# Capital Mobility and Development Take-Off

Charles Vellutini

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CAHIERS ECO & MATHS

97.58

CAPITAL MOBILITY AND DEVELOPMENT TAKE-OFF

Charles VELLUTINI

MAD
# Capital Mobility and Development Take-Off

Charles Vellutini*

June 17, 1997

## Abstract

We study the effects of capital mobility in a two-country economy with endogenous growth. In contrast with standard endogenous growth settings, this model can account for intentional private capital flows from rich to poor countries, and the impact of these flows on growth can be analyzed. Under optimistic world expectations, it can be shown that introducing capital mobility can make a country trapped by a development threshold cross the threshold and bifurcate toward sustained growth. Given the possibility of multiple equilibria with self-fulfilling prophecies, the model also emphasizes the importance of private international capital flows for growth, as opposed to public foreign finance.

**Résumé.** On étudie les effets de la mobilité du capital dans un modèle à deux pays avec croissance endogène. Contrairement aux modèles standards de croissance endogène, cette analyse permet de rendre compte des flux intentionnels de capital privé des pays riches vers les pays pauvres, et l'impact de ces flux sur la croissance peut être étudiée. Sous des conditions d'anticipations mondiales optimistes, on montre que l'introduction de la mobilité du capital peut permettre à un pays piégé par un seuil de développement de franchir ce seuil et de bifurquer vers la croissance soutenue. Compte tenu de la possibilité d'équilibres multiples avec prophécies auto-réalisatrices, le modèle met également en évidence l'importance des flux de capital privé, par opposition aux fonds internationaux publics.

Key words: capital mobility, development threshold.

Mots clés : mobilité du capital, seuil de développement.

<footnote>MAD, Université Paris I Panthéon-Sorbonne, 90 rue de Tolbiac, 75634 Paris Cedex 13. I thank Antoine d'Autume for some very helpful comments and suggestions.</footnote>
# 1. Introduction

The attempts of the recent growth literature to account for international capital flows have followed two approaches. First, capital mobility has been cast in neo-classical models of growth. This is for example the analysis of Barro, Mankiw and Sala-i-Martin (1995), and Bliss (1995). Indeed, the neoclassical model is a natural candidate for explaining international capital flows as, all other things being equal, decreasing returns to capital make countries with lower stocks of capital comparatively attractive for investment. The main limitation of these models is to retain the traditional paradigm of exogenous growth: capital flows may well be explained, but their potential influence on long-run growth is left outside the analysis.

Second, the literature on endogenous growth has described capital mobility using an expanded concept of capital that includes knowledge. The path-breaking contribution of Grossman and Helpman (1991) has been to show us how knowledge may flow internationally and how this influences long-run growth rates. Nevertheless, if we restrict our attention to physical or human capital, as opposed to a disembodied type of capital such as knowledge, it appears that standard models of endogenous growth have difficulties explaining why there are international capital flows to the poorer countries at all. This is the analysis developed by Lucas (1990), who shows that the "AK" class of models is hardly consistent with physical and human capital flows, because the more advanced countries (with a higher capital stock) also offer the best return to investment, hence the absence of capital flows from rich to poor countries.

However, indications abound that there are (increasing) movements of physical capital from rich to poor countries. See for instance the steady rise in investment to the private sector of developing countries in the statistics provided by IFC (Figure 1.1). Using another definition of flows and another group of countries, the Institute of International Finance of the World Bank reports that private finance to emerging markets amounted to about USD230 billion in 1996, up tenfold from a decade ago<fnref n="1" />. Additionally, some recent empirical studies (in particular Cohen (1994)) have supported the view that foreign finance may significantly help debtors countries improve the efficiency of their production process, which suggests that a theoretical analysis of the impact of capital inflows on the recipient country's technology could be rewarding.

The analysis we present is an attempt to account for flows of physical and human capital from rich to poor countries in a model with endogenous growth. Our main results are as follows. First, building on the research of Zilibotti (1995), and d'Autume and Michel (1993), we construct a model where flows of capital

<footnote n="1">¹Quoted in The Economist, March 1st, 1997.</footnote>
<figure><img src="image_2.png" /><figcaption>Figure 1.1: Long-Term Capital Flows to the Private Sector of Developing Countries 45 developing countries are included. Private investment in these countries accounted for 81% of private investment in all developing countries in 1994. (IFC (1995))</figcaption></figure>

from rich to poor countries are compatible with endogenous growth. Second, the model provides a theoretical setting where inflows of private capital can help an economy trapped in stagnation take off into sustained growth. This effect comes into play through the impact of capital inflows on productivity. Indeed, because technology is embodied in capital, the model can be viewed as a theory of technology transfers. As emphasized by Bernard and Jones (1996), in spite of the early theoretical contribution of Grossman & Helpman (1991), the recent empirical debate on international convergence has generally overlooked the potential role of technology transfers in the catch-up phenomenon.

After a brief review of the literature, we present the structure and assumptions of the model in Section 2. Section 3 proposes the resolution of the core sub-system of the model, and Section 4 gives the solutions of the trajectories of the stock of foreign asset and national consumptions.

## 2. Capital mobility and growth

### 2.1. Capital mobility in the neoclassical model

In a well-known paper, Barro, Mankiw and Sala-i-Martin (1995) have shown that capital mobility may increase growth in the recipient countries and lead to faster
convergence toward the steady state GDP level.

However, one striking prediction of extension of the neoclassical model of
growth to international capital mobility is that convergence (including conditional
convergence) in GNPs, a central result of the model in its autarchy version, no
longer holds. As shown in the general equilibrium models (respectively multi- and
two-country) of Bliss (1995), Béraud (1996) and Vellutini (1997), the foreign in-
vestment made by the richer country<fnref n="2" /> prevents the poorer one from reaching the
autarchy steady-state level of GNP, as in steady state a perpetual rent is paid to
foreign investors.<fnref n="3" />

This finding relies on the assumption that foreign investment has no bearing
on the technology used in the poorer country, which is clearly at odds with a direct
observation of actual foreign investment: most real-world transactions of foreign
investment do involve some transfer of technology<fnref n="4" />. Technology improvement
in the (poorer) recipient countries would therefore seem to be a natural way to
reestablish the prediction of convergence of the neoclassical model, by adequately –
and exogenously – setting the technological "jump" caused by foreign investment.
For example, assume that the pre-liberalization production function of the poorer
country is given by $y_p = A_p k_p^\alpha$, where $k_p$ is aggregate per-capita capital, $A_p$ is a
technological parameter, $\alpha$ a parameter with $0 < \alpha < 1$. Assume also that the
per capita GDP of the richer country is $y_r = A_r k_r^\alpha$, with $A_r > A_p$. If we assume,
lastly, that the technology used in the poorer country jumps to the richer country's
level as foreign investment flows in<fnref n="5" />, we may increase convergence effects in the
neoclassical model depending on the relationship of the cross-country differentials
between on the one hand the initial capital stocks and on the other hand the initial
technology levels. Note however that the poorer country would still have to pay
a perpetual rent to foreign investors. As a result, its GNPs will never converge to
the rich country's level. But the poor country's GNP with capital mobility could
potentially be larger than it would have been in autarchy.

<footnote n="2">²We call richer the country endowed with a higher stock of capital when liberalization of international capital flows begins.</footnote>

<footnote n="3">³It is never optimal to repay the principal of the international debt, nor for foreign investors to repatriate their external assets. Interest paid to lenders by the poorer country on external debt prevents the latter to reach its steady state autarchy GNP, even though its GDP steady level in the two-country economy is the same as in autarchy. In terms of welfare, however, the liberalization of capital flows is Pareto-improving. See Vellutini (1997).</footnote>

<footnote n="4">⁴See for example Khan (1991), which analyzes the patterns of foreign direct investment in China over the period 1979-1989. It is shown that transfers of technology generally accompagny foreign direct investment.</footnote>

<footnote n="5">⁵One could somewhat refine this analysis by making only the new capital more productive, although this would not affect our conclusion.</footnote>
## 2.2. Capital mobility in models of endogenous growth

It has been amply commented that endogenous growth models (for example Lucas (1988) and Romer (1990)) do not exhibit any property of convergence. It is precisely a major achievement of this research to provide us with a theoretical explanation of the observed cross-country diversity in both GNP levels and growth rates (as well as an economic explanation of sustained growth).

