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Fundación de Estudios de Economía Aplicada
Trade Liberalization, Competition and Growth by Omar Licandro* ** Antonio Navas-Ruiz DOCUMENTO DE TRABAJO 2008-03 Serie Innovación CÁTEDRA Fedea – Banco Sabadell
January 2008
* European University Institute. ** Universidad Carlos III de Madrid and GREQAM
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Jorge Juan, 46 28001 Madrid -España Tel.: +34 914 359 020 Fax: +34 915 779 575 infpub@fedea.es
Omar Licandro (European University Institute)
Antonio Navas-Ruiz (Universidad Carlos III de Madrid and GREQAM)
September 2007
Abstract
The aim of this paper is to understand whether international trade may enhance innovation and growth through an increase in competition. We develop a two-country endogenous growth model, both countries producing the same set of goods, with Örm speciÖc R&D and a continuum of oligopolistic sectors under Cournot competition. Since countries produce the same set of goods, trade openness makes markets more competitive, reducing prices and raising the incentives to innovate. More general, a reduction on trade barriers enhances growth by reducing domestic Örmsímarket power.
Keywords: Trade openess, competition and growth, R&D
JEL: F13, F43, O3
We would like to thank J.A. Erce, C. GarcÌa-PeÒalosa, G. Impulitti, R. Marimon, M. Motta, C. Ponce, G. Ottaviano, and participants to the Vienna EEA-ESEM, Faro MMGD and ASSET 2006 meetings and the EUI-MWF workshop for useful comments. O. Licandro acknowledges the Önancial support of the Spanish Ministry of Education (SEJ2004-04579/ECON). E-mails: omar.licandro@eui.eu and antonio.navas@eui.eu.
1 Introduction
During the last two decades the volume of international trade has increased enormously, among developed countries in the 80ís and extending to developing countries in the 90s. This increase in trade volumes is contemporaneous with several attempts to create regional integration agreements, as for example the European Union and the MERCOSUR. There is a common belief that these changes have turned out into a larger more competitive environment, whose consequences for Örmís productivity and growth are still subject of controversy. While the EU and the OECD countries have carried out policies to stimulate competition, some voices alert about the impact of foreign competition on the productivity of domestic Örms.
In this paper, we stress the positive impact on economic growth of the procompetitve role of international trade. Our paper is motivated by recent empirical studies both at the Örm and the industry level suggesting that the rise in competition coming from globalization has a no negligible positive impact on Örmís investments in innovation. For example, Harrison et al. (2006) evaluate the impact that di§erent reforms carried out by the European Union under the Single Market Program have had in innovation intensity. They Önd that these policies have increased both competition, measured as a reduction in markups, and innovation intensity, measured as R&D expenditures over sales, leading to productivity growth in the manufacturing sector.
In models of endogenous growth, where technological progress is the result of Örmsí private decisions, international trade a§ects the incentives to innovate through di§erent channels: The reallocative e§ects due to specialization can have an impact on growth if sectors present di§erent scopes for technological progress (Grossman and Helpman, 1991); openness to trade promotes the exchange of knowledge leading to growth, the so-called knowledge spillovers (Rivera-Batiz and Romer, 1991, and Devereux and Lapham, 1994). However, in all these studies trade openness gives little space to competition, because of the monopolistically competitive nature of markets and the assumption that innovation is carried out by potential entrants. In Rivera-Batiz and Romer, for example, markups only depend on the elasticity of substitution among varieties; moreover, openness to trade increases the market size and the number of Örms in the same proportion, leaving innovation rents unchanged. It is only under the existence of technological spillovers that innovation is fostered. Finally, their paper fails accounting for the empirical evidence cited above since it predicts constant average markups after trade liberalization and ignores, by assumption, the reaction in terms of R&D investments of incumbent Örms.
