Derivation of Gali’s Basic Model
Based on great lectures by professor Jordi Gali on Barcelona Macroeconomic
Summer School 2011. The aim of these notes is to provide me with
step by step, fool-proof derivation of basic New Keynesian model and related
analysis of monetary policy, before I forget it all.
All of this can be found in Jordi Gali’s textbook Monetary Policy, Inflation,
and the Business Cycle: An Introduction to the New Keynesian Framework.
These notes are more descriptive as to the derivation of equations, but far
less descriptive in other ways. Do read the textbook.
Intro - evidence for NK model basics In the first chapter of the textbook
textbook, there is empirical motivation for NK models. Everyone
should read it first.
The basic New Keynesian model consists of three equations:
• New Keynesian Phillips Curve, which links the inflation to the output
gap
πt = βEt{πt+1} + κyt (1)
• Dynamic IS equation, which links the output gap to the interest rate
yt = −
1
σ
(it − Et{πt+1} − rn
t ) + Et{yt+1} (2)
• and some rule for interest rate, for example the Taylor rule:
it = ρ + φππt + φyyt + vt (3)
Here variables denoted by ∼ stand for the log deviation of variable from its
natural level, that is the level that would prevail in the absence of nominal
rigidities. Variables denoted by a hat are the log deviations from steady
state. The ”natural” variables are denoted by superscript n, so that e.g. rn
t
is the natural rate of interest.
1
1 Households
The representative household solves standard problem
max E0
∞
t=0
βt C1−σ
t
1 − σ
−
N1+ϕ
t
1 + ϕ
where
Ct =
1
0
Ct(i)1−1
ε
ε
1−ε
subject to
1
0
Pt(i)Ct(i)di + QtBt ≤ Bt−1 + WtNt + Dt
and solvency constraint
lim
t→∞
Bt ≥ 0
Dt is any lump-sum income the household gets, such as profits, taxes, transfers
etc. We also need some initial condition for Bt−1.
1.1 Optimal allocation of expenditures
The household consumes continuum of goods indexed by i. To maximize
utility, the household solves
max
1
0
Ct(i)1−1
ε
ε
ε−1
while expenditures are given by
1
0
Pt(i)Ct(i)di = Zt
Lagrangian
L =
1
0
Ct(i)1−1
ε di
ε
ε−1
− λ
1
0
Pt(i)Ct(i)di − Zt
FOCs of the Lagrangian wrt to Ct(i) are
C−1
t
Ct(i)−1
ε
Pt(i)
= λ
2
Combining two together, we get
Pt(i)
Pt(j)
−ε
=
Ct(i)
Ct(j)
We can plug this into the constraint (with index j, plugging for Ct(j))
1
0
Pt(j)
Pt(j)−ε
Pt(i)−ε
Ct(i)dj = Zt
Taking all that does not depend on j out of the integral gives
Ct(i) = ZtPt(i)ε 1
1
0
Pt(j)1−ε
Using the definition of the price index
Pt =
1
0
Pt(j)1−ε
1
1−ε
we can rewrite the last term in the previous equation
Ct(i) =
Zt
Pt
Pt(i)
Pt
−ε
This expression can be inserted into the definition of Ct to get
Ct =
1
0
Zt
Pt
ε−1
ε Pt(i)
Pt
1−ε
ε
ε−1
Ct =
Zt
Pt
1
0
Pt(i)1−ε
diPε−1
t
ε
ε−1
CtPt = Zt P
(1−ε)+(ε−1)
t
ε
ε−1
CtPt = Zt
CtPt =
1
0
Ct(i)Pt(i)di
when we again used the definition of price index in the second step (third
equation).
Finally, we can combine the two above results to get
Ct(i) =
Pt(i)
Pt
−ε
Ct
3
1.2 Optimality conditions
By setting up Lagrangean of the household problem, we derive following
intertemporal condition (using e.g. derivatives wrt to Ct and Ct+1):
Qt
Pt
C−σ
t = βEt{C−σ
t+1
1
Pt+1
}
Now define Qt = exp{−it}, it is the log of nominal interest rate, because Qt
is the price of one-period bond paying 1 unit of money in time t + 1
Qt =
1
1 + it
.
Then define β = exp{−ρ} where ρ is the discount rate and
β =
1
1 + ρ
,
and πt = pt − pt−1, where pt = log Pt. From now on, small case letters will
denote logs of variables denoted by capital letters.
We will log-linearize the intertemporal condition. For log-linearization of
equations with expectations, there is a trick. Remove expectations, take logs
and then put the expectations back. This holds up to a first approximation.
We get
ct = Etct+1 −
1
σ
(it − Etπt+1 − ρ)
This equation is the Euler equation and will result into IS curve.
Intratemporal condition (derived by using derivatives wrt to Ct and Lt) in
logs is
wt − pt = σct + ϕnt = mrst.
and this equation will provide household labor supply. Notice that while
shifts in wt results in movement along the labor supply curve, shifts in ct
move the whole curve.
2 Firms
There is a [0; 1] continuum of monopolistically competitive firms, each produces
own differentiated good. Firms share production technology
Yt(i) = AtNt(i).
4
A representative firm maximizes the present value of their future profits
conditional on its inability to reset price for next k periods
max
P∗
t
∞
k=0
θk
Et{Qt,t+k (P∗
t Yt+k,t − Xt+k(Yt+k,t))}
where X is the cost function1
, Yt+k,t is explained below, P∗
t is the new, optimal
price, θ is the probability of not being able to reset price in one period and
Qt,t+k < 1 is the stochastic discount factor (explained below).
The future demand in period t + k conditional on price set in period t is
derived from household optimization:
Yt+k,t = Ct+k
P∗
t
Pt+k
−ε
On Stochastic Discounting Let’s briefly examine Qt,t+k. Think about
an asset that pays Dt+k in period t + k. In period t, it is bought for price
Qt. Household in period t gives up utility equal to
Qt
Pt
Uc,t
and gains utility in period t + k equal to
βk
EtUc,t+k
Dt+k
Pt+k
Therefore the price of the asset is
Qt = βk
Et{
Uc,t+k
Uc,t
Pt
Pt+k
Dt+k} = Et{Qt,t+kDt,t+k}
and stochastic discount factor for asset bought at time t and maturing at
time t + k is
Qt,t+k = βk Ct+k
Ct
σ
Pt
Pt+k
(4)
1
The production function does not enter the model explicitly, but it is implicitly present
here.
