A Global Analysis of Financial Market Integration ?

Masaya Sakuragawa and Hiroto Takahashi#

March 24, 2008
Preliminary, Comment Welcome

Abstract
We consider a global analysis of financial market integration by modeling a two-country
overlapping generations economy in the presence of financial market frictions, and find several
important features that have not been obtained from the local analysis around steady states.
There are a stable symmetric steady state with harmonized growth and stable asymmetric steadystates with capital flows from the poor to the rich, both of which may coexist, and out of steadystates, there are interesting non-monotone behavior of the pattern of capital movement anddevelopment. The existence of a stable asymmetric steady state does not necessarily recommend
the poor to keep capital control indefinitely, but if the stable symmetric steady state coexists,
will guide the poor to open eventually capital accounts by delaying the timing of liberalization
until arriv ing at some development stage. The existence of the unique steady state that is stable
and symmetric may or may not recommend the poor to lift capital controls promptly because thesymmetric -breaking process will occur in the transition. The concept of optimal timing forliberalization allows us to explore conditions for successful integration. Conditions for
successful integration depend on several characteristics of the levels of per-capita- income, thedistribution, enforcement technology, global savings , and TFP.

?

We are greatly thankful to Kyoji Fukao, Masahisa Fujita, Makoto Saito, Etsuro Shioji, Yukie
Sakuragawa, Tadashi Shigoka, Makoto Yano and participants in the conference held in Hitotsubashi
University and Kyoto University for helpful and valuable comments.
#
Keio University, Japan, masaya@econ.keio.ac.jp, hirotot@gs.econ.keio.ac.jp

11. Introduction
Globalization, which is meant by enhanced trade and financial integration, should promise
greater prosperities through the channels of improved allocation of resources in the world. How
global is the world economy in reality? Contrary to the conventional wisdom from theneoclassical economic theory, international integration remains limited, remarkably so in
financial markets. Since the influential paper by Feldstein and Horioka (1981), much literature
has addressed that global financial markets are far from complete.
Facing incomplete financial markets, how the world economy reaps the efficiency benefitsof financial integration is a subject of great importance. Indeed, observers and policymakers ofdeveloping countries are skeptical on the effects of capital account liberalization on economic
growth, but do not appear to intend to isolate their home country “indefinitely” from the world
financial market.
The notion of timing for financial liberalization provides a hint for thinking how financial
market integration is progressed. McKinnon (1991) addressed this question in terms of thesequence of liberalization between trade and capital markets.2 3 As a matter of fact, Japan,
Korea, and Taiwan are successful evidences. All of these countries attained miraculous growth,
but did not liberalize their capital markets until 1980s. The notion of the optimal timing forlabialization raises several questions. Whether late-developing countries should open capital
accounts? Is there an optimal timing for opening capital accounts so as to attain successful
development? From the prospect of the world economy, what determines the period forsuccessful financial integration? In this paper we address these questions from a theoreticalperspective.
In this paper, we consider a two-country overlapping generations model in the presence offinancial market frictions. Two countries are assumed to be inherently identical in production
2

Bartolini and Drazen (1997) emphasize the signaling role of capital account liberalization as acommitment to policy reforms to boost capital inflows.
3
Braun and Raddatz (2007) report that trade liberalization occurred before capital accountliberalization in 68 of the 73 countries that liberalized any of these dimensions between 1970 and
2000.

2

technologies and institutions for contract enforcement, but differ only in their initial levels of
income.
The model is standard except for one respect; entrepreneurs have no initial wealth forfinancing productive investment but can pledge up the future income (wage income in themodel) as collateral. This assumption is motivated to make the dynamical system a
one-dimensional map of the capital-labor ratio of either country and to avoid a complicateddynamical system (e.g., Boyd and Smith (1997) and Matsuyama (2004)). This assumption alsoenables us to preserve the non-monotone behavior of the interest rate in terms of thecapital-labor ratio that is the source of multiple steady states and implicitly involved in thepreceding literature. In order to reach our goal, the theoretical research is needed to provide
characterizations on the short-run and the long-run inequality of income between countries, thepatterns of capital movement, the timing of financial liberalization, and conditions for
successful integration.
We find a number of global properties that are not found in the steady state analysis. There
are a stable symmetric steady state with equality in per-capita income and stable asymmetricsteady states with an inequality in per-capita income followed by capital flows from the poor to
the rich, and both of which may coexist. Out of steady states, there are complicated but
interesting non-monotone behavior of the pattern of capital movement and development. We
show that how the process of development goes hand in hand with financial development, how
the pattern of capital movement and development evolves over time, and how and when the
globalization magnif ies and lessens the divergence in per-capita income between the rich andthe poor.
From the prospect of late-developing countries, the timing of liberalizing the home financial
market is a stringent issue for successful development. The existence of a stable asymmetricsteady state does not necessarily recommend the poor to isolate their countries indefinitely fromthe world financial market. If the stable symmetric steady state also exists, the poor can go on
the successful path by delaying the timing of liberalization until arriv ing at some development
stage. On the other hand, the existence of the unique steady state that is stable and symmetricdoes not guarantee late-developing countrie s to attain the successful development by lifting3

capital controls quickly. The world financial market involves the reversal of the direction ofcapital flows, and the poor are trapped into stagnation with capital flight in early development
stage.
The notion of optimal timing for liberalization allows us to explore conditions for successful
integration in which all countries attain the same capital-labor ratio. From the prospect offinancial integration, conditions for successful integration depend on several characteristics ofthe levels of per-capita income, its cross-country distribution, enforcement technology, global
savings , and TFP. Higher levels and more equality in per-capita income are determinants ofsuccessful integration. Furthermore, the greater TFP level, the development of contract
enforcement, or the greater global savings shortens the waiting time for successful financial
integration.
This paper is organized as follows. Section 2 reviews the related literature. Section 3 sets up
the model. Section 4 characterizes the autarkic economy and section 5 the world economy.
Section 6 analyzes the dynamic properties and provides policy implications for financial
integration.

2. Related Literature
A number of approaches have attempted to answer the question why capital does not flowfrom the rich to the poor, and sometimes even flow from the poor to the rich. Among several
research lines, one research line emphasizes the role of frictions in international financial
markets, and recently focuses on legal and institutional infrastructure and asymmetric
information problems as obstacles to smooth capital flows (e.g., La Porta et al (1997, 1998),
Levine (1998), Levine et al (2000), and others).44

Another research line has emphasized the cost of the governments’ time inconsistency problem
associated with sovereign debt. The literature includes Bulow and Rogoff (1989), Eaton andFernandes (1995), Tirole (2003), and Broner and Ventura (2007). The two research lines arecomplementary in terms of enforceability. The corporate finance approach and the sovereign debt
approach may be classified according to whether enforceability is exogenous or endogenous.
On the other hand, the research line supporting the “Lucas Paradox “(1990) has a negative stance tothe financial-friction view, arguing that poor countries also have lower endowments of factors
complementary with physical capital, and hence the large difference in capital-labor ratio will

4

The empirical literature includes Bekaert et al (2001), Reinhart and Rogoff (2004), Portes
and Rey (2005), Chin and Ito (2006), Alfaro et al (2006), and Braun and Raddatz (2007). Much
theoretical literature has explained why and how perverse capital movements and the divergencein income between the rich and the poor arise under financial market integration. The theoretical
literature includes Gertler and Rogoff (1990) , Boyd and Smith (1997), Sakuragawa and Hamada(2001) , and Matsuyama (2004), Caballerro and Krishnamurty (2001) (2006), Aghion et al(2004), and Aoki, Benigno and Kiyotaki (2006), and others.5
Gertler and Rogoff (1990) is a pioneering theoretical model that explained the perverse
capital movement using the corporate finance approach. In their static model, the difference inborrowers’ ability to rely on external finance causes capital flows to go from poor to rich. Boydand Smith (1997) developed this idea, demonstrating that financial market integration causes thedivergence in income between the poor and the rich, even with identical technologies andinstitutions for contract enforcement. Matsuyama (2004) constructed more general but tractablemodel than Boyd and Smith (1997), characterizing all the steady states, including those withbinding borrowing constraint and the one without it. The analysis of Boyd and Smith (1997) andMatsuyama (2004) depends on the poverty trap argument that involves the multiplicity of steadystates that arises from a feedback effect between the borrowers’ wealth and the aggregate
investment. But the introduction of the feedback effect makes the analysis so complicated that
their scope turned out to be limited to the local analysis around steady states. Sakuragawa and
Hamada (2001) are an exception that analyzed the effects of lifting capital controls on economic
development, in and out of steady states. They demonstrated that late-developing countries may
have an optimal timing for successful development in a model of two countries with different
institutions for contract enforcement, and with identical technological externalities in production.
We comment on other related works that link contract enforcement with international
integration. Tornell and Velasco (1992) model the lack of contract enforcement as a discrepancy
coexist with the equalization of marginal product of capital. Along this view, Caselli and Feyrer(2007) find that the cross-country difference in marginal products of capital is not so large relative tothe difference in capital-labor ratio, and support partially Lucas’s argument.
5
Caballerro and Krishnamurty (2001) (2006) and Aghion et al (2004), investigate thesmall-open-economy model of asymmetric information in order to identify perverse capital flowsthat happened in Asian emerging market countries.

