macroeconomic theory - universidade nova de...

Macroeconomic Theory

Francesco Franco

Nova SBE

March 8, 2017

Francesco Franco Macroeconomic Theory 1/31

FluctuationsDSGE

• intertemporal macroeconomic model built frommicroeconomic foundations (like RBC project)

• introduction of nominal rigidities (and required changes inmicrofoundations of the model)

• DSGE: build a model of fluctuations that combines the two


FluctuationsDSGE

• production is given by a concave production function withonly labor

Yt = F (Lt)

• Output is for consumption as capital absence impliesinvestment to be zero

Yt = Ct


FluctuationsDSGE: consumers

The consumption-saving, labor supply and money demand:discrete time version of Ramsey-Cass-Koopmans model

U =Œÿ

t=0

—tU (Ct , 1 ≠ Lt)

where the time endowment is normalized to unity and householdsdislike labor (like leisure).



The budget constraint is

At+1

= Mt + (At + WtLt ≠ PtCt ≠ Mt) (1 + it)

where A is household wealth. Assume separable utility with CESspecification where ‡ = 1/◊ and V (L) = B

“ L

“t .


Fluctuations

When solving the decentralized problem, the household optimalconditions are the usual Euler’s (itertemporal condition) equation

UC (Ct) = —UC (Ct+1

) (1 + it)Pt

Pt+1

which using our specifiation is

C

≠◊t = — (1 + rt)C

≠◊t+1

and 1 + rt = (1 + it) PtPt+1



This equation in logs using the goods market clearing conditionand defining rt = ln(1 + r) ≠ ln (1/—) gives

yt = yt+1

≠ ‡rt

a Dynamic IS (adding investment and net exports does not changethe Y ≠ r nexus)



The first order condition wrt to money, corresponds to anintratemporal optimal allocation between consumption and leisure

V

Õ (Lt) =WtPt

U

Õ (Ct)

which using our specific functional form together with marketclearing

WtPt

= BY

◊+“≠1

t


FluctuationsDSGE: Firms

• continuum of di�erentiated goods indexed by i œ [0, 1]. Eachgood produced by one firm Yi = Li . Goods market areimperfectly competitive

• labor markets are perfectly competitive• each firm faces a downard sloping demand (monopolistic

competition)

Ci =3

PiP

4≠÷

C

• show derivation and price index


FluctuationsDSGE: Firms

How do firm change tier prices/wages?1 Time dependent rules

1 Fisher model (predetermined prices)2 Taylor model (fixed prices)3 Calvo model

2 State dependent rules1 frequency e�ect2 selection e�ect


FluctuationsSimple example: predetermined prices

Assume that a fraction ◊ of the firms can aadjust their price inperiod t and that a fraction 1 ≠ ◊ have predetermined their pricefor one period. Derive optimal price setting:

P

1t = µWt

FL (L1t)

P

2t = Et≠1

5µ

WtFL (L2t)

6

define p

1t = ln(P1t) and fit = ln(Pt) ≠ ln(Pt≠1

)


FluctuationsSimple example: predetermined prices

The aggregate price level

pt = ◊p

1t + (1 ≠ ◊)p2t

derive the Phillips curve using the optimal pricing

fit = Et≠1

fit + Ÿ3

wtpt

fL (L1t)≠1

4


FluctuationsCalvo

Mainstream uses Calvo’s model, so that a firm to choose price to

maxEtŒÿ

i(—◊)i

Rt,t+i⇧t+i

and after many manipulation you obtain a NKPC

fit = —Etfit+1

+ ⁄yt


FluctuationsDSGE:NKM

Closure is obtained with a monetary rule

it = MR(fit , yt , ...)



Finally there are shocks (think of the technology, government orinterest rate shock, and many others), so that the core model iscomposed of three equations

yt = Etyt+1

≠ ‡r

nt + u

ISt

fit = —Etfit+1

+ ⁄yt + u

PCt

it = MR(fit , yt , ...) + u

MPt



• persistent versus iid ut• NKM micro ingredients: credit-market, labor-market,

goods-market, ....• Not yet a consensus


Understanding real versus new-keynesianCrowding out of G in RBC

Consider an economy made up of a large number of identical,infinite-lived households, each of which seeks to maximize

Œÿ

t=0

—t [u (Ct) ≠ v (Nt)]

with outputYt = f (Nt)

andYt = Ct + Gt


G in the neoclassical modelBenchmark case

• perfect foresight equilibrium of a purely deterministic economy• alternative fiscal policies {Gt} financed by lump-sum• The exact timing of the path of tax collections is irrelevant in

the case of lump-sum taxes, in accordance with the standardargument for “Ricardian equivalence.



