LM curve represents what’s happening to the supply and demand for

Building the IS–LM Model

IS stands for “investment’’ and “saving,’’ and the IS curve represents what’s going on in the market for goods and services. LM stands for “liquidity’’ and “money,’’

and the LM curve represents what’s happening to the supply and demand for

money. Because the interest rate influences both investment and money demand, it is the variable that links the two halves of the IS–LM model.

The model shows how interactions between the goods and money markets

determine the position and slope of the aggregate demand curve and, therefore, the level of national income in the short run.

The Goods Market and the IS Curve

The IS curve plots the relationship between the interest rate and the level of

income. It is all interest rate and income combination when the goods market is in equilibrium.

Derivation of the IS curve

Equilibrium in the goods market is established when aggregate demand equals income

𝑌 = 𝐴𝐷

𝐶 + 𝑆 + 𝑇 = 𝐶 + 𝐼 + 𝐺

Assume government spending and taxes to be zero, at equilibrium

𝑆 = 𝐼

Because investment is inversely related to the interest rate, an increase in the interest rate from r1 to r2 reduces the quantity of investment from I(r1) to I(r2). The reduction in investment, in turn, shifts the aggregate demand function downward, as in panel (b) of Figure 10-7. The shift in the aggregate demand

function causes the level of income to fall from Y1 to Y2. Hence, an increase in the interest rate lowers income.

A. INVESTMENT FUNCTION B.AGGREGATE DEMAND-OUTPUT

interest rate Expenditure

Y

r2

r1

∆𝐼

45°

Ir2 Ir1 investment Y2 Y1 Income

C. IS CURVE

Interest rate

r2

r

Y2 Y1 Income

Because an increase in the interest rate causes planned investment to fall, which in turn causes equilibrium income to fall, the IS curve slopes downward.

FACTORS THAT SHIFT THE IS CURVE

A change in autonomous factors that is unrelated to the interest rate in the goods market

1. Changes in autonomous consumer expenditure

2. Changes in planned investment spending unrelated to the interest rate 3. Changes in government spending

4. Changes in taxes

Response to a Change in Fiscal Policy

Fiscal policy is the use of government spending, taxation and borrowing to achieve economic goals

Change in government expenditure

An increase in government spending raises aggregate demand directly; a decrease in taxes makes more income available for spending

The increase in aggregate demand cause aggregate output to rise A higher level of aggregate output increases the demand for money

The excess demand for money pushes the interest rate higher. Higher interest rate leads to fall in investment and encourages savings

The rise in the interest rate eliminates the excess demand for money

Aggregate output and the interest rate are positively related to government spending and negatively related to taxes.

The Money Market and the LM Curve

The LM curve plots the relationship between the interest rate and the level of income that arises in the market for money balances. The LM curve is all interest rate and income combination when the money market is in equilibrium

At equilibrium money Money demand = money supply

The LM curve is derived by plotting the points where money demand equals supply of money. The slope of the LM curve is positive. That is at higher levels of income, equilibrium in the money market occurs at higher interest rates. The

reason for the upward slope of de LM curve is that, an increase in income from Y0

to Y1 increases money demand at a given interest rate, because the transactions

The higher interest will result in a lower speculative demand for money hence lowering the transactions component corresponding to any level of income.

A. Determination of equilibrium in the money market

Interest rate 𝑀𝑆

𝑃

I1

I0

𝐿𝑑 _{= (𝑙}

0,𝑌1, π)

𝑙𝑑 = (𝑙_0,𝑌₀, π)

M Quantity of money

B. LM CURVE

Interest rate

r2

r1

FACTORS THAT SHIFT THE LM CURVE

• Changes in the money supply

• Autonomous changes in money demand •Expected inflation

Response to a Change in Monetary Policy

An increase in the money supply creates an excess supply of money • The interest rate declines

• Investment spending and net exports rise • Aggregate demand rises

• Aggregate output rises

• The excess supply of money is eliminated

• Aggregate output is positively related to the money supply

EQUILIBRIUM IN THE GOODS AND MONEY MARKET

Interest rate

LM

E I0

IS

An increase in money supply causes LM curve to shift to the right. For the excess supply of money to be removed from the money market, the interest rate must fall. The fall of interest rate leads to increase in investment and increases output.

