Introduction Households Firms Equilibrium PS’s expectations MP instrument
Monetary Economics
Chapter 1: The Basic New Keynesian Model
Olivier Loisel
ensae
December 2020 − January 2021
O. Loisel, Ensae Monetary Economics Chapter 1 1 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
Objective of the chapter
The chapter presents the basic NK model and derives its implicationsregarding the role of expectations in the transmission of MP.
As explained in the General Introduction, the motivation for considering thismodel is threefold:
1 like RBC models, it is a (DS)GE model, so that
it is not subject − or little sensitive − to Lucas’ (1976) critique,it provides a welfare criterion to assess the desirability of policies,
2 unlike RBC models, it provides an active role for MP, due to
the inefficiency of economic fluctuations,the non-neutrality of MP in the short term,
3 it is simple.
This model will be used in Chapters 2, 3, and 6.
O. Loisel, Ensae Monetary Economics Chapter 1 2 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
From the standard RBC model to the basic NK model
As also explained in the General Introduction, the basic NK modelcorresponds to the standard RBC model in which
there is no endogenous capital accumulation (for simplicity),
there is monopolistic competition in the goods market, so that firmsare price-makers (not price-takers),
there is price stickiness in the goods market, so that
economic fluctuations are inefficient,MP is non-neutral in the short term (due to its effects on real moneybalances and the short-term real interest rate).
In particular, it is a GE model, so that its equilibrium conditions are
the first-order conditions of the private agents’ optimization problems,the constraints of these problems,market-clearing conditions.
O. Loisel, Ensae Monetary Economics Chapter 1 3 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
Which version of the basic NK model? I
There are different versions of the basic NK model, depending on
1 the nature of shocks,2 the degree of steady-state inefficiency,3 the nature of price stickiness,4 the role of money,5 the nature of labor.
Shocks ≡ stochastic exogenous variables (normalized to have a zero mean).
Steady state ≡ equilibrium in the absence of shocks.
O. Loisel, Ensae Monetary Economics Chapter 1 4 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
Which version of the basic NK model? II
1 For simplicity, I consider only two shocks:
technology shocks,shocks to the elasticity of substitution between differentiated goods.
↪→ Other shocks could be considered, whose normative implications wouldbe qualitatively similar to those of technology shocks:
consumption-utility shocks,labor-disutility shocks,government-expenditures shocks,distortive-tax shocks.
O. Loisel, Ensae Monetary Economics Chapter 1 5 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
Which version of the basic NK model? III
2 I assume that there is a fiscal authority that offers an employment subsidy(financed by lump-sum taxes), so as to partially or completely offset themonopolistic-competition distortion.
↪→ This assumption enables me to consider alternative degrees ofsteady-state inefficiency.
3 I consider price stickiness
à la Calvo (1983) (at each date, some randomly chosen firms areallowed to change their prices),not à la Rotemberg (1982) (at each date, each firm faces a resourcecost in changing its price, which is quadratic in the price-change size).
↪→ When the steady-state distortion is small, these two alternative ways ofmodeling price stickiness have qualitatively similar positive and normativeimplications.
O. Loisel, Ensae Monetary Economics Chapter 1 6 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
Which version of the basic NK model? IV
4 I first consider a cashless model, in which money implicitly serves only as aunit of account.
↪→ I show at the end of the chapter how to introduce money explicitly intothis model.
5 I assume that there is a single kind of labor as in Gaĺı (2015, Chapter 3),not several as in Woodford (2003, Chapter 4).
↪→ These two alternative assumptions have qualitatively similar positive andnormative implications.
O. Loisel, Ensae Monetary Economics Chapter 1 7 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
Time and agents
Time is discrete, indexed by t, and the horizon is infinite.
There are four kinds of agents:
a large number of households,a large number of firms,a single monetary authority,a single fiscal authority.
All of them are infinitely lived.
All households are identical, so that there is a representative household (RH).
The RH is the only agent to have a utility function, so that the basic NKmodel is a representative-agent model.
O. Loisel, Ensae Monetary Economics Chapter 1 8 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
Markets
A continuum of monopolistically competitive goods markets:
demand from households,supply from firms.
A perfectly competitive labor market:
demand from firms,supply from households.
A perfectly competitive one-period-bond market:
demand from households,supply from households.