How does capital mobility impact on this result?

The fruitful research of Grossman and Helpman (1991), thereafter G&H, occupies in this respect a prominent place. Firstly, as is well known, one important effect identified by these authors relates to international spillovers of R&D, as knowledge bears some important features of a public good (sections 6.5, 9.1 and 9.2 of G&H). This effect stimulates growth internationally, although in G&H's research convergence effects are not clear-cut because advanced countries may still retain a leading edge. Some models of convergence, which can even display leap-frogging effects, have been built on the basic insights provided by G&H. For example, Barro and Sala-i-Martin (1995, chap. 8) explore a leader-follower model that exhibits convergence properties because imitation and implementation are cheaper than innovation. Secondly, trade in goods may also influence international growth rates, through two different mechanisms: first, knowledge is assumed to flow between trading partners as they exchange goods and information; second, in two-sector economies, if the international price of the human capital-intensive good is lower than in autarchy, trade may spur growth, because more human capital will be profitably employed in R&D, a human capital-intensive activity (respectively G&H, sections 6.2 and again 9.2).

The important point to note for our purpose is that, since returns to capital are higher in the more advanced countries<fnref n="6" />, there are no intentional capital flows from rich to poor countries in this analysis (including in the leader-follower model of Barro and Sala-i-Martin): even in an extended concept of capital incorporating knowledge, flows of capital are involuntary side-effects resulting from spillovers or trade of goods. As is particularly clear from chapter 5 of G&H, that incorporates primary factor accumulation (physical and human capital), there is no room for intentional foreign investment from rich to poor countries in this line of research<fnref n="7" />.

Furthermore, as noted earlier, Lucas (1990) and also Rebelo (1992) have shown that the other family of endogenous models, which are based on constant returns to a broad concept of capital (with or without externality), yields similar results: there are no incentives to private capital flows from rich to poor countries.

<footnote n="6">⁶In this context, "advance" means accumulation of knowledge.</footnote>

<footnote n="7">⁷Conversely, in Gross and Helpman's analysis, mobility of physical capital could give rise to intentional migration of physical and human capital from poor to rich countries.</footnote>
On the other hand, if one abstracts from the North/South issue and instead considers two identical countries in steady state, the endogenous growth framework yields interesting results when uncertainty is introduced. Saint-Paul (1992) and Obstfeld (1994) have analyzed the impact of capital mobility in the presence of stochastic shocks on the returns of the different assets available. It can be shown that if risky assets are imperfectly correlated internationally, then capital mobility allows for a better diversification of risks and therefore permits the use of more specialized (and more risky) technologies (Saint-Paul) and/or higher savings (Obstfeld). In both cases, capital deepening will stimulate growth.

As a conclusion to this section, we see that contrary to intuition, we do not have a theoretical framework where private, intentional capital flows from rich to poor countries can take place from rich to poor countries, let alone endogenously improve the technology in the latter. We present an attempt to build such a model.

### 3. The model

We develop a model that seeks to reconciles two features: One, at certain levels of capital accumulation, there exist incentives (typically, cross-country differentials in return to physical capital) for intentional foreign investment. At these levels, physical capital will flow from rich to poor countries. Two, at higher levels of accumulation, economies may take-off into sustained growth, as the result of technology improvements accompanying capital accumulation. The concept of capital used here is one of a composite good that consists of the private components of a reproducible factor of production that can be thought of as human and physical capital. This definition is entirely consistent with international mobility. Indeed, most real-world transfers of physical capital are supported by elements of human capital, i.e. expatriates, technical assistance, and training.<fnref n="8" />

We build on the research by d'Autume and Michel (1993) and Zilibotti (1995), who have set the traditional issue of underdevelopment traps in closed-economy models of endogenous growth. These models provide us with the basic economic structure we need: the first phase of capital accumulation is characterized by decreasing returns to capital, while higher levels of accumulation may converge asymptotically to constant returns to capital. For convenience, we will use Zilibotti (1995) as the starting point of our analysis, as this model provides a closed-form for the production function.

<footnote n="8">⁸For example see Khan, Zafar Shah (1991) for a documented analysis on foreign investment patterns to China.</footnote>
## 3.1. The production function<fnref n="9" />

The aggregate production function is assumed to be in the form:

$$ \hat{Y}_i = \hat{Y}_i(A(K_i)\hat{K}_i, \hat{L}_i) \qquad (3.1) $$

where $\hat{Y}_i$, the aggregate GDP of country $i$, is a continuous, twice differentiable function in its two arguments. We assume that $\hat{Y}_i(.)$ is homogeneous of degree one. $\hat{K}_i$ is the aggregate stock of capital of country $i$. $A(K_i)$ is a capital-augmenting technological factor and a function of $K_i$, the per capita stock of domestic capital in country $i$, defined as $K_i = \frac{\hat{K}_i}{\hat{L}_i}$, where $\hat{L}_i$ is a non-reproducible factor of production, for example labor. $\hat{L}_i$ is considered constant in each country. We thus depart from Zilibotti by assuming that the technological factor $A(.)$ depends on the per capita stock of capital, not the aggregate stock. As will become clear, this modification is introduced in the model to allow for differences in size between the two countries: while none of them is considered "small" relative to the other, in the sense that it can influence international prices such as the interest rate, one country could still be "smaller", which will simply mean here that its population $\hat{L}_i$ is smaller. Making the technological factor $A(.)$ a function of capital per capita rather than aggregate capital can be interpreted as assuming congestion effects affecting the diffusion of technological improvements across the economy.

In per capita terms, we can write (3.1) as: $y_i = y_i(A(K_i)K_i)$, where $y_i = \hat{Y}_i/\hat{L}_i$

Defining $k_i$ as the per capita stock of capital available at the firm level, we see that $A(K_i)$ will be treated as parametric by optimizing agents, while $k_i$ will be treated as a state variable. Since we assume that the measure of firms is equal to unity, we have $K_i = k_i$, but, as in Zilibotti (1995), we maintain this notation to emphasize the externality in $A(K_i)$. When convenient, we will use the notation $f(K_i) \equiv y_i(A(K_i)K_i)$.

In this model, $A(K_i)$ can be interpreted as the state of technical efficiency directly applicable to production. Following Zilibotti (1995), we assume that $A(K_i)$ is bounded from above by the parameter $a$, which represents the frontier of scientific knowledge. Since we will maintain $a$ fixed, we assume that the technical progress captured by $A(K_i)$ consists in effect in adapting scientific breakthroughs (changes in $a$) into productivity improvements. In this model, the learning process will depend on the distance between the current state of knowledge applicable to production ($A(K_i)$) and the scientific frontier ($a$), together with the amount of per capita investment.

<footnote n="9">⁹See Zilibotti (1995) for a detailed exposition.</footnote>
One convenient specification of the learning process, which we will use in all our numerical simulations, is given by:

$$A(K_i) = \frac{a}{1 + (a/A_0 - 1)e^{-qK_i}} \quad (3.2)$$

where $q > 0$.

This specification implies increasing learning effect in the first stage of accumulation and decreasing effects at later stages, when $A(.)$ gets closer to its upper bound $a$.

We will generally assume that $A(K_i)$ verifies the following properties:

$$A(K_i) > 0, A'(K_i) > 0, \lim_{K_i \to \infty} A(K_i) = a, \lim_{K_i \to 0} A(K_i) = A_0, 0 < A_0 < a \quad (3.3)$$

It is straightforward to see that the specification (3.2) verifies properties (3.3).

