Foreign investment and convergence

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FUNDAÇÃO GETÚLIO VARGAS

FOREIGN INVESTMENT AND CONVERGENCE

DISSERTAÇÃO SUBMETIDA A CONGREGAÇÃO DA ESCOLA DE PÓS-GRADUAÇÃO EM ECONOMIA (EPGE) PARA OBTENÇÃO DO GRAU DE

MESTRE EM ECONOMIA

POR

CHARLESVELLUTINI

RIO DE JANEIRO, RJ Agosto, 1995

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Acknowledgments

I would like to express my gratitude to my teachers at EPGElFGV. In particular, Professor Renato Fragelli Cardoso assisted me throughout the development of the model and contributed greatly to its main results. The idea of this research took shape during a course on Growth Theory by Professor Pedro Calvacanti Ferreira and Professor Afonso Arinos de Mello Franco, who both provided me with very helpful insights and suggestions. AlI remaining errors are my own.

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Abstract:

In a model of two open economies with intertemporaloptimization, we characterize

optimal paths toward convergence

anti

show that the richer country achieves higher

utility than it would in autarchy, while the poorer country's convergence toward the

steady state is speeded up. However, the short-term effects offree trade andfree

capital flows on the richer economy are negative in terms of wages

anti

consumption.

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1. Introduction

The attitude of the govemments of both developed and developing countries towards foreign investment has evolved markedly over the last two decades. While some developing countries favored foreign investment early on, a good number of others, including such giants as China and India and most African and South American countries, restricted it in one way or another. Behind such restrictions often lay the fear that intemational investors' interests may not be entirely compatible with the host country's own. There is today a strong consensus and ample evidence that foreign investment, far from endangering LDC development, is a net contributor to it. Most major intemational development agencies devote a fair share of their resources to the establishment of foreign investment promotion schemes. We witness an increasing competition between LDC's to attract foreign investment from industrialized countries. Indeed, foreign investment in the poorer countries is gaining momentum. For example, as shown in Figure 1, foreign direct investment, a particular form of foreign investment that allows investors to take an active part in the management of the recipient companies, appears to be on an upward, accelerating trend.

To some extent, the poor countries' suspicions have given way to the rich countries' concem that net export of capital associated with free trade might cause trouble at home, particularly in the labor market. Rising unemployment in Europe has fueled a vigorous public debate on whether investment in low-wage countries and free trade may be detrimental to jobs and wages1.

180 180 140 120 100 80 80 40 20

Innows of foreign direct investrnent (source: lhe Economist, UNClAD)

C Rich industrial countries [J Developing countries

O~--~--~~--~--~~--~--~~--~~~~~--~~--~--~

1988 1989 1990 1991 1992 1993

Figure 1

1 See for example the series of articles published in 1995 by Maurice AlIais in Le Figaro, where the French Nobel Laureate calls for higher import tariffs on the grounds that free trade with LDC's is a chief cause of unemployment in the European Union.

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The objective of this dissertation is to explore the short-term and long-term effects of trade and capital market liberalization within a neo-classical, two-country model with intertemporal optimization. Since this study is concemed with the transition dynamics of liberalization, the model will present an additional feature which we hope will make it more realistic: investment will be irreversible in the sense that installed capital will not be moved abroad, nor consumed at home. As it slows down the convergence process -- thus avoiding a counterfactual instantaneous massive transfer of capital - it is expected that this characteristic, when combined with technological change, will be important for some predictions ofthe model, particularly concerning unemployment.

2. Literature

While foreign investment is a fact of business life, as well as a major preoccupation of professional development economists, the dynamics of its two-way capital flows, especially in the short run, have received relatively little attention from economic theory. To our knowledge, the bulk of the literature on intemational capital flows has focused on the following approaches: (i) Traditional trade theory concentrates on flows in static settings. For example, the model developed by Brewer (1985) accounts for capital flows but does not allow for a dynamic treatment. Some models, such as Hori and Stein (1977), include some dynamics, but without intertemporal optimization ofutility (savings rates are regarded as exogenous). (ii) In other, more recent, models of the open economy, for example in Blanchard and Fischer (1989, chapter 2), intertemporal optimization does take place, but only in the home country, which is considered small relative to the rest ofthe world. Grossman and Helpman (1991) also belongs to this category. (iii) More recently still, the growth literature has taken a renewed interest in open economies and has produced impressive results on how interactions between countries of comparable sizes may influence long-run growth rates, i.e., in steady state. For example, Romer and Rivera-Batiz (1991) deals with the impact of the integration of two equally sized countries and shows that integration, if it increases the world-wide exploitation of increasing retums to scale in the R&D sector, tends to augment the rate of growth. Rivera-Batiz and Xie (1993) addresses the question of the effects of integration on growth in the case of countries with different (but comparable) sizes. Grossman and Helpman (1990) shows how cross-country differences in efliciency at R&D versus manufacturing may alter long-run rates of growth. In this direction of research, the transitional dynamics are not the main concem ofthe authors, who rather concentrate on long-run effects.

