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Federal Reserve Bank of Chicago
Expectation Traps and Monetary
Policy
Stefania Albanesi, V.V. Chari and Lawrence J.
Christiano
WP 2002-04
Expectation Traps and Monetary Policy
Stefania Albanesi∗, V.V. Chari†, Lawrence J. Christiano‡§
April 12, 2002
Abstract
Why is it that inflation is persistently high in some periods and persistently low
in other periods? We argue that lack of commitment in monetary policy may bear a
large part of the blame. We show that, in a standard equilibrium model, absence of
commitment leads to multiple equilibria, or expectation traps. In these traps, expectations of high or low inflation lead the public to take defensive actions which then make
it optimal for the monetary authority to validate those expectations. We find support
in cross-country evidence for key implications of the model.
JEL Numbers: E5, E61, E63.
∗
Bocconi University
University of Minnesota and Federal Reserve Bank of Minneapolis.
‡
Northwestern University and Federal Reserve Bank of Chicago.
§
We thank three anonymous referees and an editor for useful comments. Chari and Christiano thank
†
the National Science Foundation for support. The views expressed herein are those of the authors and not
necessarily those of the Federal reserve Banks of Chicago or Minneapolis or the Federal Reserve System.
1
Many countries have gone through prolonged periods of costly, high inflation, as well as
prolonged periods of low inflation. Why do high inflation episodes occur? What can be
done to prevent them from occurring again? These are two central questions in monetary
economics.
One tradition for understanding poor inflation outcomes stems from the time inconsistency literature pioneered by Kydland and Prescott (1978) and Barro and Gordon (1983).
This literature points to lack of commitment in monetary policy as the main culprit behind
high inflation. Static versions of the models in this literature have a unique equilibrium. Inflation rates can fluctuate only if the underlying fundamentals do. In may cases, it is difficult
to see what changes in the underlying fundamentals could have generated the episodes of high
and low inflation. In infinite horizon versions of the Kydland-Prescott and Barro-Gordon
models, trigger strategies can be used to produce the observed inflation outcomes. However,
such models have embarrassingly many equilibria. It is hard to know what observations
would be ruled out by such trigger strategy equilibria.
This paper is squarely within the tradition of the time inconsistency literature in pointing
to lack of commitment as the main culprit behind the observed volatility and persistence
of inflation. We make two contributions. First, we show how the economic forces in the
Kydland-Prescott and Barro-Gordon models can be embedded into a standard general equilibrium model. Second, we find that once these forces have been embedded into a standard
model, inflation rates can be high for prolonged periods and low for prolonged periods, even
though we explicitly rule out trigger strategies. We find some support in cross-country data
for key implications of the model.
In the Kydland-Prescott and Barro-Gordon models, the key trade-off is between the
benefits of higher output from unexpected inflation and the costs of realized inflation. In our
general equilibrium model, unexpected inflation raises output because some prices are sticky.
This rise in output has benefits for households because producers have monopoly power and
the unexpected inflation reduces the monopoly distortion. In our general equilibrium model,
1
realized inflation is costly because households must use previously accumulated cash to
purchase some goods, called cash goods. The realized inflation forces households to substitute
toward other goods, called credit goods. This substitution tends to lower welfare. Thus,
by design, the general equilibrium model captures the trade-offs between the benefits of
unexpected inflation and the costs of realized inflation in the Kydland-Prescott and BarroGordon framework.
Interestingly, this way of capturing the trade-offs leads to multiple equilibria in our general
equilibrium model. Specifically, private agents’ expectations of high or low inflation can lead
these agents to take defensive actions, which then make it optimal for monetary authorities
to validate these expectations. We focus on two kinds of defensive actions. The first is that
sticky price firms set high prices if they expect high inflation and low prices if they expect
low inflation. The second is that households change the nature of payment technologies
depending on their expectations of inflation. To explain these defensive actions we briefly
describe key features of our model.
In our model, goods are produced in monopolistically competitive markets. The monopoly
power of firms causes output to be inefficiently low. A subset of monopolists set their prices
before the monetary authority selects the money growth rate, while the rest of the monopolists set prices afterward. Because of the preset, or sticky, prices, a greater than expected
monetary expansion can raise output. Such a monetary expansion tends to raise welfare
because output is inefficiently low. If sticky price firms expect inflation to be high, they take
appropriate defensive actions and set their prices correspondingly high. If the monetary authority fails to validate the expectations of firms, output will be low. A benevolent monetary
authority may find it optimal to validate firms’ expectations. Indeed, in our general equilibrium model, we show that this kind of logic holds and plays a role in leading to multiple
equilibria.
In our model, households can also take defensive actions to protect themselves against
expected high inflation. Specifically, they can choose the fraction of goods purchased with
2
cash and the fraction purchased with credit. This choice is made before the monetary
authority selects the money growth rate. Cash purchases are costly because households
forgo interest, while credit purchases require payment of a cost in labor time, which differs
depending on the type of good. In our model, as noted above, cash goods must be purchased
with previously accumulated cash, so that a monetary expansion, by raising prices, reduces
the consumption of cash goods and reduces welfare. These aspects of our model imply
that if households expect high inflation and have chosen to purchase few goods with cash,
the marginal cost of unanticipated inflation is small. The monetary authority has a strong
incentive to inflate. If households expect low inflation, however, they choose to purchase
most goods with cash and the marginal costs of unexpected inflation are high. The monetary
authority then does not have a strong incentive to inflate. These arguments suggest that
multiple equilibria are possible in our model.
This multiplicity is the reason we can account for persistent and variable inflation. We
think this multiplicity is likely to be robust across a wide range of economic models because the underlying economics is so compelling. As noted above, existing models in the
Kydland-Prescott and Barro-Gordon literature have unique equilibria. This uniqueness reflects assumptions that best response functions are linear. We have found that the best
response functions in general equilibrium models are inherently non-linear and that multiplicity occurs naturally.
Following Chari, Christiano and Eichenbaum (1998), we call this kind of multiplicity an
expectation trap because changes in private decisions induced by changes in expectations trap
policy makers into having to accommodate the expectations. Chari, Christiano and Eichenbaum (1998) show that expectation traps can occur in conventional general equilibrium
monetary models. They rely, however, on trigger strategies on the part of the monetary
authority to support such outcomes. One criticism of trigger strategies is that for folktheorem-like reasons, virtually any inflation outcome can be rationalized as an equilibrium.
In this paper, we restrict attention to Markov equilibria that rule out trigger strategies. A
3
key finding is that expectation traps occur even in the absence of trigger strategies. We show
that, generically, the economy has at least two equilibria or none at all. In our numerical
example, we find there are two equilibria. We label these the high-inflation and low inflation
equilibrium.
The expectation traps have novel implications for the properties of financial and real
variables across the high and low inflation equilibria in a stochastic version of the model
with shocks to technology and to the payment system. The interest rate response to a shock
switches sign between the high and low inflation equilibria. For example, the interest rate
is increasing in the technology shock in the low inflation equilibrium and decreasing in this
shock in the high inflation equilibrium. Output is increasing in this shock in both equilibria.
When other shocks are present, we show that this sign switch implies that the correlation
between output and interest rates is more negative in the high inflation equilibrium than
in the low inflation equilibrium. We examine cross-country data and find that within high
inflation countries, the correlation between output and interest rates is quite negative when
these countries experience high inflation episodes and is essentially zero when these countries
experience low inflation episodes. We also find that this correlation is typically positive in
low inflation economies and typically negative in high inflation countries. Our model also
implies higher volatility of nominal variables in high inflation episodes than in low inflation
episodes. This last finding is also present in the data. While a variety of other models might
imply higher volatility, it is hard to see which models would generate the change in the
magnitude and sign of the correlation between output and interest rates.
If time inconsistency problems are behind the poor inflation outcomes of many countries,
the policy implications are that setting up institutions which promote the ability of central
banks to commit to future actions can lead to large gains. Under commitment, the optimal
policy in our model has the monetary authority following the Friedman Rule and setting
nominal interest rates equal to zero. Without commitment, the economy experiences spells
of high inflation and spells of low inflation. Institutional devices which can raise welfare in
4
practice include ways of protecting central bank independence and the design of appropriate
incentive contracts for central bankers (as in, for example, Persson and Tabellini, 1993).
The plan of the paper is as follows. Section I describes our model. Section II analyzes a
restricted version of the model, in which the payment technology is exogenously fixed. The
endogenously determined payment technology case is analyzed in section III. In Section IV
we discuss cross-country evidence for key implications of the model. In Section V, we discuss
the main forces behind the expectation traps we find. The final section concludes.
I
A Monetary General Equilibrium Economy
Our economy extends and modifies the Lucas and Stokey (1983) cash-credit goods model.
Two of our modifications are intended to capture the benefits and costs emphasized in the
literature following Kydland-Prescott and Barro-Gordon. This literature points to gains of
unanticipated monetary expansion from higher output and direct costs of realized inflation.
In our model, a subset of prices are set in advance by monopolistic firms. This feature
implies that an unanticipated monetary expansion tends to raise output and welfare. We
adopt the timing assumption in Svensson (1985) by requiring that households use currency
accumulated in the previous period to purchase cash goods. This timing assumption implies
that a realization of high inflation reduces the consumption of cash goods relative to credit
goods and thereby tends to reduce welfare. Our third modification is intended to capture the
idea that when people expect high inflation, they take defensive actions to protect themselves.
Specifically, in our model each good can be paid for either with cash or with credit. To
purchase any good with credit requires a payment of an intermediation cost, which varies
across goods. For each good, households trade off the forgone interest from using cash against
the intermediation cost.1
Our infinite-horizon economy is composed of a continuum of firms, a representative household and a monetary authority. The sequence of events within a period is as follows. First,
5
the shocks to the production technology, θ, and to the payment technology, η are realized.
We refer to s = (θ, η) as the exogenous state, and we assume that s follows a Markov process.
Then households choose the fraction z of goods to purchase with cash, and a fraction µ of
firms (the sticky price firms) set their prices. These decisions depend on the exogenous state.
Let Z(s) denote the economy wide average value of z and P e (s) denote the average price
set by sticky price firms. Here, and in what follows, we scale all nominal variables by the
beginning-of-period aggregate stock of money.
