DP2016-21

Regim e-Switching Sunspot Equ ilibria ina One-Sector Growth Model with
Aggrega te Dec reasing Retu rns and Sma ll
Externalities
Takashi KAMIHIGASHI
May 17, 2016

Regime-Switching Sunspot Equilibria in a
One-Sector Growth Model with Aggregate
Decreasing Returns and Small Externalities
Takashi Kamihigashi
May 17, 2016

Abstract
This paper shows that regime-switching sunspot equilibria easily arise in a one-sector growth model with aggregate decreasing returns and arbitrarily small externalities. We construct a regimeswitching sunspot equilibrium under the assumption that the utility function of consumption is linear. We also construct a stochastic optimal growth model whose optimal process turns out to be aregime-switching sunspot equilibrium of the original economy under
the assumption that there is no capital externality. We illustrate ourresults with numerical examples.

 Research Institute for Economics and Business Administration (RIEB), Kobe Univer-sity, Rokkodai, Nada, Kobe 657-8501 Japan
Email: tkamihig@rieb.kobe-u.ac.jp

1

Introduction

In macroeconomics, sunspot equilibria are often associated with local indeterminacy, or the existence of a locally stable steady state. In the context ofgrowth models, the phenomenon of local indeterminacy has been well known
since Benhabib and Farmer (1994) and Farmer and Guo (1994). While earlier
results required unduly large degrees of increasing returns and externalities,1
local indeterminacy has been established for various settings under less objectionable assumptions, such as decreasing returns to labor (e.g., Pelloni and
Waldmann, 1998), moderate externalities (Dufourt et al., 2015), aggregate
constant returns to scale (e.g., Benhabib et al., 2000; Mino, 2001), and arbitrarily small increasing returns and externalities (e.g., Kamihigashi, 2002;
Pintus, 2006).
This paper seeks to point out the possibility of an additional mechanismthat gives rise to sunspot equilibria in an economy with aggregate decreasing
returns to scale and arbitrarily small externalities. The combination on which
we focus, aggregate decreasing returns to scale in tandem with arbitrarily
small externalities, sets a stage almost indistinguishable from the standard
neoclassical setting, posing a challenge to proponents of sunspot equilibria.
Instead of small fluctuations around a locally indeterminate steady state,
we consider large fluctuations caused by a regime-switching sunspot process.
Assuming that the sunspot process is a two-state Markov chain, we constructregime-switching sunspot equilibria in which labor supply is positive in one
state and zero in the other.
Although this type of regime-switching sunspot equilibrium is rather extreme and may not match many of the empirical regularities discussed in thelocal indeterminacy literature, there are merits to studying such equilibria in
addition to local sunspot equilibria driven by local indeterminacy. First, local
sunspot equilibria can explain only small fluctuations around a steady state,
whereas economic events of great magnitude—such as the Great Depressionof the early 20th century, Japan’s “lost decades” since the early 1990s, and
the Global Financial Crisis of 2007–2008—are characterized by large downfalls. Our model can at least generate large and sudden downfalls from asteady state. Second, our analysis suggests that regime-switching sunspot
equilibria of the type considered in this paper are widespread in models with1
See Benhabib and Farmer (1999) for a survey of earlier results. See Grandmont (1989,
1991) for a discussion of the relations between the local stability properties of a steadystate and the possibility of sunspot equilibria.

1externalities, and can even coexist with local sunspot equilibria. Our approach thus complements, rather than substitutes, the more common local
indeterminacy approach. It would be possible, in fact, to construct a modelin which two types of sunspot shocks are present and both small and large
fluctuations are endogenously generated. While the construction of such amodel would be beyond the scope of this paper, our analysis can form a basis
for further development in this direction.
We present two main results in this paper. First, assuming that the utility function of consumption is linear, we establish the existence of a regimeswitching sunspot equilibrium by recursively solving the Euler condition for
capital and the first-order condition for labor supply, and then verifying theassociated transversality condition. These conditions are easy to verify especially when the utility of consumption is linear, as none of the three aforementioned conditions depend on consumption in such a scenario. Second,
assuming an absence of capital externalities, we establish the existence of aregime-switching sunspot equilibrium by constructing a stochastic optimal
growth model whose optimal process turns out to be a sunspot equilibriumof the original economy. This latter result is somewhat similar to the observational equivalence result shown by Kamihigashi (1996). In contrast to
Kamihigashi (1996), however, we only use the three aforementioned conditions to verify that an optimal process for the stochastic optimal growthmodel can be interpreted as a sunspot equilibrium. Both results are illustrated with numerical examples.
In addition to the results on local indeterminacy previously mentioned,
this paper is also related to results on global indeterminacy (e.g., Drugeon
and Venditti, 2001; Coury and Wen, 2009) and regime-switching sunspot
equilibria (e.g., Drugeon and Wigniolle, 1996; Dos Santos Ferreira and LloydBraga, 2008).2 While our findings show the existence of regime-switching
sunspot equilibria based on global indeterminacy, they differ from the existing literature in that our model deviates only slightly from the standard
neoclassical setting under our assumptions of aggregate decreasing returns
and small externalities.3 Our model can also be viewed as a variant of theFarmer-Guo (1994) model with aggregate decreasing returns and small externalities.
2

