Fletcher School, Tufts University - WordPress.com · 2018-02-21 · Furthermore it is assumed that b+c

Fletcher School, Tufts University

Exercise 2 Short Run Output and Interest Rate Determination in an IS-LM Model

Prof. George Alogoskoufis

George Alogoskoufis, Macroeconomics

The IS LM ModelConsider the following short run keynesian model of a closed economy:

(1)

(2)

(3)

(4)

(5)

(6)

(7)

D denotes real aggregate demand, Y denotes real aggregate output and income, YD denotes real aggregate disposable income, C denotes real aggregate private consumption expenditure, I denotes real aggregate gross investment expenditure, G denotes real aggregate government purchases and T denotes real aggregate taxes, net of government transfers. A bar above a letter denotes the autonomous (exogenous) component of the corresponding variable, assumed positive and c denotes the marginal propensity to consume, assumed positive and less than unity, b denotes the marginal propensity to invest assumed positive and less than unity, and -d denotes the responsiveness of investment to the interest rate, assumed negative. Furthermore it is assumed that b+c<1.

Assuming that prices are fixed in the short run, continuous equilibrium in the output market implies that output adjusts to ensure that Y=D. Equilibrium in the money market implies that the money supply responds endogenously to changes in the interest rate of the central bank, in order to ensure that money demand is equal to the money supply.

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D = C + I +G

YD = Y −T

C = C + cYDI = I + bY − di

G = G

T = T

i = i


A. Discuss the structure of the model distinguishing between endogenous and exogenous variables, identities, behavioral equations and equilibrium conditions.

The model suggests that because prices are fixed, output Y is determined by aggregate demand, which consists of aggregate private consumption C, aggregate investment I and aggregate government purchases G.

Hence, aggregate output (income) Y is an endogenous variable, determined by the model.

Disposable income is determined by an identity, which is the difference between aggregate output and taxes (net of transfers). Since aggregate output is an endogenous variable, disposable income is also an endogenous variable.

Aggregate consumption is a positive function of disposable income and is also an endogenous variable.

Investment is a positive function of output and a negative function of the nominal interest rate and is also an endogenous variable.

Government expenditure and taxes (net of transfers) and the interest rate are determined outside the model, and are hence exogenous variables.

The definition of aggregate demand D and disposable income YD is through the first two equations which are identities.

The third and fourth equations, the consumption and investment functions, are behavioral equations, as they are meant to describe the behavior of consumers and investors. They consist of an autonomous component, and behavioral parameters, such as, c, the marginal propensity to consume, b, the marginal propensity to invest and d, the responsiveness of investment to the nominal interest rate.

The other three equations define the exogenous variables as autonomous constants.

The equality between aggregate demand and aggregate output and income is an equilibrium condition, as is the equality of money demand and the money supply in the money market.

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B. Using simple algebra, derive equilibrium output as a function of the autonomous (exogenous) components of spending and the marginal propensity to consume. Explain your findings.

Substituting equations (1) to (6) in the equilibrium condition Y=D, we get that,

or that

Solving for output, we get the IS curve as,

The LM curve is horizontal in this model, because of the fixed interest rate target of the central bank, and is given by,

Equilibrium output is a multiple 1/(1-c-b) of the autonomous components of aggregate demand. This multiple is the multiplier. The effect of every autonomous component of aggregate demand is multiplied through the increase in the disposable income of consumers, and the sales of firms, which cause second round effects on output and income through consumption and investment, which in turn cause third round effects, and so on.

Thus, equilibrium output in the short run depends positively on the components of autonomous spending and negatively on autonomous taxes net of transfers and the interest rate target of the central bank.

4

Y = D = C + I +G = C + I +G − cT + (c + b)Y − di

Y = C + I +G − cT + (c + b)Y − di

Y = 11− c − b

C + I +G − cT − di( )

i = i


C. Derive equilibrium output and the interest rate using a simple IS LM diagram. Explain your findings.

5

Real Output Y

Nom

inal

Inte

rest

Rat

e, i

IS (T0 ,G0)

E0

Y0

i0

IS (T0 ,G1)

E1

Y1

G1 > G0

LM( i0 )


D. Describe the properties of short run macroeconomic equilibrium in words.

Production in this model is always equal to aggregate demand. Any autonomous component of aggregate demand leads to a corresponding increase in output and income. The increase in income leads to an immediate increase in aggregate consumption and investment, a further increase in income, a further increase in consumption and investment, a further increase in income, and so on.

