Monetary Policy and Stock Market Boom-Bust
Cycles∗Lawrence Christiano, Cosmin Ilut, Roberto Motto, and Massimo Rostagno†
March 6, 2007

Abstract
We explore the dynamic eﬀects of news about a future technology improvement which turns out ex post to be overoptimistic. We find that it is diﬃcult
to generate a boom-bust cycle (a period in which stock prices, consumption,
investment and employment all rise and then crash) in response to such a news
shock, in a standard real business cycle model. However, a monetized versionof the model which stresses sticky wages and an inflation-targeting monetary
policy naturally generates a welfare-reducing boom-bust cycle in response to
a news shock. We explore the possibility that integrating credit growth intomonetary policy may result in improved performance. We discuss the robustness of our analysis to alternative specifications of the labor market, in which
wage-setting frictions do not distort on going firm/worker relations.
∗This paper expresses the views of the authors and not necessarily those of the European Central

Bank. We are especially grateful for the comments of Andrew Levin. We have also benefited fromthe comments of Paul Beaudry and of Giovanni Lombardo.
†
The first two authors are aﬃliated with Northwestern University and the second two with theEuropean Central Bank

1. Introduction
Inflation has receded from center stage as a major problem, and attention has shiftedto other concerns. One concern that has received increased attention is volatility inasset markets. A look at the data reveals the reason. Figure 1 displays monthly
observations on the S&P500 (converted into real terms using the CPI) for the period
1870 to early 2006. Note the recent dramatic boom and bust. Two other pronounced“boom-bust” episodes are evident: the one that begins in the early 1920s and busts
near the start of the Great Depression, and another one that begins in the mid
1950s and busts in the 1970s. These observations raise several questions. What are
the basic forces driving the boom-bust episodes? Are they driven by economic fundamentals, or are they bubbles? The boom phase is associated with strong output,
employment, consumption and investment, while there is substantial economic weakness (in one case, the biggest recorded recession in US history) in the bust phase.
Does this association reflect causality going from volatility in the stock market to
the real economy, or does causality go the other way? Or, is it that both are theoutcome of some other factor, perhaps the nature of monetary policy? The analysisof this paper lends support to the latter hypothesis.
We study models that have been useful in the analysis of US and Euro Area business cycles. We adopt the fundamentals perspective on boom-busts suggested by thework of Beaudry and Portier (2000,2003,2004) and recently extended in the analysis
of Jaimovich and Rebelo (2006). The idea is that the boom phase is triggered by asignal which leads agents to rationally expect an improvement in technology in the
future. Although the signal agents see is informative, it is not perfect. Occasionally,
the signal turns out to be false and the bust phase of the cycle begins when people
find this out. As an example, we have in mind the signals that led firms to invest2

heavily in fiber-optic cable, only to be disappointed later by low profits. Anotherexample is the signals that led Motorola to launch satellites into orbit in the expectation (later disappointed) that the satellites would be profitable as cell phone
usage expanded. Although our analysis is based on rational expectations, we suspectthat the same basic results would go through under other theories of how agents canbecome optimistic in ways that turn out ex post to be exaggerated.
Our notion of what triggers a boom-bust cycle is very stylized: the signal occurs
on a particular date and people learn that it is exactly false on another particulardate. In more realistic scenarios, people form expectations based on an accumulationof various signals. If people’s expectations are in fact overoptimistic, they come tothis realization only slowly and over time. Although the trigger of the boom-bust
cycle in our analysis is in some ways simplistic, it has the advantage of allowing us
to highlight a result that we think is likely to survive in more realistic settings.
Our results are as follows. We find that - within the confines of a set of models
that fit the data well - it is hard to account for a boom-bust episode (an episode inwhich consumption, investment, output, employment and the stock market all rise
sharply and then crash) without introducing nominal frictions. However, when we
introduce an inflation targeting central bank and sticky nominal wages, a theory ofboom-busts emerges naturally. In our environment, inflation targeting suboptimallyconverts what would otherwise be a small economic fluctuation into a major boombust episode. In this sense, our analysis is consistent with the view that boom-bust
episodes are in large part caused by monetary policy.
In our model, we represent monetary policy by an empirically estimated Taylor
rule. Because that rule satisfies the Taylor principle, we refer to it loosely as an‘inflation targeting’ rule. Inflation targeting has the powerful attraction of anchoring
expectations in New Keynesian models. However, our analysis suggests that there3are also costs. The analysis suggests that it is desirable to modify the standard
inflation targeting approach to monetary policy in favor of a policy that does notpromote boom-busts. In our model, boom-bust episodes are correlated with strong
credit growth. So, a policy which not only targets inflation, but also ‘leans against
the wind’ by tightening monetary policy when credit growth is strong would reducesome of the costs associated with pure inflation targeting.
Our results are based on model simulations, and so the credibility of the findings
depends on the plausibility of our model. For the purpose of our results here, the twocornerstones of our model are sticky wages and an estimated Taylor rule. That thelatter is a reasonable way to capture monetary policy has almost become axiomatic.
Sticky wages have also emerged in recent years as a key feature of models that fit
the data well (see, for example, the discussion in Christiano, Eichenbaum and Evans(2005).) The view that wage-setting frictions are key to understanding aggregate
fluctuations is also reached by a very diﬀerent type of empirical analysis in Gali,
Gertler and Lopes-Salido (forthcoming).
That sticky wages and inflation targeting are uneasy bedfellows is easy to see.
When wages are sticky, an inflation targeting central bank in eﬀect targets the real
wage. This produces ineﬃcient outcomes when shocks occur which require an adjustment to the real wage (Erceg, Henderson and Levin (2000).) For example, supposea shock - a positive oil price shock, say - occurs which reduces the value marginal
product of labor. Preventing a large fall in employment under these circumstances
would require a drop in the real wage. With sticky wages and an inflation-targeting
central bank, the required fall in the real wage would not occur and employment
would be ineﬃciently low.
That sticky wages and inflation targeting can cause the economy’s response toan oil shock to be ineﬃcient is well known. What is less well known is that the4interaction of sticky wages and inflation targeting in the form of a standard Taylor
rule can trigger a boom-bust episode. The logic is simple. In our model, when
agents receive a signal of improved future technology it is eﬃcient for investment,
consumption and employment to all rise a little, and then fall when expectations
are disappointed. In the eﬃcient allocations, the size of the boom is sharply limitedby the rise in the real wage that occurs as the shadow cost of labor increases withhigher work eﬀort.1 When there are frictions in setting nominal wages, however, arise in the real wage requires that inflation be allowed to drift down. The inflation
targeting central bank, seeing this drift down in inflation, cuts the interest rate tokeep inflation on target. In our model, this cut in the interest rate triggers a credit
boom and makes the economic expansion much bigger than is socially optimal. In asituation like this, a central bank that ‘leans against the wind’ when credit expands
sharply would raise welfare by reducing the magnitude of the boom-bust cycle.
The notion that inflation targeting increases the likelihood of stock market boombust episodes contradicts conventional wisdom. We take it that the conventional
wisdom is defined by the work of Bernanke and Gertler (2000), who argue thatan inflation-targeting monetary authority automatically stabilizes the stock market.
The reason for this is that in the Bernanke-Gertler environment, inflation tends torise in a stock market boom, so that an inflation targeter would raise interest rates,
moderating the rise in stock prices.
So, the behavior of inflation in the boom phase of a boom-bust cycle is the crucial factor that distinguishes our story from the conventional wisdom. To assess thetwo perspectives, consider Figure 2, which displays inflation and the stock market
in the three major US boom-bust episodes in the 20th century. In each case, in1

Our definition of the ‘eﬃcient allocations’ is conventional. They are the allocations that solve

the Ramsey problem.

5

flation is either falling or at least not rising during the initial phase of the boom.
Additional evidence is presented in Adalid and Detken (2006). They identify asset
price boom/bust episodes in 18 OECD countries since the 1970s. In their Table A1,
they show that inflation is on average weak during the boom-phase of these episodes.
In sum, the proposition that inflation is weak during the boom phase of boom-bust
cycles receives support from US and OECD data.
So far, we have stressed that integrating nominal variables and inflation targeting
into the analysis is merely helpful for understanding boom-bust episodes. In fact, if
one is not to wander too far from current standard models, it is essential.
To clarify this point, it is useful to think of the standard real business cycle modelthat emerges when we strip away all monetary factors from our model. If we take acompletely standard version of such a model, a signal shock is completely incapableof generating a boom-bust that resembles anything like what we see. Households ineﬀect react to the signal by going on vacation: consumption jumps, work goes down
and investment falls. When households realize the expected shock will not occur
they in eﬀect find themselves in a situation with a low initial capital stock. The
dynamic response to this is familiar: they increase employment and investment andreduce consumption. We identify the price of equity in the data with the price ofcapital in the model. In the simplest version of the real business cycle model this isfixed by technology at unity, so we have no movements in the stock prices. It is hard
to imagine a less successful model of a boom-bust cycle!
We then add habit persistence and costs of adjusting the flow of investment(these are two features that have been found useful for understanding postwar business cycles) and find that we make substantial progress towards a successful model.
However, this model also has a major failing that initially came to us as a surprise: inthe boom phase of the cycle, stock prices in the model fall and in the bust phase they6

rise. The reason for this counterfactual implication is simple. Investment expands in
response to the signal about future productivity because agents expect investment tobe high in the future, when technology is high. Under these circumstances, the strategy of cutting investment now and raising it in the future when technology is highis ineﬃcient because it entails heavy adjustment costs in the future. In eﬀect, highexpected future investment adds an extra payoﬀ to current investment in the formof reduced future adjustment costs. This corresponds to a reduction in the current
marginal cost of producing capital goods. In a competitive market, this reduction in
marginal cost is passed on to consumers in the form of a lower price.
So, our real business cycle model cannot simultaneously generate a rise in theprice of capital and a rise in investment, in response to a signal about future productivity. The real business cycle model has two additional shortcomings. It generates
an extremely large jump in the real interest rate and it generates very little persistence. It really only generates a boom-bust pattern in consumption, investment,
employment and output when the signal is about a shock that will occur four quarters in the future. If the signal is about a shock, say, 12 quarters in the future, agentsgo on an 8 quarter vacation and then begin working roughly in the 9th quarter. But,
as we see in Figure 1, stock market booms last considerably longer than one year.
So, while a real model takes us part way in understanding a boom-bust cycle, there
are significant shortcomings.
When we introduce monetary factors and an inflation targeting central bank,these shortcomings disappear. The monetary expansion produced in the wake of asignal about higher future productivity generates a boom in the stock price. The
monetary response is associated with very little volatility in the real interest rate, andthe boom bust cycle is highly persistent. In addition, the monetary response greatly
amplifies the magnitude of fluctuations in real quantities. Actually the boom-bust7

produced in the monetary model is so much larger than it is in the real business cycle
model that it is not an exaggeration to say that our boom-bust episode is primarilya monetary phenomenon.
A feature of the simple monetary model is that wage frictions aﬀect employmenton the intensive margin. As a result, the model is vulnerable to the Barro (1977)
critique. The problem is that in practice, variations on the intensive margin occur
in long-lasting relationships between workers and firms, and such relationships oﬀer
many opportunities to undo the eﬀects of idiosyncrasies in wage setting. To investigate whether our analysis robust to alternative specifications of the labor market
which avoid the Barro critique, we explore a recent proposal of Gertler and Trigari(2006) and Gertler, Sala and Trigari (2006). In this formulation, wage setting frictions have allocational consequences via their impact on employer eﬀorts to recruit
new workers and they do not distort on-going worker/firm relations. We find thatwith this alternative formulation of the labor market, our analysis goes through qualitatively, though the findings are less dramatic quantitatively. We conjecture that
this does not reflect a weakness in the mechanism outlined in this paper, but insteadreflects that the alternative model requires additional modifications to magnify its
monetary non-neutralities.
After analyzing the simple monetary model and its robustness to the Barro critique, we move on to the full monetary model of Christiano, Motto and Rostagno(2006). That model incorporates a banking sector and the financial frictions in
Bernanke, Gertler and Gilchrist (1999) into the simple monetary model. The resulting model is interesting for two reasons. First, we use the model to investigate therobustness of our findings for boom-busts. We feed the model the same signal about
future technology that we fed to our real business cycle and simple monetary models. We find that the full and simple monetary models behave quite similarly. The
8

second reason it is interesting to study boom-bust episodes in the Christiano, Mottoand Rostagno (2006) model is that the model has implications for diﬀerent monetary aggregates as well as credit. Discussions of boom-bust cycles often focus on the
behavior of money and credit during a boom. These discussions often emphasize theimportance of distinguishing between money and credit (see, for example, Eichengreen and Mitchener (2003)). They show, for example, that credit grew very rapidly
during the 1920s, but M2 showed weak and declining growth. Interestingly, when we
feed the signal shock to the model of Christiano, Motto and Rostagno (2006) we findthat credit rises strongly during the boom, though the predictions are ambiguous forthe monetary aggregates, with some showing strength and others weakness. This isbroadly consistent with some existing empirical studies.
Our analysis has more general implications. It is already well known that monetary policy plays an important role in the transmission of fundamental shocks. We
can add that monetary policy is also very important in the transmission of expectational shocks.
Following is a brief outline of the paper. The next section describes the realbusiness cycle version of our model in which all monetary factors have been stripped
away. Numerical simulations are used to develop the model’s implications for boombust cycles. Section 3 introduces the smallest number of monetary factors that willallow the model to successfully generate a boom-bust cycle. In this simple monetarymodel, monetary policy is assumed to follow a Taylor rule, which has been estimatedusing post war US data. This Taylor rule places positive weight on the output gap
and incorporates ‘interest smoothing’ in that the interest rate is also a function of
the lagged interest rate. The monetary policy rule is an inflation targeting rule inthe sense that the coeﬃcient on expected inflation satisfies the ‘Taylor Principle’ inbeing larger than unity. In our estimate, it is 1.95. We show that the amplitude of9fluctuation of variables in the boom-bust cycle of the simple monetary model is muchgreater than it is in the version of the model with optimal monetary policy. This isthe basis for our conclusion that the boom-bust in the monetary model is ineﬃcientin a welfare sense. Section 4 investigates robustness to the Barro critique. Section 5
presents the implications for a boom-bust model of the full model monetary model
whose pieces have been studied up to now. The paper closes with a brief conclusion.
Three appendices discuss the technical details of our analysis.

2. Real Business Cycle Model
This section explores the limits of a simple Real Business Cycle explanation of aboom-bust episode. We show that preferences and investment adjustment costs thathave become standard in successful empirical models of business cycles move us partway to a full qualitative explanation of a boom-bust episode. However, we are notsuccessful producing a rise in the price of capital in the boom phase of the cycle. In
addition, we will see that it is hard to generate a boom that is much longer than one
year. Finally, we will see that the model generates extreme fluctuations in the real
rate of interest.
2.1. The Model
The preferences of the representative household are given by:
Et

∞
X
l=0

β l−t {log(Ct+l − bCt+l−1 ) − ψL

L
h1+σ
t
}
1 + σL

Here, ht is hours worked, Ct is consumption and the amount of time that is available
is unity. When b > 0 then there is habit persistence in preferences. The resource

10constraint is
It + Ct ≤ Yt ,(2.1)

where It is investment, Ct is consumption and Yt is output of goods.
Output Yt is produced using the technology
Yt = t Ktα (zt ht )1−α ,
where

(2.2)

t represents a stochastic shock to technology and zt follows a deterministic

growth path,(2.3)

zt = zt−1 exp (µz ) .
The law of motion oft will be described shortly.

