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Working Paper 9008
STICKY PRICES, MONEY, AND BUSINESS FLUCTUATIONS
by Joseph G. Haubrich and Robert G. King
Joseph G. Haubrich is an economic advisor at
the Federal Reserve Bank of Cleveland and
Robert G. King is a professor of economics at
the University of Rochester. The authors would
like to thank Russell Boyer, Carsten Kowalczyk,
and Randall Wright for helpful comments. They
would also like to acknowledge support from the
National Science Foundation.
Working papers of the Federal Reserve Bank of
Cleveland are preliminary materials circulated
to stimulate discussion and critical comment.
The views stated herein are those of the authors
and not necessarily those of the Federal Reserve
Bank of Cleveland or of the Board of Governors
of the Federal Reserve System.
September 1990
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Abstract
Can nominal contracts create monetary nonneutrality if they arise
endogenously in general equilibrium? Yes, if (1) agents have complete
information about the money stock and (2) shocks to the system are purely
redistributive and private information, precluding conventional insurance
markets. Without contracts, money is neutral toward aggregate quantities.
However, risk-sharing between suppliers and demanders creates an
incentive for both parties to use nominal contracts. In particular, if an
increase in the money growth rate signals a rise in the dispersion of shocks
to demanders' wealth, then prices adjust only partially to monetary shocks and
money is positively associated with output.
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Introduction
Many macroeconomists believe that some form of price stickiness underlies
the observed positive association of high money growth and high real activity
at business-cycle frequencies. Often, this price stickiness is asserted to
arise from explicit or implicit contracts. Model economies that do not
include nominal contracts are consequently viewed as omitting the basic cause
of monetary nonneutrality. For example, Lucas's (1972) pathbreaking
general-equilibrium model of business fluctuations
--
which employs imperfect
aggregate information to generate monetary nonneutrality - - has been widely
criticized for excluding nominal contracts, even though no economic forces
would lead these to arise endogenously. Yet, in the past decade, few
similarly explicit model economies have been produced that (1) derive a role
for nominal contracts from underlying assumptions about the economic
environment and (2) explain the implications of contract arrangements for
money and business cycles.l
Contract theory seemingly could not justify
nominal contracts; today, the foundations of sticky prices rest more on the
cost of price changes (Rotemberg [1982], Parkin [1986]), or on the
multiplicity of rational-expectations equilibria (Azariadis and Cooper
[1985b]).
This paper provides a simple rational-expectations general-equilibrium
model in which endogenously generated contracts make a difference. That is,
under some fiscal-monetary regimes, contracts simultaneously make prices
sticky (so that they respond less than proportionately to changes in the
quantity of money) and lead to a causal positive relationship between
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contemporaneously observed money and production/effect.
Further, our model
economy is a variant of Lucas's (1972) setup. The difference is that we assume
monetary changes are neutral toward real aggregates in the absence of
contracts because economic agents accurately perceive these change^.^
These results derive from four underlying assumptions about the
preferences, technology, and information structure of a stochastic
consumption-loans model that is in most other ways identical to the
full-informationversion employed by Lucas. First, risk-averse demanders of
money are subject to idiosyncratic individual disturbances that are private
information. That is, there is a demand for insurance against idiosyncratic
disturbances, but the fact that these are private rules out the operation of
conventional insurance markets, which make payments contingent upon
verifiable losses. Second, the growth rate of money is positively
associated with the dispersion of individual disturbances. This assumption,
though not standard in formal modeling, has received attention from both
monetary theorists and policymakers. Third, prior to the realization of money
growth or individual shocks, suppliers of goods can compete by offering
alternative contracts that specify a relationship between money growth and
price adjustments. Fourth, the technology of exchange dictates that an
individual visit only one supplier after realization of aggregate and
individual disturbances.
In this environment, welfare can be improved by competitive contracts that
embody a shifting of risk, with resources being transferred between suppliers
and demanders in contingencies that involve high individual uncertainty.
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Because money growth is an indicator of the extent of individual uncertainty,
prices rise less than proportionately and production/effort expands when money
growth is high. Conversely, prices fall less than proportionately and
production/effort contracts when money growth is low.3
In our model economy, a Phillips curve emerges under two conditions:
(1)
an interaction of individual and aggregate uncertainty and (2) an
incompleteness of markets, which is due to private information. We conjecture
that our analysis illustrates a more general idea; that is, our results depend
more on the existence of market incompleteness than on the specific rationale.
