# Lectures on Monetary Theory and Policy

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Rheinische Friedrich Wilhelms Universit¨t
June 26 –July 4, 2008

Lectures on Monetary Theory and Policy

Day 2

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Instructor: Stephanie Schmitt-Groh´
Duke University, grohe@duke.edu
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Schmitt-Groh´, Stephanie and Mart´ Uribe, “Optimal Fiscal
and Monetary Policy Under Imperfect Competition,” 2004.

1. The model

• Preferences:
∞
E0         β t U (ct , ht)
t=0

• Technology: yt = zt˜t
h

• Monopolist faces demand: d(pt )Yt
• Money reduces transactions cost:

transactions cost = s(vt)ct
Ptct
vt =
Mt
s(v) satisﬁes: s(v) ≥ 0 , s (v) > 0 for v = v

• The ﬂow budget constraint of the household in period t

Ptct [1 + s(vt)] + Mt + Etrt+1 Dt+1 ≤
Mt−1 + Dt + Pt[pt Yt d(pt ) − wt˜t] + (1 − τt )Ptwt ht + Πt ,
h

• Implied Money demand function:
2            Rt − 1
vt s (vt) =
Rt
2. Analytical Results

(a) Under perfect competition Friedman rule is Ramsey opti-
mal

(b) Under imperfect competition, Rt > 1 is Ramsey optimal

3. Numerical Results
Calibration
Symbol             Deﬁnition               Value    Description
Calibrated Parameters:
β                                        0.96    Subjective discount factor
π                                        1.04    Gross inﬂation rate
h                                         0.2    Fraction of time allocated to work
sg                   g/y                   0.2    Government consumption to GDP ratio
sb                B/(P y)                 0.44    Public debt to GDP ratio
µ               η/(1 + η) √               1.2    Gross value-added markup
A        s(v) = Av + B/v − 2 AB           0.01    Parameter of transaction cost function
B                                         0.08    Parameter of transaction cost function
α                                           1    Fraction of transaction not rebated
zh                                        1.04    High value of technology shock
zl                                        0.96    Low value of technology shock
gh                                       0.043    High value of gov’t consumption shock
gl                                       0.037    Low value of gov’t consumption shock
φz                                        0.91    Prob(zt = zi |zt−1 = zi ) i = h, l
φg                                        0.95    Prob(gt = g i |gt−1 = g i ) i = h, l
Implied Parameters:
θ      U(c, h) = ln(c) + θ ln(1 − h)     2.90    Preference parameter

Note. The time unit is a year. The variable y ≡ zh denotes steady-state out-
e                    ın
put. Source: Schmitt-Groh´, Stephanie and Mart´ Uribe, “Optimal Fiscal and
Monetary Policy Under Imperfect Competition,” 2004.
Dynamic properties of the Ramsey allocation at diﬀerent degrees of market power
Variable Mean Std. Dev. Auto. corr.
µ=1
τ        18.8     0.0491     0.88
π        -3.39    7.47       -0.0279
R        0        0          0
y        0.241 0.0087        0.825
h        0.241 0.00243       0.88
c        0.201 0.00846       0.82

µ = 1.1
τ         22.6    0.0296      0.88
π         -2.81   7.52        -0.0273
R         0.59    0.00938     0.88
y         0.219   0.00781     0.826
h         0.219   0.00254     0.88
c         0.179   0.00752     0.82
µ = 1.2
τ         26.6    0.042       0.88
π         -1.46   7.92        -0.0239
R         1.95    0.0369      0.88
y         0.199   0.00701     0.829
h         0.199   0.00273     0.88
c         0.159   0.00661     0.82
µ = 1.35
τ         33      0.27        0.88
π         4.4     9.48        -0.00953
R         7.83    0.222       0.88
y         0.172   0.00595     0.837
h         0.172   0.00318     0.88
c         0.131   0.00527     0.82
Conclusion: Under the Ramsey policy

1. Under perfect competition: Rt = 1

2. Under imperfect competition: Rt > 1

3. var(πt ) high

4. var(τt ) low

5. variables inherit exogenous process of underlying shocks
2. Optimal Fiscal and Monetary Policy in Sticky Price Environ-
ments
Optimal Fiscal and Monetary Policy Under Sticky Prices

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Schmitt-Groh´, Stephanie, and Mart´ Uribe “Optimal Fiscal and Monetary Policy under

Sticky Prices” Journal of Economic Theory 114 (February 2004): 198-230.
• Optimal Fiscal and Monetary Policy in Flexible Price
Models

– Result: Optimal inﬂation volatility is very high.