We consider two specifications of adjustment costs in investment. According toone, adjustment costs are in terms of the change in the flow of investment:
¶
µ
It
)It ,
Kt+1 = (1 − δ)Kt + (1 − S
It−1

(2.4)

where
S (x) =a(x − exp (µz ))2 ,2with a > 0. We refer to this specification of adjustment costs as the ‘flow specification’. In our second model of investment the adjustment costs are in terms of thelevel of investment:
µ

¶

It
−η
Kt

¶2

It
Kt+1 = (1 − δ) Kt + It − Φ
Kt
where

µ

It
Φ
Kt

¶

1
=
2δσ Φ

11

µ

,

Kt ,

(2.5)

(2.6)

where η is the steady state investment to capital ratio. Here, the parameter, σ Φ > 0,is the elasticity of the investment-capital ratio with respect to Tobin’s q. We refer tothis specification of adjustment costs as the ‘level specification’.
Throughout the analysis, we consider the following impulse. Up until period 1,
the economy is in a steady state. In period t = 1, a signal occurs which suggests
t will be high in period t = 1 + p. But, when period 1 + p occurs, the expected

rise in technology in fact does not happen. A time series representation for t which
captures this possibility is:log t = ρ log t−1 + εt−p + ξ t ,(2.7)

where εt and ξ t are uncorrelated over time and with each other. For example, suppose
p = 1. Then, if the realized value of ε1 is high value, this shifts up the expected valueof log 2 . But, if ξ 2 = −ε1 , then the high expected value of log 2 does not materialize.
We consider the following parameterization,
β = 1.01358−0.25 , µz = 1.01360.25 , b = 0.63, a = 15.1,(2.8)

α = 0.40, δ = 0.025, ψ L = 109.82, σ L = 1, ρ = 0.83, p = 4.
The steady state of the model associated with these parameters is:CK= 0.64,= 12.59, l = 0.092
YY
We interpret the time unit of the model as being one quarter. This model is a special
case of the model estimated in Christiano, Motto and Rostagno (2006) using US data.
In the above list, the parameters a and ρ were estimated; µz was estimated basedon the average growth rate of output; β was selected so that given µz the model
matches the average real return on three-month Treasury bills; σ L was simply set to
produce a Frish labor supply of unity; b was taken from Christiano, Eichenbaum and12

Evans (2004); α and δ were chosen to allow the model to match several ratios (seeChristiano, Motto and Rostagno (2006)).
2.2. Results
Consider the line with circles in Figure 3. This line displays the response of the‘Baseline RBC’ model to a signal in period 1 that technology will jump in period 5
by 1 percent. Then, ξ 5 = −0.01, so that the impact of the signal ont is cancelled

and no change ever happens to actual technology. Note how in the figure output,
investment and hours worked all rise until period 4 and then slowly decline. The
price of capital falls despite the anticipated rise in the payoﬀ associated with capital.
This fall is discussed in further detail below, although perhaps it is not surprisingin view of the spike in the interest rate on one period consumption loans taken inperiod 4. This jump in the interest rate is extraordinarily large. In the period before
the anticipated jump in technology, the real rate jumps by more than 10 percentage
points, at an annual rate.
The solid line in Figure 3 and the results in Figures 5-8 allow us to diagnose theeconomics underlying the line with circles in Figure 3. In Figures 5-8, the circled
line in Figure 3 is reproduced for comparison. The solid line in Figure 3 displays
the response of the variables in the case when the technology shock is realized. Thisshows the scenario which agents expect when they see the signal in period 1. Their
response has several interesting features. First, the rise in investment in period 4, thefirst period in which investment can benefit from the higher expected rate of return,is not especially larger than the rise in other periods, such as period 5. We suspectthat the failure of investment to rise more in period 4 reflects the consumption
smoothing motive. Period 4 is a period of relatively low productivity, and while13

high investment then would benefit from the high period 5 rate of return, raising
investment in period 4 is costly in terms of consumption. The very high period 4
real interest rate is an indicator of just how costly consumption then is. Second,
hours worked drops sharply in the period when the technology shock is realized. The
drop in employment in our simulation reflects the importance of the wealth eﬀect
on labor. This wealth eﬀect is not felt in periods before 5 because of high interest
rate before then. Commenting on an earlier draft of our work, Jaimovich and Rebelo(2006) conjecture that this drop is counterfactual, and they propose an alternative
specification of utility in which it does not occur. An alternative possibility is thatthe sharp movement in employment reflects the absence of labor market frictions in
our model. For example, we suspect that if we incorporate a simple model of labor
search frictions the drop in employment will be greatly attenuated while the basic
message of the paper will be unaﬀected (see Blanchard and Gali (2006)).
Figure 4 allows us to assess the role of habit persistence in the responses in Figure
3. Three things are worth emphasizing based on this figure. First, Figure 4 showsthat b > 0 is a key reason why consumption rises in periods before period 5 in Figure
3. Households, understanding that in period 5 they will want to consume at a highlevel, experience a jump in the marginal utility of consumption in earlier periods
because of habit persistence. This can be seen in the expression for the marginalutility of period t consumption, λt , which is increasing in future consumption:
λt =

1
1
− bβEt.
Ct − bCt−1
Ct+1 − bCt(2.9)

Second, the early jump in the marginal utility of consumption induced by the presence of habit persistence also explains why the employment response to the technology signal is relatively strong in the presence of habit persistence (the bottom left

14

graph in Figure 4). To see this, consider the intratemporal Euler equation:
λt × MP Lt = MULt ,(2.10)

where MP Lt denotes the marginal product of labor and MU Lt denotes the marginalutility of leisure. From this expression it is clear that with a normal specification ofpreferences, it is not possible for both consumption and labor to rise in response to afuture technology shock. The rise in labor would reduce MP Lt and increase MU Lt ,while the rise in consumption would ordinarily reduce λt .2 With habit persistence,
this logic is broken because the anticipated rise in t + 1 consumption raises λt .
Third, note that the employment response without habit persistence, though weak,is positive. The reason for this is that in the absence of habit persistence, households
find it optimal to reduce consumption in order to make room for an increase ininvestment. The fall in consumption raises λt and is the reason why employment
rises in Figure 4, even when b = 0.
Figure 5 shows what happens when there are no adjustment costs in investment.
In this case, there is no cost to the strategy of simply waiting until later to raise
investment. This strategy has the advantage of permitting consumption smoothing.
One way to see this is to note how the real rate of interest hardly moves when there
are no investment adjustment costs.
Figure 6 shows what happens when we adopt the level specification of adjustment
costs. With this specification, investment falls in response to the signal. This makes
room for additional consumption which reduces λt and accounts for the weak responseof employment after the signal (bottom left graph in Figure 6). So, Figure 6 indicates
2This logic was stressed by Barro and King (1984), who argued that it would be diﬃcult tosquare the procylical movement of consumption and labor with the acyclical behavior of wages
under standard preferences.

15

that the flow specification of adjustment costs in our baseline real business cycle
model plays an important role in producing the responses in Figure 3.
A final experiment was motivated by the fact that in practice the boom phaseof a boom-bust cycle often lasts considerably more than 4 quarters. To investigate
whether the real business cycle model can generate a longer boom phase, we considered an example in which there is a period of 3 years from the date of the signal tothe bust. Figure 7 displays simulation results for the case when p = 12. Notice that
we have in fact not lengthened the boom phase very much because output, employment and investment actually only begin to rise about 4 quarters before the bust.
In addition, the model no longer generates a rise in consumption in response to thesignal. According to the figure, consumption falls (after a very brief rise) in the first
9 quarters. Evidently, households follow a strategy similar to the one in Figure 4,
when there is no habit. There, consumption falls in order to increase the resources
available for investment. Households with habit persistence do not mind following asimilar strategy as long as they can do so over a long enough period of time. With
time, habit stocks fall, thus mitigating the pain of reducing consumption.
2.3. The Price of Capital
To understand the response of the price of capital to a signal about future productivity, we study two model equations that characterize the dynamics of Pk0 ,t . One
equation is the present discounted value of future payoﬀs from capital. This is derived by focusing on the demand for capital. The other implication flows from thefact that capital is produced in the model, and corresponds to what is sometimes
referred to as the model’s Tobin’s q relation.
Let µt denote the multiplier on (2.4) and λt the multiplier on the resource con-16

straint in the Lagrangian representation of the planning problem. The first orderconditions for consumption and labor are (2.9) and (2.10), respectively. The firstorder condition with respect to Kt+1 is:£¤µt = β λt+1 α (Kt+1 )α−1 (zt+1 ht+1 )1−α + µt+1 (1 − δ) .

Note that the object on the right side of the equality is the marginal utility of an
extra unit of Kt+1 . It is tomorrow’s marginal physical product of capital, converted
to marginal utility terms by multiplying by λt+1 plus the value of the undepreciatedpart of Kt+1 that is left over for use in subsequent periods, which is converted into
marginal utility terms by µt+1 . Divide both sides of the first order condition for Kt+1
with respect to λt and rearrange:
∙
¸
µt+1
µt
λt+1
α−1
1−α
α (Kt+1 )=β(zt+1 ht+1 )+(1 − δ) .
λt
λt
λt+1
Now, recall that µt is the marginal utility of Kt+1 , loosely, dU/dKt+1 . Similarly, λt is
the marginal utility of Ct , loosely dU/dCt . Thus, the ratio is the consumption costof a unit of Kt+1 , or the price of capital, Pk0 ,t :dUt

Pk0 ,t =µt
dCt
dK= dUt+1
=
.
t
λt
dK
t+1
dCt

Substituting this into the first order condition for Kt+1 , we obtain:
Pk0 ,t = Et
where¤
βλt+1 £ krt+1 α + Pk0 ,t+1 (1 − δ) ,λtk
= α (Kt+1 )α−1 (zt+1 ht+1 )1−α .
rt+1Iterating this expression forward, we obtain:
Ã i
!
∞
X
Y βλt+jk(1 − δ)i−1 rt+i
Pk0 ,t = Et
λi=1
j=1 t+j−1
17(2.11)

Focusing on the eﬀect of the signal on future rtk ’s creates the expectation that Pk0 ,tshould jump in response to a signal about future productivity. However, we saw inFigure 3 that the real interest rate jumps in response to such a signal, and this drives
Pk0 ,t in the other direction. Since this expression highlights two conflicting forces on
Pk0 ,t , it is not particularly useful for understanding why it is that the force driving
Pk0 ,t down dominates in our simulations.
We obtain the model’s Tobin’s q relation by working the first order condition forinvestment:
¶
µ
¶
It
It
0
−λt + µt (1 − S) − µt S
It−1
It−1 It−1
µ
¶µ
¶2
It+1
It+1
0
+βµt+1 S
= 0.
It
It
µ

It

Rewriting this, taking into account the definition of the price of capital,
h
i∙ ³
´³
´2 ¸
λt+1
It+1
It+1
0
1 − βEt λt PK 0 ,t+1 S
It
It
´
³
´
³
PK 0 ,t =It
ItIt− S 0 It−1
1 − S It−1
It−1(2.12)The right side of the above expression is the marginal cost of an extra unit of capital.
This marginal cost is the sum of two pieces. We see the first by ignoring the expression
after the minus sign in the numerator. The resulting expression for PK 0 ,t is the usual
marginal cost term that occurs with level adjustment costs. It is the ratio of theconsumption cost of a unit of investment goods, dCt /dIt (which is unity), divided by
the marginal productivity (in producing new capital) of an extra investment good,
dKt+1 /dIt . To see that this is indeed the marginal cost of producing new capital,
note that this corresponds to
dCt
dIt
dKt+1
dIt

=

dCt
,
dKt+1

18

i.e., the consumption cost of capital. If we just focus on this part of (2.12), thepuzzle about why PK 0 ,t drops during a boom only deepens. This is because, withthe growth rate of investment high (see Figure 3, which shows that It /It−1 is highfor several periods), the first term after the equality should unambiguously be highduring the boom. Both S and S 0 rise, and this by itself makes PK 0 ,t rise.
Now consider the other term in the numerator of (2.12). The term in square
brackets unambiguously rises after a positive signal about future technology because
future growth in investment increases both S and S 0 . The square bracketed term
contributes to a fall in PK 0 ,t . The intuition for this is straightforward. In the wakeof a positive signal about productivity, producers of new capital understand thatinvestment will be high in the future. When there are adjustment costs, this meansthat there is an extra benefit to investing today: the reduction of adjustment costs
when investment is made high in the future. In a competitive market, suppliersof capital will respond to this by bidding down the price of capital. That is what
happens in the equilibrium of our real business cycle model.
It is similar for the central planner, who understands that a signal about positive future technology implies that building more capital today generates a future‘kickback’, in the form of reduced adjustment costs in the future. This kickback isproperly thought of as a reduction to the marginal cost of producing current capital,
and is fundamentally the reason the planner is motivated to increase current investment. This reasoning suggests to us that there will not be a simple perturbationof the real business cycle model which will generate a rise in investment and a risein the price of capital after a signal about future technology. This motivates us toconsider the monetary version of the model in the next subsection.

19

3. Introducing Nominal Features and an Inflation-Targeting
Central Bank
We now modify our model to introduce monetary policy and wage/price frictions.
In the first subsection below, we present the model. In the second subsection we
present our numerical results.
3.1. Simple Monetary Model
To accommodate price-setting, we adopt the usual assumption that a representative
final good producer manufactures final output using the following linear homogenous
technology:
Yt =

∙Z 1
0

1
λt

Yjt dj¸λf, 1 ≤ λf < ∞,(3.1)

Intermediate good j is produced by a price-setting monopolist according to the following technology:⎧⎨ K α (z h )1−α − Φz if K α (z h )1−α > Φz
t jt
t jt
t
t jt
t jt
t
, 0 < α < 1,
Yjt =
⎩ 0,
otherwise(3.2)

where Φzt is a fixed cost and Kjt and hjt denote the services of capital and homoge-

neous labor. Capital and labor services are hired in competitive markets at nominal
prices, Pt rtk , and Wt , respectively.
In (3.2), the shock to technology, t , has the time series representation in (2.7). We
adopt a variant of Calvo sticky prices. In each period, t, a fraction of intermediategoods firms, 1 − ξ p , can reoptimize their price. If the ith firm in period t cannot
reoptimize, then it sets price according to:
Pit = π̃ t Pi,t−1 ,20whereπ̃ t = π ιt−1 π̄ 1−ι .

(3.3)

Here, πt denotes the gross rate of inflation, π t = Pt /Pt−1 , and π̄ denotes steady state
inflation. If the ith firm is able to optimize its price at time t, it chooses Pi,t = P̃t to
optimize discounted profits:
Et

∞
X
¡ ¢j
βξ p λt+j [Pi,t+j Yi,t+j − Pt+j st+j (Yi,t+j + Φzt+j )] .

(3.4)

j=0

Here, λt+j is the multiplier on firm profits in the household’s budget constraint. Also,
Pi,t+j , j > 0 denotes the price of a firm that sets Pi,t = P̃t and does not reoptimize
between t+1, ..., t+j. The equilibrium conditions associated with firms are displayedin the appendix.
In this environment, firms set prices as an increasing function of marginal cost.
A firm that has an opportunity to set price will take into account future marginal
costs because of the possibility that they may not be able to reoptimize again soon.
We model the labor market in the way suggested by Erceg, Henderson and Levin(2000). The homogeneous labor employed by firms in (3.2) is produced from specialized labor inputs according to the following linear homogeneous technology:
ht =

∙Z 1

(lt,i )

1
λw

di

0

¸λw

, 1 ≤ λw .