An increase in government expenditure shift the IS curve to the right. An increase in government spending raises aggregate demand directly;

The increase in aggregate demand cause aggregate output to rise A higher level of aggregate output increases the demand for money

The excess demand for money pushes the interest rate higher. Therefore when there is any fiscal expansionary policy, interest rate increases and output increases at the new equilibrium

EXAMPLE

Consider an economy described by the following equations:

𝐶𝑑=300+0.75(𝑌−𝑇)−300𝑟

𝑇=100+0.2𝑌 𝑙𝑑=200−200𝑟 𝐿=0.5𝑌−500𝑖

a) Derive the equation for the IS curve. Graph the IS equation b) Derive the equation for the LM curve. Graph the LM equation

c) Find the short run equilibrium income and real interest rate. Illustrate the equilibrium graphically.

d) Suppose government purchases of goods and services increase by 100. What will be the effect on equilibrium income, real interest rate, consumption,

investment and real money demand?

SOLUTION

a. Equation for the IS curve

Equilibrium in the goods market requires that income equals aggregate demand/aggregate expenditures,

Y=AD. Using this condition, we have:

𝑌 = 300 + 0.75(𝑌− 100 − 0.2𝑌) − 300𝑟 + 200 − 200𝑟 + 600

𝑌− 0.75𝑌 + 0.15𝑌 = 300 − 75 + 200 − 500𝑟 + 600 0.4𝑌 = 1025 − 500𝑟

𝑌 = 2562.5 − 1250𝑟

2.05

IS curve

0 2562.5 Income

b. Equation for the LM curve

In the money market equilibrium requires equality of money demand and money supply,

𝑀𝑆

𝑃 = 𝐿

𝐿 = 0.5𝑌− 500𝑖 =133200

120 =

Note that 𝑖 = 𝜋𝑒 + 𝑟. Substituting into the money demand function we have: 0.5𝑌− 500𝑟 − 500(0.05) = 1110

0.5𝑌 = 1135 + 500𝑟

𝑌 = 2270 + 1000𝑟 Interest rate

LM

2.05

0 2270 4470 Income

-2.27

c. Short run equilibrium income and real interest rate.

We can solve for the short-run equilibrium by solving the IS and LM equations Simultaneously:

2270 + 1000𝑟 = 2562.5 − 1250𝑟

2250𝑟 = 292.5 𝑟 = 0.13

𝑌 = 2562.5 − 1250(0.13) = 2400 At equilibrium,

𝐶 = 300 + 0.75(2400 – (100 + 0.2(2400)) − 300(0.13)=1704

𝐼𝑑 _{= 200 − 200𝑟 = 200 − 200(0.13) = 174}

Interest rate LM curve

2.05

0.13 e

Is curve

0 2270 2400 2562.5 Income

d. The effect of increase in government purchase by 100 on equilibrium income, real

interest rate, consumption, investment and real money demand

The increase in government spending shifts the IS curve to the right. The new IS curve with the new level of government spending is

c. Government spending multiplier(Mg)= 1

1−𝑚𝑝𝑐(1−𝑚𝑝𝑡)=

1

1−0.75(1−0.2)= 2.5

Effect on IS income is Mg× ∆𝐺

= 2.5 × 100 = 250

𝑡ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 𝑛𝑒𝑤 𝐼𝑆 𝑐𝑢𝑟𝑣𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛

𝑌 = 2812.5 − 1250𝑟

Solving with the LM curve from above we have:

2812.5 − 1250𝑟 = 2270 + 1000𝑟

2250𝑟 = 542.5

𝑟 = 0.2411, 𝑌 = 2512.5

𝐶 = 300 + 0.75(2512.5 – (100 + 0.2(2512.5)) − 300(0.0.2411) = 1660.17

𝐼𝑑=200−200𝑟=200−200(0.2411)=151.78

𝐿=0.5𝑌−500𝑖=0.5(2512.5)−500(0.2411+0.05)=1110.7

BY JOHN