Dynamic Efficiency and Asset Bubbles
under Financial Market Friction #Masaya Sakuragawa
masaya@econ.keio.ac.jp

December 31, 2007
Abstract
This paper investigates whether the criterion for the real interest rate versus the economic
growth rate is effective in evaluating dynamic efficiency/inefficiency and understanding asset
bubbles in a dynamic economy with frictions of financial markets. In the closed economy, thiscriterion is effective for sustainability of bubbles, but not in terms of dynamic
efficiency/inefficiency. Asset bubbles coexist with capital under-accumulation and do notnecessarily improve dynamic efficiency. Asset bubbles emerge even when savings are so smallthat capital is less accumulated than the Golden Rule in the bubbleless economy. The
effectiveness of that criterion is restored in the small-open economy if the output produced byborrowers is international collateral.

#

We are gratefully thankful to Koichi Hamada, Masashi Manabe, and participants at seminar held in
Keio.
1

1. Introduction
The criterion for the real interest rate versus the economic growth rate is a problem oftheoretical and empirical importance in evaluating the efficiency of the economy andunderstanding sustainability of asset bubbles. Diamond (1965), Ihori (1978), and Tirole (1985)
demonstrate that when the return to capital is equal to the real interest rate, this criterion
becomes a benchmark for understanding dynamic efficiency and sustainability of bubbles.
However, various kinds of financial frictions, including uncertainty, transaction costs,
asymmetric information, and other incentive problems will deter this equality to continue tohold in the actual economy so that the effectiveness of this criterion is an open question. Abel etal (1989) and Bohn (1995) provide examples in which the safe interest rate is smaller than theeconomic growth rate due to risk premium but the economy is dynamically efficient.
This problem is stringent also from the empirical prospect because the current speculative
bubbles will be associated with the historically low real interest rate. Figure 1 illustrates averageinterest rates and the average economic growth rate of G7 countries (U.S., U.K., Japan,
Germany, France, Canada, and Italy).1 Remarkably, before the end of 1990s, all the interest
rates were greater than the economic growth rate except for the deposit rate, but after that period,
many of the interest rates began to be lower than the economic growth rate. Since 2000 andaround, several kinds of bubbles recurrently emerged, including IT bubbles, housing market
booming in US, the booming in the global stock markets, and appreciations in gold and oil
prices.2

1

Each of interest rates and the economic growth rate is a simple average of G-7 countries (source:
IFS). We exclude the 1991 data of German in calculating averages. Money market rate, treasury bills,
treasury bills: 3years or longer, deposit rate, and max overdraft reflect call rate, short-term rate of thegovernment bond, long-term rate of the government bond, short-term deposit interest rate, and loaninterest, respectively.
2
Additionally, behind asset bubbles in China that have been sustained for more than two
decades is the far higher economic growth rate than the interest rates.
2

Tirole (1985) demonstrates that asset bubbles arise when capital is overly accumulated in
the bubbleless economy. He argues the recurrent emergence of multiple bubbles using theterminology of “bubble substitution”, but fails in explain ing the coexistence of “global bubbles”
and weak incentive for investment relative to great global savings , which is one of importantfeatures of the current world economy.3 Caballero and Krishnamurthy (2006) develop an
insightful model of a small-open economy in which capital under-accumulation and asset
bubbles can coexist. In theirs, asset bubbles are used for collateral for financing productive
investment. Their contribution is to show the complementary role of bubbles with capital
accumulation, in contrast with Tirole where bubbles and capital accumulation are substitutable.
The purpose of this paper is to investigate the effectiveness of the criterion for the realinterest rate versus the economic growth rate in evaluating dynamic efficiency/inefficiency andunderstanding asset bubbles in the presence of financial market frictions. In doing so, weconstruct a simple model which is as close as possible to Diamond (1965), Ihori (1978), andTirole (1985) except for that there is a friction in the financial market. As the extensive literatureargues, borrowing constraints or credit rationing that arises in response to incentive problems
that prevail in financial markets leads to the breakdown of the equality between the return tocapital and the real interest rate [e.g., Stiglitz and Weiss (1981), Gale and Hellwig (1985),
Williamson (1986), Bernanke and Gertler (1989), and others]. The discrepancy between the two
rates has different results on dynamic efficiency and sustainability of asset bubbles, relative tothe standard economy.
In the closed economy, the criterion for the real interest rate versus economic growth rate iseffective for sustainability of bubbles, but not in terms of dynamic efficiency/inefficiency. Assetbubbles move the capital stock down to the smaller level than the Golden Rule so that asset
3