• Optimal Monetary Policy in Sticky Price Models (With-
out Fiscal Policy)

– Result: Optimal inﬂation volatility is very low.

•   Our Question: What is the inﬂation volatility when
prices are sticky and the optimal monetary and ﬁscal regimes
are jointly determined.

∞
M−1 + R−1B−1           τt htwt + (Rt − 1)m(Rt) − gt
=
P0          t=0              ρ0,t
Mt = Nominal money supply
Bt = Nominal Public Debt
Rt = Nominal interest rate (gross)
Pt = Price level
τt = Labor income tax rate
ht = Hours worked
wt = Wage rate
m(Rt) = Money demand function
gt = Real government spending
ρ0,t = Real interest rate between 0 and t
A Sticky-Price model: Basic elements

• Money facilitates purchases of goods

• Monopolistic competition in product markets

a
• Sticky prices ` la Rotemberg (1982)

• The government ﬁnances a stochastic stream of public con-
sumption by:

– Levying labor income taxes
– Printing money

– Issuing nominal non-state-contingent debt

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• The government is benevolent ` la Ramsey.
Households
∞
E0         β t U (ct , ht)
t=0

Ptct[1 + s(vt )] + Mt + Bt
≤ Mt−1 + Rt−1 Bt−1 + (1 − τt)Ptwt ht
                                        
˜     ˜                  ˜           2
Pt    Pt              θ  Pt
+Pt  Yt d        − wt˜t −
h           −1          
Pt    Pt                ˜
2 Pt−1

Ptct
vt =
Mt

˜
Pt
zt˜t ≥ Yt d
h
Pt
The government

Mt + Bt = Mt−1 + Rt−1Bt−1 + Pt gt − τt Ptwtht

Two Shocks

• Government purchases shocks, gt

• Productivity shocks, zt
Calibration

• The Degree of Price Stickiness θ

– The expectations-augmented Phillips curve implied
by the model
1
ˆ
ˆt = βEt πt+1 +
π                      ˆ
mct,
γ(θ)

– Use available estimates of the Phillips-curve coeﬃcient
1/γ(θ) to obtain a value for θ.

– Result: θ = 4.4.
Calibration (continued)

• Parameters of the Demand for Money

– The transactions cost function
√
s(v) = Av + B/v − 2 AB

– The implied money demand function
ct
mt =
B + 1 Rt −1
A    A R   t

– Estimates of A and B

A = 0.01   B = 0.07
– Implied interest-rate semi-elasticity:

−3 at Rt = 8%

−7 at Rt = 0%
Degree of Price Stickiness and Optimal Inﬂation Volatility
7       ← flexible prices
6

5
std. dev. of π

4

3

2

1
← baseline
0
0       2       4         6        8    10
degree of price stickiness, θ
The standard deviation of inﬂation is measured in percent per year.
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Source: Schmitt-Groh´ and Uribe (JET, 2004).
Degree of Price Stickiness and Deviations from the
Friedman Rule

4
← baseline
3.5
Nominal interest rate
3

2.5

2
← flexible prices
1.5

1
0       2       4         6        8    10
degree of price stickiness, θ

The nominal interest rate is measured as percent per year.
e
Source: Schmitt-Groh´ and Uribe (JET, 2004).
More on Dynamic Properties
of Ramsey Equilibrium
Variable Mean Std.      Dev.
Flexible Prices
τ           25.8         0.04
π           -1.8         6.80
R            1.8         0.03
Sticky Prices
τ           25.1         1.00
π          -0.16         0.17
R           3.85         0.56

Note. τ , π, and R are expressed in percentage points.
Near Random Walk Behavior of Taxes and Public Debt

Impulse response to an i.i.d. government purchases shock
Real public debt                            Tax rate
1.5                                          0.04

1                                           0.03

0.5                                          0.02

0                                           0.01

−0.5                                            0
0   2       4        6        8   10          0   2   4         6      8   10
Nominal interest rate                          Inflation
0.06                                            1

0.04                                           0.5

0.02
0
0
−0.5
0   2       4      6         8    10          0   2   4            6   8   10
Consumption                                    Output
0                                            0.6

−0.05                                           0.4

−0.1                                           0.2

−0.15                                            0

−0.2                                          −0.2
0   2        4       6       8    10          0   2   4            6   8   10

— sticky prices                                                                                            - - - ﬂexible prices
e
Source: Schmitt-Groh´ and Uribe (JET, 2004).
Accuracy of Approximation

• Flexible-Price economy admits an exact numerical solution (Schmitt-
e
Groh´ and Uribe, Journal of Macroeconomics, 2004).

