The Role of Investment Spending
In Sticky Price Models*
Charles T. Carlstrom1 and Timothy S. Fuerst2
1Federal Reserve Bank of Cleveland, Cleveland, OH 44101, USA.
2Bowling Green State University, Bowling Green, OH 43403, USA;
and Federal Reserve Bank of Cleveland.
November 2, 2000
Abstract: This paper analyzes the restrictions necessary to ensure that the policy
rule used by the central bank does not introduce real indeterminacy into the
economy. It conducts this analysis in a Calvo-style sticky price model. A key
innovation is to add investment spending to the analysis. A robust conclusion is
that to ensure determinacy the monetary authority should follow a backwardlooking rule where the nominal interest rate responds aggressively to past
inflation rates.
I.
Introduction.
The celebrated Taylor (1993) rule posits that the central bank uses a fairly simple
rule when conducting monetary policy. This rule is a reaction function linking
movements in the nominal interest rate to movements in endogenous variables (eg.,
inflation) with an elasticity denoted by . Recently there has been a considerable amount
of interest in ensuring that such rules do no harm. The problem is that by following a rule
in which the central bank responds to endogenous variables, the central bank may
introduce real indeterminacy and sunspot equilibria into an otherwise determinate
economy.1 These sunspot fluctuations are welfare-reducing and can potentially be quite
large.
This paper analyzes the conditions under which a simple Taylor rule, one that
responds only to inflation, will be determinate and hence avoid sunspot fluctuations. In
particular we ask whether the central bank should respond proactively to movements in
expected future inflation, or should they look backwards and base interest rate changes on
past movements in inflation. What about Taylor rules that respond to current inflation?
We argue that a necessary and sufficient condition for determinacy is for the monetary
authority to react aggressively to past movements in inflation. We find that this result is
quite robust.
This result is in sharp contrast to the existing literature which has argued that to
avoid real indeterminacy the central bank should respond aggressively to either expected
inflation (see Bernanke and Woodford (1997) and Clarida, Gali, Gertler (1998)) or
The term “sunspot” is in one sense misleading since these shocks are accommodated by monetary policy.
But we use the term since the central bank introduces real indeterminacy by responding to public
1
1
current inflation (see Kerr and King (1996)). These analyses are all reduced-form sticky
price models, where the underlying structural model is a labor-only economy and money
is introduced via a money-in-the-utility function (MIUF) model with a zero cross-partial
between consumption and real balances.2
Our analysis differs from those of Bernanke and Woodford (1997), Clarida, Gali,
Gertler (1998)), and Kerr and King (1996) on several dimensions.3 The most important
of which is the addition of capital and investment spending. This added margin makes
determinacy much harder to achieve with current-looking Taylor rules. In sharp contrast
to these papers we argue that essentially a necessary and sufficient condition for
determinacy is for the monetary authority to react aggressively to past movements in
inflation. Surprisingly we find that this result is quite robust. These findings are of more
than academic interest since several central banks currently use inflation forecasts as an
important part of their decision-making on policy issues.4
A recent paper by Dupor (2000) analyzed a similar sticky price environment with
investment but remarkably came to substantially different policy prescriptions. The
essential difference between the models is that his was conducted in a continuous time
model. Below we argue that adopting a continuous time model is a dangerous strategy as
it implicitly makes several strong timing assumptions. These timing assumptions involve
expectations which can be driven by sunspots.
2
Benhabib, Schmitt-Grohe and Uribe (1999a) analyze Taylor rules in a continuous time MIUF environment
with an arbitrary cross-partial Ucm and demonstrate that the conditions for determinacy depend on the sign
of Ucm. Their analysis is subject to the same two concerns we raise about the other analyses: (1) money
demand timing, and (2) the inclusion of investment spending.
3
The other difference between our model and these models is that we adopt cash-in-advance (CIA) timing
while these models adopt cash-when-I’m-done (CWID) timing where end-of-period money balances affect
utility. This difference is discussed in Carlstrom and Fuerst (1999).
4
In fact, countries with inflation targets (Canada, New Zealand, and the UK, for example) all base their
policy actions on inflation forecasts.
2
the timing of money (CIA versus CWID-timing)5, the timing of investment (is investment
instantaneous?), and the fundamental nature of sticky prices (are current or future prices
sticky?). We argue that most modelers would never have chosen, a priori, the timing
conventions implied by a continuous time if they had started with a discrete time model.
Carlstrom and Fuerst (2000a, 2000b) provide a complementary determinacy
analysis in a flexible price environment. They show that the inclusion of investment
spending has an important effect on the determinacy regions. In particular, with capital
current-looking rules are never determinate and forward-looking rules are determinate if
and only if interest rates respond passively to inflation ( < 1). These determinacy
conditions are independent of labor supply elasticity and are identical to a labor-only
economy with an infinite labor supply elasticity. We thus restrict our analysis to an
infinite labor supply elasticity.
As a final note, Carlstrom and Fuerst (2000a,2000b) demonstrate that in a flexible
price environment there is nominal determinacy if and only if the central bank responds
aggressively to lagged inflation rates. This result is identical to the real determinacy
results in sticky price models presented here, and suggests that there is a link between
real determinacy in a sticky price model and nominal determinacy in the corresponding
flexible price model. This issue is explored in Carlstrom and Fuerst (2000b).
The outline of the paper is as follows. The next section develops the basic model.
Section III provides the determinacy analysis. Section IV compares these results to the
continuous time analysis of Dupor (2000). Finally section V concludes. An extensive
appendix proves the main propositions.
5
See the discussion in footnote 3.
3
II.
A Sticky Price Model
The economy consists of numerous households and firms each of which we will
discuss in turn. We are concerned with issues of determinacy. Hence, without loss of
generality we limit the discussion to a deterministic model. As is well known, if the
deterministic dynamics are not unique, then it is possible to construct sunspot equilibria
in the model economy.
Households are identical and infinitely-lived with preferences over consumption,
real money balances and leisure given by