This paper contributes to a growing area of the microfoundations
literature that uses contract theory to model the real effects of monetary
disturbances. Not all of this literature attempts to model sticky prices.
For example, Farmer (1988) and Bernanke and Gertler (1986) pinpoint credit as
the transmission mechanism. Naturally, such studies have a distinctive
emphasis and use quite different techniques.
Much of the literature, however, does try to justify the sticky prices and
wages so central to the policy-oriented models of Gray (1976). Fischer (1977),
and Taylor (1980).
In some cases, sticky prices emerge almost as an
afterthought; in Rogerson and Wright (1988), positive money shocks create an
inflation tax and reduce wealth, which in turn affect labor supply and thus
unemployment. In contrast, the bubble (or self-fulfilling-prophecy)
literature attempts to explain sticky prices, output fluctuations, and other
business-cycle phenomena as market-based occurrences that depend on
expectations, not contracts (Azariadis and Guesnerie [1986]).
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In the broadest class, and the one in which our paper best fits, contracts
insure against risk.
As in Azariadis and Cooper (1985a) and Cooper (1988) ,
contracts produce sticky prices to provide insurance against social risk,
ensuring that risk is shared optimally across different groups.
Several salient features distinguish this work from that of Azariadis and
Cooper, both in terms of modeling techniques and results. On the technical
level, our model uses distributional rather than aggregate risk: Each
individual's position is uncertain, but the total wealth of society is not.
One advantage of this approach is that it produces sticky prices by using only
monetary shocks. Slowly adjusting prices help to insure consumers against
random monetary injections by shifting some of the risk to producers. The
risk-sharing arrangement in this study also differs from that of Azariadis
and Cooper. Here, contracts spread the risk among all agents in the economy;
this broad distribution makes sense because all parties are then risk-averse.
In Azariadis and Cooper, risk is shifted to the risk-neutral producer class
(perhaps imperfectly, because of inefficiencies that result).
As might be expected, these different modeling techniques generate new and
distinctive results. In our model, prices are sticky but not fixed; that
is, they adjust - - although not proportionately - - to changes in the money
supply. One important advantage of this approach over the fixed-price
formulation is that it generates a Phillips curve (a positive relation between
inflation and output).
Azariadis and Cooper do not even permit the money
supply to change. Our formulation, on the other hand, allows policy questions
to be considered in a natural way:
for example, how does increasing monetary
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variability change the slope of the Phillips curve? When is monetary policy
neutral?
The remainder of this paper is organized as follows: The basic structure
of our model is outlined in section I. Competitive equilibrium without
contracts is discussed in section 11, and competitive equilibrium with
contracts is developed in section 111. Section IV summarizes and concludes.
I. Structure of the Economy
In this section, we outline a stochastic consumption-loans model that
draws heavily on Lucas (1972).
In each period, N identical individuals are
born, each of whom lives for two periods.
In the initial period of the life
cycle, effort is supplied in amount n and goods are consumed in amount c. In
the latter period, goods are consumed in amount c' (a prime denotes an updated
variable).
Each individual's preferences for consumption and leisure are
given by the utility function:
U(c, 1 - n)
+ V(c'
)
Following Lucas, we assume that: (1) U is increasing in consumption and
leisure, strictly concave, and twice continuously differentiable; (2) V is
increasing, strictly concave, and twice differentiable; (3) V is restricted so
that current consumption and leisure are not inferior goods; and (4) agents'
preferences are the expected value of equation (1) under situations of
uncertainty. In addition to Lucas's preference assumptions, we require that
old-age utility exhibit decreasing relative risk aversion.
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Production takes place according t o the simple scheme used by Lucas:
One u n i t of e f f o r t y i e l d s one u n i t of output within the period, but goods a r e
not s t o r a b l e .
There a r e a large number of islands (indexed by k
productive a c t i v i t y occurs.
=
A t each date (indexed by t
1, 2,
=
... K)
0, 1
...),
i n which
it i s
physically possible t o t r a n s a c t (produce o r consume) i n only one of these
marketplaces.
I n each period, J
=
N/K agents of each generation a r e presumed
t o t r a n s a c t i n each market ( i n equilibrium).
