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Working Paper Series
Expectation Traps and Discretion
V.V. Chari, Lawrence J. Christiano
and Martin Eichenbaum
Working Papers Series
Macroeconomic Issues
Research Department
Federal Reserve Bank of Chicago
March 1996 (WP-96-5)
LIBRARY
FEDERAL RESERVE BANK
OF CHICAGO
MAY 0 9 1996
FEDERAL RESERVE
BANK OF CHICAGO
Expectation
T r a p s
a n d
Discretion^
V . V . C hari*
L a w r e n c e J. C h ristia n o * *
M a r tin E ich en b a u m * * *
M a rc h 1 9 9 6
A bstract
We argue that discretionary monetary policy exposes the economy
to welfare-decreasing instability. It does so by creating the potential
for private expectations about the response of monetary policy to ex
ogenous shocks to be self-fulfilling. Among the many equilibria that
are possible, some have good welfare properties. But, others exhibit
welfare decreasing volatility in output and employment. We refer to
the latter type of equilibria as e x p e c t a t io n tra p s. In effect, our paper
presents a new argument for commitment in monetary policy because
commitment eliminates these bad equilibria. We show that full com
mitment is not necessary to achieve the best outcome, and that more
limited forms of commitment suffice.
tW e are grateful for com m ents from B radford DeLong, B ennett M cCallum and Michael
W oodford. C h ristian o and E ichenbaum th an k the N ational Science Foundation for finan
cial su p p o rt. T h e views expressed in this paper are strictly those of the authors. T hey
do not necessarily represent the position of the Federal Reserve Bank of Chicago, or the
Federal Reserve System .
* University of M innesota and Federal Reserve Bank of M inneapolis
** N orthw estern U niversity, Federal Reserve Banks of Chicago and M inneapolis, and N B ER
*** N orthw estern U niversity, Federal Reserve Bank of Chicago, and N BER
1. I n t r o d u c t i o n
The classic literature on discretionary monetary policy attributes persistent episodes of infla
tion to central bankers’ willingness to purposefully initiate and pursue inflationary policies.
This willingness is said to reflect the appeal of inflation as a source of revenue and as a
tool for exploiting perceived short-run Phillips curve trade-offs. This paper offers a new
argument about why.inflationary episodes can arise under discretion. Our argument is that,
under discretion, policymakers can also be
pushed
into pursuing inflationary policies. This
can happen when the private sector, for whatever reason, expects high inflation. Under
these circumstances, the central banker may find it optimal to accomodate private agents’
expectations if the cost of not doing so is a recession. When the monetary authority does
accommodate, private agents’ expectations are self-fulfilling and we say the economy has
fallen into an
e x p e c t a t i o n t ra p . 1
The expectation traps emphasized in this paper arise from
agents’ expectations about how monetary policy will respond to exogenous shocks. We argue
that the notion of a policy maker caught in an expectations trap is potentially helpful in
accounting for inflationary episodes like the one endured by the U.S. in the mid 1960s and
1970s.
The expectation traps that we analyze are of two types. The first arises when agents ex
pect that monetary policy will react to shocks which do not affect preferences or technology.
If these expectations about monetary policy are self-fulfilling, non-fundamental shocks con
g e e W illiam Fellner (1976, p p .116-118) for an early discussion of th e idea th a t exp ectatio n s a b o u t
m on etary policy could be self-fulfilling by forcing th e hands of benevolent policy m akers.
2
stitute an additional set of impulses to the economy. The second type of expectation trap
can arise when agents’ expectations of how monetary policy will respond to fundamental
shocks are self-fulfilling. This can cause actual monetary policy to overreact to fundamental
shocks. Here, expectation traps amplify volatility by modifying the impact and propagation
mechanisms from fundamental shocks to the economy. They could, for example, cause a real
shock to lead to a persistent change in the inflation rate even though it would only produce
a change in the level of prices under a non-accommodative monetary policy. In either case,
expectation traps are a source of welfare-decreasing instability in the economy.
VVe articulate the preceding arguments formally using a general equilibrium monetary
model which is similar to that of Ireland (1995). The model economy is populated by a
representative household and three types of firms: a competitive producer of final goods,
a continuum of monopolists, each of whom produces an intermediate good, and a finan
cial intermediary. The household purchases the final consumption good, supplies labor to
intermediate good firms, and lends funds to the financial intermediary. The financial inter
mediary combines funds received from the households with lump-sum injections of money
from the government, and makes loans to firms. Firms need loans because they must pay
labor before they sell their output. There is a monetary authority which chooses monetary
transfers to the financial intermediary with the objective of maximizing the expected utility
of the representative household. We model discretion by assuming that the policy maker
acts sequentially through time and cannot commit to any future action.
The sequence of events within a period is as follows. First, all exogenous shocks, including
both fundamental and non-fundamental shocks, are realized. Then, intermediate goods
producers set their prices based on their expectation of the current-period money growth
3
rate. Finally, the monetary authority’s action is realized, and all other model variables are
determined, with the output of the intermediate good being demand determined.
As in Kydland and Prescott (1979), Barro and Gordon (1983) and Ireland (1995), our
model is one in which high expected money growth is socially costly while unexpected money
growth confers social benefits. High anticipated money growth is costly because it leads to
higher inflation which acts as a tax on employment. Unexpected money growth raises output
in the model because intermediate goods producers post prices before the realization of the
monetary policy action and output is demand determined. This rise in output is socially
beneficial because output is inefficiently low reflecting the existence of monopoly power in
the intermediate goods industry. This inefficiency cannot be eliminated, in equilibrium, by
monetary policy.
To establish that expectation traps can occur without commitment, we characterize the
space of sustainable outcomes for our model economy. We do this by adapting a charac
terization theorem due to Chari and Kehoe (1990,1993a, 1993b), based on Abreu (1988).
Specifically, we establish a set of restrictions on a given outcome for it to be the outcome
of some sustainable equilibrium. We then show, by way of two examples, that both types
of expectation traps discussed above are elements of the space of sustainable outcomes. In
the first example, there are no fundamental shocks. Money growth follows a first order au
toregression, with the disturbance term determined by a non-fundamental shock. We find
this example interesting for two reasons. First, it demonstrates in a particularly simple way,
the possibility of instability under discretion. Second, it offers an alternative perspective
on the empirical literature on monetary policy shocks. This literature seeks to identify an
exogenous component in monetary policy. The conventional interpretation of these shocks
4
is that they represent exogenous randomness inherent within the policy making process.
Our results suggest that this randomness may also originate from outside the policy making
process and reflect the sort of self-fulfilling expectations that can arise under discretion. ‘
In our second example of an expectations trap, there are only fundamental shocks. We
show that there is a sustainable outcome in which money growth is typically low. Occasion
ally, following a decline in the state of technology, money growth and, hence, inflation rise
for an extended period of time before returning to their typically low levels. We find this
example interesting because it illustrates another sense in which discretion makes possible
volatile equilibria. Also, the example may form the basis for a simple explanation of the
prolonged rise in inflation starting in the mid-1960’s. Specifically, it articulates the notion
that the US fell into an expectation trap in which a benevolent, sequentially optimizing
monetary authority pursued a policy which exacerbated the loss in output caused by various
fundamental shocks. Standard economic theory is at a loss to explain how a real shock
can trigger a long term rise in inflation. It is also at a loss to explain why the monetary
authority would have responded to such shocks by shifting for a period of years to a higher
money growth regime which most contemporary observers deplored as being destructive to
the economy.
