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ECONOMIC GROWTH AND CONVERGENCE IN OPEN ECONOMIES*

ROSS MILBOURNE

University of New South Wales

I. INTRODUCTION

Economic growth has re-emerged as a central theme of macroeconomic theory. Much of the debate is about the ability of the standard neoclassical growth theory of Solow (1956) and Swan (1956) to explain growth and development of a wide variety of countries including the newly industrialised countries. In this model, decreasing marginal productivity of capital implies that an economy will increase its capital per head until the marginal unit of capital generates additional output and hence additional savings to just cover depreciation and population growth. At that steady-state point the economy will grow at the (exogenous) rate of technical progress. Since the marginal product of capital is declining, the growth rate of output declines towards the steady state. Thus, conditional on savings behaviour and population growth, poorer countries should be growing at a faster rate than richer countries. This is called the convergence hypothesis.

Romer (1986) has argued that the data does not support the Solow-Swan convergence implications. He claims that (i) empirically, growth rates are a positive function of the level of economic development, and (ii) that savings and population growth rates seem to exert an influence on output growth per head. This led Romer to argue for a growth model with increasing returns to all factors. Subsequent writers have taken up this theme, and led to a class of models referred to as endogenous growth models, also referred to as the new growth theory.

In endogenous growth models, growth does not rely on an exogenous growth in the technology parameter A. Savings per head is always greater than that required to keep the capital stock per head constant, so that capital continues to accumulate and the economy experiences continual capital deepening. Equilibrium requires constant output growth and this growth is a function of savings and population growth rates. However, there are basically only two ways in which equilibrium growth can occur. If there are increasing returns to scale to all factors, population must be constant for equilibrium ( i e . no population growth). Alternatively, if there is population growth, there must be constant returns to scale to all factors and non-decreasing returns to scale to the factors which can be accumulated. The only production function which satisfies these requirements is the production function Y = AK, that is, linear in the factors which can be accumulated.

* This was first written in September 1992 and presented at seminars at the Australian National University, the University of Sydney and the University of Queensland. Thanks are due to those seminar participants, also to the Australian Research Council for financial assistance and to referees of this journal.

I A good exposition of this is Sala-i-Martin (1990a, 1990b).

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The AK model is not as restrictive as it might at first seem. Lucas (1988) has argued that the accumulation of human capital is as important as physical capital for growth. Moreover, human capital can also be accumulated. If one has human capital augmented labour as a factor and if K is a measure of overall capital, then the AK formulation may not be unrealistic. In addition to Romer (1986) and Lucas, this model has been used by Rebelo (1991), Barro (1990) and others as the basic model which endogenously generates growth.

Almost all of the endogenous growth models have been analysed in the context of a closed economy. This paper analyses the basic endogenous growth model in the context of a small open economy. It derives the conditions for growth in that context. The paper shows that there are serious problems when one extends the endogenous growth model to a small open economy and that basic endogenous growth models are vacuous in terms of predictions for growth in open economies. Evidence consistent with this is discussed in Section IV.

The paper begins by looking at the standard closed economy framework for the AK model. This non standard framework derives consumption behaviour by assuming one individual who optimises over an infinite life. This is obviously not done for realism; rather this yields the dynamic properties of consumption which drive the endogenous growth models.

11. GROWTH IN A CLOSED ECONOMY

In what follows endogenous growth is analysed in a one-sector closed economy with optimising agents along the lines of Barro (1990), Barro and Sala-i-Martin ( 1 992) and Rebelo (1991). It differs from the Solow-Swan model in two respects: consumption is generated by optimising consumers rather than consumers with a constant savings rate, and the production function exhibits non-decreasing returns to capital. The economy is assumed to consist of infinitely lived consumers (described by a representative consumer) who maximise an intertemporal utility function; and perfectly competitive firms who maximise profits. The assumption of infinitely lived consumers could be justified by appeal to each generation caring about the utility of their children along the Barro (1974) Ricardian equivalence formulation. The analysis assumes continuous time.

