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Topic 1 The I-S Curve

Introduction

The I-S curve sets out all the possible combinations of real

income (y) and the interest rate ( r ) consistent with equilibrium

in the goods market

A change in the rate of interest, r, affects equilibrium national

income via the following route:-

Where r is the change in the interest rate, I is the change in

Investment and y is the change in real national income/output.

Consider each of these 2 steps in turn

STEP 1 Investment and the Rate of Interest

Investment is the purchase of capital goods ( i.e. plant,

machinery, land, buildings etc..). Investment is flow of

expenditure per period (e.g. per quarter or per annum). If the

volume of investment in a period is greater than the erosion of

the capital stock due to wear and tear (depreciation), the capital

stock will be increasing.

Typically, firms borrow to finance investment.

At high rates of interest, the cost of borrowed money will be

high and few investment projects will be viable.

As interest rates fall, more and more investment projects

become profitable.

Therefore there is a negative relationship between r and I as setout in Figure 1

1

y

Via the

multiplier


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Note that as interest rates fall from r0 to r1, investment rises from I0 to I1To sum up Step1:- r leads to I such that an increase in r reduces

investment expenditures and a fall in r increases investment

expenditures.

Step 2:- Investment and Equilibrium Income

Consider a simple model of national income determination for a

closed economy with a government sector.

Equation 1:- Aggregate Demand

AD = C + I + G

Where AD is Aggregate Demand,

C is Consumers Expenditure or Consumption

I is the level of Investment

and G is the level of Government ExpenditureEquation 2 :- The Consumption Function

C = a0 + byD

2

I

r

I= I(r)

r0

I0

r1

I1

Figure 1 - The Investment Schedule


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Where a0 is autonomous consumption

b is the marginal propensity to consume

and yD is disposable or after tax income

Note yD = y ty where t is the income tax rate

= (1-t)y

ThereforeC = a0 + b (1-t) y

The Consumption Function is set out in Figure 2

Equation 3:- The Investment Schedule

I = I(r ) as set out above.

N.B. we will assume a linear or straight line investment schedule.

Equation 4:- Government Spending

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y

C

C= a0+b(1-t)y

a0

Slope of function is b(1-t)

Figure 2 - Consumption Function


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G = G0 i.e. government spending is exogenous it is the level

decided upon by the Government.

Equation 5 :- Equilibrium Condition

AD = y [ aggregate demand is equal to output]

Finding The Reduced Form

The above model contains 5 equations. We will simplify matters by

finding an expression for y in terms of

The structural parameters (i.e b and t)

The exogenous or autonomous variables (i.e. a0 , I(r ), and G0)

This process is termed computing the reduced form and involves

substituting Equations 2, 3 and 4 into Equation 1 and re-arranging

Thus

AD = y = C + I + G

y= a0 + [b(1-t)y] + I(r ) + G0

Note that we have y on both sides of this expression. Re-arranging

yields

y-[b(1-t)y] = a0 + I(r) + G0

Therefore

y(1-b(1-t)) = a0 + I(r) + G0

and by dividing both sides by (1-b(1-t)) we obtain Equation 6 which

is an expression for y in terms of the structural parameters and the

exogenous variables.

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Equation 6 :- The Reduced Form

y = 1/(1-b(1-t)) * [a0 + I(r) + G0 ]

where

1/(1-b(1-t)) is termed the multiplier(Equation 7)

Equation 6 tells us that the level of equilibrium output in the goods

market is the product of the levels of the autonomous/exogenous

variables times the multiplier.

From Equation 6 we can derive Equation 8

y = 1/[1-b(1-t)] * [a0 + I(r ) + G0]

Equation 8 indicates that the change in the level of equilibrium

income equals the change in the levels of the autonomous variables

times the multiplier.

Derivation of the I-S Schedule

1/ Numerical Example

Recall that we defined the I-S curve as representing all the

combinations of r and y consistent with equilibrium in the goods

market. Equation 6 gives us an expression for y in terms of a number

of variables including I(r ). We can derive the I-S curve by varying r

and examining what happens to y.

