Lecture 8

Short Run Output Determination:

The IS/LM/AS Framework

Mark Gertler

NYU

Intermediate Macro Theory

Spring 2015

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Motivation

Competitive equilbrium neoclassical model

Output always at "full employment"

Money only aects nominal variables

Di cult for model to explain:

Recessions and Depressions

The eect of monetary policy on the real economy

The eects of scal policy

1

A Model of the Short Run: Preliminaries

Start with the competitive equilibrium model and introduce 3 frictions: (i) Money

(ii) Imperfect competition

(iii) Imperfect price adjustment

(i) permits introducing nominal variables and an analysis of monetary policy (iii) permits analyzing price setting behavior by rms as a prelude to introducingimperfect price adjustment

Cant model price adjustment with perfect competition since rms take pricesas given

(i), (ii) and (iii) imply: output can be below "full employment"

monetary policy can aect the real economy2

Model Ingredients

Consumption goods only. Later we add investment and government spending.

Eventually introduce nancial market frictions.

The model includes households, rms and a central bank:

The representative household consumes a nal good Ct, supplies labor Nt, holdsreal money balances Mt=Pt, saves in the form of private bonds Bt (which, inequilibrium will be in zero net supply, since everyone is the same).

Firms are monopolistic competitors and each produce a dierentiated productYt (f) using labor Nt (f). These rms set nominal prices Pt (f). f denotes rmf:

The central bank controls the money supply Mt:

3

Resource Constraints and Money Supply

Resource constraintYt = Ct (1)

Monetary policy: Central bank sets Mt = M t:Mt =M t (2)

We next derive aggregate consumption demand and money demand by households Doing so allows constructing IS/LM model to determine output and interestrates given nominal prices

Output may be below full employment

We then derive the aggregate supply side from labor demand and supply Permits deriving the gap between output and full employment output

Permits analysis of price adjustment.

4

Household Decision Problem

Goal: Derive (i) consumption demand (ii) money demand and (iii) labor supply

Maximization problem problem: Choose Ct; Bt;Mt and Nt to solve

max1

1 C1t + am

1

1 (Mt

Pt)1 an

1 + nN1+nt +

1

1 C1t+1 (3)

Ct +Mt

Pt+Bt

Pt=Wt

PtNt (4)

Ct+1 =Pt

Pt+1

Mt

Pt+ (1 + it)

Pt

Pt+1

Bt

Pt(5)

where ; n; an; am > 0.

Money in the utility function captures convenience yield.5

Unconstrained Maximzation Problem

Choose (BtPt ;MtPt; Nt) to solve

maxf 11 (

Wt

PtNt Mt

Pt BtPt)1 + am

1

1 (Mt

Pt)1 an

1 + nN1+nt

+1

1 (Pt

Pt+1

Mt

Pt+ (1 + it)

Pt

Pt+1

Bt

Pt)1g

First order condition for labor supply

Wt

PtCt = anN

nt (6)

6

Consumption and Money Demand

First order condition for BtPt :

Ct = (1 + it)

Pt

Pt+1C

t+1 (7)

First order condition for MtPt

Ct =

Pt

Pt+1C

t+1 + am(Mt=Pt)

(8)

Absent the non-pecuniary return am(Mt=Pt); the household would not holdmoney so long as it > 0 :

If am = 0 and it = 0, bonds always dominate.

7

Consumption and Money Demand (con0t)

Household rst order condition implies relation for consumption demand

Ct = (1 + it)

Pt

Pt+1C

t+1 !

(Ct )

1 = [(1 + it)Pt

Pt+1C

t+1]

1 !

Ct =1

[(1 + it)PtPt+1

]Ct+1 (9)

Consumption demand depends inversely on (1 + it) PtPt+1 and positively on Ct+1 Intuition comes from permanent income hypothesis that we studied earlier

A rise in interest rates induces an increase in saving and a decline in Ct Desire for consumption smoothing: Ct+1 "! Ct "

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Consumption and Money Demand (con0t)

A relation for money demand follows from the rst order conditions for BtPt andMtPt:

Ct = (1 + it)

Pt

Pt+1C

t+1

Ct =

Pt

Pt+1C

t+1 + am(Mt=Pt)

Combining yields a relation for money demandMt

Pt= (

1

it+ 1)amCt

Money demand depends inversely on its opportunity cost it:

Depends positively on Ct

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Aggregate Demand: IS Curve

Aggregate DemandYt = Ct

Ct =1

[(1 + it)PtPt+1

]Ct+1

Combining yields an IS curve

Yt =1

[(1 + it)PtPt+1

]Yt+1

Given PtPt+1 and Yt+1, IS curve is downward sloping in (Yt; it) space.

An increase in it increases the real interest rate, which reduces spending.