Next, the monetary authority makes its policy decision. We denote the actual money
growth rate by x and the policy rule that the monetary authority is expected to follow by
X(s). The state of the economy after the monetary authority makes its decision, the private
sector’s state, is (s, x). Households’ and firms’ production, consumption and employment
decisions depend on the private sector’s state.
Notice that we do not include the beginning-of-period aggregate stock of money in our
states. In our economy, all equilibria are neutral in the usual sense that if the initial money
stock is doubled, an equilibrium exists in which real allocations and the interest rate are
unaffected and all nominal variables are doubled. This consideration leads us to focus on
equilibria which are invariant with respect to the initial money stock. We are certainly
mindful of the possibility of equilibria which depend on the money stock. For example, if
multiple equilibria in our sense exist, ‘trigger strategy-type’ equilibria which are functions of
the initial money stock can be constructed. In our analysis we exclude such equilibria and
we normalize the aggregate stock of money at the beginning of each period to unity.
As is customary in defining a Markov equilibrium, we begin with the decisions at the
end of the period and work our way back to the beginning of the period. Accordingly, we
first describe the end-of-period problem of households and flexible price firms given (s, x)
and future monetary policy X(s). We then describe the problem of sticky price firms and
the household’s choice of z. These problems and market clearing allow us to define a private
sector equilibrium for arbitrary x. We then describe the monetary authority’s problem and
6
define a Markov equilibrium.
A
Private Sector at the End of the Period
Here we discuss the decision problems of households and firms at the end of the period.
We begin with the household problem. In each period the household consumes a continuum of differentiated goods as in Blanchard and Kiyotaki (1987) and supplies labor. The
P
t
representative household’s preferences are ∞
t=0 β u(ct , nt ), where 0 < β < 1,
ct =
·Z
0
1
ρ
ct (ω) dω
¸ ρ1
¤1−σ
£
c(1 − l)ψ
, u(c, l) =
,
1−σ
ct (ω) denotes consumption of type ω good, lt denotes labor time, and 0 < ρ < 1.
Each good in this continuum is one of four types. A fraction µ are produced by sticky price
firms and a fraction 1 − µ are produced by flexible price firms. The sticky and flexible price
firms are randomly distributed over the goods. In addition, each good can be purchased with
cash or with credit. Let z denote the fraction of goods the household chooses to purchase with
cash. This cash-credit decision is made before households know which goods are produced
by sticky or flexible price firms, so that the cash-credit good choice is independent of the
type of firm. Thus, a fraction µz of goods are sticky price goods purchased with cash, a
fraction (1 − µ)z are flexible price goods purchased with cash, a fraction µ(1 − z) are sticky
price goods purchased with credit, and a fraction (1 − µ)(1 − z) are flexible price goods
purchased with credit. It turns out that prices for goods within each type are the same.
Utility maximization implies that the amounts purchased of each type of good are the same.
Let c11 and c12 denote quantities of cash goods purchased from sticky and flexible price firms,
respectively, and let c21 and c22 denote the quantities of credit goods purchased from sticky
and flexible price goods, respectively. Then we have that
(1)
1
c = [zµcρ11 + z(1 − µ)cρ12 + (1 − z)µcρ21 + (1 − z)(1 − µ)cρ22 ] ρ .
The household divides its labor time, l, into time supplied to goods-producing firms, n,
7
and time allocated to the payment technology according to
(2)
l =n+
η(z̄ − z)1+ν
.
1+ν
We discuss the determination of z below.
Let A denote the nominal assets of the household, carried over from the previous period.
In the asset market, the household trades money, M, and one-period bonds, B, with other
households. The asset market constraint is
M + B ≤ A.
(3)
Recall that nominal assets, money and bonds are all scaled by the aggregate stock of money.
We impose a no-Ponzi constraint of the form B ≤ B̄, where B̄ is a large, finite upper bound.
The household’s cash-in-advance constraint is
(4)
h
i
M − P e (s)µzc11 + P̂ (s, x)(1 − µ)zc12 ≥ 0,
where P e (s) denotes the price set by sticky price firms and P̂ (s, x) denotes the price set by
flexible price firms. Nominal assets evolve over time as follows:
h
i
(5)0 ≤ W (s, x)n + (1 − R(s, x))M − z P e (s)µc11 + P̂ (s, x)(1 − µ)c12
h
i
−(1 − z) P e (s)µc21 + P̂ (s, x)(1 − µ)c22 + R(s, x)A + (x − 1) + D(s, x) − xA0 .
In (5), W (s, x) denotes the nominal wage rate, R(s, x) denotes the gross nominal rate of
return on bonds, and D(s, x) denotes profits after lump sum taxes. Finally, B has been
substituted out in the asset equation using (3). Notice that A0 is multiplied by x. This
multiplication reflects that we have scaled all nominal variables by the beginning of period
aggregate stock of money and A0 is the household’s nominal assets scaled by next period’s
aggregate money stock. Next period’s aggregate money stock is simply the current stock
multiplied by the growth rate x.
Consider the household’s asset, goods and labor market decisions for a given value of
z. Given that the household expects the monetary authority to choose policy according to
8
X(s) in the future, the household solves the following problem:
(6)
v(A, z, s, x) =
max
n,M,A0 ,cij ; i,j=1,2
u(c, l) + βEs0 [max
v(A0 , z 0 , s0 , X(s0 ))|s]
0
z
subject to (1), (2), (3), (4), (5), and nonnegativity on allocations. The solution to (6) yields
decision rules, d(A, z, s, x), where
d(A, z, s, x) = [n(A, z, s, x), M(A, z, s, x), A0 (A, z, s, x), cij (A, z, s, x)],
(7)
i, j = 1, 2.
We turn now to the decision problems of firms at the end of the period. Each of the
differentiated goods is produced by a monopolist using the following production technology
y(ω) = θn(ω),
where y(ω) denotes output and n(ω) denotes employment for the type ω good. Also, θ is
a technology shock that is the same for all goods. The household’s problem yields demand
curves for each good. The fraction, 1 − µ, of firms that are flexible price firms set their price,
P̂ (s, x), to maximize profits subject to these demand curves. Because the household demand
curves have constant elasticity, firms set prices as a fixed markup, 1/ρ, above marginal cost,
W/θ, so that
(8)
P̂ (s, x) =
W (s, x)
.
θρ
Turning to the government, we assume that there is no government debt, government
consumption is financed with lump-sum taxes, and government consumption is the same for
all goods. As a result, the resource constraint for this economy is
θn = g + z [µc11 + (1 − µ)c12 ] + (1 − z) [µc21 + (1 − µ)c22 ] ,
where g denotes an exogenous fixed level of government consumption. Since there is no
government debt, bond market clearing requires B = 0, A = 1. Also, money market clearing
requires M = 1.
9
B
Private Sector at the Beginning of the Period
At the beginning of the period, after the exogenous shocks are realized, sticky price firms
set prices and households make their payment technology decision, z.
As in Blanchard and Kiyotaki (1987), sticky price firms in our economy must set their
price in advance and must produce the amount of goods demanded at that price. These firms,
like the flexible price firms, also wish to set their price as a markup, 1/ρ, over marginal cost,
W/θ. In order to do so, they need to forecast the wage rate, W. They do so by taking the
wage rate as given by the private sector equilibrium. Thus, the wage they expect to prevail
is W (s, X(s)). Thus, in equilibrium the price set by sticky price firms is given by
P e (s) =
(9)
W (s, X(s))
θρ
We now discuss the household’s payment technology decision. As noted above, each
consumption good can be purchased either with cash or with credit. For goods with ω > z̄
(where z̄ is a parameter between zero and one) the cost of purchasing with credit is zero.
Purchasing goods with ω ≤ z̄ on credit requires labor time. The household chooses a
fraction z ≤ z̄ such that goods with ω < z are purchased with cash and goods with ω > z
are purchased with credit. The labor time required to purchase fraction z of goods with
cash is given by η(z̄ − z)1+ν / (1 + ν), where ν > 0 is a parameter and η > 0 is the shock to
the payment technology. The household’s labor time, including time spent working in the
market, n, is given in (2). The household chooses z to solve the following problem:
(10)
z(A, s) = arg max v(A, z, s, X(s)).
We now define an equilibrium for each possible private sector state (s, x) and future
monetary policy rule, X(s).
Definition For each s and each x, given X(s) a private sector equilibrium is a collection of
functions P e (s), Z(s), P̂ (s, x), W (s, x), R(s, x), v(A, z, s, x), d(A, z, s, x), z(A, s) such that
the following are true:
10
1. The functions v and d solve (6)
2. The function z(A, s) solves (10) and z(1, s) = Z(s)
3. Firms maximize profits; that is, P̂ (s, x) satisfies (8) and P e (s) satisfies (9)
4. The resource constraint is satisfied at d(1, Z(s), s, x)
5. The asset markets clear; i.e., A0 (1, s, x) = M(1, s, x) = 1.
We find it convenient to define another private sector equilibrium concept. A private
sector equilibrium with a fixed payment technology is a private sector equilibrium with the
restriction that z is fixed and is not a choice variable.
C
Monetary Authority
The monetary authority chooses x to maximize the representative household’s discounted
utility:
(11)
max v(1, Z(s), s, x),
x
where v is the value function in a private sector equilibrium. Recall that a private sector
equilibrium takes as given the evolution of future monetary policy. Thus, in solving (11) the
monetary authority implicitly takes as given the evolution of future monetary policy.
D
Markov Equilibrium
We now have the ingredients needed to define a Markov equilibrium.
Definition A Markov equilibrium is a private sector equilibrium and a monetary policy rule,
X(s), such that X(s) solves (11).
Two properties of a Markov equilibrium deserve emphasis. First, the current money
growth rate does not affect discounted utility of the household starting from the next period
11
since it does not affect the next period’s state. Therefore, the monetary authority faces
the static problem of maximizing current period utility, and we only have to describe how
current money growth affects current allocations. Second, inspection of (8) and (9) shows
that P̂ (s, X(s)) = P e (s) in a Markov equilibrium. We use these properties below.
In our analysis of a Markov equilibrium, we find it convenient to define another Markov
equilibrium concept. The Markov equilibrium with a fixed payment technology is a Markov
equilibrium in which z is exogenously fixed and beyond the control of the households.