I n contrast t o Lucas, there a r e
no exogenous s h i f t s i n demand across markets (caused by a random d i s t r i b u t i o n
of t r a d e r s ) , and agents a r e f u l l y cognizant of the terms of trade i n other
markets (although t h i s information has no value i n our setup).
The importance
of market s t r u c t u r e i s explained i n more d e t a i l below.
Random money supply is the basic source of uncertainty i n our model.
Not
only i s the aggregate l e v e l of money uncertain, a s i n Lucas, but a source of
individual uncertainty i s added a s well.
O n the aggregate l e v e l , we assume
t h a t money changes through time according t o
m'
=
(2)
mx.
Here, m' i s the next period's money supply, m is t h i s period' s money supply,
and x i s the growth f a c t o r ; hence, the growth r a t e is x
i s s e r i a l l y independent with mean Z .
-
1. We assume t h a t x
Thus, over a s i n g l e period, the
money supply grows by a random f a c t o r x , which i s d i s t r i b u t e d a s proportionate
t r a n s f e r s t o the holders of money (the elder generation), who therefore spend
m'.
Those currently young w i l l take m' i n t o the next period, where they w i l l
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spend m'x'.
But in contrast to Lucas, we presume that, during the period, all
agents know the values of x and m, which are the aggregate-state
variable^.^
This individual uncertainty, introduced through monetary transfers, is the
key characteristic that distinguishes our study from that of Lucas. Each old
agent receives a transfer, T, that has a nominal value of T
=
qxm, where q
is the random shock that determines the amount of an individual's transfer.
Transfers take this complicated form in order to prevent the nonneutralities
that arise from a standard inflation tax. With a different specification,
sticky prices would still exist, but the other effects would complicate the
analysis. Within each island (and, a fortiori, in the aggregate), we require
J
that transfers in each period aggregate to zero, B Tj
j=O
=
0.
This expected
value of zero makes the transfers purely redistributive, and therefore the
uncertainty about the transfers is not aggregate but purely individual. We
further assume that q realizations are private information, so that
conventional insurance arrangements are precluded. In addition, the
distribution of the "shock," q , may depend upon money growth, x, so that the
conditional density functions of q can be written as g(q; x).
This specification captures some of the uneven distribution of monetary
injections (Friedman [1969, section 1111, Von Mises [1953, chapter VI]) and
suggests that such dispersion increases with the size of the inje~tion.~
One realistic way that this could happen is if the various financial
intermediaries react differently to monetary policies. Reserve and deposit
growth would then be differentially distributed across firms and their
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constituencies. It is likely that Gurley and Shaw (1960) originated this
argument; by the 1970s, however, even the Federal Reserve recognized
its validity (Burns [1978, p. 951).
This connection between individual uncertainty and aggregate quantity is
not standard: it destroys the simplicity of the representative-agent model.
Nevertheless, Grossman, Hart, and Maskin (1983) use this kind of relationship
to great effect. The specific interactions that we employ have often been
considered, but they have never before been formally incorporated into a
model.
Activities within each period adhere to the following sequence,
illustrated in table 1. At the beginning of a period, prior to realization of
shocks, old agents make locational decisions. In the contractual version of
our model, this is the interval in which young agents in a specific market
offer contracts in order to attract demanders. Subsequently, realization of x
and
r)
takes place, followed by production and consumption.
Table 1
Sequence of Activities within a Time Period
(1)
location decisions;
(2)
(3)
realization
production;
contracts offered
of shocks
consumption
(x, rl)
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11.
Competitive Equilibrium without Contracts
Because our analysis of the nature of competitive equilibrium without
contracts is close to that of Lucas (1972), our treatment of this subject will
be brief, developing material that will be useful in subsequent discussion.
Supply and demand for goods versus money determines the price level in our
economy. The market-clearing value of this price (in any of the K identical
islands) may be written as a function of the state of the economy (x,m):
P
=
$(x,m).
Our analysis of the nature of this equilibrium price function follows
Lucas.'
Only young agents face a nontrivial decision problem: The old
simply spend their accumulated cash balances, while the young must pick levels
of consumption (c), effort (n), and money demand/saving (A).
