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This article was downloaded by: [Universitat Politècnica de València]On: 29 October 2014, At: 11:35Publisher: RoutledgeInforma Ltd Registered in England and Wales Registered Number: 1072954Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH,UK
The Journal of InternationalTrade & Economic Development:An International andComparative ReviewPublication details, including instructions for authorsand subscription information:http://www.tandfonline.com/loi/rjte20
Can sustained growth beattained through tradingexhaustible resources forforeign research?Francisco Cabo a , Guiomar Martín-Herrán a & MaríaPilar Martínez-García ba IMUVA, Universidad de Valladolid , Valladolid , Spainb Depto. Métodos Cuantitativos para la Economía y laEmpresa , Universidad de Murcia , Murcia , SpainPublished online: 19 Nov 2012.
To cite this article: Francisco Cabo , Guiomar Martín-Herrán & María Pilar Martínez-García (2014) Can sustained growth be attained through trading exhaustibleresources for foreign research?, The Journal of International Trade & EconomicDevelopment: An International and Comparative Review, 23:2, 267-298, DOI:10.1080/09638199.2012.742131
To link to this article: http://dx.doi.org/10.1080/09638199.2012.742131
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Can sustained growth be attained through trading exhaustible
resources for foreign research?
Francisco Caboa, Guiomar Martın-Herrana* andMarıa Pilar Martınez-Garcıab
aIMUVA, Universidad de Valladolid, Valladolid, Spain; bDepto. MetodosCuantitativos para la Economıa y la Empresa, Universidad de Murcia, Murcia, Spain
(Received 23 September 2011; final version received 23 July 2012)
We analyze the existence and the stability of a sustained balancedgrowth equilibrium (SBE) in a model of two non-homogeneous tradingeconomies. A technological leader country which sells patents of newintermediate products in exchange for an exhaustible resource extractedby a technological follower trade partner. Considering a growth-essential resource, the ‘knife-edge’ assumption of exactly constantreturns to scale (CRS) to manmade inputs can be alleviated, and thescale effects associated with R&D-based growth models overcome. Afully endogenous SBE is proven to exist, although its stability turns outto be a ‘knife-edge’ possibility. The long-run equilibrium is saddle-pathstable assuming CRS in manmade inputs. Conversely, consideringincreasing returns to scale together with a completely specialized two-country trade, the equilibrium could be reached only if the twoeconomies initially guard a particular relation, described by a particularsubset of the state space.
Keywords: non-renewable natural resources; sustained economic growth;stability; international trade and fully endogenous growth; technologicalinnovation; non-scale growth model
JEL Classifications: Q56, F43, O41, C62
1. Introduction
This paper studies the sustainability of economic growth when productiondepends on an exhaustible natural resource whose use must necessarilydecrease in the long run. Undoubtedly, an endless economic growth needstechnological improvements to go beyond ‘the limits of growth’ (Meadowset al. 1972). However, the extraction of natural resources and the researchactivities do not necessarily take place in the same economy. Two stylizedfacts are observed: first, the majority of the innovative activities in the world
*Corresponding author. Email: [email protected]
� 2012 Taylor & Francis
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occur in just a few industrialized countries.1 Second, exhaustible naturalresources like fossil fuels or non-energy minerals are often located indeveloping countries.2 Thus, trade relationships between non-homogeneouscountries merge as a key element to promote economic growth. In thispaper, we analyze a bipolar two-country R&D-based endogenous growthmodel. The resource is harvested in one country, while the R&D activities,the driving forces of the economic growth, are performed in its counterpart.
The seminal growth model through the creation of new ideas by Romer(1990) assumes constant returns to scale (CRS) in manmade inputs used toinnovate. This ‘knife-edge’ assumption of exact CRS has been criticizedbecause it implies a lack of robustness (Solow 1994). Slightly increasingreturns to scale (IRS) would lead to explosive growth, while diminishingreturns (DRS) would lead to stagnation in the absence of any exogenoussupporting force. Moving away from the ‘knife-edge’ assumption of CRSmay also help to avoid the undesired strong scale effects, inherent in the firstR&D growth models, which have little empirical support (see Backus et al.1994; Jones 1995a).3 This shortcoming was partially alleviated by the semi-endogenous growth models, which assumed DRS in these inputs. However,DRS predicts the weak form of scale effects, since permanent economicgrowth must be supported by an exogenously growing factor, usuallypopulation growth. This is why these models are known as semi-endogenous(see, for example, Jones 1995b, or Eicher and Turnovsky 1999).
When the creation of manmade inputs requires a natural resource, thisresource is called growth-essential, following the definition given by Groth(2007). Under this hypothesis, any form of the controversial scale effects ongrowth (weak or strong) is ruled out (see Eicher and Turnovsky 1999 orGroth 2007). Furthermore, the introduction of an exhaustible resource,whose use necessarily declines through time, makes the assumption of IRSin manmade inputs feasible and still attains non-explosive growth. The‘knife-edge’ assumption of CRS can then be replaced by the less stringentassumption of IRS, although this could lead to instability in some cases.
In a one-sector closed economy with a central planner, Groth and Schou(2002) show that, despite the continuous declining of the exhaustibleresource, IRS lead to the instability of the equilibrium solution unless,again, an exogenous growth in population is assumed. Conversely, in amarket economy with a final output sector separated from a research sectorcharacterized by monopolistic competition and technology spillovers (i.e.with inefficiencies and externalities), the equilibrium is stable for a widerange of IRS, as Cabo et al. (2010) prove. Therefore, in a two-sector closedeconomy, unlimited economic growth occurs and it is stable without anyexogenous source of growth, making growth fully endogenous.
The research question in the present paper is whether this result can begeneralized to a two-country trade model with heterogeneous countries. Orequivalently: could a resource-endowed economy be exclusively dedicated toextractive activities, discarding innovative activities and leaving economic
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growth only to the research carried out abroad? Does such an equilibriumexist and, is it stable? As previous studies show, it is not clear whether thestability of the equilibrium is achieved more easily under international tradeor under autarky. To the best of our knowledge, this question has beenaddressed by the literature focusing on economic growth and indeterminacy(which means that there is not a single but several policies that could leadthe economy towards the same long-run equilibrium). Originally, severalworks stressed that indeterminacy arises more easily with productionexternalities and/or market inefficiencies when small open economies areconsidered (see, for example, Meng and Velasco 2004). In contrast, morerecently, Sim and Ho (2007) show that indeterminacy which may arise in aclosed economy with production externalities, turns into determinacy whenthis economy opens out to trade with another economy without externalitiesand saddle-path stability. As pointed out by Hu and Mino (2011), the keyelement which facilitates or restricts indeterminacy is the trade structure,and, in particular, the heterogeneity between the countries. In this paper, weprove that the inefficiencies and externalities which lead to determinacy inthe closed economy are not sufficient to guarantee saddle-path stability in abipolar trade model.
We characterize the feasible range of IRS within which an equilibriumsolution exists and is unique. However, convergence toward the equilibriumsolution is in general unfeasible, unless the initial conditions of the twoeconomies lie in a particular subset of the state-space (conditional stability).Thus, removing the ‘knife-edge’ condition of CRS, leads to a new ‘knife-edge’ condition, now on the initial state values.
Finally, we ask whether this instability arises only as a consequence ofthe trade structure. We prove that the equilibrium solution of the two-country trade economy is stable when the creation of new ideas follows amodel a la Romer (1990) with exact CRS in manmade inputs. As aconclusion, the source of instability is not the completely specialized tradebetween heterogeneous countries, nor the IRS in manmade inputs, but thecombination of both assumptions together.
The paper proceeds as follows. First, in Section 2, we explain the agents’behavior in all different sectors in the two economies. The sustainedbalanced path equilibrium is defined in Section 3, and its existence anduniqueness are characterized. The stability of the equilibrium is numericallyanalyzed in Section 4. The stability result in the two-country trade modelwith IRS and a growth-essential resource is compared with the standardmodel a la Romer in Section 5. The comparison is also carried outconsidering a closed economy. Section 6 concludes.
2. The model
We study a model of trade between two non-homogenous countries: atechnological leader and a follower economy endowed with a non-renewable
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natural resource. Each country produces a consumption good using labor,intermediate goods, and natural resources according to the followingtechnologies:4
Yl ¼ A
Z N
0
ðXlðjÞÞaðLYl Þ
1�a�bRbl dj;A; a; b > 0; aþ b < 1 ð1Þ
Yf ¼ ~A
Z N
0
ðXfðjÞÞ~aðLYf Þ
1�~a�~bR~bf dj;
~A; ~a; ~b > 0; ~aþ ~b < 1 ð2Þ
For variables subscript i 2 fl; fg, distinguishes between the leading andthe follower country, while a tilde stands for the follower’s parameters. Forcountry i, the amount of natural resource is denoted byRi, the amount oflabor by LY
i , while XiðjÞ represents the amount of the durable intermediategood of type j, with N the number of varieties of these intermediate goods.5
Finally, A; a;b; ~A; ~a; ~b are constant parameters.We assume that innovative activities only take place in the leading
economy. Thus, the total number of varieties is the same in both economies,and its evolution is driven by the following equation:
_N ¼ B KRDl
� �ZðLRDl Þ
1�ZNj ð3Þ
where Z 2 ½0; 1Þ;j 2 ð0; 1�, B > 0 is a constant parameter representing theexisting technology in the R&D sector, and LRD
l and KRDl are the total
labor and capital employed in this sector, exclusively located in the leadercountry.