Returning to the production function, we further assume that it verifies the following properties:

$$y_i'(A(K_i)k_i) > 0, y_i''(A(K_i)k_i) < 0 \quad (3.4)$$

$$\lim_{k_i \to 0} y_i'(A(K_i)k_i) = \infty, \lim_{k_i \to \infty} y_i'(A(K_i)k_i) = D > 0 \quad (3.5)$$

Although (3.4) states that the marginal productivity of $A(K_i)k_i$ is strictly decreasing, assumption (3.5) implies that it is bounded from below by a strictly positive quantity, another departure from the neoclassical model. A convenient specialization, which we will assume to hold in the rest of the paper, is as follows (the second term is in the Cobb-Douglas form):

$$Y_i = DA(K_i)\hat{K}_i + Z(A(K_i)\hat{K}_i)^{\theta}L_i^{1-\theta} \quad (3.6)$$

where $Z$ is a positive parameter. In per capita terms, we have:

$$y_i = DA(K_i)k_i + Z(A(K_i)k_i)^{\theta} \quad (3.7)$$

To illustrate our discussion of the economic properties of the production function, it is useful to draw the diagram of the generic market-based rate of return to capital (Figure 3.1). As decentralized agents will take $A(K_i)$ as given by the market, the competitive rate of return to capital will be:

$$r(K_i) = A(K_i)y'(A(K_i)K_i) \quad (3.8)$$
<figure><img src="image_3.png" /><figcaption>Figure 3.1: Competitive interest rate</figcaption></figure>

In order to rule out some economically meaningless cases, and defining $\rho$ as the time discount rate (with $0 < \rho < 1$), an additional assumption for the production function is that the competitive rate of return to capital be above $\rho$ as accumulation proceeds to the higher levels:

$$ \lim_{K_i \to \infty} r(K_i) = aD > \rho \qquad (3.9) $$

During the first phase of capital accumulation, the marginal productivity of capital declines, since the convexity of $y(.)$ (see the second term of the right-hand side of 3.7) then dominates technological progress brought about by the increase in $A(K_i)$. At higher levels of accumulation, as $y'_i(A(K_i)K_i)$ gets closer to its lower bound $D$, $r(K_i)$ will increase, but will be bounded by $aD$ as $A(K_i)$ tends to $a$. At the limit, the production function is therefore of the AK type. It is possible, in the general specification of the model, to have more than two solutions to the equation in $K_i: r(K_i) = \rho$. Without loss of generality<fnref n="10" />, we will assume in the rest of the paper that at most two solutions exist. We rule out the case of a unique tangent solution.

## 3.2. Autarchy equilibria

The demand-side of the economy is characterized by standard intertemporal preferences in the form:

<footnote n="10">¹⁰See Zilibotti (1995), page 1582.</footnote>
$$ \max U_i = \int_0^\infty \frac{c_i(t)^{1-\sigma}}{1-\sigma} e^{-\rho t} dt \quad (3.10) $$

where $c_i(t)$ is the per capita consumption in country $i$ at time $t$, and $\sigma$ the inverse of the constant intertemporal elasticity of substitution, with $\sigma > 0$. For simplicity, we assume no depreciation, so that the maximization is carried out under the constraint:<fnref n="11" />

$$ \dot{k}_i = y_i(A(K_i)k_i) - c_i \quad (3.11) $$

The first-order conditions resulting from this program are as follows:

$$ \frac{\dot{c}_i}{c_i} = \frac{1}{\sigma} (r(K_i) - \rho) \quad (3.12) $$

$$ \dot{K}_i = y_i(A(K_i)K_i) - c_i \quad (3.13) $$

The two equations of motion are appended with the usual transversality condition on $K_i$:

$$ \lim_{t \to \infty} c_i^{-\sigma} K_i e^{-\rho t} = 0 \quad (3.14) $$

In order to render the model analytically tractable, we finally assume (Zili-botti (1995), pages 1584 and 1588) that the two following technical conditions are satisfied. These conditions are mild and do not affect the economic meaning of the model.

1) To rule out closed orbits, it is sufficient that Bendixson's criterium be satisfied, that is, that there be no sign change in the trace of the Jacobian of the system at all points of the solutions trajectories. For this condition to be verified, it is sufficient that the following inequality be verified, which we will assume to be true:

$$ [(1 + \sigma)A(K_i) + \sigma KA'(K_i)] y'(A(K_i)K_i) > \rho, \forall K_i \in ]K^*, K^{**}[ \quad (3.15) $$

where $K^*$ and $K^{**}$ are the two solutions to the equation in $K_i : r(K_i) = \rho$ with $K^* < K^{**}$ (see Figure 3.1)

<footnote n="11">¹¹It is made use of the fact that the decentralized solution is equivalent to a centralized setting where the central planner is contrainted to treat $A(K_i)$ is parametric, thereby ignoring externalities.</footnote>
2) To guarantee the existence of an equilibrium with sustained growth we will take $\sigma$ such that:

$$ \sigma \geq 1 \qquad (3.16) $$

The resolution is this multiple-equilibria system is complex. We invite the reader to refer to Zilibotti (1995) for a full exposition of the autarchy solution, and only report results that we will later use in our comparison between autarchy and capital mobility. Under conditions (3.15) and (3.16), Zilibotti (1995) shows that two mutually excluding cases will arise, depending on the parameters of the model:

1. The development threshold case. If initial conditions – history – are such that $K_i(0) < K^*$, then the autarchy economy will converge toward fixed point $K^*$, with no growth.

2. The self-fulfilling prophecies case. This is a clear-cut case of multiple competitive equilibria. Regardless of initial conditions – that is: including for very low levels of $K_i(0)$ – the autarchy economy will take off into growth if private agents expect it to do so, because their consumption/savings behavior will then result in high accumulation, causing high returns to investment. The economy will converge to a path with sustained, constant growth. If agents do not believe in this take-off scenario, the autarchy economy will stagnate at fixed point $K^*$.

We stress that the difference between the two types of economies above comes entirely from the setting of the parameters of the model. In the rest of the paper, we choose to focus on economies of type 1 above, that is, economies that display a development threshold in autarchy. We do so because we are primarily interested in the possible impact of private capital mobility in helping economies cross the development threshold.

### 3.3. The open economy

We consider a two-country economy, each country being equipped with the same technology and the preferences described above. Before liberalization of capital flows, country 1, which we call rich by reference to its initial higher capital stock, has reached the stage of sustained growth, that is:

$$ K_1(0) > K^{**} \qquad (3.17) $$

Country 2, which we call poor, is still in the first phases of development, with

$$ K_2(0) < K^{\#} \qquad (3.18) $$
where $K^\#$ is the finite solution to the equation in $K_i$: $r(K_i) = aD$ (see Figure 3.1). We will also assume that $K_1(0)$ and/or $K_2(0)$ are sufficiently large so that

$$ \frac{K_1(0) + K_2(0)}{2} > K^* \qquad (3.19) $$

These initial conditions were chosen so that at time $t = 0$, when capital flows are liberalized, the rate of return to capital be higher in the poorer economy, triggering capital flows to the poor country. As will become clear later, condition (3.19), which is very mild considering that country 1 is in the stage of sustained growth when capital mobility in introduced, is imposed to rule out the trivial situation where the two economies may be trapped in stagnation.

Note that this simplified one-product economy does not capture any of the effects of international trade in goods. While, as demonstrated by G&H, the model therefore does miss out some important effects of openness on growth, it possesses the related advantage of isolating the specific impact of the opening of the capital account, as opposed to the liberalization of the current account. The effects of openness will stem entirely from capital mobility.

We solve the model based on competitive optimization. After decentralized optimization, we will aggregate the resulting first-order conditions, incorporating profit-maximization conditions by firms and market-clearing relationships.