Recall that
money serves as the intergenerational store of value in the
overlapping-generationsmodel; thus, money held (A) is also savings. The
young choose savings and effort to maximize expected utility:
max [U(c, 1
c,n,A
s.t. p(n
AX'
+
-
-
n)
- A
c)
q'x'm'
-
+
EV(c')lx,m]
2
c'p'
0
I 0,
where E( )Jx,m denotes an expectation of a variable conditional on x and m,
A is nominal money demand, p' is the future price level, and so on. The
first constraint arises because money is the only store of value, so
money-holding reflects the difference between current production and
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consumption. The second constraint limits the next period's consumption.
When old, the agent's money balance constrains consumption. The agent has
savings (augmented by the proportional growth of money in the next period)
Ax' and the random transfer g'x'm'.
It is useful to solve this maximization problem in two stages. First,
consider picking efficient quantities of leisure and current consumption so as
to maximize utility given a specific pattern of saving behavior. The results
of this maximization process are an indirect utility function and a
conditional demand for goods and leisure (or, equivalently, a supply of
effort) .
W(F)X = max (U(c, 1 - n)) s.t. n
c ,n
c
=
4
c
X
(-)
P
and n
-
c
X 2
- P
0
- 4n (A)
P
.(6)
Previous assumptions imply that W is twice continuously differentiable and
that 4c and
4n are
continuously differentiable. The assumption that
consumption and leisure are normal goods implies that 4'= < 0 and that
q5'n
> 0.8 Second, consider selecting an efficient savings plan (X/p)
so as to maximize
W($X
g 'x'm'
+E
v
(
F+ >Ix,m,or
P'
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where R'
=
px'/pl is the real return on money. The intertemporal efficiency
condition for this plan is simply
which states the standard first-order condition for a risky asset: equality
between current utility forgone with a unit of saving (X,p) and expected
future utility received.
Individual income uncertainty (~'x'm') may raise the demand for saving as
a "hedge," under conditions on V discussed by Sandmo (1970).
This
precautionary demand for saving is ensured if old-age marginal utility is
convex (V' " > 0), which is implied by diminishing absolute risk aversion.
That is, savings will rise with greater second-period income uncertainty as
long as the premium an individual must be paid to accept a fixed actuarially
fair bet declines with the level of future consumption (c').
Thus, in
comparison to Lucas's setup - - which involves no idiosyncratic income shocks
--
there will be more desired saving (Xp) at any rate of return R'
(pxl/p').
=
In competitive equilibrium, money supply (xm) must equal money
demand (A).
Requiring equation (7) to hold with X
-
xm, it is direct that
the price level is proportional to the money stock in competitive equilibrium;
that is, p
=
$xm.
W' (6')= -EIV1(1 + rl')$-l]lx,m
(8)
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As in Lucas (1972, theorem 2), equilibrium is unique within the class
considered here, because the left side sf equation (8) is decreasing in 11, and
the right side is increasing in 11, (see footnote 6).
Competitive equilibrium without contracts involves a neutrality of money,
again following Lucas's 1972 study, because agents have accurate information
on the money stock. Prices adjust proportionately to money shocks, and a high
x is accurately reflected in prices, p
The micro-level uncertainty
= 11,~m.~
leads to greater saving than Lucas found, however, so the price level is
lower. This reflects a greater demand for money as a hedge against future
income uncertainty. Nevertheless, realizations of these micro disturbances
have no effect on the price level, although they do reallocate consumption
across members of the elder generation.
111. Competitive Equilibrium with Contracts
At the beginning of each period, prior to the realization of aggregate and
individual shocks, we now permit the representative young agent in each market
to offer a contingent contract (it is best to view each island's suppliers as
clustered together into one firm so that no idiosyncratic demand risk is
present).
Specifically, we consider contracts that permit a demander to buy
any quantity at the price
P
=
r(x)m,
where the "price contract" (that is, the function ~ [ x ] is
) chosen by
suppliers so as to maximize their lifetime expected utility subject to
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competition. Competition among islands implies that, in choosing an island,
demanders must achieve a level of expected utility at least equal to that
achievable elsewhere
(V).
realization of x and
r],
Since a demander decides on a market prior to
the relevant constraint is thus E ( v '[ r]m
~
P P
-1
) 1
-
V.
The most general possible contract would allow an arbitrary exchange of
goods for dollars and thus specify both prices and quantities; old agents
might be unable to obtain all that they demand at the given price.