Caballero (2006) calls weak incentive for investment relative to great global savings the “asset
shortage”.
3

bubbles and capital under-accumulation coexist. Interestingly, asset bubbles emerge even whensavings are so small that capital is less accumulated than the Golden Rule in the bubbleless
economy. Asset bubbles may or may not restore dynamic efficiency, and do only if capital is far
overly accumulated in the bubbleless economy. The theoretical finding explains asset bubbles,
asset shortage, and the low real interest rate simultaneously, all of which features the current
world economy. The effectiveness of that criterion is restored in the small-open economy withthe real interest rate being constant if the output produced by entrepreneurs becomesinternational collateral. In the small-open economy, asset bubbles never crowd out capital
accumulation, but has the role of raising the real interest rate faced by domestic investors.
We comment on related works. Abel et al (1989) , Zilcha (1992), and Bohn (1995)
demonstrate that, in a stochastic model, dynamic efficiency depends on the relation between thegrowth rate and the rate of return on “risky” capital, not on the one between the growth rate andthe safe interest rate. Thus, the lower safe interest rate than the growth rate is consistent withdynamic efficiency and no-Ponzi condition.
Grossman and Yanagawa (1993) and Femminis (2002) provide endogenous growth models
in which the presence of bubbles is welfare-reducing so that the condition for dynamic
efficiency does not coincident with the one for sustainable bubbles. Ventura (2003) constructs agrowth model of financial market friction in which bubbles can promote capital
accumulation.This paper is organized as follows. Section 2 sets up the model and studies thebenchmark economy. Section 3 analyzes the closed economy when there is the friction offinancial market, and Section 4 the small-open economy. Section 5 concludes.

42. Basic Model
Let us consider an economy of overlapping generations that lasts for infinity. At each periodt = 0,1,2 ,..., ∞, the economy is populated by a continuum of ex ante identical agents that live
for two periods. Letting Nt denote the number of young people at t, the population grows at
rate n > 0 , satisfying N t = (1 + n) t N 0 = (1 + n) t . At each period the final good is produced byfirms that use labor and capital as inputs according to the constant-returns-to-scale technology
described as Yt = F ( Kt , Nt ) , where Kt and Nt are aggregate supplies of capital and labor,
and Yt is the output of the final good. That technology is described as a per-capita formby yt ≡ Yt Nt = F ( Kt Nt ,1) ≡ f ( kt ) , where kt is the capital-labor ratio and yt is the
per-capita output of the final good. f (.) is thrice continuously differentiable, increasing,concave, satisfying f ( 0) = 0 , and lim f ' ( k t ) = + ∞ . Since the production technology is
kt → 0homogeneous of degree one, output of the final good can be described in terms of the action of asingle, aggregate, price-taking firm. From the maximization problem of that firm, each input ispaid its marginal product. The rate of return to capital Rt and the wage rate Wt are determinedto satisfy Rt = f ' ( kt ) and Wt = f ( kt ) − kt f ' ( kt ) ≡ W ( kt ) , respectively. Assume that capitaldepreciates fully after one period. The final good is numeraire. The price of capital is equal to

Rt . Each of ex ante identical agents born at t maximizes log cty + β Et log cto+1 ,wherecty (cto+1 ) is consumption in the first (second ) period of life, and Et is the expectation operator.
At the first period of life, each of them supplies one unit of time inelastically in the labor market
Having received the wage repayment from the firm, each agent discovers his/her type. With
probability α ( 0 < α < 1) , he is an “entrepreneur”, while with probability 1 − α , he turns out tobe an “investor”.
Each entrepreneur has access to one linear capital investment technology that transforms

5one unit of the final good into one unit of capital after one period. On the other hand, any
investor cannot have access to the capital investment technology and earns the second-period
income only by investing the first-period income to others.
Assume that there is no enforcement mechanism to fulfill financial contracts between
debtors and creditors and hence to enforce on borrowers to repay their debt. When debtors
breach the contract and refuse to make their repayment, a portion λ ( 0 < λ < 1) of their earnings
are assumed to be forfeited by the creditor. A low λ is interpreted to capture weak bankruptcy
procedure, poor bank monitoring, and low contract enforcement, and will be associated withpoorly developed financial markets [e.g., La Porta et al (1997, 1998), Levine (1998), Levine etal (2000)]. 4 The parameter λ is thus interpreted to capture the efficiency of the broadly
defined financial system.
First of all, we investigate an economy with perfect capital market that is characterized byperfect enforcement. Letting rt +1 denote the interest rate at t+1, any of entrepreneurs earns

f ' ( kt +1) it − ( it − Wt )(1 + rt +1) by investing an amount of it in his own project, while he earns
Wt (1 + rt +1 ) by supplying his first-period income to others. Entrepreneurs are willing to start
their own projects if(2-1)

f ' ( kt +1 ) ≥ 1 + rt +1 .

We call this inequality the profitability constraint. If f ' ( kt +1 ) > 1 + rt +1 , any of them would
demand the fund indefinitely as much as possible, which drives the interest rate up. Conversely,
if f ' ( kt +1 ) < 1 + rt +1 , any of them would stop the project, which drives the interest rate down.
When agents borrow and lend indefinitely at the prevailing interest rate,(2-2)

f ' ( kt +1 ) = 1 + rt +1

4

La Porta et al (1997) find that countries with poor investor protections, measured by both thecharacter of legal rules and the quality of law enforcement, have smaller capital markets. Levine(1998) shows that financial depth is closely linked with measures of legal treatment of outside
creditors developed by La Porta et al(1997).
6

is only sustainable in equilibrium. Entrepreneurs and investors then earns 1 + rt +1 per unit
equally, and the uncertainty about his/her type is irrelevant.
Both entrepreneurs and investors save a fraction s ≡ β (1 + β ) of the first-period income.
Letting Bt denote the aggregate bubbles, which can be best thought of pieces of paper, theaggregate saving is used for financing investment in capital and purchasing bubbles, and therelation is described by(2-3)

Nt sW ( kt ) = αit Nt + Bt .

