real business cycles and classical monetary …rg king, boston university ec 541 session 2 1 real...

RG King, Boston University EC 541 Session 2 1

Real business cyclesand classical monetary economics

text reading: Gali, Chapter 2

slides based on those of Jordi Gali

document: EC541S2.tex


Questions

1. How does real activity fluctuate over time if all individuals areoptimizing, markets clear, and prices are fully flexible?

2. With conditions of monetary equilibrium, what is the behavior ofthe price level and the nominal interest rate?

Notation: upper case letters generally denote economic variables, whilethe corresponding lower case variable is a logarithm. (M,m=log(m)).This is one of two common notations in macro with the other involvingupper case letters for nominal variables and lower case variables forreal ones.


Assumptions in Gali chapter

• Perfect competition in goods, labor markets, and asset markets• Flexible commodity prices, wages and asset prices• No capital accumulation (for tractability)• No fiscal sector (for tractability)• Closed economy (for tractability)• Only asset is a nominal bond (for simplicity)

Outline of analysis

• The problem of households and firms• Equilibrium: monetary policy neutrality•Monetary policy and the determination of nominal variables


Households

Representative household solves

maxE0

∞∑t=0

βtU (Ct, Nt) (1)

subject toPtCt + QtBt ≤ Bt−1 + WtNt + Dt (2)

for t = 0, 1, 2, ... plus a solvency constraint (to be discussed further inclass).


Optimality conditions

−Un,tUc,t

=Wt

Pt(3)

Qt = β Et

{Uc,t+1Uc,t

PtPt+1

}(4)


Derivation and Interpretation

• Households adjust labor to the point where the real utility rewardto an additional unit of N , which is (W/P )Uc, is equated to thereal cost, which is Un

Utility trade-off: Uc,tdCt + Un,tdNt = 0

Budget trade-off: PtdCt = WtdNt


• Households adjust current saving to the point where the utility costof having a little bit less consumption today, Uc,t, is equated to theexpected utility from consuming a little bit more tomorrow

βEt{Uc,t+1PtPt+1

1

Qt}}

• Starting with utility consequences, if one consumes one unit lessthen one loses Uc,tdCt and tomorrow one gains βEt(Uc,t+1dCt+1)

• The budget constraint implies1 ∗ Pt+1dCt+1 = −PtQtdCt. Note the

bond is a pure discount bond, with a face value of one. Its pricetoday is Qt and the quantity purchased today is Pt

Qt


Convenient specification of utility:

U(Ct, Nt) =C1−σt

1− σ −N 1+ϕt

1 + ϕ

implied optimality conditions:Wt

Pt= Cσ

t Nϕt (5)

Qt = βEt

{(Ct+1Ct

)−σPtPt+1

}(6)

Interpretation of parameters:

1/ϕ : labor supply elasticity; 1/σ intertemporal subsitution elasticity


Log-linear versions

Exact: wt − pt = σct + ϕnt (7)

Approximate: ct = Et{ct+1} −1

σ(it − Et{πt+1} − ρ) (8)

where πt+1 is the inflation rate log(Pt+1/Pt) and where it ≡ − logQt andρ ≡ − log β (add interpretation). Note it is the continuously com-pounded nominal interest rate since the price of bond at maturity isone.


Low frequency implications

Looking across countries with different inflation rates as we did last time.

Perfect foresight steady state (with growth at rate γ):

i = π + ρ + σγ

which is the one-for-one "Fisher" relation between i and π.

Another implication is that faster growing economies should have a highera real rate

r ≡ i− π = ρ + σγ


A time series implication (Hall)

ct+1 = ct +1

σ(it − Et{πt+1} − ρ)

+[ct+1 − Et{ct+1}]

Note that the [ ] term is an expectation error. As Hall stresses, underrational expectations, this error should be uncorrelated with informa-tion used for forecasting. One of our labs will explore versions of theHall test using EVIEWS.


Ad-hoc loglinear money demand

mt − pt = yt − ηit

with η controlling effect of interest rate (expected inflation) on real bal-ances.

Reminiscent of "Cagan" money demand function discussed last time.