5
Let’s continue with the problem of the firm. FOC wrt to P∗
t is
∞
k=0
θk
Et{Qt,t+k (1 − ε)Yt+k,t + εΨt+k(Yt+k,t)
Yt+k,t
P∗
t
} = 0
∞
k=0
θk
Et{Qt,t+kYt+k,t (1 − ε) + εΨt+k
1
P∗
t
} = 0
∞
k=0
θk
Et{Qt,t+kYt+k,t ((1 − ε)P∗
t + εΨt+k)} = 0
∞
k=0
θk
Et{Qt,t+kYt+k,t P∗
t −
ε
ε − 1
Ψt+k } = 0
∞
k=0
θk
Et{Qt,t+kYt+k,t (P∗
t − Mup
Ψt+k)} = 0
We denote Mup
= ε
ε−1
the desired markup of price over the nominal marginal
costs Ψ2
. That means that the firms wants to set price such that it brings
it exactly this markup over nominal marginal cost, because this markup
maximizes profit.
The above equation is in terms of variables that do not have well defined
steady state, namely P∗
t and Qt+k,t. We express it in terms of more convenient
variables.
First, divide by Pt−1
∞
k=0
θk
Et{Qt,t+kYt+k,t
P∗
t
Pt−1
− Mup
MCt+kΠt+k,t−1 } = 0
where MC are real marginal costs and Πt+k,t−1 = Pt+k
Pt−1
is gross inflation
between period t − 1 and period t + k.
Consider this equation in zero inflation steady state (we could consider other
steady states, but algebra is much simpler here and nothing fundamental
changes). In steady state, it must be that
P∗
t
Pt−1
= 1 and Πt+k,t−1 = 1, so that
MC =
1
Mup
.
2
For simplicity, I will simplify Ψt+k(Yt+k,t) to just Ψt+k.
6
Because both MC and Mup
are fixed numbers, this always holds. From
now on, letters without subscript will denote steady state values of variables.
Similarly, from equation (4) it follows that in steady state
Qt+k,t = βk
.
2.1 Log-linearizing Phillips Curve
First, we use the ”e to the logs” trick:
P∗
t
Pt−1
= elog P∗
t −log Pt−1
= ep∗
t −pt−1
Next, we realize that because Mup
= 1
MC
, then
Mup
MCt+k =
MCt+k
MC
which in logs is the deviation of MCt+k from steady state. This deviation is
denoted by mct+k.
Now rewrite the FOC in this way:
∞
k=0
θk
Et{Qt,t+kYt+k,t ep∗
t −pt−1
− emct+k
ept+k−pt−1
} = 0.
The term in parentheses evaluates in steady state to zero. This is convenient
because now we will make first order Taylor approximation and we do not
have to care about terms wrt to Qt,t+k and Yt+k,t, as they will always be
zero3
:
∞
k=0
θk
βk
EtY [1 (p∗
t − pt−1 − 0) − 1 (mct+k − 0) − 1 (pt+k − pt−1 − 0)] = 0
∞
k=0
θk
βk
EtY [p∗
t − mct+k − pt+k] = 0
3
To be precise, they will always be something × term in parentheses = something ×
0 = 0.
7
The zeros stand for the SS value of the exponents. Now we rearrange, denote
µ = log Mup
, realize that mc = log MC = log 1
Mup = −µ, denote log nominal
marginal costs ψt = mct + pt and sum the geometric series:
∞
k=0
(θβ)k
p∗
t = Et
∞
k=0
(θβ)k
[mct+k + pt+k]
1
1 − βθ
p∗
t = Et
∞
k=0
(θβ)k
[mct+k + pt+k]
1
1 − βθ
p∗
t = Et
∞
k=0
(θβ)k
[mct+k − mc + pt+k]
1
1 − βθ
p∗
t = Et
∞
k=0
(θβ)k
[mct+k + µ + pt+k]
p∗
t =
1 − βθ
1 − βθ
µ + (1 − βθ)
∞
k=0
(θβ)k
Etψt+k
This equation can be interpreted so that the firm sets the price such that it
equals the desired markup over the probability-and-discount-weighted sum
of future nominal marginal costs.
Notice that under flexible prices (θ = 0), this equation simplifies to
p∗
t = pt = µ + ψt.
We can define log average markup in the economy and notice that under
flexible prices, the average markup is equal to desired markup:
µt = pt − ψt = µ
Now a small detour: we use the definition of price index to get
Pt = θP1−ε
t−1 + (1 − θ) (P∗
t )1−ε
1
1−ε
1 = θ
Pt−1
Pt
1−ε
+ (1 − θ)
P∗
t
Pt
1−ε
1
1−ε
We again take the first order Taylor expansion of this around zero inflation
steady state and get
pt = θpt−1 + (1 − θ)p∗
t .
8
End of detour. Lets get back to the equation for optimal p∗
t . It can be
recursively written as
p∗
t = βθp∗
t+1 + (1 − βθ)(µ + ψt).
It is easy to iterate forward4
to verify that.
Lets introduce the forward expectations lag operator5
L−1
t :
L−1
t Xt = EtXt+1.
Using the operator, we can write the previous as
1 − βθL−1
t p∗
t = (1 − βθ)(µ + ψt).
Now combining with the Taylor expansion of the price index above, we get
1 − βθL−1
t pt = (1 − θ)(1 − βθ)(µ + ψt) + 1 − βθL−1
t θpt−1
and we get rid of the p∗
t . Cool. Now we expand and rearrange:
1 − βθL−1
t pt = (1 − θ)(1 − βθ)(µ + ψt) + 1 − βθL−1
t θpt−1
pt − βθEtpt+1 = θpt−1 − βθ2
pt + (1 − θ)(1 − βθ) [µ + ψt − pt + pt]
pt − βθEtpt+1 = θpt−1 − βθ2
pt + (1 − θ)(1 − βθ) [µ − µt] + (1 − θ)(1 − βθ)pt
pt − βθEtpt+1 = θpt−1 + (1 − θ)(1 − βθ) [µ − µt] + pt + θpt − βθpt
θ(pt − pt−1) = βθ(pt+1 − pt) + (1 − θ)(1 − βθ) [µ − µt]
πt = βπt+1 −
(1 − θ)(1 − βθ)
θ
[µt − µ]
πt = βEtπt+1 − λ [µt − µ]
Where µt is average markup, under sticky prices different from desired markup
µ. If we solve this forward for better intuition, we get very important result
πt = −λ
∞
k=0
βk
Et{µt+k − µ}.
4
Iterate forward = plug expression for p∗
t+1 on the right hand side, so that you get
expression in p∗
t+2. Keep doing that till infinity,
5
We could do without the operator here, but it is cool and sexy and makes things
easier.
9
The current inflation is entirely dependent on the expectations!
Now we need to replace the markups with output. Using the production
function Yt(i) = AtNt(i) we can derive the nominal marginal costs
Ψt =
Wt
At
.