The literature does contain discussions about the link between real shocks and long
lived episodes of inflation. Many of these discussions involve ideas related to those lying
at the heart of our second example of an expectation trap. For example, these ideas play
an important role in discussions of the wage price spiral, as well as the notion that an
‘overheating’ economy, i.e. one with high sustained rates of capacity utilization, can trigger
a rise in the core inflation rate. They are also implicit in Blinder’s (1982) discussion of how
5
supply shocks could have contributed to the persistent inflation of 1970s:
‘Inflation from special factors can “get into” the baseline rate if it causes an accel
eration of wage growth. At this point policymakers face an agonizing choice—the
so-called accommodation issue. To the extent that aggregate nominal demand
is n o t expanded to accommodate the higher wages and prices, unemployment
and slack capacity will result. There will be a recession. On the other hand, to
the extent that aggregate demand is expanded (say, by raising the growth rate
of money above previous targets), inflation from the special factor will get built
into the baseline rate.’ (Blinder (1982, p. 264))
These ideas are not confined to academics. Our reading of Arthur Burns’ speeches sug
gests that his views about the genesis of the inflation of the mid 1960’s and 1970’s have much
in common with the notion that the economy was caught in an expectation trap, triggered
by transitory real shocks. The principal real shock, in his view, was the expansionary fiscal
policy (‘the forces of excess demand’) associated with the Great Society programs of the
1960s and the Vietnam war.2 As Burns put it around the time he became Chairman of the
Federal Reserve in 1970:
‘The forces of excess demand that originally led to price inflation disappeared well
over a year ago. Nevertheless, strong and stubborn inflationary forces, emanating
from rising costs, linger on...’ (Burns (1978, p.124).)
A key factor fueling rising costs, according to Burns, was the widespread expectation that
the inflation which started in the mid 1960s, would continue. He described his vision of how
inflationary expectations drive up costs, and hence prices, in a statement before the Joint
Economic Committee of the U.S. Congress in 1971:
2
B urns believed th a t crop failures and oil shocks were o th er real d isturbances th a t co n trib u ted to th e
inflation of th e 1970s
6
‘Consumer prices have been rising steadily since 1965 - much of the time at an
accelerating rate. Continued substantial increases are now widely anticipated
over the months and years ahead...in this environment, workers naturally seek
wage increases sufficiently large...to get some protection against future price ad
vances... thoughtful employers...reckon, as they now generally do, that cost in
creases probably can be passed on to buyers grown accustomed to inflation.’
(Burns (1978, p.126.)
Like the central banker in our model, Burns clearly understood that this upward pressure
on prices could not be transformed into persistent, high inflation without monetary accom
modation. As he put it in a speech in 1977:
‘Neither I nor, I believe, any of my associates would quarrel with the proposition
that money creation and inflation are closely linked and that serious inflation
could not long proceed without monetary nourishment. We well know-as do
many others-that if the Federal Reserve stopped creating new money, or if this
activity were slowed drastically, inflation would soon either come to an end or be
substantially checked.’ (Burns (1978, p. 417).)
Burns chose to accommodate agents’ inflationary expectations for the same reason that the
monetary authority in our second example does so: to do otherwise would generate a costly
recession. As he put in testimony before the Committee on Banking and Currency of the
House of Representatives on July 30, 1974:
‘...an effort to use harsh policies of monetary restraint to offset the exceptionally
powerful inflationary forces in recent years would have caused serious financial
disorder and economic dislocation. That would not have been a sensible course
for monetary policy.’ (Burns (1978, p. 171).)
Burns clearly understood the accommodation dilemna alluded to by Blinder, and judged
that the best response was to accommodate. In our terminology, he acted like a benevolent
7
central banker caught in an expectation trap. What Burns did not understand was the
critical role the institution of discretion played in setting that trap.
After our formal analysis of discretionary monetary policy, we investigate alternative in
stitutional arrangements that can eliminate the possibility of expectation traps. One solution
is full commitment on the part of the monetary authority. An obvious practical problem
with this solution is that it is hard to imagine any monetary policy authority committing to
a sequence of policy actions infinitely far into the future. More limited forms of commitment
are practical. We consider a situation in which the monetary policy authority commits at
time
t
to its time
t
4- 1 state contingent growth rate of money. We establish that expecta
tion traps cannot occur in this regime and that the equilibrium quantities in this economy
coincide with those in the Ramsey equilibrium.
The result that one period ahead limited commitment reduces the set of sustainable
equilibria to a singleton - the Ramsey equilibrium - depends on particular features of our
model economy. The intuition underlying the proof gives some insight into how it could
be extended to accommodate different environments, say, those with multiperiod wage and
price contracts.
The next section presents a formal description of our model. Section 3 discusses equilib
rium under full commitment. Section 4 characterizes the set of sustainable outcomes under
discretion, and section 5 presents our two examples. In section 6 we discuss equilibrium
under limited commitment. Finally, we offer some concluding remarks.
8
2. T h e M o d e l
We consider an infinite horizon, monetary economy with uncertainty. In each period,
t
=
0,1,2,..., the economy experiences an exogenously determined event, s{, drawn from a finite
set. Let
t.
s l = ( s 0) s i , ..., s t)
denote the history of exogenous events up to and including time
The probability of s l is given by /x(s‘). We denote the probability of s‘ conditional on sr by
fi(sl
|
s r ).
The monetary authority chooses an action,
financial intermediaries) in period
of exogenous events up to date
that private agents view
At
t,
t
= 0,1,2,.... Let
=
t.
denote the history
Throughout we assume
X t { h t- \ , s t ).
h t + 1 , h t+ 2....
( h t , s t + \, X t ( h t , s t+ i ) ) , h t+2 —
is induced by ATt(*, •) from
X ( h t~1, s t)
on
t,
and X(/ir_ 1 , s r ) , r ^
h t ~i
ht =
st , A t)
(/it+i,St+2,X t+i(/i(+i,St+2)), ... . To conserve on
notation, from here on we delete the subscript,
the number of elements in
h t = ( s l , A o , ..., A t )
as being generated according to the policy rule
We say that a set of histories
that the functions
(a lump sum monetary transfer to
and of policy actions up to time
At
if h t+! =
At
t
X.
This should not cause confusion:
are different is evident from the fact that
and hr_ 1 are different. We adopt this notational convention
for all functions of histories.
The commodity space in this economy consists of history contingent functions. That
is, allocations, prices and policy actions are expressed as functions of history. In standard
Arrow-Debreu economies, rules of this type are a function only of the history of exogenous
events. The extension considered here is necessary to accommodate the fact that the gov
ernment in our model optimizes sequentially and that, being a ‘large’ agent, its actions have
9
a non-trivial impact on private allocations and prices. At every date the government con
siders the consequences of all the current and future state-contingent actions that it could
possibly take, and selects the set of actions that it prefers. Since its objective is a function
of current and future allocations, these must be well-defined for every possible history for
the government problem to be well-posed. Histories,
ht,
can exclude past household actions
because, though individual households optimize sequentially like the government, they are
small and have no impact on aggregate allocations and prices (see Chari and Kehoe (1990)
for a further discussion.)
