Trade and Depletable Resources: The Small Open Economy

Trade and Depletable Resources: The Small Open EconomyAuthor(s): Richard HarrisSource: The Canadian Journal of Economics / Revue canadienne d'Economique, Vol. 14, No. 4(Nov., 1981), pp. 649-664Published by: Wiley on behalf of the Canadian Economics AssociationStable URL: http://www.jstor.org/stable/134821 .

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Trade and depletable resources: the small open economy

RICHARD HARRI S / Queen's University

Abstract. A two-sector, two-factor, small, open-economy, intertemporal trade model is examined in which one of the factors is an exhaustible resource. Unlike the static model complete specialization in production will occur at any date, with the economy switching at some date from resource intensive goods to less resource intensive goods. A complete set of comparative dynamic results for the model are given. One surprising result is that an increase in the price of the resource intensive good leads to a reduction in the quantity of output supplied at each date, but an increase in the length of time over which the economy specializes in resource intensive production.

Commerce international et ressources epuisables: le cas d'une petite economie ouverte. L'auteur developpe un modele de petite economie ouverte a deux secteurs et avec deux facteurs de production dont l'un est une ressource epuisable. C'est un modele qui met l'accent sur le commerce international en contexte intertemporel et dynamique. Contrairement 'a ce qui se produit en contexte statique, il y a 'a tout moment specialisation complete de la production: l'economie se d6placant 'a certains moments de la production de biens dependant plus ou moins de la ressource epuisable. Le memoire presente un ensemble de resultats de dynamique comparee 'a partir du modele. Un resultat surprenant est que une augmentation dans le prix du bien dependant le plus de la ressource epuisable entraine une r6duction de la quantite produite a chacune des periodes mais aussi un accroissement dans la longueur de la periode de specialisation de l'economie dans la production de ce bien.

INTRODUCTION

Many open economies have the feature that much of their trade occurs in depletable or non-renewable resources - in particular 'resource rich' economies tend to export their resources and import manufactured products. The theoretical work-horse of international trade, the Hecksher-Ohlin model, is ill suited to deal with trade in exhaustible resources. Its main deficiency is the inability to deal with the essential intertemporal nature of the exhaustible

This paper is a shortened version of Queen's discussion paper No. 289 which is available from the author upon request. The author is indebted to Neil Bruce, Russell Davidson, Alan Gelb, and Doug Purvis and the referees for their comments. All remaining errors are the author's own responsibility.

Canadian Journal of Economics t Revue canadienne d'Economique, XIV, No. 4 November / novembre 1981. Printed in Canada I lmprime au Canada.

0008-4085 /81 / 0000-0649 $01.50 ? 1981 Canadian Economics Association



650 / Richard Harris

resource problem. While there are some dynamic models of trade dealing with capital accumulation problems' there are relatively few dynamic models of trade which treat the exhaustible resource case.2 The purpose of this paper is to extend the two-product / two-factor static trade model to a dynamic context which allows for depletable resources. Within this model we are concerned with the pattern of production over time and the classical comparative dynamic questions related to changes in the terms of trade and factor endowments. One of our primary purposes is to investigate in what form, if any, the Stolper-Samuelson and Samuelson-Rybczinski theorems hold within the context of the model.

Second, it is of some interest to see to what extent the stories told by the Hecksher-Ohlin model remain approximately valid in an explicit dynamic treatment of resources, not only for pedagogical purposes, but also for the purposes of providing a theoretical basis for empirical work. Recently a number of empirical tests of the Hecksher-Ohlin model3 have been under- taken which explicitly incorporate natural resources (Hillman and Bullard, 1978, Vanek, 1959; Williams, 1970). The implicit assumption in these tests is that the static model is approximately suitable. Finally, the analysis is of interest because it answers, at least partially, the question as to what rules should be used for socially optimal resource depletion policies in open economies. There is, of course, an extensive literature on this question for the closed economy case4 (Symposium, 1974).

The simplifying assumptions in this paper are quite dramatic. The economy is considered to be small and thus a price-taker in world commodity markets. Financial capital is assumed to be perfectly mobile, and thus the world interest rate is taken as exogenous. The economy is required to balance its budget only in present value terms, rather than at each instant of time - in this respect the model is like Gale's (1971) and is equivalent to the case in which all agents in the economy have access to a comlete and perfect set of world capital markets. Given the assumptions of perfect certainty and perfect competition, this approach seems more reasonable than introducing arbitrary restrictions on exchange opportunities.

