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orKing raper series
Chaos, Sunspots, a n d Automatic
Stabilizers
Lawrence J. Christiano and Sharon G. Harrison
3
Working Papers Series
Macroeconomic Issues
Research Department
Federal Reserve Bank of Chicago
August 1996 (WP-96-16)
1' B R A R V
N0\ 0 5 1996
KtuERAL RESERVE
BANK OF CHICAGO
FEDERAL RESERVE B A N K
OF CHICAGO
August 1996
Chaos, Sunspots, a n d A u t o m a t i c Stabilizers*
Lawrence J. Christiano
Northwestern University, NBER, and the Federal Reserve Banks of Chicago and Minneapolis
Sharon G. Harrison
Northwestern University
ABSTRACT
W e study a one-sector growth model which is standard except for the presence of an externality
in the production function. The set of competitive equilibria is large. It includes constant
equilibria, sunspot equilibria, cyclical and chaotic equilibria, and equilibria with deterministic or
stochastic regime switching. The efficientallocation ischaracterized by constant employment and
a constant growth rate. W e identify an income tax-subsidy schedule that supports the efficient
allocation as the unique equilibrium outcome. That schedule has two properties: (i) itspecifies
the tax rate to be an increasing function of aggregate employment, and (ii)earnings are subsidized
when aggregate employment is at its efficient level. The first feature eliminates inefficient,
fluctuating equilibria, while the second induces agents to internalize the externality.
*We have benefitted from discussions with Fernando Alvarez,Jess Benhabib, Michele Boldrin, V.V. Chari,
Russell Cooper, Martin Eichenbaum, Ian Domowitz, Chris Gust, Nicola Persico, Michael Woodford,
Randall Wright, and Michelle Zaharchuk, and we are grateful to Victor Valdivia for research assistance.
Christiano isgrateful to the National Science Foundation for financial support. The views expressed herein
are those of the authors and not necessarily those of the Federal Reserve Bank of Chicago or the Federal
Reserve System.
1. In tro d u ctio n
Interest is growing in business cycle models with multiple, self-fulfilling rational expectations
equilibria,1 These models offer a new source of impulses to business cycles— disturbances to
expectations— and they offer new mechanisms for propagating and magnifying the effects of
existing shocks, such as shocks to monetary policy, to government spending, and to tech
nology.2 Although initial versions of these models appear to rely on empirically implausible
parameter values, recent vintages are based on increasingly plausible empirical foundations.3
The policy implications of the new models differ sharply from those of current main
stream equilibrium models, which emphasize shifts to preferences and technology as the
basic impulses to the business cycle. These models have been used to articulate the notion
influential early papers include Azariadis (1981), Bryant (1983), Cass and Shell (1983), Cooper and John
(1988), Diamond (1982), Farmer and Woodford (1984), Shleifer (1986), and Woodford (1986b). The first
paper to take seriously the quantitative predictions of a business cycle model with self-fulfilling expectations
isWoodford (1988). Rational expectations models with multiple equilibria have attracted attention in other
areas too. See Benhabib and Perli (1994), Krugman (1991), and Matsuyama (1991a) for a discussion in
the context of international trade and growth. See Cole and Kehoe (1996) for an analysis of the Mexican
debt crisis. See Bryant (1981) and Diamond and Dybvig (1983) for discussions in the context of models of
banking. See Boldrin, Kiyotaki, and Wright (1993) and Mortensen (1989,1991) for discussions on dynamic
models of search and matching.
2An extensive literature documents the inadequacy of propagation in standard business cycle models.
See, for example, Burnside and Eichenbaum (1995), Christiano (1988, p. 269), Cogley and Nason (1995),
Rotemberg and Woodford (1996), and Watson (1993). An early study showing how models with indetermi
nate equilibria provide increased magnification and propagation of monetary shocks isFarmer and Woodford
(1984). More recent studies include Beaudry and Devereux (1994), Benhabib and Farmer (1996), Guesnerie
and Woodford (1992), and Matheny (1994). For a recent argument that macroeconomists are short on shocks
for accounting for the business cycle, see Cochrane (1995).
3For example, the models of Benhabib and Farmer (1994), Farmer and Guo (1994) and Gali (1994a,b)
rely on increasing returns in production. (See Schmitt-Grohe 1995.) The results of subsequent empirical
research suggested that the degree of increasing returns required for the Benhabib and Fanner and Fanner
and Guo models to display sunspot equilibria is too high. (See Burnside, Eichenbaum, and Rebelo 1995 for
one such effort, and see also Benhabib and Fanner 1995 for a review of several others.) These empirical
results in turn stimulated further theoretical work: recently, Benhabib and Fanner (1995) showed that a twosector version of their model requires a much lower degree of increasing returns to guarantee the existence
of sunspot equilibria.
that economic fluctuations represent the economy’s efficient responses to shocks and to cast
doubt on the desirability of macroeconomic stabilization policy.4 In contrast, the new mod
els suggest that institutional arrangements and policy rules designed to reduce fluctuations
in output may be desirable. (See Grandmont 1986, Guesnerie and Woodford 1992, Shleifer
1986, Woodford 1986b, 1991, and the articles in the symposium summarized in Woodford
1994.)56
This paper pursues these ideas in a particular business cycle model. The model studied
is a version of the one-sector, external increasing returns model recently studied by Baxter
and King (1991), Benhabib and Farmer (1994,1995), and Farmer and Guo (1994,1995).®
Our version of this model has a significant advantage relative to the versions analyzed in the
literature. In those, analysis of the global set of equilibria is typically quite difficult, and
so researchers confine themselves to studying the set of equilibria that is local to the steady
state. By contrast, the structure of our model is such that the global set of equilibria is
transparent and can therefore easily be analyzed. It turns out that this set of equilibria is
remarkably rich, and itincludes sunspot equilibria, regime switching equilibria, and equilibria
4See Kydland and Prescott (1980) and Sargent (1979, p. 393) for a statement of the case that output sta
bilization is undesirable. The “preference and technology” literature on macroeconomics did not completely
rule out the possibility that some forms of stabilization might be desirable. Researchers who incorporated
frictions like price rigidities did see some role for activist policy. (See Fischer 1980.)
5An important early exsimple of the potential stabilizing role of institutional arrangements occurs outside
the area of business cycle analysis and is provided by the work of Bryant (1981) and Diamond and Dybvig
(1983). They showed that a spontaneous burst of pessimism on the part of depositors could trigger so
cially inefficient bank runs and that a government policy— deposit insurance— could be designed that would
eliminate this source of instability.
6As in these papers, we do not formally articulate what the source of external increasing returns is.
Examples of analyses that are explicit about the nature of external effects include Diamond (1982), Howitt
and McAfee (1988), and Romer (1986). Benhabib and Farmer (1994) suggest the possibility that there
is a way of reformulating our model so that the source of increasing returns is internal to the firm, while
leaving our basic analysis unaffected. The analysis in Romer (1987) suggests yet another possibility: that the
increasing returns may actually reflect gains from specialization. W e have not yet explored these possibilities.
2
which appear chaotic. Our analysis illustrates the potential pitfalls of focusing only on the
equilibria that are local to some steady state.
W e establish that the set of bounded solutions to a particular expectational difference
equation corresponds to equilibria for our model. This set is simple to characterize because
the difference equation is only first order and has a simple analytic representation. This
is true, despite the fact that capital accumulation and employment are endogenous in the
model. A n important feature of the difference equation is that for every initial condition,
it has two solutions. This two-branch feature of the Euler equation is an important reason
the set of equilibria for the model includes
re g im e s w it c h in g
equilibria of the type studied
by Hamilton (1989) and equilibria that appear chaotic.7
Even the efficient allocations in our model are straightforward to determine, despite the
lack of convexity in the aggregate resource constraint set due to the externality. The efficient
allocations are unique and involve no fluctuations.
W e examine the operating characteristics of two
a u t o m a tic s t a b iliz e r
tax regimes. Each
has the property that the income tax rate rises if the economy moves into a boom and
falls if it goes into a recession. Under each tax regime, the economy has a unique interior
equilibrium, in which output is constant. However, one tax regime stabilizes output on an
inefficient level of output, and the other stabilizes output on the efficient allocations. W e
show that implementing the first tax regime may increase, or even decrease, welfare.8 W e
7 For other examples of a “branching” Euler equation in infinite horizon growth models, see Benhabib and
Perli (1994) and Benhabib and Rustichini (1994).
8This possibility has been discussed by Guesnerie and Woodford (1992, pp. 383-388), Shleifer (1986),
and Woodford (1991, p. 103) in other contexts.
3
establish two things about designing a tax system that supports the efficient allocations as
the unique interior equilibrium. First, such a system must specify that the tax rate vary
with the level of aggregate economic activity. W h e n the tax rate is specified to be a fixed
constant, then there is more than one equilibrium, with the efficient one being only one
of them. Second, the efficient equilibrium in this case is determinate, so that a standard
local analysis of the set of equilibria would falsely conclude that only one equilibrium is
possible. These results draw attention to the importance of the proper design of automatic
stabilizer tax systems and point to a potential pitfall in the traditional approach to policy
design, which tends to focus on minimizing output variance.9 These results also illustrate
the potential dangers of the standard practice of focusing exclusively on local equilibria.
Finally, our model provides a convenient vehicle for articulating some econometric issues
that arise in the analysis of models with multiple equilibria. As emphasized by Woodford
(1991, p. 77), there is a widespread perception that “anything goes” with sunspot models—
any set of facts can be explained. The model in this paper can be used to illustrate that
sunspot models in principle do impose discipline on an empirical analysis.10 With one ex
ception, the econometric procedures used to analyze standard models with unique equilibria
and driven by exogenous shocks can be used to analyze and test sunspot models too. The
exception is that procedures which select parameter values by equating model first moments
and corresponding sample first moments may no longer be well-defined. This is because
9An influentialexample isthe analysis ofPoole (1970), who argues that the appropriate choice of monetary
policy regime depends on whether shocks emanate from financial markets or investment decisions. The
criterion driving the policy design in Poole’s analysis is minimization of output variance.