The definition of the aggregate capital is given by Kt +1 = α it Nt . Letting bt denote theaggregate bubbles per capita at t, (2-3) is expressed in per capita term by(2-4)(1 + n) kt +1 = sW ( kt ) − bt .

Under perfect foresight, bubbles have to earn the same rate of return as that on capital to satisfy

Bt +1 = (1 + rt +1) Bt . The aggregate bubbles per capita grow to satisfy(2-5)(1 + n)bt +1 = (1 + rt +1) bt .

Lastly bubbles have to be non-negative;(2-6)

bt ≥ 0 .

We define two kinds of equilibria. A bubbleless equilibrium is defined as an equilibrium inwhich bt = 0 for any t or bt converges to zero if bt > 0 for any t . An asymptotically
bubbly equilibrium is defined as an equilibrium in which bt does not converge to zero.5
First of all, we examine the analysis of steady states. The steady state of a bubblelessequilibrium is characterized by a pair {k , r } , satisfying (1 + n) k = sW ( k ) , and 1 + r = f ' ( k ) .
On the other hand, the steady state of a bubbly asymptotically equilibrium is characterized by apair {kGR , rGR , bGR} , satisfying (1 + n) kGR = sW ( kGR ) − bGR , 1 + rGR = f ' ( kGR ) , and rGR = n ,5Note that Tirole (1985) distinguishes between an asymptotically bubbly equilibrium and a bubbly
equilibrium by defining the latter as the one in which bt > 0 for any t .
7with bGR > 0 .
We briefly summarize the properties of the economy under perfect financial markets. If

r > n , there exists a unique bubbleless equilibrium and the interest rate converges to r , whileotherwise, there exists an asymptotically bubbly equilibrium and the interest rate converges ton [ Tirole (1985, Proposition 1)]. Ihori (1978) demonstrates that the government bond, which is
intrinsically valueless, carries the long-run capital level to the Golden Rule level of capital whenr <n.

3. The Economy with Financial Market Imperfection
We now introduce the imperfection of financial markets into the benchmark model. The
financial market is competitive in the sense that both entrepreneurs (borrowers) and investors(lenders) take the equilibrium rate rt +1 as given. Let xt denote the amount of saving that is used
for internal wealth added for the investment project. If any of entrepreneurs borrows (it − xt )and repays (1 + rt +1 ) (it − xt ) to investors honestly, he earns f ' ( kt +1) it − ( it − xt )(1 + rt +1) ,while if he breaches the promise for repayment, a portion λ of his earning is forfeited, and his
earning would be (1 − λ ) f ' ( kt +1) it . The following incentive compatibility constraint
summarizes the imperfection of the financial markets, i.e.,(3-1)(it − xt ) × (1 + rt +1) ≤ λf ' ( kt +1) it .

Equation (3-1) will be called the “borrowing constraint”. Entrepreneurs can borrow the amount
up to some fraction of the project revenue.6 One may derive the similar borrowing constraint
from a number of other incentive considerations.7 When the borrowing constraint binds with
6

Implicit in (3-1) is that entrepreneurs do not use the borrowed fund to buy bubbles. Entrepreneurs
will borrow only for capital investment because the rate of return from capital is greater than the one
from holding bubbles when (3-1) binds with equality, as argued below.
7
For example, it is possible to derive the borrowing constraint by engaging in the
costly-state-verification approach that began with Townsend (1979), and has been developed by Gale8

equality, the rate of return faced by entrepreneurs is z ( rt +1, kt +!) ≡ (1 − λ )f ' (kt +1 )(1 + rt +1),
1 + rt +1 − λf ' (k t+1 )which is assumed to be positive at this stage. Each agent that saves the amount of xt earns

xt z ( rt +1 , kt +1) if he/she is an entrepreneur, while xt (1 + rt +1) if he/she is an investor. Ex ante
identical agent turns out to choose the amount of saving xt to maximizelog( Wt − xt ) + β log xt + β (1 − α ) log( 1 + rt +1 ) + βα log z (rt +1, kt +1 ) .
Obviously the agent chooses xt = sWt since the agent’s saving behavior is independent of the
return to saving. It is useful to consider first the economy in which there are no bubbles. The
market clearing in the capital market is given by(3-2)

α (it − sW (k t )) = (1 − α ) sW ( kt ) ,Where the L.H.S. is the demand for funds by entrepreneurs, and the R.H.S. is the supply of fundby investors.
We turn to the determination of the real interest rate. At least either of the two constraints,
the profitability constraint and the borrowing constraint, should bind with equality. If theborrowing constraint is not binding, the profitability constraint should bind with equality, while
if the borrowing constraint is binding with equality, the profitability constraint may not bebinding. The above argument is summarized by(3-3)