1 Some of the original papers which consider the impact of capital accumulation on trade in an explicit dynamic model include Bardhan (1965, 1966), Borts (1964), Inada (1968), and Oniki and Uzawa (1965).

2 Some notable exceptions include Dasgupta, Eastwood, and Heal (1976), Kemp and Suzuki (1975), and Vousden (1974). For the most part, however, these papers are not concerned with the issues raised in this paper; that is, the extension of the two-sector two-factor production model to an explicit intertemporal context in which one of the factors is an exhaustible resource.

3 Stern (1975) provides a survey of much of the work done on testing the Hecksher-Ohlin model and provides a discussion of the impact on the tests of including natural resources as an explicit factor of production.

4 Dasgupta, Eastwood, and Heal (1976) consider this issue in the context of a single-sector production model in which the resource serves as a factor of production but may also be exported abroad. The country is assumed to face a price elastic world offer curve for its resource and optimally to exploit this offer curve over time. In this paper we deal with a two-sector production model, and the economy is assumed to be a price taker in all markets.



Trade and depletable resources / 651

The paper considers a model which does not allow for capital accumulation. It turns out that the model is quite similar to a model with capital accumulation, essentially because of the small country and perfect world capital market assumptions.' The paper proceeds as follows. The next section introduces notation, outlines the basic production model, and develops some properties of the national product function which are used throughout. Subsequent sections examine in detail the nature of the intertemporal equilibrium and comparative dynamic questions. A final section offers a summary and some conclusions.

STATIC PRODUCTION MODEL

In this section we outline the basic production model in the absence of capital accumulation for a small open economy. There are two sectors in the economy which produce outputs Yi and Y2 with constant returns to scale neoclassical production functions,f(xi, ki), i = 1, 2, where xi is the amount of the resource employed in the ith sector and ki is the amount of another factor employed in the ith sector. The economy is endowed with a fixed amount, k, of this other factor, which is in constant supply and is mobile between sectors. k will be referred to as 'capital' although a literal interpretation is not necessary. Let x denote the aggregate amount of the resource extracted in any period and Pt andP2 the (constant) world prices of the two traded goods. For a given flow x of the resource the production sector will maximize the value of national production; that is, it will solve the problem

(P) maxp1y1 +P2Y2,

subject to 1) yi -(xi, ki), i = 1, 2

2) xI +x2 x,k, +k2 k

3) yi . xi. ki :? O, i = 1 , 2.

Let I(p1, P2; x, k) denote the optimal value to the problem (P); we shall refer to i(*) as the national product function.6 For much of the analysis p l,P2, and k will be held constant, and thus we write the national product function simply as a function of x, 7r(x).

In characterizing the intertemporal equilibrium and in comparative dynamic exercises the properties of r will be of crucial importance. They in turn will depend on the properties of the production functions fi and the factor endowment ratio, z = xlk. At this point we introduce a fairly standard assumption.

5 The longer version of the paper explicitly considers the case of capital accumulation. The basic results, however, are essentially the same as those derived in the context of the simpler model considered here.

6 The national product function is just a special case of a restricted or variable profit function. The general properties of these functions and their uses are developed in Diewert (1974), Gorman (1968), and McFadden (1970).




x./k Z2 x

x/k

UNIT VALUE ISOQUANT IN SECTOR I

UNIT VALUE ISOQUANT IN SECTOR 2

k FIGURE 1

(A. 1) The production functions fi, i = 1, 2 do not admit factor intensity reversals.

The unit value isoquants thus exhibit 'regular' properties as in figure 1. Let z1 and Z2 denote the aggregate factor 'endowment' ratios between which diversification occurs in the static model. It will turn out that relaxing this assumption is of no great consequence for the basic analysis but it greatly simplifies notation and the number of cases which must be considered. In concluding we shall comment upon the effect of admitting factor intensity reversals. We shall denote sector 1 as the resource intensive sector and sector 2 as the 'capital' intensive sector. For example, the product YI can be thought of as a partially refined mineral produced using the raw resource input flow x1 and another factor input k1. The other product can be thought of as a more conventionally produced good requiring relatively more of capital as a factor input.