10See Dagsvik and Jovanovic (1994), Farmer and Guo (1995), Imrohoroglu (1993), Jovanovic (1989),
Sargent and Wallace (1987), and Woodford (1987,1988,1991) for farther development of this point.
4
there may be a set of possible first moments associated with any parameter configuration,
depending on which equilibrium the economy is in.
The intuition underlying the dynamics in our model is essentially the same as that de
scribed by Benhabib and Farmer (1994) and Farmer and Guo (1994). The economy is
perfectly competitive, and individual producers have linearly homogeneous production func
tions in capital and labor, which are strictly concave in each. However, economywide average
output operates as an externality in front of each firm’s production function, shifting it up
when average output is high. The latter is the key to why there are multiple equilibria in
our model and to why expectations can act as an independent source of fluctuations. If all
households act on the conjecture that the current period’s wage rate is high by supplying
more labor services to the market, then the market-clearing wage
is
high because of the
externality on labor. Similarly, ifhouseholds act on the conjecture that next period’s rental
rate on capital is high by buying more investment goods today, then their conjecture will be
validated.11
A regime which specifies that the tax rate rises with aggregate employment has the
potential to stabilize output by defeating the mechanism that gives rise to multiple equilibria.
Conjectures that the rate of return on market activity is high cannot be self-fulfilling ifthe
11This is the case for two reasons. First, the externality on capital prevents next period’s increase in
capital from directly reducing the marginal product of capital. (That the externality is strong enough for
the aggregate capital stock not to enter the marginal product of capital is the reason the difference equation
mentioned above is first order, which in turn is the reason the global set of equilibria is transparent in our
model.) Second, the externality on labor helps ensure that the increase in next period’s wage rate, occasioned
by the rise in capital next period, stimulates a large increase in employment. This indirectly helps drive up
next period’s rental rate on capital.
5
proceeds are taxed away.12
The outline of the paper is as follows. Our model and equilibrium concept are presented
in Section 2. Section 3 establishes our characterization result for the set of competitive
equilibria. Sections 4 and 5 analyze the deterministic and stochastic equilibria of the model,
respectively. Econometric issues are addressed in the context of the analysis of stochastic
equilibria. Section 6 considers the impact of an automatic stabilizing tax policy and reports
the socially optimal allocations. Section 7 concludes.
2. T h e M od el
Let
st
denote the realization of the exogenous shocks at date t.
In models with shocks
to fundamentals, st would include shifts to preferences or technology, or to government
spending. In this paper, we do not consider such shocks. In the stochastic versions of
our model,
st
represents disturbances which influence equilibrium outcomes, but which do
not affect fundamentals. W e let s*
—
( s o ,s i,
denote a history of realizations up to
and including date t. For simplicity, we only consider environments in which the number
of possible values of s t is finite for each
t.
The probability of history s‘ is denoted /xt(s‘).
To conserve on notation, from here on we delete the subscript t on /x. This should not
cause confusion: that the functions /xt (s‘) and p r (sr), r ^ t are different is evident from
the fact that the quantity of elements in s* and sr is different. W e adopt this notational
12Our argument is related to the one in Sdimitt-Grohe and Uribe (1996). They show that a procyclical
tax policy (designed to support a balanced budget) can destabilize the economy by making possible sunspot
equilibria.
6
convention for all functions of histories. The probability of st+1 conditional on
f i ( s t+1
s*
is denoted
|s*) = /i1+1(s t+1)//it (s t). W e now discuss the agents in our model and our concept of
equilibrium.
2 .1 . Households
W e assume a large number of identical households. At each s4 and
t,
the representative
household values consumption and leisure henceforth according to the following utility func
tion:
I] £
(2.1)
n(«0 ]
aJ|st
where /3c(0,1) is the discount rate,
|s l denotes histories, sJ,that are continuations of the
given history, s‘,and c(s7), n(sJ) denote consumption and labor, respectively, conditional on
history s j . The household must respect the following sequence of budget constraints:
c(*0 +
k{s>)
- (1 -
6 ) k ( s * - 1)
[1 - -r(s^)][r(s^)A:(s,“1) + w(^)n(^)] + T(^), all
where
r(sj )
tively. Also,
and
t
w (s^ )
($*)
s\
j
(2.2)
=
>
t
denote the market rental rate on capital and the wage rate, respec
is the tax rate on income,
T (si )
denotes lump-sum transfers from the
government, and k(s*) denotes the stock of capital at the end of period j , given history
s^. The household also takes fc(st_1) as given at s*. Finally, the household must satisfy the
7
following inequality constraints:
k(si ) >
for all s7 |s‘ and j
> t
O.c^) > 0 ,0 < n(s^) < 1
(2.3)
and takes as given and known the actual future date-state contingent
prices and taxes:
{ r(« 0 , w M T (« 0 ,r(« * ); J >
all s> |s %
(2.4)
W e assume that
u(c, n) = log c + <7log(l —
where
a
(2.5)
n)
> 0.
Formally, at each s* and t, the household problem is to choose (c(s^), n(s^), k ( s j );
j
> t,
all s* | s*} to maximize (2.1) subject to (2.2), (2.3), (2.4), and the initial stock of capital,
fc(st_1). The intertemporal Euler equations corresponding to this problem axe
uc(s^) =
all s> |s*, j
>
0
M si+1 I5i)ttc(«i+1){[l - r(sJ+1)]r(sj+1) + 1 - 6}
(2.6)
t, and the intratemporal Euler equations are
uc(sJ)
= t1 ” T(s,) M si)> all ^ |s‘, j >
t.
(2.7)
Here, uc(sJ) and U n { sj ) denote the partial derivatives of u with respect to its first and second
8
arguments, evaluated at c ( s j ) ,
Urn
n ( s j ).
PT
Finally, the household’s transversality condition is
K sT
(2.8)
Is t) u c ( s T ) k ( s T ) = 0.
S*>‘
The sufficiency of the Euler equations, (2.6) and (2.7), and transversality condition, (2.8),
for an interior solution to the household problem may be established by applying the proof
strategy for Theorem 4.15 in Stokey and Lucas with Prescott (1989).
2 .2 . Firms
W e assume a large number of identical firms, each of which solves a static problem at every
s*.
As a result, we can, without risking confusion, simplify the notation by deleting the s1
notation. The representative firm faces the following technology relating its output,
its capital, K
,
and labor,
Y
N ,
=
inputs and to the economywide average level of output,
f{ y , K
,N ) = y7fCaM 1-a\ 0 < 7 ,a < 1-
Y ,
to
y:
(2.9)
W e assume that
a = 1-
7.
(2.10)
The relation between the economywide average level of output and the economywide average
stock of capital, k , and labor, n, is obtained by solving y
= f( y , k, n)
for y :
(2.11)
9
given (2.10). The linearity of this function in terms of A: is essential for simplifying the
analysis to come. In addition, as discussed by Rebelo (1991), linearity allows for growth to
occur endogenously.
The firm takes y , r, and
w
as given and chooses K ,N to maximize profits:
Y - r K - w
(2.12)
N
subject to (2.9). The firms’first-order conditions for labor and capital are
I
where
/k
and
fs
n
= w, I
k
=
r
are the derivatives of / with respect to its second and third arguments,
respectively. W e assume firms behave symmetrically, so that consistency requires y
K ,n
= N .
(2.13)
= Y ,k =
Imposing these, we get
}
n
= (1 - c*)nA:, /* = cm2
(2.14)
with 7 = 2/3. With this value of 7 ,the model implies that labor’s share is 2/3, which is close
to the value estimated using the national income and product accounts (Christiano 1988).
10
2.3. Government
The income tax rate policy, r(st), is specified exogenously, and we require that the following
budget constraint be satisfied for each s*:
r(st)[r(st)k(st X) +
(2.15)
= T(s‘).
2.4. Equilibrium
The resource constraint for this economy is
c(s‘) +
k(sl)
— (1 —
6 )k(st
x) <
1)n(s*)2 = y(s‘).
(2.16)
W e then have13
Definition 2 .1 . A s e q u e n c e - o f - m a r k e t s e q u ilib r iu m i s a s e t o f p r i c e s (r(st), tu(s*); a l l s*, a l l
> 0}, q u a n t i t i e s (y(«*), c(s*), Ac(«*),n(a*); a l l s t , a l l t > 0}, a n d a t a x p o l i c y {r(st),T(st);
a l l s * ,t > 0} w i t h t h e f o l l o w i n g f o u r p r o p e r t i e s f o r e a c h t, s l :
t
• Given
t h e p r ic e s , t h e q u a n t i t i e s s o lv e t h e h o u s e h o l d ’s p r o b le m .
•
t h e p r i c e s a n d g iv e n
G iv e n
(y(s*) = fe(st-1)n(st)2},
t h e q u a n t i t i e s s o lv e t h e f i r m ’s
p r o b le m .
•
T h e g o v e r n m e n t ’s b u d g e t c o n s t r a i n t i s s a t is f ie d .
•
T h e r e s o u r c e c o n s t r a i n t i s s a t is f ie d .
W e find it useful to define an i n t e r i o r equilibrium. This is a sequence-of-markets equilibrium
in which a < n(s*) < 6 for all s* for some a and
b
satisfying 0 < a <
b <
1.
13It is easily verified that the analysis would have been unaltered had we instead adopted the date 0,
Arxow-Debreu equilibrium concept. In this case, households would have had access to complete contingent
claims markets.
11
3. C h aracterizin g E quilibrium
In the next section of the paper, we study deterministic equilibria in which prices and quan
tities do not vary with
st
and stochastic (sunspot) equilibria in which prices and quantities
do vary with s t . The analysis of these equilibria is made possible by a characterization result,
which is presented next.