O. Loisel, Ensae Monetary Economics Chapter 1 9 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
Outline of the chapter
Introduction
Households’ behavior
Firms’ behavior
Equilibrium conditions
Role of the private sector (PS)’s expectations
MP instrument
O. Loisel, Ensae Monetary Economics Chapter 1 10 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
RH’s intertemporal utility
RH’s intertemporal utility function at date 0 is
U0 ≡ E0{∑+∞t=0 βtU (Ct ,Nt)
}, with Ct ≡
[∫ 10
Ct(i)εt−1
εt di
] εtεt−1
,
where β ∈]0, 1[ is the discount factor, U is continuous and twicedifferentiable, and, for each date t,
Ct is the consumption index,
Ct(i) is the quantity of good i consumed by RH,
εt > 1 is the elasticity of substitution between differentiated goods,
Nt is the number of hours worked by RH,
Uc,t > 0, Ucc,t < 0, Un,t < 0, Unn,t < 0.
O. Loisel, Ensae Monetary Economics Chapter 1 11 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
RH’s optimization problem in words
RH chooses how much
of each good to consume,labor to supply,bonds to hold,
in order to maximize
her intertemporal utility function,
subject to
her intertemporal budget constraint,
taking as given
the price of each good,the wage,the price of bonds
(given the market structure and the large number of households).
O. Loisel, Ensae Monetary Economics Chapter 1 12 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
RH’s optimization problem formalized
Max[Ct(i)]i∈[0,1],t∈N ,
(Nt)t∈N , (Bt)t∈N
E0
{∑+∞t=0 βtU (Ct ,Nt)
}
subject to
Ct ≡[∫ 1
0Ct(i)
εt−1εt di
] εtεt−1
and
∫ 10
Pt(i)Ct(i)di +QtBt ≤ Bt−1 +WtNt + Tt for t ∈N,
taking as given
B−1, [Pt(i)]0≤i≤1,t∈N , (Qt)t∈N , (Wt)t∈N , and (Tt)t∈N .
O. Loisel, Ensae Monetary Economics Chapter 1 13 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
Notations and resolution
For each date t,
Pt(i) is the price of good i ,
Qt is the price of one-period nominal bonds(paying one unit of money at maturity),
Bt is the quantity of one-period nominal bonds held by RH,
Wt is the nominal wage,
Tt is a lump-sum component of income.
RH’s optimization problem can be solved in two steps:
1 for any given consumption index Ct , characterize RH’s choice of thedistribution of consumption across goods [Ct(i)]0≤i≤1,
2 characterize RH’s choice of the consumption index Ct and the numberof hours worked Nt .
O. Loisel, Ensae Monetary Economics Chapter 1 14 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
Distribution of consumption across goods I
Dual optimization problem:
Max[Ct (i)]0≤i≤1
[∫ 10
Ct(i)εt−1
εt di
] εtεt−1
subject to∫ 10 Pt(i)Ct(i)di = Zt .
Lagrangian:
L =
[∫ 10
Ct(i)εt−1
εt di
] εtεt−1− λ
[∫ 10
Pt(i)Ct(i)di − Zt]
.
First-order conditions (FOCs): Ct(i)−1εt Ct
1εt = λPt(i) for all i ∈ [0, 1].
O. Loisel, Ensae Monetary Economics Chapter 1 15 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
Distribution of consumption across goods II
Using these FOCs to replace Ct(i) in the definition of Ct gives λ = P−1t
where
Pt ≡[∫ 1
0Pt(i)
1−εtdi
] 11−εt
is the aggregate price index.
Replacing λ by P−1t in the FOCs gives the demand schedule
Ct(i) =
[Pt(i)
Pt
]−εtCt for all i ∈ [0, 1].
The limit case εt −→ +∞ corresponds to perfect competition.
O. Loisel, Ensae Monetary Economics Chapter 1 16 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
Consumption index and number of hours worked I
Replacing Ct(i)εt−1
εt by Ct(i)Pt(i)C−1εtt P
−1t in the definition of Ct gives∫ 1
0Pt(i)Ct(i)di = PtCt .
Therefore, the second step of RH’s optimization problem can be rewritten as
Max(Ct )t∈N,(Nt )t∈N,(Bt )t∈N
E0
{∑+∞t=0 βtU (Ct ,Nt)
}subject to
PtCt +QtBt ≤ Bt−1 +WtNt + Tt for t ∈N,
taking as given
B−1, (Pt)t∈N , (Qt)t∈N , (Wt)t∈N , and (Tt)t∈N .
O. Loisel, Ensae Monetary Economics Chapter 1 17 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
Consumption index and number of hours worked II
The FOCs of this problem are, for all t ∈N,
−Un,tUc,t
=WtPt
(labor-consumption trade-off condition),
Qt = βEt
{Uc,t+1Uc,t
PtPt+1
}(Euler equation).
Interpretation: at the optimal plan, it must be the case that
Uc,tdCt + Un,tdNt = 0 for any pair (dCt , dNt)satisfying the budget constraint PtdCt = WtdNt ,
Uc,tdCt + βEt{Uc,t+1dCt+1} = 0 for any pair (dCt , dCt+1) satisfyingthe budget constraint Pt+1dCt+1 = − PtQt dCt .