(3.5)

We suppose that this technology is operated by perfectly competitive labor contractors, who hire specialized labor from households at wage, Wjt , and sell homogenous
labor services to the intermediate good firms at wage, Wt . Optimization by labor
contractors leads to the following demand for lt,i :
lt,i =

µ

Wt,i
Wt

λw
¶ 1−λ

w

21

ht , 1 ≤ λw .

(3.6)

The j th household maximizes utility⎧´1−σq ⎫
³
P
C
t+l
t+l
⎪
⎪
∞
1+σ L
⎨
⎬
X
d
lt,j
Mt+l
j
l−t
u(Ct+l − bCt+l−1 ) − ψL
Et
β
−υ
⎪
⎪
1 + σL
1 − σq
⎩
⎭
l=0

(3.7)

d
Pt (Ct + It ) + Mt+1
− Mtd + Tt+1 ≤ Wt,j lt,j + Pt rtk Kt + (1 + Rte ) Tt + Aj,t ,(3.8)

subject to the constraint

where Mtd denotes the household’s beginning-of-period stock of money and Tt denotes
nominal bonds issued in period t−1, which earn interest, Rte , in period t. This nominalinterest rate is known at t − 1. In the interest of simplifying, we suppose that υ in(3.7) is positive, but so small that the distortions to consumption, labor and capitalfirst order conditions introduced by money can be ignored. Later, we will consider amodel in which υ is chosen to match money velocity data. The j th household is themonopoly supplier of diﬀerentiated labor, lj,t . Given the labor demand curve, (3.6),
and absent any price frictions, the household would set its wage rate, Wjt , as a fixedmarkup, λw , above its marginal cost:
Wjt = λw Pt

σLσLψL lt,jψL lt,j= λw Pt,λtλt

where λt denotes the marginal utility of consumption, defined in (2.9). In fact,
the household is subject to the same Calvo frictions faced by intermediate good
producers in setting their prices. In particular, in any given period the j th household
can reoptimize its wage rate with probability, 1 − ξ w . With probability ξ w it cannot
reoptimize, in which case it sets its wage rate as follows:
Wj,t = π̃ w,t µz Wj,t−1 ,whereπ̃ w,t ≡ (πt−1 )ιw π̄ 1−ιw .
22

(3.9)

In (3.8), the variable, Aj,t denotes the net payoﬀ from insurance contracts on therisk that a household cannot reoptimize its wage rate, Wtj . The existence of theseinsurance contracts have the consequence that in equilibrium all households havethe same level of consumption, capital and money holdings. We have imposed thisequilibrium outcome on the notation by dropping the j subscript.
The household’s problem is to maximize (3.7) subject to the demand for labor,(3.6), the Calvo wage-setting frictions, and (2.4). Households set their wage as anincreasing function of the marginal cost of working. The presence of wage frictions
leads households who have the opportunity to reoptimize their wage, to take into
account future expected marginal costs. The sluggishness of wages and the fact thathouseholds are required to satisfy (3.6) implies that in the short run, employment islargely demand-determined.
The monetary authority controls the supply of money, Mts . It does so to implement a following Taylor rule. The target interest rate is:
¶
µ
µz π
Yt
∗
Rt =
+ ut ,+ απ [Et (π t+1 ) − π̄] + αy log
βYt+(3.10)

where Yt+ is aggregate output on a nonstochastic steady state growth path and ut
is an iid monetary policy shock. The monetary authority manipulates the money
supply to ensure that the equilibrium nominal rate of interest, Rt , satisfies:
Rt = ρi Rt−1 + (1 − ρi ) Rt∗ .

(3.11)The parameter values for the model are the ones in the real business cycle model,plus the following:
λf = 1.20, λw = 1.05, ξ p = 0.63, ξ w = 0.81, ι = 0.84,
ιw = 0.13, ρi = 0.81, απ = 1.95, αy = 0.18, υ = 0.
23

(3.12)

Our monetary model is a special case of a more general model which was estimated in
US data in Christiano, Motto and Rostagno (2006). All but the first two parameters
above were estimated there.
3.2. Results
Figure 8 displays the results of the response to a signal in period 1 about a shock inperiod 13 (indicated by the ‘*’), which ultimately does not occur. The results for thereal business cycle model in Figure 7 are reproduced here to facilitate comparison.
Both models have costs in adjusting the flow of investment, as well as habit persistence in consumption. Still, the two models display strikingly diﬀerent responses tothe signal shock in period 1. First, the magnitude of the responses in output, hours,
consumption and investment in the monetary model are more than three times what
they are in the real business cycle model. Second, consumption booms in the immediate period after the signal. Third, the risk free rate moves by only a small amount,and it falls rather than rising as in the real business cycle model. Fourth, although
inflation initially rises, eventually it falls by 0.8 of one percent. Fifth, and perhaps
most significantly, the stock price rises.
The simple monetary model behaves so diﬀerently from the real business cycle
model because the monetary policy rule, (3.11), is suboptimal for our model economy.
With optimal monetary policy, the response of the allocations in the simple monetarymodel would have been virtually the same as the responses in the real business cycle
model. In particular, there would have been a boom, but only a very small one. To
understand how we reached this conclusion, consider the starred lines in Figure 9 (thesolid lines reproduce the solid lines in Figure 8 for convenience). The starred lines
were obtained by first deleting the monetary policy rule, (3.11), from the simple

24monetary model. Of course, this renders the model unable to determine values
for the endogenous variables. In eﬀect, there are now many configurations of theendogenous variables which satisfy the remaining equilibrium conditions (i.e., theresource constraint, the necessary conditions associated with optimality, etc.) From
this set of possible equilibria, we selected the Ramsey equilibrium, the one associatedwith the highest level of expected household utility. The allocations in this Ramsey
equilibrium are what is achieved with the best possible monetary policy, when we
do not place any constraints on what form monetary policy takes on.3 Note how theRamsey equilibrium responses of output, investment, consumption, hours worked,
the real interest rate, and the price of capital virtually coincide with what they arein the real business cycle. That is, had monetary policy in the simple monetarymodel been optimal, the allocations would have been essentially the same as thosein the RBC model.4
What is it about (3.11) that causes it transform what would have been a minor
fluctuation into a substantial, welfare-reducing, boom-bust cycle? A clue lies in thefact that the real wage rises during the boom under the optimal policy, while it
falls under (3.11). During the boom of the simple monetary model, employment isineﬃciently high. This equilibrium outcome is made possible in part by the low real
wage (which signals employers that the marginal cost of labor is low) and the fact that
3
4

Technical details on the computation of optimal monetary policy are provided in Appendix B.
The reasons the allocations in the RBC model and in the Ramsey equilibrium do not coincide

exactly is that there are too many frictions for monetary policy to undo. Monetary policy is only
one instrument, but there are several frictions, including sticky prices and sticky wages. Other
frictions include the distortions induced by the presence of market power in firms and households.
If these frictions were small (or, undone by a suitable choice of taxes) then if there were either
only sticky prices or only sticky wages, the Ramsey equilibrium allocations would coincide with theallocations in the RBC model.

25

employment is demand-determined. The high real wage in the Ramsey equilibrium,by contrast, sends the right signal to employers and discourages employment. Sincewages are relatively sticky compared to prices, an eﬃcient way to achieve a higher
real wage is to let inflation drop. But, the monetary authority who follows the
inflation-targeting strategy, (3.11), is reluctant to allow this to happen. Such amonetary authority responds to inflation weakness by shifting to a looser monetary
policy stance. This, in our model, is the reason the boom-bust cycle is amplified.
Thus, the suboptimally large boom-bust cycle in the model is the outcome of theinteraction between sticky wages and the inflation targeting monetary policy rule,(3.11).
To gain further intuition into the boom-bust cycle in the simple monetary model,
consider Figure 11 (there is no Figure 10 in the paper). Figure 11 shows what would
have happened if the rise in productivity that is signalled is actually realized. In
eﬀect, the figure shows what expectations agents have as they respond to the signal
about future productivity. Note that they expect a much bigger and more persistentdecline in the nominal interest rate than actually occurs in the equilibrium in Figure
8. Note too, that there is a sharp drop in employment when the technology shock isrealized, as in the real business cycle model. This is similar to what we saw in thereal business cycle analysis in Figure 3.
Figures 12 and 13 explore the role of sticky prices and wages in our analysis,
respectively, by repeating our experiment twice. In the first case, ξ p = 0.01 and allother parameters are held fixed at their baseline levels. In Figure 13, ξ w = 0.01 and allother parameters are held constant at their baseline values. The experiments convey
a strong message: sticky wages are crucial to our result, not sticky prices. Whenwages are flexible, the boom-bust in the simple monetary model closely resembles
what it is in the real business cycle model. We verified that this result is not an26

artifact of the fact that ξ p is smaller than ξ w in our baseline parameterization. For
example, when we set ξ p = 0.95 and ξ w = 0.01, the results for quantities are similar
to what we see in Figure 13.
In Figures 14 and 15 we attempt to investigate the specific role played by policy ingenerating the boom-bust cycle. Figure 14 reports what happens when the monetary
policy rule focuses less on inflation, by setting απ = 1.05. The real quantities now
fluctuate much as they do in the real business cycle model. This is consistent with
our general theme, that inflation targeting is at the heart of why we get a boom-bustin our model.
Figure 15 displays a diﬀerent way of assessing the role of monetary policy in ourequilibrium. This experiment focuses specifically on the impact of the sharp dropin the interest rate that agents expect according to the solid line in Figure 11. For
this experiment, we add an iid monetary policy shock to the policy rule, (3.11). We
suppose that the shock has the same time series representation as (2.7), except ρ = 0.
We set p = 10, and considered a signal which creates the expectation that there will
be a 100 basis point negative shock to the quarterly rate of interest in period 11. We
imposed that that shock is in fact not realized. The idea is to create a policy-induced
move in the actual interest rate, together with a prior expectation that the rate would
fall even more. Figure 15 indicates that this anticipated monetary loosening createsa substantial boom right away in the model. The boom accounts for almost all of
the monetary part of the boom-bust in the simple monetary model. The vertical
diﬀerence between the two lines in Figure 15 is roughly the size of the boom in
the real business cycle model. The results in Figure 15 suggests the following loose
characterization of our boom-bust result in the simple monetary model: one-thirdof the boom-bust cycle is the eﬃcient response to a signal about technology, andtwo-thirds of the boom-bust cycle is the ineﬃcient consequence of suboptimality in27the monetary policy rule. This suboptimality operates by creating an expectation of
a substantial future anticipated loosening in monetary policy.
In sum, analysis with the simple monetary model suggests that successfully generating a substantial boom-bust episode requires a monetary policy rule which assigns
substantial weight to inflation and which incorporates sticky wages.

4. Robustness to an Alternative Specification of the Labor
MarketThe boom phase of our analysis is characterized by a low real wage, which encourages
firms to expand employment, while workers are required to supply whatever labor isdemanded at the given wage. It is unclear exactly how one should map this labormarket specification into actual labor market arrangements, where workers spend
a substantial fraction of the time in repeated relationships with firms. Still, thesetup is vulnerable to the Barro (1977) critique. Because labor eﬀort in the model isvaried on the intensive margin, it is tempting to interpret the model as a metaphor
for actual firm-worker relationships. But, as Barro emphasized, the notion that
idiosyncrasies in the setting of wages should have allocational consequences for firms
and workers in long-lasting relationships seems implausible. There are simply too
many opportunities for people in long-lasting relationships to undo the eﬀects ofwage frictions.5
In recent years an alternative approach to labor markets has emerged, startingwith the work of Mortensen and Pissarides (1994) and, more recently, Hall (2005a,b,c)and Shimer (2005a,b). Gertler and Trigari (2006) build on this work (particularly,
the insight in Hall (2005a)) to show how the wage setting frictions that have proved so
5This point was stressed more recently by Goodfriend and King (1997).

28

useful in fitting models to macroeconomic data may be recast so as to avoid runningafoul of the Barro critique. We follow Gertler, Sala and Trigari (2006) (GST) in
incorporating a version of the Gertler and Trigari (2006) (GT) labor market model
into the monetary model of the previous section. In GST, wage setting frictions haveno impact on on-going worker employer relations because (i) they have no direct
impact on the intensive labor margin, and (ii) they do not induce worker-employer
separations since, by assumption, these occur only in response to exogenous events.6
Wage setting frictions do have allocational eﬀects in the GT and GST models, butthese eﬀects operate by their impact on firms’ incentives to recruit new workers.
The basic logic stressed throughout this paper, that wage-setting frictions andinflation targeting can trigger boom-busts, also goes through in this alternative formulation of the labor market. The low real wage in the boom phase generates anexcess of economic activity because the low real wage causes too many resources tobe allocated to recruiting workers.
4.1. The Model
The labor market in our alternative labor market model is a slightly modified versionof the GST model. GST assume wage-setting frictions of the Calvo type, while we
instead work with Taylor-type frictions.7 In addition, we adopt a slightly diﬀerent
representation of the production sector in order to maximize comparability with the6

The specification that the job separation rate is constant and the finding rate endogenous isconsistent with findings reported in Hall (2005b,c) and Shimer (2005a,b ), who report that the job
finding rate is very cyclical and the job separation rate is substantially less so.
7
We work with Taylor frictions because we want to be in a position to check the accuracy of ourlinear approximations by comparing them with higher-order approximations. It is our impressionthat the linearization strategy adopted by GST for their Calvo-specification of wage frictions does
not easily generalize to higher orders.

