DP2016-15

A Sim ple No -Bubble Theorem for
Deterministic Sequ ential Economies
Takashi KAMIHIGASHIMarch 26, 2016

A Simple No-Bubble Theorem for
Deterministic Sequential Economies∗
Takashi Kamihigashi†
March 26, 2016

Abstract
We show a simple no-bubble theorem that applies to a wide range
of deterministic sequential economies with infinitely lived agents. In
particular, we show that asset bubbles never arise if there is at least
one agent who can reduce his asset holdings permanently from someperiod onward. This is a substantial generalization of Kocherlakota’s(1992, Journal of Economic Theory 57, 245–256) result on asset bubbles and short sales constraints. Our no-bubble theorem requires virtually no assumption except for the strict monotonicity of preferences.

∗

I would like to thank an anonymous referee and the Guest Editor Jianjun Miao for
helpful comments and suggestions. Financial support from the Japan Society for the Promotion of Science (JSPS KAKENHI Grant Number 15H05729) is gratefully acknowledged.
†
Research Institute for Economics and Business Administration, Kobe University,
Rokkodai, Nada, Kobe 657-8501 Japan. Email: tkamihig@rieb.kobe-u.ac.jp.

1

Introduction

Since the global financial crisis of 2007-2008, there has been a surge of interest in rational asset pricing bubbles, or simply “asset bubbles.” Numerous
economic mechanisms that give rise to asset bubbles are still being proposed,
and the implications of asset bubbles on various economic issues are actively
discussed in the current literature; we refer the reader to Miao (2014) for ashort survey on recent developments.
In constructing models of asset bubbles, it is important to understandconditions under which asset bubbles exist or not. While conditions forexistence are mostly restricted to specific models, some general conditions
for nonexistence are known. In fact, there are various versions of the nobubble theorem that essentially states that asset bubbles never arise if thepresent value of the aggregate endowment process is finite.
Wilson’s (1981, Theorem 2) result on the existence of a competitive equilibrium in a deterministic economy with infinitely many agents can be viewed
as an early version of this no-bubble theorem. Santos and Woodford (1997,
Theorems 3.1, 3.3) established a weak and a strong version of the no-bubble
theorem for an incomplete market economy with infinitely many agents eachof whom may be finitely or infinitely lived. While the weak version shows theexistence of state prices for which no bubble arises on any asset in positive
net supply, the strong version shows that no bubble arises on any such asset
for any state prices under an additional assumption of uniformly impatient
preferences. Huang and Werner (2000, Theorem 6.1) established a version of
the no-bubble theorem applicable to an asset in zero net supply for a deterministic economy with finitely many agents whose preferences are represented
by explicit utility functions. Recently, Werner (2014, Remark 1, Theorem 1)
extended Santos and Woodford’s (1997) weak version to a complete market
economy with debt constraints (instead of borrowing constraints) as well as
their strong version to an incomplete market economy with debt constraints
under the additional assumption of uniformly impatient preferences.
While these results are based on equilibrium prices and allocations, there
are closely related results based mostly on the optimal behavior of a single
agent. For example, in a deterministic economy with finitely many agents,
Kocherlakota (1992, Proposition 3) showed that in an equilibrium with apositive asset bubble, the short sales constraints of all agents must be asymptotically binding; equivalently, asset bubbles can be ruled out if there is atleast one agent whose asset holdings can be lowered permanently from some1

period onward. A similar idea was used earlier by Obstfeld and Rogoff (1986)
to rule out deflationary equilibria in a money-in-the-utility-function model.1
The results mentioned in the preceding paragraph rely on the necessityof a transversality condition,2 and a general no-bubble result based on thenecessity of a transversality condition was shown in Kamihigashi (2001, p.
1007) for deterministic representative-agent models. Basically this result
only requires the differentiability and strict monotonicity of instantaneous
utility functions; thus it can be used to rule out asset bubbles in various
representative-agent models.
In this paper we establish a simple no-bubble theorem that can be used
to rule out asset bubbles in a considerably broader range of deterministic
models. More specifically, we consider the problem of a single agent facing
sequential budget constraints and having strictly monotone preferences. We
show that asset bubbles never arise if the agent can reduce his asset holdings
permanently from some period onward. This result uses the same idea as
those based on transversality conditions mentioned above; the contributionof this paper is to show that the result holds true under extremely generalconditions.
To demonstrate the applicability of our no-bubble theorem to generalequilibrium models, we consider a general equilibrium model with multiple
agents and multiple assets. Using our no-bubble theorem, we show substantial generalizations of Proposition 3 in Kocherlakotoa (1992).
The rest of the paper is organized as follows. In Section 2 we present asingle agent’s problem along with necessary assumptions, and formally define
asset bubbles. In Section 3 we offer several examples satisfying our assumptions. In Section 4 we state our no-bubble theorem and show some immediate
consequences. In Section 5 we present a general equilibrium model. In Section 6 we show several results on asset bubbles in general equilibrium. In
Section 7 we offer some concluding comments. Some of the proofs are included in the appendices.
1