Evaluation of the Multiplier

Let b = marginal propensity to consume = 0.8

Let t = tax rate = 0.25

We can plug these numbers into Equation 7 to find a value for the

multiplier

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Multiplier = 1/[1-b(1-t)] = 1/[1-(0.8(1-0.25))] = 1/[1-(0.8 * 0.75)]

= 1/[1-0.6]

= 1/[0.4]

= 2.5

Evaluation of the Exogenous/ Autonomous Components

The following table sets out the levels of the exogenous variables for

different rates of interest. Note that investment is the only

autonomous variable which varies with r.

r a0 I(r ) G0 A ye

5% 100 80 60 240 600

10% 100 60 60 220 550

15% 100 40 60 200 500

For r = 5% , A, the sum of the autonomous components is 240

Equilibrium y = ye = multiplier * A

= 2.5 * 240= 600

Graphing r against y yields the downward sloping I-S schedule.

2/ Graphical Derivation

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y y

AD AD = y

AD0 [r=15%]AD1 [r=10%]

AD2 [r= 5%]

r y0 y1 y25%15%10%


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Figure 3(a) indicates that falls in the rate of interest from 15% to 10% to 5%

engender increases in investment resulting in aggregate demand rising from

AD0 to AD1 to AD2. These increases in AD cause equilibrium output to risefrom y0 to y1 to y2.

Figure 3(b) sets out the relationship between r and y. As r falls investment

increases driving up equilibrium output, via the multiplier, from y0 to y1 to y2Note:- Every point along the I-S schedule represents a possible equilibrium

level of national income/output

What Determines The Slope of the I-S Curve

7

I-S

Figure 3 (a)

Figure 3 (b)


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The slope of the I-S curve depends on 2 factors

1. The slope of the Investment Schedule

2. The size of the Multiplier

Consider each in turn:-

1/ The Slope of the Investment Schedule

If investment is very sensitive to changes in interest rates, then a small change

in r will summon forth a large change in investment expenditure. This results

in a large change in equilibrium output. Thus, a flat or relatively interest

elastic investment curve results in a flat I-S curve.

If investment is not very sensitive to changes in the rate of interest, then a

small change in r will engender a relatively small increment in I and

corresponding small change in equilibrium output. Thus, a steep or interest

inelastic investment schedule produces a relatively steep I-S schedule.

Both cases are set out in Figure 4

2/ The Role of the Multiplier

8

y

r

I-S (interest inelasticinvestment)

I-S (interest

elastic investment

Figure 4


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The size of the multiplier determines the size of the change in equilibrium

output for a given A

The larger the multiplier, the larger the response of y to a given

change in r and I. Thus, the larger the multiplier, the flatter the I-S curve. The smaller the multiplier, the smaller the y given r and I.

Thus, the smaller the multiplier, the steeper the I-S curve.

Recall Equation 7 which sets out the mathematical formula for the multiplier:-

Multiplier = 1/(1-b(1-t))

Clearly, a change in either b, the marginal propensity to consume or t, the tax

rate, will cause the size of the multiplier to change. Consider each in turn:-

The effect of an Increase in the mpc on the Multiplier.

If b=0.8 and t =0.25, then we have already established that the multiplier will

have a value of 2.5.

If b rises to 0.9

Multiplier = 1/(1-b(1-t)) = 1/[1-((0.9)(0.75))] = 1/(1-0.675) = 3.08

Thus, a rise in the mpc will increases the size of the multiplier and lead to a

larger impact on real income from an interest rate induced change in

investment.

Numerical Example

r A y (mult=2.5) y (mult=3.08)

5 240 600 739.210 220 550 677.6

15 200 500 616.0

By plotting these outcomes you can establish that a rise in the mpc will cause

the I-S curve to shift out and flatten.

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The Effect of an increase in the Tax Rate on the Multiplier

Suppose b = 0.8 and the tax rate rises from 25% to 30%. For t=0.25, the

multiplier, D = 2.5.

