Translators: Market makers in merging markets

*Corresponding author. Tel.: #864-656-1242; fax: #864-656-4192.

E-mail address: [email protected] (R. Tamura).�I thank the participants at the Winter Meetings ASSA, New Orleans, the Midwest Macroeco-

nomics Meetings at Notre Dame University and Clemson University. I thank Dan Benjamin, DavidGordon, Donald Gordon, Robert McCormick, Mike Maloney, Raymond Sauer, Curtis Simon,Kevin Murphy and an anonymous referee for useful comments. I particularly thank Berc Rustem foruseful advice.

Journal of Economic Dynamics & Control25 (2001) 1775}1800

Translators: Market makers in mergingmarkets�

Robert Tamura*

Department of Economics, Clemson University, Clemson, SC 29634, USA

Received 9 January 1998; accepted 17 November 1999

Abstract

In a model with agglomeration returns to participation, there exists gains frommerging two regions that conduct business in two incompatible languages. Bilingualindividuals, translators, integrate these two regions. The speed of integration, creation ofthe initial translators, depends positively on: the maximum human capital in the tworegions, the magnitude of the agglomeration returns, and the population of the largerregion. Because the combined regions economize on the number of bilingual individuals,an economy with multiple languages is less productive than an economy with a singlelanguage, however, both economies grow at the same rate in the long run. � 2001Elsevier Science B.V. All rights reserved.

JEL classixcation: O1; O3; F1

Keywords: Translators; Regional integration; Growth; Convergence

0165-1889/01/$ - see front matter � 2001 Elsevier Science B.V. All rights reserved.PII: S 0 1 6 5 - 1 8 8 9 ( 9 9 ) 0 0 0 7 9 - 2

�This is similar to Tamura (1995), but with explicit language acquisition technology.

� In Tamura (1997a) the model solves for the e$cient accumulation of bilingual status. In thatmodel agents live forever, and once bilingual the agent always remains bilingual. Furthermore thereis no accumulation of human capital in the model.

� If human capital accumulation is no cheaper for bilingual parents to produce, why woulda society continue to be bilingual? What is not modeled here are the consumption #ows fromcultural capital. The entire stock of literature, philosophy, history, etc., not to mention the current#ow of native culture are reasons to maintain a language not used in world business trade. Onehypothesis, however, is that this #ow of &cultural consumption'must be of the same magnitude as theconsumption of market goods. If not, eventually it will become optimal to stop learning the culturallanguage and concentrate on the business language. On the other hand, introducing cultural capitalcan produce the opposite result. Some individuals learn English not for business reasons, but rather,to gain greater access to American culture, rock music, popular print, television and movies.

�See Grin (1996) for a recent survey of the economic analysis of language.

1. Introduction

Political debate on the role of language in American society has intensi"ed inrecent years. This debate is particularly heated in states that have experiencedlarge increases in immigrant population. This paper focuses on one aspect oflanguage, translators. Suppose there are two regions that produce separatelyfrom each other because of language incompatibility. Suppose that theconsumption good is produced with an agglomeration economy to marketparticipation arising from specialization. The model captures the value ofintermediaries, translators, between these two regions. Thus some individualswill become #uent in two languages, and serve to foster trade between theseformerly separate markets.� As individuals become more skilled, the paperpredicts when the bilingual individuals will be created.

Unlike Tamura (1997a) this model explicitly solves for the equilibriumtransition dynamics of bilingual status.� One feature that is ignored here is whymultiple languages might coexist forever.� The paper also shows that while allspeakers of a minority language may not become bilingual, as long as some do,there are no growth consequences from maintaining multiple languages in trade.However there will be a static loss, that will represent the idea that something isalways lost in the translation. Thus the market economizes on the number oftranslators.

This paper contributes to the growing literature on the economics of lan-guage.� These papers include Breton and Mieszkowski (1977), Lang (1986),Church and King (1993), Lazear (1995), John and Yi (1996) and Tamura (1997a).Each of these papers focused on some aspect of bilingualism. Breton andMieszkowski (1977) provide an early model examining the incentives of indi-viduals to acquire pro"ciency in a second language. They focus on the languagechoices of monolingual speakers of a minority language. Minority language

1776 R. Tamura / Journal of Economic Dynamics & Control 25 (2001) 1775}1800

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https://www.researchgate.net/publication/5051354_Language_Learning_and_Location?el=1_x_8&enrichId=rgreq-f4ed8397-3f43-480a-ad96-0d7553948895&enrichSource=Y292ZXJQYWdlOzQ4MzExMjU7QVM6OTk3Njc4NzA4MjAzNTVAMTQwMDc5NzkyMjIzNQ==

� In particular his setup is like a genetic model.

�There is strong evidence that immigrants cluster in ethnic enclaves partly to economize on theloss of cultural capital, and partly to reap the gains of previous immigrants' knowledge andexperience. See the evidence in Lazear (1995), Sowell (1981,1994,1996) and Yeh (1996).