O. Loisel, Ensae Monetary Economics Chapter 1 18 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
Consumption index and number of hours worked III
Specific functional-form assumption for the period utility:
U (Ct ,Nt) =C1−σt − 1
1− σ −N1+ϕt
1 + ϕ
where σ > 0 and ϕ > 0.
The previous FOCs then become
WtPt
= C σt Nϕt (labor-consumption trade-off condition),
Qt = βEt
{(Ct+1Ct
)−σ PtPt+1
}(Euler equation).
O. Loisel, Ensae Monetary Economics Chapter 1 19 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
Consumption index and number of hours worked IV
The labor-consumption trade-off condition can be rewritten in log-linearform as
wt − pt = σct + ϕnt ,
where, for any variable Zt , zt ≡ logZt .
A log-linear approximation of the Euler equation around a steady state witha zero inflation rate and a constant consumption level is
ct = Et {ct+1} −1
σ
(it −Et {πt+1} − i
),
where it ≡ − logQt is the log of the gross yield on one-period nominalbonds (referred to as the short-term nominal interest rate), i ≡ − log β,and πt ≡ pt − pt−1 is the inflation rate.
O. Loisel, Ensae Monetary Economics Chapter 1 20 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
Production function and price rigidity
There is a continuum of firms indexed by i ∈ [0, 1], each firm producing adifferentiated good.
All firms use the same technology, represented by the production function
Yt(i) = AtNt(i)1−α,
where α ∈]0, 1[ and At is a stochastic exogenous factor.
As in Calvo (1983), at each date, each firm may reset its price only withprobability 1− θ (independent of the time elapsed since the lastadjustment), where θ ∈ [0, 1], so that
at each date, a measure 1− θ of firms reset their prices,at each date, a measure θ of firms keep their prices unchanged,the average duration of a price is (1− θ)−1,θ is a natural index of price stickiness.
O. Loisel, Ensae Monetary Economics Chapter 1 21 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
Aggregate price level dynamics
At each date t, all firms resetting their prices will choose the same pricenoted P∗t because they face the same problem.
Therefore, Pt =[θ (Pt−1)
1−εt + (1− θ) (P∗t )1−εt
] 11−εt .
Dividing by Pt−1, one gets Π1−εtt = θ + (1− θ)
(P∗tPt−1
)1−εt, where
Πt ≡ PtPt−1 .
Log-linearization around a steady state with Πt = 1 yields
πt = (1− θ)(p∗t − pt−1).
O. Loisel, Ensae Monetary Economics Chapter 1 22 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
Firms’ optimization problem
A firm re-optimizing at date t will choose the price P∗t that maximizes thecurrent market value of the profits generated while this price remainseffective:
MaxP∗t
∑+∞k=0 θkEt{Qt,t+k
[P∗t Yt+k |t −Ψt+k (Yt+k |t)
]},
where
Qt,t+k ≡ βk(Ct+kCt
)−σPt
Pt+kis the stochastic discount factor for
nominal payoffs between t and t + k ,Yt+k |t is output at t + k for a firm that last reset its price at t,Ψt(.) is the nominal cost function at t,
subject to Yt+k |t =(
P∗tPt+k
)−εt+kCt+k for k ∈N,
taking (Ct+k )k∈N and (Pt+k )k∈N as given.
O. Loisel, Ensae Monetary Economics Chapter 1 23 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
FOC of this problem
The FOC of this problem (henceforth “firms’ FOC”) is
∑+∞k=0 θkEt{Qt,t+kYt+k |t (εt+k − 1)
(P∗t −Mt+kψt+k |t
)}= 0,
where ψt+k |t ≡ Ψ′t+k (Yt+k |t) denotes the nominal marginal cost at t + kfor a firm that last reset its price at t, and Mt+k ≡ εt+kεt+k−1 .
Under flexible prices, this FOC collapses to P∗t =Mtψt|t , so that Mt isthe “desired” (or frictionless) markup.
Dividing by Pt−1, one gets
∑+∞k=0 θkEt
{Qt,t+kYt+k |t (εt+k − 1)
(P∗tPt−1−Mt+kMCt+k |tΠt−1,t+k
)}= 0,
where Πt−1,t+k ≡ Pt+kPt−1 and MCt+k |t ≡ψt+k |tPt+k
is the real marginal cost at
t + k for a firm whose price was last set at t.
O. Loisel, Ensae Monetary Economics Chapter 1 24 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
Zero-inflation-rate steady state
At the zero-inflation-rate steady state (ZIRSS),
P∗t and Pt are equal to each other and constant over time,
therefore, all firms produce the same quantity of output,
this quantity is constant over time, as the model features nodeterministic trend,
therefore, P∗t
Pt−1= 1, Πt−1,t+k = 1, Mt+k =M≡ εε−1 , Qt,t+k = βk ,
and we note Yt+k |t = Y and MCt+k |t = MC ,
firms’ FOC then implies MC = 1M .