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Lastly, Baumol and Wolff (1986) and Barro and Sala-i-Martin (1992), among others, analyzed convergence between countries with a neo-classical production function. Here, convergence is dynamically explained by the decreasing retums displayed by the neo-classical economy. However, such convergence is analyzed in closed-economy models and takes place simply because countries share the same technology and preferences and thus ali evolve toward the same steady state, albeit separately. Barro, Mankiw and Sala-i-Martin (1995) captures the accelerating effect of international capital flows but uses a model where the home economy is small relative to the rest of the world. In this study, the neo-classical model is used in an open context with two economies that are large relative to one another.

In addition, in the above models, installed capital stock IS considered entirely

exportable.

3. The model

We will consider a one-good economy with two countries, both with a neo-classical production function and a utility function exhibiting a constant elasticity of substitution. Free trade of goods is introduced at time t = O (in this simple one-good economy, this also implies free flows of capital). Labor is immobile. Country 1 is in steady state when liberalization begins, while Country 2, being poorer in capital per capita and sharing Country 1 's technology and preferences (including a depreciation rate Ô and an impatience factor p), is below the steady state leveI. We thus take the simplifying view that underdevelopment is due to a lag in the growth paths of LDC's and the rich world. Country 2 would eventually reach the steady state, following the same optimal path that was previously chosen by Country 1, without any interaction with the latter.

We introduce a neo-classical production function with exogenous technological change in the form of a labor-augmenting factor, as follows:

Y = F(K. T

.ext )

i = 1 2 2

I "~" (1)

where

1;

is the aggregate flow of output,

K;

the aggregate stock of capital being used in a given country, ~ the labor force in that country, and x a constant positive term

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representing the exogenous technological change. We assume that the production function F(.) is strictly concave in each factor, twice differentiable, exhibits unbounded partial derivatives at the boundary, and that both factors are needed in the production process, so that we have for each K; > O and each L; > O:

We further assume that labor forces are equal in each country and grow at the constant exponential rate n, setting:

Li

==

L

=

4enl,i

= 1,2.

The consumers' instantaneous felicity function is:

l-a

C·

U(Cj )

= -'-

with O' > O,

1-0'

which exhibits the constant elasticity of substitution 1/ (T. In this definition ofutility, it is convenient to think: of each agent as a family that would tive indefinitely, or,

borrowing from the vocabulary of growth theory, as an infinite dynasty.

As in Barro and Sala-i-Martin (1992) we assume p>

n

+

(1-

a)x

to warrant stability.

Instead of assuming central planning, as is often done in the growth titerature, we

will

solve by considering the decentralized problem. Assuming central planning in the rich economy would result in different investment paths, because a central planner would take into account the impact of foreign investment on the rate of retum of the poorer economy. No such thing will happen if the economy is composed of a large number of maximizing agents, each considering himself small. Since we are primarily concemed with market economies, we naturally give priority to the decentralized solution.

We thus consider a consumer in each country, whom we assume to be the owner of a small fraction of the capital stock. This consumer will also inelastically supply firms, which operate the capital stock, with one instantaneous unit of labor. The general method we use is in two steps: we first solve the agents' maximization problems and next aggregate the resulting first-order conditions for each country. This method is equivalent to assuming that the consumer considers himself small and therefore takes the remuneration of labor and capital as given when making his consumption and

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investment decisions. We wiIl assume, ex-post, that ali consumers are identical in each economy.

The utility of each agent is defined as the discounted flow of instantaneous felicity:

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where êj (I) is the respective amount of consumption of each agent at time I.

At time zero, the two agents are endowed with different stocks of assets, and will solve:

max

Uj

with

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where a} and a] respectively denote the effective stock of domeslic assets of agent 1 and 2, and k· is the effective capital stock in the autarchy steady state given by j'(k")=n+ô+p+ax.3 _{We will assume that}_{a} and a] do not take negative values,}

thus ruling out credit.