More recently, the literature on competition and growth has developed models in which incumbents are allowed to upgrade their own technologies. The seminal work by Aghion et al. (2001) points out the escape from competition e§ect as an incentive to innovate in highly competitive environments. In a di§erent paper Peretto (1999) considers an extension of Romerís (1990) model, by adding costreduction innovations and strategic interaction among Örms. A rise in product market competition produces higher growth by reducing the number of Örms, which increases markups and makes Örms to innovate more. Therefore, there is a tradeo§ between competition and growth, since higher growth is associated with lower number of Örms and higher markups. Peretto (2003) extends the previous exercise to trade openness, and shows that it reduces both the global number of Örms and R&D costs ñdue to technological spilloversñrising the incentives to innovate.
The model in this paper is a two-country economy with R&D activities being undertaken by incumbents. Di§erent from the literature cited above, both countries produce the same set of varieties, each variety being produced by n Örms in each country under Cournot competition. When countries open to trade, it is not the mass of varieties that changes but the number of Örms competing in each variety. This approach has many important advantages for the study of the competitive e§ects of trade openness. Firstly, an increase in competition is modelled by an increase in the number of competitors o§ering the same product.1 Note that in this framework, other measures of competition like markups, market shares or market concentration reduce when the number of Örms increases.2 Second, R&D is undertaken by incumbents and innovation is Örm speciÖc, implying that the return to innovation depends crucially on Örmís size. An increase in the number of Örms has two opposite e§ects on Örmís size: A market share e§ect, by increasing the number of competitors, it reduces individual market shares; and a competition e§ect, by increasing competition it has a negative e§ect on markups, increasing the size of the market. Third, trade openness a§ects growth through the competition e§ect only, since the reduction in the domestic market share su§ered by local Örms is compensated by their participation in the foreign market.3 Finally, it is important to say that this paper studies the case of economic integration among similar economies where openness to trade intensiÖes competition within the exiting industries rather that giving access to di§erent goods produced abroad, something more frequent in cases of North-South trade.
1In Aghion et al (2001), competition is measured by the elasticity of substitution between di§erent varieties. However, as Koeninger and Licandro (2005) point out, the elasticity of substitution is an element of the environment reáecting preferences or technology. They claim that changes in the elasticity of substitution results on di§erent e¢cient allocations, which may be confounded with the associated change in competition.
2An interesting discussion about the measurement of competition is in Motta (2004).
The paper succeeds in obtaining a positive growth e§ect of trade openness as a consequence of the increase in innovation generated by a more competitive environment. Since the number of Örms a§ect innovation non-linearly, the paper shows that gains from trade are larger the less competitive countries are in autarky, as Örms would be more reactive in these environments. More generally, the paper shows that trade barriers reinforce domestic Örmsí market power leading to a decrease in innovation and growth. This paper is related to the recent work by Neary, who successfully introduces olipolistic elements into general equilibrium theory by assuming a continuum of sectors and an oligopolistic market within each sector. In this framework, the impact of trade liberalization on production and trade patterns is analyzed in Neary (2002), and the impact on wage inequality in Neary (2003), but still no work has explored its consequences for innovation and growth.
3We do not consider technological spillovers, isolating the pure e§ect of competition on innovation.
The paper is structured as follows. In section 2, we present the basic model in autarky and we try to understand what are the forces driving growth. In section 3, we allow for free trade in the case of two identical countries. Section 4 concludes.
2 Autarky
Consider an economy populated by a continuum of consumers of measure L; with instantaneous logarithmic preferences deÖned over two Önal consumption goods X and Y ,
\[\int_ {0} ^ {\infty} (\ln C _ {t} ^ {x} + \ln C _ {t} ^ {y}) e ^ {- \rho t} d t, \quad \rho > 0,\]
where represent consumption levels. Good Y is an homogeneous good.4 Good X is a Dixit-Stiglitz composite good deÖned in a continuum of industries of measure :
\[C _ {t} ^ {x} = \left(\int_ {0} ^ {N} x _ {j t} ^ {\alpha} d j\right) ^ {\frac {1}{\alpha}}, \alpha \in (0, 1),\]
where represents consumption of good j: Each individual is endowed with one unit of labour at each point in time. In order to Önance R&D activities, Örms issue shares, ; which pay a rate of return . Let us take the homogeneous good as the numeraire : The representative consumer budget constraint is given by:
4The existence of a traditional good allows for the reallocation of labor to the R&D sector without necessarily reducing labor assigned to the composite good sector. A similar result would arrive under the assumption of an elastic labor supply as in Aghion el al (2001). When the e§ect of trade openness to employment is a key issue, this alternative would preferable.