Recall that from household optimisation we get labor supply:
wt − pt = σct + ϕnt.
We know that
Nt =
1
0
Nt(i)di =
1
0
Yt(i)
At
di =
Yt
At
1
0
Pt(i)
Pt
−ε
di
which in logs becomes
nt = yt − at + dt.
Now the first order Taylor expansion of dt ≡ log
1
0
Pt(i)
Pt
−ε
di equals zero,
so that up to a first approximation
nt = yt − at.
Now we write the average markup as
µt = pt−(wt−at) = (pt−wt)+at = −σct−ϕnt+at = −σyt−ϕ(yt−at)+at = (1+ϕ)at−(σ+ϕ)yt
Under flexible prices (where µ = µt) this becomes
µ = (1 + ϕ)at − (σ + ϕ)yn
t .
Substracting, we get
µt − µ = −(σ + ϕ)yt
and now we can plug this into our Phillips curve and get its final form:
πt = βEtπt+1 + λ(σ + ϕ)yt = βEtπt+1 + κyt
For better intuition, solve this forward to get
πt = κ
∞
k=0
βk
Et{yt+k}.
Now we can see that the current inflation is a function of expected future
output gaps, but there is no role for past inflation in this model.
10
2.2 IS Curve
Take the Euler equation with the market clearing condition ct = yt:
yt = Etyt+1 −
1
σ
(it − Etπt+1 − ρ) .
Notice that this equation implies that under flexible prices the natural real
rate of interest is
yn
t = Etyn
t+1 −
1
σ
(it − Etπt+1 − ρ)
Et∆yn
t+1 =
1
σ
(rn
t − ρ)
rn
t = ρ + σEt∆yn
t+1 = ρ +
σ(1 + ϕ)
σ + ϕ
Et{∆at+1}
And now we get the dynamic IS equation
yt − yn
t + yn
t = Etyt+1 − yn
t+1 + yn
t+1 −
1
σ
(it − Etπt+1 − ρ)
yt = Etyt+1 + ∆yn
t+1 −
1
σ
(it − Etπt+1 − ρ)
yt = Etyt+1 −
1
σ
it − Etπt+1 − ρ − σ∆yn
t+1
yt = −
1
σ
(it − Etπt+1 − rn
t ) + Et(yt+1)
Solving this forward (straightforward), we obtain
yt = −
1
σ
∞
k=0
Et{it+k − πt+k − rn
t+k}
which again confirms how important are the expectations.
The monetary policy is described by Taylor rule
it = ρ + φππ + φyyt
where yt = yt − y is the deviation from steady state. Introducing ρ makes
this rule consistent with zero inflation steady state.
We can now add the ad-hoc demand for money in the form
mt − pt = yt − ηit,
11
which implies money growth
∆mt = πt + ∆yt + η∆it.
The money rule here just tells us how much money the CB has to inject into
the economy to obtain the desired interest rate.
The textbook in chapter 3 has some impulse responses and comments on the
model we just derived. Do read them.
3 Monetary policy
3.1 Efficient Natural Equilibrium
We will now assume that government provides employment subsidy so that
the price of labor is (1−τ)Wt, where τ = 1/ε, to correct for the monopolistic
nature of the market. Thus we have
yn
t = ye
t ,
where ye
t denotes the efficient level of output that would be set by a benevolent
social planner. Such a social planner would solve
max U(Ct, Nt) s.t.Ct = AtNt
FOCs of this problem are
C−σ
t = λ
Nϕ
t = λAt
and together
Cσ
t Nϕ
t = At
plug for Ct from constraint and rearrange terms to get
Ne
t = A
1−σ
σ+ϕ
t
which is the efficient level of employment. We can use this to get the efficient
level of output
Y e
t = AtNe
t = A
1+ϕ
σ+ϕ
t .
12
If we want to maximize utility, we want to minimize the deviations of the
output from the efficient output. But looking at the forward iterated Phillips
curve, this implies minimizing inflation, so we get following interest rate rule:
it = rn
t + φππt.
However, because rn
t is unobservable, this rule cannot be implemented in
practice. We generally try to get the second-best policy. To evaluate various
policies, we set up a loss function derived from the utility function6
:
min E0
∞
k=0
βt
(σ + ϕ)y2
t +
λ
π2
t
The unconditional expectations of one period utility losses are given by
L = (σ + ϕ)var(yt) +
λ
var(πt).
Turns out that Taylor rule with large weight on inflation is nearly optimal.
3.2 Inefficient Natural Equilibrium
We now drop the assumption that ye
t = yn
t and introduce third output gap
xt = yt − ye
t .
The PC now becomes
πt = βEtπt+1 + κxt + ut, ut = κ(ye
t − yn
t ).
Notice that since the ut is independent of the monetary policy (sometimes
referred to as a cost-push shock), there is suddenly a trade-off in stabilizing
inflation versus stabilizing output gap. That was not the case before. The
IS curve becomes
xt = −
1
σ
(it − Etπt+1 − re
t ) + Etxt+1
6
The derivation is quite technical. See appendix, chapter 4 of the textbook.
13
where
re
t = ρ + σEt∆ye
t+1 = ρ +
σ(σ + ϕ)
1 + ϕ
Et∆at+1.
Now the problem of monetary policy becomes
min E0
∞
k=0
βt
αxx2
t + π2
t , αx =
κ
subject to the Phillips curve providing the trade-off
πt = βEtπt+1 + κxt + ut.
For simplicity, we will assume that ut follows AR(1) process:
ut = ρuut−1 + εt.
Again, once the CB solves the problem, it uses the IS curve to determine the
interest rate
xt = −
1
σ
(it − Etπt+1 − re
t ) + Etxt+1
Monetary Policy Under Discretion We now assume that the CB does
not have any credibility and can not influence the expectations. Each period,
the CB chooses (xt, πt) to minimize
αxx2
t + π2
t , s.t.πt = κxt + vt, vt = βEtπt+1 + ut
where vt is taken as given. FOCS yield
2αxxt + 2(κxt + vt)κ = 0
αxxt + κπt = 0
xt = −
κ
αx
πt.
We can plug this expression into the Phillips curve to get
πt = βEtπt+1 −
κ2
αx
πt + ut = β
αx
αx + κ2
Etπt+1 +
αx
αx + κ2
ut.
This is a first order differential equation for πt which we can solve forward
to get
πt =
αx
αx + κ2
∞
k=0
β
αx
αx + κ2
k
Etut+k.