29model used elsewhere in the paper. In what follows, we first provide an overview andafter that we present the detailed decision problems of agents in the labor market.
4.1.1. Overview
As before, we adopt the Dixit-Stiglitz specification of goods production. A representative, competitive final good firm aggregates diﬀerentiated intermediate goods
into a homogeneous final good. Intermediate goods are supplied by monopolists,
who hire labor and capital services in competitive factor markets. The intermediate
good firms are assumed to be subject to the same Calvo price setting frictions assumed in the previous section. The novelty in the present environment is that labor
services are supplied to the homogeneous labor market by ‘employment agencies’.
An alternative (perhaps more natural) formulation would be for the intermediate
good firms to do their own employment search. We instead separate the task offinding workers from production of intermediate goods in order to avoid adding astate variable to the intermediate good firm, which would complicate the solution of
their price-setting problem. As it is, the equilibrium conditions associated with the
behavior of intermediate good firms are unchanged from what they are elsewhere inthis paper.
Each employment agency retains a large number of workers. At the beginningof the period, some workers are randomly selected to separate from the firm and go
into unemployment. Also, a number of new workers arrive from unemployment inproportion to the number of vacancies posted by the firm in the previous period.
After separation and new arrivals occur, the nominal wage rate is set.
The nominal wage paid to an individual worker is determined by Nash bargaining,which occurs once every N periods. Each agency is permanently allocated to one of30

N diﬀerent cohorts. Cohorts are diﬀerentiated according to the period in which they
renegotiate their wage. Since there is an equal number of agencies in each cohort,
1/N of the agencies bargain in each period. Agencies in cohorts that do not bargain
in the current period simply apply the wage implied for that period by the outcomeof the most recent bargaining round.
Once a wage rate is determined - whether by Nash bargaining or not - we assume
that each matched worker-firm pair finds it optimal to proceed with the relationship in that period. In our calculations, we verify that this assumption is correct,
by confirming that the wage rate in each worker-agency relationship lies inside thebargaining set associated with that relationship.
Next, the intensity of labor eﬀort is determined according to a particular eﬃciency
criterion. To explain this, we discuss the implications of increased intensity for theworker and for the employment agency. The utility function of the household in thealternative labor market model is a modified version of (3.7):
Et

∞
X
l=0

β

l−t

L
ψ1+σ
{log(Ct+l − bCt+l−1 ) − γ tLt }, γ, ω > 0,
1 + σL(4.1)

where Lt is the fraction of members of the household that are working and ψt is theintensity with with each worker works. As in GST, we follow the family household
construct of Merz (1995) in supposing that each household has a large number ofworkers. Although the individual worker’s labor market experience - whether employed or unemployed - is determined in part by idiosyncratic shocks, the household
has suﬃciently many workers that the total fraction of workers employed, Lt , as wellas the fractions allocated among the diﬀerent cohorts, lti , i = 0, ..., N − 1, is thesame for each household. We suppose that all the household’s workers are suppliedinelastically to the labor market. Each worker passes randomly from employment
with a particular agency to unemployment and back to employment according to31

exogenous probabilities described below. The household’s budget constraint is:d−Mtd +Tt+1 ≤ (1 − Lt ) Pt bu zt +Pt (Ct + It )+Mt+1

N−1
X

Wti lti ψt +Pt rtk Kt +(1 + Rte ) Tt ,

i=0

(4.2)

where Wti is the nominal wage rate earned by workers in cohort i = 0, ..., N − 1. The
index, i, indicates the number of periods in the past when bargaining occurred mostrecently. Note that we implicitly assume that labor intensity is the same in each employment agency, regardless of cohort. This is explained below. The presence of theterm involving bu indicates the assumption that unemployed workers receive a payment of bu zt final goods. The marginal cost, in utility terms, to an individual worker
who increases labor intensity by one unit is γψ σt L . This is converted to currency units
by dividing by the multiplier, λt , on (4.2) in the Lagrangian representation of thehousehold’s problem.
Let the price of labor services, Wt , denote the marginal gain to the employment
agency that occurs when an individual worker raises labor intensity by one unit.
Because the employment agency is competitive in the supply of labor services, Wt
is taken as given and is the same for all agencies, regardless of which cohort it is in.
Labor intensity equates the worker’s marginal cost to the agency’s marginal benefit:
Wt = γ

ψσt L
.
λt

(4.3)

Labor intensity is the same for all cohorts because none of the variables in (4.3) is
indexed by cohort. When the wage rate is determined by Nash bargaining, it is taken
into account that labor intensity is determined according to (4.3).

32

4.1.2. Details of the Labor Market Model
Employment agencies in the ith cohort which does not renegotiate its wage in periodt sets the period t wage, Wi,t , as in (3.9):
Wi,t = π̃ w,t µz Wi,t−1 , π̃w,t ≡ (π t−1 )ιw,2 π̄ 1−ιw,2 ,for i = 1, ..., N − 1. It is convenient to define⎧⎨ π̃ w,t+j · · · π̃ w,t+1 µj , j > 0zΓt,j =
.
⎩
1
j=0

(4.4)

(4.5)

After wages are set, employment agencies in cohort i supply labor services, lti ψt , into
competitive labor markets. In addition, they post vacancies to attract new workers
in the next period.
To understand how agencies bargain and how they make their employment decisions, it is useful to consider F (lt0 , ωt ) , the value function of the representative

employment agency in the cohort that negotiates its wage in the current period.8
The arguments of F are the agency’s workforce after beginning-of-period separations and new arrivals, lt0 , and an arbitrary value for the nominal wage rate, ωt . To
simplify notation, we leave out arguments of F that correspond to economy-wide
variables. Then,
N−1
h
X
¡0 ¢
λt+j
κ ¡ j ¢2 i jj
F lt , ωt =
lt+j
β Et
max (Wt+j − Γt,j ω t ) ψt+j − Pt+j zt+j xt+j
(4.6)
λt xjt+j2
j=0´
λt+N ³ 0N
F lt+N , W̃t+N ,
+β Et
λt

where ψt is assumed to satisfy (4.3). Also, W̃t+N denotes the Nash bargaining wage
rate that will be negotiated when the agency next has an opportunity to do so. At
8The value function for firms in other cohorts is defined analogously, and appears in the appendix.

33

time t, the agency takes this as given. The law of motion of an agency’s work force
is:
¡
¢
i+1
lt+1
= xit + ρ lti ,(4.7)

for i = 0, 1, ..., N − 1, with the understanding here and throughout that i = N is to

be interpreted as i = 0. Here, xit is the period t hiring rate of an agency in cohort i.
Expression (4.7) is deterministic, reflecting the assumption that the agency employsa large number of workers. Finally, the expression,zt+i

κ ¡ i ¢2 i
x
l , i = 0, 1, ..., N − 1,
2 t+i t+i

denotes the cost, in units of the final good, associated with a hiring rate, xit . We
include the state of technology, zt , which grows at deterministic gross rate, µz > 1,
to assure that growth is balanced in steady state.
The firm’s hiring rate is chosen to solve the problem in (4.6). It is easy to verify:¡¢
F lt0 , ωt = J (ωt ) lt0 ,(4.8)

where J (ω t ) is not a function of lt0 . The function, J (ωt ) , is the surplus that a firmbargaining in the current period enjoys from a match with an individual worker,
when the current wage is ω t .
We now turn to the worker. The period t value of being a worker in an agency
in cohort i is Vti :Vti = Γt−i,i W̃t−i ψt − γ

L
λt+1
ψ1+σ
t
i+1
+ βEt
[ρVt+1+ (1 − ρ) Ut+1 ],(1 + σ L ) λt
λt

(4.9)

for i = 1, ..., N − 1. Here, ρ is the probability of remaining with the firm in the next

period and Ut is the value of being unemployed in period t. The values, Vti andUt , pertain to the beginning of period t, after job separation and job finding has
34occurred. For workers employed by agencies in cohort i = 0, the value function isV 0 (ω t ) , where ωt is an arbitrary value for the current period wage rate,
V 0 (ωt ) = ω t ψt − γ

L
λt+1
ψ1+σ
t
1
+ βEt
[ρVt+1+ (1 − ρ) Ut+1 ].
(1 + σ L ) λt
λt

(4.10)The notation makes the dependence of V 0 on ωt explicit to simplify the discussionof the Nash bargaining problem below.
The value of being an unemployed worker is Ut :Ut = Pt zt bu + βEt

λt+1x[ft Vt+1+ (1 − ft ) Ut+1 ],λt(4.11)

where ft is the probability that an unemployed worker will land a job in period t + 1.
Also, Vtx is the period t + 1 value function of a worker who finds a job, before is is
known which agency it is found with:
Vtx =

N−1
X

i
xit−1 lt−1
Vti+1 ,mt−1
i=0(4.12)

where the total number of new matches is given by:
mt =

N−1
X

xjt ltj .

(4.13)j=0

In (4.12),ixit−1 lt−1
mt−1

is the probability of finding a job in an agency which was of type i in the previous
period, conditional on being a worker who finds a job in t.
Total job matches must also satisfy the following matching function:
mt = σ m (1 − Lt )σ vt1−σ ,
where
Lt =

N
−1
X
j=0

35

ltj ,

(4.14)

and where vt denotes the total number of vacancies in the economy. Total hours
worked is:

N−1
X

ltj .

(4.15)

mt.
1 − Lt(4.16)

ht = ψt

j=0

The job finding rate is:
ft =The i = 0 cohort of agencies in period t solve the following Nash bargaining
problem:¡¢ηmax V 0 (ω t ) − Ut J (ω t )(1−η) ,ωt(4.17)

where V 0 (ω t ) − Ut is the match surplus enjoyed by a worker. We denote the solution
to this problem by W̃t . Note that W̃t takes into account that intensity will be chosen
according to (4.3) as well as (4.4).
The resource constraint in the model must be modified to accommodate resources
paid to unemployed workers and resources used up in posting vacancies:κ X ¡ j ¢2 j
Ct + It + b zt (1 − Lt ) + zt
x l = Yt ,2 j=0 t t
N−1

u

(4.18)

where Yt is the quantity of final goods, (3.1).

After scaling the variables that grow, the equilibrium conditions associated withthe labor market can be combined with the non-labor market conditions of the modelin the previous section to solve the overall model. For complete details about how
this economy is scaled to account for growth, and for the other equations, see the
Appendix.
4.2. Quantitative Results
In the first subsection below, we report the parameterization of the model. In thesecond subsection, we discuss some properties of the equilibrium of the alternative36

labor market model. The results for boom-bust episodes are presented in the third
subsection.
4.2.1. Parameterization of Models
To accommodate the fact that the average duration of unemployment is a little overtwo months, we adopt a bi-monthly time period for the model. For comparability
with the standard sticky wage model in the previous section, we study a bimonthly
version of that model too. The parameters that the two models have in common
were set as follows:5
λf = 1.20, λw = 1.05, ξ p = , ι = 0.84, ιw = 0.13, απ = 1.95, αy = 0.18,
6−(1/6), µz = 1.0136(1/6) , b = 0.63, a = 15.1, α = 0.40,
β = 1.01358
20.025.
δ =3Here, only ξ p , β, δ, µz are adjusted relative to their values in the analysis of theprevious section (see (3.12)). The parameter, ξ p , was set to imply a mean time
between price changes of one year.9 The inertia parameter in the monetary policy
rule, ρi , was set to 0.60 in our benchmark parameterization of the model with thealternative labor market. We found that the degree of over reaction to a signal about
future productivity is sensitive to this parameter, with a smaller value making theover-reaction large.
Regarding the parameters that apply specifically to our standard sticky wage
model, we set ξ w = 0.87, which implies that wages remain unchanged on average for9

We preferred to set ξ p to a lower value. However, when we tried we encountered numericalproblems in computing the Ramsey equilibrium for the alternative labor market model. We hope
to solve this numerical problem for the next draft.

37

7.89 bi-monthly periods, or 1.3 years. The values of ψL and σ L were set as before,as specified in (2.8).
Now consider the parameters that apply specifically to the alternative labor market model. The disutility of work intensity parameter, γ, was set so as to imply avalue for labor intensity equal to unity in steady state, ψ = 1. In addition, we follow
GT in setting the hiring rate adjustment cost parameter, κ, so that the steady statevalue of the job finding rate, f, implies that jobs last on average 2.22 months, or
1.11 model periods (i.e., f = 0.9). We also follow GT in our setting of the bargaining
power parameter, η, the elasticity of matches to unemployment, σ, the degree of
convexity in the disutility from work intensity, σ L , and the constant term, σ m , in thematching function:
1
1
bu = 0.3, ρ = 0.9333, η = , σ m = 0.1, σ = , σ L = 13.5, κ = 54.42.
2
2
Following GT, the value for ρ is chosen to imply that jobs on average last about
2.5 years. Finally, we must select a value for N, the length of time between wage
negotiations. Following Dixon and Kara (2006) we set N = 15 so that the averageage of existing wage contracts in the alternative labor market model is equal tothe average age (i.e., 7.89 periods) of existing contracts in the standard sticky pricemodel.10
4.2.2. Properties of the Alternative Labor Market Model
Before evaluating the model’s implications for boom-bust episodes, we display someof its basic steady state and dynamic properties. For the dynamic properties, we findthat money has bigger real eﬀects in the standard sticky wage model than it does in
10

Dixon and Kara (2006) report that the average age of a wage contract when there is staggering

as in Taylor is (N + 1)/2.

38

the alternative labor market model. The dynamic response to a technology shock issimilar across the two models.
Consider the steady state first. At the parameter values discussed in the previous
subsection, we find:CC +Iκ
(1 − ρ)2 L
2C +I= 0.63,Kbu= 19.51, b̄ ≡
1+σ L
C +I
ww̃ψ − γ ψ= 0.11,(4.19)(1+σL )λz

= 0.026, ww̃ = 3.0, w = 0.43, w0 = 3.27, w14 = 6.97, L = 0.93.

where absence of a time subscript indicates the steady state value of the variable.
The product, ww̃, is the steady state value of the real wage paid by employment
agencies to workers, scaled by zt , and λz is the steady state value of λt Pt zt , where
λt is the multiplier on the household’s budget constraint. The object, b̄, is the ratioof the flow value of unemployment to the flow value of employment for a worker.
This ratio has attracted much attention, particularly with Hagedorn and Minovskii(2006)’s demonstration that a higher value of b̄ increases the volatility of labor market
variables. Our setting for this ratio is quite small (it corresponds to the value estimated by GST). For example, Shimer (2005a) argues that an empirically plausible
value for b̄ lies in the neighborhood of 0.4. The consumption to GDP (≡ C + I) ratio
for the model is in line with standard estimates for the US, and the capital-output
ratio corresponds roughly to the familiar US (annualized) value of 3. Our ratio of
recruitment costs to GDP of 2.6 percent is somewhat larger than the value of 0.01
The steady state unemployment rate in the model, 1 − 0.93 = 0.07, corresponds tothe value used in GT. Our steady state unemployment rate is somewhat high if wecompare it with measured unemployment. However, the concept of unemployment
in the model is broader and includes all (working age) people not working.
The object, w, is the real wage (scaled by zt ) that makes a worker indiﬀerent
39between employment (i.e., matching with an agency and staying with it until exogenously separated) and unemployment. The object, wi , is the scaled real wagethat makes an employment agency that last renegotiated its wage i periods ago justindiﬀerent between matching with a worker and not matching (see Appendix B for
details). The latter object is a function of i because it is calculated under the assumption that when the next renegotiation takes place, the firm reverts to the scaled
steady state real wage, ww̃. Our dynamic calculations assume that the real wage paid£¤
to workers lies inside the interval, wt , wit , i = 0, ..., N −1, for all t. Note from (4.19)

that this is the case in steady state. This is also true for the dynamic simulations

reported in the paper. The low value of w reflects the relatively low value of b̄ usedin the model.11
Turning to the model’s dynamic properties, we consider the dynamic eﬀects of amonetary policy shock, ut (see (3.10) and (3.11)) and of a (conventional) technology
shock, ξ t (see (2.7)). Consider Figure 16 first. It displays dynamic responses to thesame monetary policy shock in each of the bimonthly version of our standard sticky
wage model and the alternative labor market model. The response of the equilibrium
interest rate appears in the 2,5 panel of the figure. This response is slightly diﬀerent
between the two models because, according to (3.11), the equilibrium interest rate
responds to output and expected inflation, as well as to a monetary policy shock.
Note that the real eﬀects of monetary policy are substantially larger than in thestandard sticky wage model. For example, the percent response of GDP is threetimes larger in the latter model than it is in the alternative labor market model.
Similarly, the response of inflation in the alternative labor market model is larger.
The figure allows us to trace the economics of the monetary transmission mechanism
11

We attempted to raise b̄ by increasing bu . However, when we did so we ran into numericaldiﬃculties computing the Ramsey equilibria for our model. We are working to fix this.