To reiterate from the introduction, the sequence of events within a period is as follows.
First, the exogenous event, st, is realized. Then, intermediate goods producers set the prices
of their goods. Finally, the monetary authority’s action is realized and all other model
variables are determined. We now describe the problems of the agents in our economy in
detail.
2.1. F irm s
F in a l G ood F i r m s
The final good,
c ( h t ),
is produced by a perfectly competitive firm that combines a con
tinuum of intermediate goods, indexed by
'(ht) =
Here 0 < A < 1 and
denote the time
t
yi(ht)
i
€ (0,1), using the following technology:
(2.1)
[jT (y i ( h t ) ) Xd i
denotes the time
t
price of intermediate good
input of intermediate good
i.
10
i.
Let
P i(h t- \ , s t)
This price is not a function of the time
t
policy action because intermediate goods producers set their prices before
A t
is realized.
The final good producer’s problem is:
max
P { h t- i , s t ) c ( h t )
c(M,{y<(/n)}
-
f
Jo
subject to (2.1). Here, P(/it_ i,s t) denotes the time
below why this price is not a function of
A t.
(2.2)
P i ( h t - i , s t ) y i ( h t )di,
t
price of the final good. We explain
Problem (2.2) gives rise to the following input
demand functions:
P ( h t- i , s t) \
Vi(ht) = c ( h t )
(2.3)
P i { h t- l t s t ) )
In conjunction with (2.1), this implies:
1-A
X
$t) — |
which expresses the time
t
(2.4)
P %{ h t - U s t y ^ d i
price of the final good as a function of the time
t
prices of the
intermediate goods.
I n t e r m e d i a t e G ood F i r m
Intermediate good
i
is produced by a monopolist using the following technology:
Vi{h t)
=
(2.5)
9 ( s t ) n i ( h t ).
Here U i ( h t ) denotes time t labor used to produce the
i th
intermediate good and
the stochastic economy wide level of technology at time
before the date
t
government policy action,
A t,
11
t.
Recall that
0 (sl)
P i(h t- i , s t)
denotes
is chosen
is realized. We assume that the producer
must supply all of the goods demanded by the final goods producers as determined by (2.3).
Thus, at time
t,
producer i’s problem, which we refer to as
max
P r o b l e m F,
is to maximize profits:
S t)yi (ht ) — W ( h t ) R ( h t ) n i ( h t ),
(2.6)
Pi(ht-i,St)
subject to (2.3), (2.5) and taking as given the time t wage rate,
terest rate,
R ( h t ),
as well as P (h t_i, s t ), and
c ( h t ).
W ( h t ),
the gross nominal in
The firm’s unit labor costs are
W ( h t) R (h t)
because the firms need to pay workers before production, and must borrow these funds from
the financial intermediary at interest rate
R ( h t ).
Note that from the firm’s perspective (2.6) is a function of the information set (h£_ i,s ().
This is because they view
( h t - u s t , X ( h t-
At
as being generated according to
At =
St)
and
ht =
1 , s t )).
Financial Interm ediary
A perfectly competitive financial intermediary receives deposits,
and a transfer,
A t,
I ( h t ),
from households,
from the government. These funds are lent to intermediate good producers
at the gross interest rate
R ( h t ).
At the end of the period, the intermediary pays
to households in return for deposits, and distributes
R (h t) A t
R ( h t)I(h t)
to households in the form of
profits.
2.2. Household
At time r the representative household’s expected present discounted utility is given by:
OO
I s T) U { c { h t ) , n ( h t ) ) .
t —r
sl
12
(2.7)
Here
n (h
t) denotes total hours time
U (c,n )
where
tp
> 0 and
a
t
hours worked, and
= [c(l - n )*](1 ' * / ( ! - a
> —1 . The household faces the following cash constraint on its purchases:
P ( h t- U s t)c{ht) < W ( h t) n (h t)
Here
(2.8)
),
+ A/ (h t_1) -
denotes the household’s end-of-period
t
(2.9)
I ( h t ).
—1 holdings of cash.
The household’s money holdings evolves according to:
M ( h t)
=
—I ( h t )
W ( h t)n(ht) +
(2.10)
— P ( h t- i , s t )c(h t )
(2.11)
+ R { h t ) [ I ( h t ) + A t } + D ( h t ),
where
D ( h t)
denotes total time
t
profits from the intermediate goods producers.
Let
Z ( h t ) = [c(ht ) , n ( h t ) , I ( h t ), M ( h t )}.
The household’s problem, which we refer to as
contingency plan for
M ( h t- i ) .
so that
hr+i
=
is then to choose a non-negative
to maximize (2.7) subject to (2.8)-(2.10),
In solving its problem, household views
X ( h t - i , s t ),
Z ( h t)
P r o b l e m H,
( h r , sr+1, X ( h r , s r + i ) )
13
At
n ( h t) <
1 and
I ( h t) <
as being generated by the policy rule
for all r.
2 .3 . G o v e r n m e n t
At time
t,
given history
s t ),
the government chooses a current action,
At,
and a se
quence of state contingent policy rules for each future period, to maximize the representative
household’s utility, (2.7). It takes as given, M(/ir-i)
ries will be induced by its future policy rules from
takes as given the household’s contingency plan,
+ A r > 0 , r > t,
and that future histo
h t = ( h t- i , s t , A t ).
The government also
{ Z ( h T) ; r
>
t}.
We refer to this problem as
P r o b l e m G.
2.4. S u stainable E quilibrium
In order to characterize the set of equilibrium outcomes, it is useful to define a sustainable
equilibrium for our model economy.
D efinition 2 . 1 .
A s u s t a i n a b l e equilibrium is a col lect io n o f h i s t o r y - c o n t i n g e n t a ll o c a ti o n
rules Z ( h t ), p r i c i n g f u n ct io n s,
s t ), P ( h t_ i , s t), W ( h t ), R ( h t ), a n d a g o v e r n m e n t p o l i c y
rule, X ( h t - i , s t ), s uc h t h a t :
1 . for all histories, h t , the allocation rules Z ( h t ), solve P r o b l e m H,
2 . for all histories, ( h t - i , s t ), the pricing function P i ( h t _ i , s t ) solves P r o b l e m F,
3. for all histories, (/it_i, s t ), producers of the final good make zero profits so that P(/it_ j , s t )
satisfies (2.4),45
4. for all histories, h t , the goods market clears, i.e., c ( h t ) = 6 ( s t ) n ( h t ), the loan market
clears, i.e., W ( h t ) n ( h t ) = /(/it) + At, and the money market clears, i.e., M ( h t ) =
M (h t-i) + A t.
5. for all histories, (/i£_i,St), the policy rule, X(/it_ i,s t) solves
To proceed we find it convenient to scale all time
t —1
t
P r o b l e m G.
nominal variables by the end of time
stock of money, M(/it_i). Let
Pt (/it - 1; &t)
R j ( h t - i , St)
M ( h t_,)
p(/it-i,St)
P ( h t- i , s t)
M(/it-x) ’
14
d(h t)
D ( h t)
M(/it_i),
W ( h t)
w ( h t)
I{h t)
M ( h t^ y
[ t]
In addition, we define the variable
x ( h t- i , s t )
M ( h t- i )
m ( h t_i)
of money to the aggregate stock of time
X ( h t. u s t )
At
M ( h t_i)
M (h ,.
as the ratio of the household’s time
t — 1 money.