II
Analysis with Fixed Payment Technology
In this section we discuss a version of our model in which the payment technology is fixed,
in the sense that households cannot alter the value of z. We do this for two reasons. First,
this version of the model is a building block for the analysis of the model with a variable
payment technology. Second, the model with a fixed payment technology is of interest in its
own right because it is the simplest adaptation of a standard monetary model designed to
capture the frictions emphasized in Kydland-Prescott and Barro-Gordon.
In our analysis, we decompose the first-order condition associated with the monetary
authority problem, (11), into benefits and costs of inflation. Unexpected inflation has benefits
because some prices are sticky and there is a monopoly distortion. With sticky prices, higher
inflation tends to raise output, while the monopoly distortion makes higher output desirable.
These are the reasons the monetary authority in our model has a temptation to stimulate
the economy. Inflation is costly because it leads to a reduction in the relative consumption
of cash goods.
To analyze a Markov equilibrium, we first characterize a private sector equilibrium. We
then solve the monetary authority’s problem. We then show that, generically, there are at
least two Markov equilibria for the economy with a fixed payment technology.
12
A
Characterizing Private Sector Equilibrium
We now develop a set of necessary and sufficient conditions for a private sector equilibrium.
We find it convenient to adopt a change of variables. Let the relative prices of flexible and
sticky price goods q = P̂ /P e . Omitting arguments of functions for convenience, the first
order necessary conditions for household and firm optimization are:
(12)
(13)
(14)
(15)
(16)
(17)
u11
u12
u21
u22
u11
u21
u12
u22
=
=
=
=
−un =
xu21
e
P µ(1 −
z)
µ 1
,
1−µq
µ 1
,
1−µq
z
R,
1−z
z
R,
1−z
θρu22
,
(1 − µ)(1 − z)
= βEs0 [v1 (1, z, s0 , X(s0 ))|s],
where z is fixed. Here, uij denotes the partial derivative of u with respect to cij , and v1
denotes the partial derivative of v with respect to its first argument. Equations (12) and
(13) equate the marginal rate of substitution between sticky and flexible price goods to the
relative price of these goods q, and equations (14) and (15) equate the marginal rate of
substitution between cash and credit goods to the interest rate R which is their relative
price R. Equation (16) is obtained by noting that the marginal rate of substitution between
labor and consumption of flexible price credit goods is equated to the ratio of the nominal
wage to the price of flexible price goods. This ratio is simply the markup in (8).
The cash-in-advance constraint can be written as
(18)
µzc11 + q(1 − µ)zc12 ≤
1
.
Pe
A necessary condition for the household problem to be well defined is
(19)
R ≥ 1.
13
It is easy to show that the cash in advance constraint holds with equality if R > 1 and
that if the cash-in-advance constraint is slack, R = 1. These observations imply that the
appropriate complementary slackness condition is
¾
½
1
− [µzc11 + q(1 − µ)zc12 ] [R − 1] = 0.
(20)
Pe
The resource constraint is
(21)
g + z [µc11 + (1 − µ)c12 ] + (1 − z) [µc21 + (1 − µ)c22 ] = θn.
Combining (8) and (9), we have that
(22)
q(s, X(s)) ≡
P̂ (s, X(s))
= 1.
P e (s)
In equation (22) we reintroduce the dependence of variables on s and x to emphasize that P e
coincides with P̂ only when x = X(s). The conditions (12)-(22) are necessary and sufficient
for a private sector equilibrium. That is, these conditions can be used to construct the
allocation and pricing functions stated in the definition of the private sector equilibrium
above, namely, P e (s), P̂ (s, x), R(s, x), d(1, z, s, x) with W (s, x) = θρP̂ (s, x). The value
function is also straightforward to construct.
B
The Monetary Authority’s Problem
The monetary authority’s problem is static in our economy for two reasons. First, we focus
on Markov equilibria. In such equilibria, policy makers face dynamic problems only if their
decisions affect future state variables. Second, there are no state variables in our economy.
Thus, the monetary authority’s problem is simply one of choosing current money growth to
maximize current period utility.
We find it convenient to set up the monetary authority’s problem as one of choosing the
interest rate R rather than the money growth rate x. This change in instruments makes the
analysis of the variable payment technology economy much easier. As long as the cash-inadvance constraint holds with equality, the two instruments are equivalent. The equivalence
14
argument is as follows. With x as the instrument, (12)-(21) define allocation and pricing functions (cij (s, x, P e ), n(s, x, P e ), R(s, x, P e ), q(s, x, P e )). These functions evaluated at
P e (s) are the allocation and pricing functions stated in the definition of a private sector
equilibrium. Under our functional form assumptions, it is tedious but straightforward to
verify that a unique set of allocations and prices solves (12)- (21) for each x and each P e and
that a unique x is associated with each allocation. With the interest rate as the monetary
authority’s instrument, we use (12)-(16) and (18)- (21) to define allocations and prices as
functions of the interest rate,
(23)
cij (s, P e , R), i, j = 1, 2, q(s, P e , R), n(s, P e , R)
and let x be simply defined by (17). If the cash-in-advance constraint holds with equality,
under our functional form assumptions, a unique set of allocations and prices solves these
equations for each R and a unique R exists for each allocation and relative price q. Thus
the two formulations are equivalent if the cash-in-advance constraint holds with equality. If
the cash-in-advance constraint holds with inequality it is easy to see that there are many
allocations which solve (12)-(16) and (18)-(21) for given R = 1. Each of these allocations is
associated with a different value of x. In the Appendix, we prove the following lemma, which
allows us to set up the monetary authority’s problem as one of choosing the interest rate R
rather than the money growth rate x.
Lemma 1: In a Markov equilibrium, the cash-in-advance constraint (18) holds with
equality.
We now set up the monetary authority’s (static) problem. Substituting from (23) into the
utility function, we let U(s, P e , R) = u [c(s, P e , R), n(s, P e , R)] denote the utility associated
with an interest rate R, where c is defined in (1). The monetary authority’s problem is now
(24)
max U (s, P e , R),
R
subject to R ≥ 1.2 Let R(s, P e ) denote the solution to this problem.
15
C
Markov Equilibria
Here we derive a relationship between the payment parameter z and the allocations and
prices in a Markov equilibrium with a fixed payment technology. We also show that, for
given z generically at least two allocations satisfy the necessary conditions for a Markov
equilibrium. In a large class of parameterizations for our economy, we verified numerically
that the necessary conditions are sufficient for a Markov equilibrium.
The first-order condition associated with a solution to (24) is
(25)
UR (s, P e , R) = uc cR + un nR ≤ 0,
with equality if R > 1. In (25) UR is the derivative of U with respect to R and uc , un
are derivatives of the utility function with respect to c and n, respectively, and cR , nR are
the derivatives of c and n with respect to R evaluated at the allocations which satisfy the
conditions of a private sector equilibrium. In addition to conditions (12)-(21), a private sector
equilibrium must satisfy the analog of (22), namely, q(s, P e (s), R(s, P e (s))) = 1. Therefore,
in (25) the derivatives are evaluated at a value of P e such that q(s, P e (s), R(s, P e (s))) = 1.
From here on we suppress the arguments of functions, and evaluate all functions at their
equilibrium values.
In what follows, we show that (25) can be decomposed into a part that captures the
incentives to increase inflation because of the presence of monopoly power and a part that
captures the disincentives arising from the resulting reduction in cash goods consumption.
Consider the role of monopoly power. The efficient allocations with respect to the laborleisure choice in our economy satisfy
(26)
un +
θu22
= 0.
(1 − µ)(1 − z)
The first term in (26) is the marginal disutility of labor associated with increasing labor input
to credit goods production, say, and the second term is the marginal benefit from increased
credit goods consumption. In our economy the analog of (26) is (16). Note that because
16
of the presence of monopoly power, the second term in (16) is the same as the second term
in (26) multiplied by ρ < 1. As a result, the net marginal benefit of increasing labor from
its equilibrium value in our economy is positive. This distortion is due to monopoly power
and suggests that the left side of (26) is a natural measure of the monopoly distortion in our
economy. Add and subtract θu22 nR / [(1 − µ)(1 − z)] to and from (25) to obtain
·
¸
θu22 nR
θu22
UR = uc cR −
+ un +
nR ≤ 0.
(27)
(1 − µ)(1 − z)
(1 − µ)(1 − z)
The term in square brackets is our measure of the monopoly distortion. Substituting from
(16) into (27), we obtain
UR = uc cR −
(28)
(1 − ρ)θu22 nR
θu22 nR
+
≤ 0.
(1 − µ)(1 − z) (1 − µ)(1 − z)
In the Appendix, we prove the following lemma regarding the last term in (28).
Lemma 2: In a Markov equilibrium with a fixed payment technology,
(1 − ρ)θu22 nR
= f (c1, c2 )ψ MD (R, z) ,
(1 − µ)(1 − z)
(29)
where
(30)
f (c1 , c2 ) > 0 for c1 , c2 > 0,
and
(31)
1
ψ MD (R, z) = −(1 − ρ)R ρ−1
³ ρ
ρ
1
µ ψ
R ρ−1 + ψR ρ−1 + 1−µ
R ρ−1 +
ρ
³
´
+
ρ
1+ψ
ψ
z
ρ−1 + 1
+
R
1−ρ
1−z
ρ
1−z
z
>From (31) it is clear that ψ MD (R, z) satisfies the following properties:
¡ ¢
µ ψ 1−z
(32)
ψ MD (R, z) is decreasing in z and lim ψ MD (R, z) =
R→∞
1−µ ρ
1+ψ
1−ρ
z
+ ψρ
´
.
> 0.
Notice that ψ MD (R, z) does not depend on the shocks θ and η.
Now consider the disincentives to increase inflation. In the Appendix, we prove the
following lemma.
17
Lemma 3: The first two terms to the right of the equality in (28) can be written as
(33)
uc cR −
1
θu22 nR
= −f (c1 , c2 ) (R − 1) R ρ−1 .
(1 − µ)(1 − z)
Let
(34)
1
ψ ID (R) = (R − 1) R ρ−1 .
Using c2 /c1 = R1/1−ρ , we have that ψ ID (R) = (R − 1)c1 /c2 . The net interest rate R − 1
measures the extent to which cash goods consumption is distorted relative to the efficient
level. This distortion is akin to a tax (as Lucas and Stokey (1983) have argued). The base
on which this tax is levied is consumption of cash goods. Thus, one way to think of ψID is
as the product of a tax rate, R − 1, and the base of taxation, c1 , scaled by a measure of the
size of the economy, c2 . In this sense, ψ ID measures the inflation distortion. In the efficient
allocations, R = 1, and the term on the right side of (33) is zero. Inspecting (34), we have
that ψ ID ≥ 0 and
(35)
lim ψ ID (R) = ψ ID (1) = 0.