This equation encompasses two alternative approaches in R&Dendogenous growth models and, in particular, in those models dealingwith non-renewable natural resources:
. The standard model a la Romer when j ¼ 1 and Z ¼ 0. The productionof new technology is linear in labor and shows knowledge spillovers ofdegree one (see, for example, Scholz and Ziemes 1999). Thismodelization implies a ‘knife-edge’ assumption and shows strongscale effects, with little empirical support (Jones 1995a, 1995b).
. The model of a growth-essential resource when j; Z 2 ð0; 1Þ. Thecreation of new technology does not only require labor, but alsocapital stock (machinery). Because the capital stock is produced in themanufacturing sector using natural resources, new discoveries areindirectly affected by the harvested resource. In this sense, the resourceis growth-essential as stated by Groth (2007). The ‘knife-edge’assumption of CRS is not necessarily true (except when jþ Z ¼ 1)and the Jones’ critique is alleviated.
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Since both innovations and the extracted resource are needed for outputproduction in both countries, a pattern of trade may arise where the flow ofinnovation developed in the leader, in the form of patents to produce newintermediate goods, is purchased in exchange for the natural resourceextracted in the follower country.
The decentralized economy presents two markets failures. First, apositive externality captured by knowledge spillovers in the R&D sector.Second, an inefficiency in the form of monopolistic competition in theintermediate goods sectors.
2.1. Optimal equilibrium in a decentralized framework
This subsection explains the functioning of all different sectors at theequilibrium, as well as the optimal behavior of the infinitely lived households.
2.1.1. The resource sector
Assuming well-defined property rights, the extracting companies, which areowned by households in the follower country, supply the resource to finaloutput producers in the leader and the follower countries at a given marketprice, pR. The dynamics of the stock of the non-renewable resource, S, isgiven by:
_S ¼ �R; Sð0Þ ¼ S0
where R ¼ Rl þ Rf is the total harvesting of the extracting companies, whichis used by final output producers in leader and follower countries.
For tractability, the extraction is carried out at no cost.6 Then, the first-order conditions of the dynamic maximization problem of the representativeextracting firm lead to the standard Hotelling rule. This rule says that thegrowth rate of the resource price equates the rate of return on capital in thefollower economy, where the resource is being harvested:
gpR ¼ rf ð4Þ
where rf is the time-dependent interest rate in the follower country. Here andhenceforth, gx denotes the growth rate of variable x.
2.1.2. The final good sectors
Assuming perfect competition, producers of final output in the leader andfollower countries maximize instantaneous profits:
max LYl;Rl;XlðjÞYl � wlL
Yl � pTpRRl �
Z N
0
plðjÞXlðjÞ dj ð5Þ
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max LYf;Rf;XfðjÞYf � wfL
Yf � pRRf �
Z N
0
pfðjÞXfðjÞ dj ð6Þ
where wi is the salary and piðjÞ the price of the intermediate good of type j.Although labor and the intermediate goods are country specific, the resourceharvested in the follower country is used by manufacturers in botheconomies at a price pR: Variable pT is the relative value of the follower’sfinal output with respect to the leader’s. Henceforth, it will be called theterms of trade. Because quantities in different countries are measured indifferent units, variable pT allows us to scaling all the variables in terms ofYl.
Instantaneous profit maximization leads to the standard equalitiesbetween factor prices and marginal productivities:
aAðXlðjÞÞa�1ðLYl Þ
1�a�bRbl ¼ p lðjÞ ð7Þ
ð1� a� bÞ Yl
LYl
¼ wl ð8Þ
bYl
Rl¼ pTpR ð9Þ
~a ~AðXfðjÞÞ~a�1ðLYf Þ
1�~a�~bR~bf ¼ pfðjÞ ð10Þ
ð1� ~a� ~bÞ Yf
LYf
¼ wf ð11Þ
~bYf
Rf¼ pR ð12Þ
2.1.3. The R&D sector and the market of patents
The innovative sector is characterized by a large number of symmetricresearch firms operating under perfect competition. An innovative firmcreates new designs according to the function:
_Ni ¼ B KRDli
� �ZðLRDli Þ
1�ZNj
where KRDli is the capital stock the households rent to the representative
researcher at the rate of return, rl; LRDli is the labor force hired at wage wl.
There is a positive externality of the total amount of new goods which hasalready been invented, N, on the production of inventor i, who is unawarethat his new inventions increase the global stock of knowledge. Because weare assuming symmetric researchers, the increment in the total number
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of varieties can be written as in equation (3) where KRDl ¼
Pi
KRDli and
LRDl ¼
Pi L
RDli .
The representative firm in the innovative sector maximizes instantaneousprofits given by
max LRDli;KRD
lipRDi ¼ pRD B KRD
li
� �ZðLRDli Þ
1�ZNj � wlLRDli � rlK
RDli ð13Þ
where pRD are the revenues that the firm obtains from an innovation.Every new innovation is linked to the production of a particular
intermediate good in each country. Thus, an innovator in the leadereconomy who discovers a new intermediate good sells two patents, onepurchased by the producer of the intermediate good in the leader, at a pricepRDl ; and one by the producer in the follower, at a price pRD
f . Therefore,pRD ¼ pRD
l þ pRDf :
Under the assumption of perfect competition, the equilibrium ischaracterized by the equality of input prices and marginal profits
wl ¼ 1� Zð ÞpRD B KRDl
� �ZLRDl
� ��ZNj ð14Þ
rl ¼ ZpRD B KRDl
� �Z�1LRDl
� �1�ZNj ð15Þ
2.1.4. The intermediate goods sectors
The production of an intermediate good uses capital as input according tothe same one-to-one production technology often assumed in the literature:
XlðjÞ ¼ KIMl ðjÞ; XfðjÞ ¼ KfðjÞ; j 2 ½0;N� ð16Þ
where KIMl ðjÞ and KfðjÞ denote the capital used to produce the intermediate
good of type j 2 ½0;N� in the leader and the follower countries, respectively.The producer of an intermediate good purchases the patent from the
innovator. This patent has an infinite duration, and domestic andinternational protection of these patents gives the producer of eachintermediate good a monopoly power within his country borders. Themonopolistic producer of the intermediate good of type j 2 ½0;N� in countryi 2 fl; fg rents capital to households at price ri and chooses the price (orquantity) of the intermediate good, knowing the demand made by finaloutput producers (in equations [7] or [10]), in order to maximizeinstantaneous benefits, piðjÞ. Expressing the maximization problem in termsof quantities:
max XlðjÞplðjÞ ¼ plðjÞXlðjÞ � rlKIMl ðjÞ; max XfðjÞpfðjÞ ¼ pfðjÞXfðjÞ � rfKfðjÞ
s:t: : ð7Þ and ð16Þ s:t: : ð10Þ and ð16Þ
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Optimality conditions equate the marginal profits and the rate of returnon capital:
a2AðXlðjÞÞa�1ðLYl Þ
1�a�bRbl ¼ rl ð17Þ
~a2 ~AðXfðjÞÞ~a�1ðLYf Þ
1�~a�~bR~bf ¼ rf ð18Þ
Therefore, the produced amount and the price paid for any intermediategood are independent of the type j 2 ½0;N�:
XlðjÞ ¼ Xl; XfðjÞ ¼ Xf; p l ¼ rl=a; pf ¼ rf=~a ð19Þ
From equation (16), the capital stock employed by each firm in theintermediate good sector is the same for every type j:
KIMl ðjÞ ¼ KIM
l =N; KfðjÞ ¼ Kf=N ð20Þ
To purchase the patent, the monopolistic producers must issue bonds,which are bought by households obtaining an interest revenue determinedby the rate of return, ri,i 2 fl; fg. Assuming free entry into this industry, theprice for the patent in each country must equate the ongoing profits of themonopolistic producer:7
pRDl ðtÞ ¼
Z 1t
plðjÞe��rlðt; tÞðt�tÞ dt ð21Þ
pRDf ðtÞ ¼ pT
Z 1t
pfðjÞe��rfðt; tÞðt�tÞ dt ð22Þ
where �rlðt; tÞ ¼ ½1=ðt� tÞ�R tt rlðsÞ ds, and �rfðt; tÞ ¼ ½1=ðt� tÞ�
R tt rfðsÞ ds are
the average interest rates between times t and t in each economy,respectively.
The total benefits obtained by the monopolistic producers of a specificintermediate good in the leader and the follower countries are transferred tothe firm which discovered the good, located in the leader.