t 0
tU(ct,at,1-Lt),
where  is the personal discount rate, ct is consumption, at ≡ At/Pt is real cash balances
available for transactions during time t, Pt is the price level, and 1-Lt is leisure. We utilize
a MIUF environment because of its generality (see Feenstra (1986)). The utility function
is given by:
U(c,1-L)  V(c,a) – L.
As noted in the introduction, for the analytical results in this paper we will be working
with the Hansen-Rogerson indivisible labor formulation so that utility is linear in leisure.
The household begins the period with Mt cash balances and Bt-1 holdings of
nominal bonds. Before proceeding to the goods market, the household visits the financial
market where it carries out bond trading and receives a cash transfer of M t (Gt  1)
s
s
from the monetary authority where M t denotes the per capita money supply and Gt is the
4
gross money growth rate. Hence, before entering goods trading, the household has
nominal cash balances given by
At  M t  M ts (Gt  1)  Bt 1 Rt 1  Bt ,
where Rt-1 denotes the gross nominal interest rate from t-1 to t. Notice that following
Carlstrom and Fuerst (2000) we utilize cash-in-advance (CIA) timing. That is, the money
balances that aid in transactions are the money balances that the household has upon
entering goods market trading. This is in contrast to “cash-when-I’m-done timing”
(CWID) where it is implicitly assumed that the money balances that aid in transactions
are the money balances that the household has after leaving the store.
After engaging in goods trading, the household ends the period with cash balances
given by the intertemporal budget constraint.
M t 1  At  Pt {wt Lt [rt  (1   )]K t }  Pt ct  Pt K t 1  t .
Kt denoes the households accumulated capital stock that earns rental rate rt and
depreciates at rate .
Note that we are assuming that asset accumulation occurs at the
household level and that cash in advance is not needed to facilitate its purchase. The
former assumption is without loss of generality; the second assumption is quite important
as it implies that the nominal interest rate acts as a consumption (but not investment) tax.
The real wage is given by wt while tdenotes the profit flow from firms.
The first order conditions to the household’s problem include the following:
U L (t )
 wt
U c (t )
(1)
5
U c (t )   U c (t  1)[rt 1  (1   )]
(2)
U (t  1)  U m (t  1) 
U c (t )  U m (t )
  Rt  c

Pt
Pt 1


(3)
U c (t )  U m (t )
 Rt
U c (t )
(4)
Equation (1) is the familiar labor supply equation, while (2) is the asset accumulation
margin. Equation (3) is the Fisherian interest rate determination in which the nominal
rate varies with expected inflation and the real rate of interest on bonds. Notice that the
assumption of CIA timing implies that the real rate of return on nominal bonds depends
upon the marginal utility of consumption and the marginal transactions role of money.
Equation (4) is the model’s money demand function.6
As for firm behavior, we follow Yun (1996) and utilize a model of imperfect
competition in the intermediate goods market. Final goods production in this economy is
carried out in a perfectly competitive industry that utilizes intermediate goods in
production. The CES production function is given by
1
Yt  {  [ yt (i ) ( 1)/ ]di} /( 1)
0
where Yt denotes the final good, and yt(i) denotes the continuum of intermediate
goods, each indexed by i  [0,1]. The implied demand for the intermediate good is thus
6
given by
 P (i ) 
yt (i )  Yt  t 
 Pt 