Substituting (2.13) and (2.14) into the household’s intertemporal Euler equation, (2.6),
we get
g(P) =
1s,)^
)
{11 - ’V +1) M » ‘+1>2 + 1 - «>
(3-D
where
Substituting (2.14) into the household intratemporal Euler equation, (2.7), we get
c(s*) = [1 - r(st)]^n(s*)[l - n(s‘)].
(3.3)
The resource constraint implies that
c(s*) = n(s*)2 + 1 —
6 —
A(s‘).
(3.4)
Combining the two Euler equations, (3.1) and (3.3), and the resource constraint, (3.4), our
12
system collapses into a single equation in current and next period’s employment:
^2
/*(s *+1 IstM n ( s t),n(st+1);r(st+1)] = 0, all s‘, t > 0
(3.5)
* t+ l
where
v
is
v ( n , n ';
r')
1
/? [(l-T > (n ')2 + l - 6 ]
A[(n')2 + 1 -
n2 + 1 - 6 - X
6 -
(3.6)
X' ]
with
A = n2 + 1 —
6 —
(1 — r)— n(l — n).
<7
(3.7)
Here, a ' denotes next period’s value of the variable. The transversality condition, (2.8), is
equivalent to
v
v-
oT
+ 1“
, T\
)
l1 - r(®T)]2"(ar)[l -
6 -
n { s T )}]
[1 — r(sT)]^n(sT )[l — n(sr )]
n
°'
(3'8)
The basic equilibrium characterization result for this economy is
Proposition 3.1.
S u p p o se th a t r ( s ‘ ) =
0. If,
fo r a ll s t a n d t >
0,
{n(s*)} satisfies (3 .5 )
and
a <
th e n
{n(s‘)}
n(s*) <
b fo r so m e
0<
a < b <
1
c o r r e s p o n d s t o a n e q u il ib r i u m .
Proof. To establish the result, we need to compute the remaining objects— prices and
quantities— in an equilibrium and verify that they satisfy (2.6), (2.7), (2.8), (2.13), and
(2.16). A candidate set of objects is found in the obvious way. The sufficiency of the firstorder and transversality conditions for household optimization and the sufficiency of the
first-order conditions for firm optimization guarantee that these are an equilibrium.
13
The characterization result indicates that understanding the equilibria of the model re
quires understanding the
v
function. It is easily confirmed that
v
=
function in n ' for each fixed n , u . (Later, we refer to a; as the E u l e r
n ,u )
there are two possible n ’
: n ' — f u ( n ,u j )
f u (n,u>)
and
= |( 6 (71, 0 /) +
n!
u
defines a quadratic
e r r o r .)
Hence, for each
= /t(n,u;), where
^ b ( n ,u ) 2 -
4 e ( n ,u j ) }
(3.9)
fi( n ,u > )
= 5 (6(11,a/) -
y j b ( n ,u j ) 2 -
4 c ( n ,o j ) }
Here,
b ( n ,u )
(p (n )
=
=
y ( " ),(7 l; 1 .)C(n,u>)
a + q ( n ,u>)<p(n)
n 2 + 1 — 6 — ^ti(1 — n)
0n(
9Ka;)
=
1-
1 ---n(l —
1— 6
a
+
(3.10)
q ( n , u i) ip ( n )
(3.11)
n)
n)u ).
(3.12)
The function v has the shape of a saddle, as can be seen in Figure la. The intersection of v
and the zero plane (u; = 0) is depicted in Figure la as the boundary between the light and
dark region of v . This intersection defines the curves /„(•,0) and /t(-,0), which are shown
in Figure lb. W e refer to these as the upper and lower branches of the function
v.
The
lower branch intersects the 45-degree line at two points, which are denoted n 1 and n2.These
intersection points cannot be seen in Figure lb, but can be seen in Figure lc, which displays
■ n! — n
for
n
near the origin. It is easy to see from Figure la that with higher values of
a>, f i increases and
fu
decreases. The figure also indicates that for these functions to be
14
real-valued, u> must not be too big.
T he branches in th e figure are com puted using our baseline param eterization,
<7
= 2,
13 = 1.03~«, 6 = 0.02. Here, n 1 = 0.02 and n2 = 0.31. The gross growth rates of capital
(that is, A) at these two points are 0.973 and 1.004, respectively. Our assigned value of
/? is often used in the real business cycle literature. The value of a causes the m odel’s
im plication for the share o f incom e going to capital and labor to coincide w ith one estim ate
of this quantity based on the national income and product accounts. (See, for exam ple,
Christiano 1988.) In addition, this param eterization o f a facilitates som e of th e analytic
results described above. However, we have verified that the shape of the v function is not
very sensitive to the perturbations in a . The assigned value of 6 can be justified based on
our m odel’s capital accum ulation equation and on U.S. capital stock and investm ent data.
(See, for exam ple, Christiano 1988 and Christiano and Eichenbaum 1992.) The issue of how
data may be brought to bear to determ ine a value of a is addressed below.
4. D e t e r m i n i s t i c E q u i l i b r i a
We begin by considering determ inistic equilibria, in which prices and quantities depend on
t, but not on s t . To sim plify th e presentation, we drop the history notation, and we use
th e conventional tim e subscript notation instead. As we shall see, the set of determ inistic
equilibria is quite rich. For exam ple, any constant sequence {n t}, w ith tt* = n 1 or nt = n2,
satisfies th e conditions of th e characterization result and so is an equilibrium. Similarly, any
sequence w ith no € (n 1, n) and nt+1 = f i ( n t ,0 ), t > 0 is also an equilibrium, w ith n t —►n2.
15
Here, h satisfies n > n2 and n l = fi(n , 0). Figure 2 exhibits two equilibrium paths, one
starting w ith no = 0.4 and the other w ith no = 0.2. Each path converges m onotonically to
n 2.
Other determ inistic equilibria are more exotic and display a variety o f types o f regim e
switching.
For exam ple, the equilibrium employm ent policy function could be tim e non-
stationary, w ith em ploym ent determ ined by the lower branch for, say, six periods, followed
by a single-period jum p to the upper branch, followed by another six-period sojourn on the
lower branch, and so on. T he m odel has another type of regime sw itching equilibrium too,
in which th e employm ent policy function is discontinuous.
As an exam ple of the latter, consider equilibria in which em ploym ent, n', is determ ined
by the upper branch for n over one set o f intervals in (0 ,1 ) and by the lower branch over the
com plem ent of these intervals. One exam ple of this is given by
0) for
n' = f ( n ), where /( n ) =
/i(n >0)
for
n < n1
n1 < n < m 1
(4.1)
/ u(n,0) for m 1 < n < m 2
fi(n, 0)
for
m2 < n
where m l < n2 and m 2 are a chosen set of numbers. B y considering different values o f a,
(4.1) defines a fam ily of maps. As we shall see, there are elem ents in this fam ily o f maps
which exhibit characteristics that resemble chaos.14 There are several concepts o f chaos in
14For other discussions of chaos, with economic examples, see Boldrin and Woodford (1990) and Mat
suyama (1991b).
16
the literature. We consider two.
4 .1 . T o p o lo g ic a l C h a o s
We consider th e topological concept of chaos as discussed in Devaney (1989). We require
two definitions first:
D e fin itio n 4 .1 . T h e m a p f : J —* J is said to b e topologically tran sitive i f fo r a n y p a ir o f
open sets, U, V c J, th ere ex ists k > 0 such th a t f k(U) fl V
0.
Here, f l {n) = /( n ) , / 2(n) = /[ /( n ) ] , and so on. Loosely, the above definition says
that for alm ost all initial conditions, iterations on the map, / , produce an orbit (that is,
n, /( n ) , / 2(n ),...) th at visits every region, no m atter how sm all, of J. An exam ple of a map
that violates th is condition is the policy function o f the standard one-sector growth m odel.
For any initial capital stock, iterations on the policy function generate a sequence that
converges m onotonically to the steady sta te .' If the initial capital stock is below th e steady
state, then capital stocks sm aller than the initial condition and above steady state are not
visited. A second definition that is important is
D e fin itio n 4 .2 . T h e m a p f : J —* J has sen sitive depen den ce on in itial co n d itio n s i f th ere
ex ists S > 0 such th a t, for an y n € J an d a n y neighborhood N o f n, th ere e x ists y € N an d
m > 0 such th a t | / m(n) — / m(y)| > 6.
T his says th at for any initial condition, n, and any neighborhood, no m a tte r how sm all,
around n, there is at least one other initial condition whose orbit eventually differs by at
least <5 from the orbit o f n. N ote that th e parameter 6 is chosen as a function of th e map,
but it is not a function o f n or the size of the neighborhood around n. A sequence generated
by a map th at exhibits sen sitivity to initial conditions is difficult to forecast for tw o reasons.
17
First, th e slightest measurement error in th e initial conditions may result in a substantial
error o f forecast. Second, even if th e initial conditions are measured accurately, then any
slight rounding error in com puting an orbit is likely to be magnified.
Then, we have
D e fin itio n 4 .3 . L e t J b e a set. T h e m a p f : J -* J is sa id to b e ch aotic on J i f
• f h as sen sitive depen den ce on in itial conditions.
• f is to p o lo g ica lly tran sitive.
• p e rio d ic p o in ts o f f are dense in J.
B y a periodic poin t, n, we mean one for which there is some k > 0 such that n = /* (n ).
Theorem s exist th at establish conditions under which a given map is chaotic. Unfortu
nately, these theorem s require either that / be continuous (see, for exam ple, chapter 1.1 of
Devaney, 1989) or that it be piecewise continuous and differentiable w ith derivative greater
than unity in absolute value. (See Lasota and Mackey, 1985, chapter 6.) We are not aware
o f theorem s that include maps of the kind considered here. Instead, we follow th e strategy
pursued in Dom owitz and El-Gam al (1993,1994) and develop sim ulation-based evidence on
w hether our map is chaotic.