Plugging equations (16) and (20) into equations (1) and (2):
Yl ¼ AðKIMl Þ
aN1�aðLYl Þ
1�a�bRbl ð23Þ
Yf ¼ ~AðKfÞ~aN1�~aðLYf Þ
1�~a�~bR~bf ð24Þ
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From equations (23), (24), and the optimality conditions in the finaloutput sector and the intermediate goods sectors, the rates of return oncapital in each economy can be derived:
rl ¼ a@Yl
@KIMl
¼ a2Yl
KIMl
; rf ¼ ~a@Yf
@Kf¼ ~a2
Yf
Kfð25Þ
Due to the existence of imperfect competition the rate of return oncapital is lower than its marginal productivity. The capital is underpaid,compared to the competitive case. The existence of positive benefits allowsintermediate goods producers to compensate the investment in the R&Dsector.
2.1.5. Households
We assume that the capital stock as well as the bonds issued by theproducers of intermediate goods to buy patents cannot be internationallyexchanged. In consequence, the households in each economy hold thecapital stock used and the bonds issued in this particular country. Thehouseholds’ wealth in each country reads:
Vl ¼ pRDl Nþ Kl; Vf ¼
pRDf
pTNþ Kf
Because the economy is populated by a large number of identicalindividuals, the decision of a single agent regarding consumption can beinterpreted as the aggregate consumption. Households in each countrymaximize their stream of discounted utility, subject to their budgetconstraints:
max Cl
R10
ClðtÞð Þ1�s�11�s e�rt dt max Cf
R10
CfðtÞð Þ1�~s�11�~s e�~rt dt
s:t: : _Vl ¼ rlVl þ wlL� Cl s:t: : _Vf ¼ rfVf þ wf~Lþ pRR� Cf
Vlð0Þ ¼ Vl0 Vfð0Þ ¼ Vf0
ð26Þ
Consumer preferences in each country are represented by isoelasticutility functions, which depend on consumption. Cl and Cf stand for thetotal consumption in the leader and follower economies, and s and ~s are theinverse of their elasticities of intertemporal substitution of consumption.8
Thus, we are assuming identical consumer preferences within a country,although they may differ between countries. Utility flows are discounted atrates r and ~r, respectively.
Capital stock and bonds are considered perfect substitute assets thatcan be gathered together in Vi,i 2 fl; fg. These assets yield a return, ri, whichmay differ across countries. Household revenues are also increased by labor
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income. The total labor in each economy is assumed constant, L and ~L.Besides, in the follower economy households are the owners of the extractivefirms and get equal share of the profits from harvesting.
Intertemporal optimization in both countries leads to the Ramsey rules:
gCl¼ 1
sðrl � rÞ; gCf
¼ 1
~sðrf � ~rÞ ð27Þ
which define the growth rates of consumption in both countries.In a closed economy, the final output that is not consumed determines
the investment in capital stock. In an open economy, positive (negative) netexports also foster (restrain) capital growth. Net exports in the leadingeconomy are given by the total value of the patents sold to the followereconomy minus the value of the resource imported from this country. Thus,the dynamics of the capital stock reads:
_Kl ¼ Yl � Cl � pTpRRl þ pRDf
_N ð28ÞLikewise, in the follower country capital accumulation can be expressed
as:9
_Kf ¼ Yf � Cf þ pRRl �pRDf
_N
pTð29Þ
3. Existence and uniqueness of a sustained balanced growth equilibrium
In the two-country trade economy we have just described, a perfect-foresightbalanced growth equilibrium is defined as follows:
Definition 3.1. In a perfect-foresight balanced growth equilibrium, each agentbehaves optimally (taken as given the time paths of variables out of theircontrol), and the growth rates of all variables are constant.
Note that a zero or positive growth rate of the extractions, R, could notbe sustained because of the finiteness of the natural resource used in theproduction of final output in both countries. To be more specific, in thissection we focus on a sustained balanced growth equilibrium (SBE), which isdefined as follows:
Definition 3.2. A SBE is a perfect-foresight balanced growth equilibrium with apositive growing consumption in both countries, and a continuous decay in theextractions of the non-renewable resource.
Furthermore, we focus on equilibria at which trade between the twocountries is balanced. The balanced trade equation reads:
pTpRRl ¼ pRDf
_N ð30Þ
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The value of the resource traded from the follower to the leader equalsthe value of patents traded on the opposite direction. Both are measured inthe same units of output (Yl in equation [30]).
In the leading economy, labor and capital are shared between the R&Dsector and either the final output sector or the intermediate goods sector.Conversely, in the follower country, these two inputs are only employed inthe production of final output. Therefore:
LYl þ LRD
l � L; LYf � ~L
KIMl þ KRD
l � Kl; KIMf � Kf
In the equilibrium, because of the Inada conditions, all available laborand capital are employed. Thus, denoting by u and v the shares of labor andcapital employed in manufacturing, we may rewrite:
LYl ¼ uL; LRD
l ¼ ð1� uÞL;LYf ¼ ~L
KIMl ¼ vKl; K
RDl ¼ ð1� vÞKl;K
IMf ¼ Kf
Proposition 3.3. Along a perfect-foresight balanced growth equilibrium, theoutput, the consumption, the wage earnings, and the capital stock grow at thesame constant rate in each country. Furthermore, the shares of labor in theinnovative and the final output sectors and the rates of return in each countryremain constant.
g ¼ gKl¼ gKRD
l¼ gKIM
l¼ gYl
¼ gCl¼ gwl
~g ¼ gKf¼ gYf
¼ gCf¼ gwf
gu ¼ g1�u ¼ gv ¼ g1�v ¼ grl ¼ grf ¼ 0
Proof. See Appendix 1.From the production functions (23), (24), and Proposition 3.3, on a SBE
the following equalities hold
g ¼ gN þb
1� agRl; ~g ¼ gN þ
~b1� ~a
gRfð31Þ
Since production depends on the resource, an endless growth inproduction and consumption jointly with a declining resource requires acontinuous rise in knowledge. That is, the reduction in the non-renewable
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resource should be compensated by the technological progress in the form ofa larger increment in the number of varieties of intermediate goods. This isshown in equation (31), where b=ð1� aÞ and ~b=ð1� ~aÞ are the ratios of theoutput elasticities of resource and knowledge, considering the production offinal output as in equations (23) and (24), with capital, labor, and naturalresources as inputs.
Although we will come back to the standard model a la Romer, in theremainder of this section, we focus on the model of growth-essentialresource, i.e. j; Z 2 ð0; 1Þ. Along a SBE, the number of varieties ofintermediate goods, N, grows at a constant rate, although different fromthe growth rate of the economy. Likewise, the growth rates of the amount ofresource used by final output producers in each country, Rl, Rf, can bewritten in terms of the growth rates of the economies.
Proposition 3.4. Along a SBE if j; Z 2 ð0; 1Þ, then:
gN ¼Z
1� jg ð32Þ
gRl¼ 1� a
b1� j� Z1� j
g ð33Þ
gRf¼ 1� ~a
~b~g� Z
1� jg
� �ð34Þ
Proof. See Appendix 1.From Proposition 3.4, it is straightforwardly derived that a positive
economic growth together with a decrement in the use of the resource is onlyfeasible under the following condition, assumed henceforth:
Assumption 1.
1� Z < j < 1 ð35Þ
Note that producible inputs show CRS in the final output sector inboth economies and, on a SBE the use of the resource must becontinuously reduced. Then, first inequality in Assumption 1 states thatthe existence of such SBE requires IRS in the innovative sector, tocompensate the decrease in the use of the resource. That is, producibleinputs must show ‘globally’ IRS (considering the research and the finaloutput sectors together). Inequality j < 1 states that the knowledgeexternality in the research sector is less than linear as suggested by Jones(1995a).
For the total harvesting to decrease at a constant rate along a SBE, therate of reduction in extractions must be the same in both countries.
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Considering this, we can compute the gap between the growth rates of thetwo economies, which determines the growth rate of the terms of trade.
Proposition 3.5. Along a SBE, the growth rate of the terms of trade is given bythe difference between the growth rates of the leading and the followereconomies, which can be expressed as:
gpT ¼ g� ~g ¼ O; g ¼ � 1� j� Z1� j
~b1� ~a
1� ab� 1� ~a
~b
� �g ð36Þ
Proof. See Appendix 1.The producers of final output in the two economies use the same non-
renewable resource extracted in the follower region. Assuming perfectcompetition, the marginal productivity of the resource equals its price inequations (9) and (12), measured in units of YL for the leader, and in units ofYF for the follower. The price paid for the resource in the leader regionequates the price paid in the follower times the terms of trade. In consequence,the evolution of the terms of trade is given by the gap in the growth rates of themarginal productivity of the resource in leader and follower countries. Sincethe use of the resource decreases at the same rate in both economies, themarginal productivity of the resource might evolve differently exclusivelydriven by a divergence in final output production.
