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Working Paper Series
On the Implementation of Markov-Perfect
Monetary Policy
WP 09-06R
Michael Dotsey
Federal Reserve Bank of Philadelphia
Andreas Hornstein
Federal Reserve Bank of Richmond
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
On the Implementation of Markov-Perfect Monetary
Policy
Michael Dotseyyand Andreas Hornsteinz
July 14, 2011
Working Paper No. 09-06R
Abstract
The literature on optimal monetary policy in New Keynesian models under both
commitment and discretion usually solves for the optimal allocations that are consistent
with a rational expectations market equilibrium, but it does not study whether the
policy can be implemented given the available policy instruments. Recently, King and
Wolman (2004) have provided an example for which a time-consistent policy cannot
be implemented through the control of nominal money balances. In particular, they
…nd that equilibria are not unique under a money stock regime and they attribute the
non-uniqueness to strategic complementarities in the price-setting process. We clarify
how the choice of monetary policy instrument contributes to the emergence of strategic
complementarities in the King and Wolman (2004) example. In particular, we show
that for an alternative monetary policy instrument, namely, the nominal interest rate,
there exists a unique Markov-perfect equilibrium. We also discuss how a time-consistent
planner can implement the optimal allocation by simply announcing his policy rule in
a decentralized setting.
JEL Classi…cation: E4, E5, E6
Keywords: Monetary policy, Markov-perfect, determinacy, interest rate rules, money
supply rules
This is a substantially revised version of a paper previously circulated under the title “Interest Rate
versus Money Supply Instruments: On the Implementation of Markov-Perfect Policy." We would like to
thank Alex Wolman, Per Krusell, Bob King, Thomas Lubik, and Jesus Fernandez-Villeverde for comments.
The views expressed in this paper are those of the authors and do not necessarily represent those of the
Federal Reserve Bank of Philadelphia, the Federal Reserve Bank of Richmond, or the Federal Reserve System.
y
Federal Reserve Bank of Philadelphia
z
Federal Reserve Bank of Richmond
1
Introduction
Currently there is a growing literature exploring the features of optimal monetary policy in
New Keynesian models under both commitment and discretion. This work usually assumes
that the optimal policy solves a constrained planning problem where the policymaker chooses
among all allocations that are consistent with a market equilibrium. Recently, however,
attention has been paid to how to implement the optimal policy through instrument rules.
We believe that this is an important area of inquiry because the institutions responsible
for setting policies rarely have direct control over allocations. It is therefore important to
understand whether or not a planner’s allocations are obtainable with a given institutional
structure.
For the case of time-consistent policies that are Markov-perfect, King and Wolman (2004)
have examined implementation issues when the monetary authority uses nominal money balances as the policy instrument in a sticky price environment. Surprisingly, they …nd that
equilibria are no longer unique under a money-supply regime. Conditional on a given continuation allocation determined by the future policymaker, the current policymaker cannot
implement a unique point-in-time equilibrium. These multiple equilibria are supported by
strategic complementarities in the price-setting process. In particular, if a price-setting …rm
believes that all other price-adjusting …rms will set relatively high (low) prices, then it will
be optimal for the individual …rm to set a relatively high (low) price.
In this paper we clarify how strategic complementarities that are inherent to the pricesetting process interact with Markov-perfect policies. For the case of King and Wolman’s
(2004) money-supply rule, we show that multiple equilibria arise because the money-supply
rule weakens the existing strategic complementarities in the price-setting process for low
in‡ation outcomes. We then study the implementability of a Markov-perfect nominal interest rate policy, since actual monetary policy is usually implemented through interest rate
policies. We …nd that a policy that uses the nominal interest rate as the policy instrument
implements a unique point-in-time equilibrium. We obtain this result because contrary to
1
the money-supply rule, the nominal interest rate instrument uniformly strengthens strategic complementarities and thereby eliminates multiple equilibria. Finally, we brie‡y discuss
how a Markov-perfect nominal interest rate policy can also implement a unique rational
expectations equilibrium.
The comment proceeds as follows. First, we brie‡y describe the standard New Keynesian
economy used by King and Wolman (2004). Second, we describe the strategic complementarities in the price-setting process for …rms. Third, we review the King and Wolman (2004)
result that using a money-supply instrument generates multiple equilibria. Fourth, we show
that using an interest rate instrument uniquely implements the Markov-perfect allocation.