Of course in equilibrium
O’
t —1
stock
m ( h t ^ i)
= 1.
Finally, let
z ( h t)
=
(2.12)
[ c ( h t ) , n ( h t ) , i ( h t ), m ( h t )}.
In what follows, we proceed in terms of these lower case scaled variables.
3. P r i v a t e S e c t o r E q u i l i b r i u m U n d e r C o m m i t m e n t
A key objective of our paper is to characterize the set of allocations that are the outcome of
some sustainable equilibrium in the model economy. To do this it is useful to first analyze
the behavior of the economy when the government can commit to a particular policy rule.
Specifically we consider the case of commitment, by which we mean that the government
never deviates from the policy rule,
possible date
t
x ( h t _ i , s t ).
histories can be indexed by
sl
Under these circumstances, the set of all
alone. This is because, under commitment,
ht
can be expressed as a function of s 4 using the following recursion:
h-i
= 0, ho = (s°,x(/i_ 1 ,s0)),
and similarly for
h i = ( s \ x ( h - i , s 0) , x { x ( h - i , s 0) , s i ) ) ,
h 2 , ^ 3 ,..., h t .
Let z*(se) = [c*^4),
n * ( s l ),
m*(st)] denote the state contingent allocations under
commitment. In addition, let
p?(sf), p*(sf) ],
15
x*(sl)
and
d * ( s f)
denote the
pricing functions, government policy rule, and intermediate good firm profit function under
commitment, respectively.
D efinition 3.1.
A p r i v a t e s e c t o r e q u i l i b r i u m is a c o ll ec t io n o f s t a t e c o n t i n g e n t a l l o ca ti o n
rules, z * ( s t ), p r i c i n g f u nc t io n s , [^ ‘(s4), R * ( s l ), p*(s4), p*(s4)], a n d a g o v e r n m e n t p o l i c y rule,
x*(s4),
s uc h t h a t ( 1) t h e a l l o c a t i o n rules z * ( s t ) s o l v e t h e h o u s e h o l d ’s p r o b l e m , (2) the p r i c i n g
f u n c t i o n p ' ( s l ) s o l v e s t h e i n t e r m e d i a t e g o o d p r o d u c e r s ’ p r o b l e m , (3) p r o d u c e r s o f t h e final
g o o d m a k e z e r o p r o f it s, ( 4) t h e g o o d s m a r k e t clears, i.e., c*(s4) = 0(s4)n*(s4), t h e l o an m a r k e t
clears, i.e.,
tu*(s4)n*(s4) = i*(s4) + x*(s4),
a n d t h e m o n e y m a r k e t clears, i.e., m * ( s t )
= 1.
Note that a private sector equilibrium satisfies all the conditions for a sustainable equilib
rium, with two exceptions. First, it does not require optimality by the government. Second,
private sector allocation rules and pricing functions solve the private sector optimization
problems and satisfy market clearing only for histories induced by x*(s4).
We now characterize the private sector equilibrium. The first order condition of the
intermediary goods firm’s problem is:
A0 { s l )
In symmetric equilibria, p*(s4)
=
iu*(s4)
c V )V ~ A
y*(s£)y
c*(s 4) for all
A0(s4)
i,
R - ( S !).
p*(sl)
so that,
w ' j s 1)
R ' i s 1).
P m( s l )
The Lagrangian representation of the household’s problem at time 0 is:
t=
0
£ M stl5°) W CV ) ’ * V ) )
5*
-t-i^s4)
—i*(s4) 4- u>*(s4)n*(s4) —p*(s4)c*(s4)j
+ r ] (s4)[(u>*(s4)n*(s4)
4- m*(s4-1) —i*(s4) —p*(s4)c*(s4))
16
(3.1)
■ft*(s£)(i*(s£) +
where
i'(st),r j(s t)
x ( s t))
+ d*(s£) —(1 4- x(s£))m*(s£)]}
are the non-negative Lagrange multipliers on the cash constraint and the
budget constraints, respectively. In addition to the budget and cash constraints, the first
order conditions for c*(s£), n*(s£), m*(s£) and i*(s£) are:
uc(s£) = (i/(5£) + r/(s£))p*(s£),
«n(s£) = - M s‘) +
) H s l) = 0
(3.2)
v i s ^ w ' i s 1),
£ M st+V )*?(s£+1)fl’(s£+l)>
5‘+lis‘
i/(s£) + rj(s£) = J?(s£)f?*(s£),
(3.3)
(3.4)
(3.5)
where 5£+1|s£denotes the set of time £+1 histories of exogenous events s£+1 that are consistent
with s£. Also, uc(s£) and
u n(s l)
denote the marginal utility of consumption and labor at date
£. The transversality condition for the household problem is
Jim
= 0.
r_,°° yr
(3.6)
P KS1 )
These conditions can be simplified as follows. From (3.2) and (3.3), we obtain:
^ ( gt) = i p c ' j s 1)
p * ( s t)
1 —n*(st) ’
17
(3?)
according to which the household equates the real time
t
wage rate to the marginal rate of
substitution between consumption and leisure. Non-negativity of the Lagrange multipliers
and (3.5) imply
> 1.
(3.8)
Relations (3.2), (3.4) and (3.5) imply
u c(s t )
■n*(ot\
n
R
(3*)
v -"
_l_ W e t \
^
/ t+ i | t \ Uc(gt+1)
I 'r^ot+lA
^
(3.9)
According to (3.9), the household equates the marginal utility of spending an additional dol
lar on consumption to the expected marginal utility associated with investing an additional
dollar with the financial intermediary.
Recall that loan market clearing requires
iu*(s£)n*(s£)
= i*{sl)
+ x*(s£).
(3.10)
Substituting (3.10) into the scaled version of the household’s cash constraint, (2.9) yields:
p’(s£)c*(s£) = 1 -I-x*(s£).
(3-11)
c*(s£) = 0(s£)n*(s£).
(3.12)
Goods market clearing implies:
Using our assumed functional form for the utility function, equations (3.1), (3.9)-(3.12)
18
reduce to:
Y
5t+1|s*
M 5t+1lst)v (n *(st) .n *(st+1) . ^ ) . ^ ( s ‘+1).^*(st+1)) = 0,
(3.13)
where
v(n>n'l0,0'>x') = Y Z ^ N
1 - n) 1 1_<T -
(3-14)
Finally, substituting (3.11) and (3.12) into the transversality condition, (3.6), we obtain:
[0(sTK ( s T)(i - n ’(s7'))v’] 1 " = o.
(3.15)
ST
We are now in a position to characterize a private sector equilibrium.
P ro p o sitio n 3.2.