The micro-economic structure of the model is as follows: in each country, consider a representative consumer owning the national capital and renting it to firms. The technology is operated by firms and will be improved in each country as domestic accumulation proceeds, following the learning process verifying (3.3). There are no international spillovers: externalities only operate at the domestic level. Each agent provides inelastically one unit of labor to firms per unit of time. At time 0, to maximize $U_1$, the agent in the rich country (agent 1) will choose the trajectories of two variables: his consumption $c_1$ and his foreign investment, which we call $i_{12}$. These choices will be made under the following constraints:

$$ \dot{a}_1 = a_1 r_1 + w_1 + b r_2 - i_{12} - c_1 \qquad (3.20) $$

$$ \dot{b} = i_{12} \qquad (3.21) $$

where $a_1$ is the stock of domestic assets of agent 1, $b$ his stock of foreign assets; $w_1$ is the wage in country 1, and $r_1$ and $r_2$ are respectively the rental rates paid by firms in country 1 and 2 to owners of capital.

At time 0, the poor country's representative agent has only one set of choices to make to maximize $U_2$: the trajectory of $c_2$. This choice is made under the following constraint:
$$ \dot{a}_2 = a_2 r_2 + w_2 - c_2 \qquad (3.22) $$

where $a_2$ is the stock of assets of agent 2.

This specification rules out foreign investment by country 2 in country 1, so that the only assets ever possessed by agent 2 are domestic<fnref n="12" />. Given initial condition (3.18), we claim that this implies no loss of generality: it simply means that we assume that returns to capital in country 2 will always be greater or equal to returns to capital in country 1. This can be shown to be true in the fuller model (where foreign investment by country 2 in country 1 is possible), because it cannot be optimal for agents to allow the rates of return to deviate from one another once they have been equalized. This argument may be thought of as a no-arbitrage condition.

For clarity, we write the current-value Hamiltonian of agent 1:

$$ H_1 = \frac{c_1^{1-\sigma}}{1-\sigma} + \lambda_1 [a_1 r_1 + w_1 + b r_2 - c_1 - i_{12}] + \gamma_1 [i_{12}] \quad (3.23) $$

The variables of this program are as follows: $a_1$, the stock of domestic assets and $b$, the stock of foreign assets are the two state variable, while $c_1$, consumption, and $i_{12}$, foreign investment, are two control variables.

The Hamiltonian of agent 2 is:

$$ H_2 = \frac{c_2^{1-\sigma}}{1-\sigma} + \lambda_2 [a_2 r_2 + w_2 - c_2] \quad (3.24) $$

Agent 2 has only one state variable, $a_2$ (the stock of assets) and one control, $c_2$ (consumption).

Profit-maximization and market clearing conditions are as follows:

$$ a_1 = K_1, a_2 + b = K_2 $$

$$ a_1 r_1 + w_1 = y(A(K_1)K_1), a_2 r_2 + b r_2 + w_2 = y(A(K_2)K_2) $$

$$ r_i = r(K_i) = A(K_i)y'(A(K_i)K_i) $$

Combining these equations with the first-order conditions for optimization, we obtain the full system of per capita equations, for all $t \ge 0$:

<footnote n="12">¹²This setting in terms of foreign investment, where the rich agent has the initiative of international capital flows, is equivalent to a setting in terms of lending/borrowing, where both agents can choose optimally lending and borrowing respectively, these two variables between equated ex-post. See d'Autune (1996).</footnote>
$$ \dot{K}_1 = y(A(K_1)K_1) + br(K_2) - c_1 - i_{12} \quad (3.25) $$

$$ \dot{b} = i_{12} \quad (3.26) $$

$$ \dot{K}_2 = y(A(K_2)K_2) - br(K_2) - c_2 + i_{12} \quad (3.27) $$

$$ r(K_1) = r(K_2) \quad (3.28) $$

$$ \frac{\dot{c}_1}{c_1} = \frac{\dot{c}_2}{c_2} = \frac{1}{\sigma} [r(K_2) - \rho] \quad (3.29) $$

$$ \lim_{t \to \infty} K_1 c_1^{-\sigma} e^{-\rho t} = 0 \quad (3.30) $$

$$ \lim_{t \to \infty} bc_1^{-\sigma} e^{-\rho t} = 0 \quad (3.31) $$

$$ \lim_{t \to \infty} K_2 c_2^{-\sigma} e^{-\rho t} = 0 \quad (3.32) $$

Equations (3.25-3.27) derive directly from the agents' constraints. It is already apparent that a key aspect of the model is the difference between GDP and GNP. In equation (3.25), $y(A(K_1)K_1) + br(K_2)$ is country 1's per capita GNP and $y(A(K_1)K_1)$ its per capital GDP. In equation (3.27), $y(A(K_2)K_2) - br(K_2)$ is country 2's per capita GNP and $y(A(K_2)K_2)$ its per capita GDP. Equation (3.28) derives from the optimization conditions. It states that usual consequence of perfect capital mobility: rates of return to capital are equal across countries at all times.

Equation (3.29) is telling us that both agents will adjust their consumption trajectory based on the same policy rule. This is because, first, $r(K_2)$ is the rate of return that applies to all new savings from both countries<fnref n="13" /> and, second, because both agents have the same preference for consumption smoothing. Equations (3.30-3.32) are the usual transversality conditions associated with the three state variables of the competitive maximizations. While (3.30) and (3.31) obtain directly, (3.32) comes from (3.31), (3.29), and the transversality condition on $a_2$.

<footnote n="13">¹³In this model $r(K_2)$ happens to be equal to $r(K_1)$ at all times. Note however that in two-country models with constraint of investment irreversibility, where there exists an initial adjustment phase during which rates of return are different across countries, agents will still base their intertemporal consumption choice on the highest rate of return alone (that is, the one offered by the poorer country). Again, this is because this rate of return is available for any new savings from both countries.</footnote>
### 3.4. Structure of the solution

Before beginning the resolution of the model, we establish some basic results on the general structure of the solution. In the absence of constraints on international capital flows<fnref n="14" /> condition (3.28) implies that the stocks of capital in the two economies will instantaneously adjust at time zero, through an inflow of capital into country 2. However, in contrast with the neoclassical model<fnref n="15" />, there are now two possibilities to achieve condition (3.28) (see Figure 3.1). We will refer to the two possibilities as Case 1 and Case 2 respectively:

* Case 1: $r(K_1(0)) = r(K_2(0))$ and $K_1(0) > K_2(0)$

* Case 2: $r(K_1(0)) = r(K_2(0))$ and $K_1(0) = K_2(0)^{16}$

Also, to facilitate our discussion, we define the two phases below:

* Phase 1, starting from time zero, corresponds to the period when $r(K_1(t)) = r(K_2(t))$ and $K_1(t) > K_2(t)$

* Phase 2 is defined as the period when $r(K_1(t)) = r(K_2(t))$ with $K_1(t) = K_2(t)$.

We study both Case 1 and Case 2 in the next section. If we are in Case 1, the economy starts in Phase 1. One critical result of the next section is that in this case, the economy will necessarily proceed to Phase 2. If we are in Case 2, the economy jumps directly to Phase 2. As will be detailed below both cases therefore turn out to yield similar results with respect to the growth outcome of the economy.

We define the date when the system shifts from Phase 1 to Phase 2 as T. Under Case 2, T is zero.