Note that since we have introduced labor-augmenting exogenous technological change, we will often refer to effeclive variables. We denote the effective value of any aggregate variable V(I) as v == V(I) / L(I)ext and its per capita value as

v

== V(I) / L(I).

Note also that K1 and K2 are the domestic stocks of capital, regardless of the nationality of its owner. In rich Country 1, ali capital will always be owned by the residents. In Country 2, however, the total capital stock being used (K2) wiIl be the sum of the capital owned by nationals and the capital owned by Country 1. We define B as the stock of assets held by agent 1 in Country 2. Since no credit is allowed in our economy, B is always positive.

As we argued above, each agent will take prices as given, including foreign rates of returno This means that Country l's agent wiIl not consider the impact of its decisions

3 See for example Blanchard and Fischer for a detailed resolution of the model in its closed~nomy version.

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on k_2,and vice-versa. Defining i₁₂as agent 1 's investment in Country 2, rI and r₂as the

rates ofreturn in each country, w_Iand w₂as real wages, and denoting the production function in intensive form as/(.), the constraints of agent 1 's problem are:

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b

= i_l2- (n + Ô + x)b (5)

Note that in our formulation b is not automatically accrued by its remuneration br_{2 ,}as this is subject to an explicit decision of reinvestment.

Similarly, the constraint faced by agent 2 is:

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We stress that, for the sake of simplicity, we have ruled out the possibility of agent 2 investing in Country 1. It is straightforward to show, by simply constructing the above restrictions in a strictly symmetrical manner, that, given initial conditions (3), optimization requires that agent 2 never invest in Country 1.

Further, we wiU impose the foUowing upper bound on i_{12 :}

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This inequality states that agent 1 cannot use its instaUed stock of capital aI for investment in Country 2.4 _{Foreign investment is limited to agent l's available income.}

In practical terms, as in Lucas (1990), the assumption of irreversible investment rules out any instantaneous re-Iocalization of industries. This of course does not mean that re-Iocalization will not take place, but it will do so only as capital instaUed in the rich country decays. On the other hand, our hypothesis reflects the fact that such capital goods as infrastructure (for example: transport and telecommunication networks) are not likely to be moved at an acceptable cost. Overall, given the costs also implied in the re-Iocalization of non-infrastructure capital goods, we argue that irreversibility of investment is an acceptable way of modeling the observed continuity in the transfer of capital from rich to poor countries.

4 This in turn implies two assumptions: that installed capital is immobile and that there are no security markets. Another specification would be to assume this restriction to hold only in the aggregate. This would require the use of an equilibrium price ql of installed capital in country 1 in

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Lastly, we assume rational expectations, that is, in this detenninistic environment, perfect foresight.

4. Necessary conditions

Consider the discounted Hamiltonians of the respective maximization problems, for now without using constraint (7):

A I-G

H

_{2 =.L..+Â.2[a272 +w2e-}xt -ê_2e-xt -(n+ô+x)a_2]

l-a (8)

It is weIl knowns that costate variables have an interesting and meaningful economic interpretation. As they represent the marginal valuation of the associated state variable, discounted to time zero (note that we are using discounted Hamiltonians), they are often referred to as "shadow prices". Using the example of state variable b, an increase at time t of one unit in the assets held by Country 1 in Country 2 will result in a variation ofYI(t)in the utility ofCountry 1, discounted back to time zero. The shadow price of one unit of b (that is, the amount of consumption at time zero the optimizing agent would be willing to sacrifice for an increase of one unit of b at time t) is therefore Y I (t). This property of costate variables wiIl prove helpful when analyzing how agent 1 assesses the opportunity of investing abroad, as compared to domestic investment.

Next, we append to agent 1 's Hamiltonian constraint (7) with multiplier <1>1' forming the

Lagrangian LI:

A I-G

T - CI 'I [ -xt b A -xt ~ -xt ( Ô ) ]

... - - - + 1 \ . 1 a_l71+wle + ₇₂-cle -112e - n+ +x ai

l-a

+YI[~2e-xt -(n+ô+x)b] (9)

+<1>1 [a₁₇₁+wle-xt +b72 -êle-xt -~2e-xt]

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In (9), the state variables are aI and b, while the controls are

1

₁₂and êIo

We are now prepared to derive the first-order conditions that must be verified to maximize (2) subject to (3-7):