Note: some (perhaps most) economists would criticize this approach (dis-cuss)


Firms

Representative firm with technology

Yt = AtN1−αt (9)

(might stand in for CRTS production function with capital fixed over du-ration of analysis).

Profit maximization:max PtYt −WtNt

subject to (9), taking the price and wage as given (perfect competition)

(note that there is, however, no capital in this model:labor is the onlyfactor of production)


Optimality condition:Wt

Pt= (1− α)AtN

−αt

In log-linear terms

wt − pt = at − αnt + log(1− α) (10)

Sometimes described as "implicit labor demand" condition. Using thislanguage, (7) would be viewed as labor supply condition.


Equilibrium

Goods market clearingyt = ct (11)

Aggregate production relation:

yt = at + (1− α)nt


Labor market clearing

σct + ϕnt = at − αnt + log(1− α)

Asset market clearing:Bt = 0

yt = Et{yt+1} −1

σ(it − Et{πt+1} − ρ)

If we added a real bonds with a face value of one good, the equilibriumcondition Br

t = 0, since all the bonds are internal to the economy,implies

Qrt = βEt

{(Ct+1Ct

)−σ}(12)

where Qr is the real bond price at t.


Implied equilibrium values for real variables

nt = ψnaat + ϑn

yt = ψyaat + ϑyrt ≡ it − Et{πt+1} = ρ + σ Et{∆yt+1} = ρ + σψyaEt{∆at+1}ωt ≡ wt − pt = yt − nt + log(1− α) = ψωaat + log(1− α)

where ψna ≡ 1−σσ+ϕ+α(1−σ) ; ϑn ≡ log(1−α)

σ+ϕ+α(1−σ) ; ψya ≡ 1+ϕσ+ϕ+α(1−σ)

ϑy ≡ (1− α)ϑn ; ψωa ≡ σ+ϕσ+ϕ+α(1−σ)

=⇒neutrality: real variables determined independently of monetary policy(money, inflation, and the nominal interest rate do not enter no inthese equations)

=⇒ specification ofmonetary policy needed to determine nominal variables=⇒ optimal monetary policy: to be studied, but not yet clear.


Monetary Policy and Price Level Determination

Example I: An Exogenous Path for the Nominal Interest Rate

exogenous stationary process {it} with mean ρ

Et{πt+1} = it − rtwhere {rt} is determined above.Any path for inflation that satisfies

πt+1 = it − rt + ξt+1

where Et{ξt+1} = 0 for all t is consistent with equilibrium: inflation canbe subject to random shocks that are "not fundamental".


Since πt+1 = pt+1 − pt, any path for the price level which satisfiespt+1 = pt + it − rt + ξt+1

where Et{ξt+1} = 0 for all t is consistent with equilibrium.

In fact, even if there are no random shocks, the initial price level p0 cannotbe determined: only the change in the price level (inflation) is pinneddown by the equilibrium conditions.

Implied path for the money supply:

mt = pt + yt − ηitand hence it inherits the indeterminacy of pt.


Example II: A Simple Interest Rate Rule and the Taylor principle

it = ρ + φππtCombined with the definition of the real rate (and short-hand r̂t = rt−ρ):

φππt = Et{πt+1} + r̂t

If φπ > 1, unique stationary solution:

πt =

∞∑k=0

φ−(k+1)π Et{r̂t+k}

If φπ < 1, any process πt satisfying πt+1 = φππt − r̂t + ξt+1, whereEt{ξt+1} = 0 for all t is consistent with a stationary equilibrium.

price level indeterminacy


Example III: An Exogenous Path for the Money Supply {mt}

Combining money demand and Fisherian equations:

pt =

(η

1 + η

)Et{pt+1} +

(1

1 + η

)mt + ut

where ut ≡ (1 + η)−1(ηrt − yt) is pinned down by the real equilibriumAssuming η > 0 and solving forward we obtain:

pt =1

1 + η

∞∑k=0

(η

1 + η

)kEt {mt+k} + u′t

where u′t ≡∑∞

k=0

(η1+η

)kEt {ut+k}.