� In addition there is a large labor economics literature on the returns to learning secondlanguagues. These include Chiswick and Miller (1992,1994,1995,1996), Borjas (1992,1993), and Grinand Sfreddo (1996).

speakers have greater incentives to become bilingual in the dominant language,the lingua franca. Lang (1986) discusses how imperfect knowledge of an existinglanguage, say Black English versus &King's' English, can produce earningsdi!erences between populations. As in Breton and Mieszkowski (1977), Lang(1986) shows that the minority language group has strong incentives to become#uent in the lingua franca. Church and King (1993) consider a world of twomonolingual populations and a single location. There are three possible Nashoutcomes, no one becomes bilingual, the minority population becomes bilin-gual, the majority or minority population become bilingual. Which of the threeoutcomes arise is a function of the cost of acquiring second language #uency(assumed constant) and the rate at which output rises with the share of thepopulation speaking a language. The second language acquisition cost assumesthat foregone consumption produces #uency as opposed to foregone markettime. Output rises in the share of the population speaking a language becauselanguages are an example of a network technology. Linguistically homogeneoussocieties fully exploit the external bene"ts from this network technology. Lazear(1995) allows for trade amongst two separate language groups, but at lowere$ciency.� He analyzes the incentives for segmentation of the market place, or theendogenous creation of ethnic (linguistic)-enclaves.� Furthermore he studies theproblem of immigration policy. Linguistic minorities have a strong incentive tofavor immigrants with their language, at the expense of the linguistic majority. Johnand Yi (1996) examine the location and language acquisition choice of two languagegroups. In one equilibrium agents in the majority can become bilingual no matterhow large a share of the population they form. Tamura (1997a) provides a micro-foundation for the network externality in language. Tamura (1997a) shows the gainfrom linguistic homogeneity using a cross section of Summers and Heston (1991)countries. Countries that are linguistically homogeneous have higher incomes thantheir linguistically heterogeneous counterparts.�

The next section presents the basic model. Given preferences, productiontechnologies under separated production and integrated production, andhuman capital accumulation technologies, I solve for the dynamics of thetranslator population. In Section 3 the bilingual policy functions are numer-ically solved. Section 4 illustrates income divergence and convergence amongstthe speakers of two di!erent languages. Furthermore I show that upon integra-tion, all bilingual agents earn the same as monolingual agents in the model.

R. Tamura / Journal of Economic Dynamics & Control 25 (2001) 1775}1800 1777

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https://www.researchgate.net/publication/24091333_Ethnic_Capital_And_Intergenerational_Mobility?el=1_x_8&enrichId=rgreq-f4ed8397-3f43-480a-ad96-0d7553948895&enrichSource=Y292ZXJQYWdlOzQ4MzExMjU7QVM6OTk3Njc4NzA4MjAzNTVAMTQwMDc5NzkyMjIzNQ==

� It would be more realistic if the typical adult cared about the utility of his or her child, but thiswould require solving the problem with perfect foresight of the future number of translators. Thiswould make solving the problem equivalent to solving a dynamic "xed point problem. Computa-tionally, a parent that cares about the utility of his or her o!spring makes the problem much moredi$cult, see Tamura (1996). The preferences in this paper eliminates this additional "xed pointrequirement. It would be better if fertility were endogenous, as in Tamura (1999a,b), but this is left forfuture work.

Finally I show that when there exists static costs of multilingual societies, themarket economizes on the number of translators. Section 5 solves for thedeterminants of the initial bilingual generation. In Section 6 the cost of main-taining multilingual societies is shown. If translation is not perfect, then whilethe long run growth rate of an integrated multi-language economy is the same asthe long run growth rate of a single language economy of the same population,there always exists a static cost of multiple languages. The conclusion follows.

2. Model

The model of this paper is a variant of the bilingual version contained inTamura (1997a). This paper di!ers in several important ways. First, the paperfocuses on the equilibrium creation of bilingual agents, not the e$cient creationof bilingual agents. Second, the model highlights the equilibrium incentives tobecome bilingual. Third, individuals are not in"nitely lived; individuals live fortwo periods, young and old. Finally, instead of a constant level of human capital,human capital accumulates. In this section I present the model. I begin witha speci"cation of preferences. I follow with a presentation of production.Production exhibits Smithian gains from specialization. These gains manifestthemselves as an aglommeration economy in market participation.

There are a continuum of identical human capital agents inhabiting tworegions. Each region is identi"ed by their population, M and N. I assume thatM(N. The agents, while having identical levels of human capital, di!er fromeach other in a crucial way. They speak incompatible languages. Fertility isexogenous and set equal to 1.

A parent invests in the human capital of his or her children. This choice includeswhether or not to produce a bilingual or monolingual child. An adult supplieslabor inelastically; invests in his or her child's human capital, and consumes.A generation t parent cares about his or her own consumption, c

�, and the income

of his or her child, y��

.� The typical adult has preferences given by

� ln c�#� ln y

��. (1)

Children take time to educate, �. The budget constraint facing this individual isgiven by

c�"y

�[1!�

�]. (2)


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See Tamura (1992,1999a,b) for more on this agglomeration technology.

� This provides an explanation for ethnic agglomeration. Furthermore it explains why country size isnot necessarily a good "t for market size. In Romer (1990), Kremer (1993a,b), Grossman and Helpman(1991) larger markets provide greater incentives for innovative activity. Here average capital andlinguistic homogeneity limit the size of the e$cient market. Hence, China and India historically are lowhuman capital areas, and therefore, the relevant market sizes in these countries could easily be the village.Finally, I ignore integer constraints on market size and on the number of markets within a region.

