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Working Paper Series
The Pitfalls of Monetary Discretion
WP 01-08
Aubhik Khan
Federal Reserve Bank of Philadelphia
Robert G. King
Federal Reserve Bank of Richmond
Boston University
Alexander L. Wolman
Federal Reserve Bank of Richmond
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
Aubhik Khan†
Robert G. King‡
Alexander L. Wolman§
Federal Reserve Bank of Richmond Working Paper No. 01-08
October 2001
JEL Nos. E52, E51, E58, E31, E42
Keywords: discretion, time-consistency problem, optimal monetary policy,
sticky prices, multiple equilibria
Abstract
In a canonical staggered pricing model, monetary discretion leads to multiple private sector equilibria. The basis for multiplicity is a form of policy complementarity.
Specifically, prices set in the current period embed expectations about future policy,
and actual future policy responds to these same prices. For a range of values of the
fundamental state variable — a ratio of predetermined prices — there is complementarity
between actual and expected policy, and multiple equilibria occur. Moreover, this multiplicity is not associated with reputational considerations: it occurs in a two-period
model.
∗
We thank Huberto Ennis, Andreas Hornstein and Per Krusell for helpful discussions, Luca Guerrieri and
Eric Swanson for comments on a previous draft, and participants in the 2001 NBER Summer Institute and
seminars at the European Central Bank and the Federal Reserve Board for their comments. They bear no
responsibility for the contents of this paper. The views expressed here are the authors’ and not necessarily
those of the Federal Reserve Banks of Philadelphia or Richmond or the Federal Reserve System.
†
Federal Reserve Bank of Philadelphia, Aubhik.Khan@phil.frb.org.
‡
Boston University, Federal Reserve Bank of Richmond, and NBER, rking@bu.edu.
§
Federal Reserve Bank of Richmond, Alexander.Wolman@rich.frb.org.
1
Introduction
A discretionary monetary authority responds optimally to the economy’s
state. The state includes prices set by firms in the past, and those prices
were set based on expectations of what the future monetary authority would
do. Implicitly then, the monetary policy action optimally responds to the
expected monetary policy action. Under commitment, this channel is nonexistent, because the entire sequence of policy actions was determined in the
initial period. We study discretionary monetary policy in a canonical staggered price-setting model, and show that the endogeneity of policy under
discretion can lead to multiple private-sector equilibria. There can be more
than one set of beliefs about future monetary policy that, when incorporated
into current prices, induces the future monetary authority to rationalize these
beliefs.
The multiple equilibria we describe are not associated with an infinite
horizon: they occur even in a two-period model, which is the case we study
in this paper. The two-period case lets us highlight two different situations for
the monetary authority. In the final period it is straightforward to describe
the nature of private sector equilibrium and optimal discretionary monetary
policy; there is a unique optimum which varies smoothly with the single
fundamental state variable. In the initial period, matters are much more
complicated.
Just as in the literature on coordination failure in macroeconomics (Cooper
and John [1988]), multiple equilibria under discretionary monetary policy reflect a form of complementarity. But in contrast to many examples in that
literature, here the complementarity is intrinsically dynamic. The staggered
price-setting model contains two mechanisms that drive the dynamic complementarity. First, a monopolistically competitive firm will set a higher
nominal price in the current period if it expects a higher future price level,
because it will continue to charge that price in the future. Second, the firm
understands that the future monetary authority will respond to the state of
the economy, which varies with the prices set by all firms in the current period. In some circumstances, it is optimal for the future monetary authority
to aggressively increase the money stock in response to the state, thereby
making the future price level increase with the future state. This policy
endogeneity can create complementarity among price-setters, and multiple
equilibria can arise.
When certain policy actions can lead to multiple equilibria, the initial
1
period monetary authority must have beliefs about the likelihood that each
equilibrium will occur in order to solve its policy problem. We explore two
assumptions about equilibrium selection: under the first assumption it is
common knowledge that all equilibria are equally likely (uniform selection)
and under the second it is common knowledge that the best equilibrium will
always occur (Pareto selection). Under each of these assumptions, we find
that the presence of multiple equilibria considerably complicates the nature
of optimal policy. Even though maximized welfare is a smooth function of the
state, optimal policy exhibits nonstandard behavior. The policy function is
nonmonotonic and has one or more discontinuities, depending on the selection
assumption. These discontinuities arise because of multiple local maxima to
the policymaker’s objective function, with the global maximum flipping from
one local maximum to the other as the state variable changes.
Like other recent work discussed in section 6, we study a modern version
of the Kydland-Prescott and Barro-Gordon time consistency problem for
monetary policy. A form of the time consistency problem they described
naturally arises in the sticky price models that are now commonly used for
monetary policy analysis. Unlike the original models however, with explicit
staggered price-setting there is a natural state vector (here it is a scalar)
representing predetermined relative prices. The existence of a natural state
drives our results. These results, however, are not familiar from recent work.
Despite the fact that we study an ordinary model of staggered pricing with
optimizing behavior, as in King and Wolman (1999), little has been known
about the nature of discretionary monetary policy in this setting.1 The work
that has been done, for example by Dotsey and Hornstein (forthcoming) in
a model almost identical to ours, and by Clarida, Galí and Gertler (1999),
Woodford (1999) and Svensson and Woodford (1999) in models with Calvostyle pricing, has employed linear-quadratic approximation methods and a
primal approach to optimal policy.2 Those methods have not uncovered the
multiple equilibria we describe.3
1
The model in this paper is close to that of Chari, Kehoe and McGrattan (2000). Here
there is no capital, however, and firms that adjust their prices do so after current period
information is revealed.
2
We should also note that the central topic of Dotsey and Hornstein’s paper is how
different information structures affect the monetary policy problem.
3
Currie and Levine (1993) contains references to an earlier literature that derived
Markov-perfect discretionary equilibria in linear models with quadratic objectives. Oudiz
and Sachs (1984) is perhaps the first example.
2
The paper proceeds as follows. Section 2 lays out the model. Section 3
describes private sector equilibrium under arbitrary monetary policy. We use
backward induction to describe and compute equilibrium. Section 4 describes
the channels through which monetary policy can affect real outcomes in the
model. These channels are summarized by two distortions: a relative price
distortion across symmetric firms, and the markup of price over marginal
cost. Section 5 contains the results on optimal policy. In section 6 we place
our results in the context of other recent literature that emphasizes multiple
equilibria under discretion in related models. Section 7 concludes.
2
The model
The model has three sets of agents: a representative household, a continuum of monopolistically competitive firms, each producing a differentiated
consumption good, and a government that supplies money and levies lumpsum taxes. There is three-period staggered pricing: firms set their prices for
three periods, and each period one-third of all firms readjust their prices. The
model does not have any life-cycle features. While it is natural to assume
an infinite horizon for households, in order to study the issue of multiple
equilibria we find it useful to analyze a finite-horizon model.