R→∞
That is, there is no inflation distortion when the interest rate is high or low.
Substituting (29), (33) and (34) into (28), we obtain
(36)
UR = f (c1 , c2 ) [−ψ ID (R) + ψ MD (R, z)] ≤ 0
with equality if R > 1. Let ψ(R, z) = −ψ ID (R) + ψ MD (R, z) . Then a solution to
(37)
ψ(R, z) ≤ 0
with equality if R > 1 satisfies the necessary condition for monetary authority optimality. If
(36) is also sufficient, then the interest rate, R, which solves (37) corresponds to a Markov
equilibrium with fixed payment technology. Given an equilibrium value of the interest rate,
we can solve for the allocations and other prices from (12)-(16), (18) with equality, (21) and
18
(22), for each value of θ, η and z. We can then obtain the monetary authority’s policy rule
from (17).
We use the properties of the monopoly distortion function, ψ MD , in (32), and the inflation
distortion function, ψ ID , in (35), to show that, generically, there are at least two Markov
equilibria, if there are any.
Proposition 1 (Generic Multiplicity): Consider the version of our economy with a fixed
payment technology. Suppose that the monetary authority’s first order condition is sufficient
for optimality. Then, except for a set of z of Lebesgue measure zero, there are at least two
Markov equilibria, or none. Furthermore, the equilibrium interest rate does not depend on
θ or η.
Proof: A key property of the function ψ(R, z) is that it is positive for R sufficiently large.
This property follows from (32) and (35) which imply
lim ψ(R, z) = lim [−ψ ID (R) + ψ MD (R, z)] > 0.
R→∞
R→∞
Suppose first that ψ(1, z) > 0. Then, since ψ(R, z) is positive at R = 1 and positive for
large R, by continuity it follows that if ψ(R, z) is ever zero, it must generically be zero at
least twice. A non generic case occurs when the graph of ψ(R, z) against R is tangent to
the horizontal axis at a single value of R. Another nongeneric case is when ψ(1, z) = 0 and
ψ(R, z) > 0 for R > 1. Both cases are nongeneric because for an arbitrarily larger value
of z, one can see that there are multiple equilibria since ψ(R, z) is strictly decreasing in z.
Suppose next that ψ(1, z) < 0. Then, R = 1 satisfies (37) and corresponds to a Markov
equilibrium. In addition, because ψ(R, z) > 0 for R sufficiently large, continuity implies that
ψ(R, z) must be equal to zero for at least one value of R > 1.
>From (34) we have that ψ ID does not depend on θ or η. Since ψ MD does not depend
on these variables either, it follows that the equilibrium interest rate, R, does not depend on
θ or η. Q.E.D.
An example helps illustrate the results in Proposition 1. Figure 1 displays the monopoly
19
distortion, ψ MD , and the inflation distortion, ψ ID , for R ∈ [1, 4.5] and for z = 0.13 and 0.15.3
The figure shows that the first order necessary condition for monetary authority optimality
is satisfied at R = 1.38 and R = 2.07 for z = 0.13 and R = 1.10 and R = 3.17 for z = 0.15.
Thus, for z = 0.15 the inflation rate is somewhat below 10 percent in the low inflation
equilibrium and just below 217 percent in the high inflation equilibrium. To verify that the
first order condition for monetary authority optimality is also sufficient, in Figure 2a we
graph the monetary authority’s objective for z = .15, (24), as a function of R for the value
of P e corresponding to the low inflation candidate equilibrium, and in Figure 2b we graph
the corresponding objective for the high inflation candidate equilibrium. (The values of P e
are 26.3 and 165.0 for the low and high inflation equilibria, respectively.) These figures show
that the first-order conditions are indeed sufficient. They also show that the utility function
is not necessarily concave. This is why it is necessary to check monetary authority’s utility
level globally, rather than just locally.
In the numerical example, the inflation distortion has a single-peaked Laffer curve shape,
while the monopoly distortion is relatively flat. We found these properties to hold across
a range of parameterizations of the economy. The shape of the inflation distortion is reminiscent of the shape of the monetary Laffer curve in analyses where governments rely on
inflation to finance expenditures. (See, for example, Sargent and Wallace (1981).) Below we
explore the relationship between our analysis and the analysis in the monetary Laffer curve
literature.
The set of interest rates, R, and payment technology, z, which solves (37) plays a key
role in our analysis of the equilibrium with variable payment technology. We call the graph
of R against z which solves (37) the interest rate policy correspondence (henceforth, policy
correspondence for short.) The following proposition establishes properties of this correspondence:
Proposition 2 (Interest Rate Policy Correspondence): Suppose that the monetary
authority’s first-order condition is sufficient for optimality. Suppose also that for some z < z̄
20
a Markov equilibrium exists. Then, there is a critical value of z, say ẑ, such that for z < ẑ
there are no Markov equilibria, for z = ẑ there is at least one Markov equilibrium, and for
z > ẑ there at least two Markov equilibria.
Proof: First, we show that there is no interest rate less than R̄ which is an equilibrium,
where R̄ is arbitrarily large. Notice from (31) that ψ MD (R, z) → ∞ as z → 0 for all
R ∈ [1, R̄], and from (34) that ψ ID is bounded. It follows that there is some value of z, say
ẑ1 , such that for all z ≤ ẑ1 , ψ(R, z) is strictly positive. Thus, there is no equilibrium interest
rate less than R̄ for z sufficiently small. Second, we show that no interest rate greater than
R̄ can be an equilibrium. We see from (34) that ψ ID is bounded above by, say, k. Let ẑ2 be
defined by limR→∞ ψ MD (R, ẑ2 ) = 2k. Such a value of ẑ2 exists from (32). Note also that for
all z ≤ ẑ2 , limR→∞ ψ MD (R, z) ≥ 2k. By definition of a limit, some interest rate R̄ exists such
that for all R ≥ R̄, ψ MD (R, ẑ2 ) ≥ 2k − ε, where ε is, say, k/2. It follows that, for all R ≥ R̄,
ψ (R, ẑ1 ) = −ψ ID (R) + ψ MD (R, ẑ1 ) ≥ k/2 > 0. That is, there is no value of the interest rate
greater than R̄ which is an equilibrium for z = ẑ2 . Since ψ MD (R, z) is decreasing in z, there
is no value of the interest rate greater than R̄ which is an equilibrium for z ≤ ẑ2 . We have
established that there is no equilibrium if z is sufficiently small.
Next, ψ MD (R, z) is a continuous function of R and z. As z is increased from some
arbitrarily low value, there is some first value of z such that ψ(R, z) = 0 for some R.
Such a z, call it ẑ, exists by our assumption that an equilibrium exists for some z. Consider
increasing z above ẑ. Since ψ MD is strictly decreasing, the graph of ψ(R, z) against R must
intersect the horizontal axis at at least two points. Thus, for z > ẑ, there are at least two
Markov equilibria. Q.E.D.
Consistent with our theoretical findings, Figure 1 shows that the inflation distortion
does not depend on the payment technology parameter, z, while the monopoly distortion
is decreasing in this parameter. We graph the policy correspondence in Figure 3. When z
is sufficiently small, the monopoly distortion lies above the inflation distortion and there is
no equilibrium. As z increases, the monopoly distortion declines. At a critical value of z
21
the economy has a unique equilibrium and for values of z larger than this critical value the
economy has two equilibria. Notice that as z increases, the interest rate in the low inflation
equilibrium falls and that the interest rate in the high inflation equilibrium rises.
III
Analysis with Variable Payment Technology
We now characterize a Markov equilibrium in the full-blown version of our economy in
which the payment technology is variable. This equilibrium must satisfy all the conditions
of a Markov equilibrium with a fixed payment technology. It must in addition satisfy the
condition that the payment technology parameter z is chosen optimally. We have already
shown that a Markov equilibrium with fixed payment technology is characterized by the
relationship between R and z defined given by (37). Here, we show that the first order
condition for the optimal choice of z yields a second relation between R and z. The necessary
conditions for an equilibrium are completely characterized by values of R and z which satisfy
both relationships.4 In effect, we collapse the set of equilibrium necessary conditions to just
two. This simplification makes key properties of the equilibrium transparent. A simple
argument establishes that, generically, there are multiple Markov equilibria. In addition, we
are able to use the two equations to analyze the effects of exogenous shocks on equilibrium
allocations and prices.
The first-order condition associated with the household’s choice of z is
ρ
1
1 − R 1−ρ
ψη(z̄ − z)ν
=
(1 − )
ρ
1+ν .
ρ z + (1 − z)R 1−ρ
1 − n − (z̄−z)
η/(1+ν)
(38)
We can use the equations that define a private sector equilibrium, (12)-(16), (18) with
equality, (21) and (22) to substitute for labor, n, in (38). Doing so, we obtain (see Lemma
4 in the Appendix for a derivation):
ρ
(39)
h
1
z (R ρ−1
( ρ1 − 1)(1 − R ρ−1 )
ρη(z̄ − z)ν
i
=
.
ρ
(z̄−z)1+ν η
g
ψ
ψ
ρ−1
1
−
−
− 1) + ρ (R
− 1) + (1 + ρ )
1+ν
θ
22
For each z, there is at most one R that solves (39). To see this, note that the left-hand side
is increasing in R, while the right side does not depend on R. Let Rp (z, g, θ, η) denote the
value of R that solves (39). We refer to this function as the payment technology function,
or payment function, for short. The set of payment technology parameters z for which this
function is defined is developed as follows. As R → ∞, the left side of (39) converges to
(1 − ρ)/((ρ + ψ)(1 + z)), which at z = 0 becomes (1 − ρ)/(ρ + ψ). The right side of (39) at
z = 0 is ρηz̄ ν /(1 − z̄ 1+ν η/(1 + ν) − g/θ). If
(1 − ρ)/(ρ + ψ) < ρηz̄ ν /(1 − z̄ 1+ν η/(1 + ν) − g/θ),
there is some critical value of z, say z ∗ , at which the function Rp (z, g, θ, η) goes to infinity.