See Kamihigashi (2008a, 2008b) for results on asset bubbles in related models.
Various results on necessity of transversality conditions were established in Kamihigashi (2001, 2002, 2003, 2005).
2

2

2

Single-Agent/Single-Asset Framework

2.1

Feasibility and Optimality

Time is discrete and denoted by t ∈ Z+ . In this section we assume that there
are one consumption good and one asset that pays a dividend of dt units ofthe consumption good in each period t ∈ Z+ . Let pt be the price of the asset
in period t ∈ Z+ . Consider an infinitely lived agent who faces the following
constraints:
ct + pt st = yt + (pt + dt )st−1 , ct ≥ 0,s ∈ S(s−1 , y, p, d),∀t ∈ Z+ ,

(2.1)
(2.2)

where ct is consumption in period t, yt ∈ R is (net) income in period t,
st is asset holdings at the end of period t with s−1 historically given, and∞S(s−1 , y, p, d) is a set of sequences in R with s = {st }∞t=0 , y = {yt }t=0 , p =∞{pt }∞
t=0 , and d = {dt }t=0 , We present several examples of (2.2) in Subsection
3.1.
Although we consider a single agent’s problem and assume that there is
only one asset here, our results developed within this framework apply to a
general equilibrium model with many agents and many assets, as shown in
Section 6.
Let C be the set of sequences {ct }∞t=0 in R+ . For any c ∈ C, we let{ct }∞
denotethesequence
representationof c, and vice versa. In othert=0∞
words, we use c and {ct }t=0 interchangeably; likewise, we use s and {st }∞t=0
interchangeably, and so on. We define the inequalities < and ≤ on the set ofsequences in R (which includes C) as follows:c ≤ c0c < c0⇔⇔∀t ∈ Zt , ct ≤ c0t ,c ≤ c0 and ∃t ∈ Z+ , ct < c0t .

(2.3)
(2.4)

The agent’s preferences are represented by a binary relation ≺ on C. Moreconcretely, for any c, c0 ∈ C, the agent strictly prefers c0 to c if and only ifc ≺ c0 . The assumptions stated in this section are maintained until the endof Section 4 unless otherwise noted.
Assumption 2.1. dt ≥ 0 and pt ≥ 0 for all t ∈ Z+ .
Assumption 2.2. pt > 0 for all t ∈ Z+ .
3

Although the second assumption is used for most of our results, there is
an important case in which it cannot be used. In particular, if the asset is
intrinsically useless—i.e., dt = 0 for all t ∈ Z+ —then it is more than natural
to consider the possibility that pt = 0 for all t ∈ Z+ . One of our results deals
with this particular case without assuming Assumption 2.2; see Proposition
4.2.
∞
We say that a pair of sequences c = {ct }∞t=0 and s = {st }t=0 in R is a plan;a plan (c, s) is feasible if it satisfies (2.1) and (2.2); and a feasible plan (c∗ , s∗ )is optimal if there exists no feasible plan (c, s) such that c∗ ≺ c. Whenever
we take an optimal plan (c∗ , s∗ ) as given, we assume the following.
Assumption 2.3. For any c ∈ C with c∗ < c, we have c∗ ≺ c.
This assumption is satisfied if ≺ is strictly monotone in the sense that for
any c, c0 ∈ C with c < c0 , we have c ≺ c0 . Although this latter requirement
may seem reasonable, there is an important case in which it is not satisfied.
Such a case and other examples of preferences satisfying Assumption 2.3 are
discussed in Subsection 3.2.

2.2

Asset Bubbles

In this subsection we define the fundamental value of the asset and the bubble
component of the asset price in period t ∈ Z+ using the period t prices of theconsumption goods in periods t, t + 1, . . .. To be more concrete, let q0t be theperiod 0 price of the consumption good in period t ∈ Z+ . It is well known(e.g., Huang and Werner, 2000, (8)) that the absence of arbitrage implies
that there exists a price sequence {q0t } such that∀t ∈ Z+ , q0t pt = q0t+1 (pt+1 + dt+1 ),∀t ∈ N, q0t > 0,
q0 = 1.

(2.5)
(2.6)
(2.7)

In the current setting with a single asset, under Assumption 2.2, conditions(2.5) and (2.7) uniquely determine the price sequence {q0t }.
For t ∈ N and i ∈ Z+ , we define
qti = q0t+i /q0t ,

4

(2.8)

which is the period t price of consumption in period t + i. Note thatjqti qt+i=∀i, j, t ∈ Z+ ,q0t+i q0t+i+j= qti+j .
q0t q0t+i(2.9)

Let t ∈ Z+ . By (2.5) and (2.8) we have pt = qt1 (pt+1 + dt+1 ). By repeatedapplication of this equality and (2.9), we have
pt = qt1 dt+1 + qt1 pt+1
1
(pt+2 + dt+2 )= qt1 dt+1 + qt1 qt+1
1
2
= qt dt+1 + qt dt+2 + qt2 pt+2
..
.
n
X
=
qti dt+i + qtn pt+n , ∀n ∈ N.