5between social and private rate of return to capital that rises from poorly established property
rights, demonstrating that opening capital accounts, despite the high social return, leads to theslowdown of economic growth and capital flight. Greif (1994) offers interesting contrastingstudies of two groups with different internal enforcement systems, arguing that the rule-based
governance has an advantage to globalization over the relation-based governance. Acemoglu
and Zilibotti (1997) model the lack of enforcement as incompleteness risk diversification,
demonstrating that the market incompleteness, combined with technological non-convexities,
magnif ies the inequality of nations and becomes a reason for perverse capital mobility. Dixit(2003) analyzes the effect of contract enforcement on trade expansion, demon straing that in thesmall trading world human-based governance is dominant, but as the world is greater, has to be
replaced by the rule –based governance. Broner and Ventura (2007) model globalization as a
gradual improvement in technology that increases the fraction of tradable goods, demonstratingthat when countries can not commit to pay their debts, but can control enforceability,
globalization might lower domestic asset trade, worsening risk sharing and lowering welfare.
Tirole (2003) is an excellent works that demonstrates that enforceability to foreign debts affects
crucially the quality of the home financial markets.

3. The Model
The model is based on an overlapping generation model that consists of two period lived
agents. Time is discrete, and during each period t = 0, 1, ... a set of agents is born. Eachgeneration consists of a continuum of agents of unit mass. There is no population growth.
A single final good is produced using capital stock and labor as inputs. Let K t denote thecapital stock in period t , and Lt denote the labor supply in period t . Then, the output of the
final good in period t , Yt , is produced according to the production function given by

Yt = At F ( Kt , Lt ) , where F (.) exhibits the property of is a constant returns to scale
production function and At is the TFP at period t . We set At = 1 so long as unnecessary. We
denote k t ≡ K t / Lt and f ( kt ) ≡ F (k t , 1) , and assume that f (0) = 0 , f ' ( k ) > 0 > f ' ' ( k )and f ' (0) = ∞ . It is also assumed that the factor markets are competitive. The firm’s profit
maximizing leads to ρ t = f ' (k t ) and wt = f (k t ) − k t f ' ( k t ) , where ρ t is the rental rate6

on capital and wt is the real wage rate. Let W (k t ) ≡ f (k t ) − k t f ' ( k t ) , then f ' ' ( k ) < 0
implies that W ' ( kt ) > 0 holds for all k > 0 , and in addition we assume that
(A1)

W ' (0) = ∞, W ' ' ( k ) < 0.

For simplicity, capital is assumed to depreciate fully in one period.
Within each generation, agents are divided into two types, “investors” and “entrepreneurs”.
A fraction α ( 0 < α < 1) of agents are investors, each of whom is endowed with one unit oflabor in the young age. The remaining fraction 1 − α of agents are entrepreneurs, each ofwhom is not endowed with labor in the young age, but has access to a single indivisible
investment project for converting the final good to the capital good after one period; one unit of
the final good invested in this project in period t yields R units of capital in period t + 1 . In
addition, each of them is endowed with one unit of labor in the old age. After having completed
the investment project and sold the produced capital good to the final-good firm, each of them
supplies labor to the final-good firm. Both types of agents are risk neutral, and care only aboutold-age consumption. In addition, investment projects are not transferable among entrepreneurs
so that any one of them runs at most one project.
Investors receive wages by supplying one unit of labor elastically in the labor market in theyoung age, and lend all the earned income to others with a safe (gross) interest rate rt+1 whichis exogenously given to agents but endogenously determined in the model. Entrepreneurs will
start the investment project by borrowing or do nothing when young.
We assume that the legal enforcement is imperfect so that it is difficult to fulfill financial
contracts between investors and entrepreneurs. Entrepreneurs can repudiate the obligation byhiding a fraction 1 − λ ( 0 < λ < 1) of the revenue generated from the investment project sothat, in case of default, investors can seize only a fraction λ of the produced capital good. The
parameter λ captures a level of legal infrastructure for contract enforcement. Breaching thecontract yields another cost to entrepreneurs. If entrepreneurs breach the contract, they are sent
to prison, and lose the opportunity of earning wage incomes by supplying labor to thefinal-good firm. Alternatively, they may be discriminated for job opportunities for the reason of
criminal record. The latter will reflect a kind of social punishment that will be made out of court,

7the cost of which increases as the economy is rich. 6 This assumption is an artifact to motivate
the observed fact that the cost of breaching contracts is higher in richer countries in whichpeople have more chances of earning more money.
We are now ready to look at the investment decision. If potential entrepreneurs start aninvestment project by financing one unit of the final good in the financial market with a (gross)interest rate rt +1 , their old-age consumption is equal to ρ t+1 R + wt +1 − rt+1 , while otherwise,
their old-age consumption becomes wt +1 . Potential entrepreneurs are willing to produce capitalif(1)

Rf ' ( k t+1 ) ≥ rt+1 .

We refer to this inequality as the profitability constraint. However, even if (1) is satisfied, theenforcement problem may prevent entrepreneurs from financing their own projects. If
entrepreneurs repay obligations honestly, they earn Rρt +1 − rt +1 + wt +1 , while if they breach thecontract, they would obtain (1 − λ ) Rρt +1 . Anticipating the possibility of the entrepreneur’s
strategic default, the investor supplies the fund only if the incentive compatibility is satisfied sothat the entrepreneur can start the project only if(2)

λRf ' ( k t +1 ) + W ( kt +1 ) ≥ rt+1 .

This inequality implies that ( λRρt+1 + wt +1) rt +1 should exceed unity, the amount of therequirement for starting the project. We shall thus call (2) the borrowing constraint. A greater
value in λ implies the LHS of (2) to be greater, and thus weakens the incentive to default andmakes the borrowing constraint less binding. In other words, a high (low) λ reflects the high(low) level of legal protection for creditors or shareholders, and will be associated with well(poorly) developed financial markets [e.g., La Porta et al (1997, 1998), Levine (1998), Levine etal (2000)]. 7 The parameter λ is interpreted to capture the efficiency of the broadly defined
financial system, including the banking system, and stock and bond markets.
One important feature of our model is that entrepreneurs receive the wage income not in the
6

Human-based enforcement is typical in societies with underdevepled legal enforcement. See forexample, Townsend(1994), Udry(1994), and Banerjee andf Newman (1998)
7
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.

8youth but in the old age. This assumption is an artifact to motivate entrepreneurs’ incentive
compatibility to be affected by the level of their wealth that is an increasing function of thecapital-labor ratio. Following the standard overlapping generations model a la Diamond (1965) ,one might be motivated to assume that entrepreneurs use the wage income earned in the young
age as internal wealth for financing investment (e.g., Bernanke and Gertler (1989)), but we havea different way. If the latter assumption is adopted, the dynamic analysis of the world economy
with financial market integration will be far complicated. Boyd and Smith (1997) andMatsuyama (2004) had to confine their study on the local analysis around steady states. As
analyzed in details below, our modification of the model enables us to conduct the global
analysis in the world economy while preserving general features that arise from the financiallyconstrained economy.

4. The Autarky CaseIn this section, before considering the financial market integration between countries, weanalyze the general equilibrium of the case of autarky. In the case of autarky, domestic
investment is adjusted to be equal to domestic saving in equilibrium. The measure of domestic
saving is αW ( kt ) , while the measure of maximum domestic investment is 1 − α . Since each
investment supplies R units of physical capital, the maximum physical capital is (1 − α ) R .
We impose the following assumption.
(A2)αW ((1 − α ) R) < 1 − α .

The LHS of (A2) is the maximum allowable domestic saving, and hence (A2) implies thatdomestic saving is always less than the demand for funds by entrepreneurs. Under (A2), thedemand for funds exceeds the supply of funds. Accordingly, domestic investment is equal toα W ( k t ) . Since the aggregate amount of domestic investment is αW ( kt ) and each investment
supplies R units of capital, the aggregate capital stock at period t + 1 is described, in
per-capita term, as(3)

k t+1 = RαW ( kt ).

Equation (3) gives the equilibrium law of motion for capital stock in the case of autarky. (A1)9

ensures that (3) has a unique nontrivial steady state capital stock k * , defined byk * = Rα W ( k * ) . For any k 0 > 0 , k t converges monotonically to k * . The equilibrium law of
motion is illustrated in Figure 3-1.
The parameter λ does not affect the dynamics of capital, but will affect the equilibrium
interest rate. The two constraints, (1) and (2), can be summarized as(4)

rt +1 ≤ Min {Rf ' ( kt +1), λRf ' ( kt +1) + W ( kt +1)} .

The market clearing in the financial market requires the inequality in (4) to be binding withequality, and is rewritten as(5)

λRf ' ( kt +1) + W (k t+1 ) if
rt +1 ≡ r ( kt +1 ) = 
if
Rf ' (k t+1 )kt +1 < K ( λ ),kt +1 ≥ K (λ ),where K (λ ) is the threshold level of capital defined implicitly by

λRf ' ( K ( λ )) + W ( K (λ )) = Rf ' ( K ( λ )) .8 The borrowing constraint is binding if k t +1 < K ( λ ) ,while the profitability constraint is binding if k t +1 ≥ K ( λ ) .
The fraction of entrepreneurs who are able to fund the project is equal to αW ( kt ) /(1 − α ) ,while the rest, 1 − (αW (k t ) /(1 − α )) , are not. If k t +1 ≥ K ( λ ) , entrepreneurs are indifferent
between borrowing and not borrowing. However, if k t +1 < K ( λ ) , they strictly preferinvestment by borrowing. Thus the remaining 1 − αW (kt ) /(1 − α ) of entrepreneurs are denied
to receive the fund and hence the equilibrium allocation turns out to involve credit rationing.
The function Rf ' ( k t+1 ) is monotonically decreasing, while λRf ' (k t+1 ) + W ( kt +1) isdecreasing in k t +1 if k t +1 < λR and increasing if k t +1 > λR . The two curves intersect at

k t +1 = K ( λ ) . Note that K (λ ) is decreasing in λ , with K ( 0) > 0 and K (1) = 0 . Thereexists a λˆ ∈ ( 0, 1) such that K (λ ) > λR if and only if λ is less than λ̂ . In Figure 3-2

r (kt +1 ) is decreasing for all k t +1 > 0 . This case occurs when λ is high, satisfying λ ≥ λˆ . On
the other hand, in Figure 3-3 r (kt +1 ) is first decreasing for k t +1 < λR , later increasing for

λR < k t +1 < K (λ ) , and finally decreasing for k t +1 > K ( λ ) . The non-monotonicity arises
8

If the inequality does not bind, domestic investment should be equal to 1 − α .

