Endogenous Growth, Asymmetric Tradeand Resource Dependence

January 31, 2010

Abstract

Since 1980, the aggregate income of oil-exporting countries relative to that of oil-poorcountries has been remarkably constant, despite structural gaps in labor productivitygrowth rates. We rationalize this behavior in a two-country model of asymmetric tradewhere resource-poor and resource-rich economies �Home and Foreign, respectively �dis-play productivity di¤erences in R&D, but stable income shares due to terms-of-tradedynamics. Home�s income share is positively (negatively) related to the domestic (For-eign) investment rate and to the national tax on domestic resource use. This predictionis con�rmed by dynamic panel estimations for sixteen oil-poor economies. We also showthat national governments have incentives to deviate from both e¢ cient and laissez-faireallocations. In Home, increasing the tax on oil use always improves welfare through arent-transfer mechanism. In Foreign, subsidies (taxes) on domestic oil use improve welfareonly if R&D productivity is lower (higher) than in Home.

Keywords: Endogenous Growth, Exhaustible Resources, International Trade.JEL Classi�cation Numbers: F43, O40

1 Introduction

The functioning of modern economies crucially relies on the primary inputs obtained fromexhaustible natural resources. Because of the uneven distribution of endowments of fossilfuels and minerals, many industrialized economies heavily depend on imports from resource-rich countries. The relevance of this form of trade dependence is emphasized by the statistics� fossil fuels and minerals now account for 22.5% of world merchandise trade (World TradeOrganization, 2009: p.43) � and is increasingly regarded as a crucial determinant of thegrowth performance of resource-rich economies (Lederman and Maloney, 2007). Few studies,however, analyze in detail the implications of asymmetric trade between resource-rich andresource-poor countries from the perspective of modern growth theory. In this paper, weexploit the endogenous growth framework to analyze the determinants of national incomeshares, and the e¤ects of national taxes and subsidies on resource use, in a two-country world.

Our empirical reference is the economic performance of the world�s top net exporters ofoil �henceforth labelled as OEX group �relative to that of big oil-poor economies, i.e., theworld�s top importers whose domestic oil production is zero or negligible �henceforth labelledas OIM group. Since 1980, the ratio between the aggregate incomes of the two groups has

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been constant: as we show in section 2.1, the OIM share over the total income of the twogroups is approximately 73% and its time pro�le over the last three decades is remarkably�at. This result is surprising in view of the productivity di¤erentials observed during the sameperiod: since 1980, labor productivity has stagnated, if not declined, in most OEX countrieswhile it has substantially increased in the majority of OIM economies.

In line with these empirical facts, we build a model of asymmetric trade where productivitygrowth rates may di¤er across countries but the world equilibrium exhibits balanced growth.Speci�cally, we assume that a resource-poor economy (henceforth called Home) imports a�nal good and an exhaustible primary input from a resource-rich economy (henceforth calledForeign) and only exports its �nal consumption good. Both countries exhibit endogenousgrowth driven by R&D, and the productivity of research �rms is country-speci�c. In thisframework, several mechanisms may induce balanced growth at the world level. Abstractingfrom labor mobility and integrated �nancial markets, we emphasize the role of terms of trade:in the world equilibrium, national income shares are constant because the dynamics of theprices of traded goods compensate for productivity di¤erences in the respective �nal sectors.

Our �rst aim is to study the determinants of national income shares and check whetherthe theoretical predictions are supported by empirical evidence. In the model, equilibriumshares display two characteristics. First, the income share of each country is positively (neg-atively) related to the domestic (other country�s) saving rate, where the respective savingrates are positively related to the domestic rates of R&D productivity. Second, because ofthe asymmetric structure of trade, income shares di¤er from market shares in �nal produc-tion and national policies a¤ect income shares asymmetrically. In particular, an increase inHome�s tax on domestic resource use increases Home�s income share due to a rent-transfermechanism; an increase in the Foreign resource tax, instead, leaves income shares unchanged.Our panel estimations for sixteen OIM economies in relation to an aggregate group of tenOEX economies con�rm the positive (negative) relation between income shares and domestic(foreign) saving rates, as well as the positive impact of domestic oil taxes on the income shareof OIM economies.

Our second aim is to link the model predictions to the policy debate. The interactionsbetween oil-rich and oil-poor economies have a prominent role in international discussions andthe recent up-surge in oil prices revived the interest for the analysis of strategic tax policiesin this context. One crucial question is: do economies involved in asymmetric trade havepeculiar incentives to implement ine¢ cient tax or subsidies on domestic resource use? Weshow that these incentives exist and are particularly strong for oil importers. If the initialstate of a¤airs is an e¢ cient equilibrium in which all domestic market failures are internalized,Home�s government can increase domestic welfare by raising the national resource tax abovethe e¢ cient level; the Foreign government, instead, does not have any incentive to deviatefrom the initially e¢ cient world equilibrium. If the initial state of a¤airs is a laissez-faireequilibrium, productivity di¤erences come into play: in the empirically plausible case whereproductivity growth is higher in the resource-poor economy, Home�s incentive to raise theresource tax becomes stronger whereas Foreign has an incentive to subsidize domestic resourceuse.

As mentioned above, few studies analyze the implications of asymmetric trade betweenresource-rich and resource-poor countries from the perspective of modern growth theory. Thetrade-and-growth literature typically neglects asymmetric trade structures induced by unevenendowments of speci�c primary inputs. The role of terms of trade in our model is conceptually

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similar to that emphasized by Acemoglu and Ventura (2002) in a world-economy model wherea continuum of countries produces the same �nal good with heterogeneous technologies, butfree trade in intermediate goods induces a stable income distribution. In the early resourceeconomics literature, two-country models assumed that the accumulation of man-made capitalinputs was either absent (Kemp and Suzuki, 1975; Brander and Djajic, 1983) or subjectto diminishing returns (Chiarella, 1980; Van Geldrop and Withagen, 1993). The parallelliterature on endogenous growth with natural resources as inputs, pioneered by Barbier (1999)and Scholz and Ziemes (1999), generally refers to closed or small open economies: to ourknowledge, the two-country setting is only considered in two recent papers by Daubanes andGrimaud (2006) and Peretto and Valente (2010) that di¤er from the present analysis in bothaims and means.1 A common element linking the early contributions of Brander and Djajic(1983) to the present analysis and to Daubanes and Grimaud (2006) is the existence of a rent-transfer mechanism induced by national resource taxes. The argument that oil-importingcountries may use domestic taxation to extract part of the rents accruing to oil producersdates back to Bergstrom (1982), and plays an important role in our results regarding thewelfare e¤ects of discretionary national policies.

2 Growth, Trade and Resource Dependence

2.1 Empirical Facts

Our theoretical analysis focuses on exhaustible resources and can be applied to several typesof minerals and fossil fuels. The main empirical reference, however, is the relative economicperformance of oil-rich and oil-poor economies. The �rst column of Table 1 lists a group ofcountries, labeled as OEX, which comprises the seventeen top oil exporters at the world levelover the period 1980-2008.2 These economies are ranked by average yearly net exports of oilin physical terms, and satisfy two requirements: during the relevant period, each country (i)has never been a net oil importer and (ii) steadily appeared in the top exporters list in eachsingle year.3 In Table 1, the ratio between oil consumption and production in physical termshighlights di¤erent degrees of dependence and/or specialization. Angola, Oman, Norway andQatar consumed on average less than 10% of total oil production, whereas Canada exceeded70%. Due to data availability, real output growth rates for OEX countries are calculated for

1Daubanes and Grimaud (2006) use a North-South model where growth is exclusively driven by the technol-ogy of the oil-poor economy: there are no productivity gaps and terms of trade are excluded by the homogeneityof the �nal consumption good. They study national policies in the presence of perfectly integrated �nancialmarkets and characterize social optimality at the world level under the assumption that oil use generates pol-lution externalities. Peretto and Valente (2009) assume identical R&D technologies between countries in anon-scale model of endogenous growth featuring both vertical and horizontal innovations. They analyze thee¤ects of resource booms, i.e. unexpected discoveries of new resource stocks on innovation rates and relativewelfare.

2We consider seventeen countries and draw the line below Kazakhstan because the subsequent positionsare occupied by countries exporting much less oil in absolute terms. Over the 1980-1992 period, the RussianFederation would be replaced by USSR, and the 17th top exporter would be Egypt, whose average yearly netexports have been nearly one half of the preceding country, Canada. Over the 1992-2008 period, the 18thtop exporter is Colombia, with similar �gures in proportion to Kazakhstan. Indonesia and United Kingdomare excluded from the computations since they both turned from net exporters in the 1980s to net importersnowadays.

3The position of USSR between 1980-1990, not displayed in Table 1, has been taken over by the new countriesemerged from the disaggregation �especially by the Russian Federation and Kazakhstan.

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two subsets: the �rst, labeled as OEX-15, excludes Iraq and Lybia and covers the 1992-2006period; the second, labeled as OEX-13, also excludes Russia and Kazakhstan and covers the1980-2006 period.4 In the most recent period, Middle-East countries are those growing faster.

OEX Countries Oil Net Exports* Oil Cons./Prod.** Real GDP Growth***

1980-2008 1992-2008 1992-2008 1980-2006 1990-2006

Saudi Arabia 7042 8022 16.9% 1.57% 2.89%

Russian Federation - 4751 37.9% - -0.16%

Norway 1957 2752 7.2% 2.80% 2.98%

Iran 2089 2483 34.9% 3.39% 4.17%

Venezuela 2103 2432 17.8% 1.92% 2.57%

United Arab Emirates 1926 2232 13.6% 3.07% 5.04%

Kuwait 1632 1964 11.2% 2.89% 5.95%

Nigeria 1666 1901 12.8% 2.77% 3.78%

Mexico 1420 1482 57.0% 2.47% 2.86%

Algeria 1244 1422 13.6% 2.62% 2.53%

Libya 1233 1319 14.0% - -

Iraq 1236 1124 30.0% - -

Angola 639 899 4.1% 3.72% 4.44%

Oman 653 778 6.8% 6.14% 4.50%

Canada 536 752 73.1% 2.69% 2.63%

Qatar 592 751 7.5% 3.89% 7.42%

Kazakhstan - 594 29.1% - 1.36%

Table 1. Selected top oil exporters. * Average yearly net exports of oil, thousands barrels per day;

** Average yearly oil cosumption-to-production ratio; *** Average yearly growth rate of real GDP;

Sources - * and ** EIA (2009); *** World Bank (2009) and IMF (2009) for Angola and Qatar.

We contrast the OEX country sample by seventeen economies that (i) appear in the listof top oil-importers and (ii) rely on imported oil for domestic use. Speci�cally, since thelist of top importers includes top producers like the US and Brazil, we have excluded allcountries producing more than 10% of the oil they consume domestically.5 The resulting listof seventeen oil-poor, oil-importing countries is labelled as OIM and is reported in Table 2,ranked by average yearly net imports of oil in physical terms (1992-2008). The third columnshows the ratio between net imports and domestic oil consumption, which does not fall shortof 90% (Netherlands, with 89%, is the only limit case and we have decided to include it). Dataon real GDP growth are available for the whole set of countries, which we label as OIM -17,over the 1990-2006 period. The subset OIM -15, which excludes Germany and Poland, coversthe whole 1980-2006 period. Not surprisingly, the fastest-growing economies in the sample arethe so-called Asian Tigers, i.e., South Korea, Singapore and Taiwan.

4Average yearly growth rates are calculated as (yt+�=yt)1=� � 1, where yi is current price gross domestic

product in year i.5Excluded oil-importing countries (and their yearly average Net Import/Consumption ratio over 1992-2008)

are: United States (53%), China (31%), India (62%), Thailand (74%), Brazil (26%), Ukraine (77%), SouthAfrica (57%), Pakistan (81%), Australia (25%).