(2.10)(2.11)(2.12)(2.13)(2.14)

i=1

Since the above finite sum is increasing in n ∈ N,3 it follows that
pt =

∞
X

qti dt+i + lim qtn pt+n .
n↑∞

i=1

(2.15)

As is common in the literature, we define the fundamental value of theasset in period t as the present discounted value of the dividend stream fromperiod t + 1 onward:
ft =

∞
X

qti dt+i .

(2.16)

i=1

The bubble component of the asset price in period t is the part of pt that is
not accounted for by the fundamental value:
bt = p t − f t .

(2.17)

It follows from (2.15)–(2.17) thatbt = lim qtn pt+n .
n↑∞3(2.18)In this paper, ”increasing” means ”nondecreasing,” and ”decreasing” means ”nonincreasing.”

5

Using (2.9) we see thatq0t lim qtn pt+n = lim q0t+n pt+n = lim q0i pi .
n↑∞

n↑∞

i↑∞

(2.19)

Hence by (2.18) and (2.6) we have
lim q0i pi = 0
i↑∞

3

⇔

∀t ∈ Z+ , bt = 0.

(2.20)

ExamplesIn this section we present several examples of (2.2) as well as some examples
of preferences that satisfy Assumption 2.3. Some of these examples are usedin Section 4.

3.1

Constraints on Asset Holdings

The simplest example of (2.2) would be the following:∀t ∈ Z+ ,st ≥ 0.

(3.1)

This constraint is often used in representative-agent models; see, e.g., Lucas(1978) and Kamihigashi (1998).
Kocherlakota (1992) uses a more general version of (3.1):∀t ∈ Z+ ,st ≥ σ,(3.2)

where σ ∈ R. If σ < 0, then the above constraint is called a short sales
constraint.
The following constraint is even more general:∀t ∈ Z+ ,st ≥ σt ,(3.3)

where σt ∈ R for all t ∈ Z+ . Note that (3.2) is a special case of (3.3) with
σt = σ for all t ∈ Z+ .
Santos and Woodford (1997, p. 24) consider a (state-dependent) borrowing constraint that reduces in our single-asset setting to∀t ∈ Z+ ,pt st ≥ −ξt ,
6

(3.4)

where ξt ∈ R for all t ∈ Z+ . This constraint is a special case of (3.3) with
σt = −ξt /pt .
The (state-dependent) debt constraint considered by Werner (2014) andLeRoy and Werner (2014, p. 313) reduces in our current setting to∀t ∈ Z+ ,(pt+1 + dt+1 )st ≥ −ξt+1 .

(3.5)

This constraint is another special case of (3.3) with
σt = −ξt+1 /(pt+1 + dt+1 ).

(3.6)

In addition to (3.2), Kocherlakota (1992) considers the following wealthconstraint:∀t ∈ Z+ ,pt st +∞
X

qti yt+i ≥ 0,

(3.7)

i=1

which is another example of (2.2). The left-hand side above is the periodt value of the agent’s current asset holdings and future income. Note that
(3.7) is yet another special case of (3.3) with∀t ∈ Z+ ,σt = −∞
X

qti yt+i /pt .

(3.8)

i=1

See Wright (1987) and Huang and Werner (2000) for related discussion.

3.2

Preferences

Example 3.1. A typical objective function in an agent’s maximization problem takes the form∞Xβ t u(ct ),

(3.9)

t=0

where β ∈ (0, 1) and u : R+ → [−∞, ∞) is a strictly increasing function.
Suppose further that u is bounded, and define the binary relation ≺ by
c≺c

0

⇔

∞
X

t

β u(ct ) <

t=0

∞
X
t=0

7

β t u(c0t ).

(3.10)

Then ≺ clearly satisfies Assumption 2.3.
If u is unbounded below, i.e., if u(0) = −∞, then the above definition of≺ may not satisfy Assumption 2.3. In particular, given c∗ , c ∈ C with c∗ < c,we do not have c∗ ≺ c if c∗t = ct = 0 for some t ∈ Z+ and if u is bounded
above. Indeed, in this case,∞X

β u(c∗t ) =
t

t=0

∞
X

β t u(ct ) = −∞.

(3.11)

t=0

Hence the inequality in (3.10) does not hold.
Example 3.2. The above problem with unbounded utility can be avoided
by using an alternative optimality criterion. To be specific, let ut : R+ →[−∞, ∞) be a strictly
P∞ increasing function for t ∈ Z+ as above. In this case,the infinite sum t=0 ut (ct ) may not be well defined. Even if it is always welldefined, it may not be strictly increasing, as discussed above. To deal withthese problems, consider the binary relation ≺ defined by
c≺c

0

⇔

n
X
lim
[ut (ct ) − ut (c0t )] < 0,

(3.12)n↑∞ t=0

where we follow the convention that (−∞) − (−∞) = 0; see Dana and Le
Van (2006) for related optimality criteria. It is easy to see that the binary
relation ≺ defined above satisfies Assumption 2.3.
Continuing with this example, suppose that (2.2) is given by (3.1). Suppose further that each ut is differentiable on R++ , and that there exists an
optimal plan (c∗ , s∗ ) such that∀t ∈ Z+ ,c∗t > 0, s∗t = 1.