In the follower, the economy growth is driven by the technologydeveloped in the leader and is decelerated by the reduction in the use of theresource, which decreases at the same speed as in the leader region.Therefore, a faster growth in the leader country is linked with aproportionally faster growth in the follower economy. The country with asmaller output elasticity of the resource, in relative terms to the outputelasticity of knowledge, will experience a faster growth. Under equation(35), if O > 0, the ratio of output elasticities of resource and knowledge islower in the leader. This country is less dependent on the resource andtherefore, from equation (36), it will experience a faster growth than thefollower. The opposite is true for O < 0.
From equation (36), it immediately follows that the two economiessimultaneously attain a positive growth only if O is lower than one. Fromnow on, this condition is assumed to be satisfied, and can be rewritten asfollows.10
Assumption 2.
j < 1� Zþ Zb
1� a1� ~a
~bð37Þ
Since the follower country must reduce the use of the resource at thesame rate as the leader, the two economies cannot stand excessively apart, as
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stated in Assumption 2. If the leader does experience a positive growth, thenAssumption 2 guarantees that the follower will also grow. Note that whena ¼ ~a and b ¼ ~b, Assumption 2 reduces to Assumption 1.
The following proposition derives the expressions of the rates of returnson capital both in the leading and the follower countries. Once these ratesare known, the growth rates of production in the leader and followereconomies, and, hence, the growth rates of all variables are fully determinedas functions of the model parameters.
Proposition 3.6. Along a SBE, the growth rates in the leader and the followercountries are given by:
g ¼~r
f� ~sð1� OÞ ;~g ¼
~rð1� OÞf� ~sð1� OÞ ð38Þ
where
f ¼ 1� ð1� aÞð1� ~a� ~bÞð1� ~aÞb þ 1
!1� j� Z1� j
> 1
Proof. See Appendix 1.Given consumers’ preferences in one country, the corresponding Ramsey
equation in equation (27) can be interpreted as the rental price of the capitalstock, ri, i 2 fl; fg, that consumers would charge at a given growth rate ofthe economy ( g and ~g). On the productive side, the rental price of the capitalstock affects the price of the intermediate goods and the benefits of themonopolistic producers of these goods, however, it has no direct effect onthe long-run growth rate. Nevertheless, since the resource is harvested in thefollower economy, the rate of return in this economy governs the evolutionin the extraction of the resource and therefore, indirectly affects the growthrates in the leader and the follower countries along a SBE. If consumers inthe follower economy are more impatient or more reluctant to substitutepresent for future consumption (higher ~r or ~s), they would demand a higherrate of return, rf, for a given growth rate. For a higher rf, the resource wouldbe more intensively harvested until a new SBE is reached, which ischaracterized by a higher growth rate of the resource price (Hotelling rule).In an equilibrium (equations [9] and [12]), the marginal productivity of theresource must grow at a faster rate and, due to the IRS hypothesis, thiscomes together with a greater growth rate and a resource whose usedecreases more rapidly both in the leader and in the follower countries.Conversely, if consumers in the leader economy demand a higher rate ofreturn, rl, this would increase the marginal profit earned by monopolies, butwould have no effect on the rate of extraction of the resource and therefore,it would neither affect its own growth rate nor the follower’s.
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From Proposition 3.6, a necessary and sufficient condition for a positivegrowth rate in the leading economy immediately arises.
Assumption 3.
~s < 1� 1� abð1� OÞ
1� j� Z1� j
ð39Þ
Assumption 3 requires that the elasticity of the intertemporal substitu-tion in the follower country should not be excessively low. That is, they arenot too reluctant to postpone consumption.11 Savings from highly inelasticconsumers would not be sufficient to promote the necessary investment intechnological improvements. Thus, sustained growth with a limited resourcewould not be possible.
3.1. Existence and uniqueness of the sustained balanced equilibrium
The state of the two-country trade model is described by the state variablesKl, Kf, S, and N, whose behavior can be controlled by elections on variablesCl, Cf, u; Rl, and Rf, the control variables of the model. Because we want toallow for non-zero growth rates on a SBE, it will be convenient to rewritethe dynamics of our system in terms of variables that remain stationarywhen the levels of output, capital, knowledge, and consumption grow atpositive rates forever, while the extraction of the resource decreases. Notethat in a SBE, the following variables
cl �Cl
Kl;cf �
Cf
Kf; yl �
Yl
Kl;yf �
Yf
Kf; u; w � gN; and t � Rf
Rl
remain constant as stated in Propositions 3.3 and 3.4. The dynamic solutionto our model can be transformed into a system of seven ordinary differentialequations ( _cl, _cf, _yl, _yf, _u, _w; _t). Four of these variables are control-likevariables (cl, cf, u, and t) and three state-like variables (yl, yf, and w), andonce the former variables are chosen, the three remaining variables aredetermined by the state variables Kl;Kf, and N.12 In Appendix 2, we derivethe growth rates for these variables, obtaining the following system of sevendifferential equations:
gcf ¼~a2
~s� 1
� �yf þ cf �
~r~s
ð40Þ
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gyf ¼1� ~a
1� ~bwþ ð1� ~aÞ
~a~b
1� ~b� 1
!yf þ
1� ~a� ~b
1� ~bcf ð41Þ
gt ¼ �gw þ1
1� ~b~að1� ~aÞyf � ~acf� �
þ 1� ~a
1� ~b� 1þ ð1�
~aÞ~a~b
t� �
w ð42Þ
gw ¼ �Zvþ ð1� ZÞu
1� ugu þ ðj� 1Þwþ Z yl � clð Þ ð43Þ
gcl ¼a2
s1
v� 1
� �yl þ cl �
rs
ð44Þ
gyl ¼ a1� v
1� uþ 1� a� b
� �gu þ ða� 1Þ yl � clð Þ þ bgw
þ b 1� ð1�~aÞ~a
~bt
� �þ 1� a
� �w ð45Þ
gu ¼ PðuÞ Y u; tð Þwþ a� ð1� bÞZ� a2
v
� �yl � a� ð1� bÞZð Þcl
� ð46Þ
where
v ¼ KIMl
Kl¼ a2
a2 þ ð1� a� bÞ Z1�Z
1�uu
ð47Þ
x ¼ pRDl
pRD¼ 1� ð1� ZÞ b
1� a� bu
1� uð48Þ
PðuÞ ¼ xð1� uÞ1� x að1� vÞ þ ð1� a� bÞð1� uÞ þ ð1� bÞðZvþ ð1� ZÞuÞ½ � ð49Þ
Y u; tð Þ ¼ 1� a� ð1� bÞj� bð1� ~aÞ~a
~btþ að1� aÞ
bð1� xÞ
xð50Þ
A SBE exists if and only if _cl ¼ 0, _cf ¼ 0, _yl ¼ 0, _yf ¼ 0, _u ¼ 0,_w ¼ 0, and _t ¼ 0. The following proposition proves its existence anduniqueness.
Proposition 3.7. Under Assumptions 1, 2, and 3, there exists a unique SBE forthe economy with u� 2 ð0; 1Þ and c�f ; y
�f ; y�l ; w�; t� > 0. Furthermore, s > 1 or
r > rþ, where rþ is given in equation (B21), are two alternative sufficientconditions which ensure that c�l > 0.
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Proof. See Appendix 2.
4. Conditional stability with IRS and a growth-essential resource
In this section, we analyze the stability of the unique SBE found in theprevious section. That is, although the economies might eventually growindefinitely, could this sustained growth path be reached if the economy isnot initially in equilibrium?
Because the system of seven differential equations (40)–(46) has fourcontrol-like variables (cl; cf; u; t) and three state-like variables ( yl; yf; w), thestability of the SBE requires that the dimension of the stable manifold isgreater than or equal to three.
Proposition 4.1. Under Assumptions 1, 2, and 3, the sign of the determinant ofthe Jacobian matrix at the SBE is the same as the sign of Pðu�Þ, where PðuÞ isgiven in equation (49).
Proof. See Appendix 3.The sign of Pðu�Þ has been numerically computed allowing parameters
a; b; Z to take values in the interval ð0; 1Þ in a grid of 106 points. It alwaystakes a positive value.
Claim 4.2. Under Assumptions 1, 2, and 3, the dimension of the stablemanifold associated with the SBE is an even and non-null number.
According to Proposition 4.1 and Claim 4.2, the dimension of the stablemanifold is not equal to 3, which would guarantee convergence from any setof initial conditions, under unique policy functions for the four controls. Agreater dimension would mean indeterminancy in the policies that couldlead the economy towards the SBE, while a lower dimension would implydivergence of the solution except for very particular initial conditions. Tocompletely determine the dimension of the stable manifold, we rely onnumerical simulations.
Claim 4.3. As long as Assumptions 1, 2, and 3 hold, the number of negativeeigenvalues of the Jacobian matrix at the SBE is always equal to 2.