Agent 1

lL

=

n

+ Ô + x + p -

r

₂ 11

We impose transversality conditions on aI and b:

lim

by Ie-pt = O

t-+cro

Agent 2

We also impose a transversality condition:

(10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20)

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We will now aggregate these conditions in each country, by recalling that market clearing and profit maximization by firms require:

We finally transform equations (4) (6) (12) (13) (15) (17), completing our set of aggregate differential equations:

(21) (22) (23) (24) (25) (26)

Note that quantities in (21-26) are expressed in effective terms and may be interpreted in terms of national accounting. The quantity f (k_{1 )}+ bf' (k_{2 )} represents Country l's effective GNP, where bf'(k_{2 )} is the flow of income generated by the assets held in Country 2. Similarly, f(k₂)-bf'(k_{2 )}is Country 2's effective GNP. Quantities f(k) and f(k_{2 )}denote effective GDP's. Quantity i₁₂-bf'(k_{2 )}represents the net balance of payments between the two countries.

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We next tum to equations (14) and (25), which simply state that constraint (7) is binding if <1>1 is strictly positive. We will show that during some initial period, (7) will be binding. To see this, combine (23) and (24), to get

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By condition (3), we know that f' (k_{2 ) -} f' (k₁₎> O at time zero. Suppose <1>1 = O, that is, suppose that condition (7) is not necessarily binding. But this implies, by (11):

"fI = Â.

1,

that is, by (27),

~-

rI

>

o.

This contradicts (14) and (11) since we would

Â.1 "fI

then have "fI < Â.1. We conclude f'(k_{2 )}> f'(k₁₎

=>

<1>1> O and hence that there exists

an initial period during which <I> 1 (I) > O .

As is customary with dynamic problems with an inequality constraint, we complement this result by making a conjecture about the general structure of the solution. After solving, we will check that the resulting paths satisfy the optimization conditions above. Our conjecture is that there exits T such that:

<1>1(1»0, O~I<T

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We thus define

Tas

the date when conditions (7) stops being binding. This guess is plausible, as it is rather intuitive that after a certain time the flows of capital into Country 2 will decline, because convergence between the two countries is taking place and rates of retum are being brought closer to one another. Note that we have not assumed

T

to be finite. In fact, we will prove in the next section that convergence is completed in finite time and that therefore T cannot tend to infinity, but our initial guess is free from this assumption.

We now show that date Tis the exact date when convergence is completed, that is, (7) stops being binding exactly when the two domestic capital stock reach the same value. Consider a situation with f'(k_{2 )}= f'(k₁₎ and suppose <1>1> O. By (12) and (13), we have

~

-

ri

< O. By integrating this expression from time 1 to infinity, we obtain

Â.1 "fI

limInÂ.1(S)-In"'I(I)<-In"fl(I). Recalling, from (28) and (11), that we have

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lim In

"'I(S) = O, we obtain "'I> ri' which contradicts (14). We conclude, usmg .t-+co

r

I (s)

j'(k2) > j'(kl )

=>

4>1>

o,

that we have

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The result above simply means that condition (7) on the export of capital will not be binding when both stocks of domestic capital are equal. Conversely, (7) will be binding if there is a difference in the value of domestic capital stocks.

s.

Convergence in domestic capital stocks

As small agent 1 does not take into account the result of his actions on the variation of the domestic capital of Country 2, and therefore on the rate of retum offered by the latter, it is quite intuitive that convergence between the two countries will be completed within a finite period. As long as there exists a difference, however small, in the rates of retum of the two countries, agent 1 will relentlessly invest abroad, and will not invest on his own territory, letting its domestic stock of capital decay.

Formally, making use of (29), we will prove that time

Tis

finite through a phase diagram in k₂and

c

_{2 .}We first recall that for O S. 1< T, we have, by (15) and (28), kl(/)=-(n+ô+x)kl(t). This means that k_l is monotonically decreasing during this period. Further, during this initial period, (15) and (22) give: k2=j(k2)+j(kl)-CI-c2-(n+ô+x)k2. Note that when (7) is binding, we have j (k_{l ) -} c_I~ O. (18) and (26) provide us with the required differential equation in

c

_{2 '} that is, 'V I :

. 1 c2

=

-[j'(k2)-(n+ô+p)]-x

c

₂

a

(30)

Similarly, by (10) and (24), 'VI:

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Consumption in the two countries will grow at the same rate and this rate will be a function of the higher rate of retum on physical capital alone. The other rate of return, simply because it will not be chosen for investment by maximizing agents, will not be taken into account to determine the optimal path of consumption.