40

in the alternative labor market model. Thus, note how the real price of labor (thatis, Wt /Pt , the real wage paid for a unit of homogeneous labor by intermediate goodfirms to employment agencies) jumps in response to an expansionary monetary policy
shock. This jump reflects that the demand for labor rises as the demand for goods
rises in response to the expansionary monetary policy shock. Employment agencies
respond by posting higher vacancies and this leads, in subsequent periods, to an
expansion in the number of people employed (see ‘labor’). The posting of vacancies
is further encouraged by the fact that the average, across firm cohorts, of the real
wage paid to workers falls. This fall reflects the rise in inflation and the wage-setting
frictions. The intensity of work eﬀort rises in response to the rise in the price oflabor because intensity is set eﬃciently (see (4.3)). The product of intensity andlabor corresponds to hours worked. Note that the response of hours worked in thealternative labor supply model is smaller than the response in the sticky wage model.
Finally, a result that plays an important role in boom-bust dynamics is that the priceof capital rises in response to an expansionary monetary policy shock. The eﬀect isroughly the same in the two models.
We investigated the possibility that the weaker eﬀect of monetary policy shocks
in the alternative labor market model reflects our low value of b̄. To investigate this,we set bu = 2,12 which implies b̄ = 0.70 (a value suggested by Hall). The peakoutput eﬀect of a monetary policy shock rises from a little over 0.01 percent in the
low b̄ model (see Figure 16) to a little under 0.015 in the high b̄ economy. Although
the change in the value of bu has the expected qualitative eﬀect, it does essentiallynothing to close the quantitative gap in the output response of the alternative labormarket and standard sticky wage economies.
12

Given that we hold f constant, this implies κ = 20.92.

41

Now consider Figure 17, which displays the response of the two models to a standard technology shock, ξ t . The figure displays the response of the two models, whenmonetary policy is specified as in (3.10) and (3.11), as well as when it is specified
optimally (the computation of the Ramsey equilibrium is explained in Appendix C).
Consider the Ramsey equilibrium responses, which are quite similar, first. The responses of output, hours worked, and inflation are essentially the same. The initial
response of hours worked in the Ramsey equilibrium is negative, and turns positive
afterward. The composition of output diﬀers between the two models, with consumption rising relatively more strongly in the alternative labor market model. In bothmodels, the nominal rate of interest is reduced sharply in the wake of a technology
shock. The interest rate is reduced by more in the alternative labor market model,
presumably to compensate for the weaker monetary non-neutrality in that model,
which was documented in Figure 16. Note how the interest rate cuts are so largethat the zero lower bound on the nominal interest rate is violated. Strictly speaking,these are not equilibria. We report them nevertheless, as evidence on the propertiesof the models.
The responses of output and employment to a technology shock when monetary
policy is described by (3.10) and (3.11) are similar in the two models, though weaker
than in the Ramsey equilibria. The weaker responses presumably reflect that monetary policy is less expansionary in the wake of a technology shock than it is in the
Ramsey equilibria. Note how the interest rate falls less in the equilibrium than in the
Ramsey responses. As in the Ramsey equilibrium, consumption responds relatively
strongly in the alternative labor market model. A big diﬀerence in the two models
lies in the inflation rate, which falls more in the alternative labor market model.
The figures allow us to trace the economics of the transmission of a technology
shock in the alternative labor market model. Note how the price of labor falls sharply42

in the equilibrium of this model. This fall reflects the inadequacy of aggregate demand in the wake of a technology shock. As a result, there is a persistent decline
in the demand for labor by intermediate good firms. Employment agencies reactby reducing vacancies. At the same time, the real wage soars because of the wage
frictions and the drop in inflation. This provides an additional incentive for employment agencies to reduce vacancies. The result is a persistent decline in employment.
The intensity of work eﬀort falls because of the drop in the price of labor, and the
combined eﬀects result in a persistent drop in total hours worked. It is interestingthat the drop is similar across the two models, even though the rise in the real wage
in the standard sticky wage model is substantially smaller than what it is in thealternative labor market model.
4.2.3. Results for Boom-Bust
The results in Figure 18 show the dynamic response of our models to a ξ t shock that
occurs in period 1, followed by εt+p = −ξ t for p = 12. Thus, there is a signal thattechnology will improve two years in the future, a signal that in the end turns out tobe false. The figure displays the responses of the variables in our alternative labormarket and standard sticky wage models. In each case, we display the responses forthe version of the model in which policy is governed by (3.10) and (3.11), as well as
for the version in which policy is optimal. Note that the Ramsey equilibria for thetwo models are for the most part very similar (compare the circles and the stars).
Interestingly, in both models the Ramsey equilibria are characterized by a rise in thereal wage and a fall in the price of capital. Although the behavior of these variables
is qualitatively similar, there are clearly important quantitative diﬀerences. At thesame time, the ‘real wage’ in the two models clearly refers to at least slightly diﬀerent

43concepts and so comparisons of the real wage must be made with caution.
Now consider the equilibria in which policy is governed by (3.10) and (3.11).
Our earlier result that output overreacts to a signal shock in the standard sticky
wage model is reproduced in the bimonthly version of that model. The percent
increase in output, at its peak, is over three times the corresponding Ramsey increase.
The alternative labor market model also exhibits over reaction, though one that isquantitatively smaller. In the equilibrium, the percent increase in output at its peakis 0.85 percent, versus 0.47 percent in the Ramsey equilibrium. There is a similar
overreaction in hours worked. In both models, the real wage is low in the boom phase,
consistent with our diagnosis that a key problem is the failure of the equilibrium toreproduce the rise in the real wage that occurs in the Ramsey equilibrium. In thealternative labor market model, the fall in the real wage combines with a rise in theprice of capital to create an incentive to increase the hiring rate. This leads to arise in employment. In addition, the increased demand for goods leads to a rise in
the demand for labor, which produces a rise in the price of labor. This provides
additional incentive to increase the hiring rate and its leads to an increase in theintensity of work.
Recall that in the sticky wage model, there are two crucial reasons that output
over reacts to a signal about future technology: sticky wages and inflation targeting.
To investigate the role of inflation targeting, we redid the computations setting απ =1.01. As before, the overreaction does not occur with this change. In fact, output,
consumption and investment actually under react - these variables rise less than they
do in the Ramsey equilibrium. We also redid the calculations with flexible wages by
supposing that Nash bargaining occurs in each period in each agency. We foundthat responses are reduced, and more nearly resemble the responses in the Ramsey
equilibrium. For example, at its peak output rises by 0.50 percent, versus 0.47 in the44Ramsey equilibrium. Similarly, at their peaks investment, consumption and hours
worked rise 0.88, 0.34 and 0.47 percent, respectively. The corresponding Ramsey
figures are 0.98, 0.23 and 0.45 percent.13
In sum, the overreaction to an anticipated technology shock that is so pronounced
in our sticky wage model is also a feature of the alternative labor market model.
However, the quantitative eﬀects are smaller. As discussed above, our model ofboom-bust is driven by two things: a technology shock that is never realized anda monetary policy response. The alternative labor market model exhibits a similar
dynamic response to technology shocks as our standard sticky wage model. However,
the real eﬀects of monetary policy shocks are weaker in the alternative labor market
model, and this is the reason the boom-bust in the alternative labor market model isquantitatively weaker. This suggests that factors which enhance the nonneutrality ofmoney in the alternative labor market model would result in a quantitatively larger
boom-bust cycle.

5. Full Monetary Model
We consider the full monetary model of Christiano, Motto and Rostagno(2006).
That model introduces into the simple monetary model of section 3 a banking sector following Chari, Christiano, and Eichenbaum (1995) and the financial frictionsin Bernanke, Gertler and Gilchrist (1999). In the model, there are two financing
requirements: (i) intermediate good firms require funding in order to pay wages andcapital rental costs and (ii) the capital is owned and rented out by entrepreneurs,
who do not have enough wealth (‘net worth’) on their own to acquire the capital
13

The peaks in output, consumption and hours worked occur in period 12, while it occurs inperiod 11 for investment.

45

stock and so they must borrow. The working capital lending in (i) is financed bydemand deposits issued by banks to households. The lending in (ii) is financed bysavings deposits and time deposits issued to households. Demand deposits and savings deposits pay interest, but relatively little, because they generate utility services
to households. Following Chari, Christiano, and Eichenbaum (1995), the provisionof demand and savings deposits by banks requires that they use capital and labor
resources, as well as reserves. Time deposits do not generate utility services. In addition to holding demand, savings and time deposits, households also hold currency
because they generate utility services. Our measure of M1 in the model is the sumof currency plus demand deposits. Our broader measure of money (we could callit M2 or M3) is the sum of M1 and savings deposits. Total credit is the sum of
the lending done in (i) and (ii). Credit diﬀers from money in that it includes time
deposits and does not include currency. Finally, the monetary base in the model isthe sum of currency plus bank reserves. A key feature of the model that allows it tomake contact with the literature on boom-busts is that it includes various monetary
aggregates as well as credit, and that their are nontrivial diﬀerences between theseaggregates.
Figure 19 displays the response of the full model to the signal shock, and includes
the response of the simple model for ease of comparison. Note that the full model
displays a weaker response, though qualitatively the results are similar. Figure 20
displays the response of various financial variables to the signal shock. The top left
figure shows that the monetary base grows throughout the boom, and then falls in the
period it is realized that the shock will not happen. M1 growth is strong throughout
the expansion, while M3 growth falls consistently throughout the boom. Thus, thebehavior of the monetary aggregates is inconsistent. However, credit growth is strong
throughout the boom and remains high during the bust. That credit growth is con46

sistently strong throughout the boom-bust episode while monetary aggregates send
conflicting signals reflects portfolio shifts among the monetary aggregates. Moreover,as noted above, not all the expansion in credit shows up in the monetary aggregates.
Figure 21 shows what happens when we include credit growth in the policy rule,(3.11), with a coeﬃcient of 3. The strong growth of credit leads to a tightening ofmonetary policy during the boom, and causes the boom to more nearly resemble theeﬃcient response of the economy to the signal about future productivity.
In sum, we have simulated the response to a signal shock of a model with considerably more frictions than those in our simple monetary model. The basic qualitative
findings of our simple model are robust to this added complexity.

6. Conclusions
Wage setting frictions deserve to be taken seriously because they are a feature ofmodels that fit the data well. It is known that with sticky wages, a policy of targeting
inflation can lead to suboptimal results by making the real wage overly rigid. For
example, if there is a negative oil shock which requires that the real wage fall, a
policy of preventing the price level from rising will cause the real wage to be toohigh and employment too low. In this paper we have shown that the type of boombust episodes we have observed may be another example of the type of suboptimal
outcome that can occur in an economy with sticky wages and monetary targeting.
Indeed, we argued that within the current class of DSGE models that have been fit
to data, it is hard to find another way to account for boom-bust episodes.
We have not addressed the factors that make inflation targeting attractive. Theseare principally the advantages that inflation targeting has for anchoring inflation
expectations. We are sympathetic to the view that these advantages argue in favor47

of some sort of inflation-targeting in monetary policy. However, our results suggestthat it may be optimal to mitigate some of the negative consequences of inflation
targeting by also taking into account other variables. In particular, we have examined
the possibility that a policy which reacts to credit growth in addition to inflation may
reduce the likelihood that monetary policy inadvertently contributes to boom-bust
episodes.
Our argument remains tentative, even within the confines of the monetary economies
that we studied in this paper. We only showed that a monetary policy which feeds
back onto credit growth moves the response of the economy to a certain signal shock
closer to the optimal response. Such a policy would substantially reduce the violence of boom-bust episodes. However, a final assessment of the value of integrating
credit growth into monetary policy awaits an examination of how the response of
the economy to other shocks is aﬀected. This work is currently under way. The
model of Christiano, Motto and Rostagno (2006) is ideally suited for this. Not only
does it contain a variety of credit measures, so that a serious consideration of theconsequences of integrating credit into monetary policy is possible. In addition, thatmodel is estimated with many shocks, making it possible to see how integrating credit
growth into monetary policy aﬀects the transmission of shocks.

48

A. Appendix A: The Equations Characterizing Equilibriumin the Simple Monetary Model
Our simple monetary model is quite standard and the derivation of the equilibrium
conditions can be found in several places (see, especially, Yun (1996) and Erceg,
Henderson, and Levin (2000)). For the sake of completeness, we list the equationsof the model here. To define the seven equations that characterize price and wage
optimization, we must define several auxiliary variables, p∗t , wt∗ , Fp,t , Fw,t , wt∗ . The
three equations associated with optimal price setting are:

⎡

⎛

³ ι2

π t−1

π̄ 1−ι2⎢¡¢ ⎜ 1 − ξpπt1−
ξp∗t − ⎢p ⎝⎣1 − ξp
(

⎞λf
1
´ 1−λ
f

⎟
⎠

f
⎤ 1−λ
λ

λf
µ ι2 1−ι2¶ 1−λf ⎥π t−1 π̄
⎥
+ ξp
p∗t−1
⎦
πt

f

= 0,
(A.1)

)
1
¶α ³
¸ µ ι2 1−ι2 ¶ 1−λ
´1−α
λw
f
π̄
π
t
(wt∗ ) λw −1 ht−φ +βξ p Fp,t+1 − Fp,t = 0,Et λz,t (p∗t )
t
π t+1
(A.2)
∙ µ
¶α ³
¸
λf
´1−α
λ
kt−1
∗ λf −1∗ λww− φ st + (A.3)(wt ) −1 ht
λz,t λf (pt )
t
µz
⎡
⎤1−λf
⎡
⎤1−λf
1
1
³ ι2 1−ι ´ 1−λ³ ι2 1−ι2 ´ 1−λλf
π t−1 π̄2
π t π̄
ffµ ι2 1−ι2 ¶ 1−λ
1
−
ξ
1
−
ξ
p
p
f ⎢
π t+1
πt
π t π̄
⎥⎢⎥Fp,t+1 − Fp,t ⎣
= 0.
βξ p
⎣
⎦
⎦
π t+1
1 − ξp1 − ξp
λf
λf −1

∙ µ

kt−1
µz

Let π w,t denote the rate of wage inflation:
π w,t =w̃t µz π t
,
w̃t−1

49

where w̃t is the real wage, scaled by zt . The equations that relate to wage optimizationare:λ

Et {λz,t

+βξ w

µ

w
(wt∗ ) λw −1 htλw

π̃ w,t+1w̃tw̃t+1 π t+111−λw+ βξ w π̃w,t+1

λw
¶ 1−λ
(1+σL )w³

w̃tw̃t+1 π t+1

λw
´ 1−λ

w

π t+1

⎡

³

⎡

³

1 ⎢ 1 − ξw⎣
ψLπ̃ w,t+1w̃
w̃t+1 πt+1 t

1 − ξw

π̃ w,tw̃
w̃t π t t−1

1 ⎢ 1 − ξw−⎣
ψL
1 − ξw

and

⎡

⎛

⎢

⎜
wt∗ = ⎢⎣(1 − ξ w ) ⎝1 − ξw³

π̃ w,tw̃
w̃t π t t−1

1 − ξw

Fw,t+1 − Fw,t } = 0´ 1−λ1 ⎞λw

w

i1+σL
h
λw
(A.5)
Et { (wt∗ ) λw −1 ht⎤1−λw (1+σL )⎥⎦w̃t+1 Fw,t+1´ 1−λ1 ⎤1−λw (1+σL )w⎟⎠´ 1−λ1

+ ξw

w

⎥
⎦

µ

(A.4)

π̃ w,t∗w̃t−1 wt−1
w̃t π tw̃t Fw,t } = 0,

λw
¶ 1−λ

w

(A.6)
w
⎤ 1−λ
λw
⎥
⎥
⎦

.