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OIM Countries Oil Net Imports* Oil Net Imp./Prod.** Real GDP Growth***

1980-2008 1992-2008 1992-2008 1980-2006 1990-2006

Japan 5076 3113 98.0% 2.24% 1.26%

Germany - 1542 94.9% - 1.57%

South Korea 1495 1202 99.8% 6.49% 5.19%

France 1838 1105 95.4% 2.01% 1.78%

Italy 1714 999 92.5% 1.71% 1.29%

Spain 1178 797 97.9% 2.87% 2.83%

Taiwan 660 486 99.8% 6.12% 5.04%

Netherlands 702 452 89.0% 2.37% 2.39%

Singapore 499 387 99.1% 6.65% 6.20%

Belgium 534 350 98.2% 1.94% 1.90%

Turkey 474 335 91.0% 4.37% 3.87%

Poland 359 227 94.9% - 3.50%

Greece 326 225 97.7% 2.08% 2.90%

Sweden 376 218 99.4% 2.18% 2.17%

Philippines 269 189 96.9% 2.77% 3.42%

Portugal 261 179 98.9% 2.48% 2.03%

Switzerland 267 158 99.2% 1.56% 1.19%

Table 2. Selected oil-poor net importers. * Average yearly net imports of oil, thousands barrels

per day; ** Average yearly oil net imports-to-consumption ratio; *** Average yearly growth rate

of real GDP. Sources - * and ** EIA (2009); *** World Bank (2009) and IMF (2009) for Taiwan.

The �rst empirical fact concerning the two country groups refers to the behavior of grossdomestic product (GDP) and gross national income (GNI) calculated in purchasing powerparity (PPP). Computing the income shares of each group over the total income of bothgroups in each year, the resulting time paths of income shares are remarkably �at. This resultis robust to alternative PPP-adjusted measures of GNI and GDP, both in constant and incurrent prices. Figure 1, upper graphs, shows that the GDP share of OEX-13 versus OIM -15countries in 2006 are almost identical to the respective values observed in 1980, and do notexhibit notable variations over time. The same result is obtained by comparing GDP shares ofOEX-15 versus OIM -17 countries between 1990 and 2006, as well as in nominal GNI sharesdespite slight changes in the countries�sample due to data availability - see Figure 1, lowergraphs.

The second empirical fact is related to labor productivity growth. The upper graphsin Figure 2 show the time paths of labor productivity levels for OEX and OIM countries,reported by the International Labor Organization (2009) and based on the estimates of theTotal Economy Database of the Conference Board.6 Notice that labor productivity levelsare not directly comparable for a signi�cant fraction of OEX economies: due to the lack ofbasic data, labor productivity for African and Middle-East countries is calculated as ratio

6We are here referring to the "LP person GK" time series reported in the Total Economy Database. Theunderlying calculations of the Conference Board de�ne labor productivity as the ratio between PPP-adjusted(Geary-Kamis) real GDP in 1990 international dollars and number of persons engaged in work.

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between real GDP and labor force instead of employed persons, which results in highly over-estimated productivity levels. We circumvent this problem by normalizing labor productivityin 1980 equal to unity for each country and by focusing on labor productivity growth.7 Theresulting time paths are reported in the lower-left graph of Figure 2, and suggest a substantialproductivity growth di¤erential in favor of OIM economies. Apart from Canada and Norway,oil exporters have been falling behind oil-importing countries in terms of aggregate laborproductivity.

Figure 1: Upper graphs: current price (dotted line) and constant price (bold line) GDP (adjusted for PPP)shares of oil-exporters and oil-importers. Source: authors calculations on World Bank data (IMF data for

Angola 1980-1984 and Taiwan 1980-2006). Lower graphs: current price GNI shares (adjusted for PPP) of

oil-exporters (excluding Angola) and oil-importers (excluding Taiwan). Source: authors calculations on World

Bank data.

7 If employment rates are assumed to be substantially stable over time in the Middle-East and the Africancountries of the OEX sample, the time paths of normalized labor productivity reported in Figure 2 (lowergraphs) coincide with those that would be obtained from �exact�data on labor productivity levels (i.e., levelsof the ratio between real GDP and employed persons, instead of labor force).

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Figure 2: Upper graphs: Labor productivity levels for OEX countries (left) and OIM countries (right)

calculated by International Labor Organization and The Conference Board. Lower graphs: normalized (1980

= 1) labor productivity levels in all countries (left graph) and group averages OEX versus OIM (right graph).

The existence of substantial gaps in productivity growth rates suggests that stationaryincome shares may originate, at least in part, in the compensating e¤ect of the prices oftraded goods. In the next subsections, we present a general equilibrium model of endogenousgrowth in which income shares are constant because structural gaps in productivity growthare compensated by terms of trade dynamics.

2.2 The Model: General Assumptions

The world comprises two countries - or economic areas - called Home (h) and Foreign (f) andindexed by i = f; h. Each economy produces a �nal good that the residents of both countriesconsume in the same proportions due to identical preferences. In both economies, producingthe �nal good requires man-amde intermediate inputs and a natural resource extracted froman exhaustible stock, e.g. oil, which is entirely owned by Foreign. As a consequence, thestructure of international trade is asymmetric: Home only exports its �nal good whereasForeign exports its �nal good and the natural resource. The engine of growth is representedby R&D activity that expands the number of varieties of man-made intermediate products� e.g., as in Rivera-Batiz and Romer (1991). Inermediates�producers earn monopoly rents

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and the productivity of R&D �rms is enhanced by knowledge spillovers from past researchwhereby scale e¤ects are eliminated.

Since the competitive equilibrium is ine¢ cient under laissez-faire, we assume that nationalgovernments have access to three �scal instruments that allow them to correct for domesticmarket failures: a subsidy to R&D investment, a tax on �nal producers, and a tax on domesticresource use. Our primary interest, however, is not the characterization of e¢ ciency-orientedpolicies but rather the role of national resource taxes as potential strategic instruments. Inreality, the governments of several resource-poor countries impose taxes on imported resourcesin order to capture part of the rents accruing to the primary sectors of resource-rich economies.We address this issue by studying in detail the general equilibrium e¤ects of resource taxesand the welfare-tax relationship for each country.

2.3 Final Producers, Intermediate Sectors and R&D

Final Producers. In economy i, �nal output consists of Yi units of a country-speci�c con-sumption good. Assuming constant returns to scale, the �nal sector behves like a singlecompetitive �rm producing good i by means of Mi varieties of di¤erentiated man-made inter-mediate products (henceforth called intermediates), labor, and a natural resource extractedfrom an exhaustible resource stock (henceforth called resource). The production function is

Yi =

Z Mi

0(Xi (mi))

� dmi � (viLi)� R i ; (1)

where t 2 [0;1) is the time index, Xi (mi) is the quantity of the mi-th variety, mi 2 [0;Mi],of intermediate input employed in production, viLi represents labor in e¢ ciency units withvi denoting the productive e¢ ciency of each worker and Li the number of workers, and Ri isthe �ow of natural resource used in the �nal sector of country i. Elasticity parameters satisfy0 < �; �; < 1 and �+�+ = 1. Since the engine of growth is represented by increases in thenumber of intermediate productsMi, we assume for simplicity that workers e¢ ciency vi growsat the exogenous constant rate �i > 0 and that labor is supplied inelastically: Lh and Lf areconstant over time and coincide with population size in Home and Foreign, respectively. Thelaw of one price holds for all traded goods: the quantities (Yh; Yf ) are sold at the respectiveworld prices (P hY ; P

fY ) and the exhaustible resource is sold to all �nal producers at the same

world price PR. Since labor and intermediates are not traded internationally, the nominal wageand the price of each mi-th intermediate, respectively denoted P iL and P

iX(mi)

, are country-speci�c. Production costs are a¤ected by �scal policy: we denote by bi the ad valorem tax onthe purchases of intermediates and by � i the ad valorem tax on domestic resource use. Bothbi and � i are assumed to be constant in order to preserve the balanced-growth properties ofthe world equilibrium.8 The pro�t-maximizing conditions in the �nal sector read

P iLLi = �P iY Yi; (2)

PRRi (1 + � i) = P iY Yi; (3)

P iX(mi)(1 + bi) = �P iY (Xi (mi))

��1 (viLi)� R i ; (4)

8This assumption does not a¤ect the generality of our results. As shown in section 4, both e¢ cient allocationsand laissez-faire equilibria exhibit balanced growth in each instant. Decentralizing e¢ cient allocations thusrequires implementing constant taxes.

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where (4) is valid for each mi 2 [0;Mi].

Intermediate sector. Each variety is produced by a monopolist who holds the relevantpatent and maximizes instantaneous pro�ts taking the demand schedule (4) as given. Pro-duction requires �& (Xi (mi)) units of �nal good, where �& (:) is the cost function, and pro�tsread �i (mi) = P iX(mi)

Xi (mi)� P iY �& (Xi (mi)). Assuming a linear cost function �& (Xi (mi)) =

&Xi (mi) with & > 0, pro�t maximization requires

P iX(mi)= (1=�) &P iY (5)

for each mi 2 [0;Mi], and therefore symmetric quantities and pro�ts across varieties. Thisimplies that output is proportional to the number of varieties, according to the reduced Cobb-Douglas form (see Appendix)

Yi =

��2=&

� �1��

1 + bi�Mi (viLi)

�1�� (Ri)

1�� ; (6)

where 1 � � = � + guarantees decreasing marginal returns to labor and to the naturalresource.

R&D Sector. The number of intermediates�varietiesMi grows over time by virtue of R&Dactivity pursued by competitive �rms that develop new blueprints and sell the relevant patentsto incumbent monopolists. Symmetry across intermediates allows us to represent R&D �rmsas a consolidated sector earning zero pro�ts due to free-entry. Developing blueprints requiresinvesting units of the domestic �nal good purchased by the R&D sector. Assuming a lineartechnology, each unit purchased has a constant marginal productivity � taken as given byR&D �rms �equal to �i > 0 in country i. For each unit invested the R&D �rm receives fromthe national government a subsidy at constant rate ai > 0. Denoting by Zi the total amountinvested by R&D �rms, total investment in country i is Zi (1 + ai) and the increase in thenumber of varieties equals

_Mi = �i � Zi (1 + ai) : (7)

Denoting by Vi the value of each patent sold in country i, the zero-pro�t condition is9

Vi = P iY = [�i (1 + ai)] : (8)

The value of each patent sold to an incumbent producer equals the present value of futuremonopoly pro�ts, which implies the standard no-arbitrage condition

ri (t) =�i (t)

Vi (t)+_Vi (t)

Vi (t); (9)

where ri (t) is the equilibrium rato of return to private investment in country i. We assumethat the productivity of the R&D sector is a¤ected by externalities whereby the currentmarginal productivity of investment, �i, is positively in�uenced by past research e¤ort. These

9Aggregate pro�ts of the R&D sector equal Vi _Mi � P iY Zi = Vi�iZi (1 + ai) � P iY Zi, so that condition (8)maximizes R&D pro�ts for a given marginal productivity �i and implies zero pro�ts for each �rm. The samecondition is equivalently obtained assuming free entry in the R&D business for an inde�nite number of �rms.Indeed, condition (8) says that the value of each innovation must equal the private net cost paid by R&D �rms.

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externalities take the form of knowledge spillovers, exactly as in models à la Lucas (1988)where the di¤usion of public knowledge implies an un-compensated transmission of humancapital across generations. In the present context, the productivity of each R&D �rm ishigher the better the �current state of technology attained by virtue of previous research�.This concept of state-of-the-art in research is conveniently measured by the ratio between thenumber of existing varieties and current output levels, Mi=Yi. We thus posit a linear function

�i (t) � 'i � (Mi (t) =Yi (t)) (10)

where 'i > 0 is a constant proportionality factor determining the social productivity of R&D.Equation (10) implies that the growth rate of intermediates�varieties increases linearly withthe economy-wide rate of R&D investment: from (7), we have

_Mi

Mi= 'i

Zi (1 + ai)

Yi: (11)

As mentioned in Barro and Sala-i-Martin (2004: p.300-302), linear laws like (11) generallyexhibit two desirable properties. First, they eliminate scale e¤ects by making the equilibriumgrowth rate of output independent of the size of endowments. Second, they are empiricallyplausible since, in most industrialized countries, the growth rate of productivity appears to bepositively related to the ratio between R&D expenditures and output, with a proportionalitycoe¢ cient that is relatively stable over time.