See Clain-Chamosset-Yvrard and Kamihigashi (2015) for an example of a regimeswitching sunspot equilibrium in a two-country model with asset bubbles.
3
Kamihigashi (2015) shows that multiple steady states are possible even without externalities.

2

The remainder of this paper is organized as follows. In the next section we
present the model along with basic definitions and assumptions. In Section 3
we show a standard result that offers a sufficient set of conditions for a feasible
process to be an equilibrium. In Section 4 we present our main results alongwith numerical examples. In Section 5 we conclude the paper by discussing
possible extensions. Longer proofs are relegated to the appendices.

2

The Model

We consider an economy with many agents, each of whom solves the following
maximization problem:8̧
max

tct ,nt ,kt 1 u8t0Eβ t rupct q  wpnt qs

t 0

s.t. @t P Z , ct kt 1  f pkt , nt , Kt , Nt q
ct , kt 1 ¥ 0, nt P r0, 1s,(2.1)

p1  δqkt,

(2.2)
(2.3)

where ct is consumption, nt is labor supply, kt is the capital stock at the
beginning of period t, Nt is aggregate labor supply, and Kt is the aggregate
capital stock. The utility function u of consumption, the disutility function
w of labor supply, and the production function f are specified below. The
discount factor β and the depreciation rate δ satisfyβ, δ P p0, 1q.

(2.4)

In the above maximization problem, the initial capital stock k0 ¡ 0 and the8stochastic processes tKt u8t0 and tNt ut0 are taken as given. In equilibrium,
however, we have@t P Z ,Kt  kt ,Nt  nt .

(2.5)

To formally define an equilibrium of this economy, we first define a preequilibrium as a five-dimensional stochastic process tct , nt , kt , Nt , Kt u8t0 suchthat tct , nt , kt 1 u8
solvesthemaximization
problem(2.1)–(2.3)
given
k0 ¡ 0t08
and tNt , Kt ut0 . We define an equilibrium as a three-dimensional stochastic process tct , nt , kt u8t0 such that the five-dimensional stochastic process
8tct, nt, kt, nt, ktut0 is a pre-equilibrium. We also define a feasible process as3

a three-dimensional stochastic process tct , nt , kt u8t0 satisfying (2.2), (2.3),and (2.5).
We specify the functions u, w, and f as follows:c1σ  1
,
1σ
nγ 1
,
wpnq  η
γ 1
f pk, n, K, N q  θk α nρ K α N ρ .
upcq 

(2.6)
(2.7)
(2.8)

We impose the following restrictions on the parameters:
σ P r0, 1s,
θ, α, ρ, η ¡ 0,α, ρ, γ ¥ 0,α α ρ ρ ¤ 1.

(2.9)
(2.10)(2.11)(2.12)

If σ  1, then it is understood that upcq  ln c. Since σ P r0, 1s by (2.9),
u is bounded below unless σ  1. The inequality in (2.12) means thatthe production function exhibits decreasing returns to scale at the aggregate
level.4 In what follows, we use the nonparametric forms u, w, and f and theparametric forms given by (2.6)–(2.8) above interchangeably.
Let pk be the unique strictly positive capital stock k ¡ 0 such that θk α α 
δk. The capital stock p
k is the maximum sustainable capital stock. It has theproperty that for any feasible process tct , nt , kt u8t0 we have

@kt P Z ,kt ¤ maxtk0 , p
k u.

(2.13)All equilibria are therefore bounded.