��This papers assumes that in equilibrium the only group that produces bilingual agents is thesmaller M population. This is consistent with the results in Breton and Mieszkowski (1977) andLang (1986). However it di!ers from the results in Church and King (1993) and John and Yi (1996).Consider the following example from Finis Welch. In many large cities, particularly southern andsouthwestern cities, there is an incentive for some English speakers to become bilingual in Spanish.A foreman on a construction site may have strong incentives to learn Spanish to oversee Spanishspeaking laborers. However this result is consistent with Breton and Mieszkowski, and Lang if oneconsiders the relevant population set to be the foreman and the laborers. In that case English is theminority language.

Assume there are two separate regions, with adult populations M and N, whereM(N. Initially each region produces in autarky. Output in a region is given by

>��"[1!e��M

�� ]��

��

��

h� ��

�, i"M, N (3)

where �'1 represents an agglomeration economy arising from specialization;��'0, �

�'0; hM

��is the maximum human capital in region i"M, N at time t,

P��

is the size of the market or population of the region. I assume that agentsorganize production in markets that maximizes per capita income. ElsewhereI have shown that in a homogeneous human capital market the market size thatmaximizes per capita income is given by�

P��"max�i, !

�!1

ln[1!e��M�� ]�, i"M, N. (4)

In each region over time, as human capital grows the market size that maximizesper capita income increases. With growth in human capital, eventually M,N and M#N will be smaller than the optimal market size. Language incom-patibility prevents the two regions from trading with one another unless trans-lators exist. If some agents became bilingual, they could act as translators,allowing market integration. Suppose there are ¹

�bilingual individuals in the

M population.�� Integrated production yields

>�"[1!e��M

�� ]��

¹�

M��

��

h� ��

#

��

��

hH� ��

#A� ��

�

��

hK � ��

�, (5)


https://www.researchgate.net/publication/24108748_Endogenous_Technical_Change?el=1_x_8&enrichId=rgreq-f4ed8397-3f43-480a-ad96-0d7553948895&enrichSource=Y292ZXJQYWdlOzQ4MzExMjU7QVM6OTk3Njc4NzA4MjAzNTVAMTQwMDc5NzkyMjIzNQ==

https://www.researchgate.net/publication/24091414_The_O-Ring_Theory_of_Development?el=1_x_8&enrichId=rgreq-f4ed8397-3f43-480a-ad96-0d7553948895&enrichSource=Y292ZXJQYWdlOzQ4MzExMjU7QVM6OTk3Njc4NzA4MjAzNTVAMTQwMDc5NzkyMjIzNQ==

https://www.researchgate.net/publication/5051354_Language_Learning_and_Location?el=1_x_8&enrichId=rgreq-f4ed8397-3f43-480a-ad96-0d7553948895&enrichSource=Y292ZXJQYWdlOzQ4MzExMjU7QVM6OTk3Njc4NzA4MjAzNTVAMTQwMDc5NzkyMjIzNQ==

��Appendix A shows the conditions under which integrated production with translators will stillbe less productive than production under a single language.

��This information is on which intermediate tasks that they produce, and the quantity of that taskthat they produce. This specialization outcome requires that all tasks are done by only one agent.

where 0'�, �'0, A�51, h

�is the jth human capital in the N population, hH

�is

the jth human capital in the monolingual M population, hK�

is the jth humancapital of the bilingual M population, and hM

�"max��h� , hH

�, hK

��.�� Examine

the "rst term in Eq. (5). The exponent N#M!¹�#M(¹

�/M)�� shows that

a translator must serve as an intermediary between the monolingual speakers inM and N. I interpret the term 1!e��M

�� as the probability that an individual

successfully communicates his or her information to one other agent.�� Thus theterm [1!e��M

�� ]�� is the probability that all individuals suc-

cessfully communicate all their information to all other agents. Because a trans-lator intermediates between a monolingual M agent and a monolingualN agent, there are more messages being sent, hence �(0. Therefore the numberof messages sent and received by a translator is larger than the number ofmessages sent and received by a monolingual agent. The term 1!(¹

�/M)�'0

represents the loss of output from using translators. Unless the entire M popula-tion becomes bilingual, there is always &something lost in translation' betweenthe M and N populations. The "nal term in Eq. (5), in curly braces, representsthe level of resources in the economy. The "rst sum in the curly braces representsthe resources brought to the market by the N region monolingual individuals,and the second term represents the resources brought to the market by theM!¹

�monolingual agents. The "nal term represents the resources brought to

the market by the bilingual agents. The parameter A�'0 indicates that transla-

tors are useful in production directly in addition to their role as an intermediary.In fact A

�51 implies that they are more productive than a typical mono-

lingual agent with the same amount of human capital. Integrated production ise$cient if:

>��

"[1!e��M �� ]��

¹�

M��

��

h� ��

#

��

��

hH� ��

#A� ��

�

��

hK � ��

�

'[1!e��M ��]��

��

h� ��

�#[1!e��

M ��]��

��

hH� ��

�

,>��#>�

�, (6)


�� I assume that integrated production completely merges all M agents with the N agent region.I ignore partial integration between the N agents and a subset of the M agents. Partial integrationwithout translators is examined in Tamura (1999b).

��An alternative cost of bilingual education would involve just an additional "xed cost, say C, and��"1. This is left to future research. I thank Raymond Sauer for this suggestion.