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Note further that for O ~ 1 < T, we have f'(k_{2 )}> _{f'(kl )}~ (n+ô +p+xa), since k_lis monotonically decreasing, starting trom the autarchy steady state and convergence between the two domestic capital stocks is not completed yet. This implies

c

1 ₌

_c

2

~

_O

c₁

c

₂ for O~/< T.

From the differential equations above, we draw the following diagram:

J

dc2/dt=O

c2

L

r

k2

Figure 2

Note that arch O is moving downwards, as f(k_{l )}is decreasing and

c

₁~ O. This means that path A may only cross arch O once and only once. We claim that explosive path A, in which the domestic stock of capital in Country 2 is eventually decreasing, is not optimal. To see this, derive expression k₂

=

f(k₂_{)+ f(kl)-C1}

-c

₂-(n+ô +x)k₂with respect to time, to get: k~ =k₂[f'(k₂_{)-(n+ô+x)]-C1-C}₂-k1f'(k}). Using the condition on parameters p>n+(I-a)x, it is easy to see that, as k₂ eventually decreases along path A, k~ would tum negative. This in tum would imply that monotonically decreasing k₂hits zero in finite time. But then

c

₂would instantaneously drop, which violates (30). We thus infer that k2(/)~0 for O~/<T. From this and trom

k

l (I) = -(n

+

Ô

+

x)kl (I) we conclude that Tis finite. Note that the speed of

convergence between the two economies will critically depends on the depreciation rate

t5,

as this will determine the rate of decay of k} toward k_{2 .}

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Our next task is to show that once the two stocks of domestic capital (and hence the two GDP's) are equalized, both domestic economies will converge to the autarchy steady state leveI. First note that, through (29), we have k_l(t) = k₂(t) for T ~ t ~ 00,

which implies "I (t) =

k

₂(t). Put another way, once they have met, the values of k_l and k₂can never differ. This in tum, by (21) and (22), gives

(32)

Again, we will use a phase diagram to complete our proof. Consider, by (21) and (32), the differential equation in c_I and k_{l :}

T~t ~oo,

"I =/(k_{l ) -}cI -(n+ô+x)k_{l -} c2.

2 2

Together with (31), this equation results in the following diagram:

cl

J

o

"'"

r

E kl Figure 3

We

c1aim

that paths A and B are impossible. As in the analysis of figure 2, it is

important to note that arch O is moving. However, in this case, it may be moving either up or down, depending on the behavior of

c

_{2 .}Consider then the two altemative cases:

1)

~ ~

o.

Arch O in moving down. As 1:2 ₌_1:1 _,_{we are dealing with Path A only. This}

c

2 c1

path will be able to cross O once and only once. This case is thus similar to the situation presented in figure 2: the second time derivative of k_l will eventually tum negative and c_Iwill suddenly drop, which is impossible. Note that along Path A,

c

₂is

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. a l · . C2 CI all monotomc ly mcreasmg, as - = - and that therefore Arch O is monotonic y

C2 CI

shifting downwards.

2) ~ <

o.

Arch O is moving up. We are dealing with path B only. This path

will

cross O once and only once. By the transversality condition (16), we claim that this is impossible. (16) may be transformed into limkl(I)U'(cl(I»e(X-P)t =0 and (31) into

t~oo

du'(cl)/dt

=n+ô+ax+p-

_{f'(k l). As trajectory B asymptoticallyapproaches point}

u'(c

_{l )}

E, the transversality condition is violated because k_l> k· and thus u' (c_{I )} is strictly

.

mcreasmg.

The same ana1ysis may be applied to the variables k₂and

c

₂and using the fact that

c c

J. = ....!. ~ O 6, we conclude that there exists a steady state such that

c

₂

c

₁

. . ê C

k_l= k₂= J. =....!. =

o.

From this and from (23), (30) we infer that c₂ c_I

lim k_l(I) = lim k_l(I) = k·. The two domestic stocks of capital thus converge in a very

t~oo t~oo

simple way to the autarchy steady state leveI, as illustrated beIow. Path k2a represents the trajectory of Country 2' domestic stock of capital under autarchy.

k

o

_T t

Figure 4

As illustrated in Figure 4, since the consumption decision of agent 2 at each instant 1

only depends on the rate of retum 1'(k_{2 ),}together with the properties of its utility function, we argue that Country 2's convergence toward the steady state is faster that it would be in autarchy for O=::; 1 < T, that is, as Iong as Country 1 is a net foreign investor. To see this, note that during this phase, as (7) is binding, we have:

6 One would otherwise argue that arch O could shift up and down perpetually, preventing convergence.

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Compare this equation with its closed-economy version:

Since for each value of k_{2 ,}Agent 2 would make the same consumption decision regardless of Country 1 's foreign investment, we see that the term

f

(k_{1) -}

c

_1,which is simply Country 1 's net foreign investment, will accelerate the convergence of Country 2's domestic capital stock toward the steady state. Note that this is not necessarily true after time T.