14
We know that ut follows AR(1) process, so that Etut+1 = ρuut, so we can
write
πt =
αx
αx + κ2
∞
k=0
ρuβ
αx
αx + κ2
k
ut =
αx
αx + κ2
1
1 − αxβρu
αx+κ2
= αxΨut.
It follows from the FOC of the CB problem that
xt = −κΨut.
If we plug these results into a IS curve, we get an expression for the equilibrium
interest rate
it = re
t + Ψ [κσ(1 − ρu) + αxρu] ut.
This is NOT a MP rule! This is just expression for interest rate that would
prevail in equilibrium. For it to prevail, to CB must make sure that any
deviation of the πt or xt from equilibrium will be reacted to. The rule for
MP can look e.g. like this:
it = re
t + Ψ [κσ(1 − ρu) + αxρu] ut + ψπ(πt − αxΨut)
Monetary Policy Under Commitment CB pursues state-contingent
policy {πt, xt}∞
t=0 that minimizes
E0
∞
t=0
βt
(αxx2
t + π2
t ), s.t. πt = βEtπt+1 + κxt + ut
We can set up a Lagrangean of this problem with constraint variable γt:
L = −
1
2
E0
t=0
βt
αxx2
t + π2
t + 2γt (πt − κxt − βπt+1 − ut)
FOCs:
αxxt − κγt = 0
πt + γt − γt−1 = 0
for t = 0, 1, 2, ... and where γ−1 = 0. Take first difference of the first FOC
and plug into the second FOC to get
x0 = −
κ
αx
π0, t = 0
xt = xt−1 −
κ
αx
πt, t = 1, 2, ...
15
where the first equation follows from the fact that because γ−1 = 0, the
constraint in time t − 1 is irrelevant. We can think about these equations as
about targeting rules that result from the solution of the CB problem.
The second targeting rule implies
xt = xt−1 −
κ
αx
pt +
κ
αx
pt−1
and we know that in time t = 0:
x0 =
κ
αx
p−1 −
κ
αx
p0
This together gives
xt = −
κ
αx
(pt − p−1) = −
κ
αx
pt.
We can see that it is optimal for the central bank to keep price level equal to
price level target (in this case p−1). This is the case for price level targeting.
What would this price level target mean? We can add and subtract p−1 to
standard Phillips curve and plug for xt to get
pt − pt−1 = βEt(pt+1 − pt) + κxt + ut
pt = aβEt(pt+1) + apt−1 + aut, a =
αx
αx(1 + β) + κ2
which is a second order difference equation for pt. We guess the form of the
solution to be
pt = δpt−1 + ηut
and (again using the Etut+1 = ρuut) we can write
pt = apt−1 + aβ [δpt + ηρuut] + aut
pt =
a
1 − aδβ
pt−1 +
a [βηρu + 1]
1 − aδβ
ut
which implies (together with our guess, compare corresponding coefficients)
that
δ =
a
1 − aδβ
, η =
a [βηρu + 1]
1 − aδβ
.
16
We can solve for δ. Because it is a second order equation (= kvadraticka
rovnice), we choose the stable solution ( δ ∈ [0; 1]; same for η):
δ =
1 − 1 − 4βa2
2aβ
.
Now we see that pt under optimal monetary policy with commitment follows
stationary process
pt = δpt−1 +
δ
1 − δβρu
ut
which, however, implies price level targeting, even though we wanted to
stabilize inflation. The optimal trajectory for output gap is then given by
xt = δxt−1 −
κδ
αx(1 − δβρu)
ut, t = 1, 2, 3, ...
x0 = −
κδ
αx(1 − δβρu)
u0, t = 0
17
4 Wage Rigidities
Now we introduce wage rigidities into the model and see what happens.
4.1 Alternative Labor Market Specifications
With competitive labor market, we have
wt − pt = mrst, mrst = −un,t − uc,t = σct + ϕnt.
A very general way of introducing imperfections into labor market is to
rewrite the previous
wt − pt = µw
t + mrst
where µw
t is the (log) wage markup, that stands for (some) deviation/imperfection.
So why would there be a wage markup? One way to justify that is to
think about monopoly labor union (job agency) selling labor to firms. The
wages are flexible. The labor demand is given by isoelastic demand function
ND
t =
Wt
Pt
− w
.
The union maximizes the welfare of its members given by
U(Ct, Nt)
subject to the budget constraint
PtCt = WtNt + ...
where the dots stand for things the union can not influence and we do not
care about now.
Plugging for the Nt, computing FOCs wrt to Ct and Wt and putting them
together yields
Wt
Pt
= Mup
w
−Un,t
Uc,t
, Mup
w =
w
w − 1
, log Mup
w = µw
t = µw
Because of the flexible wages, the markup is constant, but the important
thing is that the markup is there. What does it do with the inflation dynamics?
In derivation of the Phillips curve, we had
πp
t = βEtπp
t+1 − λpµp
t .
18
This was derived witnout any reference to labor market. Now
µp
t = pt − (wt − at)
= at − (µw
t + mrst)
= −µw
t − (σ + ϕ)yt + (1 + ϕ)at
The last equation holds whether prices are sticky or not. We can take the
last equation a) at the natural equilibrium and b) under sticky prices and
wages, subtract and get
µp
t = −(σ + ϕ)yt − µw
t , µw
t = µw
t − µw
Now we get the previous inflation equation in the form
πp
t = βEtπp
t+1 + κpyt + λpµw
t .
Because of the non-zero last term, there is a tradeoff between stabilizing
inflation and output gap.
4.2 Enderson-Herceg-Levin Model
To model wage rigidities, we will use the model by Enderson, Herceg and
Levin. We have a [0; 1] continuum of households, each supplies his own,
unique kind of labor. Only a (1 − θw) fraction of households adjusts wage
every period. Firms use all kinds of labor and produce according to
Yt(i) = At
1
0
Nt(i, h)1− 1
w dh
w
w−1
Firms’ optimization (see the end of this section) implies following labor de-
mand:
Nt(i) = Nt
Wt(i)
Wt
−εw
.
The household sets wage to maximize
Et
∞
k=0
βk
θk
w (U(Ct,t+k, Nt,t+k)) , Nt,t+k =
W∗
t
Wt+k
− w
Nt+k,
19
where Nt,t+k is demand for labor in period t+k of household who reset price
in period t.
The FOC is wrt to W∗
t
∞
k=0
(βθw)k
Et UC(Ct,t+k, Nt,t+k)(1 − w)
Nt+k,t
Pt+k
− wUN (Ct,t+k, Nt,t+k)
Nt+k,t
W∗
t
= 0
∞
k=0
(βθw)k
Et UC(Ct,t+k, Nt,t+k)
Nt+k,t
Pt+k
W∗
t +
w
w − 1
UN (Ct,t+k, Nt,t+k)Nt+k,t = 0
Now let Mup
w = w
w−1
and let marginal rate of substitution MRSt = −
UN,t
UC,t
.