<footnote n="14">¹⁴A number of devices have been used to avoid an instantaneous massive migration of capital when economies open up. Barro, Sala-i-Martin and Mankiw (1995) rely on the assumption that only a portion of capital can be used as collateral for foreign finance. Blanchard and Fischer (1989) use adjustment costs. One could alternatively use irreversibility of investment (Vellutini (1997)). However, in order to keep the model tractable and to concentrate on our principal purpose, we do not use any of these constraints.</footnote>

<footnote n="15">¹⁵See for example Obsfeld and Rogoff (1996), section 3.6, or for a more detailed analysis Béraud (1996).</footnote>

<footnote n="16">¹⁶Note that "static" stability in each of the two cases above is ensured by the fact that at each moment each atomistic agent (in both countries) takes the level of productivity in each country, $A(K_i)$, as given and will thus ignore the impact of the changes in his own capital stock on productivity.</footnote>
## 4. The centralized model

It is possible to facilitate the analysis of the system by first isolating an autonomous sub-system of the model comprising the stocks of capital and the sum of the two national consumptions. We make use of the fact that the two consumption variables obey the same policy rule (3.29). With a constant ratio $c_1/c_2$, we see that the world consumption $c \equiv c_1 + c_2$ also follows the rule

$$ \forall t \ge 0, \frac{\dot{c}}{c} = \frac{1}{\sigma} [r(K_2) - \rho]. \quad (4.1) $$

Further, we will see in each of the following sub-sections that we can eliminate $b$ and $i_{12}$ from equations (3.25-3.27), ending up for each of the three phases of the model with an autonomous system in $c$, $K_1$ and $K_2$.<fnref n="17" />

d'Autume (1996) provides an interesting economic interpretation of this. Even though our problem is one with heterogeneous agents, this autonomous sub-system can be viewed as resulting from the problem of an hypothetical world central planner, whose objective would be to maximize $U = \int_0^\infty \frac{c(t)^{1-\sigma}}{1-\sigma} e^{-\rho t} dt$ under technological constraints, from which the variables $b$ and $i_{12}$ should be excluded, for they are not relevant to a world planner. It is by reference to this interpretation that we call the sub-system the "centralized" model.

The rest of this section focuses on solving the dynamic problem for the variables $c$, $K_1$ and $K_2$, while the next one will deal with the trajectory of $b$ and the sharing of $c$ between $c_1$ and $c_2$.

### 4.1. Phase 1

Phase 1 corresponds to Case 1 above. The technique we use to solve the system is to reduce the three-variable system into a two-variable system, using the fact that for all $t \ge 0$, we have $r(K_1(t)) = r(K_2(t))$ and $K_1(t) > K_2(t)$.

Define the real function $\mu(\cdot)$:

$$ \mu: \begin{array}{rcl} ]K^#, K^\S[ & \to & [K^\S, \infty[ \\ K_2 & \mapsto & \mu(K_2)/r(\mu(K_2)) = r(K_2) \end{array} $$

As by assumption $K^\S$ is the only minimum of $r(\cdot)$, $\mu(\cdot)$ is a well defined function on its domain, as illustrated by Figure 3.1. We give a representation of the numerical approximation of $\mu(\cdot)$ in Appendix 1. By definition of Phase 2, we have: $\forall t \in [0, T[, K_1 = \mu(K_2)$

**Proposition 4.1.** $\mu(\cdot)$ is continuous on $]K^#, K^\S[$ and differentiable on $]K^#, K^\S[$.

<footnote n="17">¹⁷See below for the explicit system in each phase.</footnote>
Proof: The proposition derives from the continuity and differentiability of $r(\cdot)$. $\mu(\cdot)$ may be viewed as the compounded function $\tilde{r}^{-1} \circ r(\cdot)$ where $\tilde{r}$ is defined as:

$$ \tilde{r}: [K^{\S}, \infty[ \to \mathbb{R}^{+} \\ K \mapsto \tilde{r}(K) = r(K) $$

Since a compounded function of continuous functions is also continuous, so is $\mu(\cdot)$ on $]K^{\S}, K^{\S}]$. However, because $\tilde{r}^{-1}$ is not differentiable at $K^{\S}$, $\mu(\cdot)$ is only differentiable on the open interval $]K^{\S}, K^{\S}[$. □

The expression of the first derivative of $\mu(\cdot)$ is found by differentiating the equation $r(\mu(K_2)) = r(K_2)$:

$$ \forall K_2 \in ]K^{\S}, K^{\S}[, \mu'(K_2) = \frac{r'(K_2)}{r'(\mu(K_2))} < 0 \quad (4.2) $$

Again, $K^{\S}$ being the only minimum of $r(\cdot)$, the sign of $\mu'(K_2)$ is always negative.

We now use $\mu(\cdot)$ to write the law of motion of $K_2$ (by (3.25) and (3.27)):

$$ \forall t \in [0, T[, \dot{K}_2(1 + \mu'(K_2)) = y(A(\mu(K_2))\mu(K_2)) + y(A(K_2)K_2) - c \quad (4.3) $$

Together with (4.1), this completes the system during Phase 1. This system is in $c$ and $K_2$ only. This system could equally be written in terms of $c$ and $K_1$, defining the inverse function $\mu^{-1}(\cdot)$ as:

$$ \mu^{-1}: [K^{\S}, \infty[ \to ]K^{\S}, K^{\S}[, \\ K_1 \mapsto \mu^{-1}(K_1)/r(\mu^{-1}(K_1)) = r(K_1) $$

We can then write the system in Phase 1 in terms of $K_1$ and $c$ (together with 4.1 and 3.28):

$$ \forall t \in [0, T[, \dot{K}_1(1 + \mu^{-1'}(K_1)) = y(A(K_1)K_1) + y(A(\mu^{-1}(K_1))\mu^{-1}(K_1)) - c \quad (4.4) $$

We stress that this equation is merely another form of (4.3). We use it to illustrate the dynamics of Phase 1 with the trajectories of both $K_1$ and $K_2$ (see Figure 4.2).

Before analyzing the dynamics of Phase 1, we need to further characterize the function $\mu(\cdot)$. We will restrict our attention to production functions that satisfy the condition:

$$ \forall K_2 \in ]K^{\S}, K^{\S}[, \mu'(K_2) = \frac{r'(K_2)}{r'(\mu(K_2))} < -1 \quad (4.5) $$

We need this condition to hold so that the quantity $(1 + \mu'(K_2))$ does not change sign. Such change in sign would be difficult to interpret as $\dot{K}_2$ would not
<figure><img src="image_4.png" /><figcaption>Figure 4.1: $r'(\mu(K_2))$ as a function of $r'(K_2)$</figcaption></figure>

be defined at $(1 + \mu'(K_2)) = 0$. This condition does not appear to be particularly severe, as can be observed from Figure 3.1, because on the one hand returns will tend to infinity when the capital stock tends to zero – with vertical slope at the limit –, and will tend to a constant return when the capital stock tends to infinity – with a horizontal slope at the limit. Although the condition can be stated simply, as it is in (4.5), actually verifying it for forms such as (3.2) and (3.6) is difficult other than by direct numerical evaluation. This is because for each given interest rate $\bar{r}$, there is no closed form for the two solutions to the equation in $K_i$: $\bar{r} = r(K_i)$, and consequently evaluating algebraically $\frac{r'(K_2)}{r'(\mu(K_2))}$ is cumbersome. Figure 4.1 illustrates the behavior of the ratio $\frac{r'(K_2)}{r'(\mu(K_2))}$, as evaluated numerically over the interval $]K^#, K^\S[$. We also represented the function $y = -x$ to illustrate that Condition (4.5) is satisfied in our benchmark numerical economy<fnref n="18" />.

**Proposition 4.2.** $\lim_{K \to K^\S} \mu'(K) = -1$

*Proof:* We first show that such limit exists. Notice that condition (4.5), which can be written as $\forall K_2 \in ]K^#, K^\S[, |r'(K_2)| > r'(\mu(K_2))$, together with the con-

<footnote n="18">¹⁸See Appendix 2.</footnote>
tinuity of $\mu(\cdot)$ in $K^\S$, which gives $\mu(K^\S) = K^\S$, that is, $|r'(K^\S)| = r'(\mu(K^\S))$, jointly imply that there exists some interval $]\epsilon, K^\S[$ with $\epsilon \in ]K^\# , K^\S[$ such that the ratio $\frac{r'(K)}{r'(\mu(K))} = \mu'(K)$ is increasing for $K \in ]\epsilon, K^\S[$. As $\mu'(K)$ is bounded on $]\epsilon, K^\S[$ by condition (4.5), we conclude that $\lim_{K \to K^\S} \mu'(K)$ exists and is strictly negative.