• The exact solution is virtually identical to the one based upon a ﬁrst-order
e
approximation (Schmitt-Groh´ and Uribe, JET, 2004).

• The sticky-price economy admits no exact numerical solution.

• The solutions based on ﬁrst- and second-order approximations are virtu-
e
ally identical (Schmitt-Groh´ and Uribe, JET, 2004).
Summary

• Under ﬂexible prices, optimal ﬁscal and monetary is charac-
terized by:

– Highly volatile and unpredictable inﬂation.

– The optimality of the Friedman Rule.

– Stationarity in tax rates and public debt.
• A small amount of price stickiness alters these predictions
quite drastically:

– Inﬂation volatility disappears.

– The Friedman rule ceases to be optimal.

– Tax rates and public debt follow near random walk pro-
cesses.
• The Real Eﬀects of Monetary Instability
Fig. 1.—Model- and VAR-based impulse responses. Solid lines are benchmark model
impulse responses; solid lines with plus signs are VAR-based impulse responses. Grey areas
are 95 percent conﬁdence intervals about VAR-based estimates. Units on the horizontal
axis are quarters. An asterisk indicates the period of policy shock. The vertical axis units
are deviations from the unshocked path. Inﬂation, money growth, and the interest rate
Fig. 1.—Continued
Would like to have a model of the real eﬀects of monetary in-
stability in which

• monetary policy aﬀects the level of real economic activity

• anticipated policy changes have real eﬀects

• monetary policy has persistent real eﬀects
Real Eﬀects of Monetary Instability: Theoretical Models

• The Calvo-Yun Model

Calvo, Guillermo “Staggered Prices in a Utility-Maximizing
Framework” Journal of Monetary Economics 12 (September
1983): 383-98.

Yun, Tack “Nominal Price Rigidity, Money Supply Endogene-
ity, and Business Cycles” Journal of Monetary Economics 37
(July 1996): 345-70.
• Consider a cashless economy. → reason that monetary dis-
turbances have real eﬀects is clearly unrelated to any money
demand considerations.

• The key assumption is that product prices are sticky. Each
good i ∈ [0, 1] is produced by a single ﬁrm in a monopolisti-
cally competitive environment.
• Firm i faces demand, ait:
−η
Pit
ait =              at       (1)
Pt

• Pit is the nominal price of good i set by the ﬁrm

• Pt is the general price level

• at denotes aggregate demand

• ﬁrm takes Pt and at as exogenously given.
• η > 1 denotes the price elasticity of demand.

• key friction: a fraction α ∈ [0, 1) of randomly picked ﬁrms is
not allowed to set the price optimally

• in the Calvo-Yun model the probability of being allowed to
reoptimize the price is independent of history.

• average number of periods since the last reoptimization:
1/(1 − α)

• ex: α = 0.75, the average number of periods since the last
reoptimization is 4, that is, the price is ﬁxed for 4 periods.
What if ﬁrm i receives a price change signal?

• It can choose Pit optimally

• It can choose a time contingent (but not non-state contin-
gent) time path {Pis}T optimally until it gets its next price
s=t
change signal at time T . For example, Cespedes, Kumhof,
and Parrado (2003) assume that a ﬁrm can pick Pit and a
constant πi such that
s−t
Pis = πi Pit
What if ﬁrm i does not receive a price change signal in period
t? Either

• the ﬁrm has to keep charging the same price as last period,
Pit = Pit−1. This case is often referred to Calvo-Yun sticki-
ness without indexation.

• the ﬁrm adjusts the price at the steady state inﬂation rate,
π∗.

• the ﬁrm adjusts prices at the lagged actual general inﬂation
rate:
Pit = πt−1 Pit−1
(ex: Christiano, Eichenbaum, and Evans; JPE 2005).

• Indexation to lagged inﬂation is only partial. For example:
χ
Pit = πt−1 Pit−1
χ measures the degree of indexation.

Cogley and Sbordone (2006) and Levin et al. (2006) typically
ﬁnd little evidence for indexation in prices. LOWW (2006):
posterior mean for χ of 0.08 [90% [0, 0.21]. Cogley and
Sbordone (2006) median 0.13 [90% [0,0.37].