Consider sen sitivity to initial conditions first. For this, we com pute the Lyapunov coef
ficient, L (n ), associated w ith the map, / , defined in (4.1). The function, L , m aps n € (0 ,1 )
into th e real line. For any fixed n € (0 ,1 ),
L(n) = l i m i ^ t o g
t= l
18
dfiO
dn
(4.2)
where n,+1 = /(n * ), i — 1, ...,T — 1, n i = ft, and d f ( n i ) / d n denotes the derivative o f / w ith
respect to n, evaluated at n = rij. To see why L is of interest, note th at the sum is equivalent
to log l^ n ^ j- If n is a periodic point o f any finite order k, then f k{n) — h. If it is a stable
periodic point, then
< 1- If T — m k , where m is an arbitrary positive integer, then
d f T {n)
dn
f t d f k(n,)
L \
dn
d f k(n)
dn
<0
(4.3)
where nj = n, and nj+1 = /* (n (), 1 = 1 ,..., m —1. This suggests th at if ra is a stable periodic
point of any period, then L (n ) m ust be negative. But if / has a stable periodic point, or a
point whose orbit intersects w ith such a point, then it violates sensitivity to initial conditions.
Thus, a negative value of L(n) indicates that one of the conditions necessary for / to be
chaotic fails. A positive value of L(n) is a necessary condition for chaos.
Figure 3 shows L(n ) for a in the range 1.25 to 2.20, w ith n fixed at 0.255.15 We truncated
the infinite sum in (4.2) at T = 2,000. We set m 1 = 0.33, m2 = 0.70. N ote th at L(h) is
positive for values of <r less than 1.5. Hence, for values of a in this range, there is evidence that
/ is characterized by sen sitivity to initial conditions. To investigate this further, consider
a = 1.25. Figures 4a and 4b show two sequences of 400 observations on hours worked,
sim ulated using / . In one case th e initial condition is 0.455, and in the other th e initial
condition is 0.454. D espite th e fact th at the initial conditions are very close, the tw o orbits
are quite different. In fact, th ey are eventually as dissim ilar as they would have been had the
15
In computational experiments not reported here, we found that the graph in Figure 3 is insensitive to
the value of fi. This insensitivity is consistent with results in Figure 5, discussed below.
19
initial conditions been far apart. B y the 70th observation, th e two series are com pletely out
o f phase. Then, by around th e 310th observation, they are alm ost identical again. A lthough
th e am plitude of th e tw o series varies som ewhat, m ost of th e differences betw een th e tw o
series has to do w ith phase. N ote how strikingly different these equilibria are from th e ones
shown in Figure 2.
Now consider topological transitivity. One way to investigate this is to com pute th e
histogram o f orbits associated w ith different initial conditions. To com pute th is histogram ,
we divide th e unit interval into 3,000 equal-width intervals and approxim ate th e histogram o f
an orbit by th e number of tim es hours worked lands in each interval in a sim ulation o f length
50,000. Figure 5 shows histogram s for orbits associated w ith four initial conditions drawn
from very different parts of the unit interval. These histogram s are sim ilar in tw o respects.
First, consistent w ith topological transitivity, each orbit appears to cover th e sam e region
of th e unit interval. In particular, let J denote the union of intervals w ith positive m ass in
Figure 5. Then J appears to be independent o f th e initial conditions. T his is consistent
w ith th e notion th at orbits associated w ith alm ost all n 6 J wander through every sm all
neighborhood of J. T he second striking feature of the histogram s in Figure 5 is th at th ey
appear to have th e sam e shape. Thus, the histogram s are consistent not only w ith th e notion
th at alm ost all orbits in J visit each subinterval in J w ith positive probability ( topological
tra n sitivity), but they are also consistent w ith th e notion th at th e probability of visitin g each
subinterval is the sam e across orbits.
20
4 .2 . S ta tis tic a l C h a o s
A second concept o f chaos, closely related to the first, is statistical (Lasota and Mackey
1985). Here, we follow the treatm ent in Domowitz and El-Gam al (1993,1994) and El-Gam al
(1991).
Let g be a density function defined on J. That is, f j g ( n ) d n = 1 and g(n ) > 0 for all
n 6 J. If we draw from g and apply the map f* : J —►J to each draw, we have a new
distribution of points on J. D enote this distribution by
Then following Dom ow itz and
El-Gam al (1993,1994), we say that / exhibits sta tistica l chaos if /* exhibits th e ergodic
property or th e m ixing property.
D e fin itio n 4 .4 . T h e m a p f : J —* J is ergodic if
fan ^
for aU 9 € G.
For regularity conditions on the lim iting density function, q, and the set of density func
tions, G , see Dom owitz and El-Gam al (1993,1994). (Obviously, G cannot include density
functions which place mass exclusively on a single periodic point.) For / to be ergodic does
not actually require th at f* settle down for large i. The m ixing property does require this.
D e fin itio n 4 .5 . T h e m a p f : J —*■J is m ixin g if
lim f l — q, for all g € G.
t—*oo
9
T he properties of ergodicity and m ixing are closely related to the notions of topological
transitivity and sensitive dependence. For exam ple, consider a density function, g, which
21
assigns positive probability to an arbitrarily selected and extrem ely sm all interval o f initial
conditions. M ixing requires that the orbits of these points eventually cover the sam e range
in J as if the initial conditions were instead drawn from a density that assigns positive
probability to every subinterval o f J.
We adopted the sim ulation-based approach of Dom owitz and El-Gam al to investigate
w hether our / map exhibits statistical chaos. Thus, we considered two g functions. One
places a uniform distribution on th e interval [0.16,0.32] and the other places a uniform
distribution on the interval [0.58,0.71].
In each case, we drew 1,000 tim es from th e g
function and com puted / 3,00° for each draw. The resulting histogram s are shown in Figure
6. There are two interesting features of these histogram s. First, to the unaided eye th ey
appear very sim ilar to each other, consistent w ith th e notion that / satisfies th e mixing
condition. Still, the differences can be reasonably substantial, as the bottom graph in Figure
6 shows. Second, th e histogram s in Figure 6 closely resemble the orbit histogram s shown
in Figure 5. T his suggests that our / map approxim ately satisfies concepts of ergodicity
in standard econom etrics textbooks (for exam ple, H am ilton 1994, pp.
46-47), in w hich
sta tistical properties o f individual sam ple realizations (that is, histogram s of orbits) coincide
w ith q = lim ^oo f g.
5. S u n s p o t E q u i l i b r i a
In th is section, we study equilibria of our m odel in which prices and quantities respond
to s t . We construct tw o equilibria to illustrate the possibilities. T he first, which we call
22
a conventional sunspot equilibrium, uses fi only. This equilibrium is constructed near the
determ inistic steady state, n2, which, as noted above, has a continuum o f determ inistic
equilibria which converge to it. Our choice of name reflects th at th is type o f equilibrium is
standard in the quantitative sunspot literature.16 The second equilibrium considered, which
we call a regim e switching sunspot equilibrium, involves stochastically sw itching betw een ft
and /„ . Our analysis o f these equilibria focuses on their welfare and business cycle properties.
For this analysis, we find it useful to use the business cycle properties o f U .S. data and o f a
standard real business cycle model as benchmarks. We conclude this subsection by making
som e observations about the econometrics of sunspot m odels in general and by discussing
the em pirical plausibility of our model.
5.1. C onventional Sunspot Equilibrium
In this equilibrium , s € R is independently distributed over tim e, w ith s = —0.06 and
s = 0.06 w ith probability 1 /2 each. These values for s were chosen so that th e equilibrium ’s
im plication for the standard deviation of Hodrick-Prescott detrended, logged equilibrium
output coincides w ith th e corresponding figure in th e data. G iven any n, next period’s hours
worked, n \ is com puted by first drawing s and then solving
n' = /,( n ,s )
(5.1)
16
Because a continuum of other nonstochastic equilibria exists near the steady state equilibrium, n2,this
equilibrium is said to be indeterminate (Boldrin and Rustichini 1994, p. 327). For a general discussion of
the link between indeterminate equilibria and sunspots, see Woodford (1986a). Examples of quantitative
analyses that construct sunspot equilibria in the neighborhood of indeterminate equilibria include Benhabib
and Farmer (1994,1995), Farmer and Guo (1994,1995), and Gali (1994a,b).
23
where ft is defined in (3.9). We set the initial level of hours worked, no, to n2. R ecall th at
n2 is th e higher of th e tw o determ inistic steady states associated w ith th e lower branch, fi.
T hat is, o f the two solutions to x = f i ( x , 0), n2 is the larger of the two.
To establish th at th is stochastic process for employm ent corresponds to an equilibrium ,
it is sufficient to verify that the conditions of the characterization result are satisfied. The
first condition is satisfied by construction, and the second is satisfied because n (s4) remains
w ithin a com pact interval th at is a strict subset o f th e unit interval. That is, let a be the
sm aller o f th e tw o values of n th at solve a = /j(a , —0.06), and let b > a be the unique value
o f n w ith th e property o = ft(b, —0.06). Here, a and b are 0.0249 and 0.9509 after rounding.
We verified th at if a < n < b, then a < vl < b for n' = /j(n , —0.06) and n' = /j(n , 0.06).
Thus, prob[ a < n' < b | a < n < 6] = 1. It follows that a < n { s l ) < b for all histories,
s ‘, w ith
> 0. T he conditions of the characterization result are satisfied, and so we
conclude that n(s*) corresponds to an equilibrium.