Second, we compute the limit by using L'Hôpital's rule.<fnref n="19" />. If $\lim_{K \to K^\S} \mu'(K) = \lim_{K \to K^\S} \frac{r'(K)}{r'(\mu(K))}$ exists then $\lim_{K \to K^\S} \frac{r'(\mu(K))}{r'(K)}$ exists. But $\frac{r'(\mu(K))}{r'(K)} = \frac{\frac{\partial r'(K)}{\partial K}}{\frac{\partial r'(\mu(K))}{\partial K}}$. We can thus safely use L'Hôpital's rule: $\lim_{K \to K^\S} \frac{r'(K)}{r'(\mu(K))} = \lim_{K \to K^\S} \frac{r'(\mu(K))}{r'(K)} \implies [\lim_{K \to K^\S} \frac{r'(K)}{r'(\mu(K))}]^2 = 1$. We conclude $\lim_{K \to K^\S} \mu'(K) = -1$. □

This proposition, although it may look rather technical, will be important in determining the slope of the locus $\dot{K}_2 = 0$ in the neighborhood of $K^\S$ (see equation (4.7)). We now turn our attention to the locus $\dot{K}_2 = 0$. Given condition (4.5), the equation of the locus in the $(K_2, c)$ space is as follows:

$$c^{\text{iso}K_2} = y(A(\mu(K_2))\mu(K_2)) + y(A(K_2)K_2) \quad (4.6)$$

This function is complex and we restrict our analysis to the minimum properties that enable us to carry out the analysis of the dynamic system in Phase 1:

**Proposition 4.3.** $c^{\text{iso}K_2}$ is a continuous and differentiable function of $K_2$ over $]K^\#, K^\S[$.

*Proof:* $c^{\text{iso}K_2}$ is a compounded function of continuous and differentiable functions $K_2$ over $]K^\#, K^\S[$. □

**Proposition 4.4.** $\lim_{K_2 \to K^\#} c^{\text{iso}K_2} = \infty$.

*Proof:* $y(A(K^\#)K^\#)$ is finite and $\mu(K_2)$ tends to infinity as $K_2$ tends to $K^\#$. □

**Proposition 4.5.** $\lim_{K_2 \to K^\S} c^{\text{iso}K_2} = 2y(A(K^\S)K^\S)$.

<footnote n="19">¹⁹We use the following form of L'Hôpital's rule:</footnote>

<footnote>If $f(\cdot)$ and $g(\cdot)$ are continuous on $]a, b]$, and differentiable on $]a, b[$ with $f(b) = g(b) = 0$ then: $\lim_{x \to b-} \frac{f'(x)}{g'(x)}$ exists $\implies \lim_{x \to b-} \frac{f(x)}{g(x)} = \lim_{x \to b-} \frac{f'(x)}{g'(x)}$</footnote>
<figure><img src="image_5.png" /><figcaption>Figure 4.2: Dynamics of Phase 1</figcaption></figure>

Proof: Follows from $\lim_{K_2 \to K^\S} \mu(K_2) = K^\S$. □

The slope of $c^{isoK_2}$ is given by:

$$ \frac{\partial c^{isoK_2}}{\partial K_2} = f'(K_2) + \mu'(K_2)f'(\mu(K_2)) \quad (4.7) $$

Given the presence of externalities, the derivative of the production function with respect to the stock of capital is of course not equal to the competitive interest rate. The study of the sign of this quantity is difficult, but fortunately we do not need it to be always negative over $]K^\S, K^\S[$ to obtain our results on the dynamics of the system. In our numerical simulation (Appendix 1), we found that $\frac{\partial c^{isoK_2}}{\partial K_2}$ is actually negative over $]K^\S, K^\S[$. But, again, the above propositions are sufficient. Finally, the system in Phase 2 may be represented as in Figure 4.2.

**Proposition 4.6.** *Phase 1 leads to Phase 2 in finite time.*

Proof: Given that closed orbits may not occur, it is sufficient to prove that neither explosive path A nor spiraling path B is feasible. Note that Figure 4.2 represents only the most direct trajectory leading to Phase 2. More complex trajectories are possible with one or several circles around the fixed point $(K^*, c^*)$ before the vertical axis in $K^\S$ is eventually hit.

Let us first consider Path A. It is noticed that Path A cannot hit the vertical axis in finite time because given that $r(K_1(t)) = r(K_2(t))$, the variable $K_2$ cannot reach the value $K^\S$ in finite time, because that would imply $K_1 = \infty$ would be reached in finite time, which is impossible. Therefore, Path A can only tend
tangentially to the vertical axis. We next rule out Path A by considering the "mirror" Path A' in the system $(K_1, c)$. To see this, first note that the slope of the locus $\dot{K}_1 = 0$ tends to the finite positive quantity $aD$ as $K_1$ tends to infinity.

20 This implies that as $K_2$ tends to $K^\#$ and $c$ tends to infinity along path A, $\dot{K}_1$ must become negative in finite time and remains negative thereafter, which violates feasibility.

Path B is ruled out by studying the linearized system in the neighborhood of $(K^*, c^*)$:

$$ \begin{bmatrix} \dot{c} \\ \dot{K}_2 \end{bmatrix} \approx \begin{bmatrix} 0 & \frac{c^* f''(K^*)}{\mu'(K^*)} \\ -\frac{1}{\mu'(K^*)+1} & \frac{\mu'(K^*) f'(\mu(K^*)) + f''(K^*)}{\mu'(K^*)+1} \end{bmatrix} \begin{bmatrix} c - c^* \\ K - K^* \end{bmatrix} $$

The trace and the determinant of the Jacobian being both positive, the eigenvalues are either real and positive or complex with positive real parts. The system is therefore locally unstable at $(K^*, c^*)$ and Path B is not feasible.

Given the properties of the locus $\dot{K}_2 = 0$ (see propositions 4.4-4.5), $K_2$ (together with the "mirror" variable $K_1$) has to hit the vertical segment in $K^\S$ in finite time. □

This proposition is telling us that the system will necessarily shift from Phase 1 into Phase 2. It is crucial, however, to note that the equations of the system in Phase 1 ((4.1) and (4.3)) provide no information on the value of the variable $c$ when the system shifts to Phase 2. In particular, we can not infer from $\lim_{K \to K^\S} \mu'(K) = -1$ that the quantity $\dot{K}_2(1 + \mu'(K_2))$ tends to zero as $K$ tends to $K^\S$, because $\lim_{K \to K^\S} \dot{K}_2$ is indeterminate.<fnref n="21" />

## 4.2. Phase 2

We have shown that under Case 1 (see page 15), the system will go through Phase 1 and will necessarily enter Phase 2 with the capital stock in each country at $K^\S$. Under Case 2, the system will jump directly to Phase 2; with condition 3.19, the capital stock in each country is above $K^*$ at the beginning of Phase 2.

The system is Phase 2 can be re-written using the relationship: $\forall t \in [T, \infty[, K_1(t) = K_2(t)$. Through (3.25) and (3.27), the law of motion of $K_2$ becomes:

<footnote n="20">²⁰Noting $c^{isoK_1}(K_1)$ this locus, we have $\frac{\partial c^{isoK_1}}{\partial K_1} = [A(K_1) + A'(K_1)K_1]y'(A(K_1)K_1) + \mu^{-1'}(K_1)[A'(\mu^{-1}(K_1))\mu^{-1}(K_1) + A(\mu^{-1}(K_1))]y'(A(\mu^{-1}(K_1))\mu^{-1}(K_1)$</footnote>

<footnote>As $\lim_{K_1 \to \infty} \mu^{-1}(K_1) = K^\#$ and $\lim_{K_1 \to \infty} \mu^{-1'}(K_1) = 0$, we have $\lim_{K_1 \to \infty} \frac{\partial c^{isoK_1}}{\partial K_1} = aD$.</footnote>

<footnote n="21">²¹The non-differentiability with respect to time of the two capital stock variables at the point where they are equalized is also a feature of the model with neoclassical technology (Vellutini (1997)).</footnote>
$$ \forall t \in [T, \infty[, \dot{K}_2 = y(A(K_2)K_2) - c/2 \qquad (4.8) $$

Together with this relationship, equation (4.1) and terminal condition (3.32) complete the system in Phase 2. Again, this equation could equally be written in terms of $K_1$ and $c$.