This fact seems to suggest that the procedure used in Barro and Sala-i-Martin (1992) does not capture the additional convergence speed provided by the migration of capital. Barro and Sala-i-Martin (1992) uses a model featuring a closed economy to

carry

out a log-linearization of the path of the domestic physical capital stock around

the steady state. This approximation is used as a basis for estimating convergence among open economies (the 48 US states). It is clear from the discussion above that the speed of convergence of open economies toward the steady state wiIl be greatly influenced by transfers of capital from rich to poor countries. Further, our discussion only dealt with the speed of convergence of the domestic capital stock toward a constant steady state leveI. As shown in figure 4, it is quite clear that for I ~ T, the speed of convergence between lhe two economies is to be markedly higher than this. However, Barro, Mankiw and Sala-i-Martin (1995) presents a model where goods are produced with three inputs: physical capital, human capital and some non-reproducible factor, that we can view as labor. As total capital is only partially mobile, the predictions of the model conforms with empirical evidence on convergence. As pointed out above, the key difference between this model and ours is that the home economy is considered small relative to the rest of the world.

6. Consumption

Before time zero (under autarchy) the rich country's representative agent is in steady state and is thus faced with a rate of retum equal to the impatience discount rate. As

the economy opens, we submit that the rich country, being faced with investment prospects that bear a rate of retum that is higher than the impatience discount rate,

will

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accept an initial consumption sacrifice to maximize its utility. Suppose that

c

₁(O) ~

c· ,

where

c·

is the autarchy steady state consumption leveI, with c·

=

j(k")-(n+ô+x)k·. (7) gives ;12(0) ~ (n+ô+x)kl(O), which means that the rich country would be investing less or the same as in a regime of free trade, even though it is faced with higher rates of returno It is straightforward to see that by saving one unit from consumption c· Agent 1 would receive in some time in the future a retum whose rate is higher than its impatience discount rate, thereby increasing its utility. We conclude that c} (O) < c· .

In order to characterize the behavior of the respective consumption paths, we will analyze the trajectory of h, the assets held in Country 2 by Country 1. Consider the behavior of the costate variable À} and y}, respectively the shadow prices of assets at home and abroad. For t < T, we have À} <

rI>

as tPJ is strictly positive. When allocating investment between his country and Country 2, the richer agent will initially choose the latter, as this decision will be rewarded along the optimaI paths with more utility. At t =

T,

these costate variables are equalized and this premium vanishes. We appear to be in a situation of indeterminacy sinceÀ} =

r}

indicates that Agent 1 is indifferent between investment at home and abroad. In fact, we will see that it is optimal for agent 1 to continue investing both locally and abroad, i.e. that it is never optimum at any time to set i}2 such that h may decrease.

To show this, we first establish a simple property that we will use later in our proof: in steady state, h is strictly positive. Suppose lim h(t) exists and is equal to zero. From

t-.oo

(21), we see that the steady state leveI consumption of Country 1 would then be equal

ê} •

to the autarchy steady state leveI. As - ~ O for alI t ~ O and c} (O) < c , agent 1 would c}

not be optimizing its utility, since his consumption would always be below the autarchy steady state leveI.

We complete the proof by considering (32) and the fact that

c} (t) =

c~

, to form the expression: c₂(t) ~

b=

6

+..1.

c ( c·)

1--}

,Vt~T j'(k₂)-(n+ô+x) 2 c}

implies

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From the stability condition on parameters p>n+(l-cr)x, we know that f'(k₂)-(n+õ +x) > n > O, so that we may represent (35) as follows:

dbldt

Figure 5

First, we show from Figure 5 that in steady state we must have

6 =

O. To see this, note that in steady state quantities k₂and c₁will tend, as demonstrated above, to constant values. The ray represented in Figure 5 will thus be fixed. Next note that we must have (1-

~

)" O, since otherwise lhe ray would eross lhe b axis below lhe origin and, as b* > O, b would grow indefinitely, which is impossible as we have

k

₂= O in steady state. We see that the only possible steady state situation is point A, since otherwise b would be driven to zero, which we ruled out, or, again, would be on an explosive path to infinity. We conclude that in steady state, we have

6

= O, that is, defining b * as the value of b in steady state,

limi₁₂=(n+õ+x)b*

t-.ao

(36)

This means that, in steady state, Country 1 will set its foreign investment to the value needed to maintain its foreign assets unchanged.