T he first-m om ent properties o f this equilibrium are reported in Table 1. T hey are sim ilar
to the corresponding properties o f the U .S. data and o f th e real business cycle m odel. T he
second-m om ent properties of this equilibrium (see Table 3) also compare favorably w ith
th e corresponding sam ple analogs, at least relative to the performance o f th e real business
cycle m odel (see Table 2). In th is context, three observations are worth stressing. F irst,
note th e equilibrium 's prediction that consum ption is sm ooth relative to output and that
productivity is roughly as volatile as hours worked. In th e latter respect, th e conventional
sunspot equilibrium actually conforms more closely to the data than does th e real business
24
cycle m odel. The real business cycle m odel im plies th at productivity is about 65 percent
more volatile than hours worked, whereas th e conventional sunspot equilibrium im plies that
productivity is about as volatile as hours worked. In th e data, productivity is about 30
percent less volatile than hours worked. Second, hours and productivity are both procyclical
in the equilibrium , as they are in the data. The equilibrium’s im plication th at productivity
is procylical reflects the increasing returns in the model.
Procyclical productivity helps
account for the fact th at equilibrium hours worked and consum ption are both procyclical in
the m odel. Finally, the model inherits a shortcom ing of standard real business cycle models
in overpredicting the correlation between productivity and hours worked. In th e data, this
quantity is essentially zero.
Some o f these properties can also be seen by exam ining the plots in Figure 7. T hey
are graphs o f th e logged and Hodrick-Prescott filtered data from the equilibrium described
above." Consum ption is sm ooth and investm ent is volatile in these graphs. In addition,
hours worked and productivity are seen to be procyclical. Overall, this sunspot equilibrium
compares quite well to th e real business cycle m odel in its ability to m im ic key features of
postwar U .S. business cycles.
5.2. R egim e Sw itching Sunspot Equilibrium
For this equilibrium , s — [s (l),s(2 )] 6 R 2, w ith s ( l) € {u, 1} and s(2) =
€ {— 0.06,0.06}.
We set the date 0 value of hours worked to n(s<j) = n 2. We use th e following reclusive
procedure to assign a level of employment to each history, s ‘, th at is logically possible, given
the specified sq. For any history, s ‘, and associated level of em ploym ent, n (s‘), let n (st+1) be
25
as follows:
n (st+ l) = f , t+l(l) (n(s*), s t+ 1(2))
(5.2)
for t = 1 ,2 ,.... We construct an equilibrium by devising a sequence o f probabilities, /*(s‘),
th at assigns positive probability only to histories, s*, for which a < n (s‘) < 6, for som e a, b
such th at 0 < a < b < 1. W hen the value of s ( l) changes along a history, we say there has
been a regim e switch.
Consider the following probabilities for s t+1(l):
0.9, n 1 < n (s‘) < n 2, n 1 = 0.0370, n2 = 0.9279
{
(5.3)
1, otherwise.
Let s(2 ) have th e sam e distribution as 5 in th e previous equilibrium .
the tw o elem ents of s are independently distributed.
We assum e that
Let a and b be defined as in the
conventional sunspot equilibrium. We verified num erically th at, under these circum stances,
if a < n (s‘) < b, then prob[a < n ( s t+l) < 6] = 1. It follows that, for all s l such th at /i(s ‘) > 0,
a < n (s‘) < 6.17 This establishes the second of th e two conditions of th e characterization
result. To establish th e first condition, note that by (5.2),
v (n (s* ),n (st+1)) = st+i(2 ), for all s*
(5.4)
17Oux specification of n1 and n2 is crucial for guaranteeing the second condition of the characterization
result. For example, with n 1 = a and n2 = b, histories, s', in which hours worked fluctuate between values
that approach 0 and 1 occur with high probability. With /i(s‘) specified in this way, the second condition of
the characterization result fails.
26
and by construction of the Euler error, s t+i(2 ),
5 3 /i(s ‘+1 | s‘)s t+i(2) = 0, for all s ‘.
(5.5)
T his establishes that the conditions of the characterization result are satisfied, and we con
clude th at n (s‘) corresponds to an equilibrium.
We now consider the dynam ic properties of the regime sw itching sunspot equilibrium.
First-m om ent properties are reported in Panel C o f Table 1, while second-m om ent properties
are reported in Panel B o f Table 3.
Regime switching is the key to understanding the
dynam ics of this equilibrium. Periodically, the economy switches to the upper branch, / u,
where employm ent is very high. The economy typically stays on the upper branch only
briefly, and when it switches down again, employment drops to a very low level: near o.
Employm ent then rises slow ly until another sw itch occurs, when the econom y jum ps to the
upper branch, and the process continues. The fact that the economy spends much tim e in
the left region of the lower branch explains why average employm ent in this equilibrium is
so low. This also explains why investm ent is, on average, negative. Regarding th e secondmom ent properties, output is substantially more volatile than it is in the data. A lso, output
displays very little serial correlation. The positive serial correlation produced by sojourns on
th e lower branch is offset by the negative serial correlation associated w ith transient jum ps
to th e upper branch. These observations are supported by th e tim e series plots o f th e logged,
H odrick-Prescott filtered data from this equilibrium, presented in Figure 8.
T he regime switching equilibrium nicely illustrates a type of sunspot equilibrium that
27
is possible. However, in contrast w ith the conventional sunspot equilibrium , th e secondm om ent properties o f th is equilibrium do not match the corresponding quantities in the
data.
5.3. E m pirical E valuation o f th e M odel
A variety o f other econom etric m ethods can be used to assess th e em pirical plausibility of
this m odel.18 One test o f the m odel analyzes the fitted values of the sunspot shocks, s. G iven
values for th e m odel param eters, these shocks can be recovered using em ploym ent data.19
For this test, we used th e data on per capita, quarterly hours worked covering the period
1955Q3-1984Q 1 studied in Christiano (1988) and Christiano and Eichenbaum (1992). T he
data, shown in Figure 9a, were converted into fractions o f available tim e worked under the
assum ption th at households’ available tim e is 15 hours per day (1,369 hours per quarter).
T he fitted values of s im ply that all quarterly U .S. observations on hours worked lie on //, th at
is, 3(1) = l throughout th e sam ple. T his complements th e findings of the second-m om ent
analysis reported above, which indicates that—w ithin the confines of th is m odel—regim e
sw itching does not improve our understanding of the aggregate data.
T he tim e series on th e fitted Euler equation error, 3(2), are shown in Figure 9c. T he m odel
requires th at th is shock satisfy (5.5). A ll dynamic m odels have at least one orthogonality
18For a formal statistical approach to the moment comparison strategy for testing undertaken in the previ
ous subsection, see the method based on the work ofHansen (1982) developed in Christiano and Eichenbaum
(1992). This analysis integrates parameter uncertainty into evaluations of the “distance” between model and
data second moments.
,9For any two consecutive observations on employment, n and n', s(2 ) = v(n, n1). Then, given s(2 ), one
finds the two values of i, ij < iu such that s(2) = v(n, x). Ifn' = xi, then s(l) = l, and s(l) = u otherwise.
28
condition like this. Generalized m ethod of moments (GMM) procedures for testin g it have
been developed and applied extensively, beginning w ith the work o f Hansen (1982) and
Hansen and Singleton (1982). These tests focus on a model’s im plication th at date t + 1
Euler errors be orthogonal to all inform ation available at date £, including a constant. The
evidence in Figure 9c indicates that this test fails: the sam ple mean of the fitted Euler error
is significantly negative ( —0.27), indicating that a nonzero constant is useful for predicting
this variable. The second-m om ent properties of the fitted values o f 5(2) are more consistent
w ith the theory. Figure 9b shows the scatter plot of consecutive values of fitted Euler errors,
and it suggests that the first-order autocorrelation of 5(2) is not significantly different from
zero. (T he point estim ate is —0.18, w ith standard error 0.0920.) Figure 9c shows th e scatter
plot of the empirical measure of hours worked at date t, n t , against th e date t + 1 fitted
Euler error, 5e+1(2). Here too, the evidence does not imply a strong relationship between
th ese variables. T he point estim ate of the correlation between these two variables is —0.20,
slightly more than tw ice the standard error of 0.09. These results are subject to tw o caveats.
First, they do not take into account sampling uncertainty in th e estim ated values of a and
S. However, th e results in Christiano and Eichenbaum (1992) suggest th at th is is very sm all
and unlikely to change the results. Second, they are based on arbitrarily settin g a = 2. A
conventional GMM approach to this would select a value for <r to ensure th at sam ple analogs
of the population orthogonality properties of the Euler errors are satisfied. For exam ple, the
value of a th at sets the sam ple average of the fitted Euler errors to zero is a = 2.72. Apart
20The standard error is l/\/115, where 115 is the number of observations in the sample.
29
from changing th e sam ple mean, this change in the value of a does not alter the properties
of the Euler errors reported above.
We conclude th at there is little evidence in the Euler errors against th e m odel and that
the upper branch appears not to be operative in the data.
Interestingly, conditional on ruling out the upper branch, / u, th e m odel im plies a re
duced form relation very much like the one found in standard m odels driven by fundam ental
shocks. For exam ple, equation (5.1) closely resembles th e equilibrium relation for employ
m ent im plied by the general equilibrium model analyzed in Sargent (1979, p. 377). T he only
qualitative difference is that in th e latter, the shock variable, a, is a com bination of distur
bances to preferences and technology, while here it is a sunspot shock, s(2 ). An im plication
is th at th e m odel can be estim ated and tested using the sam e maximum likelihood strategies
pursued in A ltug (1989), Christiano (1988), Hall (1996), and M cGrattan, Rogerson, and
Wright (1996). This observation is consistent w ith the notion that sunspot m odels sim ply
offer a new source o f shocks. From an econometric perspective, they are not qualitatively
different from m odels w ith fundam ental shocks.