In terms of expected future money growth rates

pt = mt +

∞∑k=1

(η

1 + η

)kEt {∆mt+k} + u

′

t (13)


Implied nominal interest rate:

it = η−1(yt − (mt − pt))

= η−1∞∑k=1

(η

1 + η

)kEt {∆mt+k} + vt

where vt ≡ η−1(ut + yt) is unaffected by policy (due to neutrality).

Nominal interest rate depends on expected future money growth becauseinflation depends on current and expected future money growth.


Example: Serially correlated money growth

∆mt = ρm∆mt−1 + εmt

Assume no real shocks (yt = 0).

Price response:pt = mt +

ηρm1 + η(1− ρm)

∆mt

=⇒ large price response if ρm > 0 (Cagan)

Nominal interest rate response:

it =ρm

1 + η(1− ρm)∆mt

=⇒ no liquidity effect (if ρm ≥ 0)


A Model with Money in the Utility Function

Preferences

E0

∞∑t=0

βtU

(Ct,

Mt

Pt, Nt

)Budget constraint

PtCt + QtBt + Mt ≤ Bt−1 + Mt−1 + WtNt + Dt

Letting At ≡ Bt−1 + Mt−1 :PtCt + QtAt+1 + (1−Qt)Mt ≤ At + WtNt + Dt

(1−Qt) = 1− exp{−it} ' it; opportunity cost of holding money


Optimality Conditions

−Un,tUc,t

=Wt

Pt

Qt = β Et

{Uc,t+1Uc,t

PtPt+1

}Um,tUc,t

= 1− exp{−it}

where marginal utilities evaluated at(Ct,

Mt

Pt, Nt

)• utility separable in real balances =⇒ neutrality

• utility non-separable in real balances (e.g. Ucm > 0) =⇒ non-neutrality due to effect of expected inflation on real money and,hence, on labor supply and consumption


How important is the implied non-neutrality?

Utility specification:

U

(Ct,

Mt

Pt, Nt

)=X(Ct,Mt/Pt)

1−σ

1− σ − N 1+ϕt

1 + ϕ

X(Ct,Mt/Pt) ≡[

(1− ϑ)C1−νt + ϑ

(Mt

Pt

)1−ν] 11−v

for ν 6= 1

≡ C1−ϑt

(Mt

Pt

)ϑfor ν = 1


Simulation in Walsh

Policy Rule: ∆mt = ρm ∆mt−1 + εmtCalibration: ν = 2.56 ; σ = 2 =⇒ Ucm > 0

Effects of exogenous monetary policy shock (Figs.)


Optimal Monetary Policy in a Classical Economy withMoney in the Utility Function

Social Planner’s problem

maxU

(Ct,

Mt

Pt, Nt

)subject to

Ct = AtN1−αt

Optimality conditions:

−Un,tUc,t

= (1− α)AtN−αt (14)

Um,t = 0 (15)

Optimal policy (Friedman rule): it = 0 for all t .


Intuition: the real cost of producing money here is 0 (not inproduction function) and effi ciency require that the cost of holdingmoney is equal to its production cost.

Implied average inflation: π = −ρ < 0

Implementation problem: Find the values of real balances (d∗t ) andthe real interest rate (r∗t ) which are consistent with the aboveequations when it = 0 for all dates. Since it = 0, implied equilibriuminflation:

πt = −r∗t−1

Deflation is optimal.


Implementation via an interest rate rule

it = φ(r∗t−1 + πt)

for some φ > 1. Combined with the definition of the real rate:

Et{it+1} = φit

whose only stationary solution is it = 0 for all t and for any φ.


Implementation via money growth rule

mt = dt + Ptmt −mt−1 = d∗t − d∗t−1 + πt

= d∗t − d∗t−1 − r∗t

Negative money growth occurs unless the real demand for money is rapidlyincreasing

Under both policies, there can be issues about the determination of theinitial price level, but these are eliminated if M0 is specified or someother aspect of policy sets P0.

real business cycles and classical monetary …rg king, boston university ec 541 session 2 1 real...

Documents