3

Sufficient Optimality Conditions

It follows from (2.8) that@k, n ¡ 0,

f1 pk, n, k, 0q  f2 pk, n, k, 0q  0,

4

(3.1)

In the decentralized version of the model, profits are given to consumers, who are theowners of the firms.

4where fi p, , , q is the derivative of f with respect to the ith argument. To
simplify notation, we define the following for i  1, 2:
fi pk, nq  fi pk, n, k, nq.

(3.2)

For k, n ¥ 0 we also define
g pk, nq  f pk, n, k, nq

ζk,

(3.3)

where ζ  1  δ.
The first-order condition for labor supply nt in period t is given by$'& 0 if nt P p0, 1q,1
1
u pct qf2 pkt , nt q  w pnt q ¥ 0 if nt  1,'%¤ 0 if nt  0.

(3.4)

Note from (3.1) that nt  0 is always a solution to (3.4). This observation
forms the basis for our construction of sunspot equilibria. On the other hand,
as long as kt , Kt , Nt ¡ 0, it is always optimal to choose a strictly positive
labor supply since f2 pk, 0, K, N q  8 for any k, K, N ¡ 0.
The stochastic Euler condition for the capital stock kt 1 at the beginningof period t 1 can be written as u1pctq βEtu1pct 1qrf1pkt 1, nt 1q$'& 0 if kt 1 P p0, gpkt, ntqq,¥ if kt 1  gpkt, ntq,'%¤ if kt 1  0.

ζs

(3.5)
(3.6)

We also need to consider corner solutions since one of the results shownin the next section assumes that the utility function of consumption is linear. Yet as long as labor supply in period t 1 is strictly positive withstrictly positive probability, there is a solution to (3.5) with kt 1 ¡ 0, because limkÓ0 f1 pk, nq  8 for any n ¡ 0.
The transversality condition islim β T Eu1 pcT qkT 1  0.

T

Ñ8

(3.7)

Kamihigashi (2003, 2005) provides more details on transversality conditions
for stochastic problems.
5

We see from the following result that the above first-order conditions
in conjunction with the transversality condition are sufficient for a feasible
process to be an equilibrium. The proof is a stochastic version of the standard
sufficiency argument; see Brock (1982) for a similar stochastic argument.
Lemma 3.1. A feasible process tct , nt , kt u8t0 is an equilibrium if it satisfies(3.4) and (3.5) for all t P Z and (3.7).
Proof. See Appendix A.

4Sunspot Equilibria

4.1

Common Structure

We consider a special type of sunspot equilibrium by taking a regime-switching
sunspot process tst u as given. In particular, we assume that there are two
sunspot states, 0 and 1, and that tst u is a two-state Markov chain withtransition matrix

p00 p01,
p10 p11(4.1)

where pij is the probability that st 1  j given st  i for i, j P t0, 1u. To
simplify the analysis, we assume that pij ¡ 0 for all i, j P t0, 1u. Since (4.1)
is a transition matrix, we havep00

p01  p10

p11  1.

(4.2)

In what follows, all stochastic processes (sequences) are assumed to be adapted
to the σ-field generated by the Markov chain tst u8t0 . This simply meansthat any variable indexed by t is a function of the history of sunspot statess0 , s1 , . . . , st up to period t. Since tst u is a sunspot process, it has no direct
influence on the fundamentals of the economy. An equilibrium tct , nt , kt u is asunspot equilibrium if it depends on the sunspot process tst u in a nontrivialway.
To see the possibility of a sunspot equilibrium, suppose that we have thefollowing in the maximization problem (2.1)–(2.3):#¡ 0 if st  1,Nt
 0 if st  0.
6

(4.3)

Then, provided that kt  Kt ¡ 0 for all t P Z , we must have#
¡ 0 if st  1,nt
(4.4)
 0 if st  0.
For the rest of the paper, we assume that kt  Kt ¡ 0 for all t P Z , focusing
on regime-switching sunspot equilibria satisfying (4.3) and (4.4).
Under (4.3) the first-order condition (3.4) for nt can be written as#
 0 if nt P p0, 1q,
st  1 ñ u1 pct qf2 pkt , nt q  w1 pnt q(4.5)¥ 0 if nt  1,st  0 ñ nt  0.
(4.6)
In (4.5) we have no need to consider the case nt  0 since f2 pk, 0, k, N q  8
for any k, N ¡ 0 by (2.8), (2.10), and (2.12), as mentioned above. If, on theother hand, st  0, then Nt  0 by (4.3), and nt  0 since f2 pk, n, k, 0q  0
for any k ¡ 0 and n ¥ 0.
4.2