��This assumption about the greater productivity of bilingual parents at educating children willprove useful in the analysis of incomes. In particular, this assumption implies that all individuals,monolingual and bilingual, earn the same income in integrated production.

where hM�is the maximum human capital in both regions, and hM

��and hM

��are the

time t maximum human capital in regions M and N, respectively. If there areenough translators, the two regions are integrated when integrated productionexceeds production in autarky.��

There are four cases to account for in human capital accumulation. De"ne theshare of the M population that is bilingual in period t as Z

�"¹

�/M. A mono-

lingual parent can raise a monolingual or a bilingual child. These are given bythe "rst two equations in (7). A bilingual child takes more time to educate thana monolingual child. This is represented by a lower productivity for humancapital investment time, i.e., �(Z

�)41.�� A bilingual parent can raise a mono-

lingual or a bilingual child. These are presented in the last two equations in (7).A bilingual parent is more e$cient than a monolingual parent at educatingeither a bilingual child or a monolingual child.�� Assume that these humancapital investment branches are

h��

"�h��, monolingual to monolingual,

h��

"�h��(Z

�)�

�, monolingual to bilingual,

h��

"�h�

��

�(Z��

), bilingual to monolingual,

h��

"�h�

�(Z�)�

��(Z

��), bilingual to bilingual, (7)

where ��

is time spent educating a child, �50, 0(�(Z�)41. With �50, there

is the possibility for perpetual growth.I assume the following functional forms for the return to bilingual status and

the cost of educating a bilingual child:

A(Z�)"[#Z�

�(1!)]��,

�(Z�)"�[#Z�

�(1!)], (8)

where �, , '0, ((1, �41, �', �51, and Z�"¹

�/M. The productiv-

ity of bilingual agents, A�, is bounded above by 1/. The return to bilingual


��This captures the idea that when there are more bilingual agents in the population, it is easier tohire these teaching resources for education.

��Since there is constant returns to scale in the distribution of human capital, this factor paymentsystem exactly exhausts output. Hence the agglomeration economy in market participation, thatarises from specialization, does not prevent payments based on marginal productivity.

status is a declining function of the number of translators in the M population.The reduction in human capital investment time productivity of educatinga bilingual child, �

�, depends on the number of bilingual adults in the M popula-

tion. The greater the number of bilingual adults, the lower the additionalteaching time to produce a bilingual child.�� The reduction in human capitalproduction arising from educating a bilingual child is bounded below by �.

Agents are paid the marginal product of their human capital. Each agent'searnings are the product of his or her wage per unit of human capital and theamount of human capital he or she has. The per unit human capital wage candepend on their type, i.e., if they are a monolingual M agent or a monolingualN agent, or if they are bilingual. These are represented by the "rst two lines of(9). Since there are a continuum of agents of any type, no individual agent hascontrol over his or her wage per unit of human capital.�� Earnings depend onwhether or not the two markets are integrated or not; this is the second line of(9). The third line of (9) is the segregated monolingual agent k's wage, where inequilibrium hMM

�"h

�. The fourth line of (9) shows the integrated monolingual

agent k's wage, where in equilibrium

hMM�

"�h�

for k3N,

hH�

for k3M.

The "fth line of (9) is the integrated bilingual agent k's wage, where in equilib-

rium hKMM�

"hK�

.

y�

"w�h�

,

w�

"max�w��

, w��

, w��

�,

w��

"[1!e��M ��]��

��

h� ��

��hMM � ��

, i"M, N,

w��

"��hMM � ��

,

w��

"��A(Z

�)� �hKMM � ��

��,

��"[1!e��M

�� ]��

¹�

M��

��

h� ��

#

��

��

hH� ��

#A(Z�)� �

�

��

hK � ��

��. (9)


�This assumes �=��

/�h��"0, ∀i, j, which follows from the assumption that each agent is a set of

measure 0. Therefore it takes a measurable number of agents to noticeably a!ect the rate of returnper unit of human capital for monolingual or bilingual individuals.

� I thank David Gordon and Robert McCormick for making this point clear to me.

��This is the result of log preferences, the linear accumulation technology and the assumption ofcompetitive price taking behavior of individuals, i.e., a t period parent assumes that he or she has noe!ect on w

��.

The following assumption is made regarding an agent's actions. Becauseindividuals act competitively, a parent does not take into account the externale!ect of his or her human capital investments in his or her child on the wages ofall the next generation individuals.� This assumption is relatively innocuousunder integrated production, but at the initial integration stage it is perhapsmore controversial. This will become clearer in Section 5.

In the analysis to follow, I restrict the search to only one type of equilibrium.This equilibrium contains two incompatible languages, N and M, and possiblysome translators. An alternative equilibrium would be the temporary existenceof bilingual agents, whose children become monolingual in the N language. Thisequilibrium would cause the elimination of the M language from use in com-merce. The extinction from business use of the M language arises because therate of return to human capital in any language is rising in its usage. If theM language remains in use, it is because of the existence of cultural capital,literature, popular music, philosophy, etc., written in the native tongue.�

2.1. Parental teaching time choice

In this section I show that the amount of teaching time a parent spends on hisor her child is independent of the language capability of the parent or theeventual language capability of the child. Therefore all monolingual individualswill have the same level of human capital regardless of their language, M or N,as long as their monolingual parents have the level of human capital. Considerthe problem of any parent in an integrated market. Since preferences are log, thechoice of teaching time is independent of whether or not the parent is monolin-gual or bilingual and whether or not the child is monolingual or bilingual.��Ignoring individual subscripts, a parent's problem is to choose �

�in order to:

max�ln y�#ln[1!�

�]#� ln w

��#� ln h

��. (10)

After substitution of (9) into (10), and since individuals behave competitively,�w

��/�h

��"0, the "rst order condition determining optimal teaching time is

1

1!��

"

��

, (11)


��See Appendix B for the derivation of this result.