The process only stops when aggregate output and income is such that it can satisfy both autonomous demand and consumption demand.

The interest rate affects aggregate investment negatively. Therefore, an increase in the interest rate reduces investment and aggregate demand and equilibrium income, and vice versa.

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E. Assume that government purchases rise by $1bn. What will be the effects on equilibrium output and the nominal interest rate? Assume that taxes net of transfers are reduced by $1bn. What will be the effects on equilibrium output and the nominal interest rate? What will be the effects on equilibrium output of a simultaneous increase in government purchases and taxes (net of transfers) by $1bn.

The effects on output will be determined by the relevant multiplier. There will be no effect on the nominal interest if the central bank holds the nominal interest rate at its initial value.

For an increase in government purchases by $1bn, equilibrium output will increase by 1/(1-c-b) $ billions.

For a reduction in taxes net of transfers by $1bn, equilibrium output will increase by c/(1-c-b) $ billions.

For a simultaneous increase in government purchases and taxes (net of transfers) by $1bn equilibrium output will increase by (1-c)/(1-c-b) billions.

For c=0.60, b=0.10, the government purchases multiplier is 1/0.3=3.33. The tax multiplier is 0.6/0.3=2. The balanced budget multiplier is equal to 0.4/0.3=1.333.

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George Alogoskoufis, Macroeconomics, 2017-18

F. The Effects of an Increase in Government Purchases and the Method of Financing

8

Output Y

Agg

rega

te D

eman

d D

D=C0+I0+G0-cT0+cY

45o

YE

D0

Y=D

D1

YE'

E

E'

D=C0+I0+G1-cT0+cY

YE''

E''

D=C0+I0+G1-cT1+cY

Fiscal Expansion through Borrowing

Balanced Budget Fiscal Expansion


G. Assume that the central bank reduces its nominal interest rate target by 1% (0.01). What will be the effect on equilibrium output and investment, assuming c=0.60, b=0.10, d=100 ?

The effect of a reduction in the interest rate by 0.01 is given by,

(d/(1-c-b))x0.01=333.33x0.01=3.333

This is because a reduction in the interest rate by 0.01 increases investment by 1bn, which multiplied by the multiplier of 3.333 results in a rise in output by 3.333bn.

9


H. Assume that autonomous private consumption is equal to $17bn, autonomous investment is equal to $10bn, autonomous government purchases are equal to $20bn, and that there is a balanced government budget. The interest rate target of the central bank is equal to 0.05 (5%). What is the value of equilibrium output, consumption and investment if the marginal propensity to consume c is equal to 0.60, the marginal propensity to invest b is equal to 0.10, and the responsiveness of investment to the nominal interest rate d is equal to 100? What adjustment to the interest rate target of the central bank is required in order to increase aggregate demand to the full employment level of real output YF=110?

Substituting the values given, equilibrium output is given by,

Y= (1/0.3) x (17 + 10 + 20 - 0.6x20 -100x0.05) = 30/0.3 = 100

Equilibrium consumption is given by,

C = 17 + 0.6x( 100 -20) = 65

Equilibrium investment is given by,

I=10+0.10x100-100x0.05=15

To raise aggregate demand and output from 100 to 110 billion, i.e by 10 billion, the interest rate change must be equal to,

∆i=0.3x10/100=0.03

Hence the central bank ought to reduce its interest rate target by 3 percentage points, i.e from 0.05 to 0.02.

10

Y = 11− c − b

C + I +G − cT − di( )


I. How are your answers to questions B-H modified if the central bank followed an interest rate policy rule of the form, i=i0+ e(Y-YF), where e>0 denotes the responsiveness of the interest rate target of the central bank to deviations of current real output from full employment real output YF. Where it is required assume that e=0.0005 (0.05%).

Modified Answer to B

To determine equilibrium output and the nominal interest rate we must now solve the full IS and LM model. The IS relation is given by,

The LM relation is now given by the policy rule of the central bank,

The LM curve is no longer horizontal, but upward sloping, as the central bank automatically increases the nominal interest rate when output increases relative to full employment output.

IS and LM must be satisfied simultaneously. Using the LM relation to substitute for the nominal interest rate in the IS relation, equilibrium output is determined by,

Equilibrium output in the short run depends positively on the components of autonomous spending and negatively on autonomous taxes net of transfers and the interest rate target of the central bank at full employment. It also depends positively on full employment output, as an increase in full employment output results in a fall in the nominal interest rate.