It follows that, empirically in U.S. data setting χ = 0 is not
an implausible assumption.
• the ﬁrm indexes to a combination of steady state and lagged
inﬂation:
χ
Pit = π 1−χπt−1 Pit−1
ex: Smets and Wouters (2007), posterior mean of 0.24 for
χ [90% [0.10, 0.38]. still a model with full long-run indexation, even if χ = 0
because of the assumed indexation to steady state inﬂation.
• Production Technology: yit = zt F (Kit, hit) − ψ

– F hd 1

– zt technology shock

• Firms are assumed to be price setters

• must satisfy demand at a set price: yit ≥ ait

• proﬁt maximization problem of the ﬁrm.

– φi period-t proﬁts of ﬁrm i: φi = Pit ait − utKit − wthit
t                            t   Pt
– Proﬁt maximization: E0     ∞ r P φi
t=0 0,t t t

−η
– Production must satisfy demand: ztF (Kit , hit)−ψ ≥ Pit
Pt     at

– FOCs wrt Kit and hit:

ut = mcitztFk (Kit , hit)

wt = mcitztFh(Kit , hit)
F (K /h ,1)
• wt = Fk (Kit,hit) = Fk (Kit /hit ,1)
u    F
t   h (Kit ,hit )  h   it it

• ⇒ every ﬁrm i will employ the same capital labor ratio: Kt =
ht
Kit
h   ∀i ∈ [0, 1]
it

wt
• ⇒ mcit = mct = z F (K ,h )
t h    t t

• Sveen and Weinke (JET 2005) consider a variant of the
Calvo-Yun model in which capital is ﬁrm speciﬁc. In such a
setup it is no longer true that all ﬁrms face identical marginal
costs]
Optimal price setting at the ﬁrm level

• ﬁrm i receives a price change signal

˜
• Pit denote the price it chooses.

• assume no indexation, that is, if ﬁrm i does not receive a price
change signal in period t, then its period t is: Pit = Pit−1.
˜
• Firm i chooses Pit to max
∞                               1−η                                                           −η
˜
Pit                                                           ˜
Pit
Et         αs−t rt,s Ps                    as − usKis − wshis + mcs zs F (Kis , his) − ψ −              as
s=t
Ps                                                            Ps

˜
• The ﬁrst-order condition with respect to Pit is:
∞                        −η
˜
Pit             η−1   ˜
Pit
Et          αs−t rt,sPs               as             − mcs   =0            (2)
s=t
Ps               η    Ps

˜     ˜
• Pit = Pt

η−1     ˜
Pt
• real marginal revenue                  η      Ps

• marginal costs, mcs.

˜
• Pt makes weighted average of the diﬀerence between marginal revenue
and marginal cost equal to zero.

˜     ˜
• if ﬁrm speciﬁc capital, then Pit = Pt
•   At this point, much of the literature using the Calvo-Yun apparatus, proceeds to lin-
earizing equation (2) around a deterministic steady state featuring zero inﬂation. This
strategy yields the famous linear neo-Keynesian Phillips curve involving inﬂation and
marginal costs (or the output gap).

• What if π ss = 1?

• To retain the non-linear nature of the equilibrium conditions
deﬁne two new variables, x1 and x2.
t       t

˜ −η        ˜
• Ptx1 ≡ Et
t
∞ αs−tr P
s=t    t,s s
Pt
Ps   as η−1 Ps
η
Pt

˜ −η
Pt
• Ptx2 ≡ Et
t
∞ αs−tr P
s=t    t,s s          Ps   asmcs
• So that we can write (2) as

x1 = x2
t    t                 (3)

• And now we try to write x1 and x2 recursively.
t      t
1−η
η−1         ˜
pt
t   ˜1−η
x1 = pt at     + αEt                      η
πt+1 rt,t+1 x1
t+1   (4)
η        ˜
pt+1
−η
−η                       ˜
pt          η+1
x2 = pt atmct + αEt rt,t+1
t   ˜                                   πt+1 x2
t+1     (5)
˜
pt+1
where
˜
Pt
˜
pt =
Pt
Aggregation

• Evolution of the aggregate price level Pt
1−η           1     1−η
Pt     =            Pit    di
0
α
1−η      1 1−η
=    Pit−1di +   ˜
Pit di
0           α
1−η          ˜1−η
= αPt−1 + (1 − α)Pt