\[\dot {A} _ {t} = w _ {t} + r _ {t} A _ {t} - \int_ {0} ^ {N} p _ {j t} x _ {j t} d j - C _ {t} ^ {y}, A _ {0} > 0,\]
where is the wage rate, and is the price of good
Good Y is produced by a continuum of Örms of measure one with technology:
\[C _ {t} ^ {y} = L _ {t} ^ {y},\tag{1}\]
where represents labour allocated to this sector. Sector is competitive implying that
Each good in X is produced by n Örms in an oligopolistic environment. A Örm i in produces using technology (let us omit the subscript j for simplicity)
\[q _ {i t} = z _ {i t} L _ {i t} ^ {x},\tag{2}\]
where is the stock of knowledge, which is assumed to be Örm-speciÖc. Firms in X can also invest in R&D activities leading to a reduction in marginal production costs. The R&D technology is
\[\dot {z} _ {i t} = (L _ {i t} ^ {z}) ^ {\gamma} z _ {i t}, \gamma \in (0, 1),\tag{3}\]
where represents labor allocated to R&D.5
At any point in time Örms in j decide the quantity to supply and the optimal allocation of workers to both activities, physical production and R&D, taking into consideration other Örmsí strategies. This game belongs to the family of di§erential games, or repeated games deÖned in continuos time, in which past actions a§ect current payo§s. Two di§erent concepts of perfect Markov Nash equilibria have been proposed in the literature, the open-loop and the closedloop Nash equilibrium. This paper focuses in open-loop Nash equilibria (OLNE), mainly for two reasons. Firstly, for simplicity, since under certain assumptions standard optimal control theory techniques can be applied in order to Önd OLNE, and secondly, because in models without uncertainty every OLNE is a closed-loop Nash equilibrium, (CLNE).6 In an open-loop Nash equilibrium Örms decide at time the optimal path of strategies taking other Örmsípath strategies as given. In this sense an open-loop Nash equilibrium is equivalent to a static Nash equilibrium where the possible strategies are time-paths of actions and the payo§s associated are inÖnite sum of payo§s.
Let be Örmís i strategy, where are the timepaths of output and R&D workers, and let us call ; the set of strategies of Örm i. Let be the value of Örm i when the v Örms in the market, , play strategies
5Since we are focusing on the e§ects of a pure increase in competition on growth, no technological spillovers are assumed consistently with Aitken and Harrison (1999) Öndings of little or no evidence of technological spillovers coming from international trade.
t + s
t + s,
6In a model without uncertainty the information sets, at time t and at time t + s, relevant to take the optimal decisions in are the same, therefore every open-loop is also a closed-loop Nash equilibrium. See Fehrstmann and Mueller (1984).
\[A _ {n} = \left[ a _ {1}, a _ {2}, \dots \dots , a _ {n} \right].\]
DeÖnition 1 At time is an open loop Nash equilibrium if
\[V _ {i} \left[ A _ {n} \right] \geq V _ {i} \left[ A _ {n} ^ {\prime} \right] \geq 0,\]
where
This condition implies that the optimal time path of strategies maximizes the value of Örm i taking as given other Örmsístrategies, ; and that the Örm value has to be non-negative.