Linear Utility of Consumption

One way to show the existence of a sunspot equilibrium is to use Lemma
3.1 to explicitly construct a sunspot equilibrium tct , nt , kt u satisfying (4.4).
To do so, we need to verify the Euler condition (3.5) and the transversality
condition (3.7) in addition to (4.5) and (4.6). While this is not easy to
do in general, we can explicitly construct a sunspot equilibrium using these
conditions if we assume that the utility function of consumption is linear,and hence that none of the conditions depend on consumption (except for
feasibility). We consider this special case in the following result.
Proposition 4.1. If σ  0, then a sunspot equilibrium satisfying (4.4) exists.
Proof. See Appendix B.
The sunspot equilibrium constructed in the proof of Proposition 4.1 isgenerated by the following system of equations:nt  mpkt , st q,
kt 1  mintq ppst 1 q, g pkt , nt qu,ct  g pkt , nt q  kt 1 ,
7

(4.7)
(4.8)
(4.9)

where mp, q and q pq are given by (B.5) and (B.14), respectively, in Appendix
B, and pst 1  p01 or p11 depending on st  0 or 1. Given kt ¡ 0 and st P t0, 1u,
nt is determined by (4.7), kt 1 is determined by (4.8), and ct is determinedby (4.9). With a new sunspot variable st 1 drawn according to (4.1), nt 1 isdetermined by (4.7) again, and so one.
Figure 1 depicts the functions in (4.7)–(4.9) with the following parametervalues:β  0.9, η  1, γ  0.1, p01  0.2, p11  0.8,
δ  0.05, θ  3, ρ  0.55, ρ  0.03,
σ  0, α  0.35, α  0.02.

(4.10)(4.11)(4.12)

Figure 2 shows sample paths for sunspot states, capital, labor, and consumption generated by (4.7)–(4.9). The sample path for labor supply nt closely
follows the pattern of sunspot states st , as expected from (4.4). The sample
paths for capital and consumption inherit the same pattern to a large extent.
We also find a feature specific to consumption, which rises to its highest
level when the sunspot state changes from 1 to 0 after remaining in the stateof 1 for a few periods. This is expected from the consumption function in
Figure 1. Note that this function is increasing in kt but unlike the labor andcapital functions, decreasing in st in the sense that consumption is higher
when st  0 than when st  1.

4.3

Stochastic Optimal Growth

One can conjecture that the foregoing feature of the consumption function
may not necessarily arise from the presence of externalities, but rather as aconsequence of optimal behavior. In this subsection we consider a stochastic optimal growth model without externalities and with regime-switchingproductivity coefficients. We show that the dynamics of this model are very
similar to those of the previous model. The purpose of this subsection is tosuggest the presence of a close connection between the two types of models
to facilitate the analysis of regime-switching sunspot equilibria in the next
subsection.

8

Labor (nt)

1.0
0.8
0.6
0.4
0.2
0.0

st =1
st =0

Consumption (ct)Capital (kt +1)

0 2 4 6 8 10 12 14 16 18
Capital (kt)
18
16
14
12
10
8
6
4
2
0
14
12
10
8
6
4
2
0

st =1
st =0

0 2 4 6 8 10 12 14 16 18
Capital (kt)

st =1
st =0

0 2 4 6 8 10 12 14 16 18
Capital (kt)

Figure 1: Regime-switching sunspot equilibria under (4.7)–(4.9) with parameter values given by (4.10)–(4.12)
9

Sunspot (st, t =0, ,100)
0.8
0.4
0.0

0

20

40

60

80

100

80

100

80

100

80

100

Labor (nt, t =0, ,100)
0.8
0.4
0.0

0

20

40

60

Capital (kt, t =0, ,100)
15
10
5
0
0
15
10
5
0
0

20

40

60

Consumption (ct, t =0, ,100)

20

40

60

Figure 2: Sample paths for sunspot states, capital, labor, and consumption
generated by (4.7)–(4.9) with parameter values given by (4.10)–(4.12).