2.4 Resource Extraction in Foreign

The total �ow of resource extracted in Foreign equals the sum of resource use in both coun-tries and is denoted by R (t) � Rh (t) + Rf (t) in each instant. The resource stock Q (t) isnon-renewable and is given at t = 0. Extracting �rms are competitive and take the worldprice of the resource PR as given. For simplicity, extraction costs are zero. The owners of ex-tracting �rms are households in Foreign, each of whom earns the same fraction 1=Lf of rents.Normalizing the mass of �rms to unity, the representative �rm maximizes the present-valuestream of pro�ts discounted at the domestic equilibrium interest rateZ 1

0PR (t)R (t) e

�R1t rf (v)dvdt; (12)

subject to the dynamic resource constraint _Q (t) = �R (t), with Q (0) = Q0 > 0. This solutionto this dynamic problem is characterized by the optimality conditions

_PR (t) = PR (t) rf (t) ; (13)

Q0 =

Z 1

0R (t) dt: (14)

Equation (13) is the standard Hotelling rule asserting that, along an optimal extraction path,the resource rent must grow over time at a rate equal to the rate of return to alternativeinvestments in the economy. Equation (14) is the intertemporal resource constraint requiringasymptotic exhaustion of the resource stock: leaving unexploited resources in the ground isnot optimal. Note that we would obtain identical results if we assumed that the resource stockis incorporated in private wealth: in this case, each household is endowed with a fraction 1=Lfof the initial stock and directly extracts the resource in accordance with Hotelling�s rule (13).

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2.5 Governments, Households and Trade Balance

Governments. The public sector in country i = h; f �nances public R&D expenditures bymeans of the ad valorem taxes on intermediates�purchases and resource use. Ruling out publicdebt, balanced budget is achieved in each instant by compensating possible imbalances witha lump-sum transfer Fi imposed on each household:

aiPiY Zi = FiLi + biMiP

iXXi + � iPRRi; (15)

where Fi is a lump sum tax if positive, a subsidy if negative.

Households. Each economy i is populated by Li homogeneous households. Following astandard procedure, the consumer problem is solved in two steps. First, each consumer de-cides how to allocate expenditures between imported and domestically-produced �nal goods.Second, households decide the intertemporal pro�le of consumption expenditures by maximiz-ing present-value utility subject to the dynamic wealth constraint. Denoting by cji the quantityof the good produced in country j and individually consumed in country i, the instantaneousutility function of each resident in country i reads

ui(chi ; c

fi ) = ln

h(chi )

�(cfi )1��i; (16)

where the weighting parameters � and 1�� indicate the preference taste for Home and Foreigngoods, respectively. The �rst-step problem consists of choosing consumed quantities in eachinstant by maximizing (16) subject to the individual expenditure constraint

Eci =Li = P hY chi + P

fY c

fi ; (17)

where Eci is aggregate consumption expenditure in country i. The solution to this problemyields the indirect utility function �ui = ln[! � (Eci =Li)], where ! � !(P hY ; P

fY ) is a weighted

average of �nal goods�prices (see Appendix). The second-step problem then consists of choos-ing the path of consumption expenditures that maximizes the present discounted value of thestream of utilities enjoyed over time,

Ui �Z 1

0e��t � ln[(! (t) � (Eci (t) =Li)]dt; (18)

where � > 0 is the pure time-preference rate and the path of ! (t) is taken as given by thehousehold. Since the two economies are populated by identical households, individual privatewealth consists of a fraction (1=Li) of the total value of the assets in the economy (ViMi)representing ownership of �rms. Denoting the value of individual assets by ni � (ViMi) =Li ,the dynamic constraints read

_nh = rhnh + PhL � (Ech=Lh)� Fh; (19)

_nf = rfnf + PfL + PR (R=Lf )� (E

cf=Lf )� Ff ; (20)

where rini + P iL is income from assets and labor, and PR (R=Lf ) is resource income for eachForeign resident. Each consumer maximizes (18) subject to the relevant dynamic constraintusing the path of Eci =Li as control variable. As shown in the Appendix, the resulting optimalityconditions imply

_Eci (t) =Eci (t) = ri (t)� �; (21)

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which is the standard Keynes-Ramsey rule asserting that the growth rate of consumptionexpenditures equals the di¤erence between the rate of return to assets and the time-preferencerate.

Trade. Ruling out international trade in assets, we have balanced trade in each instant

PRRh + PfY Lhc

fh = P hY Lfc

hf : (22)

Condition (22) asserts that the value of total exports of the Foreign country, resources plusexported consumption goods, equals the value of consumption goods imported from the Homecountry. This asymmetric structure of international trade is relevant for the results of themodel: country f , the resource-rich economy, exhibits a structural de�cit in the exchange of�nal goods and this implies that the allocation of domestic output between consumption andits competing uses generally induces feedback e¤ects on the trade equilibrium.

Aggregate Constraints. To simplify the notation, we denote aggregate R&D expendituresof country i as Edi � P iY Zi (1 + ai), aggregate spending in intermediates�production as E

xi �

P iY &MiXi. Adding consumption expenditures, the ttoal expenditure index of country i isde�ned as Ei � Eci + Edi + Exi . Aggregating the individual wealth constraints across house-holds and substituting the pro�t-maximizing conditions of the various sectors, we obtain (seeAppendix)

Eh � Ech + Edh + E

xh = P hY Yh � PRRh; (23)

Ef � Ecf + Edf + E

xf = P fY Yf + PRRh: (24)

These aggregate constraints imply equality between domestic expenditures and aggregate in-comes of residents. From (23), total incomes in Home equal the value of �nal output less thevalue of resource rents accruing to Foreign. In (24), resource rents are added to the value of�nal production in Foreign to obtain total Foreign incomes. In the analysis, we will denotethe aggregate income share of Home as

sh (t) �Eh (t)

Eh (t) + Ef (t)

and, accordingly, the income share of Foreign as sf (t) � 1� sh (t).

2.6 Competitive World Equilibrium

The competitive world equilibrium can be characterized as follows. First note that, in eachcountry, the equilibrium rates of return read (see Appendix)

ri =_P iYP iY

+

"� (1� �) (1 + ai)

1 + bi'i +

�

1� ��i +

1� �_RiRi

#=

_P iYP iY

+i; (25)

where i equals the central term in square brackets and is a measure of physical productivitygrowth in the �nal sector of country i. Productivity growth incorporates the e¤ect of threecountry-speci�c characteristics: the social productivity of R&D (which is crucially determinedby 'i) , the growth rate of labor e¢ ciency (�i), and the rate of increase in domestic resource

12

use ( _Ri=Ri). From (25), we can decompose the interest rate di¤erential between Home andForeign into a price component and a productivity term:

rh � rf = _P hYP hY

�_P fY

P fY

!+h � f : (26)

The price component in round brackets is a standard terms-of-trade e¤ect whereby pricedynamics interact with possible gaps in productivity growth �henceforth structural gaps �represented by di¤erences between h and f . In this regard, the current assumptions ontechnology and preferences imply that terms of trade e¤ects exactly compensate for structuralgaps in each point in time, which yields the equalization of equilibrium interest rates:

Proposition 1 In the competitive equilibrium, interest rates are equalized, rh (t) = rf (t) ineach t 2 [0;1), and

_P hYP hY

�_P fY

P fY= f � h: (27)

Interest rate equalization implies that consumption expenditures grow at the same rate inthe two countries. We now show that this in turn implies balanced growth at the world levelin each instant, and therefore constant income shares.

The key variable determining national income shares is the ratio between resource-use�ows, which we denote as � (t) � Rh (t) =Rf (t) and henceforth call relative resource use.Substituting the respective country indices in (3) we obtain the basic relation

� (t) =Rh (t)

Rf (t)=~ h~ f� P

hY (t)Yh (t)

P fY (t)Yf (t)in each t 2 [0;1) ; (28)

where ~ i � (1 + � i)�1 is the tax-adjusted resource elasticity in �nal production. Result

(28) follows from no-arbitrage over the resource price at the international level, and showsthat relative resource use equals the ratio between the values of �nal output weighted bythe respective tax-adjusted resource elasticities. Combining (28) with the balanced tradecondition, we obtain a complete characterization of the competitive world equilibrium:

Proposition 2 The world competitive equilibrium exhibits a constant relative resource use� (t) = �� in each t 2 [0;1). Output, resource use, expenditures and the mass of varieties growat rates

_Yh=Yh = h � � and _Yf=Yf = f � �; (29)_Rh=Rh = _Rf=Rf = ��; (30)_Eh=Eh = _Ef=Ef = ri � � with rh = rf ; (31)_Mi=Mi = 'i� (1� �) (1 + ai) (1 + bi)�1 � �; (32)

and income shares are constant:

sh =���~ f=~ h

�1 + ��

�~ f=~ h

� (1� ~ h) and sf =1 + ~ f

��

1 +�~ f=~ h

���: (33)

13

The balanced growth path followed by the world economy embodies some general char-acteristics of two-country models of trade. Result (29) shows that di¤erences between thegrowth rates of physical �nal output in the two countries are due to structural gaps betweenthe respective productivity indices (h 6= f ), which re�ect possible di¤erences in R&D pro-ductivity ('h 6= 'f ), in taxes on intermediates�purchases (bh 6= bf ), or in labor e¢ ciencygrowth rates (�h 6= �f ).

Income shares are constant by virtue of the terms-of-trade mechanism already outlinedin Proposition 1: price dynamics compensate for possible di¤erences in physical productivitygrowth rates, which implies balanced growth at the world level. This mechanism is conceptu-ally similar to that emphasized by Acemoglu and Ventura (2002) in a model where countriesexhibit heterogeneous AK technologies: despite the presence of productivity di¤erences, theworld income distribution is stable over time because terms of trade e¤ects in the internationalmarket for intermediates compensate for productivity di¤erences. In our model, national in-come shares are constant because structural gaps in physical productivity growth rates areo¤set by the dynamics of �nal goods prices. The fundamental di¤erence between our frame-work and standard two-country models of growth follows from the asymmetric structure oftrade and this is re�ected in equations (33). Since income shares are a¤ected by the degree ofHome�s depedence on the exhaustible resources supplied by Foreign, ~ h, the world competitiveequilibrium exhibits two peculiar features. First, each country�s income share di¤ers from itsshare of world �nal output. Second, national taxes on resource use have asymmetric e¤ects inthe two countries. We address these points in the next section. In passing, it is worth notingthat the present model exhibits closed-form solutions: the explicit time paths of resource use,physical output levels and relative prices are derived in Appendix, and will be exploited inderiving our main results.

3 National Income Shares: Theory and Evidence

3.1 Income Shares and Asymmetric Trade

The asymmetric structure of international trade implies that each country�s income sharedi¤ers from its share of world �nal output: substituting (28) in (33), we obtain

sh =(P hY Yh)=(P

fY Yf )

1 + (P hY Yh)=(PfY Yf )| {z }

Output share

� [1� ~ h]| {z }Trade dependence

; (34)

which clari�es that sh is the product of two factors. The �rst is Home�s �nal output sharewhile the second term in square brackets represents the e¤ect of trade dependence in primaryinputs �i.e., the fact that Home must use a fraction ~ h of revenues from �nal-good sales to�nance resource imports. The output share term in (34) is determined by investment ratesand, hence, by productivity parameters. As shown in Appendix, the output ratio betweenHome and Foreign reduces to

P hY Yh

P fY Yf=

�

1� � �1� If1� Ih

; (35)

Ii � 'i� (1� �) (1 + ai)� � (1 + bi)'i (1 + bi)

+�2

1 + bi; (36)

14

where Ii is one possible de�nition of domestic investment rate: the right hand side of (36)equals the fraction of �nal output invested in R&D activity (�rst term) plus the fraction ofoutput spent in producing intermediates (second term) in country i. Given (34), results (35)-(36) imply that each country�s income share is positively (negatively) related to the domestic(other country�s) investment rate and, through the investment rate, positively (negatively)related to the domestic (other country�s) level of R&D productivity. There is, however, anotherrelevant variable in�uencing the income shares, sh and sf , namely the tax rate on resourceuse in Home. The e¤ects of resource taxes on income shares and on the other variables ofinterest are brie�y described below.