Yt+1 " shifts IS curve out: PtPt+1 " shifts it in.

10

it

Yt

Yt =1(

(1+i)Pt

Pt+1

) Yt+1

I

S

Figure 1: IS Curve.

1

it

Yt

I

S

I

S

Figure 2: Increase in Yt+1

2

it

Yt

I

S

I

S

Figure 3: Increase in PtPt+1

.

3

Monetary Sector: LM Curve

Monetary SectorMt

Pt= (

1

it+ 1)amYt (10)

Mt =M t

LM Curve:M t

Pt= (

1

it+ 1)amYt

Given Pt, upward sloping in (Yt; it) space

Yt "! money demand " ! it " to reduce money demand Mt "! LM Curve shifts out: Rise in real money suppy! it down to raise moneydemand

Pt #! M tPt "! LM curve shifts out.11

it

YtL

M

MPt=(1it+ 1

)amYt

Figure 4: LM Curve

4

it

YtL

M

L

M

Figure 5: Increase in M t

5

it

YtL

M

L

M

Figure 6: Increase in Pt

6

Fixed Price IS/LM Model

Sticky price assumption:Pt+i = P t+i; i = 0; 1

! IS/LM jointly determines (Yt; it)IS:

Yt =1

[(1 + it)P tP t+1

]Yt+1

LM:

M t

P t= (

1

it+ 1)amYt

12

it

YtL

MI

S

Y et

iet

Figure 7: IS/LM Model

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Some Comparative Statics

Fixed Price IS/LM:

Yt =1

[(1 + it)P tP t+1

]Yt+1

M t

P t= (

1

it+ 1)amYt

Yt+1 " (rise in optimism)! IS curve shifts out! Yt ", it " M t " (expansionary monetary policy)! LM curve shifts out ! it # Yt " :

P tP t+1

" Increase in expected deation ! curve shifts down Yt #, it # If zero lower bound on it binds, drop in Yt increases.

Explains central bank aversion to deation.

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it

YtL

MI

S

Y et

iet

Figure 8: Impact of Increase in Yt+1

8

it

YtL

MI

S

Y et

iet

Figure 9: Increase in M

9

it

Yt

L

MI

S

Y et

iet

Figure 10: Increase in PtPt+1

with zero lower bound on i

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Flexible Price IS/LM

Suppose Pt adjusts so that Yt = Y t (full employment output)/ For now take Y t as given Shortly we introduce supply side to determine Y t :

Flexible price IS/LM model:

Y t =1

[(1 + it)PtPt+1

]Y t+1

M t

Pt= (

1

it+ 1)amY

t

Given expected deation, ex price IS/LM determines (Pt; it): Simple Quantity Theory Holds: M t "! Pt " proportionately: No eect on Y t :

14

it

YtL

MI

S

Y t

it

Figure 11: Flexible Price IS/LM model

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Aggregate Supply

Supply side needed for: Deriving full employment output Y t Describing how prices adjust over time

We rst derive aggregate labor supply curve Relates real wage to aggregate employment

We then derive rm labor demand labor demand Flexible price case: rms choose price, output and employment each period

Helps determine full employment output Fixed price case: Firms choose output and employment to meet demand

So long as it is protable

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Aggregate Labor Supply

Firm f uses the following technology to produce output Yt (f) with employmentNt(f) :

Yt (f) = AtNt (f)

Aggregating across rmsYt = AtNt (11)

Use (11) and the resource constraint (1) to eliminate Yt and Ct in the householdlabor supply curve:

Wt

Pt= anN

nt C

t (12)

= anA

tN

n+t

! Real wages vary positively with employment.

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wP

N

N s

Figure 12: Aggregate Labor Supply

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Aggregate Labor Demand

Monopolistically competitive rm chooses (Pt(f); Yt(f); Nt(f)) to solve

maxt =Pt (f)

PtYt (f)Wt

PtNt (f)

subject to (i) demand curve and (ii) production function:

Yt (f) =

"Pt (f)

Pt

#"Yt

Yt (f) = AtNt (f)

Two cases: Flex Price (long run); Fix Price (short run). Flex Price: Choose (Pt(f); Yt(f); Nt(f)) to maximizes prots

Fix Price: Choose (Yt(f); Nt(f)) to meet demand, so long as protable

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Aggregate Labor Demand: Flexible Price Cases

Use constraints to eliminate Yt(f); Nt(f) ! unconstrained problem:

maxPt(f)

t =Pt (f)

Pt

"Pt (f)

Pt

#"Yt Wt

Pt

Pt(f)Pt

"Yt

At

where the rm takes WtPt ; At; and aggregate output Yt as given.