10

Consider the following stochastic optimal growth model:

8̧

max

tct ,nt ,kt 1 u8t0Eβ t rupct q  wpnt qs

t 0

s.t. @t P Z , ct kt 1  st pkt qα α pnt qρ ρ
ct , kt 1 ¥ 0, nt P r0, 1s,(4.13)

p1  δqkt,(4.14)(4.15)

where tst u is the same two-state Markov process following (4.1). In this subsection, we define st not as a sunspot shock, but as a stochastic productivity
coefficient that directly affects production. When st  1, the aggregate production function in the maximization problem above is unchanged from thatin the previous subsection, but the externalities are internalized here. Since
output is zero whenever st  0, the problem inherits the pattern of (4.3).
Figure 3 depicts the optimal policy functions for the stochastic optimal
growth model (4.13)–(4.15) under (4.10)–(4.12). We obtain these functions
from (4.7)–(4.9) by setting
α  0.37,

ρ  0.58,

αρ0(4.16)in (4.7)–(4.9).
Note that the consumption function in Figure 3 is decreasing in st likethe consumption function in Figure 1; in fact, consumption with st  1 iseven lower than in Figure 1. We can explain this by referring to the capitalfunction in Figure 3, which shows that more capital is accumulated in thestochastic optimal growth model (4.13)–(4.15) than in the original economy
with externalities under (4.10)–(4.12). The functions in Figure 3 serve as
an example of a stochastic optimal growth model in which consumption isdecreasing in productivity, while capital and labor are increasing in productivity. This contrasts sharply with the Brock-Mirman (1972) model with i.i.d.
productivity shocks, where consumption is always increasing in productivity;
see Kamihigashi (2008, Theorem 2.1).
Figure 4 shows sample paths generated by these optimal policy functions
with productivity states identical to the sunspot states in Figure 2. The
capital and consumption paths are also similar to those in Figure 2, but thecapital path and peaks in consumption are both overall higher.

4.4

No Capital Externality

Our analysis in the previous two subsections suggests that the sunspot equilibria of the original economy are closely connected to the optimal process
11

Labor (nt)

1.0
0.8
0.6
0.4
0.2
0.0

st =1
st =0

Consumption (ct)Capital (kt +1)

0 2 4 6 8 10 12 14 16 18
Capital (kt)
18
16
14
12
10
8
6
4
2
0
14
12
10
8
6
4
2
0

st =1
st =0

0 2 4 6 8 10 12 14 16 18
Capital (kt)

st =1
st =0

0 2 4 6 8 10 12 14 16 18
Capital (kt)

Figure 3: Optimal policy functions for (4.13)–(4.15)

12

Productivity (st, t =0, ,100)
0.8
0.4
0.0

0

20

40

60

80

100

80

100

80

100

80

100

Labor (nt, t =0, ,100)
0.8
0.4
0.0

0

20

40

60

Capital (kt, t =0, ,100)
15
10
5
0
0
15
10
5
0
0

20

40

60

Consumption (ct, t =0, ,100)

20

40

60

Figure 4: Sample paths for productivity states, capital, labor, and consumption

13

of a stochastic optimal growth model. While this connection between themodels is not trivial to show in general, we can establish it fairly easily inthe absence of capital externalities. The proof of the following result utilizes
the connection.
Proposition 4.2. If α  0, then a sunspot equilibrium satisfying (4.4) exists.
Proof. See Appendix C
The condition α  0 in the above proposition means that there is no
capital externality. In the proof of Proposition 4.2, we consider the following
stochastic optimal growth model:

8̧max

tct ,nt ,kt 1 u8t0

E





β t upct q 

t 0

ρ

ρ
ρ

wpnt qs.t. @t ¥ 0, ct kt 1  st θpkt qα pnt qρ ρ
ct , kt 1 ¥ 0, nt P r0, 1s,(4.17)
ζkt ,(4.18)(4.19)

where tst u8t0 is the same two-state Markov chain following (4.1). In the
proof, we show that the Euler condition for kt 1 , the first order condition
for nt , and the transversality condition are necessary for optimality, and areequivalent to the sufficient optimality conditions for the original economy(2.1)–(2.3) with α  0. We can thus establish the existence of a sunspot
equilibrium by showing the existence of an optimal process for the above
stochastic optimal growth model.
The Bellman equation for (4.17)–(4.19) can be written as*v pkt q  max
upct q 
wpnt q βEt v pkt 1 q
c ,n ,kρ
s.t. ct kt 1  st θpkt qα pnt qρ ρ ζkt ,ct , kt 1 ¥ 0, nt P r0, 1s.
"

t

t

ρ

ρ

(4.20)

t 1(4.21)(4.22)

From the proof of Proposition 4.2, we can interpret the optimal policy functions for the above Bellman equation as constituting a regime-switching
sunspot equilibrium. To illustrate these functions, we use the same parameter values used in (4.10) and (4.11), but replace the values of σ, α, and α asfollows:σ  0.99,

α  0.37,
14

α  0.