where Pt(i) is the dollar price of good i, and Pt is the final goods price.
Intermediate goods firm i is a monopolist producer of intermediate good i. Each
intermediate firm rents capital and hires labor from households utilizing a CRS CobbDouglass production function denoted by f(K,L). Imperfect competition implies that
factor payments are distorted. With zt as marginal cost, we then have rt = ztfK(Kt,Lt) and
wt = ztfL(Kt,Lt).
As for intermediate goods pricing, we follow Yun (1996) and utilize the
assumption of staggered pricing in Calvo (1983). Each period fraction (1-) of firms get
to set a new price, while the remaining fraction  must charge the previous period’s price
times steady-state inflation. This probability of a price change is constant across time and
is independent of how long it has been since any one firm has last adjusted its price.
If  = 0 so that all prices are flexible each period, zt = (-1)/ < 1. This latter term
z ≡ (-1)/ is a measure of the steady-state distortion arising from monopolistic
competition. In the case of sticky prices ( > 0), zt will not typically equal z and will
reflect the time varying monopoly distortion. Yun (1996) demonstrates that this
distortion is given by the following log-linearized “Phillips curve”
~t  ~z t  ~t 1 ,
(5)
where t ≡ Pt/Pt-1 denotes the inflation rate from time t-1 to time t, the tildes denote log
6
In the case of “cash-when-I’m done” timing, the corresponding equations are U (t)/P = R U (t+1)/P ,
c
t
t c
t+1
7
deviations, and  = (1-)(1-)/.
A recursive competitive equilibrium is given by stationary decision rules that
satisfy (5) and the following:
U L (t )
 z t f L (t )
U c (t )
(6)
U c (t )   U c (t  1)[ z t 1 f K ( K t 1 , Lt 1 )  (1   )]
(7)
U (t  1)  U m (t  1) 
U c (t )  U m (t )
  Rt  c

Pt
Pt 1


(8)
U c (t )  U m (t )
 Rt
U c (t )
(9)
ct  K t 1  f ( K t , Lt )  (1   ) K t
(10)
Note that equations (6)-(7) and (10) are essentially the real business cycle (RBC)
conditions distorted by marginal cost and the effect of real balances on the marginal
utility of consumption. Given (9) the labor equation (6) and the capital accumulation
equation (7) can be rewritten
and U (t)/U (t) = (R -1)/R .
m
c
t
t
8
z
U L (t )
 t f L (t )
U c (t )  U m (t ) Rt
(11)
U (t  1)  U m (t  1)

U c (t )  U m (t )
 c
[ z t 1 f K ( K t 1 , Lt 1 )  (1   )]
Rt
Rt 1


(12)
In the extreme case of a rigid CIA constraint (a Leontief utility specification in
which Um = 0 in equilibrium) the monetary distortion is accentuated. Hence, we have two
distortions relative to the RBC model—the nominal rate and marginal cost. The nominal
rate acts like a consumption tax of (1+tax) = Rt, while marginal cost is a tax on factor
income of (1-zt). Equations (11) and (12) make clear that even as we move away from a
rigid CIA constraint, these two distortions are always present. If these two distortions
were held fixed, we would have the RBC model, and would thus be assured of a unique
equilibrium. Indeterminacy arises because of endogenous fluctuations in these two
distortions.
To close the model we need to specify the central bank reaction function. In what
follows we assume a reaction function where the current nominal interest rate is a
function of inflation. We will consider three variations of this simple rule:

