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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.

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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 , .

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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


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 the savings rate times the outputkapital ratio: the Harrod (1939) warranted rate of growth. It is easy to show that this is the same growth rate as would be obtained in the Solow-Swan model if the production function is given by (8).

111. GROWTH IN A SMALL OPEN ECONOMY

The preceding section illustrates the 'new growth theory' result that economies with consumers who are patient enough ( p < A ) will grow automatically via capital deepening at a constant growth rate, irrespective of technology growth or innovation.

In this section we investigate endogenous growth in an open economy. We begin by assuming that consumers maximise the intertemporal utility function (1). It is assumed that the economy is a small open economy with no impediments to capital flows. Thus each agent can borrow or lend freely at the world real rate of interest denoted r* which is assumed constant. Each agent can hold foreign debt a, at time t , that is, borrow against the rest of the world. Once again, the government is omitted from the analysis.

Net exports, IW, can be defined

nx, = f(k,)-c, -it

where i, is investment. Foreign debt (a,) changes according to

dat - * --r a , - n x , dt

Foreign debt rises by the interest on the debt but is reduced by positive net exports. Equations (16) and (17) can be combined into an equation of motion for a,

*=c,+i,+r*u,-f(k,) dt

With no depreciation, we also have

Exploiting the equivalence between the decentralised equilibrium and the command optimum, agents in the economy maximise (1) subject to (18) and (19). The Hamiltonian is

p[c, + i f +r*u, -f(k,)]+il,i, eWpr 1 ci-" - 1 H , = ~- [ 1 - 0

where p, and ill are the shadow prices of a, and k, respectively. Use of a negative multiplier for foreign debt makes the interpretation of p the shadow value of net foreign assets.

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The first order conditions are

c;" - PI = 0

+ al = o

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and

Equations (21) and (22) are H , = 0 and H , = 0 respectively. Equations (23) and (24) are conditions on the rate of change of shadow prices for the path to be optimal, and (25) and (26) are transversality conditions. Equation (25) rules out the infinite accumulation of foreign debt, that is it rules out Ponzi-games where further debt is accumulated simply to repay interest on previous debt.

Differentiating the left hand side of (23) and dividing by e-pr

and using (21) it follows that

d 1 dt d -loge, =- ( . * -p)

and therefore

L( r * -p) , cl = cneu

Simplifying (24) in like fashion


and since 1, = p, from (22), (27) and (29) imply that

r* = f ' ( k , )

This is the standard marginal productivity condition.

The implication of perfect capital markets with (30) implies that the capital stock can immediately jump, just after time zero, to where its marginal product equals the world real rate of interest. This is accomplished by net foreign borrowing (in the case of an increase in the capital stock). Thus, in an open economy, there are no dynamics for the capital stock, and all of the dynamics are on net foreign debt. The transversality conditions are written in such a way as to rule out both infinite accumulation of a, and k , (even though the marginal productivity condition would not allow this anyway).

Multiplying equation (18) by e-r*r and integrating gives

Using integration by parts on the left hand side of (31) and using (25) it follows that

Equation (32) states that the discounted value of lifetime consumption equals the discounted 'cash flow', defined as output net of investment, less initial debt. This is defined as v. Substituting from (28) into (32) it follows that

so that c g = pv

Note that p will be finite only if

- ( r * - p ) - r * < O 1 0

(33)

(35)

in which case

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Then

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We now investigate the effect of the Ak production function in this framework. It follows from (30) that

r * = A (38)

This is the first problem for endogenous growth (Ak) models when extended to an open economy. Both A and r* are exogenous; with a linear technology, the rate of interest must equal the output-capital ratio for marginal productivity conditions to hold. There is an obvious conflict between technologies which impose a constant marginal product of capital when confronted with a small open economy facing a given real rate of interest. This probably says more about the nature of technology that is assumed to generate endogenous growth than it does about real economies. However, there are additional problems for the endogenous growth model below; to show these we assume that (38) is satisfied. Clearly, if (38) does not hold agents would want to hold zero or infinite domestic capital.

Given (38),

and (35) becomes

(40) P A < - 1-0

Thus the growth rate of consumption is (110) (A -p), precisely what it was in the closed economy version. Endogenous growth in consumption will occur only if A > p.