(4.23)

With the above parameter values, the utility function u is almost logarithmic,
and the aggregate production function f pk, n, k, nq remains unchanged fromthat in (4.11).
Figure 5 shows the optimal policy functions for the Bellman equation(4.20)–(4.22) under the above parameter values. These functions are computed by numerically solving the Bellman equation using modified policy
iteration (e.g., Puterman, 2005) with 5,000 equally spaced grid points.
Figure 6 shows sample paths for sunspot states, labor, capital, and consumption generated by the functions in Figure 5. Compared to those in
Figure 2, the sample paths in Figure 6 appear less extreme due to the concavity of the utility function u. Capital accumulates while the sunspot stateis 1, and decumulates while it is 0. Consumption follows almost exactly thesame pattern as capital, while labor supply moves in the opposite directions
when st  1, as expected from the labor and consumption functions in Figure
5.

5Concluding Comments

In this paper we have shown that regime-switching sunspot equilibria easily arise in a one-sector growth model with aggregate decreasing returns
and arbitrarily small externalities. We have explicitly constructed a regimeswitching sunspot equilibrium under the assumption that the utility function of consumption is linear. We have also constructed a stochastic optimal growth model whose optimal process turns out to be a regime-switching
sunspot equilibrium of the original economy under the assumption that there
is no capital externality.
Although our results assume aggregate decreasing returns to scale, theexistence of regime-switching sunspot equilibria can easily be shown for models with increasing returns and large externalities, at least when the utility
function is linear, as in Subsection 4.2. On the other hand, the proof of
Proposition 4.2 relies on the assumption of decreasing returns; it could be
non-trivial to extend the proof to the case of increasing returns.
To conclude this paper, we discuss possible ways to extend our analysis. First, we can use a similar approach to construct more realistic sunspot
equilibria than the rather extreme equilibria considered in this paper, wherelabor supply and output are both zero when the sunspot state is zero. Consider, for example, a one-sector growth model with externalities in which the15st =1
st =0

Labor (nt)

0.8
0.6
0.4
0.2
0.0
0

5

10

15

Capital (kt)

20

25

25

st =1
st =0

Capital (kt +1)

20
15
10
5

Consumption (ct)

0

0

5

10

15

Capital (kt)

20

25

6
5

st =1
st =0

4
3
2
1
0

0

5

10

15

Capital (kt)20

25

Figure 5: Optimal policy functions for (4.20)–(4.22) and regime-switching
sunspot equilibria for (2.1)–(2.3) under (4.10), (4.11), and (4.23)
16

Sunspot (st, t =0, ,100)
0.8
0.4
0.0

0

20

40

60

80

100

80

100

80

100

80

100

Labor (nt, t =0, ,100)
0.8
0.4
0.0

0

20

40

60

Capital (kt, t =0, ,100)
16
8
0

0

20

40

60

Consumption (ct, t =0, ,100)

4.5
3.0
1.5
0.0

0

20

40

60

Figure 6: Sample paths for sunspot states, capital, labor, and consumption
under (4.10), (4.11), and (4.23)

17

first-order condition for labor supply has multiple solutions. Such a model
can easily be produced if we are allowed to assume externalities of a general form. With such a model, we can construct a sunspot equilibrium thatswitches between the multiple solutions of the first-order condition for labor
supply according to the sunspot state.
Second, although we have focused on sunspot equilibria, we can also
construct deterministic equilibria that exhibit chaotic dynamics. We cantake, for example, a deterministic sequence of states st each assigned a valueof either 0 or 1, and solve the deterministic version of the maximization
problem (4.17)–(4.19). The resulting optimal path then follows the pattern of
the sequence tst u. We can think of this as an example of symbolic dynamics;
see Kamihigashi (1999) for economic applications of symbolic dynamics.
Finally, in Proposition 4.2 we have only considered the case without capital externality. While it seems difficult to extend the same approach tomodels with capital externalities (as long as the capital depreciation rate isless than one), there is a way to deal with such models. In particular, if weallow for nonlinear discounting along the lines of Kamihigashi (2002), we canconstruct a stochastic optimal growth model whose optimal process turns
out to be a sunspot equilibrium of the original economy.