With the two states variables tied to one another and using the world consumption as a control variable, the system looks very much like autarchy. We will now compare the behavior the autarchy economy with our economy with capital mobility. To this end, we write the autarchy system:

$$ \left\{ \begin{array}{l} \dot{c}_i = \frac{1}{\sigma} [r(K_i) - \rho] \\ \dot{K}_i = y(A(K_i)K_i) - c_i \\ \lim_{t \to \infty} K_i c_i^{-\sigma} e^{-\rho t} = 0 \end{array} \right. \qquad (4.9) $$

It is apparent from this set of equations and our system in Phase 2, together with the definition of $c$ ($c = c_1 + c_2$), that the phase diagram we will use to study the dynamics of Phase 2 is identical to the phase diagram of the autarchy economy, substituting $c_i$ with $c$.

Remember that we have chosen to concentrate on autarchy economies which display development thresholds (page 11). The characterization of such autarchy development thresholds using the phase diagrams is detailed in Appendix 2. We have represented in Figure 4.3 the trajectory of the unstable manifold associated with the saddle point $(K^*, c^*)$ which is characteristic of the autarchy development threshold: it either grows for ever or hits the horizontal axis in finite time (see Appendix 2). The main point to note for our purpose is that what was a situation of development threshold in autarchy has become with capital mobility a case of multiple equilibria. This can be seen directly from the phase diagram: in autarchy, for low initial levels of capital ($K_0 < K^*$), the only feasible competitive equilibrium path leads to stagnation at fixed point $(K^*, c^*)$. This is because trajectories can never cross and because trajectories below the unstable arm associated with $(K^*, c^*)$ cannot satisfy the transversality condition.

With capital mobility, the level of capital at the beginning of Phase 2 is above $K^*$. Under Case 1, it is equal to $K^*$. Still using the phase diagram, we will now show that, depending on the value of $c(T)$, the system will therefore either converge to fixed point $(K^*, c^*)$ or take off into perpetual growth.

**Proposition 4.7.** One of the two following cases will occur:

Case A) The economy will converge to the fixed point $(K^*, c^*)$, with $\lim_{t \to \infty} c(t) = c^*$ and $\lim_{t \to \infty} K_2(t) = K^*$ (Path A in Figure 4.3)
<figure><img src="image_6.png" /><figcaption>Figure 4.3: Dynamics of Phase 2</figcaption></figure>

Case B) The economy will grow perpetually, with $\lim_{t \to \infty} \frac{\dot{c}(t)}{c(t)} = \lim_{t \to \infty} \frac{\dot{K}_2(t)}{K_2(t)} = \frac{aD-\rho}{\sigma}$ and $\lim_{t \to \infty} \frac{c(t)}{K_2(t)} = \frac{aD(\sigma-1)+\rho}{2\sigma}$ (Path B in Figure 4.3). This sustained growth path is unique.

Proof<fnref n="22" />:

First, we show the candidate Path A satisfies (4.8), (4.1) and (3.32). The Phase diagram shows that (4.8) and (4.1) are satisfied. Transversality condition (3.32) is also obviously satisfied, as both $K_2$ and its co-state variable $\lambda_2$ will tend to constants.

Second, we show that these conditions are also satisfied for the candidate Paths of case B) above. Again, refer to the phase diagram to see that (4.8) and (4.1) are satisfied. Next, we verify that (4.8) and (4.1) are compatible with characterization of the Paths of case B) above. From (4.1) and (3.9), we have $g = \lim_{t \to \infty} \frac{\dot{c}(t)}{c(t)} = \lim_{t \to \infty} \frac{1}{\sigma} [r(K_2) - \rho] = \frac{aD-\rho}{\sigma}$. We have used the fact that along this family of paths we have $\lim_{t \to \infty} K_2 = \infty$. From (4.8): $g = \lim_{t \to \infty} \frac{\dot{K}_2(t)}{K_2(t)} = \lim_{t \to \infty} \frac{f(K(t))-c(t)/2}{K(t)} = aD - \lim_{t \to \infty} \frac{c(t)}{K(t)}$, so that $\lim_{t \to \infty} \frac{c(t)}{K(t)} = \frac{aD(\sigma-1)+\rho}{2\sigma}$. To show that the considered paths satisfy the transversality condition, note that $\lim_{t \to \infty} c(t) = \lim_{t \to \infty} \Gamma e^{\frac{aD-\rho}{\sigma}t}$, where $\Gamma$ is a constant. The transversality condition is therefore satisfied, because:

$$ \lim_{t \to \infty} c^{-\sigma} K e^{-\rho t} = \lim_{t \to \infty} c^{1-\sigma} \frac{K(t)}{c(t)} e^{-\rho t} = \left[ \frac{2\sigma}{aD(\sigma-1)+\rho} \right] \Gamma^{(1-\sigma)} \lim_{t \to \infty} e^{(-\rho + \frac{aD-\rho}{\sigma}(1-\sigma))t} = 0 \text{ since } -\rho + \frac{(aD-\rho)(1-\sigma)}{\sigma} = \frac{aD(1-\sigma)-\rho}{\sigma} < 0. $$

To show the uniqueness of the perpetual growth path, it is noticed that two

<footnote n="22">²²This proof draws on Zilibotti (1996), Proposition 3.</footnote>
arbitrarily close trajectories in quadrant VI of Figure 4.3 are necessarily divergent.
To see this, for any $K_{##} > K^{**}$ define $c_{##}^{growth}$ as the corresponding consumption
level leading to sustained growth. If we choose two alternative consumption levels
$c'_{##}$ and $c''_{##}$ such that $c'_{##} < c_{##}^{growth} < c''_{##}$, then we have from (4.1) and (4.8)
(denoting the time derivatives of the corresponding trajectories with the same
superindices):

$\dot{c}'_{##} > \dot{c}_{##}^{growth} > \dot{c}''_{##}$ and $\dot{K}'_{##} < \dot{K}_{##}^{growth} < \dot{K}''_{##}$. We conclude that the three trajectories would be divergent and that therefore only the trajectory through point $(K_{##}, c_{##}^{growth})$ leads to the competitive growth path. From this we conclude that there is only one point in the $(K, c)$ space at capital level $K^{**}$ that leads to the competitive growth path, and therefore that there is only one competitive growth path.

Third, we rule out diverging paths C and D (see Figure 4.3). As in the proof
for Phase 1, we derive (4.8) with respect to time, and obtain $\dot{K}_2 = \dot{K}_2 f'(K_2) - \dot{c}$,
which becomes negative along Path C. Again, this trajectory will thus hit the
c-axis in finite time, violating feasibility. We rule out path D by showing that it
violates the transversality condition. To see this, note that along this path we have
$\lim_{t \to \infty} \frac{c(t)}{K(t)} = 0$, which implies $\lim_{t \to \infty} \frac{\dot{K}(t)}{\dot{K}(t)} = aD$. We still have $\lim_{t \to \infty} \frac{\dot{c}(t)}{c(t)} = \frac{aD-\rho}{\sigma}$. Using these limits, the transversality limit $\lim_{t \to \infty} c^{-\sigma}Ke^{-\rho t}$ turns out to be the constant $\xi\Gamma^{-\sigma} \neq 0$, where $\xi$ is a constant defined by the relationship
$\lim_{t \to \infty} \frac{\dot{K}(t)}{K(t)} = \xi e^{aDt}$. $\square$

Again, as in Phase 1, we stress that Figure 4.3 gives only the most direct tra-
jectories leading to stagnation or growth. More complex trajectories are possible
with one or several circles around the fixed point $(K^{**}, c^{**})$.