T h e a l l o c a t i o n r ul e
n*(s£)
is p a r t o f a p r i v a t e s e c t o r e q u i l i b r i u m i f a n d
o n l y i f t h e f o l l o w i n g fi ve c o n d i t i o n s a r e s atis fie d:
n* (s£) > 0,
A 1 - n * ( s £) > l
i/> n*(s£) ~
(3.13 ), (3.15),
(3.16)
and
n*(s£)
x*(s£)
1 —n*(s£) — l+ x * ( s £)'
(3.17)
P ro o f. Suppose n*(s£) satisfies the five conditions. Then we can construct the remaining
objects in a private sector equilibrium as follows. The consumption allocation rule is c*(s£) =
9 ( s t ) n ' ( s t ), the price rule is obtained from the cash constraint, (3.11). The nominal wage is
obtained from (3.7), the nominal interest rate from (3.1) and, finally, i*(s£) is obtained from
(3.10). The only conditions that remain to be checked are (3.8) and the constraint on the
household’s problem that 0 < i*(s£) < 1. The first is verified by noting that the expression to
the left of the inequality in (3.16) is the equilibrium rate of interest. To verify the constraints
on i*(s£) notice that i*(s£) = (1 + x * (s£))
—x * ( s l ). The requirement that i*(s£) < 1
is guaranteed by (3.16) and the fact, A < 1. Inequality (3.17) guarantees that 0 <
Suppose we have an equilibrium. The proof follows immediately from necessity of (3.13),
(3.15), i?*(s£) > 1, and 0 < i * ( s l ) < 1. ■
In general, the set of private sector equilibria under commitment in our economy is not
19
unique. The expectation traps that we focus on in this paper have nothing to do with this
form of multiplicity. Consequently, it is useful to proceed under assumptions which guarantee
the uniqueness of a private sector equilibrium under commitment. One such assumption is
a —
1, in which case, equation (3.13) reduces to
n*(st)
1
=
/
?
X
>
(
st+V
)
1 + £*(st+1)
A 1 —n*(s‘)
S*+1
xp
(3.18)
Consequently, if an equilibrium exists, then it is unique. This equilibrium has standard
properties: employment is increasing in A, i.e., decreasing in the degree of market power;
employment is decreasing in money growth; and it is decreasing in
xp,
which governs the
marginal utility of leisure.
To see the potential for non-uniqueness when
£*(s£) =
x.
a
^ 1, consider the case
9 (sl) =
1 and
In this case, there exists an equilibrium in which employment is given by the
deterministic analog of (3.18). However, there may be other equilibria as well, even sunspot
equilibria (see Mathene (1995) and Woodford (1986).) To eliminate these possibilities, hence
forth we restrict ourselves to the case
a —
l.3
3T o see how su n sp o t equilibria may arise w ith (J / 1, consider th e case in which th e m onetary grow th
ra te is a c o n sta n t, equal to x, a n d 0 = 1 . T hen (3.13) requires v = 0. T his im plicitly defines n ' as a
function (possibly a correspondence) from n. T h e im plicit function theorem guarantees th a t th ere exists a
differentiable function relatin g n ' to n in a neighborhood of stead y -state: n = n' = \ 3 /[xp{\ + x)], with:
dn'
dn
(1 -
a)
Here, dn'/dn = 0.77 w hen a = 7, A = 0.87, 0 = 3, 0 = 1/1.03. T h e fact th a t th is derivative is less th a n one
m akes it possible to c o n stru c t m ultiple determ inistic equilibria an d su n sp o t equilibria.
20
3.1. T h e R am sey E quilibrium
As a benchmark, it is useful to consider the equilibrium of the model under the assumption
that the government had access to a commitment technology at
t
= 0. To this end we now
define a Ramsey equilibrium.
D efinition 3.3.
A R a m s e y e q u i l i b r i u m is a c ol le ct io n o f a l l o c a t i o n r ul es f o r p r i v a t e a g e n t s ,
z ' i s 1), p r i c i n g f u n c t i o n s
[to*(s£), R*(s£), p*(s£), p*(s£)], and a
g o v e r n m e n t p o l i c y rule, x * ( s l ),
t h a t y i e l d s t h e h i g h e s t u t i l i t y f o r h o u s e h o l d s o v e r t h e s e t o f p r i v a t e s e c t o r equ il ib r ia .
P ro p o sitio n 3.4.
T h e R a m s e y e q u i l i b r i u m s atis fie s
w , (st)
P*(s£)
= A0(s£), n V )
X > ( s £+V )
1
1 + £*(st+1)
A
V>+ A’ * V ) = 1,
I A>
/?’ — 1 + £*(s£)
(3.19)
(3.20)
P ro o f. From Proposition 1, the Ramsey equilibrium maximizes discounted utility subject to
n*(s£) > 0, (3.13), (3.15), (3.16), and (3.17). With o — 1, equation (3.13) reduces to (3.18).
It follows from this that solving the Ramsey problem is equivalent to solving a sequence
of static problems subject to the constraints (3.15) - (3.17). The first expression in (3.19)
follows from the first order condition of intermediate good firms. Condition (3.16), which
requires a positive interest rate, implies
rc V ) < ,
A 1 —n * ( s l ) ~
$
(3.21)
But as long as (3.21) is satisfied, the period utility function is strictly increasing in n*(s£).
Because the left side of (3.18) is increasing in n*(s£), the solution to the Ramsey problem is to
set n*(s£) so that (3.21) holds with equality. This establishes that R * ( s l ) = 1, n * ( s l ) =
and that the right hand side in (3.18) equals one. This last result implies the first expression
in (3.20). Since n*(s{) is constant, the transversality condition is satisfied. Finally the last
expression in (3.20) follows by substituting (3.16) with equality into (3.17). ■
Notice that even absent uncertainty in technology, there are many rules for
x ' ( s t)
that
satisfy (3.20), which only requires that the time t conditional expectation of 1/ [1 + x*(s£+1)]
be equal to 1/j3. So a constant growth rate of money equal to
21
0
— 1 is sufficient but not
necessary to implement the Friedman rule. Finally, note that even in the Ramsey equilibrium,
as long as there is monopoly (A < 1), employment is less than the social optimum, 1/(1 +
tp)-
The monetary authority would like to increase employment beyond the Ramsey, but this is
not achievable by monetary policy because of the non-negativity constraint on
R ( s l ).
4. C h a r a c t e r i z i n g S u s t a i n a b l e O u t c o m e s
We now turn to the task of characterizing the set of sustainable outcomes for our model
economy. To do this we must allow for allocation rules, pricing functions and government
policy rules to be contingent on both the history of exogenous events and the history of
policy actions. Throughout we impose the restriction that x(h(_ i,st) <
x
for all (/it_ i, 5*),
i.e. we impose an upper bound on the growth rate of the money supply. See the conclusion
for an extended discussion of this assumption.
We begin by characterizing the worst sustainable equilibrium which we refer to as the
‘high inflation’ equilibrium. Consider the following candidate sustainable equilibrium in
which the government policy rule is given by
xt
denote the actual time
t
, s t)
=
policy action. For all histories
x
for all possible (/ie_ i, s4). Let
( h t _ i , s l ),
the candidate pricing
functions are,
P i ( h t - i St)
=
p { h t - \ , s t)
=
1+ x
0(5*)
ip( 1 -I- x )
4* XP
(4.1)
XP
These candidate pricing functions reflect our assumptions that (i) firms set prices in period
t
prior to the realization of the period t policy action, and (ii) they do so under the expectation
that the policy action will be
x ( h t , s t+ 1 )
=
x.