Appendix A

Proof of Lemma 3.1

  
Let tct , nt , kt u8
t0 be a feasible process satisfying (3.4)–(3.7) (with ct , nt , kt
replacing ct , nt , kt ). To simplify notation, for t P Z and i  1, 2 we define
f ptq  f pkt , nt , kt , nt q,
fi ptq  fi pkt , nt , kt , nt q.

(A.1)
(A.2)

We are to show that for any feasible process tct , nt , kt u8t0 , we have

8̧
E



t 0

8̧

β rupct q  wpnt qs  E
t



t 0

18

β t rupct q  wpnt qs ¤ 0.

(A.3)

To this end, let tct , nt , kt u8t0 be a feasible process. Fix T P N
moment. Let

for theβ rupct q  wpnt qs(A.4)∆T  E

¤E

T
¸
t

T
¸
t



t 0

t 0
T

β rupct q  wpnt qs  E

¸ t




β tu1 pct qpct  ct q  w1 pnt qpnt  nt qqu,

(A.5)

t 0

where u1 pct q is the right derivative of u at 0 if ct  0, and similarly for
w1 pnt q. We have
∆T ¤ ET¸t

β tu1 pct qrf pkt , nt , kt , nt q  f ptq

t 0

¤E

ζ pkt  kt q  pkt 1  kt 1 qs  w1 pnt qpnt  nt qquT¸
β t ru1 pct qpf1 ptq



(A.6)

ζ qpkt  kt q

t 0

tu1pct qf2ptq  w1pnt qupnt  nt q  u1pct qpkt 1  kt 1qs.

(A.7)

Recalling the first-order condition (3.4) for nt , we see that for all t P Z ,tu1pct qf2ptq  w1pnt qupnt  nt q ¤ 0.

(A.8)

Substituting into (A.7) we obtain∆T ¤ EET¸
β t ru1 pct qpf1 ptq



t 0
T 1


¸
t


β ru1 pct q

ζ qpkt  kt q  u1 pct qpkt 1  kt 1 qs

βu1 pct 1 qpf1 pt

ζ qspkt 1  kt 1 q

1q

(A.9)
(A.10)t 0

 β T Eu1pcT qpkT 1  kT 1q
T¸
1 E β tru1pct q βEtu1pct 1qpf1pt
(A.11)
1q

ζ qspkt 1  kt 1 q(A.12)t 0

 β T Eu1pcT qpkT 1  kT 1q,(A.13)

where the last equality holds by the law of iterated expectations. Recalling
the Euler condition (3.5) for kt 1 , we see that for all t P Z ,ru1pct q

βEt u1 pct 1 qpf1 pt

1q19

ζ qspkt 1  kt 1 q ¤ 0.

(A.14)

Substituting into (A.13) we obtain∆T ¤ β T Eu1 pcT qpkT 1  kT 1 q(A.15)¤ β Eu1pc qk

Ñ 0,(A.16)
where the second inequality holds since kT 1 ¥ 0, and the convergence holds
T

T

T 1

by the transversality condition (3.7). This completes the proof of Lemma
3.1.

Appendix B

Proof of Proposition 4.1

Suppose that σ  0. Then conditions (4.5) and (4.6) can be written as
st  1st  0#ñ ρθpktqα αpntqρ ρ1  ηpntqγ  0 if nt P p0, 1q,¥ 0 if nt  1,
ñ nt  0.

(B.1)
(B.2)

The Euler condition for kt 1 , (3.5), can be written as$'& 1 if kt 1 P p0, gpkt, ntqq,
ζ s ¥ 1 if kt 1  g pkt , nt q,'%¤ 1 if kt 1  0.

βEt rαθpkt 1 qα α1 pnt 1 qρ ρ

(B.3)

The transversality condition (3.7) reduces tolim β T EkT 1  0.

T

(B.4)

Ñ8

Note that (B.1) and (B.2) can be combined into
nt  mpkt , st q  st min

#

ρθ
p
kt qα α
η





+

1
γ 1 ρ ρ

,1 .