Denote MRSt+k,t = −
UN,t+k,t
UC,t+k,t
which is the MRS in period t + k of the household
that last reset wage at period t. We can rewrite the FOC in following
way
∞
k=0
(βθw)k
Et UC(Ct,t+k, Nt,t+k)Nt+k,t
W∗
t
Pt+k
− Mup
w MRSt+k,t = 0.
Note that under flexible wages, the term in square brackets implies
W∗
t
Pt+k
=
Wt
Pt+k
= Mup
w MRSt+k,t,
which means that Mup
w is the desired markup. Note also that the term in
square brackets evaluates to zero in steady state, which will again come in
handy when log-linearizing7
.
Now we can write:
∞
k=0
(βθw)k
Et UCNt+k,t ew∗
t −pt+k
− eµw+mrst+k,t
= 0
∞
k=0
(βθw)k
Et UCNt+k,t ew∗−p
(w∗
t − pt+k − w + p) − eµw+mrs
(mrst+k,t + mrs) = 0
∞
k=0
(βθw)k
Et UCNt+k,tew∗−p
[w∗
t − pt+k − mrst+k,t − (w∗
− p − mrs)] = 0
∞
k=0
(βθw)k
Et (w∗
t − pt+k − mrst+k,t − µw) = 0
7
Note also that ew∗
−p
= eµup
w +mrs
and that log Mup
w = µw = w∗
− p − mrs
20
We can continue
∞
k=0
(βθw)k
Et (w∗
t − µw) =
∞
k=0
(βθw)k
Et (pt+k + mrst+k,t)
w∗
t = µw +
∞
k=0
(1 − θw)(βθw)k
Et (pt+k + mrst+k,t)
The model assumes complete financial markets and separable utility in consumption
and labor. This implies that household consumption is independent
of previous wages, that is Ct+k,t = Ct+k. Therefore mrst+k,t = σct+k +ϕnt+k,t.
Let mrst+k denote the average marginal rate of substitution in the economy
in period t + k. We can write
mrst+k,t = mrst+k + ϕ(nt+k,t − nt+k) = mrst+k − wϕ(w∗
t − wt+k),
because the demand for individual labor of the household introduced at the
beginning of section 4.2 implies that nt+k,t = − w(w∗
t − wt+k) + nt+k.
We can rewrite the wage setting rule as
w∗
t =
∞
k=0
(1 − θw)(βθw)k
Et (µw + pt+k + mrst+k,t)
w∗
t =
∞
k=0
(1 − θw)(βθw)k
Et (µw + pt+k + mrst − wϕ(wt ∗ −wt+k))
(1 + wϕ)w∗
t =
∞
k=0
(1 − θw)(βθw)k
Et (µw + pt+k + mrst + wϕwt+k)
w∗
t =
(1 − θw)
1 + wϕ
∞
k=0
(βθw)k
Et (µw + pt+k + mrst − wt+k + wt+k wϕwt+k)
w∗
t =
(1 − θw)
1 + wϕ
∞
k=0
(βθw)k
Et µw − µw
t+k + (1 + wϕ)wt+k
w∗
t = (1 − θw)
∞
k=0
(βθw)k
Et wt+k −
µw
t+k
1 + wϕ
where µw
t+k = µw
t − µw is the log deviation of average wage markup from
steady state. Again, it is easy to verify by solving forward that this rule can
21
be recursively written as
w∗
t = βθwEtwt+1 + (1 − βθw) wt −
µw
t
(1 + wϕ)
.
Using the forward operator, we can write the previous as
1 − βθwL−1
t w∗
t = (1 − βθw) wt −
µw
t
(1 + wϕ)
.
Same as in the case of price inflation, the wage index
Wt =
1
0
Wt(i)1− w
1
1− w
implies aggregate wage dynamics in logs:
wt = θwwt−1 + (1 − θw)w∗
t .
Now combining the wage setting rule with the aggregate wage dynamics we
get
1 − βθwL−1
t wt = −(1 − θw)
(1 − βθw)
1 + wϕ
µw
t + 1 − βθwL−1
t θwt−1
and we get rid of the w∗
t .
Now we can reaarange:
1 − βθwL−1
t wt = −(1 − θw)
(1 − βθw)
1 + wϕ
µw
t + 1 − βθwL−1
t θwwt−1
wt − βθwEtwt+1 = θwwt−1 − βθ2
wwt − (1 − θw)
(1 − βθw)
1 + wϕ
µw
t
wt − βθwEtwt+1 = θwwt−1 − βθ2
wwt − (1 − θw)(1 − βθw) [µw
+ wt − µw
t − wt] +
+(1 − θw)(1 − βθw)wt
wt − βθwEtwt+1 = θwwt−1 − (1 − θw)
(1 − βθw)
1 + wϕ
µw
t + wt − θwwt − βθwwt
θw(wt − wt−1) = βθw(wt+1 − wt) − (1 − θw)
(1 − βθw)
1 + wϕ
µw
t
πw
t = βπw
t+1 −
(1 − θw)(1 − βθw)
θw(1 + wϕ)
µw
t
πw
t = βEtπw
t+1 − λwµw
t , λw =
(1 − θw)(1 − βθw)
θw(1 + wϕ)
22
This equation now replaces the wt − pt = mrst for the flexible wage case.
Let us now define the real wage gap
ωt = ωt − ωn
t = ωt − (at − µp
) ω = wt − pt
because under flexible prices the price is given as a constant markup over the
nominal marginal cost:
pt = µp
+ (wt − at) ⇒ ωn
t = wt − pt = at − µp
.
The log deviation of average price markup in the economy from steady state
under sticky prices is then
µp
t = pt − (wt − at) − µp
= −(wt − pt) + at − µp
= −ωt + at − µp
.
Going back to section 4.1 (page 18) we can see that now price Phillips curve
transforms from
πp
t = βEtπp
t+1 − λpµp
t
to
πp
t = βEtπp
t+1 + λpωt.