1 + rt+1 = min{ f ' ( kt +1), λf ' ( kt +1)

it}.
it − xt

Without loss of generality, we confine attention on symmetric equilibria in which all
entrepreneurs choose the same amount of investment. Equation (3-3), using (3-2), reduces to(3-4)

1 + rt+ 1 = min{ f ' ( kt +1 ),λf ' ( kt +1)} .
1− αand Hellwig (1985) and Williamson (1986).In addition, if banks are established as an optimal
response to the CSV problem [e.g. Diamond (1984)], rt +1 will be interpreted as the “safe” depositinterest rate, to be distinguished from the risky loan interest rate.
9

It is straightforward to see that the borrowing constraint is binding if and only if α + λ < 1 , andthen we have 1 + rt+1 =λf ' ( kt +1 ). Hereafter we impose the following in order to focus on an
1 −αinteresting case.

α + λ < 1.

Assumption 1

The fraction of entrepreneurs α is a measure of separation between creditors and debtors, andmatters when the borrowing constraint is crucial. As α → 1 , outside funds are negligible, andall investment is carried out directly by entrepreneurs, while as α → 0 , outside funds are more
important, and each of entrepreneurs has to borrow the greater amount from investors. Anotherparameter λ capture the development of the the contract enforcement mechanism as argued
above. As λ → 1 , the incentive compatibility constraint is always satisfied, and entrepreneurs
would be able to borrow as much as possible, taking rt +1 as given. As λ → 0 , entrepreneurs
would be able to borrow nothing and hence have to self-finance their investment entirely. As
either α declines or λ rises, Assumption 1 is more likely to be satisfied. Assumption 1 isintended to describe an economy with the severe borrowing constraint. Note that when
Assumption 1 is satisfied, we have 1 + rt+1 =λf ' ( kt +1)> λf ' ( kt +1 ) , which guarantees
1 −αz ( rt +1 , kt +1) to be positive.
The steady state of an economy with the binding borrowing constraint is characterized by~

the pair {k , ~r } , satisfying
(3-5)

~
k =

s
~
W (k ) , and
1+ n

(3-6)

λ
~
1+ ~
r =
f ' (k ) .
1−α10It is straightforward to see that the levels of the steady state capital stock and marginal productsof capital are the same between the two economies with and without the borrowing constraint~( k = k ). The real interest rate differs between the two economies. It follows from~
1 + r = f ' ( k ) , (3-6), k = k , and Assumption 1 that the binding borrowing constraint is asource of the declining interest rate. The possible shortage of demand for investment that arises
from the borrowing constraint has to be adjusted by decline in the interest rate to equal thepredetermined aggregate savings. 8 In other words, the marginal product of capital diverges
from the equilibrium interest rate when the borrowing constraint binds with equality.
We now turn to the analysis of the economy with bubbles when there is an imperfection offinancial markets. The market clearing in the credit market is rewritten as(3-7)

α (it − sW (k t )) = (1 − α ) sW ( kt ) − bt .

When the borrowing constraint binds with equality, (3-1) is finally expressed by(3-8){(1 − α ) sW ( kt ) − bt }(1 + rt +1 ) = λf ' ( kt +1 ){ sW (k t ) − bt} .

Equations (2-4), (2-5), (2-6), and (3-8) define an asymptotically bubbly equilibrium withbt > 0 for t → ∞ . The steady state of an asymptotically bubbly equilibrium, if it exists, is~ ~characterized by a pair {k B , b , ~rB} , satisfying
(3-9)

~
1
~
~
kB =
{W ( kB ) − b }1+ n(3-10)~~~~~{(1 − α ) sW (k B ) − b }(1 + ~rB ) = λf ' ( k B ){ sW ( kB ) − b } ,(3-11)~
rB = n , and(3-12)~b >0.

8

A number of overlapping generations models, including Azariadis and Smith (1993), Sakuragawa
and Hamada (2001), Matsuyama (2004), and others, have shown that the introduction of theborrowing constraint, regardless of whether it is either endogenously derived as a response to the
incentive problem or exogenously imposed, leads to a decline in the equilibrium interest rate.
11

We investigate general features of the bubbly equilibrium. Allowing for non-binding borrowing
constraint, (3-1) is rewritten, using (3-7) and (1 + n)k t+1 = αit , as(3-13)

1 + rt+1 ≤ λf ' ( kt +1)(1 + n) kt +1.
(1 + n) kt +1 − αsW ( kt )Following (3-3), the real interest rate is thus determined to satisfy(3-14))

1 + rt+1 = min{ f ' ( kt +1), λf ' ( kt +1)(1 + n) kt +1}(1 + n) kt +1 − α sW (kt )Given this preparation, we obtain the following.