Then the function is defined for (z ∗ , z̄]. If not, then the function is defined for (0, z̄]. Let
the domain of the function be (z̃, z̄] where z̃ = z ∗ if the above inequality holds and z̃ = 0
otherwise.
It is easy to see from (39) that Rp is decreasing in z, since the left side of (39) is increasing
in z, while the right side is decreasing in z. It is also easy to see that Rp is increasing in g/θ
and η since an increase in g/θ or η raises the right side of (39) and so increases R for a given
value of z.
Each R, z which satisfies the policy correspondence, (36), and the payment function,
(39), corresponds to a Markov equilibrium. The other allocations, prices and the monetary
authority’s policy rule can be obtained by solving (12)-(17), (18) with equality, (21) and (22).
Next, we prove a proposition that under certain conditions, there are two Markov equilibria
for our economy. We say that the policy correspondence is horseshoe-shaped if it satisfies
the following conditions: (i) there are two continuous functions, Rc1 (z) and Rc2 (z) which map
[ẑ, z̄] into the space of interest rates with Rc1 (z) < Rc2 (z), for z ∈ (ẑ, z̄], Rc1 (ẑ) = Rc2 (ẑ),
and (ii) for all z ∈ [ẑ, z̄] the solution to (37) is either Rc1 (z) or Rc2 (z), where ẑ is defined in
Proposition 2.
Proposition 3: Suppose the policy correspondence is horseshoe-shaped. Then, generi23
cally, the economy with variable payment technology has two Markov equilibria, if any.
Proof: Suppose to begin with that z̃ < ẑ. Recalling that Rp (z̄) = 1 and Rc1 (z̄), Rc1 (z̄) ≥ 1,
we can divide the proof into two cases. The first case is when Rp (z̄) < Rc1 (z̄). The second case
is when Rp (z̄) = Rc1 (z̄) = 1. Consider the first case, that is, Rp (z̄) < Rc1 (z̄) ≤ Rc2 (z̄). Now if
Rp (ẑ) > Rc1 (ẑ) = Rc2 (ẑ), then since Rp is below Rc1 and Rc2 at z̄ and above Rc1 and Rc2 at ẑ, by
continuity, Rp must intersect at least once with each Rc1 and Rc2 . Each of these intersections
corresponds to a Markov equilibrium. If Rp (ẑ) < Rc1 (ẑ) = Rc2 (ẑ) then since Rp is below Rc1
at both z̄ and ẑ, Rp and Rc1 intersect twice, if at all. The case when Rp (ẑ) > Rc1 (ẑ) = Rc2 (ẑ)
is clearly non-generic.
Consider the second case, that is, Rp (z̄) = Rc1 (z̄) = 1. Then the Ramsey policy and
allocations constitute an equilibrium. Generically, there must also be one other equilibrium.
To see this, note that, generically, if Rc1 (z̄) = 1, some neighborhood of z̄ exists such that for
all z in this neighborhood, Rc1 (z) = 1. Since Rp is strictly decreasing, it follows that for z in
this neighborhood, Rp (z) > 1 = Rc1 (z). Suppose that Rp (ẑ) < Rc1 (ẑ). Then, since Rp is above
Rc1 in a neighborhood of z̄ and below Rc1 at ẑ, by continuity Rp and Rc1 must intersect at
least once. Now suppose that Rp (ẑ) > Rc1 (ẑ) = Rc2 (ẑ). Then, since Rp is below Rc2 at z̄ and
above Rc2 at ẑ, by continuity Rp must intersect at least once with Rc2 . We have established
that in this second case, generically, there must be at least two equilibria.
Suppose next that z̃ > ẑ. Then for z near z̃, Rp is arbitrarily large and must be larger
than Rc2 . Exactly the same arguments used above can then be used to conclude that there
must be two Markov equilibria. Q.E.D.
The restriction that the policy correspondence be horseshoe-shaped is not severe. In
Proposition 2 we have shown that for each z > ẑ there are at least two interest rates
which belong to the policy correspondence. Using the implicit function theorem, these
interest rates can be represented as continuous functions of z. Thus, the assumption that
the correspondence is horseshoe-shaped only rules out the possibility that there are three or
more interest rates which belong to the correspondence. It is straightforward, but tedious to
24
extend the proof of Proposition 3 to this case. Furthermore, in all the numerical examples
we have considered, the correspondence is horseshoe-shaped.
In Figure 4, we plot the interest rate correspondence and the payment function for various
realizations of the exogenous shocks in our numerical example. In Figure 4a we plot the
interest rate correspondence and the payment function for two realizations of the production
technology shock, θ, holding the other shock at its mean value. Figure 4b displays the
analogous graph for the payment technology shock, η. These figures display four properties.
First, as we have shown in Proposition 1, the policy correspondence does not depend on these
shocks. Second, as discussed above, the payment function is decreasing in the interest rate.
Third, as also discussed above, the payment function is increasing in η and decreasing in θ.
Fourth, there are multiple Markov equilibria. Two of these are easy to see. In one, for every
realization of the shocks, the equilibrium is the one associated with the lower intersection
of the interest rate correspondence and payment function. We call this the low inflation
equilibrium. In the other, the equilibrium is the one associated with the higher intersection.
We call this the high inflation equilibrium.
Figure 4 displays an interesting sign switch phenomenon, in the sense that the interest
rate response to a shock switches sign between the high and low inflation equilibrium. For
example, from Figure 4a, we see that the interest rate is increasing in the technology shock
in the low inflation equilibrium and decreasing in this shock in the high inflation equilibrium.
We verified, for our numerical example, that in both equilibria output is increasing in the
technology shock. If technology shocks were the dominant shocks, the correlation between
output and the interest rate would be positive in the low inflation equilibrium and negative in
the high inflation equilibrium. From Figure 4b we see the sign switch for the payment shock:
the interest rate is decreasing in this shock in the low inflation equilibrium and increasing in
this shock in the high inflation equilibrium. In our numerical example, output is increasing
in the payment shock in the low inflation equilibrium and decreasing in this shock in the
high inflation equilibrium. So, if payment shocks were the dominant shocks the correlation
25
would be negative in both equilibria. It follows that in an economy with both shocks, the
correlation of output and the interest rate is negative in the high inflation equilibrium and
larger (perhaps even positive) in the low inflation equilibrium. We call this finding the
decreasing correlation implication.
Our numerical examples also show that the volatility of interest rates in the low inflation
equilibrium is substantially smaller. The reason is that the policy correspondence is flatter
at the low inflation equilibrium than at the high inflation equilibrium. We call this finding
the increasing volatility implication.
Thus far we have focused on Markov equilibria which are stationary in the sense that
they cannot depend on time. We should point out that if we add calendar time as a state
variable there are other Markov equilibria as well. For example, one such equilibrium has
the economy moving to the low inflation equilibrium on even dates and to the high inflation
equilibrium on odd dates. More interesting is the possibility of sunspot driven Markov
equilibria in which a sunspot at the beginning of each period coordinates private agents’
expectations and induces agents to pick the high or the low inflation equilibrium depending
on the realization of the sunspot. Such sunspot equilibria clearly exist and lead to volatility
in inflation rates.
IV
Interest Rates and Output in Cross-Country Data
The model’s decreasing correlation and increasing volatility implications receive support
from within-country data and cross-country data. We analyzed data from the International
Financial Statistics (2000) on output and interest rates for a number of countries. We
obtained annual data from high and low inflation countries. We defined a high inflation
country as one for which output and interest rate data are available and in which interest
rates exceed 100 percent in at least one year. Our low inflation countries are the developed
countries of Western Europe, the United States, Canada, Japan, Australia and New Zealand.
26
The list of high inflation countries is in Table 1, and the list of all countries appears in Table
2. In all cases, the tables show the relevant sample periods. The correlations reported in the
tables are based on logged, Hodrick-Prescott filtered output and Hodrick-Prescott filtered
interest rates.5
We begin with the within-country data analysis. Typically, the high inflation countries in
our sample experience episodes of high inflation and episodes of relatively low inflation. One
interpretation is that the high inflation episodes correspond to our high inflation equilibrium
and the low inflation episodes to our low inflation equilibrium. Under this interpretation,
the model suggests that the correlation between output and interest rates should be negative
in the high inflation episodes and larger in the low inflation episodes. We define episodes
of high inflation to be periods when the nominal interest rate exceeds 50 percent per year,
while we define low inflation episodes to be all other periods. Fortunately, these episodes
turned out - with minor exceptions - to be contiguous. As can be seen from Table 1, there
are seven high inflation countries. Five of these countries have had episodes of both high
and low inflation. With one exception, the correlation between output and interest rates is
higher in the low inflation episodes than in the high inflation episodes. Table 1 also reports
the mean value of the correlation between output and the interest rate for all countries
in low inflation episodes and in high inflation episodes. The correlation is −0.08 in low
inflation episodes and −0.45 in high inflation episodes. Table 1 also provides evidence for
the increasing volatility implication. In the low inflation episodes, the standard deviation of
the interest rate is 3.57, and in the high inflation episodes, this standard deviation is 350190.
For comparison purposes note that the percentage standard deviation of output is 2.34 in
the low inflation episodes and 4.57 in the high inflation episodes.
Table 2 provides cross-country evidence for the decreasing correlation and increasing
volatility implications. Table 2a shows that the mean value of the correlation between
output and interest rates is −.33 for the high inflation counties and Table 2b shows that
this average is .20 for the low inflation countries. This table also provides evidence for the
27
increasing volatility implication. The standard deviation of the interest rate is 283324 for
the high inflation countries and 1.84 for the low inflation countries. For comparison purposes
note that the percentage standard deviation of output is 4.43 in the high inflation countries
and 2.26 in the low inflation countries.
We also simulated our model and computed the correlation between output and interest
rates and the standard deviations of both output and the interest rate. The parameter
values are the same as those used in Figure 4. The autocorrelations of both shocks are 0.9,
the shocks are uncorrelated, and the standard deviations of θ and η are 0.04 and 9735.1,
respectively. We took 500 observations from our model and filtered the simulated data from
the model in the same way that the cross-country data were filtered. We found σ R = 1.90 and
σ R = 0.12 in the low and high inflation equilibria, respectively. and the standard deviation of
output is essentially the same in both equilibria. The model obviously fails to match the level
of volatility in these variables in the data. However, it is interesting that the model predicts
the interest rate is an order of magnitude more volatile in the high inflation equilibrium,
while output volatility is essentially the same. We also computed the correlation between
logged and filtered output and the filtered interest rate. That correlation is 0.013 in the low
inflation equilibrium and −0.019 in the high inflation equilibrium. These statistics from the
model are qualitatively similar to the corresponding statistics in the data.