η−1
1 = απt            p1−η
+ (1 − α)˜t                  (6)

• market clearing: ait = cit + iit
−η
Pit
ait =                   (ct + it)
Pt
−η
Pit
zt F (Kit, hit) − ψ =               (ct + it ).
Pt

• Integrating over all ﬁrms
1 Pit       −η
Kt
h t zt F      ,1 − ψ =                    di(ct + it)
ht          0  Pt

−η
• let st ≡ 0 Pit
1
Pt     di.

ztF (Kt, ht) − ψ = st (ct + it)
• express st recursively
1         −η
Pit
st =                    di
0    Pt
˜ −η
Pt       Pit
−η
= (1 − α)      +          di
Pt     α Pt
−η
−η                Pit−1
= (1 − α)˜t +
p                                 di
α  Pt
−η             −η
−η              Pt−1          Pit−1
= (1 − α)˜t +
p                                           di
Pt         α Pt−1
−η         η
= (1 − α)˜t
p           + απt st−1.
• Summarizing, the resource constraint is given by
styt = ztF (Kt , ht) − ψ            (7)

yt = ct + it                  (8)
−η        η
st = (1 − α)˜t
p     + απt st−1 ,        (9)
with s−1 given.

• The state variable st measures the resource costs induced
by the ineﬃcient price dispersion present in the Calvo-Yun
model in equilibrium. Notice that this implies that a Calvo-
Yun model has one more state variable. By comparison,
other models of sluggish price adjustment do not necessarily
have this feature. For example, in the case of Rotemberg
stickiness, there is no additional state variable.
• 2 observations are about the dispersion measure st.

1. st is bounded below by 1.
let vit ≡ (Pit/Pt)1−η . Then   1 v η/(η−1) = 1.
0 it
1  η/(η−1)
Also, by deﬁnition we have st = 0 vit      . Then, taking
into account that η/(η −1) > 1, Jensen’s inequality implies
that 1 =     1 v η/(η−1) ≤    1  η/(η−1)
= st . The lower
0 it            0 vit
bound on st implies that price dispersion is always a costly
distortion.

2. in an economy where the non-stochastic level of inﬂation
is nil, i.e., when π = 1, up to ﬁrst order the variable
st is deterministic and follows a univariate autoregressive
process of the form ˆt = αˆt−1 , where ˆt ≡ ln(st/s) denotes
s     s           s
the log-deviation of st from its steady-state value s.
Deriving the log-linear Phillips Curve

• Consider (3), (4), (5), (6). Then log-linearize them around
π ∗ = 1 to obtain the new Keynesian Phillips curve:

ˆ
πt = βEt ˆt+1 + κmct
π

(1 − α)(1 − αβ)
κ=                   >0
α

• The parameter κ is a non-linear function of the Calvo pa-
rameter α
Empirical Evidence Drawn Upon to Calibrate the Calvo
Parameter α

1. Studies that estimate the NKP directly.

Cogley and Sbordone (AER)

Sbordone (JME)

ı,
Gali and Gertler (JME) and Gal´ Gertler, and Lopez Salido

In fact, κ is estimated and then theory is used to back out α

2. Studies that estimate α by impulse response matching (lim-
ited information approach)
Christiano, Eichenbaum, and Evans (JPE, 2005)

Altig, Christiano, Linde, and Eichenbaum (2005)

3. Studies that estimate Medium-Scale Models using Bayesian
Maximum Likelihood Methods (full information approach)

Levin, Onatski, Williams, and Williams (2005)

Smets and Wouters (AER, 2007)

Again, those studies obtain an estimate of κ and then back
out α

4. Studies on the frequency of price changes in micro level data
Bils and Klenow (JPE, 2004)

Nakamura and Steinsson (QJE, forthcoming)
1
Implied Average Time (months) between reoptimizations (1−α)

Direct estimates of the Phillips Curve
Sbordone (2002)                                  10
Limited Information DSGE Estimation
Christiano, Eichenbaum, Evans (2005)             11-12
Altig, Christiano, Eichenbaum, Linde (2007)      15-16
Full Information DSGE Estimation
Smets and Wouters (2007)                         8-12
Levin, Onatski, Williams, and Williams (2005)    15-18
Micro-Level Data
Bils and Klenow (2004)                           4-5
Nakamura and Steinsson (2008)                    7-11
How to use micro evidence of the frequency of price changes to
calibrate a macro model.