We make the following assumptions about timing of actions in a given
period. At the beginning of period t, the prices set by firms in previous
periods are predetermined. A ratio of these nominal prices serves as a
real state variable. The monetary authority chooses the money supply as
a function of the state, given its beliefs about the future evolution of the
economy (that is, given its beliefs about future monetary policy). Finally,
those firms adjusting their prices in period t choose their prices, and all other
period-t variables are simultaneously determined. The state for period t + 1
is known at the end of period t.
2.1
Households
Our model has T +1 periods, where T ∈ [0, ∞), and households have standard
preferences over consumption (c) and leisure (l):
T
Ut =
j=0
β j u (ct+j , lt+j ) , T ≥ 0.
3
(1)
Note that households live for the duration of the model. Consumption is a
Dixit-Stiglitz aggregate of the differentiated products,
1
ct =
ct (z)
ε−1
ε
ε
ε−1
dz
(2)
,
0
and households supply labor (nt = 1 − lt ) in a competitive labor market for
wage wt . The household’s budget constraint is
Pt ct = Pt wt nt + Dt ,
where Pt is the price index for the aggregate consumption good and Dt is
dividend payments from firms. Efficient behavior dictates that consumption
and leisure are chosen so that the real wage equals the marginal rate of
substitution between consumption and leisure:
(3)
wt = ul (ct , lt ) /uc (ct , lt ) .
We do not derive money demand from microfoundations, but instead
assume that velocity is constant and equal to unity:
(4)
Mt = Pt ct .
By abstracting from any distortions associated with money demand, we isolate phenomena associated with sticky prices. In other work (Khan, King and
Wolman [2000]) we have found that the distortions associated with money
demand appear to be small. It would of course be interesting to study a
version of the model in which money demand is generated by money-in-theutility-function, for example, or the costly credit framework in Khan, King
and Wolman (2000).
Denoting by Pt (z) the nominal price of good z, individuals’ demands for
the differentiated consumption goods are
ct (z) =
Pt (z)
Pt
−ε
(5)
ct ,
and the price index is
1
Pt =
1/(1−ε)
Pt (z)1−ε dz
.
(6)
0
Both the demand functions and the price index are implied by optimal allocation of a given level of consumption across the differentiated products,
taking the price of each product as given.
4
2.2
Firms
Each of the differentiated product manufacturers (z ∈ [0, 1]) has access to a
deterministic technology that is linear in labor:
(7)
yt (z) = nt (z) .
Thus, firm z’s real profits in period t are given by
Pt (z)
− wt .
Pt
π t (z) = yt (z) ·
(8)
By substituting from the demand functions (i.e.- using yt (z) = ct (z)), we can
write profits as a function of relative price, the wage, and aggregate demand:
π
Pt (z)
, wt , ct
Pt
Pt (z)
Pt
=
−ε
·
Pt (z)
− wt · ct .
Pt
(9)
A firm adjusting its price in period t will charge the same price in periods
t+1 and t+2. Given that firms are owned by households, when a firm adjusts
its price it solves the following problem
max
min{2,T −t}
P0,t
j=0
βj ·
uc (ct+j , lt+j )
·π
uc (ct , lt )
P0,t
, wt+j , ct+j ,
Pt+j
(10)
the solution to which is
P0,t
=
Pt
ε
ε−1
min{J−1,T −t} j
β · uc (ct+j , lt+j ) · ct+j · wt+j · (Pt+j /Pt )ε
j=0
.
min{J−1,T −t} j
β · uc (ct+j , lt+j ) · ct+j · (Pt+j /Pt )ε−1
j=0
(11)
Note that because of the nature of price setting, the continuum of firms
sorts naturally into three groups. Let cj,t denote consumption in period t of
a good whose price was set in period t − j, and let Pj,t denote the price of
such a good. Then the consumption aggregator is
2
ct =
j=0
ε−1
1
· cj,t ε
3
ε
ε−1
,
(12)
and total labor input is
2
nt =
j=0
1
· nj,t =
3
5
2
j=0
1
· cj,t .
3
(13)
The price index is given by
2
Pt =
j=0
1
· Pj,t 1−ε
3
1/(1−ε)
(14)
.
Because individual prices remain fixed for three periods,
(15)
Pj,t−1 = Pj+1,t , j = 0, 1.
That is, a newly set price in period t − 1 becomes a one-period old price in
period t, etc.
2.3
The government, and the natural state variable
The government and the monetary authority are synonymous in our analysis.
The government chooses the level of the money supply, and changes in the
money supply are accomplished via lump-sum transfers, Ft (or lump-sum
taxes, in the case of decreases in the money supply):
Mt − Mt−1 = Ft .
(16)
This equation is the government budget constraint for an economy where
there is no interest-bearing government debt, and no distortionary taxes or
government spending.
Henceforth, we will normalize the money supply and all other nominal
variables by PD,t , an index of the prices charged in period t by firms that set
their prices in periods t − 1 and t − 2 :
PD,t ≡
1
1
1−ε
1−ε
· P1,t
+ · P2,t
2
2
1/(1−ε)
.
(17)
Note that the aggregate price index can be written as
Pt =
1
2
1−ε
1−ε
+ · PD,t
· P0,t
3
3
1/(1−ε)
.
(18)
We define the following normalized variables:
mt ≡ Mt /PD,t
pj,t ≡ Pj,t /PD,t , j = 0, 1, 2,
and
pt ≡ Pt /PD,t .
6
(19)
(20)
(21)
(22)
We will assume that the government chooses mt as a function of the
economy’s single natural state variable, which indexes the deviation between
the two predetermined nominal prices. Without loss of generality, we will
define the state variable to be the ratio of the price set by firms in the previous
period to the index of predetermined prices:
st ≡
P1,t
= p1,t .
PD,t
(23)
One might think that with two predetermined prices there would be two
state variables. However, we can normalize by one of those prices, or by an
index of them: the level of past prices is irrelevant, because the monetary
authority can choose the nominal level of Mt in an unconstrained way. Our
purpose is to study optimal monetary policy without commitment, so we
are mainly concerned with optimal choice of the function mt (s).4 However,
part of the analysis will involve studying how the economy behaves under
exogenous policy in period t, given that policy will be optimally chosen in
the future.
The above definitions and normalizations imply
pt =
1 1−ε 2
·p +
3 0,t
3
1/(1−ε)
.
(24)
The normalized price level is uniquely determined by the normalized price
set by adjusting firms.
3
Private Sector Equilibrium with Arbitrary
Policy
In order to determine optimal monetary policy, we need to be able to construct a private sector equilibrium for arbitrary monetary policy. We do this
using backward induction. This methodology makes it relatively straightforward to determine optimal policy, again using backward induction. Here,
policy in period t will be represented by a policy function mt (st ) . Under
optimal policy these functions will be chosen to maximize welfare.
4
Typically, analysis of equilibrium under discretionary optimization would involve finding a time-invariant function m (s) . In the finite-horizon cases we examine, policy functions
will depend on the horizon, hence the subscript t in mt (s).