The number of negative eigenvalues of the Jacobian matrix at the SBEhas been computed 105 times. In each iteration, parameters take valuesrandomly from uniform distribution functions within intervals:a; ~a; b; ~b; Z;j 2 ð0; 1Þ; s; ~s 2 ð0; 2Þ. The discount rates r; ~r are set equal to0.01, although a sensitivity analysis in these parameters shows no change inthe results.
Therefore, the SBE can only be attained if the initial conditions of thestate variables, Kl0;Kf0, and N0 are such that the starting point in the three-dimensional space (yl; yf; w) lies on the two-dimensional stable manifold
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converging to the saddle-point equilibrium.13 Any other initial conditionoutside this two-dimensional stable manifold would not lead the systemtowards the SBE, no matter what optimal policies in cl; cf; t, and u areinitially chosen.
Note that now, once we remove the requirement that the joint returns toscale on capital and technology in equation (3) must be exactly equal to 1, anew ‘knife-edge’ condition appears. An economy initially in disequilibriumwould not converge towards the long-run SBE except for initial values of thestates lying on the two-dimensional stable manifold.
We have considered a heterogeneous world. A two-country trade model,where the resource harvested in one country is traded in exchangefor innovation developed in its counterpart. Does this trade relationshipalways imply instability? To answer this question, we analyze the same two-country trade model under the ‘knife-edge’ assumption of CRS in manmadeinputs.
Trade and the standard model a la Romer for the creation of new ideas. Ifthe production of new technology follows the standard model a la Romer,the exhaustible resource is not growth-essential. In this case, the productionof new ideas is linear in labor and shows knowledge spillovers of degree one,j ¼ 1; Z ¼ 0 (see, for example, Scholz and Ziemes 1999):
_N ¼ BLð1� uÞN
Following the equivalent reasoning as in the case of a growth-essentialresource (in Appendix 3), the balanced growth path for the two economiescan now be described by a system of six differential equations in variables cl,cf, yl, yf, u, and t ¼ Rf=Rl. Variable w ¼ gN ¼ BLð1� uÞ is now completelydetermined by u. The system is described by equations (40), (41), (42), (44),(45), (46), assuming j ¼ 1; Z ¼ 0. Only two variables in this system, yl andyf, are state-like variables.
The equivalent numerical analysis is carried out for this standard modela la Romer with a non-growth-essential resource. This analysis shows that
Claim 4.4. The Jacobian matrix at the SBE always has at least two negativeeigenvalues, and three or four for some parameter values.
Therefore, for any set of initial conditions, there always exists at leastone path converging toward the SBE. This result allows us to reject the ideathat the completely specialized trade comes always together with instabilityproblems. If the SBE had a two-dimensional stable manifold, it wouldguarantee that the converging path is unique, whereas indeterminacy wouldarise in case of a three- or four-dimensional stable manifold. In this last case,for a given initial position of the economy, there exists an infinite number ofpaths converging to the same SBE.
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5. Comparison of stability results
In this section, we briefly comment how the introduction of the internationaltrade and the consideration of IRS in manmade inputs with a growth-essential resource affect the stability of the steady-state equilibrium. Wecompare the stability results of a closed economy with those of a globaleconomy with two trading countries. For these two scenarios, we distinguishbetween the standard model a la Romer with CRS, and a model with agrowth-essential exhaustible resource and IRS. Stability results aresummarized in Table 1.
Let us start with a closed economy and a standard R&D-basedendogenous growth model a la Romer with CRS in manmade inputs.Arnold (2000a, 2000b) proved the saddle-path stability of the Romer model,which for any initial condition guarantees the existence of a unique pathconverging to the long-run equilibrium. However, a stream of the literature(starting with Benhabib and Perli 1994; Benhabib et al. 1994) argues that theexistence of sector-specific externalities and/or inefficiencies may lead toindeterminacy which means that there is not a single but several pathsconverging to the same long-run equilibrium.14 This possibility of eithersaddle-path stability or indeterminacy when an exhaustible resource is usedas a productive input in the Romer model is highlighted by Scholz andZiemes (1999). In this paper, this benchmark scenario is modified in twodirections, both of which make it more difficult to achieve the long-runequilibrium.
First, following the Jones critique, the production of new technologydoes not need to be linear in the existing technology and, furthermore, itmay utilize capital (in our specification, this is equivalent to the assumptionof a growth-essential resource). In the Romer specification, due to thelinearity in the R&D sector, the growth rate of technology is fullydetermined by the share of labor used in this sector. In contrast, when thecapital stock enters the production of new technology, the growth rate ofnew inventions becomes a new state variable, increasing the requireddimension for a saddle-path stable manifold. Nevertheless, Eicher andTurnovsky (1999) showed the saddle-path stability in this setting. We
Table 1. Stability of the sustained balanced growth equilibrium.
Closed economy Two-trading economies
Model a la Romer Indeterminacy orSaddle-path stability(Scholz and Ziemes 1999)
Indeterminacy orSaddle-path stability
IRS and growth-essentialresource
Saddle-path stability(Cabo et al. 2010)
Conditional stability
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further assume general IRS in manmade inputs, which in a central plannersolution would lead to instability (see, for example, Groth and Schou 2002).Conversely, Cabo et al. (2010) proves that saddle-path stability is stillfeasible in a market economy which does not internalize the knowledgeexternality in the R&D sector and the inefficiency of monopolisticcompetition.
Second, stability results are re-examined, giving entrance to internationaltrade. The trade structure in our model considers fully specialized countries,which trade new technology for the natural resource. The exhaustibleresource harvested in the follower is used as a productive input in bothregions, while growth is driven by the innovation developed in itscounterpart. Thus, although the two economies are different, for theexistence of a SBE, they are constrained to reduce the use of a commonresource at the same rate in the long run. This adds a constraint that was notpresent under autarky, and hence reaching the steady-state equilibriumbecomes more difficult.
When studying a two-country trade model for the standard Romer’sspecification, our numerical analysis shows that a SBE exists and it is saddle-path stable with a stable manifold with a dimension large enough to ensureconvergence in the long run. As for a closed economy, the long-run growthpath can be reached for any initial state of the economy, and still the policiesleading the economy towards it can be unique or infinite, depending on theparameter values.
For the two-country trade model with IRS and a growth-essentialresource, a feasible SBE still exists and is unique. However, we havenumerically shown that although there exists a stable manifold convergingto this SBE, its dimension is not large enough to ensure the convergence ofthe economy towards the long-run SBE for any initial position.Convergence towards the SBE is, in general, unfeasible unless the initialstate of the system lies on the two-dimensional stable manifold (conditionalstability).
The externalities in the research sector and the inefficienciesassociated with monopolistic competition make indeterminacy feasible.Conversely, two hypotheses make stability more difficult to achieve:first, an economy with a growth-essential resource and IRS onmanmade inputs; and second, a bipolar trade between two hetero-geneous countries restricted to reduce the use of the exhaustibleresource at the same speed. Each of the hypotheses on its own is notsufficient to impede saddle-path stability. However, the two hypothesestogether lead to a lack of stability, except for very particular initialconditions. The initial position of the two economies must fulfill acertain condition regarding the initial level of technology and capitalstocks in each country.
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6. Conclusions
We analyze the sustainability of economic growth when the production offinal output requires the use of a non-renewable natural resource. Our studydoes not focus on an isolated closed economy, but on a world withheterogeneous countries, described by a two-country trade model withcomplete specialization: innovation in the technological leader country andresource harvesting in the follower.
The standard model a la Romer is characterized by the ‘knife-edge’hypothesis of CRS to producible or manmade inputs in the R&D sector.More specifically, the creation of new technology depends on labor and aknowledge externality, which is assumed linear in the existing stock oftechnology. We take a more general approach in two aspects. First, newinnovations are also dependent on the capital stock and hence indirectly onthe resource, which is called a growth-essential resource. Second, theknowledge externality is not necessarily linear and, furthermore, the jointreturns to scale in capital and technology are not necessarily constant, butinstead, we assume IRS in these manmade inputs.
In a world economy described by a completely specialized bipolartrade, and production technologies with IRS, we prove that a SBE existsand that it is unique. However, the long-run equilibrium can only bereached if the two economies happen to be initially located in the stablemanifold (a subset of the state-space), which turns out to be a new ‘knife-edge’ condition, but now on the initial state values. In a world ofheterogeneous trading countries with complete specialization, sustainedeconomic growth is a ‘knife-edge’ possibility. Either the returns to scale tomanmade inputs in the creation of new technology are constant or, underIRS, the initial conditions for the state variables turn out to lie in a subsetof the state space.