Finally, we discuss how a synthesis of the two instruments, the money supply and the nominal
interest rate, uniquely implements the Markov-perfect allocation as a rational expectations
equilibrium. A brief summary concludes.
2
The Economy
There is an in…nitely lived representative household with preferences over consumption and
leisure. The consumption good is produced using a constant-returns-to-scale technology with
a continuum of di¤erentiated intermediate goods. Each intermediate good is produced by a
monopolistically competitive …rm with labor as the only input. Intermediate goods …rms set
the nominal price for their products for two periods, and an equal share of intermediate …rms
adjusts their nominal price in any particular period. The economy we study is standard, and
to save on space we provide only an outline of the economic environment and a summary
of the equilibrium conditions. For a more detailed derivation, see, for example, King and
Wolman (2004).
2
2.1
The environment
The representative household’s utility is a function of consumption ct , and the fraction of
time spent working nt ,
1
X
t
[ln ct
(1)
nt ] ;
t=0
where
0, and 0 <
< 1. The household’s period budget constraint is
Pt ct + Bt + Mt
Wt nt + Rt 1 Bt
1
+ Mt
1
Pt 1 c t
1
+ Dt + Tt ;
(2)
where Pt is the nominal price level, Wt is the nominal wage rate, Bt are the end-of-period
holdings of nominal bonds, Tt are lump-sum transfers, and Rt
1
is the gross nominal interest
rate on bonds. The agent owns all …rms in the economy, and Dt is nominal pro…t income from
…rms. The household is assumed to hold money in order to pay for consumption purchases
Pt c t
(3)
Mt ;
and money holdings Mt are adjusted at the beginning of the period. We will use the term
“real”to denote nominal variables de‡ated by the nominal price level, which is the price of
the aggregate consumption good, and we use lowercase letters to denote real variables. For
example, real balances are mt
Mt =Pt .
The consumption good is produced using a continuum of di¤erentiated intermediate
goods as inputs to a constant-returns-to-scale technology. There is a measure one of intermediate goods, indexed j 2 [0; 1]. Production of the consumption good ct as a function of
intermediate goods yt (j) is
ct =
Z
"=(" 1)
1
(" 1)="
yt (j)
dj
;
(4)
0
where " > 1. Given nominal prices Pt (j) for the intermediate goods, the nominal unit cost
3
and price of the consumption good is
Pt =
Z
1=(1 ")
1
1 "
Pt (j)
dj
;
(5)
0
and the relative price of good j is pt (j) = Pt (j) =Pt . Producers of the consumption good
behave competitively in their markets.
Each intermediate good is produced by a single …rm, and j indexes both the …rm and
good. Firm j produces its good using a constant-returns technology with labor as the only
input,
yt (j) = nt (j).
(6)
In the labor market, each …rm behaves competitively and takes wages as given, but since
each intermediate good is unique, intermediate goods producers have some monopoly power
for their product. Intermediate goods producers set their nominal price for two periods, and
they maximize the discounted expected present value of current and future pro…ts. Since
the …rm is owned by the representative household, the household’s intertemporal marginal
rate of substitution is used to discount future pro…ts.
2.2
A symmetric equilibrium
We will study a symmetric equilibrium where all intermediate goods producers that face the
same constraints behave the same. This means that in every period there will be two …rm
types: the …rms that adjust their nominal price in the current period, type 0 …rms with
relative price p0;t , and the …rms that adjusted their price in the previous period, type 1 …rms
with current relative price p1;t . Each period, half of all …rms have the option to adjust their
nominal price. The equilibrium of the economy is completely described by the sequence of
real marginal cost, relative prices, in‡ation rates, nominal interest rates, and real balances,
4
f t ; p0;t ; p1;t ;
t+1; Rt ; mt g,
0 = (p0;t )
such that
" 1
(
t
p0;t
p0;t ) +
t+1
" 1
p0;t
t+1
t+1
1 = 0:5 p10;t " + p11;t "
t+1
=
mt =
t
=
(9)
(10)
t=
Rt
(7)
t+1
(8)
p0;t
p1;t+1
t+1
1
(11)
t+1 :
Equation (7) represents the optimal pricing equation for a …rm that can adjust its price in
the current period. The …rst term on the right-hand side is the current period marginal
pro…t, the second term is the discounted present value of next period’s marginal pro…t, and
= "= ("
1) is the markup from the static pro…t maximization problem.1 Equation (8) is
the price index equation (5) in terms of relative prices. Equation (9) relates the in‡ation rate
t+1
Pt+1 =Pt to the ratio of a price-adjusting …rm’s optimal current relative price and next
period’s preset relative price. Equation (10) relates real balances to real marginal cost, using
the household’s optimal labor supply condition, together with the fact that real balances are
equal to consumption. Equation (11) is the household Euler equation after substituting for
the marginal utility of income. For ease of exposition, we will drop time subscripts from now
on and denote next period’s values by a prime.