7
Because the horizon is finite, we will not be constructing a recursive equilibrium, nor should we expect to find a steady state equilibrium. However,
there are three time-invariant functions that will be important ingredients in
computing an equilibrium. The appendix (A.2) shows how one can use the
money demand equation and the firm-level technologies to express consumption, leisure and the future state as time-invariant functions of p0,t , mt and
st :
(25)
ct = c (p0,t ; mt ) ,
lt = l (p0,t ; mt , st ) ,
(26)
st+1 = σ (p0,t , st ) .
(27)
and
Under arbitrary policy, the only endogenous variable in these expressions is
the price set by adjusting firms in the current period (p0,t ). Note also that
these expressions can be combined with the labor supply equation (3) to
determine the real wage — and hence marginal cost — as a function of p0,t , mt
and st . All endogenous variables have been eliminated except for p0,t .
The remaining equation, needed to determine p0,t , is the optimal pricing
condition. We reproduce that equation here for convenience, substituting out
for the real wage as the marginal rate of substitution between consumption
and leisure, and normalizing the nominal variables on the left hand side:
p0,t
=
pt
ε
·
ε−1
min{2,T −t}
j=0
min{2,T −t}
j=0
β j ul (p0,t+j ; mt+j , st+j ) c (p0,t+j ; mt+j ) ·
β j uc (p0,t+j ; mt+j , st+j ) c (p0,t+j ; mt+j ) ·
Pt+j
Pt
Pt+j
Pt
ε
ε−1
(28)
(note that we use shorthand to directly express marginal utilities as functions
of p0,t+j ; mt+j , st+j , instead of indirectly through consumption and leisure).
The normalized price level on the left-hand side is a known function of p0,t
(24), and the appendix (A.3) shows how to express future inflation rates on
the right-hand side in terms of current and future p0 , m and s :
Pt+j+1
= Π (p0,t+j+1 , mt+j+1 , st+j , st+j+1 ) .
Pt+j
Note that
Pt+2
Pt
=
Pt+1
Pt
·
Pt+2
.
Pt+1
8
Proceeding further means looking for equilibrium values of p0,t that solve
the pricing equation (28). Although the details of solving the pricing equation depend on the horizon (that is, T − t), in general knowledge of p0,t and
mt+j (st+j ) , j = 0, 1, ..., T − t is sufficient to pin down all current and future
aggregate variables, which in turn pins down the optimal price for any individual firm. That is, (28) can be shown to depend only on p0,t , and thus in
principle it is straightforward to determine the set of equilibria.
3.1
The final period
In period T, the model is static. This simplifies the search for equilibrium
dramatically, because no future policy needs to be taken into account. From
(28), the optimal pricing equation in the final period is
p0,T
=
pT
ε
ε−1
·
ul (c (p0,T ; mT ) , l (p0,T ; mT , sT ))
.
uc (c (p0,T ; mT ) , l (p0,T ; mT , sT ))
Recall that the normalized price level is determined by p0,T (from (24)). Then
given the state and the policy action, both the left- and right-hand sides of
this equation vary only with p0,T . A more intuitive version of this pricing
equation results from viewing it as a best response function for an individual
firm. Let p0,T denote the price chosen by an individual firm, and p0,T denote
the price chosen by all firms. The latter pins down all aggregates, so that an
individual firm’s optimal price is a function of the price chosen by all other
firms:
ε
ε−1
·
1 1−ε 2
·p +
3 0,t
3
1/(1−ε)
ul (c (p0,T ; mT ) , l (p0,T ; mT , sT ))
.
uc (c (p0,T ; mT ) , l (p0,T ; mT , sT ))
(29)
Any fixed point of the best response function is an equilibrium value of
p0,T , for given values of mT and sT . Knowing p0,T , all other endogenous
variables can be computed using expressions derived earlier. For an arbitrary function mT (sT ) , we can then compute the equilibrium mappings
p0,T (sT ; mT ) , cT (sT ; mT ) , lT (sT ; mT ). Without imposing restrictions on the
utility function, there is no guarantee that (29) will have a unique fixed point,
so that these mappings will be functions. However, the potential for multiple
equilibria at this stage is not our focus. Thus, we will proceed assuming that
(29) does have a unique fixed point. In the example we focus on below it is
p0,T =
·
9
easy to show uniqueness. If preferences are given by u (c, l) = ln c + χl, then
(29) simplifies to
ε
p0,T =
· χ · mT ,
ε−1
which is a flat best response function, implying p0,T =
3.2
Period T
ε
ε−1
· χ · mT .
1
In period T − 1, equilibrium for a given state and a given policy action
depends on the policy function in the final period, mT (sT ) . Taking that
policy function as given and arbitrary, we again compute equilibrium by
analyzing the optimal pricing condition. Distinguishing between the prices
chosen by an individual firm and all other firms, the optimal pricing condition
can be interpreted as a best-response function:
p0,T −1 =
1
j=0
1
j=0
ε
ε−1
2
1 1−ε
· p0,T −1 +
3
3
1/(1−ε)
×
β j ul (p0,T −1+j ; mT −1+j , sT −1+j ) c (p0,T −1+j ; mT −1+j )
β j uc (p0,T −1+j ; mT −1+j , sT −1+j ) c (p0,T −1+j ; mT −1+j )
(30)
PT −1+j
PT −1
PT −1+j
PT −1
ε
ε−1
.
The right-hand side can be expressed as a function of only p0,T −1 , given sT −1
and mT −1. The first terms in the denominator and numerator, as well as
the period T − 1 price index factor, depend only on p0,T −1 , mT −1 and sT −1 .
The second terms in the numerator and denominator depend also on period
T variables. However, the policy function for period T is known and the
equilibrium correspondences for period T are known, so all period T variables
can be determined as functions of mT −1 , sT −1 and p0,T −1 . Mechanically, this
works as follows.
1. Given mT −1 , sT −1 and p0,T −1 , the law of motion for the state variable
(27) determines sT .
2. Given sT and an assumed policy function mT (sT ) for the final period,
mT is known.
3. Given sT and mT , the set of equilibrium p0,T is the set of fixed points
of the best response function for the final period (29).
10
Following these steps allows the right-hand side of the best response function for period T − 1 to be expressed as a function of only p0,T −1 , given sT −1
and mT −1 . The set of equilibrium p0,T −1 is the set of fixed points of the
best response function (30), and the equilibrium values of other variables
can be computed from (25) - (27) and steps 1 - 3. If one specifies exogenous
policy functions for period T − 1 and T , one can compute equilibrium correspondences for all T − 1 and T variables by varying sT −1 and following the
procedure just described.
This procedure is the natural extension of that used above for the final
period, but the distinction is important. In the final period, equilibrium is
determined entirely by the state and the policy action. One period earlier,
in order to solve the optimal pricing equation for p0,T −1 , we also need to
know what the policy function will be in the final period. That future policy
function determines how p0,T −1 will map — via sT — into policy actions in the
future. Those future policy actions affect future demand, marginal cost and
inflation, and therefore the optimal price for a firm in the current period
(p0,T −1 ). Optimal policy in the final period generates a particular policy
function which will be taken as given by firms — and the monetary authority
— in period T − 1.