Rt  Rss  t i
  ss



 , where   0, Rss  ss ,


where i = -1 is a backward-looking rule, i = 0 is a current-looking rule, and i = 1 is a
forward-looking rule.
9
Under any such interest rate policy the money supply responds endogenously to be
consistent with the interest rate rule. It is this endogeneity of the money supply that leads
to the possibility of indeterminacy. If the private sector responds to sunspots, then the
central bank must passively vary the money supply to keep the nominal rate on target.
These endogenous money supply movements induce endogenous movements in the
nominal rate and marginal cost distortion thus creating the possibility of real
indeterminacy.
III.
Equilibrium Determinacy.
By real indeterminacy, we mean a situation in which the behavior of one or more
real variables is not pinned down by the model. This possibility is of great importance as
it immediately implies the existence of sunspot equilibria which, in the present
environment, are necessarily welfare reducing. As noted above, there are two distortions
in this model: the nominal interest rate and marginal cost. If these two distortions were
held fixed, we would have the RBC model, and would thus be assured of real
determinacy. Hence, indeterminacy arises because of endogenous fluctuations in these
two distortions.
By definition indeterminacy results when current consumption and the real
interest rate move in opposite directions. This is true because multiple stationary paths
are possible if the paths of the endogenous variables are not explosive (i.e., ct 1  ct ).
This guarantees that the real interest rate (which equals consumption growth) will be
10
inversely related to current consumption.7 In a labor only economy this suggests that
indeterminacy is possible if and only if the real interest rate and the labor market
distortion (R-z) move together. For example, consider the case of the strict CIA
~
constraint so that the labor market equation (11) can be written as c~t  ( Rt  ~
zt ) . With
capital things are a bit more complicated because investment spending breaks the tight
link between current output and current consumption.
The endogenous movements in these two distortions (R and z) and hence
indeterminacy are affected by the Taylor elasticity . For example, for a current-looking
Taylor rule a large  dampens movements in inflation and thus marginal cost, but
magnifies movements in the nominal rate. Conversely, if  is small, nominal rate
movements are minimized while inflation and thus marginal cost movements are
magnified. This suggests that there will be problems with indeterminacy for either a very
large or a very small tau.
As we will see below a forward-looking rule guarantees that the real rate and the
labor market distortion always move together so that there is indeterminacy for all values
of . In the case of a backward-looking rule, the initial nominal rate distortion is
predetermined so that we can avoid indeterminacy with a large .
We will now discuss each Taylor timing convention in turn. Because we are
interested in highlighting the effects of capital on the determinacy conditions, in each
subsection we will first present results for a labor-only economy and then note how the
determinacy range is affected by the inclusion of the investment margin.
7
This is not quite correct as we are ignoring the effects of real balances on the real rate. However, the
basic intuition is unaffected by this modification.
11
Forward-Looking Taylor Rules:
Consider a labor-only economy in which production is linear in labor and utility is
linear in leisure. Combining the log-linearized labor condition (11), the log-linearized
Fisher (8), and the forward-looking policy rule gives
~
z t 1  ~
z t  ~t  2  ~t 1
(13)
For a given policy rule, equations (5) and (13) represent a system in zt and t.
Carlstrom and Fuerst (1999) show that under the mild assumption +  1, all forwardlooking rules are indeterminate.
With a large  the labor market distortion is dominated by movements in the
nominal rate while with a small  the labor market distortion is dominated by movements
in marginal cost. With  < 1 an increase in expected inflation lowers the real rate
~
( Rt  ~t 1  (  1)~t 1 ) but also decreases the distortion since higher expected inflation
drives up marginal cost by increasing current prices. With  > 1 an increase in expected
inflation increases the real rate but now the resulting increase in the nominal rate
dominates so that the combined distortion (R-z) also increases. This logic suggests that
for all  there is indeterminacy.8
Adding investment spending to the model does not help matters. One important
reason is because in the model with capital there is always a zero eigenvalue. To see this
substitute (8) and (9) into (7) to obtain
The only exception to this is if + < 1. In this case prices are so sticky that in a positive neighborhood
around  = 1 increases in expected inflation will drive up marginal cost enough so that the combined labor
market distortion (R-z) decreases so that there is determinacy.
8
12
Rt 1
 t 1
 f k (t  1)  (1   ).
The presence of Rt+1 versus Rt is a manifestation of CIA timing. Hence, for either a
current or a forward-looking rule the capital accumulation equation does not depend on
time t variables. This immediately suggests a zero eigenvalue.9
Since capital is the only state variable in the system a zero eigenvalue implies that
capital must immediately jump to the steady state for there to be determinacy.
Conversely, if there are sunspots, one can always construct them in a model in which
capital is set to steady-state for all periods. Hence, the conditions for determinacy in a
model with capital are at least as tight as in the model with labor. But since we always
have indeterminacy in a forward-looking rule, capital cannot make this worse. In
particular we have:
Proposition 1: Suppose that monetary policy is given by a forward-looking Taylor rule.
A necessary condition for there to be real determinacy in the Calvo sticky price model
with investment is
1 

 1 .
Equivalently this is a necessary and sufficient condition for there to be real determinacy
for at least some range of .
Proof: see the appendix.
9
The appendix formally shows the presence of a zero eigenvalue. The labor equation has time t elements
but given the assumption of linear leisure can be substituted out. However, a zero eigenvalue arises even in
13
The assumption that (1-)/ is sufficiently small is identical to the assumption
used in the labor-only case but where the labor share is 1-. As in the labor-only case, for
all plausible parameter values we have indeterminacy for all values of . If we look at the
range of very small  in which this condition does not hold, it is possible to have
determinacy, but the region is quite small. With a forward-looking rule investment
spending has basically no impact in that the economy is indeterminate under both
scenarios. As we will now see this is not true for a current-looking rule where the role of
investment is critical.
Current-Looking Taylor Rules:
In this case, the counterpart to (13) is
~
z t 1  ~
z t  (  1)~t 1
This is essentially the model analyzed by Clarida, Gali, and Gertler (1998). The only
difference is that we are using CIA timing while they used CWID timing so that our
current-looking rule corresponds to their forward-looking rule. Their analysis suggested
that to avoid indeterminacy the central bank should react aggressively (but not too
aggressively) to future inflation. In particular, Carlstrom and Fuerst (1999) demonstrate
that the necessary and sufficient condition for determinacy is
1 
2(   1)  