Allocations in this economy are suboptimal because of two distortions. The …rst distortion results from the monopolistically competitive structure of intermediate goods production: the price of an intermediate good exceeds its marginal cost. The second distortion
re‡ects ine¢ cient production when relative prices are di¤erent from one.2 The policymaker
is assumed to maximize the lifetime utility of the representative agent, taking the competitive equilibrium conditions (7)-(11) as constraints. For a time-consistent Markov-perfect
policy, the policymaker takes future policy choices as given and current policy choices are
1
2
The current and discounted future pro…ts are scaled by 1= (" 1).
For a more detailed discussion of the distortions, see King and Wolman (2004).
5
functions of payo¤-relevant state variables only. Because there are no state variables in our
example, this amounts to the planner choosing a non-contingent allocation that maximizes
the current period utility function of the representative agent. Taking future policy as given
means that the planner has no control over future outcomes, such as future relative prices
or allocations.
One usually states the problem in terms of the planner choosing the market allocation. In
this case we can view the planner choosing a vector x = (p0 ; p1 ; 0 ; ) subject to constraints
(7)-(11), and conditional on the choices of next period’s policymaker, x0 . The planner’s
choices determine the representative household’s utility through their impact on allocational
e¢ ciency and the markup. Note that the statement of the planner’s problem in terms of
the market allocation does not involve any reference to the policy instrument, z, be it real
balances or the nominal interest rate. To determine whether the Markov-perfect equilibrium
can be implemented as a competitive equilibrium, we have to characterize the feasible set
for market outcomes x conditional on the policy instrument.
3
Implementation of Point-in-Time Equilibria
In most models of monetary economies, money-supply policies lead to a unique equilibrium
with a determinate price level, whereas interest rate policies imply equilibrium indeterminacy.
Exactly the opposite is true for the simple economy we have just described. As King and
Wolman (2004) have shown, a Markov-perfect money-supply rule will imply non-uniqueness
for the point-in-time equilibrium, and as we will show, a Markov-perfect interest rate policy
will imply a unique point-in-time equilibrium. It turns out that (non)uniqueness of the
equilibrium is related to the presence of strategic complementarities in the price-setting
process and how the policy rule ampli…es or weakens these complementarities.
Before we discuss the two policy rules, we want to demonstrate that strategic complementarities are inherent to the …rms’price-setting problem. In the context of the model’s
6
monopolistic-competition framework, strategic complementarities are said to be present if
it is optimal for an individual price-adjusting …rm to increase its own relative price, p0 , if
all other price-adjusting …rms increase their relative price, p0 . To study this issue we use
a graphic representation of the individual …rm’s FOC for pro…t maximization, (7), which
states that the sum of today’s marginal pro…t, M P (p0 ; ), and tomorrow’s discounted marginal pro…t, M P (p0 = 0 ;
0
) = 0 , has to be zero. For the pro…t maximization problem to be
well-de…ned, we need the pro…t function to be concave; that is, the marginal pro…t function
M P is decreasing in the relative price. In the Appendix we also show that
Proposition 1 With constant marginal cost,
is increasing in the in‡ation rate
0
=
0
, tomorrow’s marginal pro…t, M P (p0 = 0 ;
for a neighborhood around zero in‡ation,
0
= 1.