3.2 Income Shares and Resource Taxes

Due to asymmetric trade, national taxes on resource use have asymmetric e¤ects in the twocountries: an increase in �h generally raises Home�s income relative to Foreign whereas varia-tions in � f leave income shares unchanged. Before proving these results, we recall that nationalresource taxes have counteracting e¤ects on the equilibrium level of relative resource use: from(28) and (35), we have

�� =1 + � f1 + �h

� �

1� � �1� If1� Ih

: (37)

Equation (37) shows that relative resource use also depends on teh respective investment rates,but coincides with the output ratio (P hY Yh)=(P

fY Yf ) if only if resource taxes are equal in the

two countries.The e¤ects of ceteris paribus variations in resource taxes can be analyzed for each country

on the basis of the closed form solution of the model (see Appendix). Considering variationsin Home�s resource tax, we obtain the following

Proposition 3 An increase in Home�s tax rate on domestic resource use reduces relativeresource use, reduces physical output in Home and increases physical output in Foreign,

@��

@�h< 0;

@Yh@�h

< 0;@Yf@�h

> 0;@(Yh=Yf )

@�h< 0;

the output ratio is unchanged but Home�s income share increases:

@(P hY Yh)=(P

fY Yf )

@�h= 0;

@sh@�h

> 0;@sf@�h

< 0:

The intuition behind Proposition 3 is as follows. An increase in �h raises the marginalcost of resource use in Home�s �nal sector: the contraction in resource demand lowers Home�srelative resource use as well as Home�s physical output, both in absolute and in relative terms.Despite the decline in resource use, the output ratio (P hY Yh)=(P

fY Yf ) is unchanged because

prices adjust in order to compensate for the decline in the physical output ratio:

@(P hY =PfY )

@�h= �@(Yh=Yf )

@�h: (38)

The fact that Home�s income share increases is exclusively due to trade dependence in primaryinputs: by taxing domestic resource use, the Home economy retains part of the resource rents

15

accruing to Foreign and redistributes it among Home residents. This rent transfer mechanismis evident in equation (34): since nominal output shares are independent of �h, the only e¤ectof an increase in Home�s resource tax is the reduction in ~ h, i.e. the fraction of Home�s outputused to pay resource rents to Foreign, which increases the trade-dependence term.

The e¤ects of asymmetric trade become furthermore clear when considering Foreign taxpolicy. The following Proposition establishes that an increase in � f does not a¤ect incomeshares:

Proposition 4 An increase in Foreign�tax rate on resource use increases �� and reduces For-eign physical output in both absolute and relative terms:

@��

@� f> 0;

@Yh@� f

> 0;@Yf@� f

< 0;@(Yh=Yf )

@� f> 0:

Output and income shares are not a¤ected:

@(P hY Yh)=(P

fY Yf )

@� f=@sh@� f

=@sf@� f

= 0:

Proposition 4 hinges on the fact that resource taxes in Foreign do not induce rent transferbetween countries. An increase in � f raises the marginal cost of resource use in the Foreign�nal sector, and the contraction in resource demand increases Home�s relative resource usewhile reducing Foreign physical output in both absolute and relative terms. However, taxrevenues are redistributed within the Foreign economy and this implies that world incomeshares are una¤ected by the variation in � f .

3.3 Determinants of Income Shares: An Empirical Test

Our results on the determination of income shares can be summarized as follows: The incomeshare of resource-poor economy is (i) positively related to the national tax on domestic resourceuse, (ii) positively related to the domestic investment rate, and (iii) negatively related to theinvestment rate of the resource-rich economy; the respective investment rates are, in turn, (iv)positively related to the respective levels of R&D productivity.10

We now test these predictions empirically. Using a dynamic panel-estimation technique,we check whether the income share of each oil-importing country depends positively on itsown investment rate, negatively on the average investment rate of oil-exporting countries, andpositively on domestic oil taxes. The time period is 1980-2008, and the countries for whichwe have data are sixteen OIM countries �namely Belgium, France, Germany, Greece, Italy,Japan, South Korea, Netherlands, Philippines, Poland, Portugal, Singapore, Spain, Sweden,Switzerland, Turkey � and the ten OEX countries � i.e., Algeria, Canada, Iran, Kuwait,Mexico, Norway, Oman, Saudi Arabia, United Arab Emirates, and Venezuela. This is thecountry sample which best represents the framework of our theoretical model and for whichthe relevant data are nearly completely available, except for taxes in the Philippines andSingapore.

10Prediction (i) follows from Proposition 3, whereas predictions (ii)-(iv) follows from substituting equations(35)-(36) in (34).

16

Table 3: Estimation results for income shares of oil-importing countriesArellano-Bond dynamic panel-data estimation

Endogenous variable: shareoim

(1) (2) (3) (4) (5) (6)

shareoim

shareoim (-1) 0.559*** 0.551*** 0.529*** 0.595*** 0.604*** 0.601***

(0.0940) (0.0953) (0.0990) (0.104) (0.106) (0.103)

investoim 0.482*** 0.500*** 0.554*** 0.524*** 0.516*** 0.543***

(0.108) (0.109) (0.141) (0.155) (0.158) (0.158)

investoex -0.212*** -0.235*** -0.316*** -0.347*** -0.277*** -0.333***

(0.0447) (0.0460) (0.0730) (0.0807) (0.0820) (0.0819)

eduexp 1.427*** 1.576*** 1.747*** 1.747*** 1.738***

(0.465) (0.492) (0.508) (0.506) (0.511)

oiltax 0.00568* 0.00784** 0.00734* 0.00766**

(0.00335) (0.00386) (0.00384) (0.00387)

pop 5.02e-08 1.74e-08 5.15e-08

(1.98e-07) (1.96e-07) (2.00e-07)

rdexp -0.953

(1.205)

cgovdebt 0.0171

(0.0350)

Constant 6.004 0.519 0.790 -2.733 -1.935 -4.580

(3.729) (4.160) (5.737) (9.310) (9.328) (9.932)

Observations 64 64 56 56 56 56

Nr. of countries 16 16 14 14 14 14

Wald �2 85.22 92.28 96.23 110.09 112.70 109.18

*** p<0.01, ** p<0.05, * p<0.1

Standard errors in parentheses

In order to focus on long-run e¤ects and to avoid the impact of business cycles, we build �ve-year averages; the considered periods are: 1980-84, 1985-89, 1990-94, 1995-99, 2000-04, and2005-08. To capture the dynamic development we include lags of the dependent variable. Byconstruction, the emerging unobserved panel-level e¤ects are then correlated with the laggeddependent variables, making standard estimators inconsistent. That is why the Arellano-Bonddynamic panel-data estimation is used; it provides a consistent generalized method-of-moments(GMM) estimator for the parameters of this model.

We use online data from the World Bank (2009) for the macroeconomic variables and fromthe International Energy Agency (IEA, 2009) for resource taxes. Speci�cally, incomes sharesare calculated for each oil-importing country as the ratio between its GNI level and the sum ofthe GNIs of all oil-exporting countries, which we label by shareoim. For the investment rates,we take gross capital formation as a percentage of GDP for both oil-importing and -exportingcountries. In the case of oil importers, the variable is denoted by investoim; for oil exporterswe calculate the average investment rate - with population size used as the weighting factor

17

- to get the parameter investoex. Resource taxes are measured by taxes on light fuel oil andlabelled with oiltax. Further control variables are education expenditures as a percentage ofGDP (eduexp, the investment rate for human capital), research expenditures as a percentageof GDP (rdexp, the investment rate for knowledge capital), population size pop, and centralgovernment debt as a percentage of GDP cgovdebt.

The results are presented in Table 3, which includes six representative equations [1]-[6]. Inall equations we include the (�rst) lag of the endogenous variable which is signi�cant at the1%-level in all speci�cations; this con�rms that the estimation method is appropriate. In [1]we start by testing the impact of the investment shares in both types of countries. As canbe seen from the results, the theoretical model is con�rmed by the estimations as domesticinvestment a¤ects the oil-importers�income share positively while the opposite holds true forthe impact of foreign investment rates. The next equation [2] exhibits that also the domesticinvestment rate in human capital eduexp is positive for the income share, which also holds forall the other speci�cations.

In [3], oil taxes are included. It appears as very favorable for the theoretical model thattaxation has the predicted positive sign; the signi�cance is 5% or 10% according to the speci-�cation. Thus according to the empirical results, oil-importing countries can indeed increasetheir share of total income by raising domestic oil taxes, which is a remarkable �nding.

Population size pop, i.e. the scale of the economy, has no signi�cant e¤ect in any speci�-cation, mainly because the endgenous lagged variable already captures this e¤ect. Similarly,research expenditures rdexp as well as central government debt cgovdebt have no signi�cantimpact and do not change our major �ndings. In view of these results, the model predic-tions regarding the determination of income shares appear to be consistent with the availableempirical evidence on oil-importing and oil-exporting countries.

4 E¢ ciency and Policy

All the results derived in section 2.6 hold in any competitive equilibrium de�ned at given policyinstruments. In the following subsections, we describe the characteristics of laissez-faire equi-libria and e¢ cient allocations �i.e., allocations in which domestic market failures induced byR&D externalities and monopolistic competition are neutralized by �scal authorities throughtaxes and subsidies. We then analyze the welfare consequences of discretionary variations ofnational resource taxes.

4.1 Laissez-Faire Equilibria

The laissez-faire equilibrium is represented by the case � i = bi = ai = 0 in each country. Theresulting allocation is easily derived from the previous equations. In particular, from (37),relative resource use under laissez-faire equals

��(laissez-faire) =�

1� � �1� �+

��='f

�1� �+ (�='h)

; (39)

Equation (39) shows that relative resource use is positively (negatively) related to R&D pro-ductivity in Home (Foreign) through the terms 'h and 'f . This result is peculiar to thelaissez-faire equilibrium, which is however ine¢ cient by construction. In each country, mo-nopolistic competition in the intermediate sector and knowledge spillovers in the R&D sector

18

imply that the economy misallocates domestic �nal output between R&D investment, con-sumption and production of intermediates. The following subsection describes the set of taxesand subsidies through which �scal authorities can restore e¢ ciency at the country level.

4.2 Conditional E¢ ciency

We de�ne conditionally-e¢ cient allocation for a given country an allocation in which domesticmarket failures generated by monopoly pricing and R&D spillovers are fully internalized.Formally, an allocation is conditionally e¢ cient for country i if domestic output is allocatedin such a way that it maximizes present-value utility Ui subject to the technology, income,and resource constraints faced by country i at given international prices.

At the mathematical level, the derivation of the conditionally e¢ cient allocation (CE-allocation, hereafter) for each country is very similar to the standard method of welfare max-imization in closed-economy models of endogenous growth. However, conditional e¢ ciencyand optimality are quite di¤erent concepts. In a closed economy, the centralized problem issolved by a hypothetical social planner and the resulting allocation is socially optimal becausethe planner is endowed with full control over all the elements of the allocation. In the presenttwo-country setting, instead, the CE-allocation in country i implies the maximization of do-mestic utility Ui at given international prices. On the one hand, this implies that conditionale¢ ciency in one single country does not imply optimality at the world level. On the otherhand, there is no general presumption that national governments actually wish to implementconditional e¢ ciency since this depends on the assumed information set of policymakers. Ifcountry i�s government actually takes international prices as given, implementing the CE-allocation in country i is desirable. But if country i�s government, instead, could infer all theconsequences of implementing a given allocation in country i for the world equilibrium, theallocation maximizing domestic utility Ui is generally not the CE-allocation. In other words, awell-informed policymaker may rationally choose to deviate from conditional e¢ ciency by im-plementing ine¢ cient policies that raise national welfare to the detriment of the other country.In the next subsection we show that this is indeed the case for the Home country. These con-siderations clarify the nature of our interest in conditional e¢ ciency: it is a benchmark stateof a¤airs starting from which we can study the welfare e¤ects of discretionary national policiesin isolation from the ine¢ ciencies induced by domestic market failures (i.e., monopolies andspillovers).

We denote variables evaluated in conditionally e¢ cient allocations by tildas. In Home,the CE-allocation is represented by the paths of imported resource �ows and expenditures(in consumption, intermediates�production and R&D activity), that maximize Home�s indi-rect utility subject to the �nal-good technology, the intermediate-good technology, the R&Dtechnology, and Home�s expenditure constraint:n

~Rh; ~Ech;~Exh ;

~Edh

o1t=0

= argmaxUh s.t. (1); (11); (23)

where Uh in (18) is maximized taking prices as given and the R&D externality is fully takeninto account through constraint (11). In Foreign, the CE-allocation is represented by theextraction paths of both imported and exported resource �ows, and the paths of expendi-tures (Ech; E

xh ; E

dh) that maximize utility in Foreign subject to the technology constraints, the

19

aggregate expenditure constraint, and the exhaustible resource constraint:n~Rh; ~Rf ; ~E

cf ;~Exf ;

~Edf

o1t=0

= argmaxUf s.t. (1); (11); (23) and _Q = �Rh �Rf .