2.1 Solving for the autarkic equilibrium
Consumers solve the standard optimal control problem deÖned above. The optimal conditions are
\[E _ {t} ^ {x} = E _ {t} ^ {y} = E _ {t},\tag{4}\]
\[\frac {\dot {E} _ {t}}{E _ {t}} = r _ {t} - \rho ,\tag{5}\]
\[p _ {j t} = \left(\frac {L E _ {t}}{x _ {j t}}\right) ^ {1 - \alpha} P _ {t},\tag{6}\]
where are individual expenditures in goods X, Y, respectively, i.e.,
\[E _ {t} ^ {x} = \int_ {0} ^ {N} p _ {j t} x _ {j t} d j\]
and : In the following, we use the notation to refer to both. The price index of the composite good X is given by
\[P _ {t} = \left(\int_ {0} ^ {N} p _ {j t} ^ {\frac {\alpha}{\alpha - 1}} d j\right) ^ {\alpha - 1}.\]
Firm i producing good j solves the problem:
\[V _ {i s} = \max \int_ {s} ^ {\infty} R _ {s, t} \left((p _ {j t} - z _ {i t} ^ {- 1}) q _ {i t} - L _ {i t} ^ {z}\right) d t, \qquad \mathrm{s.t.}\tag{7}\]
\[\begin{array}{r c l} {p _ {j t}} & = & {\left(\frac {L E _ {t}}{x _ {j t}}\right) ^ {(1 - \alpha)} P _ {t}} \\ {x _ {j t}} & = & {\sum_ {i = 1} ^ {n} q} \\ {\dot {z} _ {i t}} & = & {(L _ {i t} ^ {z}) ^ {\gamma} z _ {i t}, 0 < \gamma < 1} \\ {z _ {i 0}} & > & {0,} \end{array}\]
where is the usual market discount factor. Deriving Örst order conditions, rearranging terms and applying symmetry, we get:
\[q _ {t} = \theta z _ {t} l E _ {t},\tag{8}\]
\[1 = \gamma v _ {t} (L _ {t} ^ {z}) ^ {\gamma - 1} z _ {t},\tag{9}\]
\[\frac {z _ {t} ^ {- 2} q _ {t}}{v _ {t}} + (L _ {t} ^ {z}) ^ {\gamma} = \frac {- \dot {v}}{v} + r _ {t},\tag{10}\]
where is the costate associated with variable and is the inverse of the markup rate. Note that equilibrium is symmetric under the assumption that the initial stock of knowledge is equal for all Örms in all sectors, i.e. As it can be seen in the last term of equation (8), the relevant scale is the number of workers per Örm,
The left hand side of condition (10) is the marginal gain of accumulating one more unit of knowledge, and it can be decomposed in two parts: the Örst consisting on the reduction in marginal production costs, which are proportional to the quantity supplied, and the second representing learning by doing in research. Notice that the beneÖt of a cost-reduction innovation depends on the quantity produced, since it determines the amount of saved resources following such a reduction.
Given that the quantity produced determines the innovation e§ort, the way in which quantities are decided is fundamental for growth. This is in equation (8). In particular, we are interested in understanding the e§ect of a change in the number of Örms on the incentives to innovate. In our model, an increase in the number of Örms generates two di§erent, opposite forces. On the one hand, the market share of each Örm reduces, which can be seen in the last term of condition (8), since . This is the size e§ect or the market share e§ect. the other hand, the markup depends negatively on the perceived elasticity of demand Consequently, an increase in the number of Örms has a positive e§ect on quantities by increasing the inverse of the markup, represented by the Örst term on the right hand side of (8). This is the competition e§ect.
The labor market clearing condition is
\[n N (L _ {t} ^ {x} + L _ {t} ^ {z}) + L _ {t} ^ {y} = L.\tag{11}\]
The Önancial market-clearing condition implies that the aggregate asset demand is equal to the stock market value of Örms:
\[L A _ {t} = n N V _ {t}.\tag{12}\]
Finally, let us impose the market-clearing condition in sector Y:
\[L E _ {t} ^ {y} = L _ {t} ^ {y}.\tag{13}\]
2.2 Balanced growth path
A Balanced Growth Path (BGP) is an equilibrium path in which variables ; are constant and grow at a constant rate. The following proposition shows that it exists and is unique.7
Proposition 2 An interior BGP exists and is unique
Proof. Combining (3), (8), (9) and (10), under , we get
\[\theta \gamma (L ^ {z}) ^ {\gamma - 1} l E = \rho .\]
7In Appendix B, we also show that the economy jumps to its BGP at the initial time.