V
Key Features for Generating Expectation Traps
In this section, we ask which features are crucial for generating expectation traps. We
focus on four features and find that two of them play essential roles and two play more
subsidiary roles. We also ask whether introducing learning, staggered price setting, or capital
accumulation is likely to alter the results significantly.
The two essential features are the ex post benefits of higher than expected inflation and
the costs of realized inflation. The benefits of higher than expected inflation come from our
28
assumption that some prices are preset and the presence of monopoly power. The importance
of the assumption that some prices are preset can be seen by considering the case when none
are, that is, when µ = 0. Setting µ = 0 in (31), after some manipulation, we see that
ψ MD (R, z) < 0 for all R. Thus, if µ = 0, the unique equilibrium with both a fixed and a
variable payment technology has R = 1. To see the importance of monopoly power, note
that if ρ = 1, the markup of prices over marginal cost is 0. Setting ρ = 1 in (31), we see
that ψ MD (R, z) = 0 for all R. Thus, if ρ = 1, the unique equilibrium with both a fixed and
a variable payment technology has R = 1.
The cost of realized inflation in our model comes from the timing assumption under
which households must use previously accumulated currency to purchase cash goods. To
see the importance of this timing assumption, suppose instead we had adopted the timing
assumption in Lucas and Stokey (1983). Under Lucas-Stokey timing, open market operations
are conducted in the securities market at the beginning of the period. Households can use the
current monetary injection for current cash goods purchases. Mechanically, this amounts to
adding current money growth to the right side of the cash-in-advance constraint. A greater
than expected monetary expansion, therefore does not in and of itself change the mix of cash
and credit goods expansion. We should emphasize that anticipations of high inflation in the
future do change the relative amounts of cash and credit goods consumption. Basically, under
Lucas-Stokey timing, with flexible prices a one-time, unanticipated change in the quantity
of money is neutral, prices change by the quantity of the money change and all real variables
are unaffected. With sticky prices, there is a one-time increase in output and all future
real variables are unaffected. With Lucas-Stokey timing, there is no Markov equilibrium in
our model. The argument is by contradiction. Suppose there were such an equilibrium. A
monetary expansion raises output and therefore tends to raise welfare. The only cost is the
distortion in the relative prices of sticky and flexible prices. But, in any equilibrium this
relative price is one and thus changes in this relative price have a second order effect on
welfare. The monetary authority always has an incentive to raise the inflation rate. Thus,
29
there is no equilibrium.
The two subsidiary features relate to the shape of the inflation distortion function and
the monopoly distortion function. Substituting c2 /c1 = R1−ρ into (34), we see that
ψ ID = (R − 1)
c1
.
c2
We have already argued that this distortion is akin to the product of a tax, R − 1, and
the tax base, c1 . When R = 1, ψ ID = 0. As R → ∞, the behavior of ψ ID depends on the
rate at which cash goods consumption falls. In the economy in this paper, c1 goes to zero
faster than R goes to infinity, and thus the product goes to zero. In Albanesi, Chari and
Christiano (2002), we present a model in which ψ ID does not go to zero because c1 goes to
zero at the same rate as R. The fixed payment technology model in that paper has a unique
equilibrium. With a variable payment technology, however, multiple equilibria are possible.
In our model the monopoly distortion is positive for R sufficiently large. This result
implies that there are two equilibria with a fixed payment technology so that the policy
correspondence is horseshoe-shaped. In Albanesi, Chari and Christiano (2002), we show
that if the period utility function is of the following form
u(c, n) =
c1−σ
− an,
1−σ
where a is a parameter, the fixed payment technology economy has a unique equilibrium.
The policy correspondence in Figure 3 becomes a downward-sloping graph. Nevertheless,
since the payment function is also downward sloping, there can be multiple intersections and
multiple equilibria.
We also ask whether these equilibria are stable under a simple learning scheme. One
reason to do so is that if one of these equilibria is unstable, it might regarded implausible.
We will show that under one widely used learning scheme, the low inflation equilibrium is
always locally stable and the high inflation equilibrium is also stable if the payment function
is sufficiently steep. When households determine the period t payment technology, zt , they
have to form expectations of monetary policy. We assume that they expect the monetary
30
authority to set the interest rate equal to its last period value, Rt−1 . So the value of zt is
given by the payment function with Rt−1 substituted for the interest rate. We assume that
the monetary authority solves the same problem as before so that, for example, it neglects
the impact of its current policy action on agents’ expectations next period. Thus, the
monetary policymaker’s correspondence is unaffected. Suppose that in the neighborhood of
an equilibrium the payment function and the policy correspondence are given approximately
by
zt = a + bRt−1
Rt = d + ezt
where a, b, d, e are parameters. Substituting for zt we obtain
Rt = d + ea + ebRt−1 .
Thus, the local stability of the dynamical system governing the interest rate depends on
whether the absolute value of be is greater than or less than one. The slope of the payment
function is 1/b, and the slope of the policy correspondence is e. At the low inflation equilibrium, the payment function is steeper than the policy correspondence, or −1/b > −e. It
follows that the value of be is positive and less than 1, and the system is dynamically stable.
At the high inflation equilibrium, if −1/b > −e, the same argument applies and the system
is locally stable. Inspection of Figures 4a and 4b reveals that the high and low inflation
equilibria in our numerical example are locally stable under learning in the sense discussed
here. Thus, stability under learning does not provide a device for selecting equilibria in this
example.
In this paper prices are preset for one period. In principle, it is straightforward to allow
for Taylor- or Calvo- style staggered price setting so that cohorts of firms set their prices
for many periods at a time. We could also allow for capital accumulation. A particularly
31
interesting source of dynamics is to consider models in which it takes time or resources to
change z. We conjecture that in any of these extensions, outcomes similar to those described
in this paper would emerge as steady states.
VI
Conclusion
We have shown that discretionary monetary policy can account for prolonged periods of low
and high inflation. The model in this paper is a very standard monetary general equilibrium
model. Our main theoretical finding is that the model always has expectation traps. The data
provide some support for the decreasing correlation and the increasing volatility implications.
The main force driving the multiplicity of equilibria is that defensive actions taken by
the public to protect itself from high inflation reduce the costs of inflation for a benevolent
monetary authority and induce the authority to supply high inflation. This economic force
is likely to be present in a large class of monetary models. The main policy implication is
that the costs of discretionary monetary policy include not just high average inflation, but
volatile and persistent inflation as well. The gains to setting up institutions which increase
commitment to future monetary policies are likely to be high.
32
Notes
1
See Aiyagari, Braun and Eckstein (1998), Cole and Stockman (1992), Dotsey and Ireland
(1994), Freeman and Kydland (1994), Ireland (1994), Lacker and Schreft (1996), and Schreft
(1992) for payment technology models with similar features.
2
Technically, the set of interest rates should also be limited to those where (12)-(16) and
(18)-(21) have a solution. Our analysis of the monetary authority’s problem uses a first order
condition approach which only asks whether small deviations are optimal. One can use the
implicit function theorem to show that in some neighborhood of an equilibrium, (12)-(16)
and (18)-(21) have a solution. Thus, we will not have to deal with whether the allocation
functions are well defined for arbitrary interest rates.
3
In the numerical example used throughout this paper, µ = 0.1, ρ = 0.45, ψ = 1, ν = 2,
z̄ = 0.15, η = 28000.
4
In all the numerical examples we have studied, the necessary conditions also turned out
to be sufficient.
5
The smoothness parameter in the Hodrick-Prescott filter was set to 100. Each country
and sample period was treated as a distinct series for constructing the Hodrick-Prescott
filter.
33
Appendix
To prove the lemmas in the text, we use the necessary and sufficient conditions for an
interior private sector equilibrium. Using our functional form assumptions, (12)-(16) reduce
to
−1
(40)
c12 = c11 q 1−ρ
(41)
c21 = c11 R 1−ρ
(42)
c22 = c21 q 1−ρ
(43)
1
−1
η(z̄ − z)1+ν
ψ ρ 1−ρ
c c22 = θ(1 − n −
).
ρ
1+ν
We have omitted (13) because there are only three linearly independent equations in
(12)-(15). These expressions together with (20)-(22) are necessary and sufficient conditions
for a private sector equilibrium.
Lemma 1: In a Markov equilibrium, the cash-in-advance constraint (18) holds with
equality.
Proof: Suppose that the cash-in-advance constraint holds as a strict inequality in a
Markov equilibrium. We will show that there is some small deviation in the money growth
rate x from its supposed equilibrium value which raises the utility of the representative
household. Note from (20) that R = 1 for all x in some neighborhood of x. In that small
neighborhood of the equilibrium value of x, expressions (40)-(43) and (21) with R = 1 implicitly define functions cij (s, x, P e ), c(s, x, P e ), q(s, x, P e ), and n(s, x, P e ). Since (40)-(43) and
(17)-(22) are necessary and sufficient conditions for a private sector equilibrium, these functions evaluated at the equilibrium value of P e = P e (s) are the allocation rules and prices in
a Markov equilibrium. The monetary authority’s problem is maxx u(c(s, x, P e ), n(s, x, P e ))
and the first-order condition for this problem is
(44)
uc cx + un nx = 0
34
where cx and nx denote derivatives with respect to x and where all functions are evaluated
at their supposed equilibrium values. Since R = 1 and q = 1 in equilibrium, we have that
cij = c for all i, j. Using our functional form assumptions and (43) in (44), we have
uc (cx − θρnx ) = 0.
(45)
Differentiating (1) with respect to x and evaluating at cij = c, we obtain
cx = zµc11,x + z(1 − µ)c12,x + (1 − z)µc21,x + (1 − z)(1 − µ)c22,x
where cij,x is the derivative of cij with respect to x. Differentiating the resource constraint
with respect to x, we obtain
θnx = zµc11,x + z(1 − µ)c12,x + (1 − z)µc21,x + (1 − z)(1 − µ)c22,x .
Substituting for cx and θnx into (45), we have a contradiction. Q.E.D.