In Figure 1, we graph today’s marginal pro…t (solid line) and tomorrow’s marginal pro…t
(dashed line) for an individual …rm conditional on all other …rms’relative price, p0 , and a
positive in‡ation rate. The positive in‡ation rate erodes the …rm’s relative price tomorrow
and therefore the …rm will set its optimal price, p0 , above the static pro…t-maximizing relative
price,
, such that it balances today’s negative marginal pro…t against tomorrow’s positive
marginal pro…t. Now suppose that all other …rms increase their relative price. It follows
from expression (9) that tomorrow’s in‡ation rate will increase,
0
= p0 =p01 , and this will shift
tomorrow’s marginal pro…t curve up (dashed-dot line), leaving today’s marginal pro…t curve
unchanged. It is then optimal for the individual …rm to also increase its own relative price.
Thus, there is a source of strategic complementarities, independent of monetary policy. The
choice of monetary policy instrument will modify strategic complementarities through its
general equilibrium feedback e¤ect on marginal cost.
3.1
A money supply policy
We now review King and Wolman’s (2004) analysis of a Markov-perfect nominal money rule
that sets the nominal money stock in proportion to the preset nominal price from the last
7
0
) = 0,
period3
M = mP
~ 1:
(12)
Normalizing the policy rule (12) with the price level and combining it with the equilibrium
condition (10) determines marginal cost
= mp
~ 1:
(13)
King and Wolman (2004) show that for most values of the money-supply policy parameter, m;
~ the steady-state of the economy will not be unique. Since in a Markov-perfect
equilibrium without state variables the expected future policy has to be a steady state, nonuniqueness of the steady state alone suggests that the monetary policy rule may result in
indeterminacy of the point-in-time equilibrium. Suppose that we choose one of the possible
steady states as a continuation of the economy in the next period. We now show that the
choice of a money-supply instrument weakens strategic complementarities when the average
…rm chooses a low relative price, and that the complementarities persist when the average
…rm chooses a high relative price. The resulting change in shape of the optimal reaction
function, that is, the mapping from the average …rm’s relative price to an individual …rm’s
optimal relative price response, gives rise to multiple point-in-time equilibria.
Consider again the response of an individual …rm to an increase in the relative price
set by all other …rms, but now allow for the feedback of these decisions to marginal cost
coming through the money stock policy. When all other price-adjusting …rms increase their
relative price, it follows from the price index equation, (8), that the preset relative price, p1 ,
declines. From equation (13) it then follows that today’s marginal cost declines, which in
3
Prior to studying the Markov-perfect money rule, King and Wolman (2004) brie‡y discuss a monetary
policy rule that exogenously sets the nominal money stock at a constant value. For this policy a …rm’s
pricing decision is not a¤ected by other …rms decisions and the equilibrium is unique. One can also show
that for a constant money growth rule the pricing decisions are charcterized by strategic complementarity
(substitutability) if the money stock is shrinking (growing). Nevertheless, with an exogenous money stock
the current period outcome will depend on the preset nominal price, and since this price is not payo¤ relevant,
this policy is not Markov-perfect.
8
turn shifts down today’s marginal pro…t curve in Figure 1. Thus the policy-induced feedback
e¤ect reduces the individual …rm’s need to increase its own relative price in response to the
general price increase; that is, it weakens the strategic complementarities.
It is easily shown that the impact of p0 on p1 declines with p0 . Thus, strategic complementarities are weakened the most when the relative price of price-adjusting …rms is the
lowest. The resulting shape of a …rm’s optimal response function is depicted as the dashed
line in Figure 2. The graphs displayed in Figure 2 are derived for parameter values
" = 11,
rate
= 0:99,
= 1= , and assuming that next period’s policy generates a steady-state in‡ation
= 1:05. This parameterization is standard for sticky price models and implies a static
markup of 10 percent, and an annual real interest rate of 4 percent if we interpret a period
as a quarter. We can see that for low values of other …rms’relative price choice, there are
no strategic complementarities, and the reaction function is quite ‡at. If other …rms start
setting higher relative prices, then an individual …rm’s own optimal relative price starts to increase and the rate at which it responds also increases. Thus, the reaction function becomes
steeper than the 45-degree line and multiple equilibria due to self-ful…lling expectations are
possible. In the Appendix we prove the following Proposition.
Proposition 2 Suppose current and future policymakers use the same money stock rule m.
~
If m
~ 2 (m
~ 1; m
~ 2 ), then, in general, at least two point-in-time equilibria exist. If m
~ =m
~ 1 then
the point-in-time equilibrium is unique.