(3.13)

Then the standard Euler equation holds:
u0t (c∗t )pt = u0t+1 (c∗t+1 )(pt+1 + dt+1 ),∀t ∈ Z+ .

(3.14)

In view of (2.5), the price sequence {q0t } is given by
q0t =u0t (c∗t ),u00 (c∗0 )∀t ∈ Z+ .

(3.15)

The fundamental value ft takes the familiar form:
ft =

∞
X
u0t+i (c∗t+i )
i=1

u0t (c∗t )8dt+i ,∀t ∈ Z+ .

(3.16)Example 3.3. Let v : C → R be a strictly increasing function. Define thebinary relation ≺ byc ≺ c0v(c0 , c1 , c2 , . . .) < v(c00 , c01 , c02 , . . .).

⇔(3.17)

Note that (3.17) satisfies Assumption 2.3 without any additional condition
on v. For example, v can be a recursive utility function.

4

Implications of Feasibility and Optimality

4.1

No-Bubble Theorem

To state our no-bubble theorem, we need to introduce some notation. Given∗
any sequence {s∗t }∞t=0 in R, τ ∈ Z+ , and  > 0, let Sτ, (s ) be the set of∞sequences {st }t=0 in R such that(
if t < τ ,
= s∗t
(4.1)
st
∗
≥ st −  if t ≥ τ .
In other words, a sequence {st } in Sτ, (s∗ ) coincides with {s∗t } up to periodτ − 1 and is only required to satisfy the lower bound s∗t −  from period τ
onward. We are ready to state the main result of this paper.
Theorem 4.1. Let (c∗ , s∗ ) be an optimal plan. Suppose that there exist
τ ∈ Z+ and  > 0 such thatSτ, (s∗ ) ⊂ S(s−1 , y, p, d).

(4.2)

Then bt = 0 for all t ∈ Z+ .
Proof. See Appendix A.
It seems remarkable that asset bubbles can be ruled out by a simple condition such as (4.2) alone. In particular, no explicit utility function is assumed,
and the only requirement on the binary relation ≺ is Assumption 2.3, whichmerely requires strict monotonicity at the given optimal consumption planc∗ .
The idea of the proof of Theorem 4.1 is simple. In the proof, assumingthat the left equality in (2.20) is violated, we construct an alternative plan9

as follows. Let δ > 0, and let sτ = s∗τ − δ and cτ = c∗τ + pτ δ, where τ is givenby the statement of the theorem. For t 6= τ , let st be determined by thebudget constraint (2.1) with ct = c∗t . This alternative plan gives the same
consumption sequence except in period τ , where consumption is increasedby pτ δ > 0. Hence this plan is strictly preferred to the original plan (c∗ , s∗ ).
We derive a contradiction by showing that the alternative plan is feasible forsufficiently small δ > 0.
Similar constructions are used as “Ponzi schemes” by Huang and Werner(2000, Theorems 5.1, 6.1), but they are not directly linked to the nonexistenceof asset bubbles.

4.2

Consequences of Theorem 4.1

In this subsection we provide fairly simple consequences of Theorem 4.1 inthe current single-agent framework. Throughout this subsection we take an
optimal plan (c∗ , s∗ ) as given. We start with a simple result assuming thatthe feasibility constraint on asset holdings (2.2) is given by a sequence ofconstraints of the form (3.3). As discussed in Subsection 3.1, this simple
form covers various constraints on borrowing, debt, and wealth,Corollary 4.1. Suppose that (2.2) is given by (3.3) with σt ∈ R for allt ∈ Z+ . Suppose further thatlim(s∗t − σt ) > 0.

(4.3)

t↑∞

Then the conclusion of Theorem 4.1 holds.
Proof. Assume (4.3). Let  ∈ (0, limt↑∞ (s∗t − σt )). Then there exists τ ∈ Z+
such that s∗t − σt ≥ , or s∗t −  ≥ σt , for all t ≥ τ . This implies (4.2). Hencethe conclusion of Theorem 4.1 holds.
If there is a constant lower bound on asset holdings st , the above result
reduces to the following.
Corollary 4.2. Suppose that (2.2) is given by (3.2) for some σ ∈ R. Suppose
further that limt↑∞ s∗t > σ. Then the conclusion of Theorem 4.1 holds.
In Section 6 we present some consequences of the above two results in thecontext of general equilibrium and discuss them in relation to Proposition 3
in Kocherlakota (1992).
Next we present two results that apply to representative-agent models.
10

Corollary 4.3. Suppose that (2.2) is given by (3.1). Suppose thats∗t = 1.

∀t ∈ Z+ ,(4.4)

Then the conclusion of Theorem 4.1 holds.
Proof. Note that (4.4) and (3.1) imply (4.2) with τ = 0 and  = 1. Thus theconclusion of Theorem 4.1 holds.
The following proposition is immediate from the above result and (3.16).
Proposition 4.1. In the setup of Example 3.2 (including (3.14)–(3.16)), wehave∀t ∈ Z+ ,

pt =

∞
X
u0t+i (c∗t+i )
i=1

u0t (c∗t )dt+i .