22
The candidate allocation rules, interest rate and wage functions are
c(ht)
(1 +
x t)
c(h t)
n{ht)
*(«*) ’
P ( h t- i , s t) ’
w(ht)
p {h t. l , s t)i}9{st)n {h t)
=
1-
n (h t)
,
R (h t)
=
1 - (1 +
i ( h t) =
A (1 ip
x t)
n ( h t ))
n (h t)
1-
’
ipnjht)
1-
n (h t)
(4.2)
(4.3)
respectively.
By construction, these candidate rules satisfy private agent optimality and market clear
ing for all possible histories,
ht.
So to establish that the candidate equilibrium is sustainable,
we need only establish that it is consistent with optimization on the part of the government.
Consider a one-shot deviation at period t , given some history,
( h t _ i ,s 4).
Given our assumed
upper bound on money growth, the only feasible one-shot deviation is for the government to
set money growth at
xt < x
, and then to return to
x
thereafter. From (4.2), this reduces cur
rent consumption and employment. Since utility is strictly increasing in levels of equilibrium
employment below 1 /( 1 + ^ ) , this reduces current utility. Future outcomes are unaffected.
Thus, the government has no incentive to pursue a one-shot deviation for any history. Since
this includes histories in which there have been any number of deviations, it follows that
no deviation raises welfare (see Abreu (1988) and Whittle (1983).) This establishes the
sustainability of the candidate equilibrium.
To see that the candidate equilibrium is the worst sustainable equilibrium, note that
in any other equilibrium, employment at some date is necessarily higher because of (3.18).
Thus, utility must be higher too. This establishes the following proposition:
P ro p o sitio n 4.1.
T h e h i g h in fl at io n e q u i l i b r i u m is t h e w o r s t s u s t a i n a b l e e q u i l i b r i u m .
We denote by ud(s£) the highest one period utility level associated with a deviation by
23
the government:
l + x
1+ x
\
- . _________
]
ryiov 1/ If .....
-i<i<s \^p*(s£) ’ 6 ( s t ) p * ( s t ) J '
Let
x d( s l )
denote the value of
x
that achieves the optimum. In addition, let u(s£) denote
the one period utility level in the high inflation equilibrium. We are now ready to state our
main result. The following proposition establishes a set of restrictions on allocations and
prices which are necessary and sufficient for them to be the outcomes of some sustainable
equilibrium. In what follows, it is useful to define a class of policy rules characterized by a
particular trigger strategy, a
grim
trigger. Such a rule specifies that in all histories in which
there has been some deviation, x(/it_ i,s t) =
x.
such a policy rule as a
g r i m t ri g g e r equilibrium.
P ro p o sitio n 4.2.
z*(s£), u;*(s£),
Let
R " ( s t ),
We refer to an equilibrium associated with
p*(s£), p’(s£)
be
an
a rb itra ry set o f allocations
a n d p r ices . L e t x , ( s t) b e an a r b i t r a r y s e q u e n c e o f g o v e r n m e n t pol ic ie s. T h e n ,
f?*(s£),
P i ( s l ),
p*(s£), x*(s£)}
is t h e o u t c o m e o f s o m e s u s t a i n a b l e e q u i l i b r i u m
{z*(s£), w * ( s l ),
i f and o n l y if:
1. ( z m( s t ) , w * ( s t ), R * ( s t ) , p * ( s t ) , p * ( s t ) , x * ( s t )) is a p r i v a t e s e c t o r e q u i l i b r iu m ;
2.
(z , ( s t ) , w m( s t ) , R * ( s t ) , p ' ( s t ) , p , ( s t ) , x * ( s t))
OO
E
r ' E ftrM
r —t
for all
s£,
all t
s atis fie s
OO
s ‘) U ( c - ( s ' ) , n ‘ ( s ' ) ) >
«V ) +
I s') fl(sr)
Y.
r=t + 1
(4.4)
s=r
= 0 ,1 ,2 ......
P roof. We first consider sufficiency. Suppose the allocations and prices satisfy conditions 1
and 2. We construct a particular grim trigger sustainable equilibrium which supports these
outcomes. Consider the following candidate equilibrium. For all histories with no deviation,
let x(/it_ !,st) = x*(s£), let p*(/i£_x,st) =P,*(st), p i h ^ ^ S t ) = p * ( s l ), w ( h t ) = w*(s£), R { h t ) =
R , ( s t ) and let z ( h t ) = z * ( s t ). For all other histories, let x(/it_ i,s t) = x , and let Pi(ht_ i,s t),
p(/it_ i,s t) be defined by (4.1). Also, let w ( h t ), R ( h t) and z ( h t ) be defined by (2.12) and
(4.1)-(4.2) with x t replaced by x . For all histories with no deviation, these constitute a
private sector equilibrium by assumption. For all other histories, they constitute a private
sector equilibrium by the discussion leading up to Proposition 4.1. To show government
optimality, note that since the right hand side of (4.4) is the discounted utility associated
24
with a deviation by the government, no deviation can raise welfare. Thus, the candidate
equilibrium is a sustainable equilibrium, which establishes sufficiency.
We now consider necessity. Suppose (z ( h t ), w ( h t ), R ( h t ) , p i ( h t ^ u s t ) , p ( h t ^ 1, s t ), x ( h t ^ u s t ))
is a sustainable equilibrium with outcomes (z * ( s t ), w * ( s t ), R * ( s l ), pt*(s£), p * ( s l ), x * ( s t )).
Condition 1 is satisfied by the definition of a sustainable equilibrium. We establish condi
tion 2 by contradiction. Suppose 2 is violated for some s l . Consider the following devia
tion: the government sets x t = x d ( s t ), and specifies future policies according to x ( h r _ 1, s r ),
r > t. The expected present discounted value of utility under this deviation is u d ( s l ) +
/?r-t 12s=r At(5’’ I s t ) u ( s r ), where u ( s r) denotes utility in state s r . From the discussion
preceding Proposition 4.1, u ( s r ) is a lower bound on period utility when policy is set ac
cording to x ( h r - i , s r ). It follows that u ( s r ) > u(sr). Therefore, the return associated with
the deviation is greater than, or equal to, the right hand side of equation (4.4). Given our
supposition that condition 2 is violated, this means that the proposed deviation will be imple
mented at s l . Consequently, the equilibrium is not consistent with government optimization.
This establishes the contradiction. ■
We conclude this section with a brief discussion of the proposition. The proposition says
that a particular outcome is sustainable if, and only if, it is the outcome of the associated grim
trigger equilibrium. The proposition does
only
not
say that a particular outcome is sustainable
by a grim trigger equilibrium. In general, an outcome may be sustainable by a variety
of equilibria. For example, the equilibria could involve less extreme trigger strategies, or no
trigger strategy at all. The proposition is silent on the nature of the equilibria that support a
given outcome. It only provides conditions under which a particular outcome is sustainable
by
some
equilibrium. In the next section, we make these observations concrete through a
series of examples.
5. E x a m p l e s
This section reports two examples which illustrate the types of expectation traps discussed
in the introduction. In both cases, we establish that the expectation trap outcome is the out
come of some sustainable equilibrium. Our first example illustrates the type of expectation
25
trap that can arise when agents expect that monetary policy will react to non-fundamental
shocks. In addition, we discuss three equilibria that can sustain this outcome. In the sec
ond example, agents expect monetary policy to react to technology shocks in a particular
way. The example is constructed to articulate in a stylized way an interpretation of the US
inflation experience since the mid-1960s.