O. Loisel, Ensae Monetary Economics Chapter 1 25 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
Log-linearization of firms’ FOC
Log-linearization of firms’ FOC around the ZIRSS yields
p∗t − pt−1 = (1− βθ)∑+∞k=0(βθ)kEt{
µt+k +mct+k |t + (pt+k − pt−1)}
,
where µt+k ≡ logMt+k .
This equation can be rewritten as
p∗t = (1− βθ)∑+∞k=0(βθ)kEt{
µt+k +mct+k |t + pt+k}
.
Hence, firms resetting their prices choose a price that corresponds to aweighted average of their current and expected future desired markups overtheir nominal marginal costs.
O. Loisel, Ensae Monetary Economics Chapter 1 26 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
Market-clearing conditions
Market clearing in the goods markets requires, for all i and t,
Yt(i) = Ct(i).
Therefore, Yt = Ct , where Yt ≡[∫ 1
0 Yt(i)εt−1
εt di
] εtεt−1
.
Market clearing in the labor market requires, for all t,
Nt =∫ 10
Nt(i)di .
O. Loisel, Ensae Monetary Economics Chapter 1 27 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
Aggregate production function
Using the market-clearing conditions, the production function, and thedemand schedule, one gets
Nt =∫ 10
[Yt(i)
At
] 11−α
di =
(YtAt
) 11−α ∫ 1
0
[Pt(i)
Pt
] −εt1−α
di ,
and therefore the aggregate production function
yt = (1− α)nt + at − dt ,
where dt ≡ (1− α) log∫ 10
[Pt (i)Pt
] −εt1−α
di is a measure of price (and, hence,
output) dispersion across firms.
O. Loisel, Ensae Monetary Economics Chapter 1 28 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
Average real marginal cost
In the neighborhood of the ZIRSS, dt is equal to zero up to a first-orderapproximation, so that, at the first order,
yt = (1− α)nt + at
(the proof is postponed to Chapter 2).
Noting mct the average real marginal cost at t, mpnt the averagemarginal product of labor at t, and τ the constant employment subsidy, andusing yt = (1− α)nt + at , one gets
mct = log(1− τ) + (wt − pt)−mpnt= log(1− τ) + (wt − pt)− (at − αnt)− log(1− α)
= log(1− τ) + (wt − pt)−1
1− α (at − αyt)− log(1− α).
O. Loisel, Ensae Monetary Economics Chapter 1 29 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
Real marginal cost
For any firm-level variable z , let zt+k |t denote the value of z at t + k for afirm that last reset its price at t.
Using the demand schedule and the goods-market clearing condition, onegets, at the first order, the real marginal cost
mct+k |t = log(1− τ) + (wt+k − pt+k )−mpnt+k |t
= log(1− τ) + (wt+k − pt+k )−at+k − αyt+k |t
1− α − log(1− α)
= mct+k +α
1− α (yt+k |t − yt+k )
= mct+k −αεt+k1− α (p
∗t − pt+k ) = mct+k −
αε
1− α (p∗t − pt+k ).
Under constant returns to scale (α = 0), one has mct+k |t = mct+k : the realmarginal cost is independent of the output level and, hence, is commonacross firms.
O. Loisel, Ensae Monetary Economics Chapter 1 30 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
Rewriting firms’ FOC
Using the previous result, firms’ FOC can be rewritten as p∗t − pt−1
= (1− βθ)∑+∞k=0(βθ)kEt {Θ (µt+k +mct+k ) + (pt+k − pt−1)}= (1− βθ)Θ ∑+∞k=0(βθ)kEt {µt+k +mct+k}+ ∑
+∞k=0
(βθ)kEt {πt+k}= βθEt
{p∗t+1 − pt
}+ (1− βθ)Θ (µt +mct) + πt ,
where Θ ≡ 1−α1−α+αε .
Using the aggregate price level dynamics equation, one then gets
πt = βEt {πt+1}+ χ (µt +mct) ,
where χ ≡ (1−θ)(1−βθ)θ Θ.
O. Loisel, Ensae Monetary Economics Chapter 1 31 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
Natural level of output
Independently of the nature of price setting, the average real marginal costcan be rewritten, at the first order, as
mct = log(1− τ) + (wt − pt)−mpnt= log(1− τ) + (σyt + ϕnt)− (yt − nt)− log(1− α)
= log(1− τ) +(
σ +ϕ + α
1− α
)yt −
1 + ϕ
1− α at − log(1− α),
using the labor-consumption trade-off condition, the goods-market-clearingcondition, and the (approximate) aggregate production function.