In
addition, it would allow contrived uncertainty through mixed strategies and
lotteries. Thus, we compute the optimal contract over a limited - - though
broad - - class of contracts. One reason for this is that more complicated
contracting strategies are often unsustainable (Haubrich and King [1983]).
We restrict contracts mainly for tractability. A fully optimal analysis
in an already incomplete market model (OLG) would be difficult, as would the
approach of specifying the costs and information structure that would make our
contracts optimal in the broader class. Still, the nonlinearity that we allow
means that our contract should closely approximate the optimal one. In
addition, since a contract that replicates the "no contract" case of section
I1 is feasible, the optimal sticky-price contract represents an improvement.
In competitive equilibrium without contracts, the presence of a large
number of islands is inconsequential. Prices and quantities are identical in
each market. Here, suppliers in each island compete with those in other
islands in offering contracts. The presence of a large number of markets
permits us to reasonably treat
7 as not influenced by the contract
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offered by the market in question. We proceed to characterize all
Pareto-efficient contracts, without determining the split of the gains from
trade
(q) .
A young agent on an island competes with young agents on other islands by
offering contracts. If a young agent tenders a contract that provides an
expected utility of less than
7,he
attracts no money and consequently
has zero consumption in the next period.
expected utility of more than
Conversely, if the contract gives
q, everyone wants
it. We do not allow
subcontracting, so in the latter case the young agent would have to limit the
number of contracts that he accepts (because meeting them all would leave him
no leisure time), and therefore he could meet his demand for saving (money).
If the contract gives expected utility exactly equal to
7,then we
assume that the young agent obtains a proportional share (up to his demand) of
the total money supply.
ultimate equilibrium.)
(The exact rationing rule does not matter for the
In full equilibrium, each young agent maximizes
expected utility given the contract choices of the others, and the supply of
money equals the demand. We focus on the symmetric equilibrium, in which each
young agent offers the same contract.
Given the setup of the model, the indirect utility function approach of
the previous section remains helpful. But now, real saving (X/n[x]m)
depends on the contract chosen. Additionally, even though x(x) is now an
object of choice, we continue to view suppliers as treating the distribution
of future prices as invariant to their current actions, that is, taking
the form p'
=
nl(x')m', where
T'
is not an object of choice. A currently
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unborn generation chooses r'.
An efficient contract may be found by
maximizing expected young-agent utility (equation [ll.]) with respect to ~(x),
subject to the demand constraint
max E{w(-)
X
?r(x>m
It is possible to express this maximization problem as a control problem with
an integral constraint as long as x is continuously distributed (see appendix
for details).
The solution to equation (10) selects the efficient contract given the
money demand, A , and the reservation utility of the old, V.
From this
set, the individual young agent chooses optimal money demand. The total of
money demands must balance (the money market must clear).
As in the case
without contracts, the price level clears the market, adjusting to equate
demand and supply. In the contract case, this involves shifting the level of
r(x),
-
in turn changing V. A high demand for real balances by the young
means a low general price level; old agents get a lot for their money, giving
-
them high expected utility, V.
Money market clearing imposes the equilibrium condition X
=
xm.
Substituting this into the equation (10) solution, we obtain the key necessary
condition for optimal price policy, a variant of Borch's rule for
risk-sharing. That is, it must be that
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at each point on the range of x, where a is the value of the multiplier
attached to the constraint in equation (10).
This expression states equality
(in each aggregate state x) of the costs and benefits of transfers between the
contracting parties.lo
To examine how contract prices move in response to changes in x (that is,
as one moves along the range of x realizations), we implicitly differentiate
equation (11) and rearrange terms, yielding an elasticity
d log r(x)
d log x
_
dx
J
V1( (
5
=
(1 - a),
where
a
- (ar(x)
q x)dq)/(~'-o~[(l+q)~~~~]
Ix) 2 0 (also, a
< 1).
Roughly, a captures the shift in expected marginal utility induced by x
because it shifts the distribution of q. Note first that if the conditional
distribution q is independent of x, then the neutrality of money prevails in
our contract equilibrium, because a
=
0. That is, prices adjust
proportionately to changes in money and, consequently, there are no real
effects. We focus on the case where an increase in money growth (x) induces a
mean preserving spread on the distribution of individual shifts (see
Rothschild and Stiglitz [I9701 and Diamond and Stiglitz (19741). When a > 0 ,
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prices respond less than proportionately to a change in money growth because
old agents wish to purchase insurance against such aggregate states.ll
Sticky prices provide this insurance by giving the elderly more purchasing
power in states of high uncertainty. The contract shifts some of the
uncertainty's risk to the young.
