160-25i
Y=800-125i IS Function
Deriving the LM Equation
Question 2: Given the following data about the financial market of the
economy: Md = Y – 25i ; Ms = 200 Rands.
Deriving the LM equation
• Answer 2:
• Equilibrium in the financial market is when:
Ms=Md
200=0.2 Y – 25i
200+25i=Y LM function
Finding values of Y and i
Use the IS and LM functions derived in question 1 and 2 to find the equilibrium level of out put and the interest
rate.
IS function: Y=800-125i
LM function : 200+25i=Y
Answer:
Set IS=LM
200+25i =800-125i
600=125i+25i
600=150i
4=i equilibrium interest rate
Y=800-125i
Y=800-(125)4
Y=800-500
Y=300 Equilibrium output