The two maximization problems are fully described in the Appendix. The �rst result is thatif both economies exhibit conditionally e¢ cient allocations, the trade equilibrium is charac-terized by a constant e¢ cient level of relative resource use ~� equal to

~� =�

1� � : (40)

Result (40) shows that the e¢ cient level of relative resource use ~� is exclusively determinedby preference parameters, without any role for technology. As a consequence, it di¤ers from(coincides with) the laissez-faire level (39) when the technological parameters governing R&Dproductivity, 'h and 'h, are di¤erent (equal). Obviously, this does not mean that laissez-faire equilibria are e¢ cient under homogeneous technologies: when 'h = 'h and all taxesand subsidies are zero, the level of �� coincides with the e¢ cient one but the competitiveequilibrium is still ine¢ cient since domestic output is misallocated among its competing uses�i.e., consumption, intermediates�production and R&D investment �by virtue of the existingmarket failures. In fact, the second result (see Appendix for details) is that if governmentswish to decentralize the CE-allocation in the competitive economy, they must implement ane¢ cient policy consisting of the following subsidies and taxes

~ai ='i (1� �)� �� (1� �) > 0; (41)

~bi ='i� (1� �)� �

�> 0; (42)

~� i ='i (1� �)� �

�> 0: (43)

The role of subsidies to R&D investment is clearly that of internalizing knowledge spillovers:research activity generates positive externalities and must therefore be encouraged by publicauthorities through ~ai > 0. The fact that private agents exhibit ine¢ ciently low saving ratesimplies ine¢ ciently high consumption propensities and, consequently, excessive demand forthe inputs employed in �nal production.11 Since the equilibrium quantities of resources andintermediates exceed socially desirable levels, e¢ ciency is restored by imposing positive taxeson resource use and intermediates�purchases.12 It is easily veri�ed from (36) and (37) that

11The private marginal bene�ts from resource use and intermediates equal the respective marginal productiv-ities of these inputs in �nal production. The social marginal bene�ts, instead, equal the un-invested fractionsof the respective marginal productivities: see equations (??)-(??) in Appendix. The slight asymmetry betweenthe levels of the resource tax and the intermediates�tax is explained in footnote ?? below.12Since the excess demand for inputs originates in ine¢ ciently low saving rates, each input should in principle

be taxed at the same rate. In the present model, however, the e¢ cient tax on resources (42) is higher thanthe e¢ cient tax on intermediates�purchases (42). The reason is that, in the competitive economy, the excessdemand for intermediates is already partially contrasted by the monopolistic behavior of producers, whichrestricts supply. This is a general characteristic of endogenous growth models with R&D externalities. Asshown by Valente (2009) in a closed-economy model without resources, the excess-demand e¤ect (induced byine¢ ciently-low saving rates) always dominates the contrasting e¤ect of restricted supply (induced by monopolyrents) when preferences are logarithmic. Consequently, intermediates�purchases must be taxed.

20

if the governments of both countries implement the e¢ cient policy (~ai;~bi; ~� i), the equilibriumrelative resource use �� coincides with the e¢ cient level in (40). Moreover, little algebra showsthat decentralizing the CE-allocation in a market economy implies faster physical productivitygrowth relative to laissez-faire conditions. We now exploit this and the previous results in orderto address the issue of strategic taxation.

4.3 Welfare and Resource Taxation

Do the governments of Home and Foreign have particular incentives to implement ine¢ cienttaxes? In this section, we analyze the marginal e¤ects of national resource taxes on welfarefor each country. These welfare-tax relationships provide the basis for studying the poten-tial incentives that national governments have to implement discretionary policies: if �scalauthorities could take into account all the general equilibrium e¤ects of national taxes and,hence, calculate the �nal e¤ect on domestic welfare levels, they might rationally choose todeviate from laissez-faire equilibria or e¢ cient allocations. While this is a general argument,we restrict the present discussion to the analysis of potential welfare gains stemming fromvariations in national taxes on domestic resource use.

The reaction of utility levels to variations in domestic resource taxes is represented by twowelfare-tax relationship that we derive in explicit form. As shown in the Appendix, present-value utilities for each consumer in the two countries equal

Uh = {h +1

�ln

�Yh (0) �

hP hY (0) =P

fY (0)

i1��� ��ch

�; (44)

Uf = {f +1

�lnnYf (0) �

hP fY (0) =P

hY (0)

i�� ��cf

o; (45)

where {i is a constant term representing a weighted average of physical productivity growthrates in the two countries, and is therefore independent of resource taxes; the constant ��ci �Eci =(P

hY Yi) is the ratio between consumption expenditures and �nal output in country i, equal

to��ch = 1� ~ h � Ih and ��cf = 1 + ~ f�� � If ; (46)

where Ih and If are independent of resource taxes, as showin in (36). To simplify notation,we will henceforth denote the price ratio by p0 � P hY (0) =P

fY (0). Starting from (44)-(45),

utility levels can be represented as functions of the respective national taxes. As shown below,the welfare curves Uh(�h) and Uf (� f ) are concave and each exhibits a unique maximumassociated with a �nite level of the domestic resource tax, denoted as �maxi for country i. Theinterpretation of �maxi is that of a resource tax yielding maximal utility in country i for a givenstate of a¤airs �especially, for a given resource tax implemented �in the other country.

Welfare-tax relation in Home. Considering Home�s utility in (44), a ceteris paribus increasein �h a¤ects the components of Uh as follows. First, resource use in Home declines and thisreduces physical output: @Yh (0) =@�h < 0. Second, the price index p0 increases in order tocompensate for the decline in Home�s relative physical output: @p0=@�h > 0. Third, fromProposition 3, the increase in the resource tax generates rent transfer and implies an increasein Home�s income share: this implies @��ch=@�h > 0 since Home residents can increase the ratiobetween consumption expenditures and domestic �nal output other things being equal. Thisthird e¤ect will be shown to be crucial for all the results presented in this section. By virtue

21

of these mechanisms, the welfare curve Uh (�h) is concave: as shown in the Appendix, the �rstorder condition dUh=d�h = 0 is associated to a unique maximum characterized by a �nite levelof the resource tax, �h = �maxh , and by an equilibrium level of relative resource use, �� = �maxh ,exhibiting an important characteristic:

@Uh@�h

= 0 () �� = �maxh <�

1� � : (47)

Recalling (40), the e¢ cient equilibrium level of relative resource equals �1�� . Hence, result (47)

shows that the welfare-maximizing resource tax in Home is always associated to an ine¢ cientequilibrium where Home�s resource use is relatively low.

Welfare-tax relation in Foreign. Considering Foreign residents�utility in (45), a ceterisparibus increase in � f has the following e¤ects. First, resource use in Foreign declines andthis reduces physical output: @Yf (0) =@� f < 0. Second, the price index p0 declines in order tocompensate for the rise in Home�s relative physical output, and this induces a favourable terms-of-trade e¤ect on welfare since the relative price of Foreign �nal goods increases: @p�10 =@� f > 0.Third, di¤erently from Home, an increase in the Foreign resource tax leaves the consumption-to-output ratio unchanged: recalling Proposition 4, no rent transfer between countries impliesthat national income shares are independent of � f , and so is the consumption-to-output ratioin Foreign (@��cf=@� f = 0). The �rst two mechanisms nonetheless imply the concavity ofUf (� f ): as shown in Appendix, the condition dUf=d� f = 0 is associated to a �nite tax level� f = �maxf and to a speci�c equilibrium level of relative resource use, �� = �maxf , exhibiting thefollowing property:

@Uf@� f

= 0 () �� = �maxf =�

1� � : (48)

From (48), the welfare-maximizing resource tax in Foreign is always associated to an equilib-rium where relative resource use coincides with the e¢ cient level. Note, however, that thisdoes not mean that �maxf is always associated to an e¢ cient equilibrium, since relative resourceuse may be equal to �� = �

1�� also in ine¢ cient equilibria.13

Results (47)-(48) imply that if both national governments fully recognize all the general-equilibrium e¤ects of the respective taxes on resource use, the independent pursuit of domesticwelfare maximization results into con�icting objectives as each government seeks a di¤erentequilibrium level of relative resource use. This is indeed a very general conclusion since neither(47) nor (48) assume that the two economies are starting from a speci�c world competitiveequilibrium. As noted before, this result originates in the asymmetric e¤ects of national taxeson the consumption-to-output ratio: as noted in Appendix, if ��ch were independent of �h,Home welfare Uh(�h) would be maximized in association with an e¢ cient level of relativeresource use exactly as Foreign welfare. The economic mechanism behind (47)-(48) is thusthe rent transfer e¤ect generated by Home resource taxes �which allows Home residents toincrease the ratio between consumption expenditures and domestic �nal output all other thingsbeing equal �that does not arise, instead, when Foreign governments modify � f . The nexttwo Propositions apply results (47)-(48) to speci�c contexts where the initial states of a¤airs

13For example, when 'h = 'f , the laissez-faire equilibrium implies �� = �1�� (see equation (39) above) but is

ine¢ cient due to the market failures induced by R&D externalities and monopolistic competition in the twoeconomies.

22

are the benchmark regimes previously analyzed: symmetric CE-allocations and laissez-faireequilibria.

Deviations from E¢ ciency. Suppose that, in the initial equilibrium, both governmentsimplement conditionally e¢ cient allocations by means of the e¢ cient taxes (41)-(42)-(42).Given this state of a¤airs in Foreign, Home would gain from raising a positive resource tax.The opposite is not true since Foreign would not gain from deviating from the initial state ofa¤airs:

Proposition 5 If both countries start from CE-allocations, @Uf=@� f = 0 and @Uh=@�h > 0.

Proposition 5 establishes that, in an e¢ cient equilibrium, the Foreign government has noincentive to vary � f because �� = ~� already guarantees maximal domestic welfare given thisstate of a¤airs; the Home government, instead, has an incentive to deviate because �� = ~� >�maxh implies that Home welfare is ceteris paribus improved if �h is increased up to �maxh > ~�h.This result suggests that if domestic welfare represents the payo¤ of each government in apolitical game, ine¢ cient equilibria may well be the �nal outcome. Pursuing this argumentobviously requires an extended analysis of game-theoretic issues, which is beyond the scopeof this paper. We only point out that, in a one-shot sequential game where Home deviatesfrom conditional e¢ ciency by implementing �maxh > ~�h and Foreign reacts by implementingthe welfare-maximizing tax �maxf determined by the new state of a¤airs, Home�s welfare atthe end of the game is still higher than in the initial CE-allocation because the ine¢ cienciescreated by both taxes ultimately fall on Foreign residents (see Appendix).

Deviations from Laissez-Faire. The previous results do not necessarily hold if the bench-mark equilibrium is a world laissez-faire allocation, because productivity di¤erences come intoplay:

Proposition 6 If both countries start from laissez-faire allocations, then (i) 'h = 'f implies@Uf=@� f = 0 and @Uh=@�h > 0; (ii) 'h > 'f implies @Uf=@� f < 0 and @Uh=@�h > 0; (iii)'h < 'f implies @Uf=@� f > 0 while @Uh=@�h R 0.

If R&D technologies are identical in the two countries, results are similar to the case ofe¢ cient allocations. Given laissez-faire in Foreign, Home would gain from raising a positiveresource tax; given laissez-faire in Home, Foreign would not gain from deviating from laissez-faire. If R&D productivity is higher in Home, relative resource use exceeds the e¢ cientlevel: from (39), 'h > 'f implies �� > ~�. In this case, both countries have an incentive todeviate, since Foreign would gain from subsidizing domestic resource use whereas Home wouldgain from raising a resource tax. Finally, if R&D productivity is higher in Foreign, relativeresource use falls short of the e¢ cient level: Foreign would gain from raising a resource tax;Home would gain by implementing either a resource tax or a subsidy depending on how bigis the productivity gap.

In general, our interest in welfare-tax relationships stems from the fact that oil taxes areextensively used in oil-importing economies and are often interpreted as stategic policies. Thetheoretical literature made a �rst recognition of the topic after the oil shocks of 1970s, andsuggests that the rent-extraction e¤ect of oil taxes is virtually unbounded (Bergstrom, 1982).