��See Appendix B for the derivation of this result.

which can be rearranged to yield optimal teaching time:

��"

�1#�

. (12)

Thus all parents, in every time period, choose the same amount of timeeducating their children. Notice that this teaching time is completely indepen-dent of the linguistic capabilities of their children. Parents spend the sameamount of time teaching their children whether they are raising bilingualchildren or monolingual children.

2.2. Monolingual or bilingual

In this section I characterize the equilibrium language choices of parents.A parent can educate a monolingual child or a bilingual child. I show that inequilibrium all parents are indi!erent between educating a monolingual child ora bilingual child. In particular I show that all individuals, bilingual and mono-lingual, earn the same amount in the integrated market. Furthermore I showthat human capital of bilingual individuals converge to the human capital ofmonolingual individuals only if the entire M population becomes bilingual.

A monolingual parent must be indi!erent between raising a monolingualchild or a bilingual child. Since education time is constant irrespective of thischoice, equal utilities imply��

ln A��

#ln��"0 (13)

The additional earnings per unit of human capital from bilingual status, A��

,must o!set the lower amount of human capital a bilingual individual acquires,��. Recall that Z

�is the fraction of the M population that is bilingual. Given

the functional forms for returns to bilingual status, A��

, and the reductionin human capital investment productivity, �

�, the equal utility restriction

becomes

#Z��

(1!)"�[#Z��(1!)]. (14)

Now consider the choice of a bilingual parent. A bilingual parent must beindi!erent between raising a bilingual child or a monolingual child. Equalutilities imply the following:��

ln A��

#ln ��"0. (15)


Observe that the equal utilities restriction is the same as for the monolingualparent.

Given log preferences, the assumption on the accumulation technologies andproduction technologies, and assuming integrated production, Eq. (14) charac-terizes the dynamic evolution of the translator share of the M population. Allthat is left to show is when the initial set of translators will occur. This will bedetailed in Section 5.

The unique steady state bilingual share of the M population is given by

ZM "��!

1!�#�!��

. (16)

By assumption, �'. If �"1, then the entire M population becomes bilingual.However if �(1, then the bilingual population will be strictly smaller than theentire M population. Appendix B shows that the steady share of the M popula-tion that is bilingual is globally stable.

3. Bilingual policy function

This section numerically solves for the policy function of bilingual ability. Theprevious section has shown that there exists a globally stable steady statefraction of the M population that is bilingual. This fraction can be less than 1,but if �"1, then all M individuals will become bilingual.

Consider the following parameter con"guration: �"0.85, "0.25, "0.20,�"5. The steady state is ZM "0.598702855. The bilingual share of the M popu-lation policy function is graphed in Fig. 1. Observe that once bilingual indi-viduals exist, which the next section shows always occurs, then the fraction ofthe M population that is bilingual people converges to the steady-state mono-tonically from below. On the other hand, suppose for some reason there wasa stock of bilingual agents in excess of ZM , say because of a sudden in#ux ofimmigrants the period before, over time, then, some bilingual parents will raisemonolingual children. Fig. 2 contains the case where there is a single steadystate, ZM "1. The parameter set that produces this is �"1, "0.25, "0.20,�"5. Each of these examples illustrate global stability.

4. Incomes

In this section I show that the incomes of M and N monolingual individualsstart and remain equal for small market sizes, diverge for a temporary periodjust prior to integration, and then converge upon integrated production. Incomedi!erences only occur if individuals are producing in separate markets. Once themarkets merge into a single market with translators, all monolingual individuals


Fig. 1. Fraction M population children bilingual as a function of fraction M population parentbilingual.

��However I assume that the market is always at the e$cient size, P, unless P'M#N.

earn the same income. I also show that in the integrated market bilingualchildren earn the same as monolingual children. All parents are indi!erentbetween raising bilingual children or monolingual children. Since the bilingualparent spends the same amount of teaching time on education of his or her child,whether or not the child is bilingual or monolingual, consumption for thebilingual parent is independent of the language choice.

As in Tamura (1992,1995,1997a,1999a,b), per capita income is increasing inthe size of the market, up to the e$cient market size.�� The size of the market isdetermined by the number of bilingual individuals in the economy. Earnings ofthe two regions' monolingual individuals converge when integration occurs.Although bilingual individuals have less human capital than their monolingualcounterparts, they earn equal income.

Consider the monolingual individuals in M and N operating in separatemarkets. Assume that all of the initial M and N individuals began with the same


Fig. 2. Fraction M population children bilingual as a function of fraction M population parentbilingual.

��The assumption of identical human capital at the start is an innocuous one. Tamura(1991,1992,1996,1997b,1999b) provides mechanisms for human capital convergence.

level of human capital.�� Since they spend a constant amount of time educatingtheir children, their relative earnings when in separate markets are

y��

y��

"

�[1!e��

�� ]�HPH��h

�[1!e��

�� ]�HPH��h�

"1, if PH,!