4
Monetary Policy and Real Activity
Monetary policy has the potential to affect real variables in this model because some prices are predetermined. Monetary policy has an incentive to
affect real variables because monopolistic competition makes the flexible price
level of output inefficiently low. The specific channels through which monetary policy works involve relative price distortions and the markup. With
complete discounting, optimal discretionary policy would simply involve balancing these two distortions. With β > 0, the policymaker also considers the
implications of her actions for the future state, which affects future utility.
4.1
Relative price distortions
The consumption aggregate is a concave, symmetric function of each good,
and each good is produced using identical technology. For a given amount
of total labor input then, consumption is maximized by producing the same
quantity of each good. More formally, we will define the relative price
11
distortion in period t (ρt ) as the ratio of input to output:
1
ρt ≡ ·
3
2
j=0
ct
nj,t
2
=
j=0
1
· (cj,t /ct ) ≥ 1.
3
(31)
In an appendix (A.1), we derive an expression for the relative price distortion
in terms of p0,t and st :
(32)
ρt = ρ (p0,t , st ) .
If all relative prices are unity, then ρt = 1, which corresponds to the relative
price distortion being minimized. A sufficient condition for all relative prices
to be unity is that the economy be in a steady state with a constant price
level, or that prices be perfectly flexible. Furthermore, if J > 1, as it will
be throughout the paper, the only steady state with ρt = 1 is a steady state
with a constant price level. In general, if the price level varies over time, the
relative price distortion will exceed unity.
4.2
The markup distortion
With a competitive, economy-wide labor market, and technology that is linear in labor, the real marginal cost of all firms is simply given by
ψ t = wt .
(33)
The markup (µt ) of the price index over nominal marginal cost is the inverse
of real marginal cost:
µt = 1/ψ t = 1/wt .
(34)
In a flexible price economy, all firms charge the same price, and the markup is
ε
µ = ε−1
. With staggered pricing, the markup differs from µ even in a steady
state, unless the steady state involves zero inflation. See King and Wolman
(1999, pp. 363-364) for details.
Output is inefficiently low when the markup exceeds unity, as it will in
any steady state. By choosing a higher mt , the policymaker can decrease the
markup and raise output. For high enough mt though, the increase in output
comes at the expense of an increase in the relative price distortion. Furthermore, the policymaker also cares about the state with which she endows the
policymaker next period.
12
5
Equilibrium with Discretionary Policy
A discretionary equilibrium is a private sector equilibrium in which the monetary authority chooses an action that maximizes welfare, given that future
monetary authorities will also act with discretion. In terms of the earlier description of equilibrium with arbitrary policy, the monetary authority chooses
mt (st ) to maximize welfare for every st , given that mt+j (st+j ) will be chosen
to maximize welfare for j = 1, ..., T − t. One can view the policymaker as
implicitly computing the equilibria associated with all arbitrary policy actions, and choosing the action that generates the equilibrium yielding highest
welfare. If equilibrium is unique for every arbitrary policy action, then this
assessment is straightforward. If some policy actions would generate multiple
equilibria, then the policymaker needs to assign probabilities to the equilibria in order to evaluate the welfare associated with each action. In the final
period we find no evidence that any policy actions generate multiple equilibria, whereas in the previous period we find multiple equilibria even under
optimal policy for a range of the state variable.
5.1
Period T
The policy problem in the final period is
vT (sT ) = max (u (cT , lT ))
mT
(35)
subject to (25)-(26) for t = T, which express consumption and leisure as
functions of the state, the policy action, and p0,T , and subject to p0,T being
determined as the (assumed unique) solution to the optimal pricing condition
(29). The three panels of figure 1 illustrate the policy problem in period T,
for three different values of the state variable, and the three panels of figure
2 illustrate the solution as a function of the state variable.
From figure 1 it is clear that the policy problem in the final period
amounts to trading off the markup against the relative price distortion. At
low values of mT , both the markup and the relative price distortion can be
brought down by choosing a higher mT . Above a level that depends on sT ,
further increases in the money supply continue to bring about lower values
of the markup, but at the cost of a greater relative price distortion. At an
optimum, the marginal cost of a higher relative price distortion is exactly
offset by the marginal benefit of a lower markup. The nature of the trade-off
13
between the relative price distortion and the markup depends on the state
variable, because the state variable represents an inherited component of the
relative price distortion. If sT = 1.0, the policymaker in period T does not
inherit any relative price distortion, and it is feasible — though not optimal
— to eliminate the relative price distortion entirely.
One notable feature of figure 1 is that the markup is determined entirely
by the money supply: it is insensitive to the state variable. This is not a
general result, but occurs because Hansen-Rogerson preferences make the real
wage proportional to consumption. Since the markup is the inverse of the real
wage in this model, the markup is determined entirely by consumption. From
(25), consumption depends on p0,T and mT , but we saw in the previous section
that these preferences make p0,T proportional to mT . Therefore consumption
and the markup are determined entirely by mT .
Figure 2 summarizes final period outcomes under optimal policy, as a
function of the state. Welfare (vT ), in panel A, and the policy action (mT ),
in panel B, are both quasiconcave functions that are maximized at sT = 1.0.
The markup and relative price distortions, plotted in panel C, are both quasiconvex functions that are minimized at sT = 1.0. The key to understanding
these panels is to realize that there is no inherited relative price distortion
if and only if sT = 1.0. When sT = 1.0, the policymaker can eliminate the
relative price distortion and achieve the flexible price outcome by setting
mT = 1.0. This choice is suboptimal, however, because a marginal increase
in mT from 1.0 generates a lower markup and the same relative price distortion. Optimally then, even when it is feasible to eliminate the relative
price distortion, in the final period the policymaker chooses to accept a small
relative price distortion, because the high value of mT generates a relatively
low markup. For states that involve a high inherited relative price distortion,
bringing down the markup is quite costly in terms of the relative price distortion, so optimal policy chooses a low value of the money supply, thereby
accepting a higher markup.
The final period is obviously special, in that the policymaker faces no
intertemporal trade-off. In the initial period the policymaker cares about
the state with which she endows the future policymaker, because this state
affects future welfare. The final period is special in another way as well. In
the final period, firms that set their prices one and two periods earlier are
differentiated only by the prices they charge.5 In the initial period, these firms
5
It is clearly awkward to talk about firms setting their prices for three periods in the
14
influence the policy problem in different ways, because they will be resetting
their prices at different times. This final period equivalence between the two
types of firms implies that if we plotted the panels of figure 2 using P2 /PD on
the horizontal axis instead of sT = P1 /PD , the figures would look identical.
5.2
Period T
1
With the monetary policy action chosen optimally in the final period, arbitrary policy in the previous period lead to multiple equilibria. The nature of
optimal policy in period T − 1 therefore depends on the equilibrium selection
mechanism. Figure 3 displays period T −1 optimal policy — the value function
(panel A) and the policy function (panel B) — under two assumptions about
equilibrium selection.6 The circles represent optimal policy under what we
call uniform selection: if a particular value of mT −1 generates multiple equilibria, then it is assumed to be common knowledge that each equilibrium has
equal probability of occurring. The plus signs represent optimal policy under
Pareto selection: it is assumed to be common knowledge that the equilibrium
yielding highest welfare will occur with probability one. Under both selection
assumptions, the value functions are well-behaved (i.e.-quasiconcave). However, the policy functions are discontinuous: under uniform selection there
are two discontinuities, and under Pareto selection there is one discontinuity.