Substituting the latter equation, (2), (4), (8), and (13) into the labor marketclearing condition (11), we
\[f (L ^ {z}) \equiv \left(\frac {1 + \theta}{\theta}\right) \frac {\rho}{\gamma} (L ^ {z}) ^ {1 - \gamma} + L ^ {z} = l.\tag{14}\]
Since is monotonically increasing, and satisÖes the limit conditions lim 0 and , existence and uniqueness derive directly from the intermediate value theorem.
2.3 Output growth
In this economy, production in sectors Y and X do not grow at the same rate. Consistent with national accounts, let us deÖne growth by the mean of a Divisia index, meaning that the growth rate of real output is equal to the growth rate of both Önal sectors weighted by the share of each sector on nominal output. Since the homogeneous sector is not growing, and preferences are logarithmic, the growth rate of output is
\[g = \frac {1}{2} \frac {\dot {q}}{q} = \frac {1}{2} \frac {\dot {z}}{z} = \frac {1}{2} (L ^ {z}) ^ {\gamma}.\]
Technical progress only a§ects sector X, making the growth rate depend on the amount of labor allocated to research in this sector.
is the inverse of the markup and may be seen as a measure of the degree of competition. By di§erentiating (14), the growth rate can be easily shown to be increasing in . This is what we have referred before as the competition e§ect.
There is a positive relation between the degree of competition and the perceived elasticity of demand, which depends positively on both the number of Örms n and the elasticity of substitution . As we have commented before, an increase on leads Örms to increase the quantity produced. Given that innovation can be exploited in a large number of units, Örms increases innovation too. This result is the opposite to that found in monopolistic competitive models, where a rise in the elasticity of substitution decreases the markup and reduces the innovation rate. When incumbents carry out process innovation, the scale of operation becomes an important determinant of R&D decisions. The rise in the perceived elasticity of demand increases the quantity supplied and therefore the return to innovation.
3 Free trade
Let us assume that countries are identical. Since both economies are equal in factor endowments and initial stocks of knowledge no pattern of specialization from trade is observed and all the gains from trade comes from an increase in competition.
Let us assume that transportation costs are of the iceberg type; precisely, (1+) units of the product must be shipped in order to serve 1 unit abroad, where is the percentage of total production that disappears in the process of shipping. Notice that for foreign Örms selling in the domestic market, the markup in autarky has to be larger than the transportation costs, meaning that there is trade i§ Let us assume it in the next.
Under international trade, Örms are able to serve both markets so some clariÖcation about the notation must be made. Let us deÖne the quantity as the quantity supplied by a Örm located in country h to market where c; That is is the quantity supplied by the B-Örm to the A-market. Whenever only one superscript appears it indicates that the variable is deÖned for that economy, that is, would be the expenditure assigned to X by households located in country A:
Under symmetry, Örst order conditions for country A under free trade are:
\[\left(\frac {l E _ {t} ^ {x A}}{q _ {A t} ^ {A} + q _ {B t} ^ {A}}\right) ^ {1 - \alpha} P _ {t} ^ {A} \left(\frac {(n - (1 - \alpha)) q _ {A t} ^ {A} + n q _ {B t} ^ {A}}{n (q _ {A t} ^ {A} + q _ {B t} ^ {A})}\right) = (z _ {t} ^ {A}) ^ {- 1},\tag{15}\]
\[\left(\frac {l E _ {t} ^ {x B}}{q _ {B t} ^ {B} + q _ {A t} ^ {B}}\right) ^ {1 - \alpha} P _ {t} ^ {B} \left(\frac {(n - (1 - \alpha)) q _ {A t} ^ {B} + n q _ {B t} ^ {B}}{n (q _ {A t} ^ {B} + q _ {B t} ^ {B})}\right) = (z _ {t} ^ {A}) ^ {- 1} (1 + \tau),\tag{16}\]
\[1 = \gamma v _ {t} ^ {A} (L _ {t} ^ {z A}) ^ {\gamma - 1} z _ {t} ^ {A},\tag{17}\]
\[\frac {\left(z _ {t} ^ {A}\right) ^ {- 2} (q _ {A t} ^ {A} + (1 + \tau) q _ {A t} ^ {B})}{v _ {t} ^ {A}} + \left(L _ {t} ^ {z A}\right) ^ {\gamma} = \frac {- \dot {v} _ {t} ^ {A}}{v _ {t} ^ {A}} + r _ {t}.\tag{18}\]
Conditions (17), (18) are identical to conditions (9), (10) except from the fact that in (18), when computing the return on innovation, Örms take into account quantities supplied to both markets. Conditions (15), (16) determine the optimal quantities supplied in each market and are analogous to condition (8), but one for each market. Notice that Örms do not supply the same quantities to both market. B-Örms solve an identical problem and their Örst order conditions are equal to those of country A but changing the subscripts and the superscripts, from B to A
and viceversa.