The log deviation of average wage markup from steady state is
µw
t = ωt − mrst − µw
= ωt − (σyt + ϕ(yt − at)) − µw
natural equilibrium. : 0 = ωn
t − ((σ + ϕ)yn
t − ϕat) − µw
substract : µw
t = ωt − (σ + ϕ)yt
So the equation for wage inflation becomes
πw
t = βEtπw
t+1 + κwyt + λwωt, κw = λw(σ + ϕ) (5)
To the model, we need to add the wage gap identity
ωt−1 = ωt − πw
t + πp
t + ∆at (6)
How do we get this? First notice that ωt = wt − pt and ωn
t = at − µp
. Then
take first differences of
ωt = ωt − ωn
t
∆ωt = wt − wt−1 − (pt − pt−1) − (at − µp − at−1 + µp)
= πw
t − πp
t − ∆at
23
To complete the model, we need the dynamic IS curve
yt = −
1
σ
(it − Etπt+1 − rn
t ) + Etyt+1 (7)
and the interest rate rule
it = ρ + φππp
t + φwπw
t + φyπy
t + vt (8)
We can write the model as a dynamical system in the form
xt = AW Et{xt+1} + BW zt
where
xt ≡ [yt, πp
t , πw
t , ωt−1]
zt ≡ [rn
t − vt, ∆at]
Vector xt contains the endogenous state variables (all information about the
state of the system). It has three non-predetermined variables (the first three
ones), so we need AW to have three eigenvalues inside the unit circle. Vector
zt contains exogenous variables. The first one, rn
t − vt, could also be written
as a function of at, but prof. Gali chose to write it this way.
For the equilibrium to be unique, in particular case of φy = 0, we have
the following condition:
φπ + φw > 1.
We assume that the monetary disturbance follows AR(1):
vt = ρvvt−1 + εm
t .
What now follows in the lecture notes is the calibration and IRFs of the
model with sticky prices and wages. I only have that on paper, but you can
find that in chapter 6 of the textbook.
4.3 Monetary Policy design
We now have, because of sticky wages, a trade-off between stabilizing inflation
and output gap. Frictionless allocation (natural plus compensation for the
monopolistic competition) is no longer feasible, because it requires real wage
changes.
24
The second order approximation to the welfare losses8
is
L = (σ + ϕ)var(˜yt) +
p
λp
var(πp
t ) +
w
λw
var(πw
t ),
∂λw
∂θw
< 0
and obviously strict price inflation targeting is no longer optimal. Why?
With nonzero wage inflation, some wages change while others do not (some
workers can change wages, so can not). But different wages induce firms to
buy different amount of various kinds of labor, which means that the output
produced by the firms is lower than optimal.
The problem of the monetary policy is
min E0
∞
l=0
βt
(σ + ϕ)y2
t +
p
λp
(πp
t )2
+
w
λw
(πw
t )2
subject to three constraints:
πp
t = βEt{πp
t+1} + λpωt
πw
t = βEt{πW
t+1} + κwyt − λwωt
ωt−1 = ωt − πw
t + πp
t + ∆at
with associated variables ξi,t for i-th constraint. We get following FOCs:
(σ + ϕ)yt + κwξ2,t = 0
p
λp
πp
t − ∆ξ1,t + ξ3,t = 0
w
λw
πw
t − ∆ξ2,t − ξ3,t = 0
λpξ1,t − λwξ2,t + ξ3,t − βEtξ3,t+1 = 0
and we have a dynamic system
A∗
0xt = A∗
1Etxt+1 + B∗
∆at
where
xt ≡ [yt, πp
t , πW
t , ωt−1, ξ1,t−1, ξ2,t−1, ξ3,t].
We can again produce impulse responses.
8
See appendix of chapter 6 in textbook.
25
4.4 Approximately Optimal Monetary Policy
We can not achieve optimal policy. But we can get close. Lets target the
composite inflation
πt = (1 − ϑ)πp
t + ϑπW
t , ϑ =
λp
λp + λW
∈ [0; 1].
for the NK Phillips Curve
πt = βEt{πt+1} + κ˜yt, κ =
λpλW
λp + λW
(σ + ϕ).
With this composite, there is no policy trade-off and we have nearly optimal
policy, according to Woodford(2003).
5 Open Economy Extension
This section is based on chapter 7 of the textbook. It was not part of the
lectures in Barcelona. I will not follow the textbook completely, but only
describe what is needed to derive the model in Justinano, Preston (2009)9
.
Most notably, I will simplify things by assuming only one foreign economy
and I will complicate things by assuming incomplete exchange rate pass-
through.
We assume that the representative household consumes a bundle given by
Ct = (1 − α)1/η
C
η−1
η
H,t + α1/η
C
η−1
η
F,t
η
η−1
,
subject to
PH,tCH,t + PF,tCF,t = PtCt.
Here subscript H denotes domestic economy and F denotes foreign economy,
so that e.g. CH,t is the consumption of domestic goods and PF,t denotes price
index of imported goods consumed in domestic economy.
Solution to this problem yields demand functions for domestic goods and
imports. Although the algebra is similar to subsection 1.1, we will do this
once again, but in another way.
9
Monetary Policy and Uncertainty in an Empirical Small Open Economy Model, FRB
Chicago WP 2009-21
26
5.1 Domestic-foreign goods decision
Lagrangian of the household is
L = PH,tCH,t + PF,tCF,t − λ (1 − α)1/η
C
η−1
η
H,t + α1/η
C
η−1
η
F,t
η
η−1
− Ct
FOCs:
PH,t = λC−1
t (1 − α)C
−1/η
H,t
PF,t = λC−1
t αC
−1/η
F,t
Now we just realize that λ is equal to the shadow price of additional unit of
consumption, which is Pt, and rearrange to get
CH,t = (1 − α)
PH,t
Pt
−η
Ct, CF,t = α
PF,t
Pt
−η
Ct.
To get the price index, we will plug this into the definiton of Ct:
Ct = (1 − α)
1
η (1 − α)
η−1
η
PH,t
Pt
−η η−1
η
C
η−1
η
t + α
1
η α
η−1
η
PF,t
Pt
−η η−1
η
C
η−1
η
t
η
η−1
Ct = (1 − α)
PH,t
Pt
1−η
+ α
PF,t
Pt
1−η
η
η−1
CtP
(η−1) η
η−1
t
Pt = (1 − α)
PH,t
Pt
1−η
+ α
PF,t
Pt
1−η
1
1−η
5.2 Household optimization
The household’s optimization results into intertemporal Euler equation
Qt = βEt
Ct+1
Ct
−σ
Pt
Pt+1
which in log again becomes
ct = Etct+1 −
1
σ
(it − Etπt+1 − ρ),
and into intratemporal condition (in logs)
wt − pt = σct + ϕnt.
27
5.3 Some identities
Define terms of trade as
St =
PF,t
PH,t
, st = pF,t − pH,t
and log-linearize the formula for Pt to get
pt = (1 − α)pH,t + αpF,t = pH,t + αst
. It follows from the above that domestic inflation and CPI inflation are
linked by
πt = πH,t + α∆st.