Proposition 1
If the borrowing constraint is binding with equality, the asymptotically bubbly equilibrium
never achieves the Golden Rule, and attains capital under-accumulation.
Proof. The steady-state relation between k and r is expressed as(3-15)

1 + r = min{ f ' (k ), λf ' (k )(1 + n) k} ≡ min{ f ' ( k ), Λ (k )}(1 + n) k − αsW ( k )In the ( k , r ) plane, the real interest rate should be lower between the two curves. In Figure 2,

Λ (k ) is illustrated to be always lower than f ' ( k ) . When the borrowing constraint is binding,for any given r , the steady state level of capital stock has to be smaller than the one in the~benchmark economy, and hence the one in the bubbly equilibrium, k B , should be smaller thankGR . Q.E.D.

Asset bubbles never achieve the Golden Rule in the presence of the friction of financial
markets, but rather move the capital stock to the smaller level than the Golden Rule level,
eventually giving rise to capital under-accumulation. Remarkably, asset bubbles are coexistent
with capital under-accumulation, which is sharply contrasted with Diamond (1965), Ihori (1978),and Tirole (1985). The theoretical finding explains the coexistence of asset bubbles, asset

12

shortage, and the low real interest, which is a striking feature of the current world economy. The
space for parameter values under which the bubbly equilibrium exists is easily checked. As the~~saving rate s increases, k is high {from (3-5)}and f ' ( k ) is low, and hence ~r tends to be~
low {from (3-6)}. Taking f ' ( k ) as given, as either λ or α is small, ~r is more likely tobe smaller than n . Other things being equal, a higher s , a smaller λ or α is likely to leads
to the emergence of the bubbly equilibrium.
Demonstrating the dynamic properties is almost similar to Tirole (1985) and Weil (1987)
except for the derivation of the locus bt +1 = bt . The curve bt +1 = bt follows from (2-5) and(3-7), and is given by(1 + n){(1 − α ) sW ( kt ) − bt } = λf ' ((3-16)sW ( kt ) − bt){ sW ( kt ) − bt } .
1+ nThere exists a continuously differentiable function bt = Φ ( kt ) , satisfying (3-16), with thederivative;(3-17)

Φ ' ( kt ) =− λf ' ( kt +1 ) /(1 + n ) + αs(1 + n)btW ' (k t ) /{sW ( kt ) − b}2> 0.
− λf " ( kt +1) /(1 + n) + s (1 + n )αW ( kt ) {sW ( kt ) − bt }2

So long as the curve bt +1 = bt crosses the curve k t+1 = kt from below, dynamic properties are
qualitatively the same as those developed by Tirole (1985) and Weil (1987). The properties of
equilibria are summarized as follows.

Proposition 2(a) If ~r > n , there exists a unique equilibrium. This equilibrium is bubbleless and the interest
rate converges to ~r.

~(b) If f ' ( k ) > 1 + n > 1 + ~r , there exists a unique asymptotically bubbly equilibrium with~initial bubble b0 . The per-capita bubble converges to b and the interest rate converges to n .

13~In the asymptotically bubbly equilibrium, the steady-state per-capita capital, denoted k B ,

~

~

~

satisfies f ' ( k B ) > 1 + n = f ' (k GR ) , with k B < k < kGR .

~(c) If 1 + n > f ' ( k ) > 1 + ~r , there exists a unique asymptotically bubbly equilibrium with~initial bubble b0 . The per-capita bubble converges to b and the interest rate converges to n .
In the asymptotically bubbly equilibrium, the steady-state per-capita capital satisfies~~~
f ' ( k B ) > 1 + n = f ' (k GR ) , with k B < k GR < k .

A heuristic proof of Proposition 2 is as follows. Agents require that, at the stationary state, therate of return on bubbles, 1 + n , be at least equal to the rate of return on lending, 1+ ~rB , so that~it must be the case that n ≥ ~rB if b > 0. On the other hand, at the stationary state, the presenceof bubbles decreases the capital stock relative to the bubbleless equilibrium, so that we must~ ~~have k B < k and hence ~rB > ~
r if b > 0. Therefore, the necessary condition for bubbles tobe sustainable is n > ~r . Conversely, if n > ~r , bubbles absorb the aggregate savings andreduces the capital stock until the interest rate 1+ ~r is pushed up to 1 + n . Finally, if ~r > n ,it must be the case that ~
rB =

λ
~
f ' (k B ) − 1 > ~r , but then it follows from (2-5) that the
1 −αaggregate bubbles per capita should grow indefinitely, which is infeasible.

~
Asset bubbles are sustainable if and only if ~r < n . The return to capital f ' ( k ) is irrelevantto the criterion for the viability of bubbles. In contrast to Tirole (1985), asset bubbles are
sustainable even when capital is less accumulated than the Golden Rule in the bubbleless
economy is less (Proposition 2(b)). Figure 3 represents one typical configuration of theequilibrium dynamics when the steady state capital stock of the bubbleless equilibrium is

14

~

~

smaller than the one of the Golden Rule, such that k B < k < kGR . The asymptotically bubbly
equilibrium W is a global saddlepoint. All dynamic paths originating from below b0 converges
toward the bubbleless steady state D. Trajectories starting above b0 are infeasible as they alllead to the resource constraint being violated in finite periods.
We see that if there is a friction in financial markets, asset bubbles emerge for greaterparameter space than otherwise. We obtain the following

Corollary
Asset bubbles are sustainable for the greater parameter space of the saving rate s in the economy
with financial market friction than without it. The smaller is either α or λ , the parameter
space of the admissible saving rate is even greater.
Proof: Let ~s denote the saving rate that satisfies 1 + n =λf ' (k ( ~s , n)) , and let s denote
1− αthe saving rate that satisfies 1 + n = f ' (k ( s , n )) , where k = k ( s, n ) is implicitly derived from
(3-5), and increasing in s . Under Assumption 1, ~s < s holds. Furthermore ~s is decreasing ifeither α or λ decreases. Q.E.D.