For sT −1 less than approximately 1.25, there is a unique private sector equilibrium under optimal policy, and therefore ex-ante welfare and the policy
functions are identical across the two selection criteria. For higher values of
the state, the functions differ according to the selection criterion, indicating
multiple equilibria under optimal policy.7
The subsections below are devoted to explaining in some detail multiple
equilibria and discontinuities in the (optimal) policy function. Before diving into that detail, it will be helpful to have an intuitive sense of where
these results are coming from. Both the multiple equilibria and the policy
context of a two-period model. One should think about our results as describing the early
stages of solving a longer horizon model through backward induction. See the end of
section 5 for more on this issue.
6
The probability distribution across equilibria is taken as exogenous, and we consider
two distributions. Ennis and Keister (2001) discuss endogenizing the distribution. We
have not yet studied how to incorporate their approach into our model.
7
When there are multiple equilibria under optimal policy, the value function measures
ex-ante value, where the weighting criterion is applied to welfare in each of the equilibria.
15
discontinuities stem from the fact that the current policy action determines
a state variable that affects the incentives and opportunities faced by the
future policymaker.
Multiple equilibria occur because the future policy action is endogenous
with respect to current pricing behavior. A single current price setter bases
its pricing decision on expectations about the future policy action. In turn,
the future policymaker bases her policy action on the future state, which is
determined by the behavior of all current price setters (27). Intuitively this
relationship can be characterized by complementarity: if current price setters
set a higher price, the future policymaker responds to the higher state with
a higher money supply, and in turn the higher future money supply raises
the future price level, making it optimal for individual price-setters to set a
high price in the initial period. Figure 2 suggests that complementarity will
not be omnipresent, because for high values of the state variable increases in
the state lead to a lower money supply.
Discontinuities in the policy function are associated with flips from one
local maxima of the policymaker’s objective function to another. Typically,
one local maximum involves a low current relative price distortion and correspondingly high current utility, whereas the other local maximum involves
a high current relative price distortion and low current utility, but leaves the
future policymaker with a favorable value of the state, leading to high future
utility.
Multiple equilibria occur because the current policy action affects the
incentives facing the future policymaker. In contrast, policy discontinuities
stem from the fact that the set of feasible current and future utilities is not
convex. Both of these features would be absent were there no state variable
linking the present to the future.
5.2.1
The Policy Problem
For period T − 1, Figures 4.I (uniform selection) and 4.II (Pareto selection)
illustrate the objective function, the two distortions and the future state as
functions of the policy action, for three levels of the current state (sT −1 ).
Recall that we solve the model using backward induction: optimal future
policy is taken into account in determining equilibrium in period T − 1. Our
purpose in including these figures is to emphasize the existence of multiple
equilibria, and to shed some light on the discontinuities that we have already
seen exist under optimal policy. In panels B through D of figure 4, multiple
16
equilibria are reflected in there being more than one level of the distortions
and the future state for certain policy actions. In panel A, but not in the
other panels, we have imposed the relevant selection criterion in order to
generate expected welfare. Thus, discontinuities in panel A indicate points
where the number of equilibria switches from one to three or three to one. A
graphical explanation for the multiplicity will follow the discussion of figure
4.
If the current state is high enough, figure 4 shows that there are multiple
local maxima to the objective function. The policy discontinuities seen in
figure 3 are associated with the global maximum switching from one local
maximum to another. In some cases (e.g. sT −1 = 1.2, Pareto selection) two
local maxima occur with a smooth objective function, whereas in other cases
(e.g. sT −1 = 1.25, Pareto selection) multiple local maxima are associated
with jumps in the objective function. These possibilities raise two questions.
First, why are there multiple local maxima, and second, why are there discontinuities in the objective function (as opposed to the policy function)?
Figure 4 illustrates that multiple local maxima to the policymaker’s objective function occur because increasing the current policy action (mT −1 )
can have differential effects on current and future utility. Focusing on the
case of Pareto selection with sT −1 = 1.2 (this is the set of circles in figure
4.B), there is a local maximum with a low level of mT −1 , which corresponds
to a low current relative price distortion but a relatively bad future state
(that is, a future state far from 1.0). There is another local maximum with a
high level of mT −1 , in which the current relative price distortion is high, but
the future state is much closer to 1.0. Referring back to figure 2, the closer
is the future state to 1.0, the higher is welfare in the final period. Thus,
the first local maximum has high first-period welfare and low final-period
welfare, whereas the second local maximum has low first-period welfare and
high final-period welfare. Multiple local maxima can also be understood in
terms of a utility frontier: from the standpoint of the initial period monetary
authority (who cares about current and future utility, but takes as given the
behavior of future policy), the feasible set of current and future utility levels
is not convex.8
Discontinuities in the policymaker’s objective function occur as mT moves
in or out of regions where there are multiple equilibria, for either selection
8
Given that the relevant indifference curves have constant slope, nonconvexity naturally
leads to the possibility of policy discontinuities, which we do find.
17
criterion that we consider. In fact, given the regions of multiple equilibria
shown in figure 4, it would take a highly contrived selection criterion to avoid
discontinuities in the objective function.
5.2.2
Multiple Equilibria
Multiple equilibria occur for a range of T − 1 policy actions; this can be
seen in figure 4 for two values of the state variable (for a third — sT −1 = 1.1
— multiple equilibria do not occur under any policy action). That multiple
equilibria occur under optimal policy for a wide range of values of sT −1 can
be seen from figure 3.A: optimal policy and welfare depend on equilibrium
selection, which indicates multiple private-sector equilibria under optimal
policy. The multiplicity of equilibria implies some form of complementarity,
and as is often the case there is more than one way to think about that
complementarity.
Figure 5 describes the multiple equilibria in terms of what we call policy
complementarity. Holding fixed the current state and policy action, panel A
displays the future policy action as a function of the expected future policy
action. This relationship is characterized by an s-shaped function that crosses
the 450 line three times, indicating three equilibria. The complementarity
illustrated in figure 5.A is that the future policymaker chooses a higher level
of MT in response to expectations of higher MT . This is a reduced form
relationship, and panel B displays optimal policy and optimal pricing, the two
more fundamental objects which underlie the relationship. On the horizontal
axis of panel B is the future state variable, and on the vertical axis are the
actual and expected future nominal money supply.9 The dashed line is the
final period optimal policy function mT (sT ) (figure 2.B), except that the
money supply is normalized by PD,T −1 rather than PD,T . The solid line is
derived from the optimal pricing condition (30), and describes private sector
equilibrium. To understand the solid line, it is helpful to view ET −1 MT as the
“right-hand side” variable. As ET −1 MT varies, the corresponding equilibrium
price p0,T −1 can be determined as the solution to (30). And from p0,T −1 ,
the future state sT can be determined using (27). Summarizing panel B,
the dashed line describes how the future state determines the future policy
9
We refer here to the nominal money supply (MT ). What is crucial though is that we
are referring to the final period money supply normalized by initial period predetermined
prices rather than final period predetermined prices. Thus it is equivalent to refer to MT
or MT /PD,T −1 .