3.2
An interest rate policy
In this section we evaluate the bene…ts of using an interest rate instrument to implement
Markov-perfect policies. We …nd that steady states and point-in-time equilibria are unique,
despite the fact that the reaction function remains characterized by strategic complementarities. In what follows, we solve for the current equilibrium, x, conditional on current policy
z = R and future equilibrium outcomes x0 . With a …xed nominal interest rate, policy a¤ects
9
marginal cost through the Euler equation,
=
0
Rp01
p0 ;
(14)
which combines (9) and (11).
The existence of a unique steady state for a given nominal interest rate is straightforward
to show; see the Appendix.
Proposition 3 Conditional on the nominal interest rate R > 1, there exists a unique steady
state (p0 ; p1 ;
).
A point-in-time equilibrium also exists and it is unique despite the continued presence
of strategic complementarities. Indeed, the interest rate rule strengthens existing strategic
complementarities. Consider again the response of an individual …rm to an increase in the
relative price set by all other …rms, but now allow for the feedback coming through the interest rate policy. From equation (14) it now follows that today’s marginal cost increases, which
in turn shifts up today’s marginal pro…t curve in Figure 1. Thus, the policy-induced feedback
e¤ect increases the individual …rm’s need to increase its own relative price in response to the
general price increase; that is, it strengthens the strategic complementarities.4 The dash-dot
line in Figure 2 displays the reaction function for the interest rate policy conditional on the
same parameter values used for the money stock rule. In the following proposition, proved
in the Appendix, we state that as long as tomorrow’s policy does not try to implement price
stability, there will always exist a unique point-in-time equilibrium for the current period.
Proposition 4 (A) If next period’s policy choice attains an in‡ationary or de‡ationary
steady-state outcome, then (1) for any nominal interest rate for which a current period
equilibrium exists it is unique, and (2) there always exists a nominal interest rate for which
4
We note that the unique equilibrium is not obtained because the interest rate instrument introduces
commitment to the policymakers choice set. Indeed, the Markov-perfect solution of the planning problem is
obtained without even considering how the optimal allocation can be implemented, be it through a money
rule or an interest rate rule.
10
an equilibrium exists. (B) If next period’s policy choice attains a steady-state outcome with
stable prices, then (1) the current period equilibrium is indeterminate if current policy also
tries to attain the stable-price steady state R = 1; (2) no current period equilibrium exists
if R 6= 1.
Finally, the monetary policymaker can implement the Markov-perfect equilibrium as a
globally unique rational expectations equilibrium through a policy that jointly determines
the nominal interest rate and the money stock as in Carlstrom and Fuerst (2001) and Adão,
Correia, and Teles (2003). The choice of a nominal interest rate eliminates the potential for
multiple point-in-time equilibria, whereas the money rule picks the Markov-perfect equilibrium allocation among the possible solutions to the system of dynamic equations. Formally,
a choice of z = (m;
~ R) that is consistent with the Markov-perfect equilibrium determines
a unique rational expectations equilibrium as follows. From (13) we observe that choosing
m
~ =
=( p1 ) yields a monetary policy that is consistent with the Markov-perfect equilib-
rium. Using this choice we can substitute for marginal cost in the household Euler equation
(11) and together with the de…nition of in‡ation (9) and the choice of R this yields the
following linear restriction on current relative prices.
mp
~ 01
p1
1
mp
~ 1
=
)
=
:
p0
Rp01
p0
R
This restriction together with the price index equation (8) uniquely determines relative
prices. Given the unique relative prices, one obtains unique solutions for real balances and
marginal cost.5
The Markov-perfect equilibrium can only be implemented through the joint determination of the money stock and the interest rate, since, for the usual reasons, a non-contingent
5
Standard characterizations of monetary policy have the policymaker set a price (quantity) and have
the quantity (price) be determined as an equilibrium outcome. The proposed combination policy has the
policymaker choosing both, price and quantity, and therefore requires that the policymaker have complete information on the state of the world. It is not obvious if a policymaker can implement this kind of combination
policy in an environment with incomplete information.
11
nominal interest rate only policy leads to locally indeterminate equilibria. Early work of
Sargent and Wallace (1975) has shown how a non-contingent nominal interest rate policy
leads to nominal indeterminacy in ‡exible price models, and Carlstrom and Fuerst (2001)
and Adão, Correia, and Teles (2003) have pointed out that such a policy leads to real indeterminacy in sticky price models. The usual procedure to eliminate dynamic indeterminacies
arising from a …xed nominal interest rate policy — making the interest rate decision contingent on other endogenous variables; see, e.g., McCallum (1986), Boyd and Dotsey (1994),
or Carlstrom and Fuerst (1998), cannot be used to implement Markov-perfect equilibria.