The high saving rate is not necessary for asset bubbles to emerge if there is a friction in financial
markets. To the extent that the borrowing constraint is severe, asset bubbles are more likely toarise even for the small saving rate.
I now turn to the question of efficiency. 9 The argument is complicated by the finding that9

If an intra-generational transfer of income would be permitted between investors and entrepreneurs
through government intervention at the first period of their lives, an appropriate tax-subsidy scheme
will move the economy substantially to the Diamond-Tirole model. The bubbleless equilibrium isdynamically efficient if ~r > n , while otherwise, the bubbleless equilibria are dynamicallyinefficient and the asymptotically bubbly equilibrium is dynamically efficient [ Tirole (1985,
Proposition 2)].
15

the asymptotically bubbly equilibrium realizes the smaller per-capita consumption than theGolden Rule (Proposition 1). If ~r > n , the efficiency result is straightforward because thesustainable bubbles are ruled out under perfect foresight.

~If ~r < n , we have interesting findings. If f ' ( k ) > 1 + n > 1 + ~r , in the original steadystate of the bubbleless equilibrium, the per-capita consumption is less than the Golden Rule. The
introduction of bubbles moves the capital stock down to the level smaller than the Golden Rulelevel, and decreases the per-capita consumption.

~

Finally, we consider the case for 1 + n > f ' ( k ) > 1 + ~r in which capital is overly
accumulated in the bubbleless equilibrium. In Figure 4, there exists a k , under which the~

per-capita aggregate consumption is less than k GR and the same as the one in k , satisfying~~
f (k ) − (1 + n )k = f ( k ) − (1 + n )k . The welfare implications differ according to whether the~steady state of the asymptotically bubbly equilibrium k B lies greater than k or not. We
summarize the following.

Proposition 3(a) If ~r > n , the bubbleless equilibrium is dynamically efficient.

~(b) If f ' ( k ) > 1 + n > 1 + ~r , the asymptotically bubbly equilibrium does not improve efficiency.
The bubbleless equilibrium is dynamically efficient.

~(c) If 1 + n > f ' ( k ) > 1 + ~r , the asymptotically bubbly equilibrium may or may not improve~

efficiency, depending on parameter values. If k B < k , the asymptotically bubbly equilibrium
does not improve efficiency, and the bubbleless equilibrium is dynamically efficient, while if16~k B > k , it improves efficiency so that the bubbleless equilibrium is dynamically inefficient. 10

As Proposition (b) and (c) state, the inequality n > ~r is not sufficient for the bubblelessequilibrium to be dynamically inefficient so that asset bubbles arise even when the bubbleless~

economy is dynamically efficient. Additionally, as Proposition (c) states, 1 + n > f ' ( k ) is not
sufficient for the bubbleless equilibrium to be dynamically inefficient. In other words, thegreater level of capital than the Golden rule is possible under the dynamically efficient economy.
Asset bubbles can restore efficiency only when capital is far overly accumulated. This is easily~~

checked by looking at Figure 4. As k increases, k decreases, and k B is more likely to be~greater than k . In sharp contrast with Tirole (1985), neither n > ~r nor 1 + n > f ' ( k ) is not acriterion under which asset bubbles restore dynamic efficiency.

4. Introduction of a Foreign Asset
Having thus far studied the closed economy, we extend the model by assuming thatinvestors have access to a foreign asset with a constant interest rate r in order to investigate
the small-open economy. . This experiment allows us to understand to what extent generalfeatures derived in the closed economy continue to hold in the small-open economy. We impose
two further assumptions. First, domestic agents perceive bubbles to continue in the next period
in the bubbly equilibrium, but foreign people do not. This assumption allows domestic agentsonly to hold bubbles. Second, the capital good is used as “international collateral” [e.g.,
Caballero and Krishnamurthy, 2001] and foreign entrepreneurs can pledge the capital good as
collateral. The latter assumption allows domestic entrepreneurs to borrow from foreign investors.
10

The proof of © is left to Appendix.
17

As argued below, pledgeability is essential to derive main results in this section. The small-open
economy version is a simplified one of Caballero and Krishnamurthy (2006) in two substantial
ways. In theirs, bubbles are stochastic and used as collateral for financing productive
investment.
We first investigate the bubbleless economy. Since entrepreneurs find the cost of fund as r ,the borrowing constraint is described as(it − xt ) × (1 + r ) ≤ λf ' ( kt +1) it .11(4-1)