E x a m p l e 1: N o n - F u n d a m e n t a l Sh ocks
The parameter values for the model economy are given by xb — 3, A = 0.95,
P
= 1/1.03,
x
= 0.30. In each period there is a shock,
s
9t
= 1, a = 1,
€ (s(l), 5(2)}, which is drawn from
a highly persistent, symmetric, two state Markov chain, where the probability of switching
states is 0.10. The shock is non-fundamental, in that it has no impact on preferences or
technology. Still, there is a sustainable outcome, in which money growth responds to
When
s =
s(l) the money growth rate is 0 percent, and when
money is 3 percent. The Wold representation for
Xt
where
et
xt
s
s.
= 5(2), the growth rate of
is given by
—0.15 4- .9xt_i +
€t,
(5.1)
is uncorrelated with past variables, and has standard deviation 0.0092.
Several features of this example are worth emphasizing. First, the model is observationally equivalent to one in which policy is set according to (5.1). In this respect, the model
is formally identical to a standard, monetized real business cycle model (see, for example,
the ‘cash-in-advance’ model in Christiano and Eichenbaum (1996)). The interpretation of
the policy shock, however, is very different. In the literature, e( is assumed to reflect the
effects of fundamental disturbances to the policy making process, e.g., shocks to preferences
26
of policy makers. In our environment, the shocks originate outside the policy making process
and reflect the non-fundamental disturbances that can generate an expectation trap.
Second, consistent with the previous observations, the dynamic response of real variables
to changes in
xt
resemble those in standard monetary real business cycle models. For exam
ple, Table 1 indicates that employment, consumption and beginning-of-period real balances
(M
/P )
are low when
xt
is high, while
Rt
and inflation are high.
Third, Table 1 contains information relevant for verifying that the outcome in this exam
ple is sustainable. The table reports, for both states of the world, the current period utility,
it, of the household and the expected present value of its utility from tomorrow on,
v,
along
the equilibrium path. In addition, the table reports, for both states of the world, the current
period utility,
u d,
and expected value of utility from tomorrow on,
one-period deviation from the equilibrium path. In the example,
ud
vd,
associated with a
turns out to be the level
of utility associated with the socially efficient level of employment, 1/(1 + 1/>)
=
0.25. The
deviation growth rate of money, xd, needed to achieve this level of employment is larger in
the
s
= s(2) state of the world. To see why, note that achieving
the real value of end-of-period money balances, (1
+ x )/p ,
n =
0.25 requires setting
equal to 0.25. But, in the high
state of the world, intermediate goods producers anticipate a high (3 percent) growth rate
of money. Consequently, the price level is high. With
p
high,
x
must be high too. Since the
deviation induces a higher level of consumption and employment,
nevertheless sustainable because
u + v > ud
+
vd
27
u < ud.
The outcome is
in both states of the world.
M /P
x (h t-i , s t) =
0.0
II
o
oO
C
variable
?>T
eC
*o
Table 1: Non-Fundamental Expectation Trap
c
1
0.9709
71
0.2346
0.2304
0.2346
0.2237
W /P
0.9196
0.8962
u
-2.2519
-2.2536
V
-75.0892
-75.0950
ud
-2.2413
-2.2413
ua
-2.2910
-2.2910
va
-76.3650
-76.3650
(beg. of period)
Fourth, we briefly consider what sort of equilibria can sustain this outcome. One such
equilibrium is the grim trigger equilibrium used in the proof to Proposition 4.2. In results
hot reported here, we verified that a milder trigger strategy equilibrium that also sustains the
outcome is one in which a deviation triggers a shift to the maximum growth rate of money,
x,
for one period. Finally, we discuss a particular non-trigger equilibrium which sustains the
expectations trap outcome. Loosely speaking, in this equilibrium the monetary policy rule
specifies that deviations in which money growth is high in a particular state persist forever,
but deviations down have no impact on future policy. Formally, for histories, (/it_l t s(), in
28
which there has never been a deviation,
x ( h t- u s t)
0.0, if
{
s
=
( h t - i , s t ),
where
= s(l)
(5.2)
0.03, if s =
Next, consider histories,
v ( s t ),
s ( 2)
in which there was: (i) one deviation in the past, (ii) the
deviation money growth rate was a, (iii) the growth rate called for by the policy rule was
x
< a,and (iv) the state was
x ( h t- i , s t) =
where
s = s(k).
i>{s(k)) = a
In histories like this, the policy rule specifies that
and P(s(j))
= i'(s(j))
for
j
^
k.
That is, the rule
specifies that if the policy authority deviates up in a particular state, the deviation action
will be followed whenever that state recurs in the future. Histories in which there has been
more than one deviation are handled in the same way. Deviations down have no impact on
the policy rule. We verified, using numerical methods, that the outcome in example 1 is
sustainable by this equilibrium.
Taken together, these three equilibria make clear the technical role played by the grim
strategy equilibrium in the proof of the characterization result. Expectation trap outcomes
can be supported by variety of equilibria.
E x a m p l e 2: F u n d a m e n t a l Shocks.
Apart from the stochastic process governing the state of the world, s, and the state of
technology,
6t ,
the parameter values used in this example coincide with those in Example
1. The state of the world can take on values numbered 1 through 7. In state one,
while
6t
= 0.90 in the other six states. The Markov chain governing
s
9t =
1,
has the following
properties. If the economy is in state 1, it stays with probability 0.99, and it moves to state
29
2 with probability 0.01. If the economy is in states 2 through 6, the economy moves to the
next higher state with probability 0.95. It stays in the current state with the complementary
probability. If the economy is in state 7 it moves to state 1 with probability 0.95. In this
example, there is a sustainable expectation trap outcome in which money growth is 0.0,
0.05, 0.10, 0.15, 0.20, 0.15, and 0.10 in each of states 1 through 7. This outcome has the
property that money growth is typically low. Occasionally, following a decline in the state
of technology, money growth and, hence, inflation rise for aft extended period of time before
returning to their typically low levels. Figure 1 illustrates a typical realization in which
the economy is in state 1 for 5 periods, then switches to state 2 for 8 periods, and finally
returns to state 1 for 2 periods. Presumably, it is easy to construct similar examples in
which expectation traps like this are triggered by other fundamental shocks, like changes in
government purchases.
We find Example 2 interesting because it illustrates another sense in which discretion
makes possible volatile equilibria. Also, this type of example may form the basis for a simple
explanation of the prolonged rise in inflation during the mid-1960s and 1970s. Authors,
such as Fellner (1976) have argued that this rise in inflation was triggered by the simulta
neous increase in expenditures due to the expansion of the Vietnam war and Great Society
Programs. Other authors such as Blinder (1982) argue that the inflation of the 1970s was
triggered by the supply shocks of that decade. Our analysis raises the possibility that the
long lived inflation is consistent with sequential rationality on the part of private agents and
a benevolent monetary authority. It simply reflects the type of expectation trap that can
arise under discretion.