.
Notice that for the current-looking rule as we approach flexible prices (    )
the range for which the economy is determinate disappears. For plausible parameter
models without an infinite labor supply elasticity—separable preferences are a sufficient condition.
14
values (eg.,  is close to one,  and  each about 1/3) indicate that the determinacy range
is quite large with an upper bound of about 13. Hence, an aggressive (but not too
aggressive) current-looking rule is determinate in this environment.
The reason an overly aggressive current-looking Taylor rule is indeterminate can
be understood most clearly if we consider the extreme case of an inflation peg where
 This stabilizes marginal cost (see equation (5)) and makes this sticky price model
isomorphic to the corresponding flexible price economy.10 But as demonstrated by
Carlstrom and Fuerst (2000), an inflation peg produces real indeterminacy in a flexible
price model because the nominal rate (the sole labor market distortion) always moves in
tandem with the real interest rate. Intuitively, we have eliminated one distortion
(marginal cost), with the side effect of amplifying the other distortion (the nominal rate).
This amplification produces indeterminacy.
Why are some current-rules with  > 1 determinate while there is indeterminacy
for all  in the forward-looking case? Along a stable sunspot path, increases in current
inflation must be larger than the corresponding movement in expected inflation. Hence,
the implied movement in the real rate and hence consumption growth must be sharper
than it is with a forward rule. This steeply-sloped and monotonic consumption path is in
general not possible without capital.
Adding capital basically eliminates all these determinate equilibria. The essential
reason is because once again there is a zero eigenvalue implying that all determinate
equilibria must have capital immediately jumping to its steady-state. These dynamics are
10
See King and Wolman (1997) and Carlstrom and Fuerst (2000) for a discussion of the equivalence
between a sticky price model and a flexible price model under an inflation peg.
15
quite strange and would certainly call into question the wisdom of monetary policy
responding to either current or future inflation even if their equilibria were determinate.11
To understand why current looking rules with capital are almost always
indeterminate even when  > 1 suppose that to the contrary the economy were
determinate. This implies that initial investment must be given so that tomorrow capital
is at its steady state. If the economy is determinate sunspots can be ruled out because they
are explosive. Suppose there is a sunspot increase in current inflation. This increases
both Rt and zt but decreases labor since the nominal interest rate will dominate when  >
1. The decline in labor will cause both output and investment to decline. But investment
cannot decline since capital must immediately jump to its steady-state. To keep
investment from falling zt+1 and therefore t+1 must rise. For determinacy zt+1 > zt and
t+1 > t. But this will tend not to be the case since a small rise in zt+1 (a decrease in
tomorrow’s income tax) will spur investment much more than the small rise in Rt-zt (an
increase in today’s labor tax) deterred investment. Hence, zt+1 < zt so that we have a
contradiction.
More formally, the appendix proves the following:
Proposition 2: Suppose that monetary policy is given by a current-looking Taylor rule. In
the Calvo sticky price model with investment a necessary condition for determinacy is
that  be in one of the following two regions:
11
There is also a zero eigenvalue with a backward-looking rule, but in this case there are two state
variables.
16
 2(1   )a2 a2 
(1  3 )a2
  (  1)  min 
, ,
a1

 a1  
0   (  1) 
a2 (1   )

,
where a1  1   (1   )(1   ) and a2  1   (1   ) .
These regions for determinacy are remarkably narrow. Suppose that  =  = 1/3,
 = .02, and  = .99, so that a1  .35 and a2  .03. In this case, the first region is an empty
set, and the second region is 1 <  < 1.0027. In comparison to the labor-only economy,
the presence of capital makes determinacy essentially impossible.
Backward-Looking Taylor Rules:
In this case we have
~
z t 1  ~
z t  ~t  ~t 1 .
It is straightforward to show that we have real determinacy if and only if  > 1. To
understand why, assume to the contrary that we had indeterminacy when  > 1. Consider
a sunspot increase in t+1. Indeterminacy implies that z t 1  z t and  t 1   t . If  > 1
then marginal cost (zt) and inflation (t) must be inversely related (an inverse “Phillip’s
curve”). Given that the initial nominal rate distortion is fixed this implies that the labor
market distortion and the real rate are moving in opposite directions. But indeterminacy
requires that they move together which gives us our contradiction. A similar argument
17
can be used to show that when  < 1 we must have indeterminacy. Remarkably this
conclusion is not affected by the addition of investment to the model:
Proposition 3: Suppose that monetary policy is given by a backward-looking Taylor rule.
Assume that a1  (2  1)a2 . Then in the Calvo sticky price model with investment a
necessary and sufficient condition for determinacy is that  > 1.
Note: Given the above calibration  = 1/3,  = .02, and  = .99, so that a1  .35 and a2 
.03, there can only be indeterminacy for some values of  > 1 if prices are extremely
sticky,  < 0.083. Given that  = (1-)(1-)/ this implies that less than 25% of firms
adjust their prices every quarter (where  denotes the probability that a firm can adjust
prices in the current period). The above, however, is only a sufficient condition for
determinacy. Even if  < 0.05 the range for indeterminacy is very small 1.1 < 