Now, firms’ FOC implies that, under flexible prices, mct = −µt .
Therefore, the natural level of output, defined as the equilibrium level ofoutput under flexible prices and noted ynt , is such that
−µt = log(1− τ) +(
σ +ϕ + α
1− α
)ynt −
1 + ϕ
1− α at − log(1− α).
O. Loisel, Ensae Monetary Economics Chapter 1 32 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
Output gap and New Keynesian Phillips curve
Therefore, the natural level of output is
ynt =1− α
σ(1− α) + ϕ + α
[log
(1− α1− τ
)+
1 + ϕ
1− α at − µt]
.
The natural level of output does not depend on it , i.e. MP is neutralunder flexible prices.
Subtracting the two equations on the previous slide, one gets mct
+µt =(
σ + ϕ+α1−α
)ỹt , where ỹt ≡ yt − ynt is called the output gap.
Rewriting firms’ FOC, one gets the New Keynesian Phillips curve (NKPC)
πt = βEt {πt+1}+ κỹt ,
where κ ≡ χ(
σ + ϕ+α1−α
).
O. Loisel, Ensae Monetary Economics Chapter 1 33 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
Interpretation of the NKPC
The NKPC is forward-looking because, when a firm resets its price, itknows that it will not be able to change this price for some (random) time.
Therefore, the current inflation rate depends on
the current situation (term κỹt),the expected future situation (term βEt {πt+1}).
The slope κ of the Phillips curve is decreasing in θ and β: the stickier theprices or the higher the discount factor, the less prices react to the currentsituation.
As prices become flexible (θ −→ 0), the NKPC becomes yt = ynt .
O. Loisel, Ensae Monetary Economics Chapter 1 34 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
IS equation
Using the goods-market-clearing condition and the definition of the outputgap, one can rewrite the Euler equation as the IS equation
ỹt = Et {ỹt+1} −1
σ(it −Et {πt+1} − rnt ) ,
where
rnt ≡ i + σEt{∆ynt+1}
= i +σ(1− α)
σ(1− α) + ϕ + α
(1 + ϕ
1− α Et{∆at+1} −Et{∆µt+1})
is the natural rate of interest (unique equilibrium value of the ex anteshort-term real interest rate it −Et {πt+1} consistent with the output levelbeing constantly equal to its natural level).
O. Loisel, Ensae Monetary Economics Chapter 1 35 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
Set of equilibrium conditions
Given (at , µt , it)t∈N, (ỹt , πt)t∈N is determined by
the IS equation ỹt = Et {ỹt+1} − 1σ (it −Et {πt+1} − rnt ),the NKPC πt = βEt {πt+1}+ κỹt ,
for t ∈N, which implies that MP is not neutral (unless θ −→ 0).
Given (at , µt , it , ỹt , πt)t∈N, (yt , ct , nt ,wt − pt)t∈N is determined by
the definition of the output gap ỹt ≡ yt − ynt ,the goods-market-clearing condition ct = yt ,
the aggregate production function yt = (1− α)nt + at ,the labor-consumption trade-off condition wt − pt = σct + ϕnt ,
for t ∈N.
O. Loisel, Ensae Monetary Economics Chapter 1 36 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
Role of PS’s expectations I
Provided that limk−→+∞
Et {ỹt+k} = 0, iterating the IS equation forward yields
ỹt = −1
σEt
{∑+∞k=0
(rt+k − rnt+k
)},
where rt ≡ it −Et {πt+1} is the ex ante short-term real interest rate.
Using a no-arbitrage condition, one can interpret Et{
∑+∞k=0 rt+k}
as the exante long-term real interest rate.
Therefore, the output gap depends on PS’s expectations of the future pathof the short-term interest rate (through the long-term interest rate).
O. Loisel, Ensae Monetary Economics Chapter 1 37 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
Role of PS’s expectations II
Provided that limk−→+∞
Et {πt+k} = 0, iterating the NKPC forward yields
πt = κEt{∑+∞k=0 βk ỹt+k
}.
Therefore, the inflation rate also depends on PS’s expectations of the futurepath of the short-term interest rate.
So MP affects the economy not only through changes in the currentshort-term interest rate, but also through changes in PS’s expectations ofthe future path of the short-term interest rate.
O. Loisel, Ensae Monetary Economics Chapter 1 38 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
Role of PS’s expectations III
As an illustration, assume for simplicity that CB controls directly rt andfollows the rule rt = γrt−1 + ξt , where γ ∈]0, 1[ and ξt is an i.i.d. MPshock occurring at date t.
Then Et{
∑+∞k=0 rt+k}= rt
(1−γ) , ỹt =−rt
σ(1−γ) + f[(
Et{rnt+k})k∈N
]and
πt =−κrt
σ(1−γ)(1−βγ) + g[(
Et{rnt+k})k∈N
], where f and g are linear
functions.