10when λ is small, satisfying λ < λˆ .
Two conflicting effects give rise to the non-monotone feature; the effect of decreasing
returns to capital and the effect of mitigating the borrowing constraint. As the level of wealththat borrowers pledge as collateral increases, the borrowing constraint becomes less stringent.
An increase in borrowers’ wealth mitigates the borrowing constraint, allowing them to demand
more funds, and raising the interest rate up. At the low level of capital stock, the effect of
decreasing returns to capital is dominant and thus r (kt +1 ) is decreasing. As the level of capitalincreases beyond λR , the effect of mitigating the borrowing constraint begins to be dominant,and thus r (kt +1 ) is increasing. Finally, as the level of capital increases beyond K (λ ) , theborrowing constraint ceases to be binding and thus the agency cost disappears; the effect of
decreasing returns to capital makes r (kt +1 ) decreasing.
The non-monotone feature is more likely to arise if λ is small or if the income share ofcapital is high. Assuming that f (k ) = k β , the r (kt +1 ) function is not monotone when λ < β .
The non-monotone feature is not specific to this model, but arises from broad classes of agency
models of corporate finance, including the costly-state-verification approach a la Townsend(1979).

5. Equilibria in a World Economy with Financial Market Integration
Having investigated the case of autarky, in this section we study a world economy thatconsists of two countrie s that are inherently identical except for the initial stock of capital stock.
We consider the following integration. The final good is tradable, and thus borrowing andlending of the final good are allowed across countries. Capital goods are not tradable from one
country to another so that capital goods that are produced in country i (i = N , S ) have to be
used in the final-good sector of country i . Labor is immobile across countries. We impose a
simple assumption on enforcement technology. Assume that enforcement does not favor

11

domestic investors, but deals with domestic and foreign investors equally. 9 Letting k ti denotethe capital stock in country i in period t , we assume without loss of generality thatk 0S ≤ k0N .
As a counterpart of (A2), we impose the following assumption.
(A3)αW ( ktS ) + αW ( ktN ) < 2(1 − α )for all t' s . The total supply of funds is always smaller than the total demand for funds. Under(A3), the interest rate of each country should evolve as(6)

λRf ' ( kti+1) + W (k ti+1 ) ifii
rt +1 = r ( kt +1 ) ≡ Rf ' (k ti+1 )if

kti+1 < K ( λ ),kti+1 ≥ K (λ ),(i = N , S ) . The market clearing in the international financial market requires the aggregate
investments to be equal to the aggregate savings, given by(7)

k tN+1 + k tS+1 = RαW ( ktN ) + RαW (k tS ) .

In the world financial market, people of both countries face an identical world interest rate rt +1
so that we should have(8)

rtN+1 = rt S+1 = rt +1 .

Accordingly, any of the equilibria with an integrated financial market is described as asequence{k tN , ktS , rt }∞t= 0 , satisfying (6), (7), and (8), given k 0N (> 0) and k 0S (> 0) .
One remarkable departure of this paper from Boyd and Smith (1997) and Matsuyama(2004) is that our model can be reduced to a one-dimensional map of k tS or ktN .The analysis ofa one-dimensional map allows us to conduct a global analysis.
The dynamic behavior differs according to whether r (.) is monotone or not.. We firststudy the case when r (.) is monotone decreasing. Since r (.) is not differentiable at K (λ ) , we
9Actually, enforceability may depend on several dimensions, including the nationality, thedistinction between traded and non-traded goods, the distance between countries, and the state of theeconomy. Enforceability is exogenous in our model, while as motivated by sovereign debt

12

have to analyze the correspondence from ktS+1 to ktN+1 by separating the whole interval into

[ 0, K ( λ )) and [ K (λ ), +∞) , for ktS+1 and ktN+1 , respectively. It follows from the monotonicity
and the symmetric property that (8) is described as

λRf ' (ktN+1 ) + W ( ktN+1) = λRf ' ( ktS+1) + W ( ktS+1 ) for k tN+1 < K ( λ ) and ktS+1 < K (λ ) , and
Rf ' ( ktN+1) = Rf ' (ktS+1 ) for k tN+ 1 ≥ K (λ ) and ktS+1 ≥ K ( λ ) . It is straightforward to derive
k tN+1 = ktS+1 ,(9)for any ktS+1 ∈ [0,+∞ ) and ktN+1 ∈ [0,+∞ ) . Therefore, it follows from (7) and (9) thatk ti+1 (i = N , S ) evolves simply as k ti+1 = RαW ( kti ) , with k tN = ktS . A steady state ischaracterized by a pair ( k , k ) = (k , k ) , satisfying k = k and the constraint for the
N

S

*

*

N

S

fund allocation between two countries

k N + k S = RαW ( k N ) + Rα W ( k S ) .

(10)

Figure 4-0 illustrates the dynamic configuration in the ( ktS+1, ktN+1) space. The curve going
through ( k * ,0), E , and ( 0, k * ) represents the steady state relationship for the market clearing
in the world financial market, given by (10). While, PQ represents the pair ( ktS+1, ktN+1) thatsatisfies (7) given k tS and ktN , the temporal condition for the market clearing in the world
financial market.
Assume that levels of capital of two countries initially lie at A0 at period 0. On integrationof the financial markets, the equilibrium jumps at period 1 to A , an intersection of the 45 degreeline and the PQ line. Here a pair of capital stocks that will be attained under autarky, denoted A' ,is described so as to lie on PQ, satisfying k1N + k1S = Rα W ( k0N ) + RαW ( k 0S ) .
consideration, enforceability may be controlled by the government.

13

Figure 4-0A illustrates the configuration of the world financial market, which is reminiscentof MacDougall (1960), Kemp (1962), and Hamada (1966). The length of interval from ON to

OS stands for the initial stock of capital available to two countries, R times of sum of theaggregate savings in two countries, Rα{W ( k0N ) + W ( k0S )} . Thus, from (7), any point inhorizontal axis stands for the allocation of capital between two countries, where the length ofinterval from ON (OS ) to the point stands for the capital in country N(S) at period 1, k1N ( k1S ) .
The vertical axis stands for the interest rate realized in each country given r (k1i ) (i = N , S ) .
The function r (k1N ) is illustrated from ON and r (k1S ) from OS , respectively, and both
functions are illustrated in symmetry. Each function r (k1i ) is illustrated to be decreasing.
We have to specify the price adjustment rule in the world financial market. We assume theMarshallian adjustment rule in which the demand and the supply changes according to theexcess demand price. Investors rationally anticipate real interest rates of the two countries thatwill be realized if the autarkic state A' would prevail, and find the northern interest rate to belower than the southern one, as implied by r (k1N , A ) < r ( k1S , A ) , where k1i , A (i = N , S ) denotes
the level of capital that attains under autarky. On opening the financial market, northern people
keep investing their funds home, while southern people invest in the other country. Capital
flows out of country N into country S, until the interest rate differential disappears. The
equilibrium is achieved at A
After having reached A , the equilibrium converges to the steady state E . The steady state
is symmetric in the sense that stocks of capital are the same between countries. The long-run
consequence is irrespective of the initial cross-country distribution of capital. If financial
markets are integrated between both countries, the equilibrium would converge to a uniquesteady state with equality no matter how the distribution of capital stock in the event ofintegration is. Note that Figure 4-0A is depicted so that both countries face financial constraints
at the steady state, but the equilibrium may or may not involve the binding financial constraint,
depending on parameter values.
We turn to the case when r (.) is not monotone decreasing. As Figure 3-2 illustrates, r (.) is14

first decreasing for 0 ≤ kti+1 < λR , later increasing for λR ≤ kti+1 < K ( λ ) , and finallydecreasing for K (λ ) ≤ kti+1 . In addition, r (.) is not differentiable at K (λ ) . In order to apply
the implicit function theorem, we divide the whole interval [ 0, + ∞ ) into [ 0, λR) ,[ λR, K (λ )) , and [ K (λ ), +∞) . Thus the correspondence from ktS+1 to ktN+1 turns out to becomposed of nine functions, each of which is continuous and differentiable on its distinct
interval. The functions can be summarized asktS+1
if ktN+1 < λR and ktS+1 < λRSNSµ ( kt +1) if λR ≤ kt +1 < K ( λ ) and k t+1 < λR
φ ( k S ) if K ( λ ) ≤ k N and k S < λR
t +1
t +1
 t +1SNµ ( kt +1) if kt +1 < λR and λR ≤ ktS+1 < K ( λ )k tN+1 = Ψ( ktS+1) = ktS+1
if λR ≤ k tN+1 < K ( λ ) and λR ≤ ktS+1 < K ( λ ) ,φ ( k S ) if K ( λ ) ≤ k N and λR ≤ k S < K ( λ )
t +1
t +1
 t +1SNϕ1 ( kt +1) if kt +1 < λR and K ( λ ) ≤ ktS+1ϕ2 (ktS+1 ) if λR ≤ k tN+1 < K ( λ ) and K (λ ) ≤ ktS+1
 S
if K (λ ) ≤ k tN+1 and K ( λ ) ≤ ktS+1
kt +1(11)