Optimality requires:v

Õ (Nt)u

Õ (Ct)=

WtPt

f

Õ (Nt) =WtPt

using the production and the resource constraint

u

Õ (Yt ≠ Gt) = v (Yt)

where v (Yt) © v

!f

≠1 (Y )"



The “multiplier” isdY

dG

=÷u

÷u + ÷v© �

where ÷u > 0 is the negative of the elasticity of u

Õ and ÷v > 0 isthe elasticity of v

Õ with respect to increases in Y. We can see thatthe “multiplier” is necessarily smaller than 1. This means thatprivate expenditure (here, entirely modeled as non-durableconsumer expenditure) is necessarily crowded out, at least partially,by government purchases.


G in the neoclassical modelMonopolistic competition & others

We have seen that by adding monopolistic competition withflexible prices we have that

Pt = MWt/f

Õ (Nt)

so thatu

Õ (Yt ≠ Gt) = Mv (Yt)

Notice that the multiplier will be the same. This same formulationis obtained with wage stickiness, labor income tax, payroll,consumption spending or firms revenues. Just need to reinterpretthe M. Of course if M(Y ) then things change.


G in the NKM modelSticky prices or wages

• the degree to which the e�ciency wedge changes depends onthe degree to which aggregate demand d�ers from what itwas expected to be when prices and wages were set.

• Equilibrium output is thus no longer determined solely bysupply-side considerations;

• we must instead consider the e�ects of government purchaseson aggregate demand.

• monetary policy fafects real activity, and so the consequencesof an increase in government purchases depend on themonetary policy response (even unchanged).


G in the NKM modelMP: unchanged path of real interest rate

• fiscal policy: {Gt} such that Gt æ G

• only temporary variations in the level of governmentpurchases.

• monetary policy rt = r

n ∆Ct+1

= Ct

but then Ct æ C © Y ≠ G and Ct = C which implies

Yt = C + Gt

a multiplier of 1. No crowding out. No stimulus. Fairly generalresult under this MP.


G in the NKM modelMP: alternative MP

Under alternative assumptions about the degree of monetaryaccommodation of the fiscal stimulus, the size of the increase inoutput will be di�erent. Could be smaller Could be larger. Heremore model dependent. Consider the nominal marginal cost

t =Wt

f

Õ (Nt)

t = Ptv (Yt)

u

Õ (Yt ≠ Gt)

in log linear form

Ât = pt + ÷v yt + ÷u (yt ≠ gt)

where gt = Gt≠GY


Optimal Price condition

The Calvo optimal pricing condition:

p

út = µ + (1 ≠ —◊)

Œÿ

k=0

(—◊)kEt{Ât+k}

with the price level dynamics you get the usual (go back to notesmct = Ât ≠ pt

fit = —Et {fit+1

} + ⁄ \(Ât ≠ pt)

fit = ⁄Œÿ

i=0

—iEt [yt+i ≠ �gt ]

find the monetary policy to maintain a constant real interest ratein the case of an arbitrary path or government purchases close tosteady state.


G in the NKM modelMP: Taylor Rule

it = r

n + „fifit + „y (yt ≠ �gt)

and

yt ≠ gt = Et{yt+1

≠ gt+1

} ≠ 1‡(it ≠ Et{fit+1

} ≠ r

n)

Conjecture a solution (UC): yt = “y gt , fit = “figt , it = r

n + “i gt


G in the NKM modelMP: Taylor Rule

You are going to find� < “y < 1

with a normal Taylor rule

it = r

n + „fifit + „y yt

you find that“y < �


G in the NKM modelMP: no response (zero-bound)

This is a case in which it is plausible to assume not merely that thereal interest rate does not rise in response to fiscal stimulus, butthat the nominal rate does not rise; this will actually be associatedwith a decrease in the real rate of interest, to the extent that thefiscal stimulus is associated with increased inflation expectations.


G in the NKM modelMP: no response (zero-bound)

Suppose the relevant interest rate for C-S decision is di�erent frompolicy rate it

i

ct = it +�t

yt ≠ gt = Et{yt+1

≠ gt+1

} ≠ 1‡(it ≠ Et{fit+1

} ≠ r

nt )

where r

nt = r

n ≠�t . If �t is large the natural rate of interest

can become negative. Then a lower bound on it makes it

impossible to achieve zero output gap and zero inflation. In this

case the multiplier can be larger than 1. We will study it in

details in the second part.


G and WelfarePublic good can provide utility

Œÿ

t=0

—t [u (Ct) + g (Gt) ≠ v (Nt)]

in this caseg

Õ (Gt) = u

Õ (Yt ≠ Gt)

government purchases should be undertaken if and only if theyhave a marginal utility as high as that associated with additionalprivate expenditure.


Readings

(*) David Romer. Advanced Macroeconomics, 4th EditionChapter 7.


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