Therefore, the more persistent the short-term interest rate (the closer to 1 isγ), the larger the effect of the MP shock on the long-term interest rate, theoutput gap, and the inflation rate.
O. Loisel, Ensae Monetary Economics Chapter 1 39 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
In Woodford’s (2003) words I
“Successful monetary policy is not so much a matter of effective control of overnight
interest rates as it is of shaping market expectations of the way in which interest rates,
inflation, and income are likely to evolve over the coming year and later. (...)
[O]ptimizing models imply that private sector behavior should be forward-looking; hence
expectations about future market conditions should be important determinants of
current behavior. It follows that, insofar as it is possible for the central bank to affect
expectations, this should be an important tool of stabilization policy. (...) Not only do
expectations about policy matter, but, at least under current conditions, very little else
matters.
O. Loisel, Ensae Monetary Economics Chapter 1 40 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
In Woodford’s (2003) words II
[T]he current level of the overnight interest rates as such is of negligible importance for
economic decisionmaking. The effectiveness of changes in central-bank targets for
overnight rates in affecting spending decisions (and hence ultimately pricing and
employment decisions) is wholly dependent upon the impact of such actions upon other
financial-market prices, such as long-term interest rates, equity prices, and exchange
rates. These are plausibly linked, through arbitrage relations, to the short-term interest
rates most directly affected by central-bank actions. But it is the expected future path
of short-terms rates over coming months and even years that should matter for the
determination of these other asset prices, rather than the current level of short-term
rates by itself. Thus the ability of central banks to influence expenditure, and hence
pricing, decisions is critically dependent upon their ability to influence market
expectations regarding the future path of overnight interest rates, and not merely their
current level.”
O. Loisel, Ensae Monetary Economics Chapter 1 41 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
In Bernanke’s (2004b) words I
“Informal discussions of monetary policy sometimes refer to the Fed as ‘setting interest
rates.’ In fact, the FOMC does not set interest rates in general; rather, the Committee
‘sets’ one specific interest rate, the federal funds rate. The federal funds rate, the
interest rate at which commercial banks borrow and lend to each other on a short-term
basis (usually overnight) is not important in itself. Only a tiny fraction of aggregate
borrowing and lending is done at that rate. From a macroeconomic perspective,
longer-term interest rates–such as home mortgage rates, corporate bond rates, and the
rates on Treasury notes and bonds–are far more significant than the funds rate, because
those rates are the most relevant to the spending and investment decisions made by
households and businesses. These longer-term rates are determined not by the Fed but
by participants in deep and sophisticated global financial markets.
O. Loisel, Ensae Monetary Economics Chapter 1 42 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
In Bernanke’s (2004b) words II
Although the FOMC cannot directly determine long-term interest rates, it can exert
significant influence over those rates through its control of current and future values of
the federal funds rate. The crucial link between the federal funds rate and longer-term
interest rates is the formation of private-sector expectations about future monetary
policy actions. Loosely speaking, long-term interest rates embody the expectations of
financial-market participants about the likely future path of short-term rates, which in
turn are closely tied to expectations about the federal funds rate. Thus, to influence
long-term interest rates, such as thirty-year mortgage rates or the yields on corporate
bonds, the FOMC must influence private-sector expectations about future values of the
federal funds rate. The Committee can do this by its communication policies, by
establishing certain patterns of behavior, or both.”
O. Loisel, Ensae Monetary Economics Chapter 1 43 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
In European central bankers’ words
González-Páramo (2007): “expectations play an important role in the transmission of
monetary policy. Consider, for instance, the term structure of interest rates. Central
banks have a direct influence only on short-term interest rates through their monetary
policy instruments − typically an overnight call rate. However, consumption andinvestment decisions, and thus medium-term price developments, are to a large extent
influenced by longer-term interest rates, which in turn depend on private sector
expectations regarding future central bank decisions and future inflation.”
Trichet (2008): “Through their actions, central banks can directly control very
short-term interest rates. However, given that for consumption and investment decisions
the longer-term interest rates are more relevant, the whole yield curve is relevant for the
effectiveness of monetary policy. Medium- and long-term interest rates largely depend
on private expectations regarding future central bank decisions.”
O. Loisel, Ensae Monetary Economics Chapter 1 44 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
Very short-term predictability
CBs monitor markets’ expectations of their next policy-rate decision.
Sometimes, they communicate about this decision to ensure that markets’expectations are aligned with their intentions.
For instance, the ECB has used the code words “vigilance” and “strongvigilance” in its communication between 2005 and 2011 to signal a likelypolicy-rate hike two months ahead and one month ahead respectively(presumably as a trade-off between commitment and predictability).