The multiplier is now smaller than in the case where the central bank does not adjust the interest rate in response to deviations of current output from full employment output, as any increase in output brings about an increase in the nominal interest rate, which reduces investment and hence aggregate demand.

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Y = C + I +G − cT + (c + b)Y − di

i = i + e(Y −YF )

Y = 11− c − b+ de

C + I +G − cT − di + deYF( )


Modified Answer to Question C when i=i0+ e(Y-YF)

12

Real Output Y

Nom

inal

Inte

rest

Rat

e, i

IS (T0 ,G0)

i= i0+e(Y-YF)

E

YE

i0

F

iE

YF


Modified Answer to Question D when i=i0+ e(Y-YF)

Production in this model is always equal to aggregate demand. Any autonomous component of aggregate demand leads to a corresponding increase in output and income. The increase in income leads to an immediate increase in aggregate consumption and investment, a further increase in income, a further increase in consumption and investment, a further increase in income, and so on.

The process only stops when aggregate output and income is such that it can satisfy both autonomous demand and consumption demand.

The interest rate affects aggregate investment negatively. Therefore, an increase in the interest rate reduces investment and aggregate demand and equilibrium income, and vice versa.

The central bank has an interest rate target equal to i0, when current output is equal to full employment output. However, it automatically adjusts the interest rate when current output deviates from full employment output. When output is lower than full employment, the interest rate is lower, and vice versa. Thus, in short run equilibrium, when output is below full employment, the current interest rate is lower than the full employment interest rate target, and vice versa.

However, even in this case, there is no guarantee that current output will be equal to full employment output. This will only be the case if the full employment interest rate target of the central bank is low enough.

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Modified Answer to Question E when i=i0+ e(Y-YF)

The effects on output will be determined by the relevant multiplier. As in this case changes in aggregate demand and output induce changes in the nominal interest, the central bank modifies the nominal interest rate when there are changes in output and the multiplier is smaller than in the previous case.

For an increase in government purchases by $1bn, equilibrium output will increase by 1/(1-c-b+de) $ billions.

For a reduction in taxes net of transfers by $1bn, equilibrium output will increase by c/(1-c-b+de) $ billions.

For a simultaneous increase in government purchases and taxes (net of transfers) by $1bn equilibrium output will increase by (1-c)/(1-c-b+de) billions.

For c=0.60, b=0.10, d=100, e=0.0005, the government purchases multiplier is 1/0.35=2.86. The tax multiplier is 0.6/0.35=1.71. The balanced budget multiplier is equal to 0.4/0.35=1.14.

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Modified Answer to Question F when i=i0+ e(Y-YF)

15

Real Output Y

Nom

inal

Inte

rest

Rat

e, i

IS (T0 ,G0)

i= i0+e(Y-YF)

E

YE

iE''

E'

iE

YE'

IS (T0 ,G1)

IS (T1 ,G1)

G1>G0, , T1>T0 , T1-T0 =G1-G0

E''

YE''

iE'


Modified Answer to Question G when i=i0+ e(Y-YF)

The effect of a reduction in the interest rate by 0.01 is given by,

(d/(1-c-b+de))x0.01=285.71x0.01=2.857

This is because a reduction in the interest rate by 0.01 increases investment by 1bn, which multiplied by the multiplier of 2.857 results in a rise in output by 2.857bn.

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Modified Answer to Question H when i=i0+ e(Y-YF)

Substituting the values given, equilibrium output is given by,

Y= (1/0.35) x (17 + 10 + 20 - 0.6x20 -100x0.05+0.05x110) = 35.5/0.35 = 101.43

The equilibrium nominal interest rate is given by,

i=0.05+0.0005x(101.43-110)=0.0457

Equilibrium consumption is given by,

C = 17 + 0.6x( 101.43 -20) = 65.86

Equilibrium investment is given by,

I=10+0.10x101.43-100x0.0457=15.573

To raise aggregate demand and output from 101.43 to 110 billion, i.e by 8.67 billion, the interest rate target change must be equal to,

∆i=0.35x8.67/100=0.03

Hence the central bank ought to reduce its full employment interest rate target by 3 percentage points, i.e from 0.05 to 0.02.

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Y = 11− c − b+ de

C + I +G − cT − di + deYF( )

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Fletcher School, Tufts University - WordPress.com · 2018-02-21 · Furthermore it is assumed that b+c