(4.5)

A similar result is shown in Kamihigashi (2001, Section 4.2.1) for acontinuous-time model with a nonlinear constraint. It is known that astochastic version of Proposition 4.1 requires additional assumptions; see
Kamihigashi (1998) and Montrucchio and Privileggi (2001).4
Finally we consider the case of fiat money, or an asset with no dividend
payment. Since the fundamental value of fiat money is zero, its price mustalso be zero if there is no asset bubble. Hence the case of fiat money is notdirectly covered by Theorem 4.1, which requires Assumption 2.2,
Proposition 4.2. Drop Assumption 2.2 and (2.5)–(2.7) (but maintain Assumptions 2.1 and 2.3). Suppose that there exist τ ∈ Z+ and  > 0 satisfying(4.2). Suppose further that∀t ≥ τ + 1,

dt = 0.

(4.6)

Then
∀t ≥ τ,pt = 0.

Proof. See Appendix B.
4

See Kamihigashi (2011) for sample-path properties of stochastic asset bubbles.

11

(4.7)

5

General Equilibrium with Multiple Agentsand Multiple Assets

5.1

Feasibility, Optimality, and Equilibrium

Consider an exchange economy with countably many infinitely lived agentsindexed by a ∈ A, where A = {1, 2, . . . , a} with a ∈ N ∪ {∞}. There arecountably many assets indexed by k ∈ K, where K = {1, 2, . . . , k} withk ∈ N ∪ {∞}. Agent a ∈ A faces the following constraints:
X
X
(5.1)
(pk,t + dk,t )sak,t−1 , cat ≥ 0, ∀t ∈ Z+ ,pk,t sak,t = yta +
cat +
k∈K
a

s ∈S

a

k∈K
a
(s−1 , y a , p, d),(5.2)

where cat and yta are agent a’s consumption and endowment in period t; foreach k ∈ K, sak,t is agent a’s holdings of asset k at the end of period t, pk,tis the price of asset k in period t, and dk,t is the dividend payment of asset
k in period t. In (5.2), sa−1 = (sak,−1 )k∈K is agent a’s initial portfolio of all
assets k ∈ K, which are historically given, and S a (sa−1 , y a , p, d) is a set ofaa ∞∞sequences in Rk with sa = {(sak,t )k∈K }∞t=0 , y = {yt }t=0 , p = {(pk,t )k∈K }t=0 ,and d = {(dk,t )k∈K }∞t=0 .
The supply of each asset k ∈ K is given by sk ≥ 0 and is constant overtime. We assume the following for the rest of the paper.
Assumption 5.1. For any k ∈ K and t ∈ Z+ , we have dk,t ≥ 0. Furthermore, for each k ∈ K we haveX
(5.3)
sak,−1 = sk .
a∈A

Agent a’s preferences are represented by a binary relation ≺a on C. Weaa∞say that a pair of sequences ca = {cat }∞t=0 and s = {(sk,t )k∈K }t=0 in R and
Rk , respectively, is a plan; a plan (ca , sa ) is feasible for agent a if it satisfies(5.1) and (5.2); and a feasible plan (ĉa , ŝa ) is optimal for agent a if there
exists no feasible plan (ca , sa ) for agent a such that ĉa ≺ ca .
An equilibrium of this economy is a set of sequences (p, {ca , sa }a∈A ) suchthat (i) (ca , sa ) is optimal for each agent a ∈ A, (ii) for each k ∈ K andt ∈ Z+ , we have pk,t ≥ 0, and (iii) the asset and good markets clear in all
12periods:
X

sak,t = sk ,∀k ∈ K, ∀t ∈ Z+ ,

(5.4)

a∈A

X

cat =

a∈A

X

yta +

a∈A

X

sk dk,t ,∀t ∈ Z+ .

(5.5)

k∈K

Whenever we take an equilibrium (p, {ca , sa }a∈A ) as given, we assume thefollowing.
Assumption 5.2. For any a ∈ A and c̃a ∈ C with ca < c̃a , we have ca ≺ c̃a .

5.2

Asset Bubbles

As in Subsection 2.2 we define the fundamental value of an asset and the
bubble component of the asset price in period t using the period t prices of
the consumption goods in periods t + 1, t + 2, . . .. As in Subsection 2.2 welet qot be the period 0 price of the consumption good in period t for t ∈ Z+ .
It is well-known (Santos and Woodoford, 1997, (2.1); Werner (2014, (7))
that under borrowing or debt constraints (see (5.13) and (5.16)), the absenceof arbitrage implies that there exists a price sequence {q0t } satisfying (2.6)
and (2.7) such that for all k ∈ K,∀t ∈ Z+ ,q0t pk,t = q0t+1 (pk,t+1 + dk,t+1 ).