O. Loisel, Ensae Monetary Economics Chapter 1 45 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
Expectations of the next monetary-policy decision
Probability of a change of the ECB Interest Rate (Most likely rate change mentionned: +25bp)
8,80%
90,20%
1,00%
36,60%
63,20%
0,20%
88,70%
11,30%
0,00%0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
Increase No change DecreaseExpectations on the Decision of January 11, 2007 (Survey Date: December 2006)Expectations on the Decision of February 8, 2007 (Survey Date: January 2007)Expectations on the Decision of March 8, 2007 (Survey Date: February 2007)
Source: Consensus ForecastLast Survey: February 12 2006
O. Loisel, Ensae Monetary Economics Chapter 1 46 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
Short- to medium-term predictability
However, as we have just seen, what matters for the economy is much lessthe expectation of next month’s policy rate than the expectation of thewhole future trajectory of the policy rate.
This is why CBs also monitor markets’ expectations of their subsequentpolicy-rate decisions, and sometimes communicate about these decisions toensure that markets’ expectations are aligned with their intentions.
O. Loisel, Ensae Monetary Economics Chapter 1 47 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
Expectations of up to one-year-ahead policy-rate decisions
Forecasts of 3M-Euribor based on futures contracts
3,3
3,4
3,5
3,6
3,7
3,8
3,9
4
4,1
4,2
18/09
/2006
03/10
/2006
18/10
/2006
02/11
/2006
17/11
/2006
02/12
/2006
17/12
/2006
01/01
/2007
16/01
/2007
31/01
/2007
15/02
/2007
% p
er y
ear
Forward rate, March 2007 Forward rate, June 2007
Forward rate, September 2007 Forward rate, December 2007
Source: BSME.Production: POMONE.Last update: 02/03/2007.
IFO release
Gov. C. meeting
Gov. C. meeting
Gov. C. meeting
Gov. C. meeting
IFO release
IFO release
IFO release
IFO release
Gov. C. meeting
IFO release
O. Loisel, Ensae Monetary Economics Chapter 1 48 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
In Bernanke’s (2004a) words
“The fact that market expectations of future settings of the federal funds rate are at
least as important as the current value of the funds rate in determining key interest
rates such as bond and mortgage rates suggests a potentially important role for central
bank communication: If effective communication can help financial markets develop
more accurate expectations of the likely future course of the funds rate, policy will be
more effective (...).
It is worth emphasizing that the predictability of monetary policy actions has both
short-run and long-run aspects. A central bank may, through various means, improve
the market’s ability to anticipate its next policy move. Improving short-term
predictability is not unimportant, because it may reduce risk premiums in asset markets
and influence shorter-term yields. But signaling the likely action at the next meeting is
not sufficient for effective policymaking. Because the values of long-term assets are
affected by the whole trajectory of expected short-term rates, it is even more vital that
information relevant to estimating that trajectory be communicated.”
O. Loisel, Ensae Monetary Economics Chapter 1 49 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
Two possible MP instruments I
CBs have the monopoly of issuing bank notes and bank reserves.
Therefore, they control the monetary base (i.e. the supply of money).
In the basic NK model, there are two alternative MP instruments: thesupply of money and the short-term nominal interest rate.
One of the simplest way to introduce money in this model is to make realmoney balances enter the utility function in a separable way.
In this case, the corresponding FOC leads to the following log-linearizedmoney-demand equation: mdt − pt = σyt−itν , where ν ≡ −
UmmMUmP
> 0.
The money-market-clearing condition then gives mst − pt = σyt−itν .
O. Loisel, Ensae Monetary Economics Chapter 1 50 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
Two possible MP instruments II
Money then plays a residual role, as it appears only in latter equation.
If the instrument is ms , then CB chooses ms , and the price on the moneymarket (interest rate i) adjusts so that md (i) = ms .
If the instrument is i , then CB chooses i and adjusts ms so thatms = md (i).
In the presence of money-demand shocks (i.e. shocks added to themoney-demand equation) that CB does not observe in real time, i may bepreferable to ms as an instrument.
This is because unlike ms , i isolates the real variables from these shocks.
O. Loisel, Ensae Monetary Economics Chapter 1 51 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
Two possible MP instruments III
In practice, before the 2008-2009 crisis (i.e. for conventional MP), the MPinstrument was i :
a MP committee chose a value for i without considering theadjustment in ms necessary for i to reach this value,a specialized staff of the CB adjusted ms for i to reach this value.
The basic NK model is consistent with this practice.
However, the MP instrument has not necessarily been i since the 2008-2009crisis, as unconventional MP (quantitative or credit easing) has, on someoccasions and in some places, pushed ms higher than the value consistentwith the market interest rate being close to the policy rate chosen.