30
6. L i m i t e d C o m m i t m e n t
We now show that a simple institutional change in the way monetary policy is conducted
eliminates the expectation traps associated with discretionary monetary policy. The change
is that we impose a form of limited commitment upon the monetary authority, in which it is
required to commit to a state-contingent action one period in advance. We formally model
this by specifying the time
t
sequence of events as follows. First, the exogenous event,
realized. Second, intermediate goods producers set the time
the monetary authorities’ date
other date
t
of date
ht
t
is
t
t
st,
is
prices of their goods. Then,
4- 1 action, contingent upon St+i, is realized. Finally, the
model variables are determined. In this modified version, the history at the end
= (h t - i , S t , x ( s t + l )), where x(st+1) denotes the monetary authority’s time t + 1
state-contingent action.
P ro p o sitio n 6.1.
U nder lim ite d co m m itm e n t, th e set o f sustainable o u tc o m e s coincides
w i t h t h o s e in t h e R a m s e y e q u i l i b r i u m .
R em ark 1. Recall t h a t
AJ a n d t h e g r o w t h .rate
in a R a m s e y e q u i l i b r iu m , e m p l o y m e n t , n R , is g i v e n b y n R
= A/(t/>+
o f t h e m o n e y s a t j s h e s ^^t+i M st+1 I st) i+x(7~q = h T h u s > in a
R a m s e y e q u i l i b r i u m , a l l o c a t i o n s and p r i c e s ar e u n i q u e l y d e t e r m i n e d a n d m o n e t a r y p o l i c i e s
are d e t e r m i n e d o n l y in an e x p e c t e d val ue sense.
P roof. In the Ramsey equilibrium, employment is independent of the state of nature. The
first step in the proof is to show that in any sustainable equilibrium, the date t consumption
and employment allocations depend only upon the date t + 1 state-contingent monetary
policy actions. To see this, first recall that, in a sustainable equilibrium, for all histories,
the continuation outcome induced by the sustainable equilibrium must solve the consumer’s
optimization problem and satisfy market clearing. The analog of (3.18) is:
n(ht)
A1 -
n(ht)
\ sl )
5t+l
1
1 + x(st+1)
(6.1)
It follows that, for all histories, the employment allocation at date t depends only upon the
state-contingent actions of the monetary authority at date t 4-1. The second step in the
proof is by contradiction. Consider a sustainable outcome such that n ( h t ) ^ n R for some h t .
31
Consider the following sequence of deviations by the government: x(sr+1) = ^ —1 for r > t .
Under this deviation, n ( h r ) = n R for r > t. But, equations (3.16) and (6.1) imply that n ( h t)
is strictly less than n R. Since utility is strictly increasing for levels of employment less than
1/(1 + xp), this deviation raises utility. This establishes the contradiction. ■
To understand this result, it is useful to recall why expectation traps are possible when
there is no commitment. Two conditions that are necessary for expectation traps to arise
are: (i) private agents’ actions depend upon their expectations about future monetary policy,
and (ii) private agents suffer a loss if their expectations are not validated. When these
conditions are satisfied and the monetary authority is benevolent, then it is possible for
expectations to drive monetary policy. Many equilibria can arise, because there is nothing in
the environment to uniquely pin down agents’ expectations. Limited commitment eliminates
expectation traps by forcing the monetary authorities to commit before private agents make
their decisions. In this way, expectations are uniquely pinned down.
We conclude this section by discussing how the previous proposition would have to be
modified to accommodate different environments. For example, suppose agents committed
to nominal wages or prices
K
periods in advance. To eliminate expectation traps in this
environment, we anticipate that monetary policy would have to commit
K
4-1 periods in
advance. In environments in which decisions depend on the entire future history of monetary
policy, for example, when agents have the option to invest in physical capital, then it may
not be possible to eliminate all expectation traps with limited commitment.
7. C o n c l u d i n g R e m a r k s
This paper studied the operating characteristics of monetary policy when the monetary au
thority cannot commit to future policies. Our main finding is that discretion exposes the
32
economy to expectation traps. In addition, we argue that alternative institutional arrange
ments for the conduct of monetary policy which impose limited forms of commitment on
policy makers can eliminate the possibility of expectations traps.
The existing literature emphasizing the benefits of commitment has not recognized that
expectation traps of the sort analyzed in this paper can occur can occur under discretion.4
Perhaps the paper closest in spirit to ours is Cole and Kehoe (1995), who construct a model
of sovereign debt and show that, under discretion, default can occur in response to a non fun
damental shock. In the debt default literature, Calvo (1988) and Chari and Kehoe (1993a,b)
show that, under discretion there can be multiple equilibria, even with a finite horizon.
Neither of these papers discusses the possibility of excessive volatility.
The sunspot literature has discussed extensively the possibility of excessive volatility.5
This literature shows that there are economies in which private agents respond to nonfun
damental shocks and also shows that private agents can respond excessively to fundamental
shocks (see Beaudry and Devereux, 1994). In this literature the excessive volatility arises
from the structure of preferences and technology of private agents. In our environment, it
arises from the structure of policy making.
We conclude by discussing a shortcoming of our model, and then arguing that the basic
findings of this paper are likely to be robust to a variety of plausible remedies. In our
model we assumed there is an exogenously given upper bound,
x,
on the money growth
4See for example, Kydland and Prescott (1977), Calvo (1978), Fischer (1980), Barro and Gordon (1983),
Stokey (1991), Chari and Kehoe (1990,1993a,b) and Ireland (1995). A reference to the possibility of ex
pectations traps in which monetary policy responds to nonfundamental shocks appears in Rogoff (1989, p.
245-46). However, Rogoff refers to this in the context of trigger-strategy equilibria, while we show that these
equilibria can be supported without trigger strategies.
5See Azariadis (1981), Benhabib and Farmer (1994,1995), Bryant (1983), Cass and Shell (1983), Cooper
and John (1988), Farmer and Guo (1994), Gali (1994a,b), Matsuyama (1991), Shleifer (1986) and Woodford
(1986,1991).
33
rate. Our characterization theorem requires the existence of a worst sustainable equilibrium.
In our model economy there is no worst equilibrium absent an exogenous upper bound on
the money growth rate. This reflects the fact that, in our environment, there is only a
contemporaneous benefit, and no contemporaneous cost, from an unexpected increase in
money growth. To avoid the need to specify an exogenous upper bound on money growth,
we must introduce features which associate contemporaneous costs with high money growth.
One such cost that has been widely emphasized is that increases in the inflation rate induce
resource misallocation by changing the cross-sectional distribution of relative prices. One
simple way to capture this resource misallocation is to introduce staggered price setting
along the lines of Taylor (1980). To see this, consider the case where each period, half the
intermediate goods producers set a price for their goods which will prevail for two periods.
Under these circumstances, the higher is inflation, the greater the difference is between
the two sets of prices. This results in an inefficient distribution of production among the
intermediate goods producers. If we assume that the firms setting prices in any particular
period do so after the current period realization of the monetary action, then this obviously
creates a contemporaneous cost to increasing the money growth rate, and this cost is greater
at higher inflation rates. We expect that expectation traps can arise under discretion in this
environment too: suppose the subset of firms setting prices in a particular period expect
money growth to be high in the next period. They will find it in their interest to set prices
high. Then, in the next period the monetary authority faces a choice: either validate the
expectations of the previous periods’ price setters, or incur a recession. As in the environment
of this paper, it may choose to validate the expectations, thus ensnaring the economy in an
expectations trap.
34
FIG U R E 1
employment
output
years
years
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@FedFRASER