Carlos Carvalho, Heterogeneity in Price Stickiness and the Real
Eﬀects of Monetary Shocks, The B.E. Journal of Macroeco-
nomics: Vol. 2 : Iss. 1 (Frontiers), 2006.
• What if ﬁrm speciﬁc capital?

• The following table is taken from Altig et al. 2005
TABLE 4 of ACEL: IMPLIED AVERAGE TIME (Quarters) BETWEEN
RE-OPTIMIZATION 1/(1 − α)

Model                                     Firm-Speciﬁc    Homogeneous
Capital Model   Capital Model
Benchmark                                     1.51            5.60
Benchmark (6 lag VAR)                         1.12            3.13
Monetary Shocks Only                          3.59            3.64
Neutral Technology Shocks Only                1.00            1.00
Embodied Technology Shocks Only               1.46            6.10
Low Cost of Varying Capital Utilization       2.33            2.34
Intermediate Markup: 1.04                     2.24            5.52
High Markup µ = 1.20                          3.48            5.04
Inﬂation Persistence

• simpler model: no capital accumulation and

F (hit) = zthit
• equilibrium
zt ht   =   st (ct + gt )                                                      (10)
st   =       η
απt st−1 + (1 − α)˜−η
pt                                               (11)
1    =     η−1
απt   + (1 − α)˜1−η
pt                                                  (12)
1−η
η−1              ˜
pt
x1
t     =   p1−η (ct + gt )
˜t                      + αEt                    η
πt+1 rt,t+1 x1
t+1
η              ˜
pt+1
−η
˜
pt
x2
t     =   p−η (ct + gt )mct + αEt rt,t+1
˜t                                             η+1
πt+1 x2
t+1
˜
pt+1
x1t   =   x2
t                                                                 (13)
Uh (ct , ht )
−                 =   wt                                                                 (14)
Uc (ct , ht )
Uc (ct , ht )   =   λt                                                                 (15)
λtrt,t+1      =   βλt+1 /πt+1                                                        (16)
1
Rt     =                                                                      (17)
Et rt,t+1
wt    =   mct Fh(ht )                                                        (18)

plus one equation specifying the monetary policy.
• a log-linear approximation to the equilibrium dynamics around
a non-stochastic steady state with zero inﬂation.

• Let a hat over a variable denote the log deviation from steady
state:
ˆ
xt ≡ ln(xt /x)

• functional form for the utility function:
c1−σ   (1 − h)1−˜
ω
U (c, h) =      +                   ˜
σ, ω > 0
1−σ       1−ω ˜
ˆ        ˆ
πt = βEt πt+1 + κ(ω + σ)ˆt
c                    (19)
ˆ                 ˆ
−σˆt = Rt − σEtˆt+1 − Et πt+1
c             c                                (20)

ˆ       ˆ
Rt = απ πt + t

Let’s write our problem as:

                           
Et ˆt+1
π            β −1     −β −1 κ(ω + σ)
ˆt
π        0
=  απ −β −1         κ(ω+σ)
        +   1
Etˆt+1
c             σ        1 + σβ            ˆt
c        σ t

How to solve this?
First note that this model has (up to ﬁrst order) no endogenous
state variable. This means that, if the model has a unique solu-
tion, then we should expect that as of period t = 1 all variables
must be back at the steady state. Formally, in response to the
monetary policy shock, 0 = 1 we have:

E0 ˆ1
π        0
=
E0ˆ1
c        0
This observation allows us to solve the model:
                           
0         β −1     −β −1 κ(ω + σ)
ˆ
π0       0
=  απ −β −1         κ(ω+σ)
        +   1
0          σ        1 + σβ            ˆ0
c        σ 0

This is just a system of 2 equations in 2 unknowns and we can
solve:
1
ˆ0 = −
c                      0
απ κ(ω + σ) + σ

κ(ω + σ)
ˆ
π0 = −                   0
απ κ(ω + σ) + σ
and then we can back out:
σ
ˆ
R0 =                    0
απ κ(ω + σ) + σ
ˆ       ˆ
It follows that if 0 > 0, then R0 > 0, π0 < 0, and ˆ0 < 0.
c

• Model predicts that monetary policy shock has only a one-period
eﬀect.

• By contrast, in the data, eﬀects of monetary policy are de-
layed and highly persistent.