where µ ( k tS+1 ) is decreasing, φ ( k tS+1 ) is increasing, ϕ1 ( ktS+1) is increasing, and ϕ 2 ( ktS+1 ) is
decreasing, with µ ( λR) = λR ,,lim

ktN+ 1 →K ( λ ) −µ −1 (k tN+1 ) = N lim

kt +1 → K ( λ ) +lim

ktS+ 1 →K ( λ ) −µ ( ktS+1 ) = S lim

φ −1 (k tN+1 ) = K (λ ) ,lim ϕ1−1 (k tN+1 ) = Nlim ϕ 2−1 ( ktN+1 ) = φ (λR) ,ktN+ 1 →λR −andkt +1 →λ R +lim

ktS+ 1 →K ( λ ) +kt + 1→ K (λ )+ϕ1 ( ktS+1 ) = ϕ1 ( K (λ )) ,lim

ktS+ 1 →K ( λ ) −φ (k tS+1 ) = K (λ ) , φ ( K (λ )) = K (λ ) ,ϕ 2 ( k tS+1 ) = ϕ 2 ( K ( λ )) = K (λ ) ϕ2 ( K ( λ )) = K (λ ) . The derivation is left toAppendix. Figure 4-1 illustrates the correspondence Ψ (.) , which is composed of both the 45
degree line and the ellipse.
NS

Any of steady states is defined as a pair {k , k } , satisfying (10) and(12)

k N = Ψ (k S ) .

15

If Ψ ( k ) = k , a steady state is a symmetric equilibrium, while otherwise, a steady state isSS

an asymmetric equilibrium. It finally follows from (7) and (11) that the system can be reduced
to a one-dimensional map of k tS , which is given by

k tS+1 + Ψ ( ktS+1 ) = Rα {W ( k tS ) + W ( Ψ (k tS ))} .

(13)

Then there exists a correspondence from k tS to k tS+1 that is continuous and differentiable
except for at most countable points:

k tS+1 = Φ ( ktS ) ,

(14)
with
(15)

Φ ' ( kts ) ≡dkts+1W ' ( kts ) + W ' (Ψ (k ts )) Ψ' ( kts )
=
R
α
,
dkts
1 + Ψ ' ( kts+1)

with Ψ ' ( k ts+1 ) ≠ −1 , where Ψ ' (.) = φ ' (.) for ( ktS+1 , ktN+1 ) ∈ [0, K (λ )) × [ K ( λ), ∞ ) ,Ψ ' (.) = µ ' (.) for ( ktS+1 , ktN+1 ) ∈ [0, λR) × [λR, K (λ )) or ∈ [ λR, K ( λ )) × [0, λR ) ,Ψ ' (.) = ϕ1 ' (.) for ( ktS+1 , ktN+1 ) ∈ [ K ( λ ), ∞ ) × [0, λR) , and Ψ ' (.) = ϕ2 ' (.) for( ktS+1 , ktN+1 ) ∈ [ K (λ ), ∞ ) × [λR, K (λ )) .
We should comment on a possible singular case of Ψ ' ( ktS+1 ) = −1 . We cannot generally
rule out a case when the system traverses Ψ ' (.) = −1 in the asymmetric region with k tN+1 ≠ ktS+1 .
In order to focus on the analysis with well-behaved equilibria , we impose the following two
technical conditions.

Condition 1: µ ' ( ktS+1 ) < −1 if k tS+1 < λR and µ ' ( ktS+1 ) > −1 if k tS+1 > λR
Condition 2: φ ' (.) > −1 and ϕ2 ' (.) < −116Condition 1 is satisfied when the production function is the Cobb-Douglas form. Condition 2 issatisfied for plausible parameters when the production function is the Cobb-Douglas form. The
proof is technical and available on request. Specifically, we have µ ' (ktS+1 ) = −1 at ( Rλ , Rλ ) .
But on the 45 degree line ( k tN+1 = ktS+1 ), k tS evolves according to k tS+1 = RαW (ktS ) 10 so that the
singular problem is innocent at the symmetric region.
These two conditions are useful to investigate dynamic properties. If Ψ (.) is increasing,Φ (.) is increasing so that the dynamic path shows either a monotone convergence to a steadystate or a monotone divergence from the steady state. If Ψ (.) is decreasing but its derivative isΦ ' ( kts ) = Rαgreater than -1, as implied by Condition 2,

W ' ( kts ) + W ' ( Ψ (k ts )) Ψ' ( kts )1 + Ψ ' ( kts+1)

should be positive since W ' ( ktS ) > W ' ( Ψ( ktS )) for k tS < ktN = Ψ (k tS ) so that Φ (.) is
increasing. However, if Ψ (.) is decreasing with the derivative of less than -1 , as implied by
Condition 1, Φ (.) may be increasing or decreasing. If Φ (.) is decreasing around a steadystate, the dynamic path will fluctuate through cycles. Hereafter we use “monotone” in the sensethat the system does not show cyclical behavior.
Let k S denote any steady state to satisfy k = Φ (k ) . We obtain the following.
S

S

Proposition 1~

Let k denote a smallest steady state. Under Conditions 1 and 2, if Φ ' ( k ) > 0 for~

any k tS ∈ ( k − ε ,+∞) with any small ε , any dynamic path in the system is expressed as amonotone convergence or divergence from any steady state.
Proof: We prove this by illustrating the case with three steady states. For other cases, theprocedure for the proof is essentially the same and deleted. First, we prove the simple case of
10

Using

lim µ ( ktS+1 ) = Rλ , we have

ktS+ 1→ Rλlim µ ' ( ktS+1) = Slim {1 µ ' ( ktS+1 )} , where l’Hopital’sS

kt +1 → R λrule is used for equality. Given that µ (.) is decreasing,17

k t +1 → Rλlim µ ' ( ktS+1) = −1 .

S

kt +1 → R λk tN = ktS . If k tN = ktS , the system can be reduced to k ti+1 = Rα W ( k ti )(i = N , S ) so that k tS(or k tN ) converges monotonically to a symmetric steady state E .
Next we prove the significant case of k tN ≠ ktS in three steps.
(Step 1): For any k tS ∈ Ω1 ≡ {k tS | K (λ ) ≤ φ ( ktS ) and k tS < K (λ )} , Φ (.) is bounded, sothat Φ (.) is continuous. We next show that as k tS → K (λ ) , k tS+ 1 < K (λ ) . As k tS → K (λ ) ,k tS+1 + ktN+1 = Rα {W ( ktS ) + W (φ ( ktS ))} → 2 RαW ( K ( λ )) since

lim φ ( k tS ) = K (λ ) . It

ktS →K ( λ )follows from (A1) and K ( λ ) > k * that 2 RαW ( K (λ )) < 2 K (λ ) , which implies that as

k ts → K (λ ) , k tS+1 + k tN+1 < 2K (λ ) . Then there exists a pair ( k tS+1 , k tN+1 ) with k tN+1 > k tS+1
which satisfies k tN+1 = Ψ (k tS+1 ) and k tS+1 + k tN+1 < 2K (λ ) . It is obvious fromk tS+1 + k tN+1 < 2 K (λ ) and k tN+1 > k tS+1 that k tS+1 < K ( λ ) should be met.11 In Figure 4-2A,
there is no steady state in Ω1 . Thus, when Φ (.) is continuous and increasing, as depicted in
Figure 4-2D, Φ (.) has to be below the 45 degree line over this interval so that k tS is
monotone decreasing.
(Step 2): For any k tS ∈ Ω 2 ≡ {k tS | λR < ktS < K ( λ ) and µ ( ktS ) < λR} or∈ Ω 3 ≡ {ktS | K ( λ ) ≤ ktS and ϕ 1( k tS ) < λR} , Φ ' (.) is bounded so that Φ (.) is continuous.
As k tS → λR , k tS+1 + ktN+1 = Rα {W ( ktS ) + W ( µ ( ktS ))} → 2RαW ( λR) sincelimµ ( k tS ) = λR . It follows from (A1) and λR < k * and (A1) that 2 RαW (λR ) > 2λR ,S

kt →λR11

If k tN > k tS and there exist multiple pairs ( k tS+1 , k tN+1 ) which satisfy (7) and (11), the selectedequilibrium will be a pair ( k tS+1 , k tN+1 ) which satisfies k tN+1 > k tS+1 . For the discussion on theequilibrium selection, see section 6.

18

which implies that as k tS → λR , k tS+1 + k tN+1 > 2λR . Then there exists a pair ( k tS+1 , k tN+1 ) with

k tN+1 < ktS+1 which satisfies k tN+1 = Ψ (k tS+1 ) and k tS+1 + k tN+1 > 2λR . It is obvious fromk tS+1 + k tN+1 > 2λR and k tN+1 < ktS+1 that k tS+1 > λR is met. In Figure 4-2A, there is a uniquesteady state in Ω 2 , denoted by G , and no steady state in Ω 3 . Thus, when Φ (.) is
continuous and increasing, as depicted in Figure 4-2E, Φ (.) has to cross the 45 degree line
only once at k tS = kG from above. Let k G denote the level of capital stock in country S
corresponding to at the steady state G . Then as depicted in Figure 4-2E, k tS is monotone
increasing if k tS < kG and monotone decreasing if k tS > kG .
The dynamic behavior of k tS over this interval is a mirror image of that of ktN for

k tN ∈ {ktN | λR < ktN < K (λ ) and µ −1 (k tN ) < λR} or ∈ {ktN | K (λ ) ≤ ktN and
φ −1 ( k tN ) < λR }. Analogously from the above argument, ktN converges monotonically to
F over this interval.
(Step 3): k tS should be related to ktN through (11). For k tS ∈ {ktS | ktS < λR andK ( λ ) ≤ φ ( ktS )} , k tS should be monotone decreasing, and for k tS ∈ {ktS | ktS < λR and
λR < µ ( ktS ) < K (λ )} , k tS should converge monotonically to k F .Q.E.D.