The main result of the paper is to call into question the sustainability ofeconomic growth when this is dependent on the use of an exhaustibleresource (like fossil fuel). Economic growth is sustained if the economy iscapable of innovating at a fast enough rate under the hypothesis of CRS inproducible inputs. Considering IRS instead does not necessarily impedesustained balanced growth. However, sustainability becomes an issue if weescape from the closed economy scenario and analyze a global economycharacterized by the bilateral trade between a country endowed with anexhaustible natural resource which does not innovate, and a technologicallyadvanced country. The closer the trade structure to this completelyspecialized trade and the more likely the assumption of IRS to manmadeinputs, the more difficult it is for the world economy to reach a SBE. In aworld economy characterized by IRS to manmade inputs and completelyspecialized trade, a SBE is difficult to reach. By contrast, if either returns toscale are constant or economies are closed to trade, adequate policies can be
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implemented to put the economy on a SBE. The study of the stability of theSBE for a non-completely specialized trade is an interesting topic for furtherresearch.
Acknowledgements
The authors have been partially supported by MICINN and JCYL under projectsECO2008-01551/ECON, ECO2011-24352 and VA001A10-1. We thank two anon-ymous reviewers for their helpful comments.
Notes
1. According to Coe et al. (1997), in 1990, industrial countries accounted for 96%of the world’s R&D expenditure; and 90% of the world’s patents originate inthe US, Japan, Germany, France, and the UK.
2. 90% of the world oil reserves are concentrated in 15 countries (only two ofwhich are industrialized countries). Source: US Energy Information Adminis-tration. Independent Statistic and Analysis. http://www.eia.doe.gov/emeu/international/contents.html
3. As Jones (2005) defines, ‘strong scale effects’ arise when the growth rate of theeconomy is an increasing function of scale (which typically means population).In models that exhibit ‘weak scale effects’, the level of per capita income in thelong run is affected, to some extent, by the size of the economy.
4. All variables depend on the time argument, omitted when no confusion canarise. Parameters will be highlighted.
5. Although the number of varieties of intermediate goods, N, is an integer, it isapproximated here (as often done in the literature, see, for example, Barro andSala-i-Martin 2004) by a continuous variable, obtaining essentially the sameresults.
6. This is a usual assumption in this literature (see, for example, Scholz andZiemes 1999; Groth and Schou 2007). It is equivalent to a constant cost per unitof extracted quantity, with pR being then the net resource price.
7. Ongoing profits of the monopolistic producer in the follower country are scaledby the terms of trade, pT, so the price of the patent in this economy, pRD
f ðtÞ, ismeasured in units of Yl, just like as the price of the patent in the leader.
8. For s ¼ 1 or ~s ¼ 1, a logarithmic utility function is considered.9. Notice that the dynamics of the capital stock in each ‘economy is expressed in
terms of this economy’s output.10. For O < 0, Assumption 2 trivially holds under Assumption 1. Conversely, for
O > 0, Assumptions 1 and 2 merge to define a non-empty interval for j:1� Z < j < 1� Zþ Z b
1�a1�~a
~b:
11. From Assumptions 1 and 2, it follows that the upper bound for ~s inequation (43) is greater than 1. Therefore, this condition is not excessivelyrestrictive.
12. As defined in Mulligan and Sala-i-Martin (1991, 1993), state-like and control-like variables are transformations of the original state and control variablesthat, contrary to the original ones, do not grow on a balanced path.
13. This two-dimensional stable manifold is defined by the SBE and thecorresponding components (yl; yf; w) of two eigenvectors associated with thetwo negative eigenvalues.
14. This would explain different realizations for similar countries starting fromidentical initial conditions, which still converge to the same steady state.
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References
Arnold, L. 2000a. ‘‘Endogenous Technological Change: A Note on Stability.’’Economic Theory 16: 219–26.
Arnold, L. 2000b. ‘‘Stability of the Market Equilibrium in Romer’s Model ofEndogenous Technological Change: A Complete Characterization.’’ Journal ofMacroeconomics 22, no. 1: 69–84.
Backus, D.K., P.J. Kehoe, and F. Kydland. 1994. ‘‘Dynamics of the Trade Balanceand the Terms of Trade: The S-Curve.’’ American Economic Review, 84, March.
Barro, R., and X. Sala-i-Martin. 2004. Economic Growth. 2nd ed. Cambridge, MA:MIT Press.
Benhabib, J., and R. Perli. 1994. ‘‘Uniqueness and Indeterminacy: On the Dynamicsof Endogenous Growth.’’ Journal of Economic Theory 63: 113–42.
Benhabib, J., R. Perli, and D. Xie. 1994. ‘‘Monopolistic Competition, Indeterminacyand Growth.’’ Ricerche Economiche 48: 279–98.
Cabo, F., G. Martın-Herran, and M.P. Martınez-Garcıa. 2010. ‘‘ExhaustibleResources and Fully-Endogenous Growth with Non-Scale Effects.’’ Cahier duGERAD. G-2010-33.
Coe, D.T., E. Helpman, and A.W. Hoffmaister. 1997. ‘‘North–South R&DSpillovers.’’ Economic Journal, Royal Economic Society 107 (440): 134–49.
Eicher, T.S., and S.J. Turnovsky. 1999. ‘‘Non-Scale Models of Economic Growth.’’Economic Journal 109: 394–415.
Groth, C. 2007. ‘‘A New-Growth Perspective on Non-Renewable Resources.’’ InSustainable Resource Use and Economic Dynamics, edited by L. Bretschger and S.Smulders, 127–63. Dordrecht: Springer
Groth, C., and P. Schou. 2002. ‘‘Can Non-Renewable Resources Alleviate the Knife-Edge Character of Endogenous Growth?’’ Oxford Economic Papers 54: 386–411.
Groth, C., and P. Schou. 2007. ‘‘Growth and Non-Renewable Resources: TheDifferent Roles of Capital and Resource Taxes.’’ Journal of EnvironmentalEconomics and Management 53: 80–98.
Hu, Y., and K. Mino. 2011. Globalization and Volatility Under Alternative TradeStructures. KIERWorking Papers, 791, Kyoto University, Institute of EconomicResearch.
Jones, C.I. 1995a. ‘‘R&D-Based Models of Economic Growth.’’ Journal of PoliticalEconomy 103: 759–84.
Jones, C.I. 1995b. ‘‘Time-Series Test of Endogenous Growth Models.’’ QuarterlyJournal of Economics 110: 495–525.
Jones, C.I. 2005. ‘‘Growth and Ideas.’’ In Handbook of Economic Growth, edited byP. Aghion and S. Durlauf, 1065–111. Amsterdam: North-Holland
Meadows, H.H., D.I. Meadows, J. Randers, and W.W. Benhrens. 1972. The Limitsto Growth. New York: Universe Books.
Meng, Q., and A. Velasco. 2004. ‘‘Market Imperfections and the Instability of OpenEconomies.’’ Journal of International Economics 64: 503–19.
Mulligan, C.B., and X. Sala-i-Martin. 1991. A Note on the Time-Elimination Methodfor Solving Recursive Dynamic Economic Models. NBER Technical WorkingPaper no. 116. The National Bureau of Economic Research, Cambridge, MA.
Mulligan, C.B., and X. Sala-i-Martin. 1993. ‘‘Transitional Dynamics in Two-SectorModels of Endogenous Growth.’’ Quarterly Journal of Economics 108: 739–73.
Romer, P.M. 1990. ‘‘Endogenous Technical Change.’’ Journal of Political Economy98: 71–102.
Scholz, C.M., and G. Ziemes. 1999. ‘‘Exhaustible Resources, Monopolistic Competi-tion, and Endogenous Growth.’’ Environmental and Resource Economics 37: 169–85.
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Sim, N., and K.W. Ho. 2007. ‘‘Autarkic Indeterminacy and Trade Determinacy.’’International Journal of Economic Theory 3 (4): 315–28.
Solow, R.M. 1994. ‘‘Perspectives on Growth.’’ Journal of Economic Perspectives 8:45–54.
Appendix 1. Proofs of Propositions 3.3–3.6
Proof of Proposition 3.3
The shares of labor and capital devoted either to manufacturing, u; v, or toinnovation, 1� u; 1� v, are bounded, and hence, they remain constant along abalanced growth equilibrium, gu ¼ g1�u ¼ gv ¼ g1�v ¼ 0. Furthermore from equation(27), the rates of return on capital have to be constant along a balanced growthequilibrium, grl ¼ grf ¼ 0, in order to have constant growth rates for consumption inthe leader and follower countries.
Taking into account equations (28) and (29), along a balanced growthequilibrium, the following equalities must be fulfilled:
g ¼ gKl¼ gYl
¼ gCl; ~g ¼ gKf
¼ gYf¼ gCf
:
The evolutions of wages in the leader and follower economies immediatelyfollow from the labor marginal productivity in equations (8) and (11): gwf
¼ ~g andgwl¼ g.Furthermore, from the equilibrium conditions in the R&D sector equations (14)
and (15) and knowing that along a balanced growth equilibrium g1�u ¼ grl ¼ 0andgwl
¼ g, then:
gpRDþ ZgKRD
lþ jgN ¼ g ðA1Þ
gpRDþ ðZ� 1ÞgKRD
lþ jgN ¼ 0 ðA2Þ
Therefore gKRDl¼ g, and since gKl
¼ g along the balanced growth equilibrium,then in the leading economy, the shares of the capital stock employed in themanufacturing and in the R&D sectors along the balanced growth equilibrium bothgrow at the same rate: gKRD
l¼ gKIM
l¼ gKl
¼ g.