N ext, note that before the steady state is attained, k₂and c₁are both growing, which (from equation (35» means that the ray in Figure 5 will continuously shift as illustrated :

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b

1

db/dt

Figure 6

It is easy to see that all points left ofaxis b eventually lead to b· = O, which is impossible. Points located on the right side ofaxis b are the only possible ones, and b is growing

until

the steady state is reached. In steady state, b will reach the limiting case where the economy is on the b axis, as illustrated in Figure 6. We conc1ude that we have, for aII t,

6

~

o.

This means that agent 1 will never reduce the stock of assets held in Country 2.

We thus can characterize the two consumption paths in steady state. From equations (21-22) and (36):

• •

•

c} =f(k )-(n+8+x)k +(p+x(a-1»b =c +(p+x(a-1»b (37)

• •

•

'2 = f(k ) - (n+8+x)k - (p+ x(a-1»b = c - (p+x( a-1»b (38) Country 1 will consume a fixed and perpetuai rent, of vaIue(p + x( a-1»b ., earned by its foreign assets above the autarchy steady state consumption leveI. Conversely, Country 2 will pay for the acceleration of its convergence toward the steady state with a pennanent remittance ofthe same amount from its long-term consumption.

Figure 7 represents our main results on the dynamics of consumption, with c2a denoting the optimaI growth path of Country 2 in autarchy. Shaded areas represent the positive or negative impact offree trade as compared to autarchy.

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c

o

_T t

Figure 7

We stress that the characterization of Agent 1 's initial consumption sacrifice depends crucially on the assumption that agents tive indefinitely. In a model where agents would die, the pay-back period during which the reward of the initial sacrifice is paid may not be short enough for the older agents to reduce their consumption as shown above.

With all endogenous variables analyzed, we close this section by noting that our initial guess on the structure of the solution is consistent with ali necessary conditions.

7. Welfare analysis

In

our model, it is easy to show that Country 1 will benefit from liberalization in terms of welfare. The argument we use to prove this is rather simple: as i₁₂ is a control of agent 1, choosing autarchy by setting this variable to zero ali along is an available option. Since utility is what is being maximized by agent 1, we infer from the results of the preceding sections that agent 1 's utility is greater in free trade than it is in autarchy. This reasoning must of course be considered in the light of the underlying assumption on rational expectations.

Using the Mean Value Theorem, we see that Country 1, as compared to autarchy, will increase its GNP at all time by investing abroad. Consider the static impact of an increment Ilb of b. There exists a real number S E(k} -l1b,k}) such that

f(k})- f(k} -l1b) = f'(s)l1b. But ~ +Ilb S.k} -Ilb so that /,(s) ~ f'(k₂+l1b) and (39)

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Note that, as shown above, the steady state value of k} is the same in autarchy and with free capital flows. One sees that the sum of Country 1 's domestic product and of the remuneration ofhis foreign assets is higher than its autarchy GNP at all time.

Similarly, we show that Country 2 will increase its GNP during the first period of convergence (t:S T). By the Mean Value Theorem, we know that for any

a

₂ and any posltlve b there exits a real number s E (a2 ,a2 + b) such that

f(a2 +b)- f(a2)

=

f'(s)b. But f'(a2 +b):s f'(s), which gives

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This inequality shows that the increase of domestic product caused by the inflow of b into Country 2 is greater than the remuneration of b, or, that the increase in the remuneration of labor in Country 2 due to b more than compensates any dividendslinterest paid to Country 1. Both Country 2's domestic and national products will go up as soon as the economy opens.

The utility of the two countries compares to autarchy in an asymmetric and complementary way: the rich country will accept an initial consumption sacrifice and will be recompensed in the long run by a perpetual constant stream of consumption above its former autarchy steady state leveI. Country 2 will be paying this stream to Country 1 forever, though not before receiving the benefits of a faster convergence toward its (now lower) consumption steady state.