Lemmas 2 and 3 are established using (40)-(43), (20) with equality, and (21) to construct
the functions cij (s, P e , R), q(s, P e , R) and n(s, P e , R), differentiating these functions with
respect to R and evaluating the derivatives at q = 1. Mechanically, we first drop n from the
system by substituting out for n in (43) using (21). Then, we differentiate (40)-(42) and
simplify to obtain one equation in c11,R and qR . We use (18) to obtain another equation in
these variables. We can then evaluate all the other derivatives.
Substitute for n from (21) and for c from (1) into (43), to obtain
ψ
[zµcρ11 + z(1 − µ)cρ12 + (1 − z)µcρ21 + (1 − z)(1 − µ)cρ22 ] c1−ρ
22
ρ
η(z̄ − z)1+ν
= θ − g − z [µc11 + (1 − µ)c12 ] + (1 − z) [µc21 + (1 − µ)c22 ] − θ
.
1+ν
Differentiating with respect to R we get
(46) z [µc11,R + (1 − µ)c12,R ] + (1 − z) [µc21,R + (1 − µ)c22,R ]
£
¤ 1−ρ
ρ−1
ρ−1
ρ−1
+ψ zµcρ−1
1 c11,R + z(1 − µ)c1 c12,R + (1 − z)µc2 c21,R + (1 − z)(1 − µ)c2 c22,R c2
ψ
+ (1 − ρ)cρ c−ρ
2 c22,R = 0,
ρ
35
where all derivatives are evaluated at a value of P e such that q = 1. Here, c1 = c11 = c12 and
c2 = c21 = c22 when q = 1. Now, differentiate (40)-(42) with respect to R to obtain
c12,R = c11,R −
(47)
(48)
c21,R
(49)
c22,R
c1
qR
1−ρ
ρ
c1 R 1−ρ
= c11,R R
+
1−ρ
c2
= c21,R −
qR .
1−ρ
1
1−ρ
Differentiating (18) with equality and substituting for c12,R from (47), we obtain
¶
µ
c1
qR + (1 − µ)zc1 qR = 0.
µzc11,R + (1 − µ)z c11,R −
1−ρ
Simplifying, we obtain
(50)
qR =
1−ρ
c11,R .
ρ(1 − µ)c1
>From (47)-(49) and (50), using (c2 /c1 )1−ρ = R, we obtain
(51)
(52)
µc11,R + (1 − µ)c12,R = c11,R −
(1 − µ)c1
qR = c11,R (1 − 1/ρ),
1−ρ
µc21,R + (1 − µ)c22,R = c21,R −
(1 − µ)c2
qR
1−ρ
ρ
1
(53)
R 1−ρ
c1 R 1−ρ
)+
= c11,R (1 −
ρ(1 − µ)
1−ρ
and
ρ
(54)
c22,R = c11,R (1 − 1/ρ)R
Substitute from (50)-(54) into (46) to obtain
"
1
1−ρ
c1 R 1−ρ
.
+
1−ρ
ρ
zc11,R (1 − 1/ρ) + (1 − z) c11,R (1 − 1/ρ)R
1
1−ρ
"
c1 R 1−ρ
+
1−ρ
1−ρ
+ψzcρ−1
1 c2 c11,R (1 − 1/ρ) + ψ(1 − z) c11,R (1 − 1/ρ)R
ρ
1
R 1−ρ
c1 R 1−ρ
ψ
)+
) = 0.
+ (1 − ρ)cρ c−ρ
2 (c11,R (1 −
ρ
ρ(1 − µ)
1−ρ
36
#
ρ
1
1−ρ
c1 R 1−ρ
+
1−ρ
#
Grouping terms, we obtain
·
µ
¶¸
µ ¶ρ
1
1
1
1
c11,R
c
z + (1 − z)R 1−ρ + ψzR + ψ(1 − z)R 1−ρ + ψ
R 1−ρ 1 −
c1
c2
ρ(1 − µ)
·
µ ¶ρ ¸
ρ
1−z ψ c
ρ
(1 + ψ)
+
R 1−ρ .
= −
ρ−1
1 − ρ ρ c2
Finally, we obtain the following expression:
(55)
ρ
1−ρ
h
(1 + ψ) 1−z
+
1−ρ
ψ
ρ
³ ´ρ i
ρ
c
R 1−ρ
c2
c11,R
´
³ ´ρ 1 ³
´
=³
1
1
ψ
1
c
c1
1−ρ
1−ρ
1−ρ
z + (1 − z)R
+ ρ (1 − ρ) c2 R
+ ψzR + ψ(1 − z)R
−1
ρ(1−µ)
We use these derivatives to obtain cR and nR . Differentiating (1) with respect to R, we
obtain
(56)
£
¤
ρ−1
ρ−1
ρ−1
cR = c1−ρ zµcρ−1
1 c11,R + z(1 − µ)c1 c12,R + (1 − z)µc2 c21,R + (1 − z)(1 − µ)c2 c22,R
Substituting from (51) and (52), we obtain
ρ
1
cR
c1 R 1−ρ
ρ−1
ρ−1
1−ρ +
=
c
zc
(1
1/ρ)
+
(1
z)c
(c
(1
1/ρ)R
).
−
−
−
11,R
11,R
1
2
c1−ρ
1−ρ
Collecting terms:
cR =
c1−ρ cρ−1
2 c1
·
¸
¶
´µ
ρ
1
c11,R ³
1
z
1
−
+
R 1−ρ .
zR + (1 − z)R 1−ρ
1−
c1
ρ
1−ρ
Differentiating the resource constraint we obtain nR :
θnR = z [µc11,R + (1 − µ)c12,R ] + (1 − z) [µc21,R + (1 − µ)c22,R ] .
or, after substituting from (51) and (52) and collecting terms:
(57)
´
ρ
1
1 ³
c1
R 1−ρ .
θnR = c11,R (1 − ) z + (1 − z)R 1−ρ + (1 − z)
ρ
1−ρ
We now prove Lemma 2.
37
Lemma 2: In a Markov equilibrium with a fixed payment technology,
(1 − ρ)θu22 nR
= f (c1, c2 )ψ MD (R, z) ,
(1 − µ)(1 − z)
(58)
where f (c1 , c2 ) > 0 for c1 , c2 > 0,and ψ MD (R, z) is given in (31).
Proof: From (57), using (c2 /c1 )1−ρ = R, we obtain
#
"
¶
µ
(1 − ρ1 )
ρ
1
(1 − z) /z c1
1 − z 1−ρ
+
R 1−ρ
θnR =
c11,R z (1 − ρ) 1 + (
)R
1−ρ
z
(1 − ρ1 ) c11,R
#
"
¶
µ
1
1
c
c2 (1 − ρ )
(1
z)
/z
1
z
−
−
1
=
c11,R z (1 − ρ) R ρ−1 + (
) +
R−1 .
1
c1 1 − ρ
z
c
(1 − ρ ) 11,R
Substituting in (29) and using the result that for our functional forms u22 /(1−µ)(1−z) =
³ ´1−ρ
uc cc2
, we obtain
·
¸
1
(1 − ρ)θu22 nR
ρ
c1 −1
1−z
ρ−1
R }
= f (c1 , c2 ) −(1 − ρ)R
){(1 − ρ) −
−(
(1 − µ)(1 − z)
z
(1 − ρ) c11,R
= f (c1 , c2 )ψ MD (R, z) ,
where
f (c1 , c2 ) = uc c2
µ
c
c2
¶1−ρ
c11,R
1
( − 1)
z
ρ
c1
and
(59)
1
ψ MD (R, z) = −(1 − ρ)R ρ−1 + (
ρ
c1 −1
1−z
){
R − (1 − ρ)}.
z
(1 − ρ) c11,R
Consider the term in parenthesis in (59). When we use (55), this term is
³ ´ρ 1 ³
´
1
1
1
z + (1 − z)R 1−ρ + ψzR + ψ(1 − z)R 1−ρ + ψ cc2 R 1−ρ ρ(1−µ)
−1
h
³ ´ρ i ρ
− (1 − ρ)
1−z
(1 + ψ) 1−ρ
R 1−ρ
+ ψρ cc2
³ ´ρ 1 ³
³ ´ρ 1
´
ψ
1
c
1−ρ
z + ψzR + ψ c2 R
− 1 − (1 − ρ) ρ cc2 R 1−ρ
ρ(1−µ)
h
³ ´ρ i 1
=
ψ
c
(1 + ψ) 1−z
R 1−ρ
+
1−ρ
ρ
c2
³ ´ρ 1
µ ψ
z (1 + ψR) + 1−µ ρ cc2 R 1−ρ
³ ´ρ i 1 .
= h
ψ
1−z
(1 + ψ) 1−ρ + ρ cc2
R 1−ρ
38
Substituting for c/c2 in this expression and then substituting in (59), we obtain:
1
ψ MD (R, z) = −(1 − ρ)R ρ−1
i
´
h
³ 1
ρ
ρ
µ ψ
ρ−1 + 1 − z
(1 − z) R ρ−1 + ψR ρ−1 + ( 1−z
)
zR
1−µ ρ
z
³
´
.
+
ρ
ψ
ρ−1 + 1 − z
zR
(1 + ψ) 1−z
+
1−ρ
ρ
Dividing the numerator and denominator of the term in braces by 1 − z and rearranging, we
obtain:
1
ψ MD (R, z) = −(1 − ρ)R ρ−1
We have proved the lemma. Q.E.D.
´
³ ρ
³ 1
ρ
µ ψ
ρ−1
ρ−1
+ ψR
+ 1−µ ρ R ρ−1 +
R
³
´
+
ρ
1+ψ
ψ
z
ρ−1
+ ρ 1−z R
+1
1−ρ
1−z
z
´
.
Lemma 3: The first two terms to the right of the equality in (28) can be written as
(60)
uc cR −
1
θu22 nR
= −f (c1 , c2 ) (R − 1) R ρ−1 .
(1 − µ)(1 − z)
Proof: Using our functional forms, we obtain
"
µ ¶1−ρ #
θu22 nR
c
uc cR −
= uc cR − θ
nR .