Our final test of th e m odel focuses on its im plications for the aggregate production
technology, (2.11). To assess the plausibility of this form ulation, we plot th e log o f output per
unit o f capital versus th e log of per capita horns worked in Figure 10. (See th e observations
marked *.) T he output and capital stock data used are the quarterly data covering th e period
1955Q 3-1984Q 1 studied in Christiano (1988) and Christiano and Eichenbaum (1992). T he
data do indicate a positive relation, but it is not as strong as the one im plied by th e m odel, in
30
which l o g ( y / k ) = 2 log(n). One way to assess th e plausibility of the m odel is to compare this
line w ith th e least squares line fitted through the U .S. data points. A factor com plicating
the comparison w ith this data is th at the m odel does not contain a theory o f th e error term
in this relation—clearly one is needed, given the wide dispersion of the U .S. data points.
Now, suppose the errors reflect technology shocks, which could easily be incorporated into
the analysis. Then, assum ing equilibrium labor responds positively to technology shocks, a
standard sim ultaneity bias argument im plies that the slope of th e least squares line is biased
upward, as an estim ate of the power on hours worked in the aggregate production function.
(See Klenow 1992.) Thus, conditional on interpreting the dispersion o f data points in Figure
10 as reflecting the effects of technology shocks, we conclude that the data in th at figure
constitute a rejection of the very high power on hours worked in the production function of
this paper.21
To summarize the results so far, the analysis in this section shows that— at least in the
context o f our model— it is far from true that “anything goes” em pirically w ith sunspot
m odels. T he m odels can be tested using standard econometric m ethods— GMM procedures
for comparing sam ple and m odel-based second mom ents, GMM procedures for testin g Euler
equations, and standard maximum likelihood procedures. A lthough th e conventional sunspot
equilibrium does a surprisingly good job of accounting for business cycle phenom ena, in the
end, its strong increasing returns assum ption is rejected by the data.22
2 1 This
complements findings in Burnside, Eichenbaum, and Rebelo (1995) and in the references they cite.
For an attempt to extend our analysis to a version of the model with a smaller externality, see Guo and
Lansing (1996).
22
31
5.4. W elfare A n alysis
We approxim ated th e expected discounted u tility for our equilibria using a M onte Carlo sim
ulation m ethod. For th e conventional sunspot equilibrium and th e regim e sw itching sunspot
equilibrium , th e expected present discounted utilities axe —378.21 (0.24) and —570.58 (1.77),
respectively (numbers in parentheses are M onte Carlo standard errors).23 To understand
the im pact on u tility o f variance in the Euler error, s(2 ), we also com puted expected
u tility for a high variance version o f our conventional sunspot equilibrium . In this case,
s(2) € { —0.55,0.55}. T he expected present value of u tility for this equilibrium is —363.35
(2.14).
T he present discounted level of u tility associated w ith the constant em ploym ent
determ inistic equilibrium at ri2 is —378.49. We refer to this equilibrium as th e con stan t
em ploym en t equilibrium.
To compare these welfare numbers, we converted them to consum ption equivalents. T hat
is, we com puted the constant percentage increase in consum ption required in th e constant
em ploym ent equilibrium to make a household indifferent between th at equilibrium and an
other given equilibrium. T he results are shown in Table 4. They indicate th at going from the
constant em ploym ent equilibrium to the regime sw itching sunspot equilibrium is equivalent
to a 289 percent perm anent drop in consum ption. Going to the conventional sunspot equi
librium is equivalent to a 0.9 percent permanent rise in consum ption, and going to th e high
23
For each equilibrium, we drew 1,000 histories, s‘,each truncated to be of length 2,500 observations.
Subject to the initial level of employment being n2 always, we computed consumption and employment along
each history. For each equilibrium, we computed 1,000 present discounted values of utility, vi,...,vioooOur Monte Carlo estimate of expected present discounted utility, v, is the sample average of these: v =
iwo Ei^° vi- The fact that we use a finite number of replications implies that v is approximately normally
distributed with mean v and standard deviation <r</v/l,0 0 0 ,where <7 < isestimated by the standard deviation
of vi,viooo* W e refer to 0i/\/1,000 as the Monte Carlo standard error.
32
variance version o f th at equilibrium is equivalent to an 11.2 percent rise in consum ption.
An interesting feature o f these results is th at, despite concavity in th e u tility function,
increasing volatility in s(2) raises welfare. This reflects a trade-off betw een two factors.
First, other things being the sam e, a concave utility function im plies th at a sunspot equilib
rium is welfare-inferior to a constant, determ inistic equilibrium (concavity effect). However,
other things are not the same. T he increasing returns means that by bunching hard work,
consum ption can be increased on average w ithout raising the average level of em ploym ent
(bunching effect). W hen the volatility of the model economy w ith initial em ploym ent n2 is
increased by raising the volatility of s(2 ), then the bunching effect dom inates th e concavity
effect. W hen volatility is instead increased by allowing regime switches, then th e concavity
effect dom inates. In interpreting these results, it is im portant to recognize th at they say
nothing about th e nature of the efficient allocations. A ll of the equilibria th at we consider
are inefficient, because o f th e presence of the externality in production.
6. P o l i c y A n a l y s i s
We now consider the im pact of various policies on the set of equilibria. We consider two
countercyclical tax policies th at reduce the set of interior equilibria to a singleton in that
output is a constant. We refer to the first as a pure stabilizer because it does not distort
margins in equilibrium. T he second tax policy introduces just the right distortions so that
the equilibrium supports the optim al allocations. We show that, for a tax policy to isolate
the efficient allocations as a unique equilibrium, it is necessary that the tax rate vary in the
33
right way w ith th e state o f the economy. For exam ple, under a constant tax rate policy, th e
equilibrium is not unique. Interestingly, th e equilibria are isolated in this case, so th at th ey
would escape detection under the usual procedure of analyzing local equilibria.
6.1. A P ure A u tom atic Stabilizer
In this section, we display a particular procyclical tax rate rule which reduces th e set o f
equilibria to a singleton w ith a* = n2 for all t (the constant em ploym ent equilibrium ). T he
tax policy has th e property th at in equilibrium, the tax rate is always zero and thus does not
distort any margins. Given our previous results for the constant em ploym ent equilibrium ,
this tax rate rule improves welfare relative to the regime sw itching sunspot equilibrium , but
actually reduces welfare relative to the conventional sunspot equilibrium . T he possib ility
th at stabilization of a sunspot by government policy m ight reduce welfare should not be
surprising, given that both the sunspot equilibrium and the n2 equilibrium are inefficient.
Consider th e following tax rate:
r(n ) = 1 - —
n
(6.1)
where n denotes econom ywide average employment and n2 is th e higher o f th e tw o n on sto
chastic steady state em ploym ent levels. (See Figure lb .) N ote th at this tax rate is zero w hen
aggregate em ploym ent is n2. It turns positive for higher levels of em ploym ent and negative
for lower levels.
Let v ( n , n ' ) denote (3.6) after substituting out for r(n ) from (6.1). It is easily verified
34
th a t, fo r each v alu e o f n , th e re is a t m o st one n ' th a t solves v ( n , n ') = 0. T h is is g iv en b y
n' = /( n ) =
n2 - AT(n)(l - 6)
n2[1 -f aif(n )]
where
K (n )
= -^r(l a [ ti)
n),
A(n) = n2+ 1 -
6 -
^n2(l - n).
a
T he function, / , and its derivative, / ' , have the property that at n = 1,
/(1 ) = 1, / '( ! ) =
0
a n 2 + 1 —6
2 - 6
< 0
since a n 2 < 1. Figure 11 shows / under our baseline parameter values. For convenience, th e
two branches o f v = 0, f u and //, are also displayed.
There are three things worth em phasizing about / . First, it cuts the 45-degree line from
below at n = n2, and it intersects the horizontal axis at a positive level of em ploym ent. T his
im plies th at there is no infinite sequence, nt, t = 0 ,1 ,2 ,..., w ith no < n 2 and n t — f { n t - 1 ),
such th at
> 0 for all t. Since satisfaction of th e Euler equation, v = 0, is a necessary
condition for an interior solution to the household problem, it follows that there is no interior
equilibrium w ith no < n2. Second, a sequence of employm ents, nt, t = 0 ,1 ,..., which has th e
property nt = /(n t_ i) and no > n2, has the property nt —►1 as £ —►oo. Appealing again to
th e necessity of the Euler equation, we conclude th at there is no interior equilibrium w ith
no > n 2. Third, nt = n2 for all £ satisfies the Euler and transversality conditions and so
corresponds to an interior equilibrium. Thus, th e only determ inistic interior equilibrium is
35
the one that corresponds to n t = n2 for t = 0 ,1 ,.... That sunspot equilibria are also ruled
out follows from the fact that the Euler equation cuts the 45-degree line from below and
from the arguments in Woodford (1986a). These remarks establish
P rop osition 6.1. For t h e b a selin e p a r a m e te r iz a tio n a n d u n d e r th e t a x p o lic y in (6 .1 ), th e r e
is a u n iq u e in te r io r e q u ilib r iu m w ith n t
= n 2 fo r a ll t.
Note that under the tax rate policy considered here, rt = 0 in equilibrium. Evidently,
the mere threat to change tax rates is enough to rule out other equilibria. This feature of
fiscal (and monetary) policies designed to select certain equilibria is common in models with
multiple expectational equilibria. (See, for example, Boldrin 1992, p. 215 and Guesnerie and
Woodford 1992, p. 380-382.)
6.2. O ptim al A llocation s
The efficient allocations correspond to a fictitious planner’s choice of investment, employ
ment, and consumption to maximize discounted utility subject to the resource constraint.
We reproduce the utility function here for convenience:
°
t=0 **
togl1 - n (3*)]}-
(6-2)
The resource constraint is
c(s*) + fc(s‘) - (1 - S W s * - 1) < k i s '- 'f t n i s * ) ) 2, for all t, s*.