Thus, the expected effect of changing x/.lt.(x) on old-agent utility
involves the interaction of the proportionate redistribution of money (1
+
q)
and its marginal utility value (V') in equation (13) above. Prices will be
sticky if a > 0. The denominator is unambiguously positive from the
definition of W and V in equations (4) and (5).
sign of a depends on (1
+
With positive prices, the
q)~'g~dq.
Sandmo's (1970) results on the theory of saving under uncertainty are
pertinent to the interpretation of this condition; that is, E[(1
+ q)V(c)]
is
exactly the expected utility reward for investing at the random gross return
(1
+
q).
He notes that, at a given level of saving, increases in the
dispersion of q may either lower or raise the reward, even when V' is convex.
This ambiguity reflects two offsetting economic elements.
Individuals will
want to save less to protect the income that they have, but they will also
want to save more as protection against a "rainy day." That is, first, an
increase in the level of interest-rate uncertainty leads an agent to feel less
inclined to expose current resources to the possibility of loss; Sandmo
identifies this effect formally with a negative substitution effect on saving.
Second, greater uncertainty about interest rates leads to an increased
potential for low consumption, which is highly valued. Thus, in a manner
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formally identical to the effect of income uncertainty noted above, there is a
positive impact on saving on this account. As a result, as x induces a mean
preserving spread on q , it will raise E( (1 + q)Vf [(I
--
at a given
X
-.(XI
+
.
rl)-L]
)
(x)
as long as preferences are such that saving will rise
with interest-rate uncertainty; that is, individuals are not too willing to
substitute for old-age consumption.
Specifically, following Diamond and Stiglitz (1974), we require that
U(C)
=
-cVtI(c)/[V'(c)]
> 1 and that
a (c) decreases with consumption, which
ensures that savings will rise with an increase in interest-rate
uncertainty.12 In that case, there will be a contract specifying sticky
prices and a positive relationship between money growth and output.
Figures la and lb show the relationship between money growth, contract
prices, and effort/production in our economy. Equation (13) also demonstrates
that the model displays a variant of Lucas's (1973) hypothesis on the
Phillips-curve slope, because greater variability in the growth component (x)
reduces the responsiveness of output to monetary shocks. Sticky prices
probably do not uniquely support the insurance allocation. Other mechanisms
can insure the elderly, such as a social security program with direct
(incentive-compatible)payments that are linked to the monetary growth rate.
Such schemes will not dominate the sticky-price contract, but merely support
the same allocation in different ways.l3 We believe that the
sticky-price contract is the best approximation of a real institution.
Furthermore, the empirical predictions arising from our model
--
connecting
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monetary uncertainty with output and prices - - should in principle allow
researchers to determine whether or not our mechanism provides the insurance.
Thus, if transfers are in fixed nominal terms, the fact that money growth
induces an increase in the dispersion of individual shocks can lead to the
Phillips-curve response illustrated in figure lb, although it requires
stronger restrictions on preferences than Lucas (1972) uses. In particular,
we require that agents are relatively unwilling to substitute away from
old-age consumption. With these preferences in place, increases in money
growth are unmatched by proportional increases in the price level
--
a result
of the contract. Thus, an increase in money will provoke a positive output
response.
Intuitively, what the contract does is protect consumers (the old) from
the uncertainty and risk associated with random money injections. Without
contracts, producers (the young) bear none of that risk because they adjust
prices proportionally. With contracts specifying sticky prices, however,
producers do bear some of the risk. When the money supply is high, for
example, they must produce more and work harder.
IV. Summary and Conclusions
This theoretical investigation was conducted under two guiding principles.
First, the analysis of sticky prices must be conducted in a
general-equilibrium setting in order to ensure consistent behavioral responses
and to lay the groundwork for an examination of policy alternatives in
accordance with the Lucas (1976) critique. Typical, sticky, nominal-price
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stories such as Fischer (1977) postulate nominal contracts, exogenously
imposing a pattern of arrangements on the labor market of an otherwise
neoclassical model. No specific gains result from nominal contracting at the
private or social level identified in Fischer's or Taylor's (1979, 1980)
models.
