Catch-up-in-turn-in-a-multi-country-international-trade-model-with-learning-by-doing-and-invention_2003_Journal-of-Development-Economics.pdf

catch up in turn in a multi country international trade model with learning by doing and invention tomoyuki nakajima department of economics brown university box b providence ri 02912 usa received 1 july 2000 accepted 1 september 2002 abstract we consider how the world income distribution evolves over time when both learning by doing and invention are the engines of economic growth and when countries interact with each other through international trade the main findings are i catch up is possible ii in the three country model transitional dynamics are much richer than in the two country model in particular catch up may occur in turn there may be a first stage in which the middle income country grows fastest closing the gap with the richest country but increasing the income difference with the poorest followed by a second stage where the poorest country catches up d 2003 elsevier science b v all rights reserved jel classification f10 o11 o31 keywords catch up international trade learning by doing invention 1 introduction how the cross country income distribution evolves over time has been an important question in macroeconomics in the last few decades there have been many theoretical contributions relating to this question on one hand there are models that imply at least partial convergence among countries for example in a neoclassical growth model decreasing marginal product of capital enables a poorer country to grow faster in an endogenous growth model if imitation is less costly than invention again a poorer country could grow faster krugman 1979 barro and sala i martin 1995 grossmand and helpman 1991 on the other hand there are also theories that generate divergence 0304 3878 03 see front matter d 2003 elsevier science b v all rights reserved doi 10 1016 s0304 3878 03 00070 1 tel 1 401 863 3807 fax 1 401 863 1970 e mail address tomoyuki nakajima brown edu t nakajima journal of development economics 72 2003 117 138 or non convergence in a neoclassical growth model if the production function is not concave there might be multiple steady states to what steady state a country converges depends on the initial capital stock dechert and nishimura 1983 in an endogenous growth model if knowledge is country specific convergence may not occur grossmand and helpman 1991 if learning by doing is the source of productivity growth interna tional specialization through international trade may lead to divergence young 1991 also ventura 1997 shows that with international trade ramsey model may not lead to convergence he shows that it depends on the elasticity of substitution between capital and labor however the actual evolution of the income distribution across countries does not appear to be as simple as monotonic convergence or divergence for example table 1 shows the growth rates in european countries for the past 100 years 1875 1980 countries are divided into three groups according to their income levels in 1875 the income gaps between the three groups in 1980 are much smaller than those in 1875 the per capita income in the middle income group was 60 of that in the high income country uk in 1875 and it became 110 in 1980 similarly the per capita income in the low income group was 53 of the uk s in 1875 and 90 in 1980 so over this period the income gaps between the three groups fell substantially however the transition was not monotonic it is true that the gap between the richest country and the middle income groups has almost monotonically declined over this period but the gap between the richest country and the low income group did not start to decrease until 1950 and the gap between the middle income group and the low income group increased significantly until 1960 1 the purpose of this paper is to develop a model in which catch up is possible in the long run but which generates non monotonic transition paths this is done by considering both a factor that leads to convergence and a factor that tends to prevent it specifically the key feature of the model is that both learning by doing and invention are the engines of economic growth2and that countries interact with each other through international trade here the factor that leads to convergence is that poorer countries are less constrained by 1 what happened among the european countries is more than catch up because some of them have become richer than the uk in other words there has been leap frogging which will not be discussed in this paper 2 for empirical evidence for the importance of learning by doing see bahk and gort 1993 lucas 1993 also argues its relevance in explaining the fast growth in the east asian countries table 1 average growth rates of european economies 1875 1980 groups1875 19001900 19251925 19501950 19601960 19701970 1980 uk1 170 850 852 372 261 76 middle income group 1 361 380 824 604 062 55 low income group 1 070 890 803 455 302 76 source lucas 1998 middle income group france germany low countries and scandinavian countries low income group rest of western europe t nakajima journal of development economics 72 2003 117 138118 the pace of invention than richer countries and the factor that leads to divergence is international division of labor and learning by doing with this model we show that although catch up is possible in the long run the transitional dynamics may not be monotonic that is even when the income gaps between countries get smaller eventually they may increase at first for example in the three country model it can be either the middle income country or the low income country that grows fastest in the transition to the balanced growth path it is even possible that the low income country grows slowest at first although it eventually grows faster hence the income gap between the low income country and the middle income country and the income gap betweenthelow incomecountryandthehigh incomecountrymayexpandinitially inthose