8. Labor market and rigid real wages

To evaluate the effects of an imperfect labor market, we will now specialize the production function into a Cobb-Douglas technology, setting:

Y =

_I AKa(L.ext)l-a _I _I _, O<a < 1 ₍₄₁₎

We will show that in this model, the effects of the liberalization on wages and employment are unambiguously strong. As the rich country lets its domestic stock of capital decay and invests in Country 2, real wages will decline in Country 1 and go up in the poor economy. This is easily seen by recalling that wages at competitive equilibrium are equal to the marginal product of labor, that is:

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dY a

W;(/)

=

dL~

=

A(I-a)k; (42)

where W; denotes real wages. Noting that the derivative of real wages with respect to

k; is strictly positive, we establish the result above. Real wages in the two countries will evolve exactly as shown in Figure 4 for domestic stocks of capital: during the fust period of convergence, they will decline in Country 1 and rise in Country 2, and will rise in both countries from time T onwards.

We next assess the impact of an imperfect labor market in Country 1 by assuming that real wages are rigid and hence can never be lower

than

their levei before liberalization. This is admittedly an extreme assumption but as pointed out by Brewer (1985), "it may, for example, provide useful insights into the effect on advanced countries of multinational company investment in low wage countries (mobile capital) ifworkers in advanced countries resist wage reductions (fixed wages)". It is straightforward to show that in this case unemployment may be building in the rich country during the fust phase ofliberalization (/:S T), depending on the parameters ofthe model. To see this, note that we can express the demand for labor as a function of the domestic stock of capital (by (41

»:

Further, setting w₁(I) = w₁(O), 'VI ~ O gIves

"I =-(n+ô+x)kl,/:S T and k_l

=

K~,

weobtain

Le

i'f

l-a - = - ô + x ( - )

E{

a

(43) "d "

li

=

KI + x( 1-a) .

E{

KI

a

Reca1ling (44)

This expression appears to make good economic sense, as when real wages are fixed and capital decays at rate Ô, the variation of demand for labor will depend on how strongly technological progress x compensates for such decay. Technological progress

xis pondered by the labor's share in total GDP, as it is labor-augmenting.

Recalling that the supply of labor grows at the constant rate

n,

we see that liberalization combined with fixed real wages may result in unemployment if and only if

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l-a -ô+x(--)<n

a

(45)

Further, in the presence of technological change, we show that even if (45) is satisfied, unemployment wiIl recede entirely in the long run. There is no permanent transfer of jobs from rich to poor countries. We use the fact that, as displayed in figure 4, at some

time after T, Country I 's domestic stock of capital will start growing again. In steady

°d

state, K} will growat the constant rate x + n. 7 (43) then gives

~

=

n+~.

As the

~

a

supply oflabor grows at rate n, the economy will eventually retum to full employment. Once full employment is recovered, real wages will rise again, since the demand for labor will be greater than supply.

9. Conclusions

One general result of the model is that short-term and long-term effects of liberalization must be carefully distinguished.

Conceming employment or wages, the short-term effects of liberalization are likely to be negative for the rich country. Real wages will decline, or, if they are artificially maintained at their autarchy levei, unemployment may develop. However, with technological progress, the rich economy always retums to full employment. There is no permanent transfer of jobs.

Further, the model shows that there may exist economic forces that will push down the rich country's consumption in the first stages of liberalization. This is because a comparatively rich, dynamically optimizing agent will accept an initial consumption sacrifice to take advantage of the higher retums available abroad. The only short-term indication that the rich country will be globally better-off with liberalization is that GNP grows as soon as economies open.

Globally, liberalization will bring a net positive variation of utility to the rich country, as optimal intertemporal distribution of production between two economies exhibiting concave technologies will deliver intertemporal benents.

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For the poor country, short-tenn etTects are positive, as convergence toward the steady state is accelerated from time zero. Wages will quickly go up, together with GDP and GNP. After some finite date, the poor country will pennanent1y be paying a fixed remuneration on the rich country's assets on its territory, but the principal of this debt will never be repaid.

The model also shows how open large economies equipped with concave technologies converge over time. Convergence between the two domestic stocks of capital -- and thus GDP's -- will be completed in finite time, as small optimizing agents will invest abroad as long as there exists a ditTerence in rates of returno The rich country's national stock of capital -- and thus its GNP -- will pennanent1y stay above the poorer country's.

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References

Allais, Maurice. (1995) "La question majeure aujourd'hui",

Le

Figaro, May 29, 1995;

"Le

chômage

et

l'ordre public - Que faire ?",

Le