First order necessary condition (marginal revenue = marginal cost)

(1 ")"Pt (f)

Pt

#"Yt "

WtPt

At

"Pt (f)

Pt

#"1Yt = 0

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Aggregate Labor Demand: Flexible Prices (cont)

price markup over marginal cost ! Rearranging rst order condition:

Pt (f)

Pt= (1 + )

WtPt

At

1 + =1

1 1=" Firm sets price Pt(f)Pt as a markup over marginal case Markup varies inversely with demand elasticity ":

As "!1 (perfect competition), ! 0 : i.e. price = marginal cost.

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Flexible Price Equilbrium Employment

Pt (f)

Pt= (1 + )

WtPt

At

Since all rms are identical: Pt (f) = Pt !

1 = (1 + )

WtPt

At!

At = (1 + )Wt

Pt

! Marginal product of labor At = markup over real wage WtPt Use aggregate labor supply curve to eliminate WtPt :

At = (1 + )anA

t (N

t )

n+

Nt exible price equilibrium employment (i.e. full employment).

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Flexible Price Equilbrium Employment and Output

Nt and Y t determined by: Labor market equilibrium

At = (1 + )anA

t (N

t )

n+

Production function

Y t = AtNt Eliminating Nt

At = (1 + )anAnt (Y

t )

n+ !

Y t = (1

(1 + )an)

1

n+A

1+n

n+t

= 0! Y t ; Nt = competitive equilbrium values Y ot ; Not > 0! Y t ; Nt < Y ot ; Not

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WP

N

Y

N

NoN

At

N

Y

NS

AwP

Aw/P = 1 +

Figure 13: Labor Market Equilibrium: Flexible Prices. N is ht eflexible price equilibriumamount of labor, while N o is ht ecompetitive equilibrium amount.

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Aggregate Labor Demand: Fixed Price Case

With xed prices the rm produces to meet demand so long as it is protable: Protable so long as markup t(f) = 0 (i.e. price marginal cost).

Since Pt(f) xed at P t(f); t(f) varies

P t(f)

Pt= (1 + t(f))

WtPt

At;

Symmetric equilibrium: All rms charge P t(f)! P t(f) = Pt !

1 = (1 + t)

WtPt

At!

At = (1 + t)Wt

Pt

t varies inversely with WtPt : since price xed, markup falls as marginal cost rises.

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Aggregate Demand and the Markup: Fixed Price Case

Given Yt: Nt and t determined by Labor market equilibrium

At = (1 + t)anA

t (Nt)

n+ (13)

Production function

Yt = AtNt (14)

(13) and (14) !inverse relation between Yt and t (countercyclical markup):1

1 + t= an

Y

n+t

A1+nt

(15)

(As we show later), ination varies inversely with t Firms raise prices when markups low, and vice-versa

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Fixed Price IS/LM/AS Model

IS Curve:Yt =

1

[(1 + it)P tP t+1

]Yt+1

LM Curve:

M

Pt= am

1 1

(1 + it)

!1Yt

AS Curve:

1

(1 + t)= an

Y

n+t

A1+nt

IS/LM determines (Yt; it): Given Yt; AS determines t t then aects ination, as we show later.

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Flexible Price IS/LM/AS Model

[(1 + it )PtPt+1

] exible price (i.e. natural) real rate of interest

it exible price nominal rate (given PtPt+1) IS Curve:

Y t =1

f[(1 + it ) PtPt+1]gY t+1

LM Curve:

M

Pt= am

1 1

(1 + it )

!1Yt

AS Curve:

1

(1 + )= an

Yn+t

A1+nt

25

WP

N

Y

N

i

YY e

ie

NoNNe

At

NNe

Y

Ye

NS

Y

LMIS

Figure 14: Price Rigidity Case

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General IS/LM/AS Model

IS Curve:Yt =

1

f[(1 + it) PtPt+1]gYt+1

LM Curve:

M

Pt= am

1 1

(1 + it)

!1Yt

AS Curve:

1

(1 + t)= an

Y

+nt

A1+nt

Two polar cases: Fix Price: Pt xed ! IS/LM determines Yt; it and AS determines t Flex Price: xed ! AS determines Y t and IS/LM determines Pt; it

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Fixed Price IS/LM/AS with Output Gap

IS Curve:Yt =

1

[(1 + it)P tP t+1

]Yt+1

LM Curve:

M

Pt= am

1 1

(1 + it)

!1Yt

AS Curve (after combining AS curves for x and ex price models):

1 +

(1 + t)= (

Y tYt)+n

"Markup gap" 1+(1+t)

varies inversely with output gap

As we show later, ination varies positively with 1+(1+t)

and hence with YtYt

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Some Comparative Statics

Yt+1 "! Yt "; it "; YtYt#; t #

M t "! Yt "; it #; YtYt"; t #

Y t # (supply shock)!Y tYt#; t #

Later we will show that ination moves inversely with t and thus inversely withY tYt

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