cases catch up occurs in turn in the sense that there is a first stage in which the middle income country grows fastest closing the gap with the richest country but increasing the income difference with the poorest followed by a second stage where the poorest country catches up the rest of the paper is organized as follows in section 2 we present our baseline three country model we analyze the autarky case in section 3 the two country case in section 4 and the three country case in section 5 final remarks are in section 6 2 the model our model builds on the previous models developed by young 1991 1993 3time is continuous the world economy consists of three countries with different productivity levels 4labor is the sole factor of production and it is completely immobile across countries countries have identical population size l in each country individuals are endowed with one unit of labor at each point in time and supply it inelastically hence the amount of labor supplied at each point in time is l in every country there is a continuum of goods that have ever been or ever will be invented they are arranged along the real line r the level of productivity to produce a good varies both across countries and over time there are two sources of economic growth one is expansion of the set of producible goods through invention and the other is improvement in the level of productivity to produce each good through learning by doing for simplicity we consider a very simple form of invention let k t be the index of the most sophisticated good that has been invented up through time t we assume that once a good is invented anyone in any country has an access to the information on how to produce it hence all countries have the same set of producible goods l k t this could be interpreted as that imitation is costless we also assume that invention occurs exogenously k t c 1 where c is an exogenously given parameter as we will see later because the world knowledge frontier k t tends to restrict the product mix in the richest country its growth is directly constrained by the pace of invention on the other hand the product mixes in 4 it is straightforward to extend the model to a general n countrycase 3 related models include stokey 1988 1991 and boldrin and scheinkman 1988 among others t nakajima journal of development economics 72 2003 117 138119 poorer countries does not face such a restriction this is what makes it possible for a poorer country to grow faster than the richest country along transition in our model while the set of producible goods is the same across countries countries differ in their productivity levels since labor is the sole factor of production the production technology is described by how much labor is required to produce each good let ai s t be the amount of labor necessary to produce one unit of good s at time t in country i hence an improvement of the productivity to produce good s in country i corresponds to a decline in ai s t over time to simplify the argument we assume that learning by doing is the only source of productivity improvements and that the labor requirement function ai s t takes the following form ai s t exp s for s a l ti t exp s 2ti t for s a ti t k t 8 2 notice that technology of country i at time t is described by two parameters k t and ti t k t determines the set of producible goods ti t determines the level of productivity for each producible good an increase in tiimproves productivity in country i by shifting the labor requirement function ai s downward for example consider good s as tiincreases the productivity to produce good s in country i goes up as long as titi t in country i will enhance the productivity of producing other goods through an increase in ti but it does not spillover across countries since all countries share the same set of producible goods l k t countries differ only in terms of the productivity levels ti t i 1 2 3 assume that country 1 has the most advanced technology initially and country 3 the least t1 0 t2 0 t3 0 t nakajima journal of development economics 72 2003 117 138120 we also assume that the speed of invention c satisfies c w 2 4 because otherwise the long run growth rate of the world economy would become independent of the speed of invention we assume that goods are not storable and that there are no international financial markets so we do not need to specify intertemporal preferences consumers have identical preferences let ci s t denote the consumption of good s at time t of each individual in country i we assume that his her utility flow is given by u ci t z k t l nci t nh ci s t nci t n ds 5 where nci t n z k t l ci s t ds 6 and h x x x2 2 7 this form of the utility function has two important characteristics first since hv 0 0 12 k z k l p s f s ds z k l h f s ds 13 as well as eqs 10 and 11 note that the consumer consumes only a subset of available goods because the marginal utility is bounded hv f s 1 f s v1 thus there exists a threshold price level p such that a good is consumed in a strictly positive amount if and only if its price is less than p the first order condition eq 12 implies that the consumption density f s is given by f s maxfk p p s 0g svk the labor requirement function eq 2 implies that the set of goods which are consumed in equilibrium takes the form of s f s 0 z a that is z is the worst commodity consumed and a is the best commodity note that they satisfy z t avk and a z p and a a minf p a k g 14 it follows that the consumption density f s is written as f s maxfk a z a s 0g svk 15 thus the amount of consumption of good s c s ncnf s is proportional to the relative productivity of producing that good a z a s given k and t the equilibrium levels of z a f s and k are determined by solving eqs 10 13 and 15 given these variables ncn is given by the budget constraint eq 11 l is by the first order condition eq 12 it is straightforward to show that there exists a threshold value h such that the world knowledge constraint binds i e a k if k tvh and it