Proof of Proposition 3.4
Along a SBE, the number of intermediate goods grows at a constant rate andtherefore, gN ¼ g _N. Thus, from equation (3), the growth rate of the leading economyand the growth rate of knowledge must be proportional as equation (32) states.
From equation (31), the growth rate of the non-renewable resource used formanufacturing in each country can be written as:
gRl¼ 1� a
bg� gNð Þ; gRf
¼ 1� ~a~b
~g� gNð Þ
Taking into account equation (32), the growth rates of the use of these resourcescan be expressed as functions only of the growth rates of final outputs, given byexpressions (33) and (34).
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Proof of Proposition 3.5
Taking into account that along the SBE, the growth rates of resource uses (Rl;Rf)and resource harvesting (R) must be the same, equality g� ~g ¼ Og immediatelyfollows from equations (33) and (34). The growth rate of the terms of trade is derivedfrom equations (9) and (12).
Proof of Proposition 3.6
From equations (A1), (A2), and (32):
gpRD¼ gpRD
f¼ gpRD
l¼ 1� Z� j
1� jg < 0
Log-differentiating equations (9):
gpT þ gpR þ gRl¼ g
Taking into account the growth rates of variables pR and Rl, the growth rate ofthe terms of trade can be expressed as:
gpT ¼ 1� 1� Z� j1� j
1� ab
� �g� rf ðA3Þ
From equations (36) and (A3), the rate of return on assets in the follower countrycan be written as a function of the leading economy’s growth rate. Thus, fromequations (27) and Proposition 3.3, the rates of return in both countries are related asfollows:
rf ¼ fg ¼ frl � rs
Furthermore, taking into account equations (27) and (36), the rates of return oncapital in leader and follower countries read:
rl ¼s~r
f� ~sð1� OÞ þ r; rf ¼f~r
f� ~sð1� OÞ
and the expressions in the statement of the proposition follow.
Appendix 2. Derivation of the differential equations system (40)–(46) and
proof of Proposition 3.7
Derivation of the differential equations system (40)–(46)
Taking into account the balanced trade equation in equation (30), from equations(28) and (29):
gKl¼ yl � cl; gKf
¼ yf � cf ðB1Þ
From equation (27) together with equations (B1) and (25), equations (40) and(44) follow.
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Dividing equation (14) by equation (15), and equation (8) by the first expressionin equation (25), gives:
1� ZZ
1� v
1� u¼ 1� a� b
a2v
uðB2Þ
which determines v as a function of u, as given in equation (47).Let us define a new variable x ¼ pRD
l =pRD. Taking into account equations (9) and(30), it is obtained that
Yl ¼1� xb
pRD_N ¼ 1� x
bpRDB ð1� vÞKlð ÞZðð1� uÞLlÞ1�ZNj ðB3Þ
Using equation (B3) in the first expression in equation (25) and taking intoaccount equation (15), x is obtained as a function of v. Considering equation (B2),expression (48) is obtained.
On differentiating equations (12) and (24), and taking into account the Hotellingrule equation (4):
gRf¼ 1
1� ~b~a yf � cf� �
þ ð1� ~aÞw� ~a2yf �
ðB4Þ
where equations (B1) and (25) have been used.On differentiating equation (24) and using equation (B4), equation (41) follows.The differentiation of equations (21) and (22) leads to
gpRDl¼ rl �
plxpRD
; gpRDf¼ gpT þ rf �
pfð1� xÞpRD
pT ðB5Þ
From the balanced trade equation in equation (30) taking into account thesecond expression in equation (B5):
gRl¼ wþ gw �
ð1� ~aÞ~a~b
wt ðB6Þ
From equations (B4) and (B6), equation (42) follows.
From equation (3), it immediately follows that gN ¼ B ð1� vÞKlð ÞZðð1� uÞLÞ1�ZNj�1; and equation (43).On differentiating equation (23) and taking into account the expressions (B1) and
(B6) it follows that
gyl ¼ ða� 1Þðgv þ yl � clÞ þ ð1� a� bÞgu þ bgw þ b 1�~að1� ~aÞ
~b
� �þ 1� a
� �w
ðB7Þ
Equation (B2) establishes thatð1� uÞgv ¼ ð1� vÞgu, then, from equation (B7),equation (45) follows.
From gpRDl
in equations (B5) and (48), an expression for gpRDis obtained.
Likewise, log-differentiating equations (8) and (14) another different expression forgpRD
is obtained. From the equalization of these two expressions, equation (46)follows.
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Proof of Proposition 3.7
From the differential equations (43) and (44), the steady-state values cl and w� can bewritten as:
w� ¼ Zj� 1
rs� a2
sv�y�l
� �; c�l ¼
rs� a2
sv�� 1
� �y�l ðB8Þ
where v� denotes the value of variable v satisfying equation (47) when u equates u�.From equation (45) and plugging the expressions in equation (B8), if w� is
different from zero:
t� ¼ð1� aÞ 1þ j�1
Z
� þ b
b~að1� ~aÞ~b ðB9Þ
The expression above is always positive under Assumption 1.
From equation (40), c�f as function of y�f is obtained:
c�f ¼~r~s�
~a2
~s� 1
� �y�f ðB10Þ
From equation (41), on replacing the expression above, the followingrelationship between the steady-state values of variables w and yf can be derived:
w� ¼~a2
1� ~a~bþ 1� ~a� ~b
~s
!y�f �
1� ~a� ~b1� ~a
~r~s
ðB11Þ
From equation (42) plugging equations (B9), (B10), and (B11), after some easycomputations, the steady-state value y�f can be derived:
y�f ¼~r~a2
1þ 1�ab 1þ j�1
Z
� 1�~a�~b1�~a
DðB12Þ
where
D ¼ 1� ~sþ 1� ab
1þ j� 1
Z
� �1�
~bð1� ~sÞ1� ~a
!
The numerator of expression (B12) is positive by Assumption 1. An upper boundfor ~s must be imposed to ensure that D is also positive:
~s < �s; where �s �1þ 1�a
b 1þ j�1Z
� 1�~a�~b1�~a
1� 1�ab 1þ j�1
Z
� ~b
1�~a
After some straightforward algebra, it can be proved that this upper boundmatches that in Assumption 3. Therefore, under Assumptions 1, 2, and 3, y�f isalways positive.
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Once y�f is known, from equations (B10) and (B11):
w� ¼~rD
ðB13Þ
c�f ¼~r
~sDD� 1� ~s
~a2
� �1þ 1� a
b1� ~a� ~b1� ~a
1þ j� 1
Z
� � !" #ðB14Þ
The expression in brackets in equation (B14) can be rewritten as:
~s~a2
1� ~a2 þ 1� ab
1þ j� 1
Z
� �~a2~b1� ~a
þ 1� ~a� ~b1� ~a
" #( )> 0
Therefore, since D > 0, both w� and c�f are also positive.From equation (46), on replacing equations (B8) and (B9) and using equations
(47) and (48), after tedious computations, v�can be written as:
v� ¼a2 j�1
Z ðs� 1Þ � 1�
w� � rh i
ða2 þ bZÞ j�1Z ðs� 1Þ � 1
� w� � r
h i� ð1� aÞaZw�
ðB15Þ
The expression ðj� 1Þðs� 1Þ=Z� 1 is negative under Assumption 1, andtherefore, v� 2 ð0; 1Þ.From equation (B8), c�l can be written as a function of w�:
c�l ¼j� 1
Zw� þ v�
a2r� s
j� 1
Zw�
� �
The expression above is positive if and only the following inequality applies:
v� > a2� j�1
Z w�
r� j�1Z sw�
¼ a2� j�1
Z~rD
r� j�1Z s ~r
D
Equivalently, on replacing equation (B15), after some simplifications, it reads:
j�1Z ðs� 1Þ � 1
� ~r� rD
ða2 þ bZÞ j�1Z ðs� 1Þ � 1
� ~r� rD
h i� ð1� aÞaZ~r
>� j�1
Z ~r
rD� j�1Z s~r
The numerator and the denominator in the LHS of inequality above arenegative. Hence, this inequality reads:
j� 1
Zðs� 1Þ � 1
� �~r� rD
� �rD� j� 1
Zs~r
� �
< �j� 1
Z~r ða2 þ bZÞ j� 1
Zðs� 1Þ � 1
� �~r� rD
� �� ð1� aÞaZ~r
�
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After some algebra, the above inequality can be expressed as:
� j� 1
Z
� �2
ðs� 1Þ~r2ðs� a2 � bZÞ � rDð~rþ rDÞ
þ j� 1
Z~r ð2s� 1ÞrDþ s~r� ð1� aÞaZ~r� ða2 þ bZÞð~rþ rDÞ �
< 0 ðB16Þ
Then, the following two inequalities are sufficient conditions to ensure c�l > 0:
ðs� 1Þðs� a2 � bZÞ > 0 ðB17Þ
ð2s� 1ÞrDþ s~r� ð1� aÞaZ~r� ða2 þ bZÞð~rþ rDÞ > 0 ðB18Þ
The first inequality equation (B17) is satisfied if s < a2 þ bZ or s > 1. It can beeasily proved that s < a2 þ bZ is incompatible with the fulfillment of inequalityequation (B18). Therefore, sufficient conditions (equations [B17]–[B18]) need toimpose s > 1.