18
action, and the solid line describes how the expected future policy action
determines the future state. Panel A combines these elements to express the
optimal policy action as a function of the expected policy action. If firms
expect the future money supply to be high, they set a high price, which
translates into a high value of the future state. Encountering a high value of
the state variable, the future policymaker responds with a high value of the
money supply, validating expectations. Figures 3 and 4 show that multiple
equilibria occur under optimal policy only for high values of the state variable.
When the state variable is low, at the optimal policy action there is a unique
fixed point to the policy response function illustrated in figure 5.
One can also interpret the multiple equilibria in terms of a more conventional notion of complementarity among price-setting firms. Complementarity in pricing is induced by the nature of policy. Figure 6.A displays the
best-response pricing function of an individual firm in period T − 1, conditional on the same initial-period state and policy action used in figure 5 (the
best-response function is given by (30)). As one would expect from figure 5,
this best-response function is s-shaped and has three fixed points. Panel A
illustrates the multiplicity in terms of p0,T −1 . Panel B plots the relationship
between sT and p0,T −1 given by (27), allowing one to verify that the equilibria illustrated in figure 6 for p0,T −1 correspond to those illustrated in figure
5 for sT . Obviously there is a connection between the forms of complementarity represented in figures 5 and 6. Figure 6 shows that an individual firm
responds to higher prices of all other firms by setting a higher price. This
occurs because the higher price set by all other firms raises the future state
variable, inducing the monetary authority to choose a higher money supply
in the future. Figure 5 emphasizes the role of policy: expectations of a higher
future money supply raise the future state variable, inducing the monetary
authority to validate those expectations.
5.3
Optimal Policy Under Commitment
The multiple equilibria illustrated in figures 5 and 6 are inherently related to
discretion. Because there is no exogenous uncertainty in the model, optimal
policy under commitment involves choosing in the initial period the best
path for the nominal money supply. Thus, under commitment the final period
nominal money supply (normalized by PD,T −1 ) is only a function of the initial
state (sT −1 ). Referring to figure 5.B, this means that under commitment the
optimal policy curve (dashed) would be a horizontal line. The optimal pricing
19
curve would be unchanged, so there would be only one possible equilibrium.
The essence of commitment is that in the initial period the policymaker
promises that in the final period she will not react to sT , the endogenous
state variable. A credible promise not to react to the final period state
eliminates the mechanism that under discretion leads to multiple equilibria.
5.4
Robustness
There are of course a number of robustness exercises it would be interesting
to conduct. We have already undertaken some of them. As stated earlier,
our results on multiple equilibria and discontinuities under discretion are
inherently related to the existence of a state variable, representing a predetermined relative price. When prices are set for two periods rather than
three, there is no state variable. Wolman (forthcoming) describes the unique
steady state under discretion with two-period staggered pricing.
In this paper results are presented for preferences with an infinite labor supply elasticity. It is natural to wonder whether the results extend to
other preference specifications. We have studied examples of Cobb-Douglas
preferences and found qualitatively similar results.
Given that we assume three-period staggered price-setting, it would be
desirable to extend the horizon to more than two periods. We have done
this, and found that similar multiplicities and discontinuities occur in earlier
periods. However, these very features make it difficult to compute solutions
and to explain results.
We intend to pursue at least two other robustness exercises. First, with respect to money demand, do the results carry over to settings where the money
demand function is derived from a standard assumption such as money-inthe-utility function? Based on our work on optimal policy with commitment,
we are inclined to believe the answer is yes. Second, although the staggered
price-setting framework we use is more appealing at a micro level, the literature on monetary policy has leaned more towards Calvo pricing. It would
therefore be interesting to know whether our results carry over to the Calvo
setting. Because there will still be a state variable, it seems likely that similar
results will obtain with Calvo pricing.10
10
Articles mentioned in the introduction that describe discretionary policy under Calvo
pricing use linear quadratic approximations and a primal approach.
20
6
Related Literature
The nature of equilibria under discretion in sticky-price models is sensitive
to specific assumptions about price setting. In a wide class of models, monopolistic competition makes the level of output inefficiently low, and with
some prices predetermined, a surprise monetary expansion will produce a real
expansion as well. When all prices are predetermined, as in Ireland (1997),
there is no cost to the monetary authority of a surprise expansion. The only
Markov-perfect equilibrium in that environment involves the highest possible inflation rate.11 In Albanesi, Chari and Christiano (2000), all period t
prices are set after period t − 1 variables have been observed, but a fraction
of firms set their prices before the monetary authority moves. There is thus
a cost to surprise inflation: it creates a relative price distortion. However,
this relative price distortion is invariant to the expected rate of inflation.12
In our model, some firms choose their prices contemporaneously with (or before) period t − 1 variables. This feature leads to a relative price distortion
even in steady state (as long as the price level is not constant), and the magnitude of this distortion varies with the steady state rate of inflation. The
monetary authority thus faces the following trade-off in our model. With
output inefficiently low, the discretionary policymaker would like to create a
surprise expansion. The cost of a surprise is that it can exacerbate relative
price distortions. In equilibrium there can be no surprises, so in equilibrium the marginal benefit of a surprise in terms of increased output (reduced
markup) must be exactly offset by the marginal cost in terms of increased
relative price distortion. Because steady inflation raises relative price distortions, intuitively there ought to be a constant rate of inflation at which the
discretionary policymaker is content not to create surprises. Wolman (forthcoming) shows that such a steady state does exist when prices are set for two
periods. With three-period pricing however, a state variable is introduced
11
Ireland’s model also contains nontrivial distortions associated with money demand,
whereas our model does not. However, the discussion here would also apply to a version
of his model without money demand distortions. Benigno and Benigno (2001) discuss a
two-country version of such a model. They show that strategic considerations make price
stability a discretionary equilibrium under certain conditions.
12
Albanesi, Chari and Christiano study an environment with nontrivial money demand
distortions, and these play an important role in determining the Markov-perfect equilibria
of their model. We conjecture that absent money demand considerations, their model
would have no interior Markov-perfect equilibrium — the monetary authority would always
want to create a surprise expansion.
21
that leads to the unusual behavior we have described.