does not i e ah the intuition is clear if k t is very large the country is far behind the world knowledge frontier and hence the set of goods produced there is not constrained by it ah the world knowledge constraint does not bind at is exactly one half the learning function eq 3 then implies that if k t h t t w 2 zc k t therefore if k tzh then the country grows at the rate g t w 2 t t and k t decreases over time if k tt2 0 we will see to what extent catch up is possible in our economy and how it is made possible by our assumption that the t nakajima journal of development economics 72 2003 117 138124 technological leader is constrained by the progress of world knowledge k t labor in country 2 the poor country is taken as the numeraire w2 t 1 bt 4 1 instantaneous equilibrium with two countries the state of the world economy at a given point in time is given by the three variables the productivity levels of the two countries t1and t2 and the level of world knowledge k these three variables determine the instantaneous equilibrium as in the previous section we omit time index t when we discuss the instantaneous equilibrium assume that t1 t2 as shown in the next two subsections t1 0 t2 0 implies that t1 t t2 t for all t perfectly competitive goods markets imply that the price of good s equals the lower marginal cost p s minfw1a1 s a2 s g 19 the labor requirement function eq 2 implies that a1 s a2 s for svt2and a1 s t2 therefore it must be the case that country 1 is richer w1z1 and that country 1 produces goods that have been invented more recently the instantaneous utility maximization problem of the representative consumer in country i is given by max ncin ffi s g z k l ncinh fi s ds subject to 1 z k l fi s ds 20 and wi z k l p s fi s ncinds 21 notice that consumers in different countries are different only in terms of their labor income wi as in the autarky case the first order conditions for the utility maximization problem are given by hv fi s vli kip s if fi s 0 22 ki z k l p s fi s ds z k l h fi s ds 23 as well as eqs 20 and 21 it is straightforward to show that the consumption density fi and the multipliers ki and li are all independent of the wage rate wi and hence they are identical across t nakajima journal of development economics 72 2003 117 138125 countries in what follows we omit the country subscript i from these variables note that since f1 s f2 s all s consumers in different countries consume the same set of goods with the same proportions they differ only in the scale ncin c1 s c2 s nc1n nc2n w1 for all goods s which are consumed in positive amounts as in the autarky case it follows from first order condition eq 22 that consumption density f s satisfies f s maxfk p p s 0g 24 then eqs 2 and 19 imply that the set of goods which are consumed in equilibrium takes the following form fs p s v pg z j h a where j and h may coincide as in the autarky case z denotes the worst good consumed and a the best good consumed unlike the autarky case the set of consumed goods might be disjoint j1 or w1 1 that is whether or not country 1 has a strictly higher income ii whether h j or h j that is whether or not the set of consumed goods is disjoint and iii whether a1 h j and a k people consume a disjoint set of goods z j v h k the rich country produces goods sa h k and the poor country produces sa z j equilibrium b is the case in which w1 1 h j and a k people consume the connected set z k the rich country produces j k and the poor country z j equilibrium c is the case in which w1 1 h j and a k people consume z a the rich country produces j k and the poor country z j equilibrium avis the case in which w1 1 h j and a1 h j and a k people consume z a the rich country produces j a and the poor country z j as is shown below when a k there is no equilibrium in which w1 1 5 a formal analysis of the instantaneous equilibrium is given in appendix a t nakajima journal of development economics 72 2003 117 138126 what form of equilibrium emerges depends on the differences between the level of world knowledge k productivity of the rich country t1 and productivity of the poor country t2 let huk t1 vut1 t2 notethatalowerhimpliesthattherichcountryismore constrainedbytheworldknowledge and a higher v implies that the poor country is farther behind the rich country thus with a lower h itismore likelythatthe world knowledge constraint binds a k in thiscase ifthe productivity difference between the two countries v is sufficiently small then there will be noincomedifferencesothatw1 1andtheequilibriumwillbe oftypec when vgets larger than a certain threshold value then w1becomes greater than 1 and equilibrium of type b emerges whenitgetsverylarge equilibriumturnsintotypea thecaseinwhichtheworld knowledge constraint is not binding a k is similar except that w1is always greater than 1 in this case this follows from our assumption of the same population size l1 l2 l suppose by way of contradiction that w1 1 as well as al a contradiction 4 2 catch up is not possible without the world knowledge constraint now let us turn to dynamics we first consider the case in which there is no world knowledge constraint that is k 0 l we shall see that in this case the rich country always grows at least as fast as the poor country so that catch up is not possible 6 let k 0 l so that the set of goods consumed is never constrained a t k t for all t thus at each point in time the instantaneous equilibrium is either of type avor of type bv there is a critical value v such that when v t zv equilibrium of type avemerges and when v t v equilibrium of type bvdoes as in the autarky case we use eq 16 to define the growth rate of real gdp for each country it is straightforward to show that the growth rate of country i satisfies gi t 2 w ti t 2 25 it follows that a balanced growth path is a dynamic equilibrium in which both countries technology improves by the same amount over time t1 t t2 t 26 for all t or equivalently v t 0 along such a path the relative wage rate w1 t is constant as before we use