�!1

ln[1!e�� ]

(min�M, N�,

[1!e�� ]�M��h

�[1!e��

�� ]�HPH��h

�

(1, if N'PH,!

�!1

ln[1!e�� ]

'M,

[1!e�� ]�M��h

�[1!e��

�� ]�N��h

�

(1, if PH,!

�!1

ln[1!e�� ]

'max�M, N�.

(17)


The "rst line of (17) shows that when markets are smaller than M, all individualsearn the same income. When market sizes exceed M, the smaller populationregion, M individuals earn less than N individuals in their separate markets.Thus human capital accumulation leads to income divergence.

Now consider earnings of M and N agents when the markets merge intoa single market with translators. Observe that their earnings converge immedi-ately upon integration of the two separate markets,

y��

y��

"

[1!e�� ]��

�� Z�

��(N#M!MZ

�)h� �

�#A� �

�hK � ��

��h�

[1!e�� ]��

�� Z�

��(N#M!MZ

�)h� �

�#A� �

�hK � ��

��h�

"1. (18)

Now consider bilingual agents. The "rst thing that can be shown is that allbilingual agents have the same human capital, independent of the type of parentthey come from, bilingual or monolingual. Human capital of a bilingual child isgiven as a function of what generation his or her ancestors became bilingual:

hK�"

��h

��

�"��h��

�� if t!1 was monolingual,

�hK��

��

��

�"�(�h��

��

�)��

��

�"��h��

�� if t!1 was bilingual,

�hK��

��

��

�"��hK ��

��

��

��

��

�,

"��(�h��

��

�)��

��

��

��

�"��h��

��, if t!2 was bilingual,

2

��h��

�� if 1 was bilingual.

(19)

Therefore all bilingual adults in period t have the same human capital,independent of when they or their ancestors became bilingual. If Z

�4ZM , then

from (13), the income of bilingual individuals is equal to the income of anymonolingual individual. Now consider the case where Z

�'ZM . Since a bilingual

parent can raise a monolingual child at lower cost than a monolingual


��This enters the realm of expectations and coordination. If agents all believe that no one will bebilingual, or that a set of measure 0 becomes bilingual, then no one has an incentive to raisea bilingual child. If at the same time, agents believe that a measurable number of bilingual agents willexist next period, then some may have an incentive to raise a bilingual child. Thus expectations mayproduce multiple equilibria. Coordination is a natural response to this problem. That is to say,a coalition or a government would have a language policy that improves on the decentralizedequilibrium. This is beyond the scope of this paper.

parent can raise a monolingual child, Appendix B shows that the equal utilitycondition for bilingual parents implies that earnings of bilingual children frombilingual parents are equal to the earnings of monolingual children frombilingual parents. This condition can be written as

A(Z��

)�(Z�)�hK

��

�(Z��

)"

�hK��

�(Z��

). (20)

Observe that the human capital of a monolingual child from a bilingual parentis equal to the human capital of a monolingual child from a monolingual parent.Thus all individuals earn the same amount whether they are monolingual orbilingual.

As a result of (13), output in the market with translators relative to a marketwith a single language is given by

>��

>��

"

[1!e��M�� ]��

�� Z�

��N#M��h

�[1!e��M

�� ]��N#M��h

�

"[1!e��M�� ]��

� ��Z�

�(1. (21)

Recall that by assumption �(0, and '0, then the gap between a market withtranslators and a market with a single language falls over time, because humancapital rises, but is bounded by Z�

�. Therefore production with translators is the

same as production under a single language only when all of the M populationbecomes bilingual.

5. Creation of initial bilingual agents

In this section the incentive to create the "rst group of bilingual M agents isexamined. What makes this more di$cult to analyze is that any individual agentis assumed to be a set of measure 0. Thus if a single agent in t were bilingual,Z

�"0, and there would be no productivity forthcoming from translators.

Therefore the creation of bilingual agents requires that a measurable quantity beproduced. How this coordination occurs is left unanswered.�� As a result, thispaper considers the initial creation of translators as occurring at the "rst datethat per capita output is higher under integrated production and that it is


individually rational for some agents to raise a bilingual child, given that theyknow a positive measure of bilingual children will exist next period.

If there are no bilingual agents in period t!1, then from (14) Z�

is either 0 or((�!)/(1!))� �. All that is left to show is whether or not integrated produc-tion is bigger than segregated production (aggregate e$ciency), and integratedearnings are greater than segregated earnings (individual rationality). Let Z

�be

the initial share of the M population that becomes bilingual. Compare integ-rated and segregated production at t:

>��

"�(N#M!MZ�)#MZ

�(A

��)� ��

�Z�

��h

��[1!e��M

�� ]��

��

"�N#M��Z�

��h

��[1!e��M

�� ]��

��

>��

"N��h��

�[1!e��M�� ]�#M��h

��[1!e��M

�� ]�, (22)

where the second line arises because individual rationality implies A��"1.