This approach is not applicable, since, by de…nition, decisions in Markov-perfect equilibria
can depend only on payo¤-relevant state variables and not other endogenous variables, be
they lagged or contemporaneous. A feasible way to obtain a locally unique rational expectations equilibrium for the interest rate rule is to restrict the solution to be in accord
with McCallum’s (1983) minimum state variable solution. Since there are no state variables,
the minimum state variable solution must be the steady state, which we have shown to be
unique for the interest rate policy, both in real and nominal terms. We also note that in
an economy like ours with ‡exible prices, it is well known that the minimum state variable
solution still displays nominal indeterminacy. This di¤erence indicates another important
distinction between ‡exible and sticky price environments.
4
Conclusion
In this comment we have analyzed the importance of the monetary policy instrument in
decentralizing a time-consistent planner’s optimal policy. In that regard, our work is part
of a growing literature investigating the implementation of optimal plans. We have shown
that whether a planner uses a money instrument or an interest rate instrument is crucial for
determining if optimal Markov-perfect allocations can be attained via the appropriate setting of the instrument. King and Wolman (2004) were the …rst to alert us to the non-trivial
12
rami…cations of decentralization. They produced a surprising result of signi…cant impact,
namely, that decentralization is a non-trivial problem. With regard to using a money instrument, implementation of the optimal allocation is unattainable. A time-consistent planner
using a money instrument could not implement the allocations chosen by a planner who was
able to directly pick allocations. In fact, they showed that steady states and equilibria were
not unique at the optimal in‡ation rate. Since, in reality, no central bank picks allocations,
this result presents a challenge for understanding just how a time-consistent central bank
might operate. Here we have shown that it does not. A planner using an interest rate instrument can achieve the Markov-perfect allocations of the standard time-consistent planning
problem. The result occurs for two key reasons. The interest rate instrument pins down
future in‡ation in ways unobtainable using a money instrument and, in so doing, increases
the degree of strategic complementarity that arises from the monopolistically competitive
price-setting problem itself.
References
[1] Adão, B., I. Correia and P. Teles, 2003. Gaps and triangles, Review of Economic Studies
70 (4), 699-713.
[2] Carlstrom, C.T. and T.S. Fuerst. 1998. Price-level and interest-rate targeting in a model
with sticky prices, Federal Reserve Bank of Cleveland Working Paper 9819.
[3] Carlstrom, C.T. and T.S. Fuerst. 2001. Timing and real indeterminacy in monetary
models, Journal of Monetary Economics 47, 285-298.
[4] Boyd, J.H. and M. Dotsey. 1994, Interest rate rules and nominal determinacy, manuscript.
[5] King, R. G., A.L. Wolman, 2004. Monetary discretion, pricing complementarity, and
dynamic multiple equilibria, Quarterly Journal of Economics 119, 1513-1553.
13
[6] McCallum, B.T. 1983. On non-uniqueness in rational expectations models: An attempt
at perspective, Journal of Monetary Economics 11, 139-168.
[7] McCallum, B.T. 1986. Some issues concerning interest rate pegging, price-level determinacy, and the real bills doctrine, Journal of Monetary Economics 17, 135-160.
[8] Sargent, T. and N. Wallace. 1975. Rational expectations, the optimal monetary instrument, and the optimal money-supply rule, Journal of Political Economy 83, 242-254.
14
Appendix
A
Proof of Prop 1. Strategic Complementarities
The optimal relative price of a price-setting …rm satis…es the FOC for pro…t maximization
(7). With constant marginal cost, = 0 , and positive in‡ation this implies
p0
1
p0 =
0
(A.1)
since the marginal pro…t function is decreasing in p0 . The derivative of the …rm’s marginal
pro…t tomorrow with respect to in‡ation is
@M P (p0 = 0 ; ) =
@ 0
0
= ("
1)
p0
" 1
0
2
p0
1
0
(A.2)
02
Thus, tomorrow’s marginal pro…t is increasing in in‡ation if and only if
2
>
p0
0
(A.3)
:
Note that with zero in‡ation the optimal relative price satis…es p0 =
positive markup, > 1, we get
2
>
= p0 :
. Since we have a
(A.4)
By continuity condition (A.3) is satis…ed for a neighborhood around zero in‡ation.