(5.6)

Although this need not be true for all assets simultaneously under short sales
constraints, for each asset k ∈ K there exists a price sequence {q0t } satisfying(5.6) (as in Subsection 2.2). To simplify the exposition, for the rest of thepaper, we take as given an asset h ∈ K satisfying the following.
Assumption 5.3. There exists an asset h ∈ K such that ph,t > 0 for allt ∈ Z+ .
This assumption uniquely determines the price sequence {q0t } satisfying(5.6) and (2.7) (i.e., q00 = 1). As in Subsection 2.2, we define qti for all t ∈ N
and i ∈ Z+ using (2.8). We use these sequences to define the fundamental
value of asset h and the bubble component of ph,t as in Subsection 2.2:∞
X

qti dh,t+i ,(5.7)

bh,t = ph,t − fh,t .

(5.8)

fh,t =

i=1

13

Following the arguments for (2.10)–(2.20), we have
lim q0i ph,i = 0
i↑∞

5.3

⇔

∀t ∈ Z+ , bh,t = 0.

(5.9)

Examples of Constraints on Asset Holdings

In this subsection we fix an agent a ∈ A and present some examples of (5.2).
As in Subsection 3.1, the simplest example would be the following:∀t ∈ Z+ , ∀k ∈ K,

sak,t ≥ 0.

(5.10)To discuss more general constraints, for t ∈ Z+ and k 0 ∈ K, we define
saK\k,t = (sak0 ,t )k0 ∈K\{k} .

(5.11)

Consider the following constraint:asak,t ≥ σk,t(saK\k,t ),∀t ∈ Z+ , ∀k ∈ K,

(5.12)awhere σk,t
: Rk−1 → R ∪ {−∞} for all k ∈ K and t ∈ Z+ . Note that (5.10)
a
is a special case of (5.12) with σk,t(saK\k,t ) = 0 for all k ∈ K, sK\k,t ∈ Rk−1 ,and t ∈ Z+ .
The (state-dependent) borrowing constraint considered by Santos andWoodford (1997, p. 24) can be written in the current setting asX∀t ∈ Z+ ,pk,t sak,t ≥ −ξta ,(5.13)
k∈K

where ξta ∈ R for all t ∈ Z+ . This constraint is a special case of (5.12) with
 a P
a

− ξt + k0 ∈K\{h} pk0 ,t sk0 ,t if p > 0,k,ta
(5.14)
σk,t(saK\k,t ) =
pk,t

−∞
otherwise.
a
In fact, with σk,tdefined as above, (5.13) is equivalent to∀t ∈ Z+ ,asah,t ≥ σh,t(saK\h,t ),(5.15)

since ph,t > 0 for all t ∈ Z+ by Assumption 5.3. In other words, it suffices toimpose (5.15) only for one asset h ∈ K; the other constraints given by (5.12)
are redundant.
14

The (state-dependent) debt constraint considered by Werner (2014) andLeRoy and Werner (2014, p. 313) can be written in the current setting asXa.
(5.16)∀t ∈ Z+ ,(pk,t+1 + dk,t+1 )sat ≥ −ξt+1
k∈K

Given Assumption 5.3, this constraint is another special case of (5.15) withPa
+ k∈K\{h} (pk,t+1 + dk,t+1 )sak,tξt+1
aσh,t (sK\h,t ) = −.
(5.17)
ph,t+1 + dh,t+1
A multi-asset version of the wealth constraint considered by Kocherlakota(1992) can be written as∀t ∈ Z+ ,X

pk,t sak,t +∞Xqti yt+i ≥ 0.

(5.18)

i=1

k∈KThis is yet another special case of (5.15) with
P
P∞ i
a
q
y
+
t+i
t
i=1
k∈K\{h} pk,t sk,taa
σh,t (sK\h,t ) = −.
ph,t(5.19)We should mention that (5.16) and (5.18) are also special cases of (5.12)awith σk,tdefined similarly to (5.14).

6

General Equilibrium Results

In this section we take an equilibrium (p, {ca , sa }a∈A ) as given, and consider
conditions to rule out asset bubbles in the general equilibrium setting introduced in the previous section.
We start by extending Theorem 4.1 to the current general equilibrium
setting. For this purpose, we need additional notation. Given any sequencekasa = {(sak,t )k∈K }∞t=0 in R , τ ∈ Z+ , and  > 0, let Sh,τ, (s ) be the set ofksequences {(sk,t )k∈K }∞t=0 in R such thatsk,t = sak,t , ∀t ∈ Z+ if k 6= h,(= sah,t
if t < τ ,sh,ta≥ sh,t −  if t ≥ τ .
Note that (6.2) takes the same form as (4.1).
15

(6.1)
(6.2)

Theorem 6.1. Suppose that there exists an agent a ∈ A such that for someτ ∈ Z+ and  > 0, we have
Sh,τ, (sa ) ⊂ S a (sa−1 , y a , p, d).