IV.
A Comparison with a Continuous Time Analysis.
In a recent paper Dupor (2000) conducts a similar determinacy analysis in a
continuous time MIUF model. The environments appear to be the same: prices are sticky
as in Calvo (1983), output is produced using both labor and capital, and preferences are
linear over labor. However, Dupor reaches a quite different outcome: he reports that  <
1 is necessary for determinacy.
In this section we will explore the reasons for the different results. Dupor’s
(2000) model is given by the following equations:
18
U L (t )
 z (t ) f L (t )
U c (t )
(20)
c(t )  c (r (t )   )
(21)
f k (t )  r (t )  
(22)
R(t )  r (t )   (t )
(23)
 (t )  P( (t ), z (t ))
(24)
k(t )  f (k (t ), L(t ))  k (t )  c(t )
(25)
Equation (20) is the labor choice equation; (21)-(22) are the capital accumulation
equations; (23) is the Fisher equation; (24) is the Calvo pricing relationship, and (25) is
the resource constraint. These match up with our conditions 5-10 except that we include
the money demand curve (9).
To understand the differing results across models, it is essential to consider the
discrete time counterpart to this continuous time model. There are many subtle modeling
differences between Dupor and the current work, and all of these differences relate to
timing. Evidently, the cumulative effect of these differences leads to Dupor’s (2000)
unique result. The central timing differences are as follows:
The first thing to note is that given the Fisher equation the discrete time analog to
(t) is t+1, expected inflation. This implies that the immediate comparison to our model
should be a forward not a current-looking rule. This however, does not solve the problem
since we argue that all forward-looking rules are indeterminate.
Second, as noted by Carlstrom and Fuerst (1999), the discrete time analog to this
model is a MIUF model with cash-when-I’m-done timing. That is, the money balances
19
that aid in transactions are the money balances left after leaving goods market trading.
This is a peculiar timing convention. If we adjust our model to CWID timing, then as
mentioned earlier a current-looking rule with CIA timing is equivalent to forward-looking
Taylor rule with CWID timing. This implies that the relevant comparison to our model is
a current-looking rule. But this does not explain the differing results as we argue that all
current-looking rules are indeterminate.
Third, what is the meaning of Dupor’s Calvo pricing equation (23)? Given that
the discrete time analog to (t) is t+1 this implies that the discrete-time analog to Dupor’s
Calvo pricing equation (23) is actually
 t  2   t 1  P( t 1 , zt ) .
What is unusual here is that t is not part of the time-t pricing equation! According to
this expression, there is no restriction on contemporaneous movements in t. That is
prices are only sticky in the future. This is why Dupor notes that there is always nominal
indeterminacy in the model—there is nothing to pin down t. This is quite peculiar. We
have a sticky price model in which the current price is entirely flexible so that i.i.d.
movements in the current money supply and current price level have no real effects.
Current prices are flexible but future prices are costly to adjust. Only anticipated inflation
has real affects.
Fourth and finally, equations (21)-(22) equate consumption growth between t and
t+1 to the marginal product of capital in time-t. There is no counterpart to this
convention in a discrete time model since the real rate of interest is linked to the future
marginal product of capital. This difference is particularly relevant as Dupor notes that
his results are driven by the linkage between the current marginal product of capital and
20
the real rate of interest between today and tomorrow.12 If we just adjust our equilibrium
conditions in the discrete time model for these timing changes it is easy to show that we
obtain Dupor’s result.
In our view, it is hard to defend any of these modeling conventions. This is not
meant to be a criticism of Dupor (2000) per se, but instead a criticism of continuous time
aggregative monetary models. The essential nature of money is a timing problem—the
timing of receipts is not synchronous with the timing of expenses. If we can ignore
timing, then we can ignore money. But if we wish to model money then we must think
carefully about timing. Similarly if we think prices are sticky we have to think carefully
about what this implies in a continuous time model. In the above continuous time model
today’s prices are completely free while expected inflation is sticky.
Some argue that we do not know the answer to many of these timing questions
and therefore we should ignore these central timing issues. Continuous time models lure
the modeler down this path. But by ignoring these timing issues, the modeler is actually
making very strong implicit assumptions about them—assumptions that would never
have been chosen a priori if the modeler had started with a discrete time model.
V.
Conclusion.
The central issue of this paper is to identify the restrictions on the Taylor interest
rate rule needed to ensure real determinacy. A standard result in the literature is that an
aggressive response to either forward or current inflation is necessary and sufficient for
12
The importance of this can be understood by recalling that the zero eigenvalue associated with current
looking rules results because the capital accumulation equation , Rt 1   t 1  f k (t  1)  (1   ) , does
not depend on time t.
21
determinacy. This standard result derives from reduced form MIUF models. The
essential point of this paper is that this result does not stand up to more careful structural
modeling.
In our view, these models ignore two central issues: (1) the appropriate timing for
money demand modeling, and (2) the role of investment spending. This paper, in turn,
considers both of these issues and overturns the standard result. We argue that the
monetary authority must react aggressively to past inflation to ensure determinacy.
We believe that both of these issues are of the first importance. Basing policy
advice on a model that does not include these issues seems quite premature. We have
argued elsewhere (Carlstrom and Fuerst (1999)) for a reassessment of the timing
conventions used in monetary models. We will not review those arguments here.
As for investment spending, the role of investment across the business cycle has a
long tradition in monetary economics so that ignoring it seems like a bad idea. We have
demonstrated above that in the case of current-looking policy, inclusion of the investment
choice dramatically shrinks the region of determinacy. The end result is that for anything
but the most extreme parameter values this model with Calvo stickiness implies that
monetary policy must respond aggressively to past inflation to generate determinacy. In
short, there is a clear danger to any policy that is current or forward-looking. The only
way to ensure real determinacy is to respond aggressively to past movements in inflation.
noting that this “extra” equation at time t
22
Appendix
Proposition 1: Suppose that monetary policy is given by a forward-looking Taylor rule.
A necessary condition for there to be real determinacy in the Calvo sticky price model
with investment is
1 