In order to complete the deÖnition of an equilibrium allocation, market clearing conditions need to be added:
\[n N (L _ {t} ^ {x h} + L _ {t} ^ {z h}) + L _ {t} ^ {y h} = L, h = \{A, B \},\tag{19}\]
\[L A _ {t} ^ {h} = n N V _ {t} ^ {h}, h = \{A, B \},\tag{20}\]
\[L (E _ {t} ^ {y A} + E _ {t} ^ {y B}) = L _ {t} ^ {y A} + L _ {t} ^ {y B}.\tag{21}\]
3.1 Balanced growth path
A balanced growth path for this economy is a symmetric equilibrium in which variables are constant and variables grow at a common constant rate (we have omitted some supraindexes).
Proposition 3 Under , a balanced growth path exists and is unique
Proof. See Appendix A.
As shown in the appendix, equilibrium conditions can be reduced to the following equation in
\[f ^ {*} (L ^ {z}) = \left(\frac {1 + \theta^ {*}}{\theta^ {*}}\right) \frac {\rho}{\gamma} (L ^ {z}) ^ {1 - \gamma} + L ^ {z} = l,\tag{22}\]
which is in fact the same equation than in autarky but with given by:
\[\theta^ {*} = \frac {(2 n - 1 + \alpha) (2 (1 - \alpha) (1 + \tau) + \tau^ {2} (1 - \alpha - n))}{n (2 + \tau) ^ {2} (1 - \alpha)}.\tag{23}\]
The question is whether the growth rate of technological progress is higher under free trade than under autarky or in another terms, whether
Proposition 4 Under :(The growth rate under free trade is always higher than in autarky)
Proof. See Appendix A.
Trade openness has no e§ect on the r.h.s. of equation (22) because neither local resources, nor the local number of producers changes. In other words, the increase in the number of competitors in each country has no size e§ect, since Örms are at the same time selling in both countries. However, the increase in the number of competitor has an e§ect through competition. In the extreme case , the markup takes the same functional form as in autarky, but with 2n instead of n as the number of competitors. A reduction in markups puts the competition e§ect at work as already explained in the previous section. Even if Örms are selling less in their domestic market than under autarky, the global quantity they supply is larger, because of the competition e§ect. Therefore, openness to international trade leads to more innovation and growth. Proposition 4 shows that the competition e§ect also works under trade frictions. It is important to notice that this result is not driven by any scale e§ect, since the number of workers per Örm l is equal in both cases, under autarky and trade openness. Now, we proceed to discuss some comparative statics.
Transportation costs are a barrier for foreign competitors reinforcing the market power of domestic Örms and making the competition e§ect less e§ective, as shown in the proposition below.
Proposition 5 An increase in transportation costs has a negative impact on the rate of innovation
Proof. It can be easily shown by di§erentiating (23) with respect to .