(6.3)

Then bh,t = 0 for all t ∈ Z+ .
Proof. Let pt = ph,t , dt = dh,t , and bt = bh,t for all t ∈ Z+ . Then (2.5)–(2.7) hold, and Assumptions 2.1–2.3 follow from Assumptions 5.1–5.3. Since(ca , sa ) is optimal for agent a and (6.3) implies (4.2) with (c∗ , s∗ ) = (ca , sa ),it follows that the conclusion of Theorem 4.1 holds.
The following result is essentially a restatement of Corollary 4.1 in thecurrent general equilibrium setting with multiple assets.
Proposition 6.1. Suppose that there exists an agent a ∈ A such that (5.2)
ais given by (5.15) for some σh,t
: Rk−1 → R for all t ∈ Z+ . Suppose furtherthata(saK\h,t )) > 0.
lim(sah,t − σh,t

(6.4)

t↑∞

Then the conclusion of Theorem 6.1 holds.
aProof. This follows from Corollary 4.1 with σt = σh,t(saK\h,t ) for all t ∈Z+ .

As discussed in Subsection 5.3, various constraints on borrowing, debt,
and wealth can be written in the form of (5.15). Hence Proposition 6.1
applies to economies with such constraints.
In what follows, we present some results that can be regarded as generalizations of Proposition 3 in Kocherlakota (1992). We discuss the relation
between his and our results after showing our results. For the rest of thissection, we maintain the following assumption.
aAssumption 6.1. For each agent a ∈ A, there exists a sequence {(ηk,t)k∈K }∞
t=0
k
in
R
,
we
have
in Rk such that given any sequence s = {(sk,t )k∈K }∞t=0
a
sk,t ≥ ηk,t
, ∀k ∈ K, ∀t ∈ Z+⇔

s ∈ S a (sa−1 , y a , p, d).

(6.5)

This assumption means that the feasibility constraint on asset holdings
for each agent, (5.2), consists of sequences of constraints of the form (3.3)
for all assets. The following result is immediate from Proposition 6.1.
16

Corollary 6.1. If there exists an agent a ∈ A such thata) > 0,
lim(sah,t − ηh,t

(6.6)

t↑∞

then the conclusion of Theorem 6.1 hold.
To state the next result, we define the following for each a ∈ A:a
η ah = lim ηh,t
.

(6.7)

t↑∞

Corollary 6.2. If there exists an agent a ∈ A such thatlim sah,t > η ak ,

(6.8)

t↑∞

then the conclusion of Theorem 6.1 holds.
Proof. Let a ∈ A satisfy (6.8). This strict inequality implies thatlim sah,t > −∞,t↑∞η ah,t < ∞.

(6.9)

Hence
a
lim(sah,t − ηh,t
) ≥ lim sah,t − η ah,t > 0,t↑∞

(6.10)t↑∞where the second inequality holds by (6.8). Thus the conclusion of Theorem
6.1 holds by Corollary 6.1.
If there exists a constant lower bounded on the asset h for each agenta ∈ A, the above result reduces to the following.
Corollary 6.3. Suppose that∀a ∈ A, ∃ηha ∈ R, ∀t ∈ Z+ ,aηh,t
= ηha .

(6.11)

Suppose further that there exists an agent a ∈ A such thatlim sah,t > ηha .
t↑∞Then the conclusion of Theorem 6.1 holds.
17(6.12)

Kocherlakota (1992, Proposition 3) in effect shows a special case of Corollary 6.3 under the following additional assumptions: (i) there is only one asset(i.,e k = 1); (ii) the binary relation ≺a of each agent a ∈ A is represented by(3.10), where u depends on a ∈ A and is denoted as ua : R+ → [−∞, ∞) but
β is common to all agents; (iii) for each a ∈ A, ua is continuously differentiable on R++ , strictly increasing, concave, and bounded above or below by
zero; and (iv) the optimal plan (ca , sa ) of each agent a ∈ A satisfies∀t ∈ Z+ , cat > 0,∞X
β t ua (cat ) < ∞.

(6.13)(6.14)

t=0

Corollary 6.3 shows that none of Kocherlakota’s additional assumptions
is needed under Assumption 5.2, which is implied by his assumptions. Hence
Corollary 6.3 is a substantial generalization of Proposition 3 in Kocherlakota(1992). He uses the extra assumptions mostly to derive a transversality
condition, which is crucial to his approach. By contrast, our results are
based on our simple no-bubble theorem, Theorem 4.1, which can be provedby an elementary perturbation argument that fully exploits the structure ofsequential budget constraints.

7

Concluding Comments

In this paper we showed a simple no-bubble theorem that applies to a wide
range of deterministic economies with infinitely lived agents facing sequential budget constraints. In particular, we showed that asset bubbles can beruled out if there is at least one agent who can reduce his asset holdings
permanently from some period onward. This is a substantial generalizationof Kocherlakota’s (1992) result on asset bubbles and short sales constraints;
our no-bubble theorem requires virtually no assumption except for the strict
monotonicity of preferences.
Although we also developed some results on asset bubbles in a generalequilibrium setting, all of them are based on the optimal behavior of a single
agent. Additional results can be shown by using our results in conjunction
with other arguments based on market-clearing and aggregation.

18

Appendix A

Proof of Theorem 4.1

Let (c∗ , s∗ ) be an optimal plan. It suffices to verify thatlim q0i pi = 0,

(A.1)

i↑∞

which implies the desired conclusion by (2.20). Suppose by way of contradiction thatlim q0i pi > 0.