We will now introduce the law of one price gap
ΨF,t =
εtP∗
t
PF,t
where εt is the effective nominal exchange rate and P∗
t is the price index in the
foreign economy. This gap captures the fact that the prices of imported goods
do not move one to one with prices of identical goods in the foreign economy.
One reason for that could be monopolistically competitive importers, who
absorb the exchange rate fluctuations into their markups. Log-linearize to
get
ψF,t = et + p∗
t − pF,t.
Combine with the definition of the terms of trade to get
st = et + p∗
t − ψF,t − pH,t.
Next, define the real exchange rate as
Qt =
εtP∗
t
Pt
, qt = et + p∗
t − pt, log Qt = qt.
It now follows that
qt = et + p∗
t
=ψF,t+pF,t
−pt = ψF,t +pF,t −pt = ψF,t +pF,t −pH,t −αst = ψF,t +(1−α)st
28
5.4 International risk sharing
Assuming that agents in foreign and domestic economy share preferences,
complete financial markets imply that the price of one period bond is equal
over economies and that the marginal utility is equal over economies, so that
βEt
Ct+1
Ct
−σ
Pt
Pt+1
= Qt = βEt
C∗
t+1
C∗
t
−σ
P∗
t
P∗
t+1
εt
εt+1
.
Employing the definition of the real exchange rate, this becomes
Ct = C∗
t
Ct+1
C∗
t+1
Qt
Qt+1
1/σ
We will iterate forward. I’ll show just the first step.
Ct = C∗
t
C∗
t+1
Ct+2
C∗
t+2
Qt+1
Qt+2
1/σ
C∗
t+1
Qt
Qt+1
1/σ
= C∗
t
Ct+2
C∗
t+2
Qt
Qt+2
1/σ
.
We will end up with something like this:
Ct = ϑC∗
t Q
1/σ
t
where ϑ is a constant generally dependent on initial conditions. Take logs
and plug for qt to get
ct = c∗
t +
1 − α
σ
st +
ψF,t
σ
.
Justiniano and Preston (2009) employ different utility function for households
and the final forms of equations are different. See appendix.
5.5 Uncovered Interest Parity
Assuming complete financial markets, the price of one-period riskless bond
denominated in foreign currency is εtQ∗
t = Qt,t+1εt+1. Remember that Q∗
t =
29
exp{−i∗
t }. Combine this with domestic bond pricing equation Qt = EtQt,t+1.
Remember that Qt = exp{−it}:
exp{it}Qt,t+1 = 1 = exp{i∗
t }
εt+1
εt
Qt,t+1
exp{it}Qt,t+1 = exp{i∗
t }
εt+1
εt
Qt,t+1
Et{Qt,t+1 exp{it} − exp{i∗
t }
εt+1
εt
} = 0
Log-linearizing around perfect-foresight steady state yields familiar UIP condition
(in logs):
it = i∗
t + Et{∆et+1}.
The same equation can be also derived intuitively. Assume an agent in the
domestic economy that has one unit of money and thinks about investing it.
She can either invest it in domestic asset and in the next period she gets
1
Qt
= eit
Alternatively, she can convert her money into foreign currency and invest 1
εt
units of foreign currency in foreign asset. In the next period she gets
ei∗
t
εt
units of foreign currency which equals to
ei∗
t
εt
εt+1
units of doestic currency. Because we assume complete financial markets,
arbitrage ensures that these two yields need to be equal.
5.6 Firms
The firms in home economy use production technology
Yt = AtNt
30
so that log real marginal costs expressed in terms of domestic prices are
mct = wt − pH,t − at
mct = (wt − pt)
=σct+φnt
+(pt − pH,t) − at
mct = σct + φ nt
=yt−at
+αst − at
mct = σct + φyt + αst − (1 + φ)at
Justiniano and Preston (2009) assume households with habit in consumption
in their utility function. In that case, the equation for the real wage employed
in the second step becomes
Wt
Pt
=
Nφ
t
(Ct − hCt−1)−σ
and if log-linearized, we have
wt − pt = φnt +
σ
1 − h
(ct − hct−1).
Thus the marginal costs equal
mct = (wt − pt)
= σ
1−h
(ct−hct−1)+φnt
+(pt − pH,t) − at
mct =
σ
1 − h
(ct − hct−1) + φ nt
=yt−at
+αst − at
mct =
σ
1 − h
(ct − hct−1) + φyt + αst − (1 + φ)at
All variables here are in log difference from steady state.
Firm’s optimization problem results in following price setting rule in logs10
:
¯pH,t = µ +
1 − βθ
θ
∞
k=0
(βθ)k
Et{mct+k + pH,t+k}
10
Recall that now we are only talking about firms producing domestic goods, that’s why
pH,t and not pt
31
where ¯pH,t is the newly set optimal price11
In subsection 2.1 (see especially page 8-9), we obtained following three rela-
tions
πt = βEtπt+1 − λ(µt − µ)
µt = pt − ψt = −mct
µ = mc
These relations were derived without any assumption about closed economy
and continue to hold for the open econoemy case. Thus, we can now combine
them to get
πH,t = βEtπH,t+1 + λmct.
This, together with the expression for mct defines the dynamics of inflation.
5.7 Equilibrium
Goods market clearing in the domestic economy requires that the whole
output of each good is consumed either in the domestic, or in the foreign
economy
Yt(j) = CH,t(j) + C∗
H,t(j)
=
PH,t(j)
PH,t
−
CH,t +
PH,t(j)
PH,t
−
α
PH,t
P∗
t εt
−η
C∗
t
=
PH,t(j)
PH,t
−
(1 − α)
PH,t
Pt
−η
Ct +
PH,t(j)
PH,t
−
α
PH,t
P∗
t εt
−η
C∗
t
=
PH,t(j)
PH,t
−
(1 − α)
PH,t
Pt
−η
Ct + α
PH,t
P∗
t εt
−η
C∗
t
where the second equality rests on the assumption of identical preferences
across economies that ensures that the foreign demand for exports is derived
in the same way as demand for imports:
C∗
H,t(j) =
PH,t(j)
PH,t
−
C∗
H,t =
PH,t(j)
PH,t
−
α
P∗
H,t
P∗
t εt
−η
C∗
t
11
Previously, the newly set optimal price was denoted by a star. But now star denotes
foreign economy, so we will use the bar.
32
. We also assume no nominal rigidities in exports, so that P∗
H,t = PH,t.