 1 .
Equivalently this is a necessary and sufficient condition for there to be real determinacy
for at least some range of .
Proof: The relevant equilibrium conditions are given by (5)-(10). Given the assumption
that utility is linear in leisure the first order conditions (6) and (7) can be written as
z t xt
L
U c (t )
,where xt  t
(1   )
Kt

(A1)


U c (t )  U c (t  1) z t 1 xt11  (1   )
(A2)
Substituting (A1) and equation (A1) scrolled forward one period into (A2) yields



z t xt   z t 1 xt1 z t 1 xt11  (1   ) .
(A3)
We can express this as
xt+1 = F(xt,zt+1,zt).
(A4)
The resource constraint provides another equation:
K t 1  K t xt1 (1  ) K t  ct .
(A5)
23
The money demand curve implies that real balances depend only on ct and Rt. Using
(A1) we then have that ct depends only on Rt, zt and xt. Using the policy rule we can then
write (A5) as
Kt+1 = G(xt,t+1,Kt,zt).
(A6)
As for the Fisher equation (8), the money demand curve implies

R 
U c (t ) U c (t  1) t 1 
 t 1 

Using (A1) and the policy rule, we can express this Fisher equation as
t+2 = H(t+1,xt,xt+1,zt,zt+1).
(A7)
In summary, we have three equations (A4), (A6), and (A7). The pricing equation (5)
yields zt = P(t, t+1). Using this to eliminate zt and zt+1 from the above yields the
following linearized Euler equations
xt+1 = F(xt,t, t+1, t+2)
Kt+1 = G(xt,t,t+1,Kt)
t+2 = H(xt,xt+1,t,t+1).
Let wt denote the vector [xt, t+1, t, Kt] so that the linearized system can be expressed as
Awt+1 = Bwt
where A and B are 4x4 matrices with elements given by the derivatives of F, G, and H.
After inverting A, we are left with the matrix A-1B which has four eigenvalues. Since
there is only one state variable in this system (Kt), we need three explosive eigenvalues
for determinacy. One eigenvalue is zero, while another is given by
1
s
1 s
1
24
where s is the steady-state consumption-to-output ratio. The remaining quadratic
equation is given by
J 2 q 2  J 1q  J 0
where
J 2   a2    a1
J1  [ (a1    )  (1   )a2 ]
J 0  a2   
a1  1   (1   )(1   ) and a2  1   (1   ) .
Recall that for determinacy we need both roots of J to be outside the unit circle. The
above implies J(0) > 0, J’(0) < 0, and
J (1)   (  1)(1   )a2
Since J(0) > 0 and J’(0) = J1 < 0 a necessary condition for determinacy is for J(1) > 0, or 
> 1. There are two cases to consider:
Suppose first that the two roots of J are real:
 J1
 1 for determinacy. Solving this inequality yields the
2J 2
If the roots are real we need
following necessary and sufficient condition for determinacy (assuming the roots are real)
1 
a1 
1 

2a1  
a2
u
(A10)
For this interval not to be empty we have the following necessary condition for
determinacy (with real roots)
25
1 


a1  
 1 .
a2
(A11)
That is, if this condition is satisfied there exists some range of  such that we can have
real determinacy.
Suppose instead that the roots are complex:
If the roots are complex, the norm of the complex roots is equal to
J0
. For determinacy
J2
we need this norm to exceed unity and for  > 1. The implied restriction is
1 

1 

a2
.
a1
Once again this interval is non-empty only if (A11) holds. Hence, a necessary condition
for determinacy is (A11). QED
Proposition 2: Suppose that monetary policy is given by a current-looking Taylor rule.
Then in the Calvo sticky price model with investment a necessary condition for
determinacy is that  be in one of the following two regions:
 2(1   )a2 a2 
(1  3 )a2
  (  1)  min 
, ,
a1
a