We distinguish equilibria between four different cases roughly according to the number ofsteady states. Figure 4-2A illustrates one typical configuration of three steady state equilibria .
This case arises if λR < k * < K (λ ) so that all countries are borrowing-constrained in thesymmetric steady state.12 Also in the asymmetric steady states all countries are
If k * < λR , the analysis is simple. Then there exists a unique steady state that is globally stable.
This case arises if either α or A is small relative to λR .
1219borrowing-constrained. This case corresponds to the case studied by Boyd and Smith (1997).
Around F , Ψ (.) is decreasing with the derivative of less than -1 so that Φ ' (.) may bepositive or negative. The equilibrium around any of asymmetric steady states may exhibit cycles.
Figure 4-2H depicts the case when Φ (.) is decreasing at k F so that k ts fluctuates though
cycles over the interval[ Φ 2 ( k c ), Φ (k c )] . There necessarily exists a 2-period cycle around thesteady state.13 14 Any dynamical path lies on the 45 degree line or goes along the ellipse.
Anyway, any path on the ellipse arrives at either of near the asymmetric steady states, but neverto arrive at the symmetric one.
We have another configuration of three steady state equilibria [Figure is not illustrated]. In
the asymmetric steady state poor country is borrowing-constrained, but the rich does not, withk S < λR < K (λ ) < k N . This case corresponds to the case investigated by Matsuyama (2004) asa “symmetry-breaking” case. Around F , Ψ (.) is increasing so that Φ (.) is increasing.
Under Conditions 1 and 2, any dynamic path in the system is expressed as a monotone behavior,as demonstrated in Proposition 1. The two asymmetric steady states, F and G , are stable, andthe unique symmetric steady state, E , is also stable.
Figure 4-2B illustrates five steady states equilibria . This case arises if K (λ ) < k * so that all
countries are borrowing-unconstrained in the symmetric steady state . In the asymmetric steadystate poor country is borrowing-constrained, but the rich does not, with k S < λR < K (λ ) < k N .
13

The 2-period cycle may be stable or unstable. , depending on the .One method for characterizing
global properties of the 2-period cycle is to check Schwarzian derivative. If the Schwarzian
derivative is negative, the cycle is stable and has several nice properties, but the sign isindeterminate.
14

Demonstrating the existence of any other higher-order cycles will be complicated and we do

not pursue in this paper. Matsuyama (1999) develops a model of growth through cycles andinvestigates the possibility of higher-order cycles than 2-period cycle. Grandmont (1986) provides
an accessible review of chaos.

Assuming that f (k ) = k β , the Φ (.) function is increasing over the relevant interval sothat the system is monotone if λ >0.04 for β = 0.3 and if λ >0.11 for β = 0.4 given thatthe smallest (net) interest rate is zero.
17

20

Under Conditions 1 and 2, any dynamic path in the system is expressed as a monotone behavior,as demonstrated in Proposition 1. The symmetric steady state, E , is stable. Noteworthy, amongfour asymmetric steady states, F and G , are stable , while the other two, H , and J , areunstable. The steady state H (or J ) is a threshold across which equilibria diverges to different
stable steady states. The equilibrium that lies at the left of H converges to the asymmetricsteady state F , while the equilibrium that lies at the right of H will finally converge to thesymmetric steady state E . The monotone dynamic behavior requires that λ is not too
small.17
Figure 4-2C illustrates the case with a unique steady state case. This case arises if

K (λ ) < k * . But the dynamic process is far different from the case when the r (.) function ismonotone, as depicted in Figure 4-0. There are multiple paths that converges to the steady state,

E , One path is on the 45 degree line, while others on the ellipse. The path on the ellipse
exhibits a perverse behavior. First, k tS decreases while ktN increases, later k tS and ktN bothincrease, and finally k tS increases while ktN decreases.
Different configurations of equilibria emerge, depending on the world-market-clearing
condition at the steady state (10) and the ellipse (11) , both of which depend on parameter values.
The world-market-clearing condition at the steady state (10) depends on the TFP A , theproductivity of the project R and the population of investors α , a direct measure of the
global saving. 19 As either of these parameters gets greater, (10) tends to expand outward from
19

Consider the Diamond’s standard overlapping generation economy with two-period-lived

agents. Assume that the preference of an agent of country i ( = N , S ) is represented bylog Cty + β i log Cto+1 , where Ct (Ct+1) denote consumption when they are young (old), and βiyois the subjective discount factor of a person of a country i . Then the saving rate of a country i is

βi (1 + β i ) . The aggregate fund allocation in the world economy is described as
k tN+1 + ktS+1 =βNβS
W (k tN ) +W ( ktS ) . Comparing this equation and (7), it is obvious to see
1+ βN
1 + βS

that α is the direct measure of a country’s saving rate.

21

the origin because an increase in either A, R , or α makes the aggregate capital stock
available to both countries greater. The ellipse (11) depends on R and the fraction ofconfiscation of the revenue by investors λ , a measure of financial development. 20 As λbecomes smaller, the ellipse get greater with λR getting smaller and K (λ ) getting greater.
Conversely, as λ becomes greater, the ellipse get smaller, and finally disappears for thethreshold λ̂ .21
It turns out that as either A, λ (the measure of contract enforcement) or α (the measureof global savings) is small, the equilibria with three steady states are more likely to occur, as
depicted in Figure 4-2A. As either of the three parameters increases or all, the equilibria withfive steady states are likely to emerge, as depicted in Figure 4-2B. Finally, if all parameters
increase further, the equilibria with a unique steady state are more likely to occur, as depicted in
Figure 4-2C.

6. Global Analysis and Financial Integration
Having studied equilibrium dynamics of the world economy with financial integration, weturn to which of the equilibria is selected and in which direction capital flows along theequilibrium path. As in the previous section, we focus on the case when Conditions 1 and 2 aresatisfied.
Figure 5-2A illustrates the case with three steady states. This case is more likely when the
TFP is small, institutions for contract enforcement are less developed, and the world savings
20

If the TFP is taken into account, the aggregate saving-investment relation is describedas k tN+1 + k tS+1 = RαAt W ( k tN ) + Rα AtW ( ktS ) , while the real interest rate is expressed as

λRAt f ' ( kti+1) + AtW ( kti+1 ) if kti+1 < K (λ ),. It is obvious from the latterr = r(k ) ≡ RAt f ' ( kti+1)
if kti+1 ≥ K ( λ ),equation that the no-arbitrage condition is independent of A .
21 The effect of an increase in R on (11) is ambiguous; an increase in R increases not only
λR but also K (λ ) .
i
t +1

i
t +1

24

Note that as long as Condition 1 and 2 are satisfied, there are at most three pairs which satisfy (7)and (11).

22

rate is low. Assume that both countries open capital accounts at some period T in thedeveloping stage when kTN and kTS ( kTN > kTS ) satisfy

2λR < Rα {W (kTN ) + W ( kTS )} < 2 K ( λ ) . These inequalities say that the PQ line representing
the market clearing in the world market, (7), lies at the northeast of ( λR, λR) and at thesouthwest of ( K (λ ), K ( λ )) . At period T + 1 there exist three intersections between the PQ
line and (11). Two of the intersections, B1 and B3 , lie at the intersection of PQ and the locus

k tN+1 = µ (k tS+1 ) , while the rest of them, B2 , lies at the intersection of PQ and the 45 degree line.
Now look at B' on PQ that represents the levels of capital that would be attained underautarky at period T +1. Note that the point B' is inside the ellipse. The corresponding
configuration of the world financial market is depicted in Figure 5-2B, where there are threeintersections. The three intersections of r (k TN+1 ) and r (k TS+1 ) are denoted as B1 , B2 andB3 , respectively, and each of them corresponds to the same symbol in Figure 5-2A.24
At the autarkic state B' , investors will rationally anticipate the northern interest rate to behigher than the southern one to meet r (k TN+,1A ) > r ( kTS+, 1A ) since k tN+1, A > µ ( k tS+,1A ) . Following
the Marshallian adjustment rule, capital flows out of country S into country N , until theinterest rate differential disappears. Thus, the equilibrium is achieved at B1 at period T+1. Solong as autarkic economies reach somewhere on PQ with the northwest of the 45 degree line,
the integrated equilibrium is achieved at B1 . Under this rule, the symmetric equilibrium B2 isunstable, and B3 is locally stable.25 In Figure 5-2A, after having arrived at B1 , theequilibrium approaches to an asymmetric steady state F along k tN+1 = µ ( ktS+1 ) . In theasymmetric steady state F , capital flows from country S country N.26 In the asymmetric
25

Although we have argued that the equilibrium is settled down to B1 , we may not exclude thepossibility for either B2 or B3 to be chosen as equilibria if an alternative adjustment rulewould be specified. If investors react perversely to the interest rate differential, B2 or B3 might
be viable as equilibrium accompanied by sudden capital reversal, which will be attributed not tofundamental reasons but to sunspot or confidence in the market. Some perturbation in confidence
will lead to a sudden and great reversal of capital flows. In the case of Asian financial crisis, thecollapse of confidence in the market triggered the reversal of capital.
26
This result follows from (A1) and the fact the level of capital in country N is larger than k *23steady state F , the marginal product of capital in country S is higher than the other, but thegreater agency cost in the southern financial market deters the inflow of capital to country S.
One answer to Lucas (1990) paradox is that the difference in the efficiency in the financial
market makes it impossible for returns to capital to be equalized between countries. However, it
is not easy to understand the direction of capital flows on the equilibrium path. In fact, we
obtain the following lemma.