36
(6.3)
T h is p ro b lem sim plifies g reatly . T h u s, u sin g th e ch ange o f v aria b le in (3.2) a n d th e id e n tity
OO
5ZY
.P
t=0 st
1
(6.4)
^ s‘) !<>**(»'')
|lO g
ko
+ /? £
p
W
P
loS *(«*) |
the objective function can be written
(6.5)
In (6.5), consumption has been substituted out using the (scaled) resource constraint after
replacing the weak inequality in (6.3) by a strict equality. Notice that the objective in (6.5)
is separable across dates and states. This has two implications. First, unsurprisingly, the
efficient allocations are insensitive to sunspots. Second, the efficient levels of employment
and capital accumulation do not exhibit cycles. It is trivially verified that this result is
independent of the curvature on leisure in the utility function, the degree of nonconvexity on
labor in the production function, and the degree of homogeneity on capital in the resource
constraint.24 Thus, for example, increasing the gains from bunching production, by raising
the power on labor above 2, and reducing the associated costs, by making utility linear in
leisure, still does not imply that the efficient allocations exhibit cycles.
24
Lack of cycling in the efficient allocations also obtains for utility functions which are homogeneous of
degree 7 ^ 0 in consumption. See the Appendix for further discussion.
37
With our specification of preferences, optimizing (6.5) requires that the planner maximize,
for each t , s*,
log[n2 + 1 - 6 —A] +
0
log A + <xlog[l —n]
(6.6)
1 - 0
by choice of n and A, subject to
0 < A < n 2 + l —6, 0 < n < 1.
(6.7)
The objective, (6.6), is not concave, because of the nonconcavity in the production function.
However, for fixed n, (6.6) is strictly concave in A, and its optimal value is readily determined
to be A = 0(n 2 + 1 —6). Substituting this into (6.6), the criterion maximized by the efficient
allocations becomes
Y ~ g loS(n2 + 1 - 6) + & log(l - n)
(6.8)
after constant terms are ignored. The constraint on this problem is 0 < n < 1. There are
two values of n that set the first-order condition associated with maximizing (6.8), and the
larger of the two is the global optimum. This is given by n°, where
n
2
2 + o { \ — /3 )’
.
<r(l - /?)(! - 6)
2 + a ( l —($)
(6.9)
With the baseline parameter values, n° = 0.98, which implies that the optimal value of A is
1.94, or 94 percent per quarter. The fact that equilibrium employment is so high reflects the
38
fact th at th e efficient allocations internalize the externality in the production function.
It is easily verified th at th e tax rate which supports n° as an equilibrium is r = —2. It
is not surprising th at this involves a subsidy, since the tax m ust in effect coax individuals
into internalizing the positive externality associated w ith production. Consider first th e case
in which th e tax rate is sim ply fixed at r = —2 for every n. Let v (n, n') denote (3.6) after
substituting out for r = —2. In effect, reducing r from zero to -2 pushes the saddle in Figure
la down, so th at the w = 0 plane now covers the seat of the saddle. T he consequences can
be seen in Figure 12a, which displays the values of n! that solve v(n, n') = 0 for n € (0 ,1 ).
N ote the region of values for n for which there are no values of n' that solve v(n , n') = 0.
In the other regions, there are generally two values of n' th at solve this equation for each
n. Interestingly, th e unique intersection of these points w ith th e 45-degree line, at n°, is
associated w ith a slope greater than one.
A s a result, the equilibrium associated w ith
n ° ,n ° ,n 0, ... is determ inate. However, there is at least one other equilibrium, fi,n 0,n 0, ....
(See Figure 12a for n .) Evidently, the constant tax rate policy does not guarantee a unique
equilibrium.
One way to construct a tax regime that selects only the desirable equilibrium follows the
strategy taken in the previous subsection. Thus, consider
Evidently, w ith this policy, r(n °) = —2, so that there is an equilibrium associated w ith th is
tax policy which supports the efficient allocations. Also, it is easily verified that— following
39
the sam e reasoning as in th e preceding subsection—th e Euler equation has only one branch.
In addition, we found for th e baseline parameter values th at th is branch is m onotone, and
it cuts th e 45-degree line from below. It follows by th e logic leading to Proposition 6.1 th at
there is a unique interior equilibrium.
7. C o n c l u s i o n a n d D i r e c t i o n s for F u t u r e R e s e a r c h
We have displayed a m odel environment which rationalizes im plem enting a tax regim e which
is procyclical in the sense that if aggregate employment were to rise, th e governm ent stands
com m itted to raise the tax rate. In this sense, th e environment seem s to rationalize th e
im portance assigned by m acroeconom ists before the 1970s to devising autom atic stabilizer
tax system s.25 B ut, since a properly constructed tax regime elim inates fluctuating equilibria,
the actual tax rate is constant. We expect this basic result to survive in versions of our
m odel w ith fundam ental shocks. Thus, if there were technology shocks, we conjecture th at
the optim al tax rate regim e would move procyclically w ith sunspot shocks, but would not
vary w ith technology shocks. Assum ing an efficient tax regime elim inates sunspot equilibria,
the optim al “autom atic stabilizer” tax rate would then no t be procyclical in equilibrium .
An interesting question for future research would be to investigate w hat happens when th e
tax regim e cannot respond differently to fluctuations due to sunspots and to fluctuations
due to technology shocks. Possibly, under these circum stances an efficiently constructed ta x
regim e w ould exhibit procyclical behavior in equilibrium. Another interesting question for
25 For
a discussion of “automatic stabilizers,” see Christiano (1984).
40
future research would explore the robustness of our result th at a properly constructed tax
regim e necessarily stabilizes fluctuations. We have shown th at th is is so under a particular
hom ogeneity assum ption on the resource constraint. B ut, standard m odels do not satisfy
this condition.
41
Table 1: First-M om ent Properties
n
k/y
c/y
i/y
growth in k
growth in y
Panel A: U .S. D ata
0.23
10.62
0.73
0.27
1.0047
1.0040
Panel C: Real Business C ycle M odel
0.23
10.64
0.73
0.27
1.0040
1.0040
Panel B: Conventional Sunspot
0.309
0.745
10.46
0.255
1.0045
1.0046
Panel D: Regime Switching Sunspot
0.094
5.17
298
-4 .1 7
0.989
4.74
Note: Entries in the table axe the mean of the indicated variable. U .S. data results are
taken from Christiano (1988). Statistics based on m odel econom ies are com puted using
100 artificial data sets of length 114 each. Entries in th e table are an average of 100.
42
Table 2: Second-M oment Properties
xt
Correlation of y t w ith x t+T
Ox/Oy
r = 2
r = 1
T = 0
r = —1
T = —2
Panel A: U .S. D ata
y
0.02
0.65
0.86
1.00
0.86
0.65
c
0.46
0.48
0.66
0.78
0.76
0.61
i
2.91
0.33
0.56
0.71
0.68
0.57
n
0.82
0.69
0.81
0.82
0.66
0.41
y /n
0.58
0.12
0.32
0.55
0.55
0.53
y/n,n
0.70
-0 .1 7
-0 .0 7
-0 .0 3
0.21
0.33
Panel B: Standard Real Business Cycle
y
0.02
0.51
0.74
1.00
0.74
0.51
c
0.55
0.59
0.78
0.98
0.69
0.44
i
2.37
0.45
0.70
0.99
0.76
0.55
n
0.38
0.40
0.67
0.98
0.77
0.57
y/n
0.63
0.57
0.78
0.99
0.71
0.47
y/n,n
1.65
0.61
0.77
0.94
0.61
0.33
Note: R esults axe taken from Christiano and Todd (1996, tables 2 and 3). Panel B results
are based on 2,000 artificial observations sim ulated from a standard real business cycle
m odel. In both panels, prior to com puting statistics, data were logged and HodrickPrescott filtered. T he m odel corresponds to the one in this paper, w ith a = 3.92,
7 = 0, 6 = 0.021, a = 0.344, and a production function has the form Y = K a (zn)^l ~a\
w ith z — z_iexp(A ), and A ~ 7/iV (0.004,0.0182). The last two rows o f each panel
report the standard deviation of productivity (y /n ) relative to that of hours (n ). T he
correlations reported there are ccrrr[(y/n)t , nt_T].
43
Table 3: Second-M oment Properties, Sunspot Equilibria
xt
II
r = 2
H-*
Correlation o f y t w ith x t+T
T
= 0
T = —1
T
= -2
Panel A: Conventional Sunspot
y
0.02
0.35
0.63
1.00
0.63
0.35
c
0.33
0.58
0.72
0.87
0.44
0.13
i
3.13
0.26
0.57
0.99
0.66
0.40
71
0.51
0.22
0.54
0.98
0.66
0.42
y /n
0.52
0.46
0.69
0.98
0.57
0.27
y/n,n
1.02
0.49
0.68
0.91
0.44
0.11
Panel B: Regime Switching Sunspot
y
0.78
-0 .0 7
-0 .0 7
1.00
-0 .0 7
-0 .0 7
c
0.06
0.25
0.30
0.35
-0 ,4 2
-0 .3 5
i
na
na
na
na
na
na
n
0.54
-0 .1 1
0.11
0.99
-0 .0 1
-0 .0 3
y/n
0.47
-0 .0 3
-0 .0 2
0.99
-0 .1 3
-0 .1 2
y/n,n
0.88
0.01
0.02
0.96
-0 .1 9
-0 .1 7
Note: M odel statistics axe com puted in th e sam e wav as for Panel B in Table 2. T he
notation “na” appears in the investm ent colum n of Panel B, because gross investm ent
is often negative.
44
Conventional Sunspot I
0.9%
Conventional Sunspot II Regime Switching
11.2%
-289%
Note: This is the constant percentage decrease in consumption required for households in
the indicated equihbrium to be indifferent between that equilibrium and the constant
equilibrium at n = n2. Let v denote the discounted utility associated with the constant
employment level. Let v denote the discounted utility associated with one of the other
equilibria. Then, the number in the table is 100[exp((l —0 ) ( v — v ) ) —1].