The representative consumer is assumed to maximise

Where cr is instantaneous consumption and p is the constant rate of time discount.

1995 ECONOMIC GROWTH AND CONVERGENCE IN OPEN ECONOMIES 311

Consumers are assumed to hold equity, and thus own the capital stock. They receive a wage plus the return on capital. For the purpose of comparing the differences in closed versus open economies, population growth and depreciation are irrelevant, and to simplify the algebra they are assumed to be zero. Normally, (1) is maximised subject to a budget constraint involving capital accumulation, with capital accumulation determined by the profit maximisation problem for firms. However it proves expositionally useful to note that in the structure of the model here (with no externalities) the decentralised equilibrium is the same as the command optimum. The command optimum involves maximising (1) subject to

In ( 2 ) the left hand side represents output per head, which is a function capital per head (k,) at time t . 2 The right hand side of ( 2 ) represents the demand for output from consumers (c , ) and producers (investment, which in the absence of depreciation, adds to the capital stock).

The Hamiltonian of the maximisation problem can be written

H , = {-+Ar[f(k,)-cr]}KPr c:-" - 1 1 - 0

from which the first order conditions are

cr-a = a,

%=A,[p-f'(k,)] dt

with the transversality condition

and the initial condition

k , = kO.

It is also the case that we require k, t 0. Most specifications off(k) (including that of this paper) imply this and this condition is not formally used in what follows.

Other factors of production, such as human capital, are ignored to simplify exposition. These omissions do not alter the results.

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From (3) it follows that

-oiogc, = log a, so that, differentiating with respect to t

d d 1 dA, dt dt A, dt

-o--logc, = -logar = - -

Thus, from (4) and (6)

To generate endogenous growth we assume that the production function is

Y, = f(4) = Ak,

Then d 1 -loge, = -(A - P). dt o

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(7 )

(9)

Integrating (9) it follows that

Thus consumption grows through time if A > p ; that is, if the productivity parameter or outputkapital ratio is greater than the rate of time preference. In economies with relatively impatient agents (high p), consumption will be high initially and will then decline.

Growth in consumption is sufficient to generate growth in income. Given (8). ( 2 ) can be written

_. dkf =Ak,-c, dt

dkf - - _ or k, dt k,

It has been standard in the endogenous growth literature to concentrate on the steady-state properties of the model; this steady state features positive and constant growth even though there is no growth in A. For comparison with the open-economy version, the steady-state paths of the variables are derived. As Sala-i-Martin (1990a, 1990b) has shown, this features a constant growth rate of k,, so that from (1 1 ) c,lk, must be constant; this implies that k, must grow at the same rate as c , .

I995 ECONOMIC GROWTH AND CONVERGENCE IN OPEN ECONOMIES 313

Then we can write

It follows from ( I 1) and (12) that

1 -(A- p ) = A - 5 O ko

so that A(o - 1) + p

co=kO( ) and thus

A ( o - I ) + p L ( A - ~ ) t c , = ( 1.I eu (14)

Equations (12) to (14) define the time path of capital, output and consumption as a function of the initial level of capital, kO, and the parameters of the model. If A ~ p , growth is generated even though A (the level of technology) is constant. Growth is endogenous and relies on continual capital deepening. Moreover, the growth rate is independent of the initial level of capital, and thus the initial level of income.

The equilibrium growth rate, (llo)(A - p), can be related to the implicit savings rate. The savings rate, s, can be written

It follows from (13) and (14) that

A ( o - l ) + p - A-p s = l - -- OA OA

Thus

so that the growth rate of the economy is

6= SA

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Economies whose agents are relatively patient (low p) have high savings rates and equivalently high growth rates. Moreover growth does not slow down; there is no necessary convergence of growth rates. Note also that the growth rate in (15) can be written as t