We characterize the properties of ir(x) in the following

Lemma I 7T(x) is a continuously differentiable function, monotone increasing and concave on R+.7 In addition: 1. ir(x) is twice continuously differentiable on (0, xl) and (x2, + oo) with 7T"(x)

<0,

2. 7r'(x) = constant for x E (xI, x2), 7r'(x) is non-increasing on R+, 3. lim r(x) = lrn P2f2(x, k) lrn '(x) = lim P2fX2(x, k)

x-O x-O x-O x-O

lim r(x) lim pP f1(x, k) lim 7'(x) = lim pt f '(x, k), x+ 00 x~o x 00 x+ 00 co

7 The notation used throughout the paper is as follows: i'(x) denotes d7r(x)/dx and i"(x) denotes d2T(x)/dx2; R+ denotes the positive real line;fxi(xi, ki) denotes the partial derivative aft(xi, k1)/Oxi andfki(xi, ki) likewise;i denotes the time derivative, dx/dt.



Trade and depletable resources I 653

VA

77(x)

NATIONAL PRODUCT

o / , ~~~~FUNCTION

/ XI X2

FIGURE 2

4. forx E [0, xl], ir(x) = p2f2(x, k),

forx E [X2, + co), 7r(x) = p 1f l)x, k). Most of the properties follow immediately from the properties of the

traditional textbook trade model. The linearity of the national product function over the interval [xt, X2] comes from the fact that factor prices are constant within the diversification cone. A typical ir() function is graphed in figure 2.

INTERTEMPORAL PRODUCTION EQUILIBRIUM

Given a constant world interest rate r > 0, and perfect foresight on the part of producers, an intertemporal competitive equilibrium in the production sector will correspond to the solution of the problem

(pv) max e rt7r[x(t)]dt,

subject to

(i) x(t)dt < s, and

(ii) x(t) -> O, all t, where x(t) is a piece-wise continuous function, and s > 0 is the stock of resources available at t = 0.

This is a conventional control problem, and we form the Hamiltonian H, as

H = ert7r(xt) - xXt + ,tte-txt, (1)

where

A 0 0 A(f xtdt-s 0

/J- t ? 0 /I tXt =0.




Note that X is independent of t. A necessary condition for an optimal policy, if one exists, is that

X = e-r'[,t + Ir'(xt)] for all t. (2)

Some basic characteristics of an optimal deletion and production pattern are summarized in

Proposition 1 1. Along an optimal program, if one exists, either xt > 0 all t, or there exists a

finite T> 0 such that xt > 0 for 0 s t < T and xt = 0 for t : T.

2. A sufficient condition for xt > 0 all t is lrn fX2(x, k) = + oo. (3) x-O

3. The stock of resources is exhausted in finite time, or asymptotically exhausts, that is, X > 0.

The proof of proposition 1 is an adaption of the methods used by Dasgupta and Heal (1974) and is omitted for the sake of brevity.8

Let n _rn 7m'(x) - lim P2I2(x, k). Then x fio x-O

Proposition 2 If xo > x2 along an optimal policy, if one exists, there exists a t* > 0 such that x(t*) = x2 and

lim x(t) = x'. t-t* +

For all t 7 t*, x(t) is continuous.

Proof: Recall that X = et7r'(xt) for xt > 0. Thus provided xt > 0, and rr"(xt) # 0, xt is a continuous function of t. For given X define t* as the solution to X =

e rt' ir'(x2); using lemma 1 (1), ifx S x2, then x(t) is continuous for all t S t* and x(t*) = x2. As 7T'(x) non-increasing, for t > t* and 0 < xt < x2, 7T'(Xt) > IT'(X2) = IT'(x 1), or xt < x 1. Therefore,

lim x(t) = x'. t-+t* +

Thus the discontinuity in x(t) at t *. To show that x(t) is continuous for all t # t*, it remains only to show that x(t) is continuous at T, the exhaustion time, if finite exhaustion occurs. Suppose a transition occurs at T with

lim x, > O and XT=O- t-T-

Then

lim e-&7T7(Xt) = [T + 7'(xT)]e'T, (4) t-T-

8 The proofs are provided in the longer version of the paper.



Trade and depletable resources /655

x, i

x2

t

FIGUREl3 FIGURE 3

or

v (lim XI) -T + -rr'(x74) t-T-

But recall that as

lim xt > O, 7r' lim x < n t(T- t-T-

and thus (4) implies )T < 0, which is not possible. Q.E.D.