• This simply version of Calvo Yun + Taylor rule is a poor ex-
planation of the observed eﬀects of monetary policy shocks.
Optimal Monetary Policy in the Calvo-Yun Model: With a
subsidy
• Calvo-Yun model with capital accumulation

• and ﬁscal policy.       Why? we want to allow for a subsidy to ﬁrms to induce them

to produce at the competitive rather than the (lower) monopolistic level. The subsidy

is ﬁnanced with lump-sum taxation.

• We will show: optimal policy calls for constant prices at all
times, that is, the optimal volatility of inﬂation is equal to
zero.

• How to show this result?
• Introduce proportional income tax (subsidy)

ct + it + τtL = (1 − τtD )[wt ht + ut Kt] + φt
˜    (21)

• τtD = income tax rate;      τtL = lump-sum taxes

• the ﬁrst-order conditions associated with the household’s
problem now are
Uc(ct , ht) = λt ,                (22)

−Uh (ct , ht) = wt (1 − τtD )λt ,         (23)
D
λt = βEt λt+1 (1 − τt+1 )ut+1 + (1 − δ)          (24)
• Production side is as before
ztF (Kit , hit) − ψ

• Demand for good i, denoted ait ≡ cit + iit
ait = (Pit/Pt)−η (ct + it),

• Proﬁts of ﬁrm i
Pit
φit ≡     ait − ut Kit − wthit.   (25)
Pt

• present discounted value of proﬁts
∞
Et         dt,s Psφis.
s=t
• ﬁrms must satisfy demand at posted prices
−η
Pit
ztF (Kit , hit) − ψ ≥              at .   (26)
Pt

•
mcitzt Fh(Kit, hit) = wt
and
mcitzt Fk (Kit , hit) = ut .

• as before: mcit , are identical across ﬁrms.
The Fiscal Authority

• ﬁscal policy is passive in the sense that the government’s
intertemporal budget constraint plays no role for the deter-
mination of the equilibrium real and nominal allocation. (Of
course, the size of τtD is important for equilibrium outcomes.)
Competitive Equilibrium

A competitive equilibrium is a set of processes ct, ht, λt , wt,ut,
˜
mct, Kt+1, it , yt , st, pt and a no Ponzi game condition that
satisfy, given initial values for π−1, K0, s−1, stochastic processes
for τtD and πt and the exogenous stochastic process zt:
Kt+1 = (1 − δ)Kt + it          (27)
Uc(ct , ht ) = λt      (28)
−Uh(ct, ht) = wt(1 − τtD )λt         (29)
D
λt = βEtλt+1 (1 − τt+1 )ut+1 + (1 − δ)            (30)
mctzt Fh(Kt , ht ) = wt       (31)
mctztFk (Kt, ht ) = ut        (32)
1−η
1 = απt−1 + (1 − α)˜1−η    pt       (33)
1
yt = [ztF (Kt , ht ) − χ]         (34)
st
yt = ct + it        (35)
st = (1 − α)˜−η + απt st−1
pt           η
(36)
∞                       s
s−t Uc(s)
˜ −1−η
Pt                        η − 1 Pt˜
Et         β                        −1
πk αs−t            (cs + is ) mcs −                 =0      (37)
s=t
Uc(t)                       Ps                           η Ps
k=t+1
s
˜                      −1
˜
Pt /Ps = pt           πk    (38)
k=t+1
• Does there exist a monetary-ﬁscal policy regime that would
result in a real allocation that is Pareto eﬃcient?

• Yes: Set for all t ≥ 0
πt = 1
and set
1
τtD = −
η−1

• all equilibrium conditions are the same as in the case of ﬂex-
ible prices and NO imperfect competition. Thus the alloca-
tion would be Pareto optimal.
• What if no subsidy? Even without the subsidy, optimal mon-
etary policy in the Calvo-Yun model involves zero inﬂation in
the stochastic steady state of the Ramsey equilibrium (see
Woodford 2003)
Implementation of the Optimal Allocation

How to implement the optimal policy, πt = 1?

• Isn’t Implementation Trivial?