B
Proof of Prop 2.
Money Rule
Non-uniqueness of PITE with
Suppose that today’s and tomorrow’s policymakers choose the same policy rule within the
set of feasible policy rules, m
~ =m
~ 0 2 (m
~ 1; m
~ 2 ). It is straightforward to show that this policy
is consistent with the existence of two steady-state equilibria (King and Wolman (2004)).
We now show that even conditional on choosing future behavior to be in accord with one of
the two possible steady states, p01 = p1 and 0 = , there exist two point-in-time equilibria
in the current period. An individual …rm’s optimal relative price is determined by the pro…t
maximization condition, (7),
0"
+
p0 =
;
(B.1)
1 + 0" 1
conditional on today’s marginal cost and tomorrow’s marginal cost and in‡ation rate. Together with the policy rule (13) and the de…nition of the in‡ation rate (9), the reaction
function simpli…es to
1
(p1 =p0 ) + (p0 =p1 )"
p0 = p0
m
~
1 + (p0 =p1 )" 1
15
1
= g (p0 ; p1 ) :
(B.2)
In equation (B.2) the left-hand side price p0 is interpreted as an individual …rm’s optimal
relative price in response to the expected aggregate relative prices, p0 and p1 , on the righthand side. Note that the price index equation (8) implies that p1 is a decreasing function
of p0 . For parameter values and policy choice such that
m
~ = 1, we can interpret g as the
reaction function and Figure 3 can be used to visualize the argument below.
One can show that the “reaction” function g in terms of the relative price p0 intersects
the 45-degree line at p0 = 1 and is above (below) the 45-degree line when p0 is less than
(greater than) one,
8
9
8
9
< < =
< > =
= 1:
g (p0 ; p1 ) = p0 for p0
(B.3)
:
;
:
;
>
<
As p0 becomes large, g(p0 ; p1 ) converges to the 45-degree line from below,
lim g (p0 ; p1 ) = p0 :
p0 !1
(B.4)
With some some additional algebra, one can show that the derivative of the g function at
p0 = 1 is
1
(p1 )1 "
@g (p0 ; p1 )
jp0 =1 =
:
(B.5)
@p0
1 + (p1 )1 "
We can now show that for m
~ 2 (m
~ 1; m
~ 2 ) the LHS and the RHS of expression (B.2) will in
general intersect twice. On the one hand, from the properties of the steady state it follows
that since m
~ > m
~ 1 , that is,
m
~ > 1, the slope coe¢ cient of the LHS linear expression
in p0 is less than one. Thus the LHS de…nes a line through the origin below the 45-degree
line. On the other hand, the RHS of (B.2) intersects the 45-degree line at p0 = 1 and stays
above (below) the 45-degree line whenever p0 is less than (greater than) one. Furthermore,
as p0 becomes arbitrarily large the RHS of (B.2) converges to the 45-degree line from below.
Since the LHS is strictly below the RHS for p0
1, the two curves do not intersect in
this range. We know that at least one intersection point exists, since we consider policy
rules that are consistent with the existence of a steady state, and the steady-state price is a
solution to the reaction function (B.2). Thus, there must be an intersection point for p0 > 1.
If m
~ =m
~ 1 , then we know that a unique non-in‡ationary steady state with p0 = 1 exists,
and this steady state also satis…es (B.2). For this case, the LHS is the 45-degree line and the
RHS has a unique intersection with the 45-degree line at p0 = 1. Furthermore, from (B.5)
it follows that the slope of the RHS at p0 = 1 is negative. With a marginally larger value of
m,
~ the slope of the LHS becomes less than one, and there will be at least two intersections
with the RHS to the right of p0 = 1.