Lemma 1(a) Suppose that (7) intersects with (11) at ( k tS+1 , k tN+1 ) ∈ [ 0, λR) × [ λR, K ( λ )) . If an
equilibrium pair of autarkic economies ( k tS+,1A , k tN+1, A ) satisfies k tN+1, A > ( <) µ ( ktS+,1A ) , theequilibrium interest rates of the two countries satisfy r (k tN+1, A ) > ( <)r (k tS+,1A ) under autarky.
(b) Suppose that (7) intersects with (11) at ( ktS+1 , ktN+1 ) ∈ [0, K (λ )) × [ K ( λ), ∞ ) . If an autarkic
pair ( k tS+,1A , k tN+1, A ) satisfies k tN+1, A < ( >)φ ( k tS+,1A ) , the equilibrium interest rates satisfyr (k tN+1, A ) > ( <)r (k tS+,1A ) under autarky.

Proof: We prove only part (a), because part (b) can be proved in a similar manner.
Pick up a pair ( ktS+1, k tN+1 ) satisfying k tN+1 = µ (k tS+1) and k tS+1 < λR ≤ ktN+1 < K (λ ) . We thenpick up a pair under autarky ( ktS+,1A, ktN+1, A ) satisfying k tN+ ,1A > µ (k tS+,1A ) , by increasing k tN+1 ,starting from ( ktS+1, k tN+1 ) , while fixing k tS+1 . We have

λRf ' (ktN+1 ) + W ( ktN+1 ) > λRf ' ( µ (k tS+1 )) + W ( µ ( ktS+1 )) = λRf ' ( ktS+1) + W ( ktS+1) , where theinequality arises from k tN+ ,1A > µ (k tS+,1A ) and the fact that λRf ' (.) + W (.) is increasing in k tN+1 ,while that in country S is smaller than k * (e.g., Boyd and Smith (1997, Proposition 5)).

24

and the equality arises from the definition of µ (.) and k tS+,1A = ktS+1 . It finally follows thatr (k tN+1, A ) > r ( k tS+,1A ) . Q.E.D.

Although we prove only the region of k tN > ktS , by symmetry we can develop a similarargument for the region of k tN < ktS . Hence we have the following corollary from Lemma 1
and the symmetric argument.

Corollary 1
If an autarkic pair ( ktS+,1A, ktN+1, A ) is inside (outside) the ellipse, equilibrium interest rates in two
countries satisfy r (k tN+1, A ) > ( <)r (k tS+,1A ) under autarky.

It is straightforward to have the following.

Proposition 2
Suppose that an equilibrium pair ( ktS , ktN ) satisfies k tN = Ψ ( ktS ) . Then if the state ofwould-be autarkic economies at period t + 1 , ( ktS+,1A, ktN+1, A ) , is inside (outside) the ellipse, capital
flows from country S (N) to country N(S).

The position of the would-be autarkic state at period t + 1 relative to the ellipse determines the
direction of capital flows at period t + 1 . When the levels of capital are not so large in all thecountries, the autarkic state is likely to lie inside the ellipse, and capital is likely to flow out ofcountry S to country N. Proposition 2 also implies that as long as (7) intersects with the ellipse,

k tN+1 > k tS+1 will be preserved.
Looking at Figure 5-2C, we see the direction of capital movement on the path to F . The
25

arrows in solid line indicate where the state of would-be autarkic economies is located at next
period given that the equilibrium is on the ellipse. When ( k tS , k tN ) lies at [ 0, k * ) × [0, k * ) ,the state of would-be autarkic economies at period t + 1 , ( ktS+,1A , k tN+1, A ) , lies at the northeast of
the current state,27 while when ( k tS , k tN ) lies at [ 0, k * ) × [k * , ∞ ) , ( ktS+,1A , k tN+1, A ) lies at thesoutheast of the current state. It is obvious that all the arrows approach toward ( k * , k * ) . In
Figure 5-2C, all the arrows are described to be directed inside the ellipse at any point on

k tN+1 = µ ( ktS+1 ) around F . Thus, Proposition 2 predicts that on almost all the path to theasymmetric steady state F , capital flows out of country S to country N.
If the integration occurs at the early developing stage, with kTN and kTS being so small
to satisfy Rα{W ( kTN ) + W (kTS )} < 2λR or at the stage, with kTN and kTS being great tosatisfy 2 K (λ ) < Rα {W (kTN ) + W ( kTS )} (although the latter case is unlikely), the intersectionis uniquely realized on the 45 degree line, and the equilibrium converges to the symmetricsteady state E .28 However, the symmetric steady state and the path on the 45 degree line areunstable in the following sense. In Figure 5-2B, the initial equilibrium is achieved at, forexample, B2 , but for a small perturbation the financial market requires to be cleared at either of
the asymmetric equilibria, B1 and B3 because r (.) is upward sloping around B2 . Once theequilibrium diverges from the symmetric equilibrium, it never goes back, but eventually falls
into any one of the asymmetric steady states. If a small perturbation is allowed for along thedevelopment path, the equilibrium eventually falls into one of the asymmetric steady states.
The symmetric steady state is seldom achieved; financial integration leads to an increased
inequalit y of per-capita income between countries with capital flows from a poor to a wealthy
27

This is because (A1) implies that k ti < kti+, A1 < k * holds if k ti < k * , while k ti > kti+, A1 > k *holds if k ti > k * .
If 2 λR < Rα {W (k TN ) + W (k TS )} < 2 K (λ ) is satisfied and the stocks of capital are accidentallythe same at the timing of integration, as shown in Figure 5-2A, the equilibrium at period t+1 isachieved at B2 , and converges to the symmetric steady state E .
2826

country. 29 The “symmetric -breaking” behavior of the world financial market has already been
extensively argued by many researches, including Boyd and Smith (1997), Sakuragawa and
Hamada (2001) and Matsuyama (2004).
Figure 5-3A illustrates the case with five steady states. This case is more likely when the
TFP is fairly high, institutions for contract enforcement are fairly developed, and the worldsavings rate is at the intermediate level. Now suppose, for example, that under autarky C ' will
be reached in period T + 1 (note that k TN > kTS ). Then there exist three intersections between
the PQ line representing (7) and (11), denoted, C1, C2 , and C3 . In this case,r (k tN+1, A ) > r ( k tS+,1A ) holds (Lemma 1(b)), and hence at the instant of integration southernpeople have an incentive to invest in country N. Thus at period T+1 the equilibrium is achieved
at C1 . Having arrived at C1 , the equilibrium converges to an asymmetric steady state F witha flight of capital from country S to country N. In Figure 5-3C, arrows are typically directed
inside the ellipse at almost all the points on the path converging to F so that capital flows outof country S to country N on almost all the path to the asymmetric steady state F . When theworld economy arrives at F , it turns out that the globalization leads to an increased inequalityof per-capita income between countries.
However, we have a different consequence of financial integration in Figure 5-3A. Consider
the case when both countries are sufficiently wealthy that the autarkic equilibrium reaches D ' .
At D ' , r (k tN+1, A ) > r ( k tS+,1A ) holds (Lemma 1(b)), and hence at the instant of integration
southern people have an incentive to invest in country N. The outflow of capital from country S
to country N shifts the equilibrium to D1 . However, once the equilibrium arrives at D1 , theequilibrium goes on k tN+ 1 = φ ( ktS+1 ) toward I , and after passing through I converges to thesymmetric steady state E . Figure 5-3C shows that on the path to I , arrows are directed
outside the ellipse, and thus capital tends to flow from country N to country S. Figure 5-3B
represents the configuration of the world financial market on the symmetric path. The aggregate
29

Note that poor countries then receive the gain of possessing foreign assets so that the GNP

gap is not so large as GDP.