Since each agent is paid his or her marginal product, individual earnings aregiven by

y��

"�N#M��Z�

��h

��[1!e��M

�� ]��

�� ,

y��

"N��h��

�[1!e��M�� ]�,

y��

"M��h��

�[1!e��M�� ]�. (23)

Individual rationality implies aggregate e$ciency. Assume that it is individuallyrational for each individual in M and N to integrate. This implies that the "rstline of (23) is larger than either the second or the third lines of (23). Multiplyingthe "rst line of (23) by M#N, and the maximum of the second and third lines of(23) by M#N maintains the inequality. But M#N times the maximum of thesecond and third lines of (23) is larger than aggregate segregated production.Thus if earnings of workers in integrated production exceeds earnings ofworkers in segregated production, the initial creation of bilingual agents occurs.

If the optimal market size under a single language is less than M#N, thenobviously the creation of a single market with both regions will not occur. Nowassume that the e$cient market size is larger than M#N. Then the questionbecomes, &when is it optimal to merge two regions into a single market thatrequires translators?' Each region's human capital grows at rate ��/(1#�).Since N5M by assumption, earnings will always be larger in the N region,once the region's population becomes the binding constraint on market size.Integration occurs when integrated production workers earn more thanN workers.

y��

"�N#M��Z�

��h

��[1!e��M

�� ]��

��

'N��h��

�[1!e��M�� ]�"y��

��. (24)


�� I assume that the left-hand side is strictly positive. This requires that the gains from specializa-tion, �, are reasonably large, and the static loss from translation, , is not too large.

Dividing by both sides by �h��

�; taking logs of each expression, and using theapproximation that ln[1!x]+!x produces

(�!1) ln [M#N]#� ln Z�!(N#M!MZ

�#MZ��

�)e��M

��

'(�!1) ln N!Ne��M�� . (25)

Simplifying the above expression produces

(�!1) ln �M#N

N �#� ln Z�

M!MZ�#MZ��

�

'e��M�� . (26)

Clearly as human capital accumulates, the right-hand side goes to 0.�� If theleft-hand side is positive, and since it is independent of time, there will alwaysexist a date when two regions will merge into a single economy. If the left-handside of (26) is bigger than 1, then integration occurs immediately, for any positivelevel of human capital, subject to the restriction that ln[1!x]+!x is a goodapproximation. More interestingly suppose the left-hand side of (26) is positive,but less than 1. Since hM

�grows at rate ��/(1#�), replacing into (26) and taking

logs and rearranging produces

ln�!ln�(�!1) ln �M#N

N �#� ln Z�

M(1!Z�#Z��

�) ��!ln �

�!�

�ln h

��

ln��

1#��(tH. (27)

The larger the M population, relative to the N population, the slower the merge.The larger N, the earlier the merge will take place. The greater the returns tomerging, greater �, the faster the merge will take place. The smaller the loss dueto translation, smaller �, the earlier a merge will occur. The greater the initialhuman capital stock, h

, the earlier the integration will occur. The larger the

values of ��

and ��, the earlier integration will occur. The faster the growth rate

of human capital, ��/(1#�), the earlier integration will occur. For the para-meter values assumed: N"300,000,000; M"199,999,999; �

�"0.05; �

�"10;

�"0.85; "0.2; "0.25; "0.8; �"!0.25; �"5, �"4.5; �"0.5; �"5;h�"1, the implied tH"32.37 generations. The steady state fraction of the

M population that is bilingual is ZM "0.598702855. This accords with thedynamic solution below.


��Tamura (1999a,b) presents models with endogenous population growth and endogenoushuman capital in a similar model, but without the language incompatibility problems. Per capitaincome growth is more rapid with endogenous population growth than with "xed population.

6. Dynamic path of bilingual world

In this section I present a time series solution of the model with bilingualindividuals. Two initially monolingual regions exist, for example the US andEurope, and they speak two di!erent languages. Over time, bilingual individualsarise. I compare the per capita incomes of the two regions, with the per capitaincomes of a monolingual world with the same population. There exists a staticloss to maintaining two languages, unless all M agents become bilingual. How-ever the world with translators is wealthier than a world without translators.

The numerical solutions are quite simple to produce. Human capital in themodel grows at the constant rate of ��/(1#�). In each period, the programdetermines the optimal market size given by (4). When the optimal market sizeexceeds N, it becomes a question as to when the two markets will integrate.Integration occurs when (26) holds. The initial fraction of the M population thatis bilingual is given by Z

�. From that time period onward, (14) characterizes the

time path of the bilingual share of the M population.The parameter set for the solution is: N"300,000,000; M"199,999,999;

��"0.05; �

�"10; �"0.85; "0.2; "0.25; "0.8; �"!0.25; �"5, �"

4.5; �"0.5; �"5; h�"1. Given these parameters, the optimal market size is

below either region's population for 29 generations. In generation 30, theoptimal market size exceeds M, but is less than N. For generations 31 and 32,the market size of each region is given by the population of each region.Integration occurs in generation 33.

Fig. 3 presents log per capita incomes of a single language economy withpopulation M#N. The steeper portion of the graph represents two features ofgrowth, human capital accumulation and the increase in the size of the optimalmarket. For generation 33, the optimal market size no longer grows, and thepopulation of the combined regions becomes the binding constraint.��

Fig. 4 shows the per capita income of agents in the two language M andN regions relative to the single language economy. While the optimal marketsize is smaller than either M or N, relative incomes are the same as in a singlelanguage world. However when the optimal market size exceeds M, the relativeincomes of M individuals begins to fall. Since N'M, individuals in theN region do not fall relative to the single language economy until the nextgeneration. Beyond this point, as long as the two markets are separate, relativeincomes fall rapidly. In generation 33, the "rst set of bilingual individuals areproduced and the two markets are merged. From then on, there is a partialrecovery in relative incomes, reaching a stationary level of about 66%.