Equation (4-1) is rewritten, using (1 + n) kt +1 = αit , as

(

(4-2)

1+ n
1+ n
kt +1 − sWt ) × (1 + r ) ≤ λf ' (k t+1 )kt +1 .
ααThe borrowing constraint is binding with equality when two inequalities, (1 + n) kt +1 > αsWt
and 1 + r > λf ' ( kt +1) , are satisfied. If either of the two is not met, the equilibrium is notborrowing constrained. Thus without loss of generality, we focus on an economy with

) (
)
)
(
kt > k = max{ k , k } , where (1 + n) k = αsW ( k ) and 1 + r = λf ' ( k ) . Given kt > k , (4-2)
determines the evolution of k t . From the application of the implicit function theorem, there
exists a continuously differentiable function(4-3)

k t+1 = Ω (k t ) ,with Ω ' (.) =αs (1 + r )W ' ( kt )> 0 for k t ∈ ( k ,+∞) . Given k 0 , k t(1 + n){1 + r − λf ' (k t+1 ) − λf " (k t +1)}

converges to the steady state level of the capital stock, denoted k (r ) , with k ( r ) = Ω( k ( r )) .12~
We easily find that k ( r ) > k if and only if r < ~r . The equilibrium involves the capital inflow

Letting ~r denoting the domestic interest rate, if ~r < r , domestic investors find it more
beneficial to invest abroad, and the domestic interest rate rises until it is equal to the world interest
rate r , while ~r > r , entrepreneurs find it more beneficial to fund from foreign investors, and thedomestic interest rate falls until it is equal to r .
12There may possibly exist multiple k (r ) ’s, but we do not refer to that possibility because themultiplicity of equilibria is out of focus in this paper.
1118

or outflow. The capital inflow of (1 + n) kt +1 − sWt is positive (negative) if r < ~r (r > ~r ).
We next investigate the asymptotically bubbly equilibrium. Bubbles evolve as(4-4)

bt +1 =

1 + rtB+1
bt .
1+ n

As argued below, entrepreneurs have no incentive to hold bubbles, and so only domestic
investors hold bubbles.,(4-5)(1 − α ) sWt = bt ,if rtB+1 > r , or (1 − α ) sWt ≥ bt if rtB+1 = r . Note that as will be made clear below, rtB+1 may
be higher than r .
We primarily investigate an interesting case of n > r , the case of which corresponds to theone in which bubbles are viable in the closed economy. Since then the return from bubbles isgreater than the one from lending at least near the steady state, there exists an equilibrium withsustainable bubbles.
We distinguish between two cases of (i) 1 + n > 1 + r > 1 + ~r and (ii) 1 + n > 1 + ~r >1 + r .
Consider first the case for 1 + n > 1 + r > 1 + ~r . Equations (4-3), (4-4), and (4-5) define a(asymptotically) bubbly equilibrium with bt > 0 for t → ∞ . Equation (4-5) is implied by the
conjecture that the rate of return of bubbles is higher than that of lending to entrepreneurs atleast near the steady state. The determination of equilibrium is recursive. Equation (4-3)
determines the evolution of k t . Given k t , (4-4) determines bt . Finally, the sequence of bt
determines rtB+1 . The steady state is characterized by a pair {k B ( r ), b( r ), r B} , satisfyingk B ( r ) = Ω ( k B ( r )) , r B = n , and (1 − α ) sW ( k B (r )) = b . Domestic investors hold bubbles
and never lend to entrepreneurs. Entrepreneurs borrow entirely from foreign investors.
Entrepreneurs never hold bubbles by borrowing because bubbles are not effective as collateralfor foreign borrowing. This argument supports the equilibrium.

19

This economy enjoys the capital inflow of (1 + n) kt +1 − αsWt ( > 0). Bubbles lead to thereversal of capital flows from “out” to “in”. Despite the presence of bubbles, the level of capitalremains unchanged ( k ( r ) = k B ( r )) . It turns out that the same amount as channeled fromdomestic savings into bubbles is compensated for by the inflow of capital.
We now turn to the efficiency problem. The per-capita consumption is written as(4-6)

C = { f (k ( r )) − (1 + n )k ( r )} + ( r − n) × outflow .13

The first term remains unchanged before and after the introduction of bubbles. The second term
represents the “net” inflow of capital. In an economy with r < n , reducing capital outflows or
the reversing the direction of capital flow from “out” to “in” improves efficiency. The
bubbleless economy is dynamically efficient in the sense that bubbles can reduce capitaloutflows, and thus improve efficiency dynamically.
In the case for 1 + n > 1 + ~r > 1 + r , all results continue to hold except for one point. In the
bubbleless economy also, the economy enjoys the inflow of capital. By the presence of bubbles,
the economy faces the larger amount of capital inflows. Of course, efficiency result is preserved.
Note that whether f ' ( k ) is greater than 1 + n or not is irrelevant to general features because

k is entirely determined by parameters in this small-open economy. Consider finally the caseof n < r . The efficiency result is trivial because bubbles are ruled out under perfect foresight.
We summarize the following.