45
A . A p p e n d i x : Linearity of Policy R u l e s U n d e r H o m o g e n e i t y
In this appendix, we establish efficiency for a policy of the form, fct+i = A*kt and n* = n*,
where A*,n* are fixed numbers. We do this for a class of economies in which the resource
constraint is homogeneous in capital and in which preferences are homogeneous in consump
tion. Our result parallels that in Alvarez and Stokey (1995), except their environment does
not explicitly allow for variable hours worked.
Consider the following planning problem:
max
V j9‘«(c(1nt) ,0 < / ? < 1
(A.1)
t_ o
subject to the following feasibility constraints:
fco > 0 is given, 0 < c* < F(fct,k t + lf nt), 0 < n* < 1,
k t +1
> 0, for £ = 0 ,1 ,2,....
We assume that F is homogeneous:
F ( k , k', n ) =
n)> where /(A , n) = F ( 1, A, n), A =
xj;>
0.
(A.2)
In terms of A and n, the constraints on the planner axe:
B =
{A,n : 0 < n < 1, 0 < A, and /(A ,n ) > 0}.
That is, the planner’s feasible set is the set of infinite sequences, {At,
Xt , n t e B for each t > 0. We place the following assumptions on / :
/ : B —*■R , continuous, decreasing in A, and increasing in n.
such that
(A.3)
Also,
there exists a largest value of A, A > 0, such that /(A, 1) > 0
*pr<i.
(A.4)
46
and
there exists
0
< h <
We place th e following assum ptions on
u(c, n) = <Pg{n)/ 7 ,
7
^
0
1
such th at / ( 1 , n) >
0
.
(A .5)
u:
, g (n ) >
0
, g is continuous and decreasing.
(A .6 )
We have the following proposition:
P r o p o s itio n A .l . I f
(i) th e functions F an d u s a tisfy (A .2)-(A .3), an d (A .6 ),
(ii) (A .4 ) holds when 7 > 0 , a n d (A .5) h olds when 7 < 0 .
then, a p o lic y o f th e follow ing form solves (A .l):
kt+i = A*fct, n t = n * , t > 0, for fixed (n*, A*) G B .
P r o o f . W rite
u {c,n )
=
( f ( X , n ) ) y g ( n ) / q f . A lso,
kT"
* = 1,2,.
=
Sim ple su b stitu tio n establish es
v(ko) =
max
= k'tf’w
{*i+i,nt}£o t=0
where:
00
w
(/(A t,rtt))7 ff(ut)
max
{(At,nt)€B)t«0 t=0
(A .7)
7
We establish —00 < w < 0 0 . W hen 7 < 0, then u is bou n ded a b o ve b y zero an d so trivially,
w < 0 0 . For th e case 7 > 0 , consider th e (infeasible!) p o lic y o f a p p lyin g th e en tire tim e
en dow m en t b o th to la b o r effort and to leisure, an d o f a p p ly in g all o f o u tp u t b o th to con
su m p tio n and to in vestm en t. T h e value o f th is p o lic y is w = (/(0, l ))7 <7(0) / £7 ( 1 —/JA7^ ) ! .
We h ave w < 0 0 , since w < w < 0 0 . To establish — 0 0 < w when 7 > 0 n o te s im p ly th a t
47
u is b o u n d ed b elo w b y zero in th is case. For th e case 7 < 0, n o te th a t th e feasible policy,
At = 1 , n* = n, for t > 0 has retu rn k ^ w , w here w = f ( \ , h ) ' 1g ( n ) / [7 ( 1 —/?)], so th a t
— 0 0 < U) < w .
W e h a ve esta b lish ed th a t w is a fin ite scalar. B y w ritin g (A .7 ) o u t explicitly, on e verifies
th a t w satisfies th e follow ing expression:
w = (Ao,no)€5 {^(/(* o ." o )) 7 * (" o) h + 0 (Ao)
* 1
J •
(A .8 )
L e t A* a n d n* d en o te values o f Ao an d n 0 th a t so lve th e above m a x im iza tio n p ro b lem . T h e
resu lt follow s from th e fa ct th a t th ese so lve a p ro b lem in which th e o b je ctiv es an d co n strain ts
are in d ep en d en t o f ko. Q .E .D .
R em a rk 1 . T h e p r o o f for th e class o f u tility fu n ction s u(c, n) = log(c) + g(n ) is a trivia l
p e rtu rb a tio n on th e argum ent in th e tex t.
R em a rk 2 . When 7 > 0 , then th e fixed p o in t p ro b lem in (A .8 ) can b e show n to be th e
fixed p o in t o f a contraction m apping. In th is case, w in (A .7) is th e o n ly solu tion to (A .8),
an d th e con traction m a p p in g th eorem p ro vid es an ite r a tiv e algorithm for co m p u tin g w , A*,
and n*. W hen 7 < 0, th e m a p p in g im p lic itly defined in (A .8 ) is n o t necessarily a contraction.
A lv a re z a n d S to k e y (1995) s u p p ly ite ra tiv e schem es for co m p u tin g w in th is case.
48
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Figure la: The v(n ,n') Function
Figure lb: Contour:
v {n ,n ')
=0
Figure 1c: Close-up of Figure 1b
n (Hrs. T o d a y )
Note: Figure la is a three-dimensional view of the function v in equation (3.6), computed using the standard
param eter values. The dark and light regions identify the parts o f v that are less than and greater than zero,
respectively. Figure lb shows the values of n' that set v(n,nr) to zero, given n. H e re ,/ a n d /u denote the
low er and upper branch functions defined in (3.9), respectively. Also, n 1 and n2 denote the points w here//
crosses the 45-degree line. Figure 1c displays /(/z)-/z from Figure lb for values of n in a neighborhood of
the origin. It shows t h a t / first cuts the 45-degree line from below, at n l , and then again from above, at n2.
Figure 3: Testing Sensitivity to Initial Conditions
Figure 4a: Fraction of Available Time Worked
Figure 4b: Figure 4a, continued
hours worked
Figure 2: Two Equilibria on the Lower Branch
occurrences per interval
occurrences per Interval
occurrences per Interval
Figure 5: Histograms of Four Orbits
250.00 -r
n (0) s 0.25
200.00 -150.00
100.00
50.00
0.00
■«yOao — C0i0r-*0**— TT' OooOCMt Or ^ O — e O' Oa p QCN^ ' QO* — com
d d d d d o d d d d d d d d d d d d d d d d d d d d d d d
n (0) = 0.59
0.00
'^■vOoO'—
O LCONri^wOM —c o’
»3
M ij ^O T
P ^r O
>0
— N
COIO
>
---CN(N
n,cOoafOoO,7C ,«
^ 'i n~ iCnOi*oQfC) O;0Q' C0 M' 0^ <
) '0O* N
N
d d d d d o d d d d d d d d d d d d d d d d d d d d d d d
250.00 T
occurrences per Interval
d d d d d o d d d d d d d d d d d d d d d d d d d d d d d
250.00 T
’T
c oCvSoCr '^ pI*N' (—0 5e '0C«a<o0 o,?cTNIi'o7r ^^ O
T ''O^
—NcoiO
' -' O
- c>O
- '(—
NW
^ l' —
0 lc O'
A wQc
i oDQCNT
'0O'0
0'0
NN
d d o d d d d d d d d d d d d d d d d d d d d d d d d d d
Note: Four histograms of 50,000 iterates on/, defined in (4.1), using standard parameter values, except 0^=1.25. The
iterates are differentiated according to the initial condition on n, as indicated.
Figure 6: Density Functions Induced b y /
250.00 T
| 200.00 c
n (0) in interval (0.16, 0.32)
& 150.00 CO
©
c© 100.00
K 50.00
0.00
■<T>oO'-comcooCN'«3,
>o^ — c o^,,!rCotOiO«OiO'0'0'0<)r^rvf^
maoQCN^r^O'-eooaoocsco
— — •— CMCMCNCMcOCOCO«OCO^T^,
d d d d d d d d d d d d d d d d d d d d d d d d o d d d
TT'0«X>
C O i O ^ O ' - ^ ' O O O O C MV
f ^!T' <3r
O ' —CO'
Qc OQCN^
' QC^
*— — — —
cNCM<NCNCMcocoeoeo^r
TrO
^T,
mu5mu535'
0' 0' OOf
' N. r—
' >.com
r'*
d d d d d d d d d d d d d d d d d d d d d d d d d d d d o
Difference Between the First Tw o Histograms in Figure 6
QCMTT'OO*'—
M ri ^O
cococococoMc
’
So
Ti
TO
r o^ OT QmCmMm^mrv^OQ'^O---Oc'oOm'«OO^O rC ^
d d d d d d d d d d d d d d d o d
Note: The top two graphs show the density of iterate number 3,000 on the map/defined in (4.1), with initial
condition drawn uniformly from the indicated interval.
F ig u r e 7 : L o g g e d a n d H P F ilt e r e d D a ta fro m C o n v e n t io n a l S u n s p o t E q u ili b r iu m
Note: These graphs depict a realization of length 114, when we use our baseline parameterization.
Figure 8: Logged, and H P Filtered Data from Regime Switching Sunspot Equilibrium
Log HP Filtered y and n
Log HP Filtered y and c
Log HP Filtered y and y/n
Note: These graphs depict a realization of length 114, when we use our baseline parameterization.
Figure 9a: U.S. Data, Time Worked
Figure 9b: Euler Errors
Figure 9c: Scatter Plot
Figure 9d: Scatter Plot
1
Euler error (t)
*
0.5
*
*
X*
*
* ,
0
s*
*
Ig *
-0.5
*
*
*
2
<D
v_
*
*
u .
.32
LU
*
*
-0.5
X
’ -1
-0.5
0
0.5
Euler error (t-1)
1
0.22
0.23
0.24
hours worked (t-1)
0.25
log, output per unit of capital
Figure 10: Hours Worked Versus Output Per Unit of Capital
Figure 11: Euler Equation, v(n,n')=0, for Taxed and Untaxed Economies
n-n
@FedFRASER