 1
0   (  1) 
a2 (1   )

,
where a1  1   (1   )(1   ) and a2  1   (1   ) .
Proof: The linearized Euler equations are given by
xt+1 = F(xt,zt,zt+1)
26
Kt+1 = G(xt,t,Kt,zt)
zt+1 = H(xt,xt+1,t+1,zt)
t+1 = P(t,zt).
Let wt denote the vector [xt, zt, t, Kt] so that the linearized system can be expressed as
Awt+1 = Bwt
where A and B are 4x4 matrices with elements given by the derivatives of F, G, H and P.
After inverting A, we are left with the matrix A-1B which has four eigenvalues. Since
there is only one state variable in this system (Kt), we need three explosive eigenvalues
for determinacy. Once again one eigenvalue is zero, while another is given by
(1   ) 
1
1
1 s
where s is the consumption-to-output ratio. Hence, a necessary and sufficient condition
for determinacy is that the remaining two roots be outside the unit circle. The relevant
quadratic equation is given by
J 2 q 2  J 1q  J 0
where
J 2   a2
J1   (  1)a1  (1   )a2
J 0  a 2    (  1)
a1  1   (1   )(1   ) and a2  1   (1   ) .
The above implies
J (1)   (  1)(a1   ) .
27
If  < 1 then J(1) < 0 and J(0) > 0 which means one root is in (0,1). Hence, a necessary
condition for determinacy is  > 1. Under this restriction, J(1) > 0 so that additional
necessary conditions are J(0) > 0 and J(-1) > 0. These put an upper bound on :
 2(1   )a2 a2 
, 

 a1  
 (  1)  min 
(A12)
To make further progress and tighten the bound more closely, we must consider the two
cases where the solutions to J are real or complex.
Suppose first that the two roots of J are real:
We first note that J is quadratic and convex. In this case (assuming the roots are real,
A12, and  > 1) we have the following two potential regions of determinacy:
 J1
 1 and
2J 2
 J1
 1 , or
2J 2
(1  3 )a 2
  (  1)
a1
and
(A13)
0   (  1) 
a2 (1   )
a1
(A14)
Combined with (A12) and  > 1 these two regions of determinacy (assuming the roots are
real) are
0   (  1) 
a2 (1   )
a1
and
(A13)
 2(1   )a2 a2 
(1  3 )a2
  (  1)  min 
, .
a1
a



 1
28
(A14)
Suppose instead that the roots are complex:
The norm of these roots is given by
J 0 a2  (  1)
.

J2
a2
Since J0 > 0, this yields the following necessary and sufficient condition for determinacy
(if the roots are complex):
0   (  1) 
a2 (1   )

.
(A15)
Combining the real and complex regions and noting that  < a1 a necessary condition for
determinacy is that  be in one of the following two regions:
 2(1   )a2 a2 
(1  3 )a2
  (  1)  min 
, ,
a1

 a1  
0   (  1) 
a2 (1   )

.
QED
Proposition 3: Suppose that monetary policy is given by a backward-looking Taylor rule.
Assume that
 a1  (2   1)a2 . Then in the Calvo sticky price model with investment a
necessary and sufficient condition for determinacy is that  > 1.
Proof: The linearized Euler equations are given by
xt+1 = F(xt,zt,zt+1)
Kt+1 = G(xt,t-1,Kt,zt)
zt+1 = H(xt,xt+1,t+1,zt)
29
t+1 = P(t,zt).
Let wt denote the vector [xt, zt, t, t-1, Kt] so that the linearized system can be expressed
as
Awt+1 = Bwt
where A and B are 5x5 matrices with elements given by the derivatives of F, G, H and P.
After inverting A, we are left with the matrix A-1B which has five eigenvalues. Since
there are two state variables in this system (Kt, and t-1), we need three explosive
eigenvalues for determinacy. Once again one eigenvalue is zero, while another is given
by
(1   ) 
1
1
1 s
where s is the consumption-to-output ratio. Hence, a necessary and sufficient condition
for determinacy is that two of the remaining three roots be outside the unit circle. Hence,
there must be one real root within the unit circle. The relevant cubic equation is given by
J 3 q 3  J 2 q 2  J1q  J 0
where
J 3   a2
J 2   a1  (1   )a2
J1  a2      a1
J 0    
a1  1   (1   )(1   ) and a2  1   (1   ) .
30
The above implies J(0) < 0 and
J (1)   (  1)(a1   ) .
For determinacy we need exactly one real root in (0,1). Hence,  > 1 is necessary for
determinacy. The assumed condition in the proposition implies that J is concave at 1.
Hence, the remaining two roots, either real or complex, must be outside the unit circle.
QED
31
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