(A.2)

i↑∞

Then since q0i pi > 0 for all i ∈ Z+ by Assumption 2.2 and (2.6), it follows
that there exists b > 0 such thatq0i pi ≥ b.

∀i ∈ Z+ ,(A.3)

Equivalently, we have 1/pi ≤ q0i /b for all i ∈ Z+ . Multiplying through by diand summing over i ∈ N, we obtain
∞
X
di
i=1

pi

≤

∞
X
q i di
0

i=1

b

=

f0
< ∞,
b

(A.4)

where the equality uses (2.16).5
Let τ ∈ Z+ and  > 0 be given by (4.2). For each δ ∈ (0, ) we construct
an alternative plan (cδ , sδ ) as follows:(
if t 6= τ ,
c∗t
δ
(A.5)
ct =
c∗τ + pτ δ if t = τ ,∗if t ≤ τ − 1,s t
δst = s∗τ − δ(A.6)
if t = τ ,[yt + (pt + dt )sδt−1 − c∗t ]/pt if τ ≥ τ + 1.
It suffices to show that (cδ , sδ ) is feasible for δ > 0 sufficiently small; for then,we have c∗ ≺ cδ by (A.5) and Assumption 2.3, contradicting the optimalityof (c∗ , s∗ ).
5

An arguments similar to (A.4) is used by Montrucchio (2004, Theorem 2).

19

Note that (cδ , sδ ) satisfies (2.1) by construction. Hence by (2.1) we have∀t ≥ τ + 1,

pt (s∗t − sδt ) = (pt + dt )(s∗t−1 − sδt−1 ).

(A.7)

For t ≥ τ define
δt = s∗t − sδt .

(A.8)

Note that δτ = δ by (A.6). We have pt δt = (pt + dt )δt−1 for all t > τ by(A.7). Thus for any t > τ we havept + dt
pt + dt pt−1 + dt−1
δt−1 =
δt−2 = · · ·
pt
pt
pt−1
t
∞
Y
Y
pi + dipi + di
=δ
≤δ
,
pi
pi
i=τ +1
i=1

δt =

(A.9)
(A.10)

where the equality in (A.10) holds since δτ = δ, and the inequality in (A.10)
holds since dt ≥ 0 for all t ∈ Z+ by Assumption 2.1.6
To show that (cδ , sδ ) is feasible, it suffices to verify that δt ≤  for allt ≥ τ ; for then, we have s ∈ S(s−1 , y, p, d) by (4.2) and (A.8). For thispurpose, note from (A.4) that


∞
∞
di
f 0 X di X≥≥
ln 1 +
b
p
pi
i=1 i
i=1
!


∞
∞
X
Y
pi + dipi + di
=
ln
= ln
.
pi
pi
i=1
i=1
It follows that∞Ypi + di
i=1

pi

< ∞.

(A.11)(A.12)(A.13)

Using this and recalling (A.9)–(A.10), we can choose δ > 0 small enoughthat δt ≤  for all t ≥ τ . For such δ, (cδ , sδ ) is feasible, contradicting the
optimality of (c∗ , s∗ ). We have verified (A.1), which implies the conclusionof the theorem.
6

An argument similar to (A.9)–(A.10) is used by Bosi et al. (2014).

20

Appendix B

Proof of Proposition 4.2

Let τ ∈ Z+ and  > 0 be as in (4.2). Suppose by way of contraction thatpτ 0 > 0 for some τ 0 ≥ τ . Without loss of generality, we assume that τ 0 = τ =0;7 i.e.,p0 > 0.

(B.1)

First suppose that∀t ∈ N,pt > 0.

(B.2)

0Then Assumption 2.2 holds. We construct {q0t }∞t=0 by (2.5) with q0 = 1.
Then (2.5)–(2.7) hold. Since Assumptions 2.1–2.3 and (2.5)–(2.7) now hold,
Theorem 4.1 applies. But note from (4.6) and (2.16) that∀t ∈ Z+ ,ft = 0.

(B.3)

Hence by Theorem 4.1, we have bt = 0, i.e., pt = ft = 0, for all t ∈ Z+ . This
contradicts (B.2).
We have shown that (B.2) cannot be true. In other words, there must bet ∈ N such that pt = 0. Let T be the first T ∈ Z+ withpT > 0,pT +1 = 0.

(B.4)

Such T must exist by (B.1). We construct an alternative plan (c, s) as follows:(
if t 6= T ,
c∗t
(B.5)
ct =
∗
cT + pT  if t = T ,(if t 6= T ,
s∗t
st =
(B.6)
∗
sT −  if t = T .
According to this plan, the agent sells the asset when its price is strictly
positive, and buys it back when it is free. It is easy to see from (4.2), (2.1),
and (B.4) that (c, s) is feasible. But then we have c∗ ≺ c by (B.5) andAssumption 2.3, contradicting the optimality of (c∗ , s∗ ).
We have shown that we reach a contradiction whether (B.2) holds or not;thus we must have p0 = 0.
It is no loss of generality to assume that τ 0 = τ = 0 since we only consider variables
in and after period τ 0 .
7

21

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23