O. Loisel, Ensae Monetary Economics Chapter 1 52 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
ECB policy rates and EONIA
100
By setting the rates on the standing facilities, the Governing Council effectively determines the corridor within which the overnight money market rate can fluctuate. Chart 4.2 shows the development of key ECB interest rates since January 1999 and how the interest rates on the standing facilities have provided a ceiling and a floor for the overnight market interest rate, measured by the euro overnight index average (EONIA).
Chart 4.2 shows that, in normal times, the EONIA has generally remained close to the rate on the MROs, thus demonstrating the importance of these operations as the main monetary policy instrument of the Eurosystem. This behaviour changed in October 2008, when the Eurosystem adopted non-standard measures to counter the
negative effects of the intensification of the financial crisis (see Box 5.1 and Chapter 5). Chart 4.2 also shows that the EONIA exhibits a pattern of occasional spikes. This pattern is related to the Eurosystem’s minimum reserve system, as explained further in Section 4.3.
Finally, Chart 4.2 shows that the differences between the standing facility interest rates and the rate on the MROs were kept unchanged between April 1999 and October 2008 at ±1 percentage point. The width of the corridor was then temporarily narrowed to ±0.5 percentage point, before being widened again to ±0.75 percentage point in May 2009, when the Governing Council decided to set the rate for the MROs at 1.0%.
Corridor of standing facility
interest rates
EONIA, key ECB interest
rates and the minimum
reserve system
Chart 4.2 Key ECB interest rates and the EONIA since 1999
(percentages per annum; daily data)
0
1
2
3
4
5
6
7
0
1
2
3
4
5
6
7
1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010
marginal lending rate deposit rate EONIA main refinancing/minimum bid rate
Source: ECB.1) Before 28 June 2000 MROs were conducted as fixed rate tenders. Starting with the operation settled on 28 June 2000, and until the operation settled on 15 October 2008, MROs were conducted as variable rate tenders with a pre-announced minimum bid rate. Since the operation settled on 15 October 2008, MROs have been conducted as fixed rate tenders with full allotment. This procedure is scheduled to remain in place at least until the maintenance period ending on 12 July 2011. The minimum bid rate refers to the minimum interest rate at which counterparties may place their bids (see Section 4.4).
Source: ECB (2011).
O. Loisel, Ensae Monetary Economics Chapter 1 53 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
ECB MRO rate and EONIA
0
0,5
1
1,5
2
2,5
3
3,5
4
4,5
5
01/09/2008 01/12/2008 01/03/2009 01/06/2009 01/09/2009
%
eonia main refinancing operations rate (fixed rate or minimum bid rate)
O. Loisel, Ensae Monetary Economics Chapter 1 54 / 55
Introduction Households Firms Equilibrium PS’s expectations MP instrument
Targeted and effective Fed Funds rates
0
0,5
1
1,5
2
2,5
3
3,5
09/02/2008 12/19/2008 04/01/2009 06/15/2009 08/29/2009
%
effective targeted
O. Loisel, Ensae Monetary Economics Chapter 1 55 / 55
IntroductionObjective of the chapterFrom the standard RBC model to the basic NK modelWhich version of the basic NK model? IWhich version of the basic NK model? IIWhich version of the basic NK model? IIIWhich version of the basic NK model? IVTime and agentsMarketsOutline of the chapter
HouseholdsRH's intertemporal utilityRH's optimization problem in wordsRH's optimization problem formalizedNotations and resolutionDistribution of consumption across goods IDistribution of consumption across goods IIConsumption index and number of hours worked IConsumption index and number of hours worked IIConsumption index and number of hours worked IIIConsumption index and number of hours worked IV
FirmsProduction function and price rigidityAggregate price level dynamicsFirms' optimization problemFOC of this problemZero-inflation-rate steady stateLog-linearization of firms' FOC
EquilibriumMarket-clearing conditionsAggregate production functionAverage real marginal costReal marginal costRewriting firms' FOCNatural level of outputOutput gap and New Keynesian Phillips curveInterpretation of the NKPCIS equationSet of equilibrium conditions
PS's expectationsRole of PS's expectations IRole of PS's expectations IIRole of PS's expectations IIIIn Woodford's (2003) words IIn Woodford's (2003) words IIIn Bernanke's (2004b) words IIn Bernanke's (2004b) words IIIn European central bankers' wordsVery short-term predictabilityExpectations of the next monetary-policy decisionShort- to medium-term predictabilityExpectations of up to one-year-ahead policy-rate decisionsIn Bernanke's (2004a) words
MP instrumentTwo possible MP instruments ITwo possible MP instruments IITwo possible MP instruments IIIECB policy rates and EONIAECB MRO rate and EONIATargeted and effective Fed Funds rates