Albanesi, Chari and Christiano also stress the existence of multiple equilibria without reputational considerations. The nature of multiplicity that
we are describing is different from that in Albanesi, Chari and Christiano.13
There, multiplicity appears in the current optimal policy action: there are
generally two current policy actions which can be equilibria (i.e.-optimal),
and which one occurs is indeterminate. Given the policy action, however, all
other variables are uniquely determined. Here, there is generally a continuum
of policy actions that can be equilibria, depending on the probabilities associated with the three equilibria under exogenous policy. These probabilities
are not given by fundamentals. Given some common set of beliefs about the
probabilities of different outcomes under exogenous policy, the policymaker
has a unique optimal action, but that action does not uniquely pin down
current outcomes. Practically speaking, neither the inflation rate nor output
is pinned down by the choice of money growth. In addition, the mechanism
behind multiplicity is quite different than that in Albanesi, Chari and Christiano. There, multiple equilibria are related to the nature of money demand
(in particular, multiplicity occurs when money demand is sufficiently interest
elastic), and there are no endogenous state variables. Here money demand is
completely interest inelastic, and multiple equilibria are due to the presence
of an endogenous state variable.
Dedola (1999) studies discretionary policy in a Rotemberg-style model of
pricing, and finds multiple equilibria. Dedola models money demand using
a cash-in-advance constraint, and like Albanesi, Chari and Christiano, the
multiple equilibria are related to the money demand specification.
The introduction referred to several papers that do not find multiple
equilibria under discretion, in models quite similar to ours.14 Two features
are important in explaining the different conclusions we reach. First, those
papers take a primal approach, meaning that the policymaker is assumed
to be able to select allocations (we assume the policymaker is able to select
only the money supply). This approach would seem to directly eliminate the
multiplicity described here. Second, they approximate the constraints with
linear functions and the objective with a quadratic function. In the infinite
13
As in this paper, the multiple equilibria in Albanesi, Chari and Christiano do not
derive from reputational considerations. In contrast, the multiplicity in Chari, Christiano
and Eichenbaum (1998) is reputation-driven.
14
Clarida, Gali and Gertler (1999), Woodford (1999) and Svensson and Woodford (1999),
Dotsey and Hornstein (forthcoming).
22
horizon settings of those papers, it is possible for this L-Q approach to find
multiple steady states. However, for a particular steady state the dynamics
will necessarily be unique. In the finite horizon, non-primal framework of
this paper, an L-Q approximation would eliminate the multiplicity: from
figure 5 it is clear that multiple equilibria are related to a nonlinearity in the
constraints.
Krusell and Smith (2001) study a model of time-inconsistent preferences,
and show that there are a continuum of Markov-perfect equilibria to the game
that represents the consumption-savings problem of an infinitely lived agent
who cannot commit to future actions. They also suggest that multiplicity
may be endemic to a large class of problems in which there is no commitment.
Our results support their suggestion, and can even be thought of as indicating
that things are worse than they suggested. For we find multiple equilibria in
a finite horizon model, whereas the multiplicity they discuss arises with an
infinite horizon.
7
Conclusion
In a simple staggered price-setting model we find that discretionary monetary policy leads to multiple equilibria. Inflation and real allocations are
not pinned down by fundamentals alone. The nature of the multiplicity is
such that even for a given value of the money supply, the outcome can be
indeterminate. From a positive perspective, we conclude that lack of commitment may help to explain fluctuations in inflation and real activity. From
a normative perspective, multiplicity under discretion provides an additional
argument for a monetary commitment mechanism. Our results also suggest
that caution should be used in applying popular approximation methods to
staggered-pricing models. These methods have not uncovered the multiple
equilibria that we find.
Multiple equilibria occur under discretion because there is a form of policy
complementarity. Expected future policy influences current pricing decisions,
which determine a state variable to which future policy responds. In effect
then, policy responds to expected policy. For a nontrivial range of values of
the initial state, this response involves complementarity and multiple equilibria: if firms expect a high money supply in the future, they set a high price
in the current period. The monetary policymaker in the future then finds
it optimal to validate this high price with a high money supply. Likewise,
23
expectations of a low money supply are also self-fulfilling. Feedback from
expected policy to policy itself is of course present in a large class of discretionary policy environments. The likelihood of such mechanisms leading to
multiple equilibria is an open question.
24
References
[1] Albanesi, Stefania, V.V. Chari and Lawrence Christiano (2000), “Expectation Traps and Monetary Policy,” manuscript, ftp://ftp.igier.unibocconi.it/wp/2001/198.pdf.
[2] Barro, Robert and David B. Gordon (1983), “A Positive Theory of Monetary Policy in a Natural Rate Model,” Journal of Political Economy 91,
589-610.
[3] Benigno, Gianluca and Pierpaolo Benigno (2001), “Price Stability in
Open Economies,” http://homepages.nyu.edu/~pb50/FNash4.pdf.
[4] Chari, V.V., Lawrence Christiano and Martin Eichenbaum (1998), “Expectation Traps and Discretion,” Journal of Economic Theory 81, 462492.
[5] Chari, V.V., Patrick Kehoe and Ellen McGrattan (2000), “Sticky Price
Models of the Business Cycle: Can the Contract Multiplier Solve the
Persistence Problem?” Econometrica 68, 1151-1179.
[6] Clarida, Richard, Jordi Galí and Mark Gertler (1999), “The Science of
Monetary Policy,” Journal of Economic Literature 37, 1661-1707.
[7] Cooper, Russell and A. Andrew John (1988), “Coordinating Coordination Failures in Keynesian Models,” Quarterly Journal of Economics
103, 441-463.
[8] Currie, David and Paul Levine (1993), Rules, reputation and macroeconomic policy coordination, Cambridge University Press.
[9] Dedola, Luca (2000), “Credibility and the Gains from Commitment in
a General Equilibrium Model,” University of Rochester Ph.D. thesis.
[10] Dotsey, Michael and Andreas Hornstein, forthcoming, “Is the Behavior
of Money Useful to a Discretionary Policymaker?” Journal of Monetary
Economics.
[11] Ennis, Huberto and Todd Keister (2001), “Optimal Policy with Probabilistic Equilibrium Selection,” Federal Reserve Bank of Richmond
Working Paper 01-03, http://www.rich.frb.org/pubs/wpapers/.
25
[12] Ireland, Peter (1997), “Sustainable Monetary Policies,” Journal of Economic Dynamics and Control 22, 87-108.
[13] Khan, Aubhik, Robert G. King and Alexander L. Wolman (2000), “Optimal Monetary Policy,” Federal Reserve Bank of Richmond Working
Paper 00-10, http://www.rich.frb.org/pubs/wpapers/.
[14] King, Robert G. and Alexander L. Wolman (1999), “What Should the
Monetary Authority Do When Prices Are Sticky?” in Monetary Policy
Rules, John Taylor, ed.
[15] Krusell, Per and Anthony A. Smith Jr. (2001), “ConsumptionSavings
Decisions
with
Quasi-Geometric
Discounting,”
http://fasttone.gsia.cmu.edu/tony/quasi.pdf.
[16] Kydland, Finn E. and Edward C. Prescott (1977), “Rules Rather than
Discretion: The Inconsistency of Optimal Plans,” Journal of Political
Economy 85, 473-492.
[17] Oudiz, G. and J. Sachs (1984), “Macroeconomic Policy Coordination
Among the Industrial Countries,” Brookings Papers on Economic Activity 1, 1-64.