Proposition 4
In the small open economy, there exists an equilibrium with sustainable bubbles if and only if

r < n . Furthermore, bubbles improve efficiency dynamically and hence the bubbleless13

Strictly expressing, we have Ct = f ( k t ) − (1 + n) k t+1 + (1 + r ) outflowt − (1 + n )outflowt+1 ,but the steady state expression is enough for the welfare analysis because k t = k (r ) continues to
hold.
20

economy is dynamically inefficient if r < n .

The efficiency result is quite different from the one in the closed economy. Even in the presenceof financial market friction, the smaller real interest rate than the economic growth rate is anecessary and sufficient condition under which the bubbleless economy is dynamicallyinefficient, and the presence of bubbles improves efficiency. The efficiency result is coincident
with Caballero and Krishnamurthy (2006).
Finally note that this finding is dependent entirely on the assumption that the capital good
becomes international collateral. If otherwise, we find quite different arguments. The presenceof bubbles crowds out domestic capital formation because entrepreneurs now can not rely on
foreign borrowing for funding their projects. Entrepreneurs have to offer at least the same rateof return as the one of bubbles so that the economy attains the same allocation as the closed
economy. The direction of efficiency is ambiguous. In (4-6), the economy incurs the loss fromthe first term because bubbles strengthen capital under-accumulation, but obtains the gain fromthe second term because capital outflows become zero.

Conclusion
In this paper we studied whether the criterion for the real-interest-rate versus
the-economic-growth-rate is effective in evaluating dynamic efficiency/inefficiency andunderstanding rational bubbles in a dynamic economy with frictions of financial markets. We
find that it is effective in the small-open economy, but not in the closed economy. Particularly,
we find that when the real interest rate is endogenously determined in the model, the criterion
for sustainable bubbles and the one for dynamic efficiency differ, and that difference generate sseveral interesting features that are not found in the standard economy of Diamond (1965), Ihori

21

(1978), and Tirole (1985).
This study may be a small step to study global bubbles and the historically low real interest
rate, and furthermore “global imbalance” in the world economy.14 To answer whether the global
bubbles make the world economy more efficient is also an important question. Theoreticalfindings in this paper suggest that the small open economy is better off by creating bubbles
while the closed economy may not. The conflicting results on efficiency suggest that theconclusion depends on what kinds of role of bubbles we will focus on. In this paper we focus on
the two aspects of bubbles, one is the role of raising the real interest rate, and the other ofcrowding out capital accumulation. As pointed out by Caballero and Krishnamurthy (2006), if
the role of bubbles as collateral for financing productive investment is also focused on, we may
have different results.
The third role of bubbles may be one key element to answer a question on whether asset
bubbles and capital accumulation are substitutable or complementary. Given the resource
constraint, asset bubbles will crowd out capital accumulation, on one hand, but on the other
hand, if asset bubbles are used as collateral for borrowing, they will promote capital
accumulation. The more enriched model will provide a foundation that helps us to understandbubbles in the presence of financial market frictions and hence the current world economy.

Appendix
The proof of the former part of © is similar to that of (b), and is omitted. I make the proof of the~~latter. By the introduction of bubbles, k t monotonically decline from k to k B .
Letting CT denote the per-capita aggregate consumption at period T at

14

Caballero (2006) suggests the approach of the financial-market-friction view to be promising tosolve “global imbalance”.
22

~
~
~
kT = k , CT (1 + n) −1 = f (k ) − (1 + n) k . The per-capita aggregate consumption at period at~period T + 1 becomes CT +1 (1 + n) −1 = f (k ) − (1 + g )kT +1 > CT (1 + n) −1 so long as~
k GR < kT +1 < k . Furthermore, CT + 2 (1 + n) −1 = f (kT +1 ) − (1 + n) kT + 2 > f ( kT +1) − (1 + n) kT +1~~~> f ( k ) − (1 + n) k = CT (1 + n) −1 if k GR < k T + 2 < k T +1 < k . This process continues so long as

kT + N > kGR for N = 1,2,.... .
The introduction of bubbles makes the equilibrium path of capital monotonically decline.
There exists some period T + Nˆ when kT + Nˆ is less than kGR . For any k S , satisfying~k < k B < … < kS +1 < k S < … < k GR , the followings are obtained; C (1 + n) − 1 = f (k ) − (1 + n) k

< f (k S ) − (1 + n) k S < f (k S ) − (1 + n) k S +1 = CS (1 + n) −1 .Over the whole process from the
bubbleless steady state toward the new steady state, the aggregate consumption is greater than~the one in the bubbleless steady state so long as k B > k . Q.E.D.

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25

Figure 2
Capital under-Accumulation and Borrowing Constraint

1+ r

f ' (k )Λ (k )

1+ n

~
kB

26

k GR

k

Figure 3

Equilibrium Dynamics

bt = bt +1

bt

~
b

W

k t = k t+1
b0

~
kB

k0

Figure 4

DG~
k = f '−1 ( ~
r)

k GR

Per-capita Consumption and Dynamic Efficiency

f (k ) − (1 + n) k

~
k kB

k GR

27

~
k

k

kt