It turns out that, unlike the case of the closed economy, the growth rates of output and capital will not necessarily equal that of consumption. Define the growth rate of y , as y Since y = Ak it follows that capital and output must have the same growth rates. Thus we can write

k, = koeY'

yr = AkoeY'

and search for the equilibrium value of y In this case


For v to be finite, y< r* = A. Then, (44) becomes

v=kO-aO.

Equilibrium growth can thus be represented by

k, = koeY'

y t = AkoeY'.

and it follows that I a-I)A+p 0 ; (A-f)t nx, = ( A - y)koeY' - [ ( ]( k - ao)e

0

(45)

(47)

with the condition that y< A, which is the only restriction on the growth rate of output.

It immediately follows that there are an infinite number of steady state growth rates of output in a small open economy endogenous growth model. Moreover, there may not be any growth at all: y= 0 is a perfectly valid solution to the model. Thus a non-decreasing return to reproducible factors production model does not necessarily generate growth when extended to a small open economy.

Before proceeding with this issue, we should note the conditions under which this model predicts growth in consumption. From equations (46) and (40) it follows that growth in consumption will occur only if

p < A < - P 1 - 0

If o is close to zero, this places incredible, almost knife-edge properties on the parameters of the model. Equation (48) is an additional restriction necessary for endogenous growth of consumption in open economies.

Why does a production function with constant returns to reproducible factors generate growth in a closed economy, but not in an open economy? The answer is that in the closed economy models, savings has nowhere else to go except into the accumulation of more capital. Thus positive net savings always generate capital deepening in endogenous closed economy growth models.3 As soon as there is any alternative to the accumulation of domestic capital, the endogenous growth model prediction breaks down. In the limit, free trade in capital disconnects capital accumulation and the savings rate, so that in the absence of other assumptions the 'AK' production function does not necessarily generate growth. In this framework, a theory is required as to why capital is attracted to some regions, and at a faster rate, than others.

Net of depreciation and population growth.


IV. EMP~RICAL EVIDENCE

The fact that endogenous growth models do not predict growth and therefore have nothing to say about convergence in open economies sits more easily with the empirical evidence. First of all, there is much debate about the empirical evidence on cross-country differences in growth. Barro (1991) and Mankiw, Romer and Weil (1992) also show that the basic cross section results give no convergence and in fact weak divergence, in growth rates. However, allowing for initial levels of human capital, the same data set gives ‘conditional Convergence’, that is convergence given countries begin with the same level of human capital. Barro and Sala-i-Martin (1992) empirically find convergence among the US states (which are presumably 50 small open economies); one would expect these states to have begun with approximately the same levels of human capital and other socio-economic indicators.

There is also a debate about convergence during different time periods. Ambromovitz (1986) finds insignificant convergence since 1973 and Baumol and Wolff (1988) find slight divergence in growth rates since 1973. Dowrick and Nguyen (1989) argue that these results are affected by cycles and that one obtains continued convergence since 1973.

In regards to scale economies, Backus, Kehoe and Kehoe (1990) (hereafter BK2) find scale effects in manufacturing, but not in aggregate output, for a cross section of countries. The findings indicate that there are no scale effects or intensity effects from research and development, and thus presumably no scale effects from human capital. This does not support the models of Romer (1990) which relies upon human capital scale effects, or Lucas (1988) which relies upon the intensity of accumulation of human capital.

Meshing these results together presents some difficulty when one also considers the following. While Barro and Sala-i-Martin (1992) find convergence for the US states, Neri (1994) using a similar approach finds divergence for the Australian states since 1976.4 It is hard to reconcile these results unless there is no unambiguous prediction on economies of scale and growth from open economy models. Manufacturing has been far more important in the US states than Australia: if anything, there should be no convergence in the US, according to the endogenous growth models, but convergence in Australia, where manufacturing is a much smaller proportion of total output, according to the findings of BK’. Moreover, the divergence in Australia corresponds to relatively poor performance in those states with higher relative manufacturing sectors (Victoria and South Australia), that is, in the only sectors which should be experiencing scale effects. This empirical evidence seems to suggest that there is little connection between economies of scale, growth and Convergence.

4 Maxwell and Hite (1992) find a similar result with a different methodology.


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