To characterize the cross-sectional average utility of households, we require an additional variable related to wage dispersion, wt+ :⎡⎛⎢⎜ 1 − ξw(1−
ξ)
wt+ = ⎢
w ⎝
⎣

³

π̃w,t
w̃
w̃t π t t−1

1 − ξw´ 1−λ1 ⎞λw (1+σL )
w

⎟
⎠

50

+ ξw

µ

π̃ w,t+w̃t−1 wt−1
w̃t πt¶λw (1+σ L )1−λww⎤ λ 1−λ(1+σ )
w

⎥
⎥
⎦

(A.7)

L

.

The fact that real marginal cost, st , is the real wage divided by the marginal productivity of labor is represented as follows:Ã!αλwµz (wt∗ ) λw −1 ht
w̃t
st =
,
(1 − α) t
kt−1

(A.8)

where ht denotes the unweighted average of household employment, and the adjustλw

ment, (wt∗ ) λw −1 ht , is required to convert household average work, ht , into eﬀective
average labor eﬀort. Also, kt denotes the capital stock, scaled by zt−1 . A similar
adjustment is required to define an aggregate resource constraint:
½ µ
¶α h
¾
λf
i1−α
λ
kt−1
∗ λf −1∗ λww(wt ) −1 ht−φ .
ct + It = (pt )
t
µz

(A.9)

The capital accumulation equation is:kt − (1 − δ)µ−1
z kt−1 =¸∙
S 00 µ2z It
21−− 1) It ,(2 It−1(A.10)

where we interpret It as investment scaled by zt .
The equation that defines the nominal rate of interest is:Et {β1
π t+1 µz

λz,t+1 (1 + Rt ) − λz,t } = 0,

and the marginal utility of consumption, λz,t , is:
∙
¸
µz
1
= 0.
+ bβ
Et λz,t −ct µz − bct−1µz ct+1 − bct(A.11)(A.12)

Here, λz,t has been scaled by multiplying by zt . The period t first order condition
associated with the choice of kt+1 is:
⎤
⎡
⎛ ¡
⎞1−α
¢ λλw−1
∗
w
ht+1
µ w
1 ⎣
⎠
st+1 + qt+1 (1 − δ)⎦ = 0.
−λz,t + Et λz,t+1 β
α t+1 ⎝ z t+1
µz qt
kt

(A.13)

51

The first order necessary condition associated with the optimal choice of It is:∙¸
S 00 µ2z ItIt
2
00 2 It(A.14)− 1) − S µz (− 1)(
Et {λzt qt 1 −2 It−1
It−1
It−1
¶2
µ
It+1
00 2 It+1−λzt + βλzt+1 qt+1 S µz (− 1)
} = 0.
ItIt
The cross-sectional average of household period utility is:L)µ ∗ ¶ λwλ(1+σ
w −1
¡
¢
w
ψ
L
t
L
u ct , ct−1 , wt∗ , wt+ , ht = log(ct µz − bct−1 ) −h1+σ.
t
1 + σ L wt+(A.15)

The appearance of wt∗ /wt+ reflects that household labor eﬀort varies across the population with the variation in individual wages. When σ L > 0 this matters for averageutility, and wt∗ /wt+ provides the necessary adjustment. When σ L = 0 then utility islinear in hours and no adjustment is required. Accordingly, in this case (A.6) and(A.7) coincide, so that wt∗ /wt+ = 1.
The 15 endogenous variables to be determined are:π t , ct , ht , kt+1 , p∗t , Fp,t , st , Rt , wt+ , wt∗ , Fw,t , w̃t , λz,t , It , qt .
We have listed 14 equations above, (A.1)-(A.14). When the equations are augmented
by a monetary policy rule, then we can solve for our 15 endogenous variables. This
corresponds to what we call our simple monetary model. Alternatively, the 14 equations can be augmented by the first order conditions associated with the Lagrangian
representation of the Ramsey problem. The paper discusses the dynamic properties of the resulting Ramsey equilibrium. Technical notes related to this appear in
Appendix C.

B. Appendix B: Equilibrium with Alternative Labor Market
This section displays the equilibrium conditions used to solve the version of the modelwith Gertler-Trigari labor markets. It then provides formulas for the steady state.
52

B.1. Equilibrium Conditions Needed to Solve the Model
To define J in (4.8), it is convenient to first denote the growth rate of employment
from t to t + j for an agency that renegotiates in period t :Ωt,j =
Then,

l=0
⎪
⎪
⎩

¢

t+l + ρ

j>0

1

j=0

.

)
h
i
¡
¢
λt+j
κ
2
max
(Wt+j − Γt,j ω t ) ψt+j − Pt+j zt+j xjt+jΩt,j(B.1)
βj
J (ω t ) = Et
j
λt
2
xt+j
j=0
´
λt+N ³N+β Et
J W̃t+N Ωt,N .
λt
N
−1
X

(

⎧ j−1
Y¡
⎪
⎪
⎨
xl

After some algebra, it can be verified that optimality associated with xit implies:
´
³
¢2
κ¡
i+1
Wt+1 − Γt−i,i+1 W̃t−i ψt+1 +Pt+1 zt+1 xi+1
t+1 +ρPt+1 zt+1 κxt+1 ],
2
(B.2)

λt+1
Pt zt κxit = βEt[λtfor i = 0, 1, ...N − 2. Similarly, optimality for the hiring decision of a firm that is in
the last period before the current contract expires implies:
Pt zt κxN−1
= βEt
t

´
¢2
λt+1 ³κ¡[ Wt+1 − W̃t+1 ψt+1 + Pt+1 zt+1 x0t+1 + ρPt+1 zt+1 κx0t+1 ].
λt
2
(B.3)

Note the familiar result here, that the value of the increase in period t+1 employment(i.e., the expression on the right side of the equality in (B.2) and (B.3)) resulting
from an increase in period t hiring is captured by a function of the hiring decision int + 1.
The derivative of J with respect to ωt , evaluated at W̃t , is, after taking into
account the envelope conditions associated with xit+i ,53

Jw,t = −{ψt +N−1X

β j Etj=1

λt+jΩt,j Γt,j ψt+j },
λt

(B.4)

where Jw,t denotes the derivative of J (ωt ) evaluated at ω t = W̃t .
It is easily verified that the marginal value to the worker of the wage is:Vw,t = ψt +N−1X(βρ)j Et Γt,j

j=1

λt+j
ψ ,
λt t+j(B.5)

where Vw,t denotes the derivative of V 0 (ω t ) in (4.10) with respect to ω t , evaluated atω t = W̃t . The diﬀerence in the way the agency and the worker discount future wages
reflects their asymmetric positions. Both take into account the way the wage contract
evolves over time, with Γt,j . However, the worker includes ρ in the discounting, totake into account the fact that it could separate. The firm is bound to the contract
for all periods of the contract, and so ρ does not enter its discounting. However,the number of workers covered by the contract will change over time as new workers
arrive and old workers separate. This factor is accounted for by the presence of Ωt,j .
The first order necessary condition associated with the Nash bargaining problem,(4.17) is:£¤ηVw,t Jt = (1 − η) Vt0 − Ut Jw,t ,(B.6)

where Jt and Vt0 denote J (ωt ) and V 0 (ω t ) evaluated at ω t = W̃t .
Before the preceding equations can be used to solve the model, they must be
scaled. We adopt the following scaling of variables:
Jt
W̃t
Wt
, λzt t+1 = λt+1 zt+1 Pt+1 , wt =, w̃t =
,
zt Pt
Wt
zt Pt
Vti
Ut
Vx
x
=
, Uz,t =, Vz,t
= t .
Pt zt
Pt zt
Pt zt

Jz,t =iVz,t54

The following relations, which make use of (4.5), are useful:⎧⎨ π̃w,t+j ···π̃w,t+1 j > 0
Γt,j W̃t
π t+j ···π t+1= Gt,j wt w̃t , Gt,j =⎩Pt+j zt+j1
j=0Γt−i,i+1W̃t−i = Gt−i,i+1 wt−i w̃t−i
Pt+1 zt+1
Γt−i,i W̃t−i= Gt−i,i wt−i w̃t−i .
Pt ztDivide (B.1) by Pt zt :Jz,t = Et

N−1
X

βj

j=0

+β N Etλz,t+j hκ ¡ j ¢2 ix(w̃t+j − Gt,j wt w̃t ) ψt+j −
Ωt,j
λz,t
2 t+j

(B.7)

λz,t+NJz,t+N Ωt,N .
λz,tEquations (B.2) and (B.3) can be written
κxit = βEtκ ¡ i+1 ¢2
λz,t+1x[(w̃t+1 − Gt−i,i+1 wt−i w̃t−i ) ψt+1 ++ ρκxi+1
t+1 ],
λz,t
2 t+1

(B.8)

for i = 0, 1, ...N − 2, and
κxN−1
= βEt
t

λz,t+1
κ ¡ 0 ¢2x[(w̃t+1 − wt+1 w̃t+1 ) ψt+1 ++ ρκx0t+1 ].
λz,t
2 t+1

(B.9)

The conditions for Vti are:i= ψt Gt−i,i wt−i w̃t−i − γVz,tLλz,t+1
ψ1+σ
t
i+1
+ βEt
[ρVz,t+1+ (1 − ρ) Uz,t+1 ], (B.10)(1 + σ L ) λz,tλz,tfor i = 0, ..., N − 1. Here, the weight in front of wt−i w̃t−i is understood to be unity,
when i = 0. The condition for Jw,t , in terms of scaled variables, is:Jw,t = −{ψt +N−1X

β j Etj=155

λz,t+jGt,j Ωt,j ψt+j }.
λz,t(B.11)

The condition for Ut becomes:
Uz,t = bu + βEt

λz,t+1x[ft Vz,t+1+ (1 − ft ) Uz,t+1 ].
λz,t(B.12)

The condition for Vtx becomes:
x
=
Vz,t

N−1
X

j
xjt−1 lt−1
j+1
Vz,t
.
m
t−1
j=0

(B.13)

The condition for eﬃcient intensity becomes:w̃t = γ

ψσt L
.
λz,t

(B.14)

The following six conditions are unaﬀected, but repeated here for convenience:
mt =

N−1
X

xjt ltj ,(B.15)

j=0

mt = σ m (1 − Lt )σ vt1−σ ,N−1X(B.16)

ltj ,(B.17)

ltj ,(B.18)

mt,
1 − Lt(B.19)

ht = ψt

j=0

Lt =

N
−1
X
j=0

ft =
and

¡
¢ i
lti+1 = xit−1 + ρ lt−1,

(B.20)

for i = 0, ..., N − 1. The expression for Vw,t becomes:
Vw,t = ψt +N−1X(βρ)j Et

j=156

λz,t+jGt,j ψt+j .
λz,t(B.21)

Finally, the Nash Bargaining condition becomes:£ 0¤ηVw,t Jz,t + (1 − η) Vz,t− Uz,t Jw,t = 0.

(B.22)The previous 12 + 3N equations contain the following 15 + 3N endogenous variables:ix, xit , Vz,t, mt , vt , wt , w̃t , Uzt , lti , ft , λz,t , ht , π t , ψ t , Lt .
Jz,t , Jw,t , Vw,t , Vz,t

(B.23)

Regarding the equilibrium conditions associated with the rest of the model, we
drop the equations in Appendix A which pertain to wage setting, i.e., (A.4), (A.5),(A.6), (A.7). The 8 variables in that section, that are not listed in (B.23) are:p∗t , Fp,t , kt , st , ct , It , Rt , qt .
The 11 equations that remain are (A.1), (A.2), (A.3), (A.8), (A.10), (A.11), (A.12),(A.13), (A.14), (3.11) with wt∗ ≡ 1 andλfκ X ¡ j ¢2 j
ct + It + (1 − Lt ) b +xt lt = (p∗t ) λf −1
2 j=0

N−1

u

½ µ
t

kt−1
µz

¶α

¾

h1−α
−φ
t

.

(B.24)

Thus, we have 23 + 3N equations in 23 + 3N unknowns.
B.2. Equilibrium Conditions on the Bargaining Set
In the previous section we assumed that agencies and individual workers only separate
when it happens for exogenous reasons. Implicitly, we assumed that the scaled wage,
wti ≡

Wti
,
zt Pt

paid by an employment agency which has renegotiated most recently i periods in the£¤past is always inside the bargaining set, wit , wit , i = 0, 1, ..., N − 1. Here, wit has theproperty that if wti > wit then the agency prefers not to employ the worker and wit
57

has the property that if wti < wit then the worker prefers to be unemployed. We now
describe our strategy for computing wit and wit .
The lower bound, wit , sets the surplus of a worker in an agency in cohort i tozero. By (B.10):
Uz,t = ψt wit − γLλz,t+1
ψ1+σ
t
i+1
+ βEt
[ρVz,t+1+ (1 − ρ) Uz,t+1 ],(1 + σ L ) λz,tλz,tfor i = 0, ..., N − 1. In steady state, this is
w = Uz + γ

ψ1+σL− β[ρVz + (1 − ρ) Uz ],(1 + σ L ) λz

where a variable without time subscript denotes its steady state value. We now
consider the upper bound, wit , which sets the surplus of an agency in cohort i tozero, i = 0, ..., N − 1. From (B.7)
0 = Et

N−1−i
X

β

j=0

+β N−i Et

j λz,t+j

λz,t

h¡
¢
κ ¡ j ¢2 iixw̃t+j − Gt,j wt ψt+j −
Ωt,j
2 t+j

λz,t+N−i
Jz,t+N−i Ωt,N −i ,λz,tfor i = 0, ...., N − 1. In steady state:iκ
1 − β N−i h2iw̃ − w − (1 − ρ) + β N−i Jz ,
0=
1−β
2

or,

i
h
1−β
κ
wi = w̃ − (1 − ρ)2 −Jz ,2
1 − β i−N

which is a monotonically increasing function of i.

For the dynamic economy, the additional unknowns are the 2N variables composed of wit and wit for i = 0, 1, ..., N − 1. We have an equal number of equations tosolve for them.
58

B.3. Steady State
Our strategy for computing the steady state replaces the job finding rate, f, as an
unknown, by κ. In addition, we impose that the steady state value of intensity isψ = 1, and we solve for the value of γ that is consistent with this restriction.
B.3.1. Labor market equations
In steady state, lti = llj for all i, j, t, l, so thatx = 1 − ρ, Ωt,j = 1.
Expression (B.7) reduces to:
Jz =

i
1 h
κ
w̃ − ww̃ − (1 − ρ)2 .
1−β2(B.25)

The first order conditions, (B.8) and (B.9) are all identical in steady state, and given
the value of x obtained above (as well as ψ = 1), they reduce to:κ (1 − ρ) = β[w̃ − ww̃ +κ
(1 − ρ)2 + ρκ (1 − ρ)],2or,w = 1 + κ (1 − ρ)

1
(1 − ρ) + (βρ − 1) β1
2w̃

(B.26)

which gives w as a function of w̃.
The derivative of the marginal value of a worker is, in steady state:

Jw = −
Equation (B.10) becomes:
Vz = ww̃ − γ

1 − βN.
1−β

1+ β[ρVz + (1 − ρ) Uz ],(1 + σ L ) λz
59

(B.27)

where the i superscript on Vz has been dropped because all the equations in (B.10)
are the same, and so they have the same solution. Equation (B.13) implies:
Vzx = Vz .