4.3. Global dynamics

The above results have enabled us to describe the global dynamics of the system.
First, Starting from an autarchy with threshold effects, capital mobility gives way
to a situation of multiple equilibria. The initial level of world consumption $c(0)$
will be chosen in function of expectations leading to either stagnation or sustained
growth. This is true for both Case 1 and Case 2, in other words, whether the
economy jumps directly to Phase 2 (with capital at the beginning of phase 2
above $K^*$) or first goes through Phase 1 (with capital at the beginning of phase 2
equal to $K^*$). However, it is important to stress that this result depends critically
on initial conditions (3.17-3.19). In particular, in this model capital mobility will
be of no avail in suppressing threshold effects if the poor country has reached its
stagnant steady state when capital liberalization begins, for returns to capital will
then be lower than in the country with sustained growth.

Second, the expectations determining $c(0)$ and therefore the final outcome —
perpetual growth or stagnation — are formed at the world level: the two countries will stagnate or grow together. Since we have dealt with per capita quantities all along, the case for the larger country leading world expectations appears to be a realistic one. If we consider that the labor force of the poorer economy is significantly smaller than the one in the rich country, then the case of the former being pulled out of stagnation by foreign finance – a case of contagious optimism – looks stronger.

## 5. Foreign assets and national consumptions

We now turn to the trajectories over time of $b$ (the stock of foreign assets) and the two national consumptions $c_1$ and $c_2$. Define the constant $m \equiv \frac{c_1(t)}{c(t)}$, $\forall t$. Following d'Autume (1996), we will use agent 1's intertemporal budgetary constraint to calculate $m$ which will then determine the respective national consumptions $c_1$ and $c_2$. From (3.25) and (3.26), we have:

$$ \forall t \ge 0, \dot{b} = y(A(K_1)K_1) + br(K_2) - \dot{K}_1 - mc \quad (5.1) $$

This equation simply tells us that foreign assets will accumulate as a difference between net revenue in country 1 (after allowing for domestic investment) and domestic consumption. Using this relationship as a differential equation in $b(t)$, we obtain the following expression for $b$:

$$ b(t) = \int_{0}^{\infty} \exp \left[ - \int_{0}^{v} r(K_{2}(s))ds \right] [mc - y(A(K_{1})K_{1}) - \dot{K}_{1}] dv \quad (5.2) $$

Using the initial condition $b(0) = 0$, we then can compute the value of the constant $m$:

$$ m \int_{0}^{\infty} \exp \left[ - \int_{0}^{v} r(K_{2}(s))ds \right] c(v)dv = \int_{0}^{\infty} \exp \left[ - \int_{0}^{v} r(K_{2}(s))ds \right] [y(A(K_{1})K_{1}) - \dot{K}_{1}] dv \quad (5.3) $$

This relationship does enable us to calculate (numerically) $m$, since the trajectories of all the time-variant quantities ($K_1, K_2, c$) are known to us. From the definition of $m$ and the knowledge of the trajectory of $c$, we obtain $c_1$ and $c_2$. The trajectory of $b$ is obtained by (5.2). The usefulness of these equations, which are exactly identical to their counterparts in the model with neoclassical technology (see Vellutini (1997)), would be most visible through the numerical calculation of the trajectory of $b$, a task that we have left to further research since this calculation is not central to the purpose of the analysis we present in this note.
# 6. Conclusions

In this model, competitive capital inflows may cause a country in an early stage of development to be pushed into sustained growth, even though in autarchy that country would have been trapped in stagnation. The self-fulfilling prophecies of the open multiple-equilibria economy are jointly formed at the world level, which gives rise to the possibility of contagious optimism, a case that looks stronger than contagious pessimism if the rich country (with autarchy growth) has a significantly larger population.

Further, because of the crucial role of expectations, the model suggests that inflows of private capital may be more important for growth than the same quantities of inflows of public capital, for example from bilateral or multilateral donors. This is because private, competitive capital flows may act as a channel through which optimism may be propagated to domestic savers and other international investors. In contrast, foreign public finance, not being based – or believed to be based – on utility maximization and rational expectations, is not likely to influence these expectations.

This is a sharp departure from the findings of the standard models of endogenous growth – in which there are simply no incentives for private capital to flow from rich to poor countries. But even in this model, if a country has already reached a stagnant steady state, they will be no incentives for foreign private capital to flow in, and liberalization of the capital account will be of no use in pulling that country out of stagnation. In that case, which boils down to autarchy stagnation, the distinctive role of public investment (domestic or foreign) is more apparent.
## References

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d'Autume, Antoine (1996). "Note sur les modèles à deux pays". Mimeo, MAD, Université de Paris 1 Panthéon-Sorbonne.

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<figure><img src="image_7.png" /><figcaption>Figure 6.1: Competitive interest rate (numerical calculation)</figcaption></figure>

Appendix 1: Numerical calculations

To verify numerically certain assumptions of the model and as a first step
to a full numerical simulation, the following benchmark values for supply-side
equations 3.2 and 3.7 and preferences 3.10 were used:

$$
a = 0.075; A_0 = 0.015; q = 0.01; D = 1; \theta = 0.2; Z = 5; \rho = 0.05; \sigma = 1.
$$

These values are derived from the values used in Zilibotti (1995). As in that
paper, they are "somewhat arbitrary" and are not the result of a full calibration
exercise. Figure 6.1 illustrates the supply-side of our numerical economy. We also
represent a numerical evaluation of $\mu(K_2)$.
<figure><img src="image_8.png" /><figcaption>Figure 6.2: The Function $\mu(K_2)$</figcaption></figure>

## Appendix 2: Classification of autarchy economies

Following Zilibotti (1995), the behavior of the unstable manifold associated with the fixed point ($K^*, c^*$) is used as a criterion to classify autarchy economies into (i) economies with threshold effects; (ii) multiple equilibria economies. It is noticed that the unstable manifold can either grow for ever or hit one of the axes in finite time. The trajectory cannot connect the saddle point with itself because we have ruled out closed orbits. When the unstable manifold hits the vertical axis, we have an economy with multiple equilibria. When it grows for ever or hits the horizontal axis, we have a case of development threshold. To see this, observe first that the unstable arm of the saddle point divides the plane into two regions corresponding to the basin of repulsion of the fixed point ($K^{**}, c^{**}$) and the basin of repulsion of the origin. This is because the system has three fixed points (the origin, ($K^*, c^*$) and ($K^{**}, c^{**}$)) and that both the origin and ($K^{**}, c^{**}$) are locally unstable, while ($K^*, c^*$) is a saddle point whose basin of repulsion is no other than the unstable arm associated to it. In this analysis it is of course crucial to remember that trajectories cannot cross. We then use the phase diagrams below to illustrate the two possible types of economy.

1) The threshold economy. As illustrated by Figure 6.3, we have two sub-cases: if the unstable arm hits the horizontal axis, it is straightforward that there
<figure><img src="image_9.png" /><figcaption>Figure 6.3: Threshold autarchy economy</figcaption></figure>

exists a set of initial conditions ($K_0 < K^*$) such that the only feasible competitive
equilibrium path leads to the saddle point. If the unstable arm grows for ever,
then sustained growth is equally impossible for low level of initial capital, because
all trajectories below the unstable arm (see proof of proposition 4.7) violate the
transversality condition. It is interesting to note that in this case also we have
the issue or multiple equilibria for initial value of capital above $K^*$. It is precisely
this case that turns out to be relevant for our two-country economy with capital
mobility.

2) The multiple equilibria economy. As illustrated in Figure 6.4, when the unstable arm hits the vertical axis the basin of repulsion of the fixed point ($K^{**}$, $c^{**}$) is an area bounded by the unstable manifold and a sub-set of the vertical axis. As trajectories do not cross, this implies that paths leading to sustained growth must "come" from the origin<sup>23</sup>. In other words, we have a case of multiple equilibria for arbitrary small initial levels of capital.

<footnote n="23"><sup>23</sup>See Zilibotti (1995) for a proof that there exists a competitive equilibrium trajectory leading to sustained growth.</footnote>
<figure><img src="image_10.png" /><figcaption>Figure 6.4: Multiple equilibria autarchy economy</figcaption></figure>