These papers do demonstrate the important effects of nominal contracts,
however. Without an explicit framework that generates contracts endogenously,
it is possible that such sticky-price models are internally consistent, since
factors motivating a demand for a specified wage contract may also restrict
employment or consumption decisions. Moreover, these results are devoid of
predictions about how contracts will change in the face of variations in the
economic environment. Second, in our view, the analysis of sticky nominal
prices requires explicit consideration of a monetary economy. There
must be elements of real uncertainty associated with monetary movements if
nominal price stickiness is to be explained as a result of contractual
arrangements that arise for risk-allocating reasons. Other recent work also
adheres to these principles. To the extent that menu-cost and
multiple-equilibrium models lead to different testable implications, they
indicate the necessity of ascertaining the true cause of price stickiness.
Different sources will create different macroeconomic implications,
reemphasizing the point made above about the inadequacy of models that impose
contracts exogenously.
With these guiding principles, we opted to study a stochastic
consumption-loansmodel that is a minor variation on Lucas (1972).
In this
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setup, monetary growth was assumed to be positively related to the dispersion
of individual transfer payments. Although money was neutral toward real
aggregate quantities when an exogenous restriction was placed on contracts,
neutrality did not continue to prevail when the restriction was lifted: The
sticktness partially insured consumers against random money injections by
shifting some of that risk to producers. Rather, competitive contracts
specified price stickiness
adjustment in prices
--
--
in the sense of less-than-proportionate
and, consequently, a positive relationship between
production and money growth. Thus, our model economy provides a
counter-example to Barro's (1976) conjecture that efficient competitive
contracts necessarily reduce the dependence of output on nominal money growth.
Finally, our model economy incorporates some of the features that McCallum
(1982) identifies as central elements of business fluctuations. Suppliers set
prices (contingency plans) in advance of the realization of demand. High
money growth does lead to high output
--
a result, one can argue, of prices
that do not adjust enough. At the same time, our model is not obviously
Keynesian; that is, important social costs of nominal contracting are not left
uncontemplated in private arrangements.
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APPENDIX
In this appendix, we obtain the optimal contract for our model economy by
solving an integral-constraint control problem. To do this, we rely on
methods provided by Takayama (1985, chapter 8, section C) in his discussion of
Hestenes' theorem.
Recall that the island's objective is to maximize young-agent utility
subject to the demand constraint that requires old-agent utility to at least
equal that achievable elsewhere. The problem is to choose the price function
(or, in particular, n[x]) that maximizes
E(v((~
+
q)A))
71 (x)
2
E(w(-]
X
) subject to
dx>m
v.
If we let h(x) be the density function of x and g(q; x) be the
conditional density function of
E ,
the objective and constraint each take the
form of an integral. Specifically, the constraint may be written as
Forming the Hamiltonian according to Takayama's methods, we get
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where
p
and a are multipliers.14 Since our problem is a variable
right-side endpoint problem (that is, T [ Z ] is not specified in
advance), we can set
p
- 1 in equation
(A2) without loss of generality.
Maximizing the Hamiltonian with respect to the control, ~(x),we obtain the
necessary condition
This implies the key condition (Borch's rule), which, after imposing X
=
mx,
becomes
W'[L)
(x)
=
aEI(1 + t))~'[(l
+
.)*)I
1x9
(A3)
which is equation (11) in the main text.
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24
FOOTNOTES
1.
In a modification of Lucas's (1972) setup that incorporates
entrepreneurs and relatively risk-averse workers, Azariadis (1978)
demonstrates that endogenous labor market risk-allocating
arrangements - - which require an enforceable contingent contract - - may
enhance the real effects of imperfectly perceived nominal disturbances.
Efficient ex ante arrangements in Azariadis's model do not permit real
quantities (hours worked or total compensation) to depend on
contemporaneously perceived monetary disturbances.
It is useful to establish some terminology concerning monetary
neutrality. The traditional view (Patinkin [1965, chapter IV]) is that a
money change is neutral only if all real variables for all individuals
are left unaltered in equilibrium. Our focus is on economies in which
monetary events are interconnected with uninsurable redistributive
events at the individual level, necessarily violating the Patinkin
definition of neutrality. We employ a weaker neutrality concept - invariance of aggregate real variables - - throughout our discussion.