23

Our results may be linked to those of a related paper by Brander and Djajic (1983) showingthat oil importers tend to impose strategic tari¤s in response to monopolistic behavior of oilproducers. In this respect, our results are di¤erent in four respects. First, we show that oil-poor economies have an incentive to raise the domestic tax starting from both laissez-faire ande¢ cient allocations, not just in response to Foreign actions. Second, in Brander and Djajic(1983), the oil-rich economy tends to restrict the resource supply to the oil-poor countrywhereas, in our model, the Foreign government does not have incentives to deviate from thee¢ cient level of relative resource use. Third, if Home begins a tax war in our model, Foreigndoes not react by restricting its oil supply to Home, but rather by reducing its domestic oildemand: if Home enacts a strategic increase of �h which reduces �� below the e¢ cient level,the response of Foreign is to increase � f because this increases �� and brings it towards thee¢ cient level. Fourth, the only case in which Foreign has an independent incentive to subsidizedomestic oil consumption is when there is a structural productivity gap in favor of Home: thisresult is speci�c to the present model and, to our knowledge, is novel to the literature.

5 Conclusion

Since 1980, the aggregate income of oil-exporting countries relative to that of oil-poor countrieshas been remarkably constant, despite structural gaps in labor productivity growth rates. Werationalized this behavior in a two-country model of asymmetric trade where growth ratesare endogenously determined by R&D activity, and productivity di¤erences between countriesare compensated by terms-of-trade dynamics. The model predictions regarding the basicdeterminants of income shares are supported by empirical evidence: the share of each oil-pooreconomy is positively (negatively) related to the domestic (average foreign) investment rateand to the national tax on domestic resource use. Our results regarding the marginal e¤ectsof taxation are also relevant for the current policy debate. If the initial state of a¤airs is ane¢ cient equilibrium in which all domestic market failures are internalized, the governmentof an oil-importing country could increase domestic welfare by raising the national resourcetax, whereas this potential incentive does not arise for the government of the oil-exportingcountry. If the initial state of a¤airs is a laissez-faire equilibrium, instead, incentives arecrucially a¤ected by productivity di¤erences: if productivity grows faster in the oil-importingcountry, the government of the oil-exporting country has an incentive to subsidize domesticresource use.

A Appendix

Monopoly rents and derivation of (6). Maximization of�i (mi) =�P iX(mi)

� &P iY�Xi (mi)

subject to (4) yields

Xi (mi) = Xi =

"�2 (viLi)

� R i& (1 + bi)

# 11��

; (A.1)

�i (mi) = �i = (1� �)P iXXi: (A.2)

Plugging (A.1) back in (4) yields (5). From (1), symmetric varieties imply Yi =MiX�i (viLi)

� R i ,where we can substitute (A.1) to obtain (6).

24

Optimal resource extraction in Foreign: derivation of (13)-(14). Given the objec-tive (12) and constraint _Q = �R, the current value Hamiltonian associated with the extractionproblem is PR (t)R (t)� � (t)R (t), where � (t) is the dynamic multiplier. The conditions foroptimality in extraction read

PR (t) = � (t) ; (A.3)

_� (t) = rf (t)� (t) ; (A.4)

limt!1

� (t)Q (t) e�R1t rf (v)dv = 0; (A.5)

Plugging (A.3) in (A.4) we have Hotelling�s rule (13). Integrating (A.4) and substituting theresulting expression in the transversality condition (A.5) yields limt!1 � (0)Q (t) = 0, whichimplies that, for any positive initial price � (0) = PR (0), the resource stock must be exhaustedasymptotically,

limt!1

Q (t) = 0:

Integrating the resource constraint between t = 0 and in�nity, and imposing limt!1Q (t) = 0yields (14).

Static consumer problem. Maximizing the utility index (16) subject to the expenditureconstraint (17) for each country i = f; h, we obtain

cfi =chi =

1� ��(P hY =P

fY ); (A.6)

Eci =Li =1

�P hY c

hi and E

ci =Li =

1

1� �PfY c

fi ; (A.7)

�ui = ln

("��1���

�1��(P hY )

�(P fY )1��

#(Eci =Li)

); (A.8)

where (A.6) is the �rst order condition in each country, (A.7) results from plugging (A.6) backin constraint (17), and (A.8) is the indirect utility function obtained from substituting theoptimal quantities (A.7) back in the utility index (16). Denoting the term in curly brackets in(A.8) as ! = !(P hY ; P

fY ), we can write �ui = ln [! � (Eci =Li)]. For future reference, notice that

(A.7) implyP hY Lfc

hf = �Ecf and P

fY Lhc

fh = (1� �)E

ch: (A.9)

Dynamic consumer problem: derivation of (21)-(??). The current-value Hamil-tonian of the dynamic consumer problem is

ln(! � (Eci =Li)) + �i ��rini + P

iL � (Eci =Li)� Fi

�;

where � is the dynamic multiplier associated to the wealth constraint; we neglect resourcerents for Foreign residents (i = f) since this does not a¤ect the solution. The optimalityconditions read

Li=Ei = �i and _�i = �i (�� ri) ; (A.10)

0 = limt!1

�i (t)ni (t) e��t: (A.11)

25

Equation (21) directly follows from (A.10).

Aggregate constraints: derivation of (23)-(24). Substituting ni = (ViMi) =Li andViri = �i + _Vi from (9) in (19) and multiplying both sides by Li we obtain

Vh _Mh = �hMh + PhLLh � Ech � FhLh:

Plugging Vi _Mi = P iY Zi from (7)-(8) and �iMi =MiXi�P iX � &P iY

�from (A.2), we obtain

P hY Zh + Ech + P

hY &MhXh =MhP

hXXh + P

hLLh � FhLh

Plugging FiLi = aiPiY Zi � biMiP

iXXi � � iPRRi from (15), we have

P hY Zh (1 + ah) + Ech + P

hY &MhXh =MhP

hXXh (1 + bh) + P

hLLh + �hPRRh

From conditions (2) and (4), we can substitute P iLLi = �P iY Yi and MiPiXXi (1 + bi) = �P iY Yi

to obtainEch + P

hY Zh (1 + ah) + P

hY &MhXh = (�+ �)P

hY Yh + �hPRRh;

where we can plug �+ � = 1� together with condition (3) to obtain

Ech + PhY Zh (1 + ah) + P

hY &MhXh = P hY Yh � PRRh: (A.12)

Substituting Edh � P hY Zh (1 + ah) and Exh � P hY &MhXh we obtain (23). Repeating the above

steps for the Foreign economy with constraint (20), and recalling that R�Rf = Rh, we obtain(24).

Equilibrium interest rates: derivation of (25). In equilibrium, the pro�t to patent-value ratio equals

�iVi= �i

(1 + ai) (1� �)P iXXiP iY

= 'i �(1 + ai) (1� �)�

1 + bi; (A.13)

where the central term is obtained from the ratio between pro�ts in (A.2) and patent-value in(8); the last term is then obtained by substituting �i with (10) and plugging

�P iXMiXi

�=�P iY Yi

�=

�= (1 + bi) from (4). Equations (8) and (10) also imply Vi =�P iY Yi

�= ['i �Mi (1 + ai)] so that

_Vi (t)

Vi (t)=

_P iYP iY

+_YiYi�

_Mi

Mi: (A.14)

Substituting (A.13) in (9), the interest rate in country i equals

ri = 'i� (1� �) �1 + ai1 + bi

+_P iYP iY

+_YiYi�

_Mi

Mi: (A.15)

Time-di¤erentiating (6) we obtain

_YiYi=

_Mi

Mi+

�

1� ��i +

1� �_RiRi: (A.16)

Plugging (A.16) in (A.15) we obtain equation (25) in the text.

26

Proof of Proposition 1. Proposition 1 hinges on a preliminary result that is related toexpenditures allocation in Home. De�ne the expenditure-to-output ratios in country i as

��ci � Eci =�P iY Yi

�; ��di � Edi =

�P iY Yi

�and ��xi � Exi =

�P iY Yi

�: (A.17)

From (5) we have ��xi =�P iY &MiXi

�=�P iY Yi

�= �

��P iXMiXi

�=�P iY Yi

��, where we can substi-

tute�P iXMiXi

�=�P iY Yi

�= �= (1 + bi) from (4) to obtain

��xi =ExiP iY Yi

=�2

1 + biin each i = h; f . (A.18)

From (21), the growth rate of ��ci � Eci =�P iY Yi

�equals

���ci��ci= ri (t)� ��

_P iYP iY

�_YiYi= 'i

� (1� �) (1 + ai)1 + bi

�_Mi

Mi� �;

where we have substituted ri (t) by (A.15). Plugging Edi � P iY Zi (1 + ai) in (11) and using��di � Edi =

�P iY Yi

�, the growth rate of varieties equals _Mi=Mi = 'i��

di , which can be substituted

in the above expression to obtain

���ci��ci= 'i

� (1� �) (1 + ai)1 + bi

� 'i��di � �: (A.19)

For future reference, notice that (A.18)-(A.19) are valid in both economies. Now considercountry Home. Exploiting (A.17), and de�ning the tax-adjusted resource elasticity ~ h � (1 + �h)

�1, the aggregate constraint (23) can be written as

��ch + ��dh + ��

xh = 1� (PRRh) =

�P hY Yh

�= 1� ~ h; (A.20)

where the last term follows from (3). From (A.20), we can substitute ��dh = 1� ~ h � ��ch � ��xhin (A.19), and eliminate ��xh by (A.18), obtaining

���ch��ch= 'h��

ch + 'h

� (1� �) (1 + ah) + �21 + bh

� 'h (1� ~ h)� �; (A.21)

where all terms to the right hand side except ��ch are constant. Since 'h > 0, equation (A.21)is globally unstable around the unique stationary point: ruling out by standard argumentsexplosive dynamics �which would be associated to unbounded dynamics in the propensity toconsume �we thus have

��ch = (1� ~ h)�'h�� (1� �) (1 + ah) + �2

�� � (1 + bh)

'h (1 + bh)in each t: (A.22)

By (A.18) and (A.22), constant values of ��ch and ��xh imply a constant ��

dh which, from (A.20),

equals

��dh = 1� ~ h � ��ch � ��xh ='h� (1� �) (1 + ah)� � (1 + bh)

'h (1 + bh): (A.23)

27

Constant expenditure-output ratios imply that the growth rate of P hY Yh coincides with thebalanced growth rate of all expenditure shares _Ech=E

ch =

_Edh=Edh =

_Exh=Exh , which is determined

by the Kenyes-Ramsey rule (21). Moreover, since (23) and (3) imply that the ratio

Eh=PhY Yh = (1� ~ h) (A.24)

is constant, the balanced growth rate in Home is

_EhEh

=_EchEch

=_P iYP iY

+_YhYh= rh � �: (A.25)

Now substitute (A.9) in (22) to obtain

PRRh + (1� �)Ech = �Ecf : (A.26)

Substituting PRRh = ~ hPhY Yh from (3) and Ech = ��

chP

hY Yh from (A.22) in (A.26) we get

Ecf =1

�[~ h + (1� �) ��ch] � P hY Yh; (A.27)

where all the terms in square brackets are constant, implying that the ratio Ecf=�P hY Yh

�is

constant over time. Since P hY Yh grows at the same rate as Ech by (A.25), we have _Ecf=E

cf =

_Ech=Ech. By the Keynes-Ramsey rules (21) this implies equal interest rates in the two countries,

rh = rf . Imposing rh = rf in (26) yields (27). �

Proof of Proposition 2. Using the de�nition Rh = �Rf and condition (3) for countryi = f , constraint (24) implies

Ef = P fY Yf + PRRh = P fY Yf + �PRRf = P fY Yf�1 + ~ f�

�: (A.28)

Recalling de�nitions (A.17), result (A.28) and the central term in (24) imply ��cf + ��xf + ��

df =

1 + ~ f�, where we can substitute ��xf = �2(1 + bf )

�1 from (A.18) to obtain

��df = 1 + ~ f� ��2

1 + bf� ��cf : (A.29)

Plugging (A.29) in (A.19) for country i = f we obtain

���cf��cf= 'f

� (1� �) (1 + af )1 + bf

� 'f�1 + ~ f� �

�2

1 + bf� ��cf

�� �: (A.30)

Now divide both sides of (A.27) by P fY Yf and solve for ��cf � Ecf=(P

fY Yf ) to obtain

��cf =1

�[~ h + (1� �) ��ch] �

P hY Yh

P fY Yf=1

�[~ h + (1� �) ��ch] �

~ f~ h�; (A.31)

where we have used (28) to get the last term. Now de�ne

� � 1

�+1� ��

� ��ch

~ h> 1: (A.32)