27

world savings are now great so that the function r (ktN+1 ) shifts to the left and the functionr (ktS+1 ) to the right, compared to Figure 5-2B. The intersection is unique and determined at I
to allocate capital equally between countries.
The direction of capital flows differs along the developing stage, depending on the levelsof capital stock of both countries. A look at non-monotone configuration of r (k ) illustrated in
Figure 3-3 helps our understanding behind this mechanism. When the level of capital stock
realized under autarky lies around λR , the corresponding interest rate tends to be lower thanthat of the other country, and thus the country’s saving flows out abroad by openness. However,once the autarkic level of capital stock reaches the level near K (λ ) , the interest rate tends to behigher than that of the other, and thus the country enjoys capital inflows by openness.
To which steady state the world economy will converge depends on the distribution of thecapital stock when integration occurs. From the above argument, if the PQ line goes across thewestern region relative to H on the steady-state saving-investment relation (10), theintegration will move the equilibrium to an asymmetric steady state, while otherwise, theintegration will eventually move the equilibrium to the symmetric steady state. Thus, as
depicted in Figure 5-3D, the shaded region illustrates the pair of levels of autarkic capital stock
under which the symmetric steady state is finally realized. The downward curve is a set of pairs( k tS , k tN ) satisfying k HN + k HS = RαW (ktN ) + RαW ( ktS ) , where ( k HS , k HN ) is the pair oflevels of capital corresponding to H . If the levels of capital in both countries are high, theintegration tends to move the world economy to the symmetric steady state with harmonized
growth. Furthermore, given the aggregate capital of both countries constant, if the inequality issmall, the integration tends to be successful. 30
From the prospect of late-developing countries, the capital account liberalization promotes
development only if some threshold level of income has already been attained. As has raised bymany observers and government officials, including McKinnon (1991), this theoretical finding
explains the important role of the timing of lifting capital account for late-developing countries.
30

Since the curve with shade is convex to the origin, any PQ line is more likely to intersect with the28Late-developing countries can go on the successful path by delaying the timing of liberalization
until arriv ing at some development stage.
Conditions for successful financial integration depend on the levels and the distribution of
per-capita income across countries. Other things being equal, higher levels of per-capita income
and more equal per-capita income are determinants of successful integration. Furthermore,
several parameters, A (the TFP level), λ (the measure of contract enforcement), or α (themeasure of global savings) will affect the point H , and so the downward curve depicted in
Figure 5-3D, and thus the period for successful integration. Other things being equal, a greater
value of either A , λ , or α makes the aggregate savings necessary to attain successful
integration, k HN + k HS = RαW (ktN ) + RαW ( ktS ) , smaller, and hence shorten the waiting time forsuccessful financial integration.
Some empirical evidence is consistent with the theoretical finding. Chinn and Ito (2006)
investigate whether financial openness leads to financial development using panel data covering
108 countries over the period of 1980-2000, and find that financial openness promotes
development of domestic financial markets only if a threshold level of legal development has
been attained. Bekaert et al, (2001) find that financial liberalization tends to promote economic
growth and the effect is greater for countries with high education levels in their sample of 30
emerging countries.
Episodes also abound. It was only in the 1980s that the governments of fast-growing East
Asian countries, including Japan, Korea, and Taiwan, lifted capital account completely. The
Japanese government announced liberalization of capital control in 1979, but the Japaneseliberalization was not genuine toward 1980s. In Taiwan, capital transactions were decontrolled
only as recent as in 1987. In Korea, liberalization was gradually made toward 1980s, but
restrictions on capital movements are not yet completely removed. Although capital controls arein principle aimed at restricting capital outflows, some countries experienced rapid and massive
inflows of capital soon after the removal, including Italy, Spain, New Zealand, and Uruguay.
Bartolini and Drazen (1997) provided these episodes in order to emphasize their signaling
area of the shaded region with inequality being small.

29

argument of capital account liberalization as a commitment to policy reforms, but thesecountries may have attained the stage in which the reversal of capital flows will happen.
Finally, Figure 5-4A illustrates the case with the unique steady state. This case is more
likely when the TFP is high, institutions for contract enforcement are well developed, and theworld savings rate is high. Figure 5-4A illustrates the typical case showing that the early lifting
capital accounts makes the period necessary to arrive at the steady state longer. To see this,
suppose for example that two economies will reach at Z ' in period T + 1 if they are inautarky. At Z ' , r (k TN+,1A ) > r ( kTS+, 1A ) holds (Lemma 1(b)), and hence southern people have an
incentive to lend to country N. The equilibrium at period T+1 is achieved at Z1 . Once theeconomy arrives at Z1 , it goes on k tN+ 1 = φ ( ktS+1 ) until it reaches I , and afterward, on the 45
degree line, eventually converges to the symmetric steady state E . Figure 5-4B shows that on
the path reaching I , capital flows first from country S to N, later reverses the direction ofcapital flows S; on all the points to path from I to E , capital flows cease.
By financial integration, the convergence to the symmetric steady state will occur surely,
but will give rise to an increased inequality of nations in the transition. On early financial
integration, the catching-up country first experiences capital outflows, later a reversal of capital
flows, and finally capital inflows. The “trickle -down effect” from wealthy countries reverses the
direction of capital flows. Once fast-growing countries have achieved some stage ofdevelopment, the increased global savings leads to a decline in the world interest rate , which inturn, stimulates borrowing and investment of late-developing countries, thus promoting thesecountries to catch up with fast-developing countries.31
From the prospect of late-developing countries, a best strategy for faster development is tokeep capital accounts closed until the world economy under autarky reaches the state aroundZ2 . Conditions for successful financial integration depend on the levels of per-capita income
across countries, but now not the distribution. Further development of institutions for contract
enforcement, captured by a smaller λ , will make the ellipse smaller, and the period forsuccessful integration is earlier.
31

Drazen develops an alternative theory behind which the capital reversal happens.

30

The illustrated the pattern of capital flows is similar to the one of Latin American countries
past twenty years. In the 1980s, a number of Latin American countries experienced capitaloutflows, involving capital flights, but in the 1990s, capital begins to go back to these countries.
Some economists argue that the reversal of capital movement in this region arises from thedecline in the world interest rate driven by the change in the global saving-investment relation
not from any domestic institutional change.32

7. Trade Openness and “Order of Liberalization”
The problem of the optimal timing for lifting capital account has been traditionally argued
in terms of the sequence of liberalization of first trade and secondly capital flows (e.g.
McKinnon (1991)). Braun and Raddatz (2007) report that trade liberalization occurred before
capital account liberalization in 68 of the 73 countries that liberalized any of these dimensions
between 1970 and 2000. Chin and Ito (2006) find that trade liberalization becomes aprecondition for capital account liberalization.
Although some observers address the sequential openness, the rational behind the sequential
liberalization is not necessarily clear. According to the standard trade theory, trade openness andfinancial market openness should be, and the sequence is irrelevant. In order to investigate thisproblem, particularly to explore the possible implications for sequential opening, we extend themodel to allow for capital goods to be tradable.
Assume that among capital goods produced by entrepreneurs, a fraction µ ( 0 < µ < 1) of them
are tradable, while the remaining are not so that some of capital goods produced in country

i can be used in the final-good firm in country j (i ≠ j ) . Many capital goods appear to be
nontraded (e.g., railways). Accordingly, the stock of capital financed and produced in country i~at period t , denoted k ti , is distinguished from that available in country i at period t ,denoted kti . Letting mt denote the amount of capital goods that are exported from country N
32

See Calvo (1996), for example.

31

~

~

to S, by definition we have k tN = k t N − mt and ktS = k tS + mt . Together with the definitions,the market clearing in the world financial market finally reduces to (7).33 The trade of capital
goods implies the reallocation of capital goods between countries, and thus represented by ashift on the same PQ line. Since the trade of capital goods is motivated by the difference in theprice of the capital good, the amount of capital goods traded mt +1 satisfies f ' ( ktN+1) = f ' ( ktS+1) ,~or mt +1 = µkt N+1 if f ' ( ktN＋1) < f ' ( ktS+1) . Additionally, the no-arbitrage conditionr (ktN+1 ) = r ( ktS+1) should be met.
If the trade realizes the equality f ' ( ktN+1) = f ' ( ktS+1) , the trade openness brings the world
economy on the symmetric path, whereas otherwise, it tends to shift the capital allocation on the
PQ line in the asymmetric region. If the liberalization of capital accounts is made together, thetrade openness never influences the development path. The allocation of capital goods made bytrade is offset by capital movement. In the case of simultaneous liberalization, trade and capital
flows are substitutable.
We next turn to the sequential openness. We find an interesting case in the case of fivesteady states. In Figure 6-1A, at period T two closed economies arrive at I ' that is outside thecurve with shadow, but moves along PQ from I ' to I " with trade openness and with no liftof capital control.

Once the world economy have arrived at I " that is inside the curve withshadow, lifting capital control brings the world economy finally on the path to the symmetricsteady state E .
Developing countries can attain faster growth first by trade liberalization and secondly bycapital account liberalization. The benefit of the sequential openness arises when the timing oflifting capital control matters for development. The sequential problem becomes an urgent
policy issue only if some threshold level of financial development has been attained. 34~~

Note that k t+N1 + ktS+1 = ktN+1 + ktS+1 .
34
If technological spillovers through the flow of traded goods are also taken into account,developing countries are more likely to gain the benefit from sequential openness.
33

32

8. Conclusion
We have considered a global analysis of financial market integration, and found several
important features that have not been obtained from the local analysis around steady states.
There are a stable symmetric and stable asymmetric steady states, both of which may coexist,
and out of steady states, there are interesting non-monotone behavior of the pattern of capital
movement and development. The timing of integration that attains harmonized growth depends
on several characteristics including the stage of economic and legal developments, income
inequality, and global savings. The concept of optimal timing for liberalization allows us to
explore conditions for successful integration.
This paper is extended in several directions. A slight modification of the model can explain
the persistent current account imbalance between the U.S. and some Asian countries, so-called,“great imbalance”. In the growing economy the capital account surplus (deficit) is a reflectionof capital outflows that will arise in a setup of countries with different institutions for contract
enforcement. A modeling of that kind might lead to a conclusion that if China progresses thecapital account liberalization following the request of the U.S., the US-China imbalance
becomes even greater with further outflows of capital from China.
The introduction of foreign direct investment (FDI) will be a promising avenue to extend
this model. FDI may be identified as equity participation to the project associated with thetransfer of the control right of the firm, that should be distinguished from “security investment”
involving the participation to the project as debtholders. How the richer movement of capital
including both security and direct investments will change the development path and contributeto successful integration is an interesting issue to be explored.
The difference in the pledgeability among goods will be an interesting avenue to be
examined. The pledgeability may differ according to how technological change is embodied orwhether the output is a traded or a non-traded good. Following the model, the parameter λmight differ between the traded and non-traded goods, or among many goods that differ in theextent of embodiment of technological change.