Inequality equation (B18) can be rewritten as:
s >ð1þ a2 þ bZÞrDþ ða2 þ bZÞ~rþ ð1� aÞaZ~r
2rDþ ~r¼ s
It can be proved that s > a2 þ bZ and
s < 1, ð1� a2 � bZÞðrDþ ~rÞ � ð1� aÞaZ~r > 0:
It is easy to see that ð1� a2 � bZÞ � ð1� aÞa > 0, and therefore, s < 1. Hence,s > 1 is a sufficient condition for a positive c�l > 0.
Another sufficient condition ensuring the positiveness of c�l can be derived fromthe inequality equation (B16). The LHS of the inequality can be rewritten as aquadratic polynomial in s:
�1s2 þ �2sþ �3 ðB19Þwhere
�1 ¼ �j� 1
Z
� �2
~r2 < 0
�2 ¼j� 1
Z~r
j� 1
Z~rða2 þ bZþ 1Þ þ 2rDþ ~r
� �
�3 ¼ �rD ~rþ rDþ j� 1
Z~rð1þ a2 þ bZÞ
� �
� j� 1
Z~r2 ð1� aÞaZþ ða2 þ bZÞ 1þ j� 1
Z
� �� �
From the expressions of �2 and �3 and taking into account Assumption 1, it iseasy to see that �2 > 0 implies �3 > 0. Therefore, because �1 is negative, �3 negativeis a sufficient condition to ensure that equation (B19) is negative for any positivevalue of s, and therefore, c�l is guaranteed to be positive.
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Coefficient �3 can be rewritten as a quadratic polynomial inr,�3 ¼ D1r2 þ D2rþ D3; where
D1 ¼ �D2 < 0
D2 ¼ �~rD 1þ j� 1
Zð1þ a2 þ bZÞ
� �
D3 ¼ �j� 1
Z~r2 ð1� aÞaZþ ða2 þ bZÞ 1þ j� 1
Z
� �� �> 0
Coefficient D3 has been proved to be positive under Assumption 1. With thisinformation, and after some algebra, the following equivalence can be easilyestablished:
r > 0 ^ �3 < 0ð Þ , r > rþ > 0; ðB20Þ
where
rþ ¼~r2D�Xþ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX2 � 4ð1� aÞðj� 1Þa
q� �ðB21Þ
and
X ¼ 1þ j� 1
Zð1� a2 � bZÞ
Recapitulating, we have found two alternative sufficient conditions to ensurec�l > 0, either r > rþ or s > 1.
Appendix 3. Proof of Proposition 4.1
The Jacobian matrix of the differential equations system (40)–(46) at the steady statec�f , y
�f , t
�, w�, c�l , y�l , and u� is the following 7� 7 matrix:
J� ¼ J1�; J2�; J3�; J4�; J5�; J6�; J7�ð Þ0
which seventh file is given by
J7� ¼ Pðu�Þu� 0; 0; a73; Y u�; t�ð Þ; ð1� bÞZ� a; a76; a77ð Þ
where
a73 ¼ �b~að1� ~aÞw�
~b; a76 ¼ �ð1� bÞZþ a� a2
v�
a77 ¼ð1� a� bÞZ
1� Zy�l
u�ð Þ2þ ð1� aÞað1� ZÞ
1� a� bw�
x�ð Þ2ð1� u�Þ2
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and the six previous files are
J1� ¼ c�f 1; ~a2~s � 1; 0; 0; 0; 0; 0
� J2� ¼ y�f
1�~a�~b1�~b
; a22; 0; 1�~a1�~b
; 0; 0; 0�
J3� ¼ t� � ~a1�~b
;~að1�~aÞ1�~b
;~að1�~aÞ
~bw�; a34; Z; �Z; 0
� þ Cðu�Þ
u� J7�
h iwhere
J4� ¼ w� 0; 0; 0; j� 1; �Z; Z; 0ð Þ � Cðu�Þu� J7�
h iJ5� ¼ c�l 0; 0; 0; 0; 1; a57ð ÞJ6� ¼ y�l 0; 0; a63; a64; aþ bZ� 1; 0ð Þ þ �ðu�Þ
u� J7�
h i
a22 ¼ ð1� ~aÞ~a~b
1� ~b� 1
!
a34 ¼1� ~a
1� ~b� jþ
~að1� ~aÞ~b
t�
a57 ¼ �ð1� a� bÞZsð1� ZÞ
y�l
u�ð Þ2
a63 ¼ a73; a64 ¼ b j�~að1� ~aÞ
~bt�
� �þ 1� a
CðuÞ ¼ Zvþ ð1� ZÞu1� u
�ðuÞ ¼ að1� vÞ þ ð1� a� bÞð1� uÞ � b Zvþ ð1� ZÞu½ �1� u
Given that c�f , y�f , t�, w�, c�l , y�l , and u� are positive values, the sign of thedeterminant of the Jacobian matrix is the same as the sign of the product of Pðu�Þand the following determinant:
1 ~a2~s � 1 0 0 0 0 0
1�~a�~b1�~b
a22 0 1�~a1�~b
0 0 0
� ~a1�~b
~að1�~aÞ1�~b
ð1�~aÞ~aw�~b
a34 Z �Z 0
0 0 0 j� 1 �Z Z 00 0 0 0 1 a2
sv� � 1 a570 0 a63 a64 1� a� bZ �ð1� a� bZÞ 00 0 a73 Y u�; t�ð Þ ð1� bÞZ� a a76 a77
����������������
����������������
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where some simplifications, which do not change the sign of thedeterminant, have been done (J1�=c
�f ; J2�=y
�f ; J3�=t
� �Cðu�ÞJ7�=u�; J4�=w�þCðu�ÞJ7�=u�; J6�=y�l ��ðu�ÞJ7�=u�; J5�=c�l and J7�=ðPðu�Þu�Þ). By addingthe fourth file, J4�, to the third one, J3�, and the fifth column, J�5, to the sixthone, J�6, the determinant reduces to
1 ~a2~s � 1 0 0 0 0 0
1�~a�~b1�~b
a22 0 1�~a1�~b
0 0 0
� ~a1�~b
~að1�~aÞ1�~b
ð1�~aÞ~aw�~b
~b�~a1�~bþ ~að1�~aÞ
~bt� 0 0 0
0 0 0 j� 1 �Z 0 00 0 0 0 1 a2
sv� a570 0 a63 a64 1� a� bZ 0 00 0 a73 Y u�; t�ð Þ ð1� bÞZ� a � a2
v� a77
����������������
����������������
Now, by adding ð1�a�bÞZað1�ZÞv�y�
l
u�ð Þ2 J�6 to the seventh column, J�7, and expanding thisdeterminant along the seventh column, in first place, and along the sixth column insecond place, it follows:
ð1� aÞað1� ZÞ1� a� b
w�
x�ð Þ2ð1� u�Þ2ð�a2Þsv�
1 ~a2~s � 1 0 0 0
1�~a�~b1�~b
a22 0 1�~a1�~b
0
� ~a1�~b
~að1�~aÞ1�~b
ð1�~aÞ~aw�~b
b�a1�~bþ ~að1�~aÞ
~bt� 0
0 0 0 j� 1 �Z0 0 a73 a64 1� a� bZ
������������
������������After some algebra
ð1� aÞað1� ZÞ1� a� b
w�ð Þ2
x�ð Þ2ð1� u�Þ2ð�a2Þsv�
ð1� ~aÞ~a~b
1 ~a2~s � 1 0 0 0
1�~a�~b1�~b
a22 0 1�~a1�~b
0
� ~a1�~b
~að1�~aÞ1�~b
1 1�~a1�~b
0
0 0 0 jþ Z� 1 �Z0 0 �b b jþ Z� 1ð Þ 1� a� bZ
������������
������������
Expanding the determinant by the third column, we have:
� ð1� aÞa3ð1� ZÞ1� a� bð Þsv�
w�ð Þ2
x�ð Þ2ð1� u�Þ2ð1� ~aÞ2~a3b
~bð1� ~bÞ
� 1� 1
~s
� �1� j� Zð Þ 1� a
b
~b1� ~a
þ Z
" #� 1� a
b1� j� Z
~s
( )
Since the term between the squared brackets is equal to ð1� jÞð1� OÞ,Assumption 3 assures that the determinant is positive. Therefore, the sign of thedeterminant of the Jacobian matrix is the same as the sign of Pðu�Þ.
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