(B.5)

Substituting into the left-hand side of (B.3) we obtain
Et rαθpkt 1 qα α1 pnt 1 qρ ρ E#trαθpkt 1q
 p01hpkt 1q
p11 hpkt 1 q

 mpk

α α 1

ζ
ζ

ζs

t 1 , st 1

if st  0,
if st  1,

20

q

ρ ρ

ζs

(B.6)
(B.7)
(B.8)

wherehpk q  αθk α α1 mpk, 1qρ ρ#   min αθ ρθηρ ργ 1 ρ ρkpα α1qpγ 1q ρ ργ 1ρρ, αθk α α1

(B.9)

+
.

(B.10)

Both expressions in the curly brackets are strictly decreasing in k by (2.10)
and (2.12) (note that pα α  1qpγ 1q ρ ρ α α  1 ρ ρ ¤ 1
by (2.12)). Thus hpq is strictly decreasing, which implies that the inverseh1 pq exists. Indeed, for z ¡ 0 we have$  & z η
h1 pz q  min% αθ ρθρ ρ
γ 1 ρ ρp
 qp qγ 1 ρ ρα α 1 γ 1 ρ ρ

,

z
αθ

,
.

-.

1
α α 1

(B.11)

Note that
lim hpk q  8.

Ó

k 0

(B.12)

Substituting (B.6)–(B.8) into (B.3) we obtain$'& 1 if kt 1 P p0, gpkt, ntqq,
β rps 1 hpkt 1 q ζ s ¥ 1 if kt 1  g pkt , nt q,'%¤ 1 if kt 1  0,t

(B.13)

where pst 1  p01 or p11 depending on st  0 or 1. For p ¡ 0 define

1  βζ
q ppq  h1.
βp(B.14)

Note from (B.12) that we can rule out the case kt 1  0 in (B.13). Hence we
can write (B.13) as
kt 1  mintq ppst 1 q, g pkt , nt qu.

(B.15)

We construct a process tct , nt , kt u8t0 recursively as follows: given kt ¡ 0
and st P t0, 1u, let
nt  mpkt , st q.
21(B.16)

Determine kt 1 by (B.15). Let
ct  g pkt , nt q  kt 1 .

(B.17)

Draw st 1 according to (4.1). Determine nt 1 by (B.16), and so on. By
construction, this process is feasible and satisfies (B.1)–(B.3). It also satisfies(B.4) by (2.13). Thus it is a sunspot equilibrium. The conclusion of the
proposition now follows.

C

Proof of Proposition 4.2

Suppose that α  0. Consider the stochastic optimal growth model (4.17)–(4.19). The Euler condition for kt 1 is written as u1pctq βEtu1pct 1qrst 1αθpkt 1qα1pnt 1qρ ρ$'& 0 if kt 1 P p0, gpkt, ntqq,¥ if kt 1  gpkt, ntq,'%¤ if kt 1  0.

ζs
(C.1)

This is equivalent to the equilibrium Euler condition (3.5) for kt 1 for theoriginal economy (2.1)–(2.3) with α  0 and (4.3). The first-order condition
for nt for the above stochastic optimal growth model is given by
u1 pct qst pρ

$'& 0 if nt P p0, 1q,
ρ ρ 1
α
ρ ρ1
 ρ w pntq '¥ 0 if nt  1,
ρqθpkt q pnt q%¤ 0 if nt  0,

(C.2)

which simplifies to$'& 0 if nt P p0, 1q,
1
α
ρ ρ1
1
u pct qst ρθpkt q pnt q
 w pntq '¥ 0 if nt  1,%¤ 0 if nt  0.

(C.3)

This is equivalent to (3.4) with α  0 and (4.3). The transversality condition
for the above problem is identical to (3.7).
Conditions (C.1) and (C.3) are necessary for optimality by standard arguments. The transversality condition (3.7) is also necessary by the argument
22

of Kamihigashi (2005, Section 6).5 Given that the sunspot variable st is
discrete, we can easily establish the existence of an optimal process for the
optimal stochastic growth model (4.17)–(4.19) by a standard argument (e.g.,
Ekeland and Sheinkman, 1986). Let tct , nt , kt u8t0 be an optimal process for(4.17)–(4.19). Then by the above argument, the process satisfies (3.4)–(3.7).
Thus by Lemma 3.1, the process is an equilibrium of the original economy(2.1)–(2.3). Since it depends on st in a nontrivial way, it is a sunspot equilibrium.

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