(B.28)

Uz = bu + β[f Vzx + (1 − f ) Uz ].

(B.29)

L = h = Nl,(B.30)m = (1 − ρ) h.

(B.31)

Equation (B.12) becomesEquation (B.17) implies

so that equation (B.15) reduces to:Also,m = σ m (1 − L)σ v 1−σ .
Also,
f=
Also,
Vw =

m
.
1−L

(B.32)

1 − (βρ)N.
1 − βρ

The first order condition associated with the Nash problem is:
ηVw Jz + (1 − η) [Vz − Uz ] Jw = 0,
B.4. Non-labor market equations
In steady state,p∗t = 1.
60

(B.33)

Also,
λz
Fp =
and
λz λf
Fp =
so that

h³ ´α
k
µz

h³ ´α
k
µz

i
h1−α − φ1 − βξ p1−αh

1 − βξ p

i
−φ s
s=

λz
=

,

h³ ´α
k
µz

1−α

h

1 − βξ p

i
−φ

1
.
λf

and
π̃ w = π̄.
The marginal cost equation reduces to:
1
w̃
=
λf
1−α

µ

µz h
k

¶α

.

The resource constraint in steady state is:¶αkh1−α − φ
µz
µ ¶α
1
k
=
h1−α ,
λf µz

κc + I + (1 − L) b + (1 − ρ)2 L =2uµ(B.34)

by the condition that profits are zero in steady state. That is profits in steady state
are, before scaling:
Pt Yt − Pt st (Yt + φzt ) .
Dividing by Pt zt , we obtain in steady state:(λf − 1) Yz = φ,where Yz is c + I (i.e., it is gross output scaled by zt ). From the capital accumulation
equation:£¤k 1 − (1 − δ)µ−1
= I.
z
61

The nominal interest rate is, in steady state:
1+R =

πµz
.
β

Also,
λz =

1 µz − bβ.
c µz − b

The investment intertemporal equation gives:
µ
¶1−α
µz
1
µz h
+ 1 − δ.
=αβkλf(B.35)

The adjustment for the investment equation is:
q = 1.
B.5. Steady State Solution
As noted above, we treat f as known and we set ψ = 1 (so that h = L). These two
restrictions allows us to solve for γ and κ. The value of κ is found by a non-linear,
one-dimensional search. Fix an arbitrary value of κ. First compute Jw , Vw , R and x
using the trivial expressions above. Use (B.35) to compute the labor-capital ratio,
h/k :
h
1
=
k
µz

½

1
∙
¸¾ 1−αλf µz− (1 − δ).
α βThen, from the law of motion for investment:
1−δ
I
=1−
.
k
µz
Solve for w̃ using
w̃ =
λf

1−α
³ ´α .

62µz hkNow, fix an arbitrary value of γ. From the eﬃciency condition on intensity and w̃,compute λz :
w̃ =

γ
.
λz

Given λz andλz =1 µz − bβ,c µz − bcompute c. Adjust γ until the resource constraint:
¸
¶
µ
∙ µ
¶α
c
k
κ
1−δ
1
1−h u
2 h
,
− (1 − ρ)=−b − 1−k
λf µz h2kkµz
is satisfied.
Use (B.26) compute w and use (B.25) to compute Jz . Then use (B.27), (B.28)
and (B.29) to solve for Vzx , Vz , and Uz . After using (B.28) substitute out for Vzx ,these equations are:
Vz =
Uz =

ww̃ − γ (1+σ1L )λz + β (1 − ρ) Uz
1 − βρ

,ww̃ − γ (1+σ1L )λz + β (1 − ρ) Uz
bu
βf
+
,
1 − β (1 − f ) 1 − β (1 − f )1 − βρ

so that
Uz =

bu +


βf ww̃−γ(1+σL )λz1−βρ

1 − β (1 − f ) − βf β(1−ρ)
1−βρ.

(B.36)

Equation (B.36) can be used to solve for Uz and then (??) is solved or Vz . Combining(B.31) and (B.32) we obtain the steady state level of employment:
L=h=

f
.
f +1−ρ

Then, (B.31) and (B.30) are solved for m and l. Given h, Fp , k, c, λz and I can becomputed. For Fp , the steady state zero-profit condition must be applied. Adjust κ
until (B.33) is satisfied.
63

C. Appendix C: Ramsey-Optimal Policy
We find the Ramsey-optimal allocations for our economy using the computer codeand strategy used in Levin, Lopez-Salido, (2004) and Levin, Onatski, Williams andWilliams (2005). For completeness, we briefly describe this strategy below. Let xt
denote a set of N endogenous variables in a dynamic economic model. (According
to Appendix A, in our simple monetary model, N = 15.) Let the private sector
equilibrium conditions be represented by the following N − 1 conditions:
X µ (st+1 ) ¡ ¡ ¢ ¡
¢
¢
t
t+1
f
x
s
,
x
s
,
s
= 0,
,
s
t
t+1
µ (st )
t+1 t
s

(C.1)

|s

for all t and all st . Here, st denotes a history:

st = (s0 , s1 , ..., st ) ,and st denotes the time t realization of uncertainty, which can take on n possible
values:st ∈ {s (1) , ..., s (n)}¡ t¢= prob[st ],µ s

so that µ (st+1 ) /µ (st ) is the probability of history st+1 , conditional on st .
Suppose preferences over x (st ) are follows:
∞
X
t=0

βt

X ¡ ¢ ¡ ¡ ¢ ¢µ st U x st , st .

(C.2)

st

In our simple monetary model, U is given by (A.15). The Ramsey problem is tomaximize preferences by choice of x (st ) for each st , subject to (C.1). We express theRamsey problem in Lagrangian form as follows:
⎧
⎫
⎪
⎪
∞
⎨
⎬
X X ¡ ¢X µ (st+1 ) ¡ ¡ ¢ ¡
¡
¡
¢
¢
¡
¢
¢
¢
t
t
t
t
t
t+1
U x s , st + λ s,
max
f x s ,x sβ
µ s
, st , st+1⎪
| {z } t+1 t µ (st ) |
{z
}⎪
⎩
⎭
t=0
st
s
|s
1×N−1

64

N−1×1

where λ (st ) is the row vector of multipliers on the equilibrium conditions. Consider
a particular history, st = (st−1 , st ) , with t > 0. The first order necessary condition
for optimality of x (st ) is¡ ¢ X µ (st+1 ) ¡ ¡ t ¢ ¡ t+1 ¢¡ ¡ ¢ ¢¢
U1 x st , st + λ st
f1 x s , x s(C.3), st , st+1t|{z
} | {z } t+1 t µ (s ) |
{z
}
s
|s
1×N
1×N −1
N−1×N¡ t−1 ¢ ¡ ¡ t−1 ¢ ¡ t ¢¢−1
f2 x s0, x s , st−1 , st = |{z}+β λ s| {z }|
{z
}
N−1×N

1×N−1

1×N

after dividing by µ (st ) β t . In less notationally-intensive notation,

U1 (xt , st ) + λt Et f1 (xt , xt+1 , st , st+1 ) + β −1 λt−1 f2 (xt−1 , xt , st−1 , st ) = 0.
The first order necessary condition for optimality at t = 0 is (C.3) with λ−1 ≡ 0. The
time-consistency problem occurs when the multipliers associated with the Ramsey
problem are non-zero after date 0. Following the Ramsey equilibrium in such a futuret requires respecting λt−1 . However, if λt−1 6= 0, then utility can be increased at t byrestarting the Ramsey problem with λt−1 = 0. When we study Ramsey allocations,
we assume there is a commitment technology that prevents the authorities fromacting on this incentive to deviate from the Ramsey plan.
The equations that characterize the Ramsey equilibrium are the N − 1 equations,(C.1), and the N equations (C.3). The unknowns are the N elements of x and
the N − 1 multipliers, λ. The equations, (C.3) are computed symbolically using thesoftware prepared for Levin, Lopez-Salido, (2004) and Levin, Onatski, Williams andWilliams (2005). The resulting system of equations is then solved by perturbation
around steady state using the software package, Dynare.
To apply the perturbation method, we require the nonstochastic steady statevalue of x. We compute this in two steps. First, fix one of the elements of x, say the
inflation rate, π. We then solve for the remaining N −1 elements of x by imposing the65N − 1 equations, (C.1). In the next step we compute the N − 1 vector of multipliers
using the steady state version of (C.3):£¤
U1 + λ f1 + β −1 f2 = 0,

where a function without an explicit argument is understood to mean it is evaluatedin steady state.14 Write
= U10
£
¤0
X = f1 + β −1 f2Y

β = λ0 ,so that Y is an N × 1 column vector, X is an N × (N − 1) matrix and β is an(N − 1) × 1 column vector. Compute β and u as−1

β = (X 0 X)

X 0Y

u = Y − Xβ.
Note that this regression will not in general fit perfectly, because there are N − 1‘explanatory variables’ and N elements of Y to ‘explain’. We vary the value of π
until max |ui | = 0. This completes the discussion of the calculation of the steadystate and of the algorithm for computing Ramsey allocations.
14

This step is potentially very cumbersome, but has been made relatively easy by the software

produced for Levin, Lopez-Salido, (2004) and Levin, Onatski, Williams and Williams (2005). Thissofware endogenously writes the code necessary to solve for the multipliers.

66

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[18] Gali, Jordi, Mark Gertler and David Lopes-Salido, forthcoming, ‘Markups, Gaps
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[19] Gertler, Mark, Luca Sala and Antonella Trigari, 2006, ‘An Estimated Monetary
DSGE Model with Unemployment and Staggered Nominal Wage Bargaining,’
manuscript, New York University.
[20] Gertler, Mark and Antonella Trigari, 2006, ‘Unemployment Fluctuations with
Staggered Nash Bargaining,’ manuscript, New York University.
[21] Goodfriend, Marvin and R. G. King, 1997, ‘The New Neoclassical Synthesis andthe Role of Monetary Policy,’ NBER MacroeconomicAnnual 1997, Cambridge,
MA:MIT Press, 231-283.
[22] Hagedorn, Marcus and Iourii Manovskii, 2006, ‘The Cyclical Behavior of Equilibrium Unemployment and Vacancies Revisited,’ mimeo, University of Pennsylvania.
[23] Hall, Robert, 2006, ‘The Labor Market and Macro Volatility: A Nonstationary
General Equilibrium Analysis,’ mimeo, Stanford University.
[24] Hall, Robert, 2005a, ‘Employment Fluctuations with Equilibrium Wage Stickiness„ American Economic Review, 95(1), 50-65.
[25] Hall, Robert 2005b, ‘Employment Eﬃciency and Sticky Wages: Evidence from
Flows in the Labor Market,’ Review of Economics and Statistics 87(3), 397-407.
[26] Hall, Robert 2005c, ‘Job Loss, Job Finding, and Unemployment in the U.S.
Economy over the Past Fifty Years,’ NBER Macroeconomics Annual, M.Gertler
and K. Rogoﬀ editors, MIT Press, 101-137.
69

[27] Jaimovich, Nir, and Sergio Rebelo, 2006, ‘Do News Shocks Drive the Business
Cycle?,’ Northwestern University.
[28] Merz, Monika, 1995, ‘Search in the Labor Market and the Real Business Cycle,’
Journal of Monetary Economics 36, 269-300.
[29] Mortensen, Dale and Christopher Pissarides, 1994, ‘Job Creation and Job Destruction in the Theory of Unemployment,’ Review of Economic Studies 61,
397-415.
[30] Shimer, Robert 2005a, ‘The Cyclical Behavior of Equilibrium Unemployment,
Vacancies, and Wages: Evidence and Theory,’ American Economic Review
95(1), 25-49.
[31] Shimer, Robert 2005b, ‘Reassessing the Ins and Outs of Unemployment,’ mimeo,
University of Chicago.
[32] Yun, Tack, 1996, ‘Nominal Price Rigidity, Money Supply Endogeneity, and Business Cycles,’ Journal of Monetary Economics, 37(2): 345 - 370.

70

Figure 1: S&P500 Divided by CPI (Shiller)
1600

1400

1200

1000

800

600

400

200

1880

1900

1920

1940

1960

1980

2000

Fig 2A: Standardized inflation and stock price in interwar periodFig 2B: Standardized inflation and stock price in 1950's to 1970's period

2.5
3

2
1.5

2

1
1

0.5
0

0

-0.5
-1

-1

-1.5
-2

1924

1926

1928

1930

1932

1955

Fig 2C: Standardized inflation and stock price in interwar period
4Inflation

3

S&P500
2

1

0

-1
1980

1985

1990

1995

2000

2005

1960

1965

1970

1975

Figure 3: Real Business Cycle Model with Habit and CEE Investment Adjustment Costs
Baseline - Tech Shock Not Realized, Perturbation - Tech Shock Realized in Period 5
OutputInvestment

Consumption

0.9

0.7

0.6

0.8

0.6

0.5
0.7

0.4

0.4

0.6

percent

percent

percent

0.5
0.5

0.3

0.4

0.2

0.3

0.1

0.2

0.3
0.2
0.1
0

2

4

6

8

2

4

6

8

2

4

t

t

t

Hours Worked

Riskfree rate with payoff in t+1 (annual)

P

6

8

6

8

k'

1000
1.5

0.4
800

0
-0.2

1

600
percent

basis points

percent

0.2

400
200

0.5
0

0

-0.4

-0.5
-200

-0.6
2

4

6
t

8

2

4

6
t

8

2

4
t

Perturbed RBC Model
Baseline RBC ModelFigure 4: Real Business Cycle Model without Habit and with CEE Investment Adjustment Costs
Technology Shock Not Realized in Period 5
OutputInvestment

0.3

Consumption

0.5

0.2

0.45

0.15

0.4

0.1

0.15

0.35

percent

0.2

percent

percent

0.25

0.3

0.05
0

0.25
-0.05

0.1

0.2
-0.1
0.15

0.05

-0.15
2

4

6

8

2

4

6

8

2

t

t

Hours Worked

Riskfree rate with payoff in t+1 (annual)

P

6

8

6

8

k'

1000

0.5

-0.2

0.45
800
basis points

0.35
0.3
0.25
0.2

-0.3
-0.4

600

percent

0.4
percent

4

t

400

-0.6
200

0.15
0.1

-0.5

-0.7

0

-0.8

0.05
2

4

6
t

8

2

4

6
t

8

2

4
t

Perturbed RBC Model
Baseline RBC ModelFigure 5: Real Business Cycle Model with Habit and Without Investment Adjustment Costs
Technology Shock not Realized in Period 5
OutputInvestment

Consumption

0.5

0.3

0.2

0.4

0.25

0.3

0.15

0.2
percent

0.1
0

0.05

0.05

-0.1

0

-0.2

-0.05

-0.3

-0.1

2

4

6

0

2

4

6

8

2

4

t

t

Hours Worked

Riskfree rate with payoff in t+1 (annual)

P

1000

0.4

800
basis points

0.2
0.1
0

6
t

8

8

6

8

k'

0
-0.2

600
400

-0.4
-0.6

0
4

6

0.2

200

2

-0.05

t

0.3
percent

-0.4

8

0.5

-0.1

0.1

percent

percent

0.1

percent

0.2
0.15

-0.8
2

4

6
t

8

2

4
t

Perturbed RBC Model
Baseline RBC ModelFigure 6: Real Business Cycle Model with Habit and with Level Investment Adjustment Costs
Technology Shock Not Realized in Period 5
OutputInvestment