• sticky price economy without capital the nominal interest
rate in the Pareto optimal allocation can be found as
λt
Rt =        λ
βEt πt+1
t+1
And because under optimal policy πt+1 = 1 at all times and
in all states, one could consider the following

Rt = ˆt − Et ˆt+1
ˆ    λ       λ
• Let’s consider the economy without capital studied earlier.
We will introduce a subsidy (so that we can prove that πt = 1
and τtD = 1/(1 − η) is the optimal policy.) Also assume that
government purchases follow an exogenous AR(1) process.
The equilibrium conditions of that model are given by:

• In that case, a stationary competitive equilibrium is a set
of processes ct , ht, λt , wt, mct, st, pt , πt , rt,t+1 , x1, and x2
˜                   t       t
for t = 0, 1, . . . that remain bounded in some neighborhood
around the deterministic steady-state and satisfy, given the
initial value s−1 = 1 and exogenous stochastic processes gt
and zt, the following set of equations:
ztht = st(ct + gt)                                               (39)
η               −η
st = απt st−1 + (1 − α)˜t
p                                       (40)
η−1           1−η
1 = απt    + (1 − α)˜t
p                                         (41)
1−η
1−η         η−1         ˜
pt                      η
x1 = pt (ct + gt)
t   ˜                + αEt                         πt+1 rt,t+1 x1
t+1
η        ˜
pt+1
−η
−η                              ˜
pt          η+1
x2
t      ˜
= pt (ct + gt)mct + αEt rt,t+1               πt+1 x2
t+1
˜
pt+1
x1
t   = x2
t                                                     (42)
Uh (ct, ht)
−               = wt                                                     (43)
Uc(ct , ht)
Uc (ct, ht)= λt                                                        (44)
λt rt,t+1 = βλt+1 /πt+1                                               (45)
1
Rt =                                                           (46)
Et rt,t+1
wt = mctFh(ht)                                                 (47)
plus one equation specifying the monetary policy and one
equation specifying ﬁscal policy. The latter two conditions
are meant to pin down Rt and τtD .

• If we solve this economy under the assumption that πt = 1
and τtD = 1/(1 − η), then we obtain from the log-linearization
of equations (45) and (46)

Rt = ˆt − Et ˆt+1
ˆ    λ       λ                    (48)

• Therefore we could consider as the two missing equations:

τtD = 0
ˆ          and    Rt = ˆt − Et ˆt+1
ˆ    λ       λ
• The question is whether those two ﬁscal and monetary policy
rules would render the equilibrium unique and would imple-
ment πt = 1 a.d.a.c.

• Notice that in the economy without capital λt and Etλt+1
are just functions of the state, therefore the monetary policy
rule (48) would, in perfect foresight, amount to an interest
rate peg.

• But we know that under a pure interest rate peg equilibrium
is indeterminate in perfect foresight. (Recall that we proved
above that if απ = 0 in the Taylor rule, then the perfect
foresight equilibrium is indeterminate. So the proposed ’rule’
would not uniquely implement the Pareto optimal allocation.
• Hence, the strategy of postulating the solution for the nom-
inal intersest rate as a policy rule fails to implement the
optimal policy uniquely. And we need to ﬁnd another way to
implement the optimal policy.

• Implementation with a Taylor Rule?

Could one implement the optimal policy, πt = 1 with a
Taylor-type interest rate feedback rule? The answer to this
question is: No. To see this, consider any shock that changes
ˆ
the real rate, Rt − Et ˆt+1. Because πt+1 = 1 under the opti-
π
mal policy, the nominal rate has to be up to ﬁrst order equal
to the real rate. At the same time the Taylor rule would im-
ply that the nominal interest rate is forever constant. This
proves that a Taylor Rule can never implement the Pareto
Optimal allocation.

• Woodford (2001) therefore suggests to modify the Taylor
rule to include what he calls the natural rate of interest,
which is the real rate that would obtain in the absence of
price rigidities.

Let’s consider the optimality condition:
λt+1
λt = βRt Et
πt+1
Notice that if in the Pareto optimal allocation Et λt+1/λt
varies, which it surely does, then it must be the case that Rt
also varies.
• Therefore, Woodford suggests a modiﬁed Taylor rule that
does:
ˆP
Rt = Rt O + απ πt
ˆ              ˆ                  (49)
where
ˆP
Rt O = Et Zt+1    where    Zt+1 = λt /βλt+1

• Do we know whether equilibrium will be unique under this
rule?

• The Model without Capital Yes. For the following reason.
Again we use the fact that ˆt − Et ˆt+1 are only functions
λ     λ
of the state and hence have no inﬂuence on determinacy.
What will therefore control the determinacy of equilibrium is
the coeﬃcient απ . Above, we have shown that in the Calvo-
Yun model without capital, the equilibrium is locally unique
if and only if
απ > 1

It follows that the modiﬁed Taylor rule given in equation (49)
uniquely implements the Pareto optimal allocation if and only
if απ > 1.

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