3.
Our model thus illustrates a general principle (discussed in more detail
by Haubrich [1983]) concerning price movements in model economies that
have (1) incomplete insurance due to private information and (2)
contractual exchange contingent on aggregate variables. The principle
is that aggregate disturbances may have different qualitative effects on
near-representative-agenteconomies with and without contracts if these
aggregate shocks alter the dispersion of individual circumstances.
Grossman, Hart, and Maskin (1983) also discuss the role of aggregate
shocks as signals of the unobservable individual disturbances upon which
we focus here. However, they pinpoint economies in which asymmetric
information between firms and workers is key, but do not explore the
neutrality of money to any important degree.
4.
Smith (1985) takes a different approach, using contracts to add
uncertainty.
5.
This notation, though standard in Lucas (1972), may be a bit confusing.
Here, m is the inherited stock of money
and m' is the period
t stock of money (spent by the old in t), while x is the period t money
shock.
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6.
Von Mises (1953) assigns an important role to the distribution of
monetary injections, even suggesting that, with such dispersion, prices
adjust less than the quantity theory would predict. He states:
"This increase in the stock of money, as we have seen,
starts with the original owners of the additional
quantity of money and then transfers itself to those that
deal with these persons, and so forth... at first only
certain economic agents benefit and the additional
quantity of money only spreads gradually through the
whole community .... There is no increase in the available
stock of goods; only its distribution is altered . . . . It
is true that the prices paid for these commodities were
higher than would have corresponded to the earlier
purchasing power of money; nevertheless, they were not so
high as to make full allowance for the changed
circumstances. Europe had exported ships and rails,
metal goods and textiles, furniture and machines, for
gold which it little needed." (pp. 208-211)
7.
We follow Lucas (1972) in restricting attention to the stationary price
functions and considering only monetary equilibria. We now know that
even this is a broad class, so we consider only "fundamental"
equilibria, ruling out "sunspots" - - equilibria with stationary random
prices unrelated to the environment's intrinsic uncertainty (see
Azariadis and Cooper [1985b] and Azariadis and Guesnerie [1986]). Some
of these can depend on the economy's entire history.
8.
Following Lucas, we use the prime symbol to denote a derivative,
although it also represents updated variables.
9.
McCallum (1984) notes that this result derives from two facts: (1)
money growth is permanent and (2) the proportionate distribution of new
money effectively gives money a positive nominal return. This rules out
nonneutrality due to inflationary finance.
10.
The Borch-Arrow condition, equation (ll), also equilibrates demand and
supply, because it determines the price, ~(x), that clears this market.
In most cases, since the young meet increased demand for goods by
working more, the market clears. Rationing takes place only if a very
low T(x), and resulting high demand by the old, would require more than
24 hours of work (n > 1). An Inada condition prevents this. If the
marginal utility of the young approaches infinity when either
consumption or leisure approaches 0, the left side of equation (I), W',
also approaches infinity. The right side must then increase, implying
lower consumption by the old. Intuitively, a nonzero probability of
rationing shifts too much risk to the young, and the price of output
rises until the old demand less.
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11.
In Haubrich (1983), changes in an aggregate-state variable alter the
level of efficient risk-pooling in the banking model developed by
Haubrich and King (1983). Here, by contrast, the aggregate-state
variable alters the extent of efficient risk-shifting. In both cases,
it is central that the aggregate shock have implications for the
dispersion of individual circumstances.
12.
With convexity, this implies that both relative and absolute risk
aversion must decrease with consumption. Lucas (1972), with a somewhat
different problem, assumes o(c) > 0.
13.
Smith (1985) takes a similar position. In that paper, nominal contracts
provide lotteries, which remove a nonconvexity. Other methods could
provide the lotteries, but, as in our paper, Smith concentrates on
explaining the observed contract.
14.
If x had a discrete distribution, p and a would be a series of
Lagrange multipliers - - one pair for each point in the x distribution.
However, a continuous x distribution permits us to discuss marginal
changes more readily, although it requires the control problem.
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Figure l a
Money and Prices under Contracting
Price
level
growth
@lm)
SOURCE:
Authors ' calculations .
Money growth (x)
Figure l b
Money and Output under Contracting
Output
level
6')
SOURCE: Authors' ca1 cul ations
.
Money growth (x)
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@FedFRASER