28

Since ��ch is constant by (A.22), � is also constant and (A.31) implies

��cf = �~ f� and

���cf��cf=_�

�: (A.33)

Substituting the second expression in (A.33) into (A.30) we obtain

_�

�= 'f (�� 1) ~ f � � +

'f�� (1� �) (1 + af ) + �2

���'f + �

�(1 + bf )

1 + bf; (A.34)

where all terms to the right hand side except � are constant. Since 'f (�� 1) ~ f > 0, equa-tion (A.34) is globally unstable around the unique stationary point: ruling out by standardarguments explosive dynamics �which would be associated to unbounded dynamics in thepropensity to consume in Foreign �we thus have � (t) = �� in each t 2 [0;1), where �� is thesteady-state level (??) obtained by imposing _� = 0 in (A.34):

�� ��'f + �

�(1 + � fK)� 'f

�� (1� �) (1 + af ) + �2

�~ f (�� 1)'f (1 + �

fK)

: (A.35)

From (28), a constant � implies

_P hYP hY

�_P fY

P fY=_YfYf�_YhYh

(A.36)

where we can substitute (27) and (??) to obtain (29). Since PRRh = ~ hPhY Yh, the Hotelling

rule _PR=PR = rh in (13) and result (A.25) imply that PRRh grows at the rate rh � �, so that_Rh=Rh = ��. This, in turn, implies _Rf=Rf = �� because � = Rh=Rf is constant, whichproves (30). From (A.28), a constant � also implies that Ef grows at the same rate as P

fY Yf ,

which coincides with the growth rate of Eh and P hY Yh by (A.36) and (A.25). We thus have (31).From (A.33), we can substitute ��cf = �~ f

�� in (A.29) to obtain ��df = 1 � �2

1+bf� (�� 1) ~ f��,

where we can eliminate (�� 1) ~ f�� by means of (A.35) to obtain

��di ='i� (1� �) (1 + ai)� �(1 + bi)

'i(1 + bi)(A.37)

in country i = f . But (A.23) implies that (A.37) is also valid for country i = h. From ((11)),both countries exhibit _Mi=Mi = 'i��

di , so that result (A.37) implies (32). Finally, from (23)

and (24), Home�s income share can be written as sh =PhY Yh(1�~ h)PhY Yh+P

fY Yf

, where we can plug (28) to

obtain (31). �

Closed form solution. Equation (6) and result (29) imply that physical �nal output incountry i equals

Yi (t) =(�2=&)

�1��

1 + bi�Mi (0) (vi (0)Li)

�1�� (Ri (0))

1�� � e(i��)t; (A.38)

29

where Mi (0) and vi (0) are exogenously given. The only jump variable is the initial resourceuse Ri (0), which can be determined from the solution of the optimal extraction problem:14

Rh (0) =��

1 + ���Q0 and Rf (0) =

1

1 + ���Q0: (A.39)

Equations (A.38)-(A.39) yield explicit solutions for the time paths of physical output in thetwo countries and, in particular, imply that Home�s physical output relative to Foreign is aconcave function of relative resource use in each point in time:

Yh (t)

Yf (t)= ��

1�� � 0 � e(h�f)t; (A.40)

where we have de�ned 0 ��Mh(0)Mf (0)

�1+bf1+bh

��vh(0)Lhvf (0)Lf

� �1���, which is given at t = 0.

Derivation of (35)-(36)-(37). substituting the de�nition Ecf = ��cfPfY Yf in (A.27) we

haveP hY Yh

P fY Yf=

���cf~ h + (1� �) ��ch

: (A.41)

Substituting ��cf = 1 + ~ f�� � ��xf � ��df from (A.29), ��ch = 1 � ~ h � ��xh � ��dh from (A.20), andPhY Yh

P fY Yf=

~ f~ h�� from (28), we obtain

~ f~ h�� =

�(1���xf���df )+�~ f��(1��)(1���xh���dh)+�~ h

, which can be solved for �� to get

�� =~ h~ f�

�(1� ��xf � ��df )(1� �)

�1� ��xh � ��dh

� : (A.42)

Plugging PhY Yh

P fY Yf=

~ f~ h�� back into (A.42) yields (35), where Ii � ��xi +��di and (36) follows directly

from (A.18) and (A.37). Since ~ i = (1 + � i)�1, result (A.42) also implies (37).

Proof of Proposition 3. From (37) we have @��@�h

< 0. Equations (A.38)-(A.39) imply@Yh@��

> 0 and @Yf@��

< 0 and therefore @Yh@�h= @Yh

@��� @��@�h < 0 and

@Yf@�h

=@Yf@��� @��@�h > 0. These results

in turn imply @(Yh=Yf )@�h

< 0. From (35), we have @ (PhY Yh)=(P

fY Yf )

@�h= 0. Plugging this result

into (34), we have @sh@�h

=(PhY Yh)=(P

fY Yf )

1+[(PhY Yh)=(PfY Yf )]

� @@�h

(1� ~ h) > 0 due to @~ h@�h

< 0. This implies@sf@�h

= @(1�sh)@�h

< 0. �

Proof of Proposition 4. From (37) we have @��@�f

> 0. Equations (A.38)-(A.39) imply@Yh@��

> 0 and @Yf@��

< 0 and therefore @Yh@�f

=@Yf@��� @��@�f > 0 and

@Yf@�f

=@Yf@��� @��@�f < 0. From (35),

we have @ (PhY Yh)=(P

fY Yf )

@�f= 0, which also implies @sh@�f

= 0 because ~ h is independent of � f , and

therefore @sf@�f

= 0. �14Since R = Rh + Rf and � = ��, the intertemporal resource constraint (14) can be written as Q0 =R10Rf (t)

�1 + ��

�dt and directly integrated to obtain Rf (0) in (A.39), from which Rh (0) can be obtained as

��Rf (0).

30

Conditional e¢ ciency in Home. By de�nition, the CE-allocation in Home solves

maxfEch;Exh ;Edh;Rhg

Z 1

0e��t � ln((!=Lh) � Ech)dt subject to

Yh = MhX�h (vhLh)

� Rh ;

Exh = P hY &MhXh;

P hY Yh = Ech + Edh + E

xh + PRRh;

_Mh = Mh'h �hEdh=(P

hY Yh)

i;

where ! = !(P hY ; PfY ) is taken as given and symmetry across varieties is already imposed

without any loss of generality. The �rst constraint is the �nal-good technology (1), the secondis the intermediate-good technology with linear cost, the third is (23), the fourth is the R&Dtechnology (11) with knowledge spillovers taken into account. Recalling that �dh � Edh=(P

hY Yh)

and combining the �rst three constraints, the problem becomes maxfEch;Xh;�dh;RhgR10 e��t �

ln((!=Lh) � Ech)dt subject to

P hYMhX�h (vhLh)

� Rh �1� �dh

�= Ech + P

hY &MhXh + PRRh; (A.43)

_Mh = Mh'h�dh; (A.44)

where the controls are�Ech; Xh; �

dh; Rh

and the only state variable is Mh. The current-value

Hamiltonian is

ln [(!h=Lh) � Ech] + �0h �Mh'h�dh +

+�00h �hP hYMhX

�h (vhLh)

� Rh �1� �dh

�� Ech � P hY &MhXh � PRRh

iwhere �0h is the dynamic multiplier associated to (A.44) and �

00h is the static multiplier attached

to (A.43). The optimality conditions read

@

@Ech= 0! 1

Ech= �00h; (A.45)

@

@Xh= 0!

�1� �dh

��P hY Yh = P hY &MhXh; (A.46)

@

@�dh= 0! �0hMh'h = �00hP

hY Yh; (A.47)

@

@Rh= 0!

�1� �dh

� P hY Yh = PRRh (A.48)

��0h � _�0h =@

@Mh! ��0h � _�0h = �0h'h�

dh + �

00hP

hY

�YhMh

�1� �dh

�� &Kh

�;(A.49)

31

and imply15

~Eh =h1�

�1� �dh

�i� P hY Yh; (A.50)

~Exh = ��1� �dh

�� P hY Yh; (A.51)

~Ech = ��1� �dh

�� P hY Yh; (A.52)

Edh = �dh � P hY Yh: (A.53)

Substituting (A.46) and (A.47) in (A.49) we have

_�0h�0h= �� 'h

h1� �

�1� �dh

�i: (A.54)

Time-di¤erentiating (A.47) and using (A.54) we have

_�00h�00h= �� 'h (1� �)

�1� �dh

��_P hY Yh

P hY Yh;

where we can substitute �00h = 1=Ech from (A.45) to obtain

_EchEch

��

P hY Yh

P hY Yh= 'h (1� �)

�1� �dh

�� �: (A.55)

From (??) we have_EchEch�

�PhY YhPhY Yh

= � _�dh1��dh

which can be combined with (A.55) to get

_�dh = ��1� �dh

�� 'h (1� �)

�1� �dh

�2: (A.56)

Equation (A.56) is globally unstable around its unique steady state: ruling out explosivedynamics by standard arguments, the conditionally-e¢ cient rate of investment in R&D is

~�dh ='h (1� �)� �'h (1� �)

and 1� ~�dh =�

'h (1� �)(A.57)

in each point in time. Substituting (A.57) in (A.51)-(A.52) we obtain

~�xh =��

'h (1� �)and ~�ch =

��

'h (1� �): (A.58)

Conditional e¢ ciency in Foreign. Following the same preliminary steps of the Homeproblem, the CE-allocation in Foreign solves

maxfEcf ;Xf ;�df ;Rh;Rfg

Z 1

0e��t � ln((!=Lf ) � Ecf )dt subject to

P fYMfX�f (vfLf )

� Rf �1� �df

�= Ecf + P

fY &MfXf � PRRh; (A.59)

_Mf = Mf'f�df ; (A.60)

_Q = �Rh �Rf (A.61)

15Plugging (A.48) in constraint (23) we have (A.50). Plugging (A.46) in technology Ekh = P

hY &MhKh yields

(A.51). Plugging (A.46) and (A.48) in (A.43) we have (A.52). Equation (A.53) is determined residually by~Edh = ~Eh � ~Ek

h � ~Ech.

32

where (A.59) follows from (24) and, di¤erently from Home, we have the resource constraint(A.61) and also exported resources Rh as an additional control. The state variables are Mf

and the resource stock Q. The Hamiltonian is

ln�(!=Lf ) � Ecf

�+ �0f �Mf'f�

df +

+�00f �hP fYMfX

�f (vfLf )

� Rf �1� �df

�� Ecf � P

fY &MfXf + PRRh

i+

+�000f � (�Rh �Rf )

where �0f is the dynamic multiplier associated to (A.60), �00h is the Lagrange multiplier attached

to (A.59), and �000f is the dynamic multiplier associated to (A.61). The �rst order conditionsread

@

@Ecf= 0! 1

Ecf= �00f ; (A.62)

@

@Xf= 0!

�1� �df

��P fY Yf = P fY &MfXf ; (A.63)

@

@�df= 0! �0fMf'f = �00fP

fY Yf ; (A.64)

@

@Rh= 0! �00f � PR = �000f (A.65)

@

@Rf= 0! �00f �

�1� �df

� P fY Yf = �000f Rf ; (A.66)

��0f � _�0f =@

@Mf! ��0f � _�0f = �0f'f�

df + �

00fP

fY

�YfMf

�1� �df

�� &Kf

�;(A.67)

��000f � _�000f =@

@Q! ��000f � _�000f = 0: (A.68)

Notice that, from (A.65)-(A.66) and de�nition Rh = �Rf , we have

PR ~Rf =�1� �df

� P fY

~Yf ; (A.69)

PR ~Rh =�1� �df

� ~� � P fY ~Yf ; (A.70)

so that expenditures equal16

~Ef =h1 +

�1� ~�df

� ~�i� P fY ~Yf ; (A.71)

~Exf = ��1� ~�df

�� P fY ~Yf ; (A.72)

~Ecf =�1� �+ ~�

��1� ~�df

�� P fY ~Yf ; (A.73)

~Edf = ~�df � PfY~Yf : (A.74)

16Plugging (A.70) in (24) yields (A.71). Plugging (A.63) in technology Ekf = P fY &MfKf yields (A.72).

Plugging (??) and (A.70) in (A.59) we have (A.73). Equation (A.74) is determined residually by ~Edf = ~Ef �

~Ekf � ~Ec

f .