33

References
Acemoglu, D., and F. Zilibotti, 1997, Was Prometheus Unbounded by Chance? Risk,
Diversification, and Growth, Journal of Political Economy 105, 709-751.
Alfaro, L, S. Kalemli-Ozcan, and V. Volosovych, 2006, Why doesn’t capital flow from rich topoor countries? An empirical investigation, forthcoming in Review of Economic Statistics
Bartolini, L., and A. Drazen, 1997, Capital-Account Liberalization as a Signal, American
Economic Review 87, 138-54.
Bernanke, B. and M. Gertler, 1989, Agency Costs, Net Worth, and Business Fluctuations,
American Economic Review 79, 14-31.
Boyd, J.H. and, B. D. Smith, 1997, Capital Market Imperfections, International Credit Markets,
and Nonconvergence, Journal of Economic Theory 73, 335-64.
Braun, M, and C. Raddatz, 2007, “Trade liberalization, capital account liberalization and thereal effects of financial development,” Journal of International Money and Finance 26,730−761.
Broner, F.A., and J.Ventura, 2007, Globalization and Risk Sharing,
Bekeart, G, C.R. Harvey, and C. Lundblad, 2001, Emerging Equity Markets and Economic
Development, Journal of Economic Development 66, 465-504.
Buiter, W. H., 1981, “Time Preference and International Lending and Borrowing in an
Overlapping-Generations Model,” Journal of Political Economy 89, 769-97.
Bulow, J., and K. Rogoff, 1989, A Constant Reconstructing Model of Sovereign Debt, Journalof Political Economy 97, 155-78.
Caballero, R. J., and A. Krishnamurthy, 2006, “Bubbles and capital flow volatility: causes andrisk management”, Journal of Monetary Economics 53, 35-53.
Calvo, A.G., L. Leiderman, and C. M. Reinhart, 1996, “Inflows of capital to Developing
countries in the 1990s,” Journal of Economic Perspective 10, 123-39.
Caselli, F, and J. Feyrer, 2007, “The marginal product of capital,”
Economics 121, 535-567.

34

Quarterly Journal ofChinn, M.C., and H. Ito, 2006, “What matters for financial development? Capital controls,
institutions, and interactions,” Journal of Development Economics 81, 163-192.
Dixt. A., 2003, Trade expansion and contract enforcement,

Journal of Political Economy 111,

1293-1317.
Eaton, J., R. Fernandes, 1995, “Sovereign Debt” in Grossman and Rogoff, eds., Handbook ofInternational Economics, Vol.3, Amsterdam, North Holland, p2031-77.
Grandmont, J,M., “Periodic and aperiodic behavior in discrete one-dimensional dynamicalsystem,” in Contributions to mathematical economics: In honor of Gerard Debreu, ed. by
W.Hildenbrand and A.Mas-Colell. Amsterdam: North-Holland
Gertler, M., and K. Rogoff, 1990, North-South lending and endogenous domestic capital market
inefficiencies, Journal of Monetary Economics 26, 245-266.
Greif, A., 1994, Cultural beliefs and the organization of society: a historical and theoreticalreflection on collectivist and individualist societies, Journal of Political Economy 102,912-50.
Hamada, K., 1966, Strategic affects of taxation on foreign investment income, Quarterly
Journal of Economics 80, 361-75.
Kemp, M.C., 1962, Foreign Investment and the National Advantage, Economic Record 38,
56-62.
La Porta, Rafael, Florencio Lopez-de-Silanes, Andrei Shleifer, and Robert W. Vishny.

“LegalDeterminants of External Finance.” Journal of Finance 52, 1997, 1131-50.
La Porta, Rafael, Florencio Lopez-de-Silanes, Andrei Shleifer, and Robert W. Vishny. “Law and
Finance.” Journal of Political Economy, 1998.
Levine, R., 1998, The Legal Environment, Banks, and Long-Run Economic Growth, Journal of
Money, Credit, and Banking 30.
Lucas, L. E., “Why doesn't capital flow from rich to poor countries?” American Economic
Review, Papers and Proceedings 92 (1990), 92-96.
MacDougall, G.D.A., 1960, The Benefits and Costs of Private Investment from Abroad: A
Theoretical Approach, Economic Record 36, 13-35.
Matsuyama, K., 1999, Growing through cycles, Econometrica 67, 335-47.

35

Matsuyama, K., Financial Market Globalization, Symmetric -Breaking and Endogenous
Inequality of Nations, Econometrica 72, 853-884.
McKinnon, R. I., 1973, Money and Capital in Economic Development, Washington, D.C.:
Brookings Institution
McKinnon, R. I., 1991, The Order of Economic Liberalization: Financial Control in theTransition to a Market Economy, Johns Hopkins Press
Portes, R. and H. Rey, 2005, “The determinants of cross border equity flows,” Journal ofInternational Economics 65, 269-96.
Reinhart, C. M., and K.S. Rogoff, 2004, “Serial default and the “Paradox” of rich-to-poor
capital flows”,

American Economic Review, Papers and Proceedings 106, 53-58.

Sakuragawa, M. and K. Hamada, 2001, “Capital Flight, North-South Lending, and Stages of
Economic Development,” International Economic Review 42, No.1: 1-26.
Tirole, J., 2003, Inefficient Foreign Borrowing: A Dual-and Common-Agency Perspective,
American Economic Review 93, 1678-702.
Tornell, A., and A. Velasco, 1992, “The Tragedy of the Commons and Economic Growth: Why
does Capital Flow from Poor to Rich Countries?” Journal of Political Economy 100,
1208-31.
Townsend, R., 1979, “Optimal contracts and competitive markets with costly state verification” ,Journal of Economic Theory 31, 265-93.

36

Figure 3-1
k t +145°k*k t +1 = RαW (k t )

0

kt

k*

k0

Figure 3-2
r (kt +1 )λRf ' (k t +1 ) + W (k t +1 )Rf ' (k t +1 )k t +1K (λ ) λ R

Figure 3-3
r (k t +1 )λRf ' (k t +1 ) + W (k t +1 )Rf ' (k t +1 )λR

K (λ )

38

k t +1

Figure 4-0
ktN+1

P
K (λ )

ktN+1 = ktS+1
E

k

*

A'
A0

λR

A

λR

k

K (λ )

*

ktS+1

Q

Figure 4-0A
r (k1N )r (k1S )A'r (k1S , A )A
r1

r (k1N , A )

ON

k1N

k1S

Rα {W (k 0N ) + W (k 0S )}39OSFigure 4-1
ktN+1
ktN+1 = φ (ktS+1 )ktN+1 = φ (ktS+1 )ktN+1 = ktS+1

K (λ )

ktN+1 = ϕ 2 (ktS+1 )ktN+1 = µ (ktS+1 )

λR

k

N
t +1

ktN+1 = ϕ1 (ktS+1 )= µ (k )S
t +1

K (λ )

λR

40

ktS+1

Figure 4-2A
ktN+1

M
K (λ )

F
E
k

*

λR

L
G

λR

k*

ktS+1

K (λ )

Figure 4-2B
ktN+1

H

F
k*
K (λ )

E

I

J

λR
G

K (λ ) k *λR
41k tS+ 1

Figure4-2C
ktN+1

E
k*
I

K (λ )λR

λR

K (λ )

ktS+1

k*Figure 4-2D
k tS+1

Φ ( k tS )

Λ

λR

k*

42

k (λ )k tS

Figure 4-2E
k tS+1Φ (ktS )Λ

λRk * k FSκ (λ )Γk tS

Figure 4-2F

ktS+1

Φ (ktS )

Λ

λR

kF

43

kH

K (λ )ktSk*Figure 4-2G
ktS+1

Φ (ktS )λR

K (λ )

k*

kG

ktS

Γ

Figure 4-2H
k ts+1Φ (kts )Λ

λR Φ 2 (k c ) k c

44kF

Φ(k c )

Γk ts

Figure 5-2A
ktN+1
P
M
K (λ )

F
k*
rT +1

λR

E

B' '
B1

B'
B2

L
G

B3

λR

k

ktS+1

K (λ )*

Q

Figure 5-2Br (k TN+1 )r (k TS+1 )

B' '

B3
rT +1

ON

B'

B1

B2

k TN+1r (kTN+,1A )r (kTS+, 1A )k TS+1

Rα {W (k TN ) + W (kTS )}45

OSFigure 5-2C
ktN+1

M

K (λ )
F
E

k

*

λR
L
G

λR

k

46

*

K (λ )

ktS+1

Figure 5-3A
ktN+1

P'

P
C1

F
H

k*

K (λ )

C'

D1

E

D'

I
D2

C2

D3
J

λR
C3

Q'

G

Q
ktS+1

K (λ ) k *λR

Figure 5-3B
r (kTS+1 )r (k TN+1 )

I

rT +1

ON

kTN+1 = kTS+1
Rα {W (k TN ) + W (kTS )}47

OS

Figure 5-3C
ktN+1

H

F

E

k*
K (λ )

I

J

λR
G

λR

K (λ ) k

ktS+1*

Figure5-3D
ktN+1

F

E

H

k*
K (λ )

J

λR
G

λR

K (λ ) k *
48

ktS+1

Figure5-4A
ktN+1
P'
P

k*

W1

K (λ )

W'
Z1

E
I

Z'
Z2

λR

Q'
Z3

λR

K (λ ) k *

ktS+1
Q

Figure5-4B
ktN+1

E
k*
K (λ )I

λR

λR

K (λ )

49

k*

ktS+1

Figure6-1A
ktN+1

P

H
F
E

I'
k*
K (λ )

I ''

J

λR

G

Q

λR
λR

K (λ ) k *50