Finally, the di§erence between R&D investment in both regimes, autarky and free trade, is small when the number of Örms is large. This is due to the fact that n has a non-linear impact on produced quantities; while for a small number of Örms the reduction in quantities due to free trade is important, for a large number of Örms it has a very small impact. For example, the gains from trade completely vanish when competition in autarky is at the largest value compatible with positive trade. Remind that there is trade if and only if , or equivalently i§ . It is easy to see that if The fundamental reason is that Örmsíresponse to the increase in competition due to trade openness is strong when the autarky level of competition is low, while in a competitive autarky economy, the response of Örms to an increase in competition is quite small since they already have very little market power.
4 Conclusions
This paper develops an endogenous growth model with Örm speciÖc innovations, Cournot competition on a continuum of oligopolistic markets and free trade between identical economies. It shows that international trade induces growth in participant countries through an increase in competition. The paper di§ers from the literature by constructing an environment where openness to trade generates a pure increase in competition and makes Örms to innovate more to proÖt from the associated increase in market size. This research reinforces the view that at least for the case of developed countries trade openness enhances innovation and growth through an increase in competition.
By restricting the analysis to identical economies, the present paper may be seen as a contribution to the understanding of the growth e§ect of regional integration agreement among similar countries, as it is the case of France and Germany in the European Union and to some extend Brazil and Argentina in the MER-COSUR. A natural extension will be the study of economies with di§erent initial conditions (i.e. di§erent technological levels) or di§erent factor endowments. This would make possible the study of the interaction between developed and developing economies, as it is the case of Mexico and the US in NAFTA or the accession of Ireland and Spain to the EU. Di§erences in the initial stock of knowledge determine the initial di§erences in marginal costs and market shares; di§erences in market size depend upon di§erences in factor endowments. The innovation path of both economies will be determined by how these two forces interact.
As a complement to that extension we can explore how di§erent trade policies a§ect the results. The model will predict that unilateral trade policies will reduce growth in the liberalizing country since the increase in competition coming from this policy is o§set by the creation of an artiÖcial comparative advantage to foreign Örms. However, preferential trade liberalization agreements, will enhance growth in the liberalizing countries reducing growth in the protectionist country due to the fact that, the reduction of trade barriers between the two countries increases competitiveness of these Örms in both economies with respect to Örms in a third country.8
Another interesting extension would allow for sectorial di§erences. In this case, it would possible to identify sectors having larger gains from trade. Considering, for simplicity, intersectorial independence, we suspect that the less competitive sectors will have larger gains from trade.
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A Free Trade
Proposition 3 Under , a balanced growth path exists and is unique
Proof. Under symmetry, and , for all
t. Taking condition (15) for both countries, we get
\[\frac {(n - 1 + \alpha) q _ {t} + n \breve {q} _ {t}}{n q _ {t} + (n - 1 + \alpha) \breve {q} _ {t}} = \left(\frac {1}{1 + \tau}\right),\]
implying
\[\breve {q} _ {t} = \frac {(1 - \alpha) (1 + \tau) - n \tau}{1 - \alpha + n \tau} q _ {t},\tag{24}\]
which requires
\[\tau < \frac {1 - \alpha}{n - 1 + \alpha}\]
to and be simultaneously positive.