In figure 3 we graph a possible resource depletion path. Over the interval [0, t*] only the resource intensive sector produces, and the resource depletion rate continuously declines. At t* the economy switches instantaneously from producing YI, the resource intensive good, to Y2 the less resource intensive good, and the rate of resource extraction jumps discontinuously from X2 to xI. For all t > t* only the capital intensive good is produced. Consequently we have the following somewhat surprising result. For any configuration of factor endowments (s, k) the economy will never diversify in the production of both goods. This finding is, of course, completely contrary to the conventional Hecksher- Ohlin model, where diversification is the key to many of the standard results in static trade theory.

The intuition behind the intertemporal specialization result is in the observation that in this model at any instant of time the asset price of the resource, vt, is fixed, and hence factor prices are given. The ratio of endowment flows, xlk, however is an endogenous variable to the economy as a whole and adjusts to clear the factor markets at the given factor prices. Given a positive interest rate, the ratio of factor prices vtIwt rises continuously over time to ensure asset market equilibrium. In the two-sector model this implies factor prices will never be consistent with diversified production in both sectors, except at that instant at which factor prices equal the static factor prices consistent with production in both sectors. Thus what drives intertemporal specialization is the Hotelling asset market equilibrium condi-




tion, together with competition in product and factor markets forcing price equal to unit cost. It is not the case, for example, that the net intertemporal production possibilities set of the economy is linear, the usual explanation for specialization in the static trade models. An immediate corollary of this result is that factor price equalization, in the sense that factor prices are equal at each date, will not occur unless countries have both identical technologies and identical endowments.

It is interesting to characterize the dynamics of the equilibrium resource/ capital ratio, z over the intervals [0, t*1 and [t*, T1. Differentiating 7r'(xt) - Xert and substituting, we get

-=rfxilfxxi,i= I fort t*,i=2,fort>t*. (5)

Letting gi(z) =_fi(z, 1) and Ei(z) = zgi'(z)Igi(z), (4) can be written as

z/z = -(rlk) cr(z)/[l - Ei(Z)], (6)

where o&(z) is the elasticity of factor substitution in the ith sector, Ei(z) is the elasticity of the output-capital ratio with respect to changes in the resource- capital ratio. (6) highlights the role played by the elasticity of substitution in an equilibrium (and optimal) depletion pattern.

Let ci(v, w) be unit cost function9 for the ith sector where v is the resource price and w is the rental rate on capital. As vt = Xert, the resource price rises at the rate of interest until t = T. The rental rate wt is determined by the competitive price equal cost condition,

Pi-=ci(vt, wt), i =Ifor t -<t*, i =2for t >t*. (7)

As vt rises or the resource-capital ratio falls, wt continuously falls. Thus, in the absence of capital accumulation, we would expect to observe a continuously falling rental rate on capital; more specifically from (6) we have

Wt -ztvt for all t - T, (8)

or

1wt r[gi(z)gi'(z)z] (9)

The limiting behaviour of the economy depends crucially on the production function in sector 2, the less resource intensive sector. In this sector, following Dasgupta and Heal (1974), we say that a resource is essential if

lim f2(x, k)-=O, for k > 0, (10) x-O

9 The cost function ci(v, w) is dual to the production function of the ith sector,fl(xi, ki). Woodland (1977) has shown how production equilibrium in the standard non-joint production, static trade model may be characterized entirely in terms of the properties of the sectoral cost functions.




that is, it is impossible to produce without the resource. A resource is inessential if

f2(0, k) > 0 for k>O. ( 1)

Now we have the following proposition.

Proposition 3 If

lim p2f 2(x, k) n < + oo, x-O

then along an optimal policy, if one exists, resources are depleted in finite time.

Proof: Suppose to the contrary, xt > 0 for all t : 0. Then as X ert1r'(xt), we get that xt -> 0, and 7T'(xt) -- oo as t -> oo. But iT'(xt) = pJ 2(xt, k), for t sufficiently large, and this is bounded above by n, giving a contradiction. Q.E.D.