Consumption

0.5

0.3

0.4

0.25

0.3

0.2

0.15

0.2

percent

0.2

percent

percent

0.25

0.1

0.15
0.1

0

0.1

0.05
-0.1

0.05

0

-0.2
2

4

6

8

2

4

6

8

2

t

t

Hours Worked

Riskfree rate with payoff in t+1 (annual)

P

6

8

6

8

k'

1000

0.5

-0.1

0.45
800
basis points

0.35
0.3
0.25
0.2

-0.2
-0.3

600

percent

0.4
percent

4

t

400

-0.5
-0.6

200

0.15

-0.4

-0.7

0.1

0

0.05
2

4

6
t

8

-0.8
2

4

6
t

8

2

4
t

Perturbed RBC Model
Baseline RBC ModelFigure 7: Real Business Cycle Model with Habit Persistence and Flow Adjustment Costs
Perturbed - Technology Shock Expected in Period 13, But Not Realized
OutputInvestment

Consumption

0.35
0.6

0.2

0.5

0.15

0.3

0.2
0.15

0.4

percent

percent

percent

0.25

0.3

0.1

0.2

0.05

0.1

0.1
0.05
0

0

-0.05
5

10

15

20

5

10

15

20

5

t

t

Hours Worked

Riskfree rate with payoff in t+1 (annual)

Pk'

15

20

15

20

1000

0.5

-0.1
800

-0.2

0.3
0.2

-0.3

600
percent

basis points

0.4
percent

10

t

400

-0.4
-0.5
-0.6

200

-0.7

0.1

-0.8

0

-0.9

0
5

10

15
t

20

5
Perturbed RBC Model
Baseline RBC Model

10

15
t

20

5

10
t

Figure 8: RBC and Simple Monetary Model
Expectation of Technology Shock in Period 13 Not Realized
OutputSimple Monetary Model
Real Business Cycle Model

Investment

1

Consumption
0.6

1.5

0.6
0.4

percent

percent

percent

0.8
1
0.5

0.4
0.2

0.2
0
5

10

15

20

5

Hours Worked

15

20

5

basis points

500

0
10

15

20

5

Inflation (APR)

basis points

0
-0.2
-0.4
-0.6
-0.8
10

15

15

20

20
0
-20
-40
-60
-80

20

15

20

0.2
0
-0.2
-0.4
-0.6
-0.8

20

5

10

real wage

Nominal rate of interest (Ret+1, annual)

0.2

5

10

0
percent

5

15

Pk'

1000

0.5

0

10

Ex post realized real Ret+1/S t+1 (annual)1
percent

10

percent

0

-0.1
-0.2
-0.3

5

10

15

20

5

10

15

20

Note: subscript on nominal rate of interest indicates date of payoff. Re is graphed at date t. S indicates gross change in price level from t-1 to t.
t+1tFigure 9: Simple Monetary Model and Associated Ramsey Equilibrium

1
0.8
0.6
0.4
0.2
0

5

10

15

1.5
1
0.5
0

20

5

hours worked

10

15

0.5

12
10
8
6
4
2

15

20

5

net inflation (APR)

10

15

2.2
2
1.8
1.6

12
10
8
6
4

10

15

20

5

10

15

20

15

20

-0.5

-1

20

14

10

0

5

net nominal rate of interest

2.4

5

0

price of capital

10

15

20

real wage
percent deviation from ss

10

0.2

5

percent deviation from ss

annualized, percent

1

5

0.4

20

14

0

0.6

net real rate of interest

Annualized, percent

percent deviation from ss

consumption
percent deviation from ss

investment
percent deviation from ss

percent deviation from ss

outputRamsey Equilibrium of Simple Monetary Model
Simple Monetary Model

0.2
0.1
0
-0.1
-0.2
5

10

15

20

Figure 11:
Response of Simple Monetary Model and Perturbed Model to Signal Shock
Perturbation: Technology Shock Realized in Period 13
OutputSimple Monetary Model

Investment

Consumption
0.7

1

1.5

0.6

0.6
percent

0.8

percent

percent

Perturbed Simple Monetary Model

1

0.5
0.4
0.3

0.4

0.5
0.2

0.2
5

10

15

20

5

Hours Worked

10

15

20

5

10

15

20

15

20

P k'

e

Ex post realized Rt+1/S t+1 (annual)0

Annual, percent, deviation from ss

5

10

15

Inflation, S

percent

0.5

0.4

50
0

20

5

10

15

20

Nominal Interest Rate, Ret+1 (annual)

t

0
0
-0.5
-50
-1
-100

-1.5
5

10

15

20

0.2
0

-50

5

10

15

20

5

percent deviation from ss

basis points

percent

1

10

Real wage
0.6
0.4
0.2
0
-0.2
5

10

15

20

Note: subscript on nominal rate of interest indicates date of payoff. Ret+1 is graphed at date t. S t indicates gross change in price level from t-1 to t.

Figure 12:
Response of Simple Monetary Model and Perturbed Model to Signal Shock
Perturbation - [p = 0.01
OutputPerturbed Simple Monetary Model
Simple Monetary Model

Investment

Consumption

1.5
1

1.5

percent

1

percent

percent

2

1

0.6
0.4

0.5
0.5
10

15

0.2

20

5

Hours Worked

10

15

20

5

10

basis points

1.5
1
0.5

15

20

15

20

P k'

e

Ex post realized Rt+1/S t+1 (annual)01
percent

5

percent

0.8

-100

0.5

-200
0

10

15

Inflation, S

20

5

10

15

20

Nominal Interest Rate, Ret+1 (annual)

t

0.5

0
-50

0

-100
-150

-0.5

-200
5

10

15

20

5

10

15

20

5

percent deviation from ss

Annual, percent, deviation from ss

5

10

Real wage
0
-0.2
-0.4
5

10

15

20

Note: subscript on nominal rate of interest indicates date of payoff. Ret+1 is graphed at date t. S t indicates gross change in price level from t-1 to t.

Figure 13:
Response of Simple Monetary Model and Perturbed Model to Signal Shock
Perturbation - [w = 0.01
OutputPerturbed Simple Monetary Model
Simple Monetary Model

Investment

1

Consumption
0.6

1.5

0.6
0.4

percent

percent

percent

0.8
1

0.4

0.5

0.2

0

0

0.2
0
10

15

20

5

Hours Worked

20

5

0.5

100

15

Inflation, S

50
0

5

10

15

20

5

Nominal Interest Rate, Ret+1 (annual)

1

100

0.5

50

0

0

-0.5

-50
5

10

15

20

0
-0.2

20

t

20

0.2

percent deviation from ss

10

15

P k'

-50

0
5

10

e

basis points

percent

15

Ex post realized Rt+1/S t+1 (annual)

1

Annual, percent, deviation from ss

10

percent

5

5

10

15

20

10

15

20

Real wage
0.4
0.2
0
-0.2
5

10

15

20

Note: subscript on nominal rate of interest indicates date of payoff. Re is graphed at date t. S indicates gross change in price level from t-1 to t.
t+1tFigure 14:
Response of Simple Monetary Model and Perturbed Model to Signal Shock
Perturbation - DS = 1.05
OutputPerturbed Simple Monetary Model
Simple Monetary Model

Investment

1

Consumption
0.6

1.5

0.6
0.4

percent

percent

percent

0.8
1

0.4

0.5

0.2

0

0

0.2
0
5

10

15

20

Hours Worked

10

15

20

5

50

20

0.3
percent

basis points

0.5

15

P k'

e

0

0.2
0.1
0

-50

-0.1

0
10

15

Inflation, S

20

5

10

15

N t

15

20

5

Nominal Interest Rate, Ret+1 (annual)

t

0.2
0
-0.2
-0.4
-0.6
-0.8
5

10

20
0
-20
-40
-60
-80

20

b

i t

i

5

l

t

10

fi t

t i di15tpercent deviation from ss

5

Annual, percent, deviation from ss

10

Ex post realized Rt+1/S t+1 (annual)1
percent

5

20

d t

f

ff Re i

10

15

20

Real wage
0.2
0
-0.2
5

h d td t t

10

15

i di

20

t

h

i

i

l

Perturbed Simple Monetary Model
Simple Monetary Model

Figure 15:
Response of Simple Monetary Model and Perturbed Model to Signal Shock
Perturbation - 100 Basis Point Negative Policy Shock to Interest Rate in Period 11 that is Not Realized
OutputInvestment

Consumption

1
1.5

0.6

0.6

percent

percent

percent

0.8
1

0.5
0.2

0.2
10

15

20

5

Hours Worked

10

15

20

5

10

1.2
basis points

1
0.8
0.6
0.4

15

20

P k'

e

Ex post realized Rt+1/S t+1 (annual)
0.4

50
percent

5

percent

0.4
0.3

0.4

0

0.2
0
-0.2

-50

0.2

-0.4
10

15

Inflation, S

20

5

10

15

20

Nominal Interest Rate, Ret+1 (annual)

t

0.5
50
0
0
-0.5

-50
5

10

15

20

5

10

15

20

5

percent deviation from ss

5

Annual, percent, deviation from ss

0.5

10

15

20

Real wage
0
-0.1
-0.2
-0.3
5

10

15

20

Alternative Labor Market Model

Figure 16: Response to a monetary policy shock

Bimonthly Version of Simple Monetary Model

0.2
0.1
0
5

10

-3

average real wage

20

0
-10
-20
5

10

15

0

20

percent deviation from ss

0.02

15
10
5

5 10 15 20

5price of capital

2.36
0.04
2.34
0.02

2.32

0

2.3
5 10 15 20

vacancies
0.3
0.2
0.1
0
5

10 15 20

5

0.04
0.03
0.02
0.01
0
5 10 15 20

net inflation (APR)
2.38

0.06

0.05

10 15 20net nominal rate of interest
Annualized, percent

0.3

x 10

0.04

20

price of labor

15

0.06

hours worked

consumption

10

15

5
4.95
4.9
4.85

20

5

10 15 20intensity

x 10

-3

labor

0.02

0.01

0
5 10 15 20

percent deviation from ss

15

percent deviation from ss

percent deviation from ss
percent deviation from ss

10

percent deviation from ss

0.01

0.08

x 10

percent deviation from ss

percent deviation from ss

0.02

percent deviation from ss

percent deviation from ss

0.03

5

-3

investment

output1050
510 15 20

Equilibrium, Alternative Labor Market Model
Ramsey, Alternative Labor Market Model
Equilibrium, Bimonthly Version of Simple Monetary Model
Ramsey, Bimonthly Version of Simple Monetary Model

Figure 17: Response to a contemporaneous positive technology shock

15

0
-5
-10
-15
-20
10

15

20

average real wage
1.5
1
0.5
0
5

10

15

0
-0.5

20

price of labor

5

0.5

20

5

percent deviation from ss

percent deviation from ss

10

1

10

15

percent deviation from ss

0

1.5

1
0.8
0.6
0.4
0.2
0

20

5

price of capital

10 15 20

net inflation (APR)

4
2

3
2

0

1
-2

0
5

10

15

20

vacancies
50-5510 15 20

5

10

15

-0.5
-1
-1.5

0
-0.5
-1

5 10 15 20

5 10 15 20

5
0
-5

20

intensity-1.5

0

net nominal rate of interest
Annualized, percent

0.2

2

hours worked

5

percent deviation from ss

0.4

percent deviation from ss

0.6

consumption

percent deviation from ss

percent deviation from ss

0.8

5percent deviation from ss

investment

percent deviation from ss

percent deviation from ss

output1

10 15 20

labor
0.2
0
-0.2

5 10 15 20

Equilibrium, Alternative Labor Market Model
Ramsey, Alternative Labor Market Model
Equilibrium, Bimonthly Version of Simple Monetary Model
Ramsey, Bimonthly Version of Simple Monetary Model

Figure 18: Response to a signal (not realized) of future technology shock

0
5

10

15

percent deviation from ss

percent deviation from ss

5

0
15

0
5

10

10

1

20

price of labor

5

2

20

-0.2
-0.4
-0.6
-0.8
5

10

15

20

0.4
0.2
0

20

5

-1

3
2

-2
5 10 15 20
vacancies

5

10

15

20

0

5

10 15
intensity

0.8
0.6
0.4
0.2
0
5

10

15

20

15

20

30
20
10
0

20

1

10

net nominal rate of interest

-1.5

-5

0.5

5

4

-0.5

0

1

10 15 20

0

5

1.5

net inflation (APR)

percent deviation from ss

percent deviation from ss

percent deviation from ss

015

price of capital

average real wage
0.2

10

percent deviation from ss

0.5

3

2

Annualized, percent

1

4

hours worked

0.6

percent deviation from ss

1.5

consumption
percent deviation from ss

investment
percent deviation from ss

percent deviation from ss

output5

10 15 20
labor

0.5
0.4
0.3
0.2
0.1
0
5

10

15

20

Figure 19: Response of Full and Simple Monetary Model to Signal ShockOutputInvestment
0.8

0.6
0.4

percent

1.5

0.8

percent

percent

1

Consumption

1
0.5

0.4
0.2

0.2
5

10

15

20

5

10

15

20

5

Hours Worked

Real Wage

Ex post realized real Ret+1/S t+1 (annual)basis points

0
-0.1
-0.2
-0.3

10

15

20

5

10

t

P

15

50
0
-50

20

5

10

t

15

20

t

Inflation (APR)k'Nominal Interest Rate (Re , annual)
t+1

0.2
0.4
0.3
0.2
0.1
0

0

20
0
-20
-40
-60
-80

basis points

percent

20

t

0.1

-0.2
-0.4
-0.6
-0.8
5

15

t

1.2
1
0.8
0.6
0.4
0.2
5

10

t

percent

percent

0.6

10

15
t

20

5

10

15
t

20

5

10

15
20
t
Perturbed Monetary Model
Baseline Monetary ModelFigure 20: Behavior of Money, Credit and Net Worth in Full Monetary Model
Base growth (APR)(currency plus bank reserves)

M1 growth (APR)(currency plus demand deposits)

M3 Growth (APR)(M1 plus savings deposits)

20

7

8

15

6

6

5

4

4

2

3

0

2

-2

1

-4

10
5
0
-5
5

10

15

20

5

Total credit growth (APR)(working capital loans plus loans to entrepreneurs)

15

20

Net Worth

0.7
3

10

0.6

2.5

0.5

2

0.4

1.5

0.3

1

0.2
0.1

0.5
5

10

15

20

5

10

15

20

5

10

15

20

Figure 21: Response of Full Monetary Model and Perturbed Model to Signal Shock
Perturbation - Monetary Policy Response to Credit Growth
Investment

Outputpercent

percent

0.6
0.4
0.2

5

10

15

20

5

10

15

20

5

10

15

20

t

t

Hours Worked

Real Wage

Ex post realized real Ret+1/S t+1 (annual)

0.2
percent

0.6
0.4
0.2
0
10

15

0
-0.2

20

5

10

t

15

40
20
0
-20
-40
-60
-80

20

5

10

t

15

20

t

Inflation (APR)
k'

0.8
0.6
0.4
0.2
0
10

15
t

Nominal Interest Rate (Re , annual)

0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
20

t+1

basis points

P

5

0.4

t

0.8

5

0.6

0.2

basis points

percent

0.4
0.2

percent

0.8

0.8

0.6

percent

Consumption

0

-20
-40
-60
-80

5

10

15
t

20

5

10

15
20
t
Perturbed Monetary Model
Baseline Monetary Model