Under symmetry, and for all and t, implying that the inverse demand function (6) for any variety produced in any country becomes
\[p _ {t} = \left(\frac {L E}{n (q _ {t} + \breve {q} _ {t})}\right) ^ {1 - \alpha} P _ {t} = \frac {l E}{q _ {t} + \breve {q} _ {t}},\]
the last equality follows from the deÖnition of the price index P: Substituting the latter condition and (24) in (15) and rearranging terms, it follows
\[q _ {t} = \left(\frac {(1 - \alpha + n \tau) (2 n - 1 + \alpha)}{n (2 + \tau) ^ {2} (1 - \alpha)}\right) z _ {t} l E\tag{25}\]
\[\breve {q} _ {t} = \left(\frac {((1 - \alpha) (1 + \tau) - n \tau) (2 n - 1 + \alpha)}{n (2 + \tau) ^ {2} (1 - \alpha)}\right) z _ {t} l E.\tag{26}\]
At the balanced growth path, from (5), and from (3). From (17),
(18), (25) and (26), we obtain:
\[\gamma (L ^ {z}) ^ {\gamma - 1} \theta^ {*} l E = \rho\]
where, by analogy with the autarky case,
\[\theta^ {*} = \frac {(2 n - 1 + \alpha) (2 (1 - \alpha) (1 + \tau) + \tau^ {2} (1 - \alpha - n))}{n (2 + \tau) ^ {2} (1 - \alpha)}.\]
From the labor market clearing condition (19),
\[L ^ {x} + L ^ {z} + \frac {L ^ {y}}{n N} = l.\]
From (21) and (4), ; from (2), . Substitution and by their expressions in (25) and (26), we get
\[f ^ {*} (L ^ {z}) \equiv (\frac {1 + \theta^ {*}}{\theta^ {*}}) \frac {\rho}{\gamma} (L ^ {z}) ^ {1 - \gamma} + L ^ {z} = l,\]
i.e., is the same equation as in the autarkic model but with instead of . Interiority and uniqueness of the solution is therefore ensured by looking at the autarkic balanced growth path proof.
Proposition 4 Under
Proof. From the deÖnition of and ,
\[\begin{array}{r c l} \theta^ {*} - \theta & = & \frac {(2 n - 1 + \alpha) (2 (1 - \alpha) (1 + \tau) + \tau^ {2} (1 - \alpha - n))}{n (2 + \tau) ^ {2} (1 - \alpha)} \\ & & - \frac {n - 1 + \alpha}{n}, \\ & = & \frac {(1 - \alpha) ^ {2} (1 + \tau) + \tau^ {2} n (1 - \alpha - n)}{n (2 + \tau) ^ {2} (1 - \alpha)}. \end{array}\]
It can be easily shown that the r.h.s. is decreasing in , with when is at its maximum value
B Stability analysis under autarky
Let us combine equations (2), (8) and (11) to get
\[L ^ {x} = \theta l E\]
\[\frac {1 + \theta}{\theta} L ^ {x} + L ^ {z} = l,\]
implying
\[E = \frac {l - L ^ {z}}{(1 + \theta) l}.\]
By logdi§erentiation,
\[\frac {\dot {E}}{E} = - \frac {L ^ {z}}{l - L ^ {z}} \frac {\dot {L} ^ {z}}{L ^ {z}}.\]
From (5),
\[r = \rho - \frac {L ^ {z}}{l - L ^ {z}} \frac {\dot {L} ^ {z}}{L ^ {z}}.\tag{27}\]
Adding (3) and (10), we get
\[\frac {\dot {z}}{z} + \frac {\dot {v}}{v} = r - \frac {q}{z} \frac {1}{z v} = r - \gamma \theta l (L ^ {z}) ^ {\gamma - 1} E = r - \frac {\theta \gamma}{1 + \theta} (L ^ {z}) ^ {\gamma - 1} (l - L ^ {z}).\tag{28}\]
The second equality emerges after substituting and by their expressions in (8) and (9), respectively, and the last one after substituting the expression for computed just above. Di§erentiating (9)
\[\frac {\dot {z}}{z} + \frac {\dot {v}}{v} = (1 - \gamma) \frac {\dot {L} ^ {z}}{L ^ {z}}.\]
Substituting it and (27) in (28), we get
\[\frac {\dot {L} ^ {z}}{L ^ {z}} = \left(\rho - \frac {\theta \gamma}{1 + \theta} (L ^ {z}) ^ {\gamma - 1} (l - L ^ {z})\right) \left(\frac {l - L ^ {z}}{(1 - \gamma) l + \gamma L ^ {z}}\right).\]
The sign of the second term in the r.h.s. is positive since . The unique interior steady state, let us denote it by , is obtained by equalizing the Örst element of the r.h.s. to zero (condition (14)). As it can be easily seen, for and for , implying that the interior steady state is unstable. Consequently, the only rational expectation equilibrium is for all