[18] Svensson, Lars E.O. and Michael Woodford (1999), “Implementing Optimal Policy through Inflation-Forecast Targeting,”
http://www.princeton.edu/~svensson/papers/swiftabs.htm.
[19] Wolman, Alexander L. (1999), “Sticky Prices, Marginal Cost, and the
Behavior of Inflation,” Federal Reserve Bank of Richmond Economic
Quarterly, http://www.rich.frb.org/eq/abstract.cfm?articleID=64
[20] ______, forthcoming “A Primer on Optimal Monetary with Sticky
Prices,” Federal Reserve Bank of Richmond Economic Quarterly.
[21] Woodford, Michael (1999), “Optimal Monetary Policy Inertia,” NBER
working paper 7261.
26
A
Appendix: Derivations
In the appendix we allow for more general staggered pricing behavior; ω j will
denote the fraction of firms in period t charging a price set in period t − j.
In the body of the paper we impose ω 0 = ω 1 = ω 2 = 1/3.
A.1
Relative price distortion as function of p0,t and st
Starting from (31), using the demand functions, we can rewrite the relative
price distortion in terms of relative prices instead of quantities:
J−1
ρt =
pεt
j=0
ω j · p−ε
j,t
=
ω 0 p1−ε
0,t + (1 − ω 0 )
=
ω 0 p1−ε
0,t + (1 − ω 0 )
ε/(1−ε)
ε
1−ε
−ε
−ε
ω 0 p−ε
0,t + ω 1 p1,t + ω 2 p2,t
−ε
ω 0 p−ε
0,t + ω 1 st + ω 2
−ε
P2,t
P1,t
s−ε
.(36)
t
Above we have written the relative price distortion as a function of p0,t , st and
P2,t
. By manipulating the definition of st , we can express PP2,t
as a function
P1,t
1,t
of st , and thus express the relative price distortion solely as a function of p0,t
and st :
ω1
1 − ω0
st =
sε−1
=
t
stε−1
P2,t
P1,t
P2,t
P1,t
−
1−ε
ω1
1 − ω0
ω1
1 − ω0
=
=
ω1
+ 1−
1 − ω0
+ 1−
P2,t
P1,t
·
ω1
1 − ω0
ω1
= 1−
1 − ω0
1 − ω0
1 − (ω 0 + ω 1 )
1 − ω0
1 − (ω 0 + ω 1 )
27
1/(ε−1)
so
P2,t
P1,t
1−ε
·
1−ε
·
P2,t
P1,t
sε−1
−
t
sε−1
−
t
1−ε
ω1
1 − ω0
ω1
1 − ω0
1
1−ε
.
(37)
Now sub this last expression into (36):
ρt = ρ (p0,t , st ) , where
ω 0 p−ε
0,t
A.2
+
s−ε
t
ρ (p0,t , st ) = ω 0 p1−ε
0,t + (1 − ω 0 )
ε/(1−ε)
1 − ω0
ω2
ω1
1 − ω0
ω1 + ω2
stε−1
−
×
ε
ε−1
.
(38)
Time-invariant functions for consumption, leisure,
and evolution of the state variable
For consumption, substitute the definition of the normalized price index into
the money demand equation:
c (mt , p0,t ) = mt /pt
= mt · ω 0 p1−ε
0,t + (1 − ω 0 )
1/(ε−1)
.
For leisure, use the firm-level technology and the definition of the relative
price distortion:
l (mt , p0,t ; st ) = 1 − nt
= 1 − ct ·
ω j (cj,t /ct )
= 1 − c (mt , p0,t ) · ρ (p0,t , st ) .
28
The state variable in period t + 1 can be written as a function of period t
values of the state variable and the price chosen by adjusting firms:
P1,t+1
PD,t+1
P0,t
=
PD,t+1
st+1 =
P0,t
=
ω1
1−ω0
1−ε
· P0,t
+ 1−
ω1
1−ω 0
1−ε
P1,t
1
1−ε
=
ω1
ω1
+ 1−
1 − ω0
1 − ω0
P1,t
P0,t
1−ε
p0,t
st
ε−1
st+1 =
ω1
ω1
+ 1−
1 − ω0
1 − ω0
1
ε−1
1
ε−1
.
(39)
In the text, we refer to the right hand side of (39) as the function σ (p0,t , st ) .
A.3
Cumulative inflation rate
The cumulative inflation rate between periods t and t + j is relevant because
it determines the real deterioration of a nominal price that was set in period
t. It is easiest to work with the cumulative inflation rate if we view it as
the product of one-period inflation rates ( PPt+1
). The first step is to use the
t
money demand equation for t + 1 to eliminate Pt+1 :
Pt+1
Mt+1 1
=
· .
Pt
ct+1 Pt
Next, multiply and divide by PD,t+1 , and use the definition of the normalized
money supply:
Pt+1
Mt+1 1 PD,t+1
=
Pt
PD,t+1 ct+1 Pt
mt+1 PD,t+1
=
·
.
ct+1
Pt
29
Now replace Pt with its definition, and divide the numerator and denominator
by PD,t+1 , recalling the definition of st+1 :
Pt+1
mt+1
PD,t+1
=
·
1−ε
1−ε 1/(1−ε)
Pt
ct+1
ω 0 P1,t+1 + (1 − ω 0 ) PD,t
mt+1
1−ε
=
· ω 0 s1−ε
t+1 + (1 − ω 0 ) (PD,t /PD,t+1 )
ct+1
1/(ε−1)
(40)
.
To proceed further, express PD,t /PD,t+1 as a function of st and st+1 :
PD,t /P1,t
PD,t
=
PD,t+1
PD,t+1 /P1,t
PD,t /P1,t
=
PD,t+1 /P2,t+1
(1/st )
=
(PD,t+1 /P1,t+1 ) (P1,t+1 /P2,t+1 )
st+1 P2,t+1
=
·
; then use (37) to get
st P1,t+1
st+1
·
=
st
1 − ω0
1 − (ω 0 + ω 1 )
sε−1
t+1
1
1−ε
ω1
1 − ω0
−
, (41)
and substitute this last expression into (40):
mt+1
Pt+1
=
·
Pt
ct+1
ω 0 s1−ε
t+1
mt+1
=
·
ct+1
ω 0 s1−ε
t+1
st+1
+ (1 − ω 0 )
· h (s)
st
+ (1 − ω 0 )
st+1
st
1−ε
h (s) =
Thus, we have
ε−1
s
−
ω1
1 − ω0
Pt+1
= Π (p0,t+1 , mt+1 , st , st+1 ) ,
Pt
30
1/(ε−1)
h (s)
where
1 − ω0
1 − (ω 0 + ω 1 )
1/(ε−1)
1−ε
.
1
1−ε
.
(42)
where
sε−1
t
Π (p0,t+1 , mt+1 , st , st+1 ) ≡
mt+1
1
· ω0+
·
c (p0,t+1 ; mt+1 ) st+1
1 − ω0
(1 − ω 0 ) sε−1
t+1 − ω 1
1 − (ω 0 + ω 1 )
31
(43)
1/(ε−1)
.
@FedFRASER