Note: this does not depend upon the resource's being essential or inessential. (Iff2 iS CES with o- > 1, then inf2 the resource is inessential but

lrn fx2(x, k) + oo, x-Oo

thus resources are not depleted in finite time. Suppose the resource is essential; if the technology is CES with o- < 1, then

lim fx2(x, k) < + oo x-O

as x -3 0, and resources are exhausted in finite time). In an open economy whether a resource is essential or inessential is of little consequence to the issue of finite versus asymptotic depletion. What does matter is the behaviour of the marginal product of the resource in the capital intensive sector, as propositions 2 and 5 demonstrate. This is not surprising, since given the assumption of a perfect capital market, zero levels of production do not imply zero levels of consumption in the small open economy.

COMPARATIVE DYNAMICS

We now examine how various parameter changes affect prices and resource allocation over time. Formally this is an exercise in comparative dymanics as in Oniki (1973), but owing to the structure of the problem, in particular fixed world prices, we can treat the problem in a fairly straightforward manner.

Let us rewrite the national product function as IT(pi, x). In (PV) a first- order condition at any t is 7r,(pi, x) = v, which we can invert to get the supply function x(pi, v) (taking appropriate account of the 'flat' in the m( * ) function).




In examining the comparative dynamics we focus on the two intervals [0, t*] and (t*, T1, and how various parameter changes affect the diversification time t*. Recall that

i,(p1, x) = p1f I (x, k) for tE [O, t*],

IT,(P2,x) =p2L2(X ,k) for t E(t*, T).

In addition, we make use of the relations,

T (P1, X2) = Xert*, (13)

xT(P 2, 0) = x2(O, k)=n = XerT (14) ft* rT

x(pl, v ?)dt + x(p2, v,)dt- s. (15)

(13) characterizes the switch point t*; (14) characterizes the depletion date T, when exhaustion occurs in finite time; and (15) is the total resource constraint on resource inputs over the life of the economy, and T may of course be infinite.

Changes in the price of the resource intensive good Differentiating (12) and using the condition Tx (pi, x) = Xert, we get

fxldp1 + plfxxldxt = dXert = dvt for t , t*

p2f x2dxt = dXert dvt for t> t*, (16)

and therefore

dt = (-er -xe Pfx for t < t*, dpl dpi er X

dx,_ dA rt(17) dpl d- er/p2 fx for t > t*.

Differentiating the resource constraint gives us

(X 2- x1)dt* + t( j 1AX dt)dpl + (f avjertdt)dA = ds, (18)

where we have used the fact that XT = 0 if T finite, and if T = +oo, small changes in parameters do not affect the optimality of an asymptotic depletion policy. (Note also that Ox(p2, vX)I&p1 = 0 if t > t*, from (16) as dp2 = 0). Differentiating (13) implies fx1(x2, k)dp1 + p1fx1(x2 k)dx2 = dXert + rertXdt*, or, using Xert = x

dt* = I I r' f dx2 1 dA (19) dpj rp1 f d1A dp-

Finally, differentiation of (14) gives

dX -rXdT. (20)




In order to sign many of our results it is necessary to determine how the critical resource level x2, varies with a change in p l. Recall that because x2 is competitive (cost minimizing) at factor prices (v*, w*) we have 10

x2k = cll(v*, w*)c2l(v*, w*) -(V*IW*), 4' < 0,

where (v*, w*) satisfy the unit cost conditions. An increase in pI raises the ratio v *1w * by the Stolper-Samuelson theorem and hence lowers the critical resource level, x2.

Proposition 4 An increase in the price of the resource intensive good decreases the resource/capital factor endowment ratio at which the switch point t* occurs.

Noting that ds = 0, and substituting (19) into (18) gives 1I dA (A t rx dA __ ax

(2- xt)a - x rAxd)- + ii erdtd d-t, rAdp1 ~~ av -dp1l aoPi where

I Ip vxx dp; a 1 + m>d2 0.