For t = 0.3

D = 1/(1-(0.8*0.7)) = 1/(1-0.56) = 1/(0.44) = 2.27

A lower multiplier results in equilibrium output being lower for any rate of

interest. The higher ye , the greater the drop in ye ,(ye), for a given reduction

in the size of the multiplier.

The change in Equilibrium Output given a Change in the Multiplier

We have seen that

1. A rise (fall) in the mpc will increase (decrease) the multiplier

2. A rise (fall) in the tax rate will decrease (increase) the multiplier

Note further that, if the multiplier rises by x% then equilibrium output for a

given r will also rise by x%.

Proof

Let ye1 = D1*A and ye2 = D2*A

Where A is the sum of the autonomous components

and D is the multiplier and D1< D2

ye2 - ye1 = (D1*A) (D2*A) = A (D1 D2)

The percentage change in y is given by

[(ye2 - ye1)/ y

e1] * 100 = [A*(D1 D2)] / [A*D1] *100

= [(D1 D2)/ D2] *100

As ye increases, an x% increase in the multiplier will result in a

correspondingly greater difference between ye1 and ye2 . i.e. the I-S curve

flattens

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Numerical Example

Consider a 20% rise in the size of the multiplier (e.g. from 2.5 to 3.0)

r A Ye0 (d = 2.5) Ye1 (d=3.0) y20.0 40 100 120 2017.5 50 125 150 25

15.0 60 150 180 30

12.5 70 175 210 35

10.0 80 200 240 40

7.5 90 225 270 45

5.0 100 250 300 50

Plotting this data in Figure 5 shows the I-S curve flattening

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y

r

I-S (low multiplier)

I-S (high multiplier)

Figure 5


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What Causes the I-S Curve to Shift

A rise in any of the autonomous components for a given rate of interest will

cause a parallel shift in the I-S schedule. Suppose the multiplier, D= 2.5 and

A rises by 20. Equilibrium output will rise as follows

r A1 y1 A2 y25 240 600 260 650

10 220 550 240 600

15 200 500 220 550

This could result from

a0 = 20 [upward shift in the consumption function]

I(r) = 20 [outward shift in the investment function]

G = 20 [increase in government spending ]

Government spending can be increased at the behest of the government. We

will discuss shifts in a0 and I(r ) later when we consider consumer and business

confidence.

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Graphical Derivation

A rise in either a0 or G0 (or any autonomous component of investment) willcauseA to rise, at any given rate of interest r. If A rises from A0 to A1,

then y = D* A will rise from y0 to y1 and from y2 to y3.

A rise in A results in a higher equilibrium y for any given rate of interest

(i.e. the I-S curve shifts out)

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y

y

ADAD = y

AD [r=10%, A=A1]

AD[r= 5%, A=A

1]

r

y0

y1

y2

5%

15%

10%

I-S [ A = A1]

AD[r= 5%, A=A

0]

AD [r=10%, A=A0]

I-S [A = A0]

y3


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Key Learning Objectives

After reading this handout you should be able to

Derive the I-S schedule, both graphically and mathematically

Understand the economics underlying it- i.e A change in the rate of interest

causes investment to change. A change in the volume of investment will alter

aggregate demand and thus, via the multiplier, equilibrium national output.

Thus, changes in r cause movements along the I-S curve.

Determine that the slope of the I-S curve depends on

1. The interest elasticity of investment . The more responsive investmentis to a change in the rate of interest (i.e. the more interest elastic investment

is) the greater the change in investment for a given change in interest rates

and thus the greater the impact on equilibrium national output for a given

r . Thus, the greater the interest elasticity of investment, the flatter the

slope of the I-S schedule

2. The size of the multiplier. The greater the size of the multiplier, the

greater the change in equilibrium output for a given change in investment.

Thus, the larger the multiplier the flatter the slope of the I-S curve

Determine the factors that cause the I-S curve to shift. The I-S curve is derived

by varying r and establishing what happens to equilibrium y. This exercise

assumes that a0, G0 (and any autonomous component of investment demand)

are held constant. If any of these autonomous components change, then the I-S

curve will shift. (e.g a rise in G0 will cause the I-S curve to shift out).

Jim Stevens

OCTOBER 2004.

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