We will now log-linearize the market clearing condition. Bear in mind that
the variations of
PH,t(j)
PH,t
−
are only of the second order.
yt = (1 − α)

−η( pH,t
=pt−αst
−pt) + ct

 + α


−η( pH,t
=pF,t−st
−( p∗
t + et
=ψF,t+pF,t
)) + c∗
t



yt = (1 − α) [ηαst + ct] + α [η(ψF,t + st) + c∗
t ]
yt = (2 − α)ηαst + (1 − α)ct + ηαψF,t + αy∗
t
Here we assume that c∗
t = y∗
t , an assumption that is reasonable when considering
large and almost closed foreign economy.
33
A.1 Justiniano and Preston
In Justiniano and Preston, the utility function is specified as
U(Ct, Nt) = E0
∞
t=0
βt
˜εG,t
(Ct − hCt−1)1−σ
1 − σ
−
N1+ϕ
t
1 + ϕ
.
FOC wrt to Ct is
βt
˜εG,t(Ct − hCt−1)−σ
= λtPt
Therefore the complete markets assumption becomes
˜εG,t+1(Ct+1 − hCt)−σ
˜εG,t(Ct − hCt−1)−σ
Pt
Pt+1
=
˜ε∗
G,t+1(C∗
t+1 − hC∗
t )−σ
˜ε∗
G,t(C∗
t − hC∗
t−1)−σ
P∗
t
P∗
t+1
εt
εt+1
˜εG,t+1
˜εG,t
−1/σ
(Ct+1 − hCt)
(Ct − hCt−1)
=
˜ε∗
G,t+1
˜ε∗
G,t
−1/σ
(C∗
t+1 − hC∗
t )
(C∗
t − hC∗
t−1)
Qt
Qt+1
−1/σ
(Ct+1 − hCt)
(Ct − hCt−1)
=
˜ε∗
G,t+1
˜εG,t+1
−1/σ
˜εG,t
˜ε∗
G,t
−1/σ
(C∗
t+1 − hC∗
t )
(C∗
t − hC∗
t−1)
Qt
Qt+1
−1/σ
(Ct+1 − hCt)
(Ct − hCt−1)
=
˜εG,t+1
˜ε∗
G,t+1
1/σ
˜ε∗
G,t
˜εG,t
1/σ
(C∗
t+1 − hC∗
t )
(C∗
t − hC∗
t−1)
Qt+1
Qt
1/σ
After iterating forward, we get
Ct − hCt−1 = (C∗
t − hC∗
t−1)Q
1/σ
t
˜εG,t
˜ε∗
G,t
1/σ
for log-linearization, we can rewrite that as
Cect
− hCect−1
= (C∗
ec∗
t − hCec∗
t−1 )(Qeqt
)1/σ EeεG,t
Eeε∗
G,t
1/σ
where Q, C and E are respective steady state values. In symmetric equilibrium,
it is true that Q = 1 and C = C∗
. First, the Taylor expansion of the
left hand side:
LS (1 − h)C + Cct − hCct−1
34
Now, the right hand side
(1 − h)CQ1/σ E
E
1/σ
+ CQ1/σ E
E
1/σ
c∗
t − hCQ1/σ E
E
1/σ
c∗
t−1
+
1
σ
(1 − h)CQ1/σ−1 E
E
1/σ
Qqt +
1
σ
(1 − h)CQ1/σ1 E1/σ−1
E1/σ
EεG,t
−
1
σ
(1 − h)CQ1/σ1
E1/σ
E−1/σ−1
Eε∗
G,t
We can substract the steady state values from both sides, employ identities
from above and divide by C, which leaves us with
ct − hct−1 = c∗
t − hc∗
t−1 +
1 − h
σ
qt +
1 − h
σ
εG,t −
1 − h
σ
ε∗
G,t.
This is the final log-linearized form. The Euler equation is log-linearized in
similar fashion. We start with
Qt = βEt










(Ct+1 − hCt)
Ct − hCt−1)
=At





−σ
Pt
Pt+1





.
Note that Qt = e−it
here is different from Qt above, it is not the real exchange
rate and it is not equal to 1 in steady state. In fact, if you evaluate the Euler
equation in steady state, you see that β = Q. The Taylor expansion of LHS
yield
Q + Qqt = Q − Qit = Q − βit
Now for the RHS:
β − βσ
A−σ−1
A−σ
P
P
Cct+1 + βσA−σ
Aσ−1
hCct−1 + β
A
A
−σ
Ppt
P
−
−β
A
A
−σ
P
P2
Ppt+1 − βσ
A
A
−σ−1
−hCA − AC
A2
ct
= β − β
σ
1 − h
ct+1 + β
σ
1 − h
hct−1 + βpt − βpt+1 − β
σ
1 − h
(−h − 1)ct
= β − β
σ
1 − h
(ct+1 − hct) + β
σ
1 − h
(ct − hct−1) − βπt+1
35
Equate LHS and RHS, substract steady state values and divide by β to get
σ
1 − h
(ct+1 − hct) =
σ
1 − h
(ct − hct−1) + (it − πt+1).
I forgot to add the demand shocks. They multiply the consumption, so that
they end up exactly as price level P. We then get
σ
1 − h
(ct+1 − hct) =
σ
1 − h
(ct − hct−1) + (it − πt+1) + (εG,t+1 − εG,t).
A.2 Labor demand
Each household supplies his own, unique kind of labor denoted by h. The
firms hire labor in bundles given by CES aggregate
Nt(i) =
1
0
Nt(h, i)
εw−1
εw dh
εw
εw−1
.
When deciding about hiring, the problem of the firm is to maximize the
amount of labor hired given the level of wage expenditures:
max Nt(i) s.t.
1
0
Nt(h, i)Wt(h)dh = Zt.
FOCs wrt to Nt(h, i) yield
Nt(h, i)− 1
εw + λWt(h) = 0.
Due to symmetry of firms in equilibrium, we can drop the firm index i.
Combining two together, we get
Nt(h)
Nt(k)
=
Wt(h)
Wt(k)
−εw
We can use this to plug for Nt(h, i) into the constraint
1
0
Nt(k)
Wt(h)
Wt(k)
−εw
Wt(h)dh = Zt
Nt(k)
1
Wt(k)−εw
W1−εw
t = Zt
Nt(k) = Zt
1
Wt
Wt(k)
Wt
−εw
36
We now use this to write Nt as
Nt =


1
0
Zt
1
Wt
Wt(k)
Wt
−εw
εw−1
εw
dk


εw
εw−1
Nt =
Zt
Wt
1
W1−εw
t
1
0
Wt(k)1−εw
dk
εw
εw−1
Nt =
Zt
Wt
W1−εw
t
W1−εw
t
dk
εw
εw−1
NtWt = Zt
Combining the two previous results, we get the demand schedule for labor
Nt(i) = Nt
Wt(i)
Wt
−εw
.
37