Solving for dXAdp, gives

d J a dt (X2 x')a dp1 axt e ertdt -(x2- x')/rA (21)

Now 9xIep1 - -fxj/p1fxxl > 0 for t S t* and axlav = /pifxx < 0. Consequently the numerator and denominator are both negative in (21), implying dXIdp1 > 0. Furthermore, rewriting (17) for t < t* yields

dxt = e[tAp -pd ] (Ai) A PfXxx. (22)

The sign of (22) is independent of t and depends upon the sign of (p1I/)(dX/dp1) - 1. By the continuity of the extraction path, since dx2Idp1 < 0, for some small E> 0, dxtldpl < 0 where t = t*- E. But if this is true, then it is true for all t < t *. Therefore (p I/X)(dX/dp1) > 1. Summarizing, we have the following proposition.

Proposition 5 An increase in the world price of the resource intensive good has the following effects. 10 cli(v, w) denotes the partial derivative aci(v, w)lOv and C2A(V, w) denotes the partial

derivatives Oci(v, w)IOw, i = 1, 2. Shepherd's lemma in conjunction with the assumption of constant returns to scale implies that Oci(v, w)lOv equals the cost minimizing demand for the resource input in the ith sector per unit of output produced in the ith sector. A similar statement holds with respect to the capital input.




1. It increases the resource rents for all t both absolutely and relative to the world price p 1.

2. The rate of resource extraction falls for all t # t*, and output in the producing sector falls for any t # t*

3. The resource/capital ratio falls for all t # t*. 4. If resources initially exhaust in finite time, then the time to depletion

decreases.

The effect of an increase in the price of the resource intensive good on the other factor price is not quite so straightforward. Recall thatp 1 = c 1(vt, wt) for t - t* and P2 = c2(vt, wt) for t > t* by the specialization proposition. Differentiating these equations gives

dpw C 72(Vt, WT) dp) =-Z,(vt, wt) dpt < 0 for t > t*. For t - t *, however, the effect of an increase in p 1 on the absolute level of wt appears to be ambiguous. The effect on the factor price relative to P1, however, depends upon the property of the production function in the resource intensive sector. Using the marginal product condition wtlp =

fkI(xt, k) for t - t*, we have that

d(wtlpl) fkx(xt, k) for t < t*, dpl ~~~dpl

so that the sign depends upon whether the marginal product of capital in the resource intensive sector is increasing or decreasing in the resource / capital ratio.

Proposition 6 1. For t > t* an increase in the price of the resource intensive good causes

both the absolute and relative return to capital to fall. 2. For t - t * an increase in the price of the resource intensive good causes the

real return to capital (that is, the rental rate on capital relative to the world price of the resource intensive good) to rise or fall asfkX I is greater than or less than zero.

It might be useful at this stage to compare these results to the traditional Stolper-Samuelson results. They are not strictly comparable of course, because of the intertemporal nature of the problem and the completely specialized production pattern. However, in a modified form the Stolper- Samuelson result stands intact. An increase in the price of the resource intensive good gives rise to an increase in the real resource price at any date. The real reward to capital unambiguously falls in the capital intensive time interval. In the resource intensive interval the effect on real reward of capital depends on the production function of the resource intensive good.

From proposition 5 we have that for any t t t*, the rate of resource




xj A

x2 21

xLi

XI~~~~

xl -

I X

t* t* T T FIGURE 4

extraction falls, and furthermore if depletion occurs in finite time then the time to depletion decreases with an increase in the price of the resource intensive good. It follows then, that since the stock of the resource is unchanged, it must be the case that t* has increased and over the interval (t*, t* + dt*) the rate of resource extraction has increased. A diagram of the effect of an increase in pi on the extraction path illustrates what occurs (see figure 4).

The effect of an increase in Pi on the output pattern is quite interesting. While the increase in p I actually diminishes the level of output over the initial period of production, it has the effect of increasing the length of time the resource intensive good is produced and diminishing the length of time (in the finite depletion time case) the capital intensive good is produced.

Proposition 7 An increase in the price of the resource intensive good increases the length of time for which the resource intensive good is produced.

How does one explain what is, at first glance, a peculiar supply response, in that an increase in price actually lowers the quantity supplied? In understand- ing this result it is useful to keep in mind that the resource intensive good is produced on two margins: the intensive margin or variation in quantity supplied at any date prior to the switch point t* and the extensive margin or the length of time over which the good is produced. An increase in the price of the good at all dates in the future raises the return on both the intensive and the extensive margin, so there is ajoint supply effect. However, the added value of production on the extensive margin is sufficiently large that it actually leads to a reduction in the quantity supplied on the intensive margin, owing to the overall resource constraint.