(61)
(1 − µ)(1 − z)
c2
Substituting for θnR from (57) and cR from (56) into (61), we obtain
"
¶
µ ¶1−ρ #
´µ
ρ
1
1 − z 1−ρ
1
c11,R ³
c
1−ρ
+
R
zR + (1 − z)R
1−
uc cR − θ
nR = uc [
c2
c1
ρ
1−ρ
µ ¶1−ρ
´ 1−z ρ
1
1 ³
c11,R
c
R 1−ρ ]c1
(1 − ) z + (1 − z)R 1−ρ −
−
c1
ρ
1−ρ
c2
µ ¶1−ρ
1
c11,R
c1
c
= uc
c2 z(1 − )
(R − 1)
c1
ρ c2
c2
1
= −f (c1 , c2 )(R − 1)R ρ−1
where
µ ¶1−ρ
c11,R
c
1
f (c1 , c2 ) = uc
c2 z( − 1)
.
c1
ρ
c2
We have proved the lemma. Q.E.D.
39
Lemma 4: Equation (38) reduces, in a private sector equilibrium, to (39):
³
´
´³
ρ
1
ρ−1
1
1
R
−
−
ρ
ρη(z̄ − z)ν
h³ 1
³ ρ
´
´
´i
=³
1+ν η
z R ρ−1 − 1 + ψρ R ρ−1 − 1 + (1 + ψρ )
1 − (z̄−z)
− gθ
1+ν
Proof: Using (43) in (38), we obtain:
ρ
µ
¶
1
θρη(z̄ − z)ν
1 − R ρ−1
1−
=
.
ρ
ρ z + (1 − z)R ρ−1
(c/c2 )ρ c2
(62)
We use the resource constraint, (21), and (43) to obtain an expression for c2 in terms of c1 /c2
and z. Rearranging (43) we obtain:
Ã
(z̄ − z)1+ν η
θn = θ 1 −
1+ν
!
ψ
−
ρ
µ
c
c2
¶ρ
c2 .
Substituting this into the resource constraint, taking into account cρ = zcρ1 + (1 − z)cρ2 , and
rearranging, we obtain:
c2 =
z cc12
³
´
1+ν η
θ 1 − (z̄−z)
−g
1+ν
³ ´ρ
.
ψ
c1
+ ψz
+
(1
z)(1
+
)
−
ρ
c2
ρ
Substituting for c2 in (62), we obtain:
ρ
ρ−1
ν
1
1−R
θρη(z̄ − z)
(1 − )
= ³ ´ρ
×
ρ
ρ z + (1 − z)R ρ−1
z cc12 + 1 − z
z cc12 +
ψz
ρ
³ ´ρ
+ (1 − z)(1 + ψρ )
´
³
.
1+ν η
g
θ 1 − (z̄−z)
−
1+ν
c1
c2
After rearranging and making use of R = (c1 /c2 )ρ−1 , we obtain (39). Q.E.D.
40
References
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[2] Albanesi, Stefania, V.V. Chari, Lawrence J. Christiano, 2002, ‘Expectation Traps and
Monetary Policy’, Working Paper Version.
[3] Barro, Robert J., and David B. Gordon, 1983, ‘A Positive Theory of Monetary Policy
in a Natural Rate Model,’ Journal of Political Economy 91 (August): 589-610.
[4] Blanchard, O.J. and N. Kiyotaki, 1987, ‘Monopolistic Competition and the Effects of
Aggregate Demand,’ American Economic Review, Vol. 77, No. 4, 647-666.
[5] Chari, V. V., Lawrence J. Christiano and Martin Eichenbaum, 1998, ‘Expectation Traps
and Discretion,’ Journal of Economic Theory , vol. 81 No. 2, pp. 462-492.
[6] Cole, Harold, and Alan Stockman, 1992, ‘Specialization, Transactions Technologies and
Money Growth,’ International Economic Review, vol. 33, no. 2, pp. 283-298.
[7] Freeman, Scott and Finn Kydland, 1998, ‘Monetary Aggregates and Output,’ American
Economic Review vol. 90, no. 5, December, pp.1125-35.
[8] Ireland, Peter, 1994, ‘Money and Growth: An Alternative Approach,’ American Economic Review,’ Vol. 84, No. 1, 47-65.
[9] International Financial Statistics, 2000. International Monetary Fund, Washington D.C.
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Inconsistency of Optimal Plans,’ Journal of Political Economy, vol. 85, no. 3, June,
pages 473-91.
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Journal of Monetary Economics, vol. 38, no. 1, pp. 3-23.
41
[12] Lucas, Robert E., Jr., and Nancy L. Stokey, 1983, ‘Optimal Fiscal and Monetary Policy
in an Economy Without Capital,’ Journal of Monetary Economics, vol. 12, pp. 55-93.
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Journal of Political Economy, vol. 93, no. 5, October, pp. 919-44.
42
Table 1: Evidence from High Inflation Economies
Country
Low Inflation
ρ(y, R)
mean R
σR
Argentina
−0.49
8.15
3.28
Brazil
−0.31
21.39
Brazil
−0.80
Chile
σR
Low Inflation
High Inflation
Period
Period
ρ(y, R)
mean R
σy
2.34
−0.57
9.24∗105
3.95
24.45∗105
1992 - 2000
1980 - 1991
3.28
4.06
0.03
2362.59
3.08
3713.38
1963 - 1980
1981 - 1995
25.16
1.11
3.46
NA
NA
NA
NA
1996 - 2000
−0.11
25.85
3.03
6.37
−0.65
73.89
7.83
26.04
1984 - 2000
1977 - 1983
Israel
0.28
21.93
1.70
3.47
−0.68
245.70
1.30
241.92
1972 - 2000
1979 - 1987
Peru
0.46
28.78
1.65
1.72
−0.58
846.35
7.77
1483.92
1995 - 2000
1986 - 1994
Turkey
NA
NA
NA
NA
−0.41
68.15
3.13
18.98
NA
1987 - 2000
Uruguay
NA
NA
NA
NA
−0.30
88.10
4.92
16.31
NA
1976 - 2000
−0.08
21.88
2.34
3.57
−0.45
132516
4.57
350190
NA
Column mean
σy
High Inflation
References
[1] Notes: In this table y denotes the logged, Hodrick-Prescott filtered level of output, R denotes
the Hodrick-Prescott filtered interest rate, ρ(y, R) is the correlation between y and R, σ y is the
standard deviation of y mutiplied by 100 and σ R is the standard deviation of the interest rate.
All data are from the International Financial Statistics.
NA
NA
Table 2a: Full Sample Evidence from High Inflation Economies
Country
ρ(y, R)
mean R
σy
Argentina
−0.59
5.28∗105
4.78
1.98∗106
1980 - 2000
Brazil
0.03
946.05
3.51
2304.49
1963 - 2000
Chile
−0.36
39.86
5.53
15.17
1977 - 2000
Israel
−0.24
113.47
1.66
154.95
1979 - 2000
Peru
−0.46
519.32
7.48
1128.99
1979 - 1993
Turkey
−0.41
68.15
3.13
18.98
1987 - 2000
Uruguay
−0.30
88.10
4.92
16.31
1976 - 2000
Mean, High Inflation
−0.33 (-0.29)
75677
4.43
283324
0
σR
Period
NA
Table 2b: Full Sample Evidence from Low Inflation Economies
Country
ρ(y, R)
mean R
σy
σR
Period
Australia
0.54
8.89
1.74
1.92
1970 - 2000
Austria
0.48
6.09
1.78
1.55
1967 - 1998
Belgium
0.32
5.22
1.79
1.48
1953 - 1998
Canada
0.40
8.36
2.48
2.45
1975 - 2000
−0.31
9.81
1.82
2.06
1972 - 2000
Finland
0.18
9.68
4.03
1.94
1978 - 2000
France
0.09
6.96
1.48
1.60
1950 - 1998
Germany
0.54
5.38
2.58
1.91
1960 - 2000
−0.04
5.79
4.07
1.75
1960 - 1998
Ireland
0.15
10.65
2.51
2.27
1971 - 1999
Italy
0.09
11.28
1.53
2.08
1969 - 2000
Japan
0.24
6.21
3.06
1.69
1957 - 2000
New Zealand
0.48
11.11
2.61
2.10
1985 - 2000
−0.25
9.44
1.87
1.48
1972 - 2000
Spain
0.33
11.60
2.18
2.40
1974 - 2000
Sweden
0.01
8.75
2.16
2.20
1966 - 2000
Switzerland
0.43
3.40
2.64
1.59
1969 - 2000
United Kingdom
0.03
7.78
2.28
2.20
1969 - 2000
United States
0.20
6.15
2.15
1.78
1955 - 2000
0.20 ( 0.20)
8.03
2.26
1.84
NA
Denmark
Netherlands
Norway
Mean, Low Inflation
Notes: In this table y denotes the logged, Hodrick-Prescott filtered level of output, R denotes the
Hodrick-Prescott filtered interest rate, ρ(y, R) is the correlation between y and R, σ y is the standard deviation
1
of y mutiplied by 100 and σ R is the standard deviation of the interest rate. All data are from the International
Financial Statistics.
2
0.35
Figure 1: Marginal Benefits and Marginal Costs for Monetary Authority
0.3
Monopoly Distortion, low z
Inflation Distortion
Benefits and Costs
0.25
0.2
0.15
0.1
Monopoly Distortion, high z
0.05
0
1
1.5
2
2.5
R
1
3
3.5
4
4.5
Figure 2a: Utility for Deviations from Low Inflation Equilibrium
0.1922
0.192
0.1918
utility
0.1916
0.1914
0.1912
0.191
0.1908
1
1.05
1.1
1.15
R
2
1.2
1.25
1.3
Figure 2b: Utility for Deviations from High Inflation Equilibrium
0.1766
0.1764
0.1762
utility
0.176
0.1758
0.1756
0.1754
0.1752
1
1.5
2
R
3
2.5
3
3.5
Figure 3: Interest Rate Policy Correspondence
3.5
3
R
2.5
2
1.5
1
0.125
0.13
0.135
z
4
0.14
0.145
0.15
3
2.8
2.6
2.4
Payment
Function,
Low θ
Policy Correspondence
R
2.2
2
1.8
1.6
Payment Function, High θ
1.4
1.2
1
0.125
0.13
0.135
z
5
0.14
0.145
0.15
Figure 4b: Markov Equilibrium With Payment Technology Shocks
3
2.8
2.6
2.4
2.2
Policy Correspondence
R
Payment Function, High η
2
1.8
Payment Function, Low η
1.6
1.4
1.2
1
0.125
0.13
0.135
z
6
0.14
0.145
0.15
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