C
Proof of Prop 3. Uniqueness of Steady State with
Interest Rate Rule
Equations (14) and (9) determine the unique steady-state in‡ation rate
= R:
16
(C.1)
Equations (8), (9), and (C.1) uniquely determine the steady-state relative prices
p 0"
1
= 0:5 1 +
" 1
and p1 = p0 = :
(C.2)
From equation (7) we obtain the steady-state marginal cost
=
D
" 1
11+
1+
"
p0 :
(C.3)
Proof of Prop 4. (Non)uniqueness of PITE with
Interest Rate Rule
The current equilibrium is de…ned by the two equations (14) and (7), which map the current
period relative price p0 to current period marginal cost : Rewriting (7), we have
1 0
R p01
= f1 (p0 ) =
= f2 (p0 ) =
1
p0
(p0 + A0 p"0 ) ;
(D.1)
(D.2)
0
where A0 = (p01 )1 " 1
, and next period’s variables are evaluated at their steadyp01
state values, p1 and
as determined by (C.1), (C.2) and (C.3). An intersection of the two
functions represents a potential equilibrium.
The two functions always intersect at p0 = 0, but p0 = 0 is not a feasible outcome
since the price index equation (8) together with p1 positive implies a lower bound p0 for the
¯
optimal relative price. Both functions are strictly increasing at p0 = 0,
1
+ ( )"
@f1
=
@p0
R 1 + ( )"
@f2
1
=
1 + A0 "p"0 1 :
@p0
(D.3)
(D.4)
The function f2 is strictly concave (linear, strictly convex) if A0 < 0 (A0 = 0, A0 > 0),
@ 2 f2
1 0
=
A " ("
2
@p0
1) p"0 2 :
(D.5)
The sign of the term A0 depends on the in‡ationary stance of next period’s steady-state
17
policy. From (7) we get
A0 =
1 "
(p1 )
1 "
=
(p1 )
=
(p1 )1
"
(
1
(
1
# )
1
1 1 + ( )" 1
" p0
1+ ( )
p1
)
" 1
01 + ( )
1 + ( )"
"
1
1 + ( )"
(D.6)
:
The …rst equality uses the steady-state expression for next period’s marginal cost (C.3), and
the second equality uses the steady-state expression for next period’s in‡ation rate (C.2).
Thus, A0 is negative (positive) if next period’s policy is in‡ationary,
> 1 (de‡ationary,
0
< 1), and A = 0 if next period’s policy implements price stability,
= 1.
If next period’s policy is in‡ationary and an intersection between f1 and f2 exists for
positive values of p0 , the intersection point is unique since the function f1 is linear and the
function f2 is strictly concave. The two functions intersect for positive p0 if at p0 = 0 the
function f2 is steeper than f1 ,
@f1
1 1
+ ( )"
1
@f2
=
=
" <
@p0
R 1+ ( )
@p0
:
(D.7)
p0 =0
This condition can always be satis…ed for a su¢ ciently large nominal interest rate R 1. In
other words the policymaker can always …nd an interest rate for which the functions intersect.
Recall that there is a lower bound for feasible relative prices p0 , so the policymaker has to
choose an interest rate that implies a su¢ ciently large value¯ for the relative price p0 . A
policymaker can always …nd such an interest rate, since he can always replicate the steady
state by choosing R = R . Thus there exists a choice for R such that an equilibrium exists
and it is unique. An analogous argument applies if next period’s policy is de‡ationary.
If next period’s policy implements price stability, that is,
= 1= and p1 = 1, then
the only policy for today that is consistent with the existence of an equilibrium is a nominal
interest rate such that R = 1. But then equations (D.1) and (D.2) are satis…ed for any
feasible combination of (p0 ; ) such that
p0 > p0 and = p0 = .
¯
If current policy is in‡ationary or de‡ationary, R 6= 1, then the only solution to equations
(D.1) and (D.2) is p0 = 0. But p0 = 0 is not a feasible outcome, so no equilibrium exists.
18
Figure 1: Strategic Complementarities
Marginal Profit
Marginal Profit Tomorrow MP(p0/π ′,ψ ′)/π ′
π′ ↑ ⇒
Own Price p0
p0
μ ψ =1
ψ↑⇒
Marginal Profit Today MP(p ,ψ)
0
19
Figure 2: Reaction Functions for Money Stock and Interest Rate Rules
Reaction Function
1.35
Money Stock Rule
Interest Rate Rule
1.3
Own relative price choice p0(t)
1.25
1.2
1.15
1.1
1.05
1
0.95
0.95
1
1.05
1.1
1.15
1.2
Other firms’ relative price choice p0(t)
20
1.25
1.3
1.35
@FedFRASER