33

Before deriving the explicit value of ~�df we show that the e¢ cient relative resource use ~� isconstant over time. From the balanced trade condition (A.26), we have PRRh + (1� �)Ech =�Ecf where we can use (A.52) and (A.73) to eliminate E

ch and E

cf , respectively, and also use

(A.48) to eliminate PRRh, obtaining

1� ~�dh1� ~�df

� PhY~Yh

P fY~Yf=��1� �+ ~�

� + (1� �)� ; (A.75)

where tildas denote conditionally-e¢ cient values. Taking the ratio between (A.48) and (A.70)we have

~� =1� ~�dh1� ~�df

� PhY~Yh

P fY~Yf: (A.76)

Combining (A.76) with (A.75) we obtain

~� =�

1� � �1� � + �

=�

1� � : (A.77)

Result (A.77) implies that ~� is constant. Now go back to (A.67) and substitute (A.63)-(A.64)to re-write it as

_�0f�0f= �� 'f

h1� �

�1� ~�df

�i: (A.78)

Time-di¤erentiating (A.64) and substituting (A.62)-(A.60), we obtain

_�0f�0f= �

_EcfEcf

+

�P fY Yf

P fY Yf� 'df ~�df

which can be combined with (A.78) to obtain

_EcfEcf

�

�P fY Yf

P fY Yf= 'f

h(1� �)

�1� ~�df

�i� �: (A.79)

Since ~� is constant by (A.77), time-di¤erentiation of (A.73) yields_EcfEcf�

�P fY Yf

P fY Yf= � _�df

1��df.

Plugging this result in (A.80) we obtain the usual equilibrium relation (see (A.56) above forHome) which can be solved for the steady-state level

~�df ='f (1� �)� �'f (1� �)

or 1� ~�df =�

'f (1� �): (A.80)

Substituting (A.80) in (A.72)-(A.74) we obtain

~�xf =��

'f (1� �)and ~�cf =

��1� �+ ~�

�'f (1� �)

: (A.81)

Derivation of (40). Equation (40) is proved in (A.77).

34

Derivation of (41)-(43). E¢ cient taxes are obtained by equalizing e¢ cient and equilib-rium values of (�xi ; �

di ; �

ci ). First, results (A.58) and (A.81) imply ~�

xi =

��'i(1��)

in both coun-tries. Imposing the equality between the e¢ cient values ~�xi and the competitive-equilibriumvalues ��xi =

�2

1+biderived in (A.18), we obtain the e¢ cient tax on intermediates�purchases ~bi in

(42). Second, results (A.57) and (A.80) imply ~�di ='i(1��)��'i(1��)

in both countries. Imposing the

equality between ~�di and the competitive-equilibrium values ��di =

'i�(1��)(1+ai)��(1+bi)'f (1+bi)

derived

in (A.37), and substituting ~bi by (42), we obtain the e¢ cient subsidy ~ai in (41). Now considerHome: from (A.58) we have ~�ch =

��'h(1��)

whereas (A.23) implies ��ch = 1 � ~ h � ��dh � ��xh.Setting ~�ch = ��

ch and imposing that ��

xh = ~�

xh and ��

dh = ~�

dh by virtue of (41)-(42), we obtain

��

'h (1� �)= 1� ~ h � ��dh � ��xh = 1� ~ h �

'h (1� �)� �'h (1� �)

� ��

'h (1� �)

where the last term follows from ~�dh and ~�xh derived in (A.57) and (A.58). Rearranging terms

and solving for ~ h we obtain ~ h =�

'h(1��), which implies the e¢ cient resource tax for Home

~�h ='h (1� �)� �

�: (A.82)

The optimal resource tax in Foreign ~� f then follows from (A.76). Since 1 + ~�h = 1 � ��dhby (A.82), the only way to satisfy �� = ~� in equations (28) and (A.76) is to set 1 + ~� f =1 � ��fh =

1�'h (1� �), which proves (43). It can be easily veri�ed that, residually, (41)-(43)

imply ��cf = ~�cf .

Derivation of (44)-(46). De�ning the constant ��i � (�=Li) (1��� )

1�� and recallingthat Eci = ��

ciP

iY Yi by de�nition (A.17), present-value utility (18) reads

Ui =

Z 1

0e��t � ln

"��i

��ciPiY Yi

(P hY )�(P fY )

1��

#dt: (A.83)

Plugging the respective country indices, we obtain

Uh =

Z 1

0e��t � ln

���h

�P hY =P

fY

�1����chYh

�dt and Uf =

Z 1

0e��t � ln

h��f

�P fY =P

hY

����cfYf

idt:

Substituting P hY (t) =PfY (t) = [P

hY (0) =P

fY (0)]e

(f�h)t and Yi (t) = Yi (t) e(i��)t and collect-

ing the terms to isolate P hY (0) ; PfY (0) ; Yh (0) ; Yf (0) and (��

ch; ��

cf ), we can de�ne the constants

{h �Z 1

0e��t � ln

he[h��+(1��)(f�h)]t

idt+

1

�ln��h;

{f �Z 1

0e��t � ln

he[f��+(�(h�f )]t

idt+

1

�ln��f ;

and rewrite Uh and Uf as in (44)-(45). The �rst expression in (46) follows from (A.22) andIh � ��xh + ��dh. Rewriting (A.29) as ��cf = 1+ ~ f� � ��xf � ��df and substituting If � ��xf + ��df weobtain the last expression in (46).

35

Utility-Tax relationship in Home. Setting t = 0 in (28) and solving for p0 =P hY (0) =P

fY (0) we obtain

p0 = � � 1 + �h1 + � f

Yf (0)

Yh (0)=

�

1� � �1� If1� Ih

� �10 � ���

1�� ; (A.84)

where the last term follows from (A.40). Equation (A.84) implies

@p0@�h

� 1p0= �

1� � �@��

@�h� 1��: (A.85)

From (A.38)-(A.39) we also have

@Yh (0)

@�h� 1

Yh (0)=

1� � �@��

@�h� 1��� 1

1 + ��: (A.86)

Di¤erentiating (44) we have �@Uh@�h=

@��ch@�h

1��ch+ (1� �) @p0@�h

1p0+ @Yh(0)

@�h1

Yh(0), where we can plug

@��ch@�h

1��ch= �@~ h

@�h1��chfrom (46) together with (A.85)-(A.86) to obtain

�@Uh@�h

= �@~ h@�h

1

��ch+

1� � �@��

@�h� 1����1

1 + ��� (1� �)

�: (A.87)

From (37) we have @��@�h

1��= @~ h

@�h1~ hand (A.87) can be written as

�@Uh@�h

= �@~ h@�h

� 1~ h��~ h��ch�

1� � ��1

1 + ��� (1� �)

��: (A.88)

Recalling that @~ h@�h1~ h= � 1

1+� and substituting ��ch by (46) in the above expression, we obtain

�@Uh@�h

=

1 + �h

�1

(1 + �h) (1� ~ h � Ih)� 1

1� � ��1

1 + ��� (1� �)

��: (A.89)

De�ne the two terms in curly brackets as �a (�h) � 1= [(1 + �h) (1� ~ h � Ih)] and �b (�h) �1

1�� �h11+��

� (1� �)i, respectively. We have

�@Uh@�h

=

1 + �h

h�a (�h)��b (�h)

i; (A.90)

where�a (�h) is strictly increasing and�b (�h) is strictly decreasing in �h. Moreover, lim�h!1 �� =0 implies that

lim�h!1

�a (�h) = 0 and lim�h!1

�b (�h) =�

1� � > 0: (A.91)

Result (A.91) combined with @�a=@�h < 0 and @�b=@�h > 0 implies two results. First, therecan only exist a unique �nite value ��h satisfying �a

���hR�= �b

���hR�. Second, the value ��hR

satisfying �a���hR�= �b

���hR�is associated with a maximum in Uh because (i) �a = �b implies

@Uh@�h

= 0, and (ii) @�a=@�h < 0 and @�b=@�h > 0 imply that@Uh@�h

> 0 for any �h < ��hR as well

as @Uh@�h< 0 for any �h > ��hR.

36

Utility-Tax relationship in Foreign and derivation of (48). From (37), ~ f�� isindependent of � f . As a consequence,

@��cf@� f

1

��cf=@~ f

��

@� f

1

��cf= 0: (A.92)

Result (A.84) implies@p0@� f

� 1p0= �

1� � �@��

@� f� 1��: (A.93)

From (A.38)-(A.39) we have

@Yf (0)

@� f� 1

Yf (0)= �

1� � ���

1 + ��� @��

@� f� 1��: (A.94)

Di¤erentiating (45) we have �@Uf@�f=

@��cf@�f

1��cf� � @p0@�f

1p0+@Yf (0)@�f

1Yf (0)

; where we can plug (A.92),

(A.93) and (A.94) to obtain

�@Uf@� f

=

1� � �@��

@� f� 1������

��

1 + ��

�: (A.95)

From (37) we have @��@�f

� 1�� > 0, which implies that Uf (� f ) is concave in � f . Setting to zerothe term in brackets in (A.95), the maximum is characterized by condition (48) in the text.

Proof of Proposition 5. In a symmetric CE-allocation we have �� = ~� = �1�� . In

Home, this implies @Uh=@�h > 0 because �a (�h) > �b (�h) in (A.90). In Foreign, this implies@Uf@�f

= 0 from (A.95). �

Proof of Proposition 5. In a laissez-faire equilibrium, �� is given by (39). If 'h = 'f wehave �� = �

1�� and @Uf=@� f = 0 and @Uh=@�h > 0 are proved exactly as in Proposition 5. If'h > 'f , we have �� >

�1�� , which implies @Uh=@�h > 0 because �

a (�h) > �b (�h) in (A.90),

and @Uf@�f

< 0 from (A.95). If 'h < 'f , we have �� <�1�� , which implies

@Uf@�f

> 0 from (A.95)whereas @Uh=@�h is generally ambiguous in (A.90). �

References

Acemoglu, D., Ventura, J. (2002). The World Income Distribution. Quarterly Journal ofEconomics 117 (2): 659�694.

Barbier, E.B. (1999). Endogenous growth and natural resource scarcity. Environmental andResource Economics 14: 51�74.

Barro, R., Sala-i-Martin, X. (2004). Economic Growth. Second Edition. Cambdridge MA:MIT Press.

Bergstrom, T. (1982). On Capturing Oil Rents with a National Excise Tax. AmericanEconomic Review 72 (1): 194-201.

37

Brander, J., Djajic, S. (1983). Rent-Extracting Tari¤s and the Management of ExhaustibleResources. Canadian Journal of Economics 16: 288-298.

Chiarella, C. (1980). Trade between resource-poor and resource-rich economies as a di¤er-ential game. In Kemp, M. and Long, N. (eds.), Exhaustible resources, optimality andtrade, pp. 199-246. Amsterdam: North-Holland.

Daubanes, J., Grimaud, A. (2006). On the North-South E¤ects of Environmental Policy:Rent Transfersm Relocation and Growth. IDEI Working Paper 477.

EIA (2009). International Statistics. Energy Information Administration, www.eia.gov.

IMF (2009). International Statistics. International Monetary Fund, www.imf.org.

ILO (2009). Labor Productivity Statistics from the Conference Board: Total Economy Data-base. International Labor Organization, www.ilo.org.

Kemp, M., Suzuki, H. (1975). International Trade with a Wasting but Possibly ReplenishableResource. International Economic Review 16 (3): 712-732.

Lederman, D., Maloney, W.F. (2007). Trade Structure and Growth. In Lederman, D. andMaloney, W.F. (eds). Natural Resources �Neither Curse nor Destiny. Washignton DC:The World Bank.

Peretto, P., Valente, S. (2010). Resource Wealth, Innovation and Growth in the GlobalEconomy. CER-ETH Economics Working Paper Series n.10/124, ETH Zürich.

Rivera-Batiz, L., Romer, P. (1991). Economic Integration and Endogenous Growth. Quar-terly Journal of Economics 106 (2): 531-555.

Scholz, C., Ziemes, G. (1999). Exhaustible resources, monopolistic competition, and endoge-nous growth. Environmental and Resource Economics 13: 169�185.

Van Geldrop, J.H., Withagen, C. (1993). General equilibrium and international trade withexhaustible resources. Journal of International Economics 34: 341-357.

World Bank (2009). World Development Indicators. Washington DC: The World Bank.

World Trade Organization (2009). International Trade Statistics 2009. Geneve: WTO Pub-lications.

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