We now turn to an examination of changes in factor endowments. In particular, we examine the effect of a change in the stock of depletable resources on the intertemporal competitive equilibrium and examine to what




extent the basic result of the Samuelson-Rybczinski theorem remains true. Differentiating the resource constraint (15) gives us

z T

{ 1T x ertdt}dA + (x2 _ xl)dt* = ds, (23)

where T is possibly equal to +00 and x, = 9x/Iv < 0, for all t such that xt > 0. Differentiating (13) and noting that x2 is determined independently of s, gives

dA ert* + Arert* d 0, ds ds

or,

dt* I dA -ds- rA 'ds

24

Substituting in (23) we solve to get

dA 12 1

ds I j{ [jo Xve rdt - (x2-x) < 0. (25)

Finally, from the terminal condition we get

dA + rA d = 0. (26) T ds

Summarizing these is

Proposition 8 An increase in the endowment of the stock of the depletable resource causes 1. a decrease in resource rents for all t 2. an increase in the time to depletion (provided depletion occurs in finite

time) 3. an increase in the length of the period over which the resource intensive

good is produced.

From the first-order conditions we have that

dxt _ 1 dAert > O, for t t*,

-d- =If" (x t) d s

proving

4. an increase in s increases the rate of extraction and the level of production for all t # t*.

The Samuelson-Rybczinski result does hold true in a modified sense. 11 An increase in the stock of the resource leads to a higher level of production in the resource intensive industry, and to a longer period of specialized production in the resource sector. It is also true however that the less resource intensive

11 Murray Kemp and Ngo Van Long (1979) derive a similar result.




sector expands for all t > t*. However, provided depletion occurs in finite time we have that d(T - t*)Ids = 0, and therefore we get

Proposition 9 A change in the stock of the depletable resource has no effect on the length of time the economy produces the capital intensive good, provided depletion occurs in finite time.

Consequently, we can say that at least in one sense the expansion of the resource base tends to expand the resource intensive sector relatively more than the capital intensive sector. It is also true, of course, from the price equal unit cost condition we have that an increase in the stock of the resource necessarily raises the real rental rate on capital for all t. Therefore, clearly factor price equalization does not occur. Proposition 9, however, is quite interesting, because it does show that the capital intensive production period is independent of the resource stock, and consequently economies with different resource bases will have equal capital intensive production periods.

C O N C L U S I O N

In this paper we examined the properties of a dynamic two-sector Hecksher- Ohlin type model of a small open economy incorporating exhaustible resources. The major conclusions one can derive from studying this model are as follows. First, the pattern of production in the dynamic model with exhaustible resources does not resemble that found in the static model. In particular, specialization in production will always occur; non-specialization is the key to many of the results in the static model and the basis for much of the empirical work done. It appears that the static trade model may not be appropriate for theory or empirical work when one of the significant factors of production is an exhaustible resource. Second, the classic comparative static theorems, the Stolper-Samuelson and Rybczinski theorems, have their comparative dynamic counterparts in the model considered here, although factor price equalization will not obtain in any sense. A number of comparative dynamic results were derived. One of the most surprising was that an increase in the price of the resource intensive good leads to a decrease at each moment in time in the level of production of whichever good is being produced. The economy, however, spends a longer period of time producing the resource intensive good.

If factor intensity reversals are admitted, the analysis becomes more complicated in that the economy will switch back and forth between producing in the two sectors as time goes on. However, the economy will always specialize in production. The comparative dynamics become more complicated with multiple switch points. It is not clear how the results of this paper will generalize. The other and possibly more crucial assumption of the paper is that the economy is a price-taker in world markets. A next step is to




consider a two-country general equilibrium model with capital accumulation in which all prices, including interest rates, are endogenous.

REFERENCES

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Dasgupta, P., R. Eastwood, and G. Heal (1976) 'Resource management in a trading economy.' Working paper no. 65, Institute for Mathematical Studies in the Social Sciences, Stanford University

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Trade and Depletable Resources: The Small Open Economy

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