Fig. 3. Log per capita output in single language economy.

Fig. 4. Per capita income of M (bottom curve) and N (top curve) residents relative to single languageeconomy.


Fig. 5. Market size in single language economy.

In Fig. 5 the log of the optimal market size in the single language economy ispresented. For the "rst 31 generations, the market size is less than M#N. Forthe remaining time periods, the optimal market size is constant and equal toM#N. Fig. 6 presents the relative market sizes for the M and N regions. Again,since the population of region M is lower than the population of region N, itbecomes a binding constraint on individuals earlier.

The annualized growth rates of per capita income in the single languageeconomy is contained in Fig. 7. I assume that a generation is 20 years. Observethat as human capital accumulates and the optimal market size expands, thegrowth rate of per capita income accelerates. Once the optimal market sizeexceeds the combined population of the M and N regions, the per capita incomegrowth rates falls. The long run generational growth rate is 1.5, which isapproximately 2% per year.

Fig. 8 contains the di!erence between the annualized per capita incomegrowth rates for each region, M and N, with the annualized per capita incomegrowth rates for the single language economy presented in Fig. 7. Notice that themaximum absolute value of the deviations in growth rates are about 4.5% peryear. This is true for both the period when the single language economy isgrowing more rapidly, and when the economy with translators is growing morerapidly. Since the M region is smaller than the N region, growth rates for


Fig. 6. Market size of M and N regions relative to single language economy.

Fig. 7. Annualized growth rates in single language economy.


Fig. 8. Annualized growth rate di!erences of M and N regions with single language economy.

M residents fall below the single language economy "rst. However upon integra-tion the growth rates of the M residents far exceed the growth rates of theN residents. Although there are persistent income di!erences between thetranslator economy and the single language economy, there are no long runsystematic di!erences in growth rates of these two economies.

7. Conclusion

This paper has produced a model of translators. A two region world, one withM agents and the other with N agents, M(N, each speaking incompatiblelanguages. The model shows that when there is an agglomeration economy tomarket participation, arising from gains from specialization, there exists anincentive for some of the M agents to become bilingual. The incentive to becomebilingual is larger: (1) the higher the average level of human capital in themarket, (2) the greater the agglomeration economy, (3) the larger N is relative toM, and (4) the smaller the static cost of translation.

While the two regions are segregated, incomes diverge between monolingualM and N speakers if their market sizes di!er. When integration occurs, allindividuals earn the same income. Because it costs more to raise a bilingual childthan a monolingual child, bilingual individuals earn more per unit of human


� It is possible that multiple languages can be productive relative to a single language economy. Iflanguages are productive by themselves, re#ecting regional speci"c information that cannot bereproduced or translated into a single language, then output with translators may be greater thansingle language production. This is beyond the scope of this paper.

capital than their monolingual counterparts. Bilingual individuals have lesshuman capital than either monolingual individuals. In equilibrium these twoe!ects o!set each other and bilingual individuals earn the same amount asmonolingual speakers. The world economizes on the number of translators.Only if there is no long run cost of becoming bilingual, �"1, will all M agentsbecome bilingual. If not all M agents become bilingual and when translation isnot perfect, '0, there exists a static cost of maintaining multiple languages.

Appendix A

In this appendix the integrated production technology with translators isscrutinized. In particular conditions are found in order that monolingual integ-rated production is always at least as productive than an economy with the samehuman capital distribution, but with multiple languages and translators.� Sinceincome of translators and monolingual individuals are identical in the mergedmarket, output in a world with a single language and output with translators aregiven by

>��

"[1!e�� ]��M#N��h

�,

>��

"[1!e�� ]��

�� Z�

��M#N��h

�. (A.1)

By assumption '0, and �(0, and therefore Z�

�41, and M(1!Z

�#

Z��

)'M. Thus output with translators is always less than output witha single language, except when all M individuals are bilingual.

Appendix B

In this appendix, the equal utilities conditions are examined for both mono-lingual parents and bilingual parents. Consider a generation t monolingualadult. He or she could raise a monolingual child or bilingual child. Thedi!erence between these two choices provide the following utility di!erential:

;(h��bilingual child)!;(h

��monolingual child)

"ln y�#ln[1!�

�]#� ln y��

��!ln y

�!ln[1!�

�]!� ln y��

��

"

��

�ln A(Z��

)#ln �(Z�)#ln �#ln h

�#ln ��


!

��

�ln �#ln h�#ln ��

"

��

�ln A(Z��

)#ln �(Z�)�. (B.1)

In equilibrium a monolingual parent must be indi!erent between the twopossible choices. Setting the di!erence equation to 0 in the "nal line producesthe result in the paper.

Now consider a generation t bilingual adult. He or she could raise a monolin-gual child or a bilingual child. The di!erences between these two choices providethe following utility di!erential:

;(h��bilingual child)!;(h

��monolingual child)

"ln y�#ln[1!�

�]#� ln y��

��!ln y

�!ln[1!�

�]!� ln y��

��

"

��

�ln A(Z��

)#ln �(Z�)!ln �(Z

��)#ln �#ln h

�#ln ��

!

��

�!ln �(Z��

)#ln �#ln h�#ln ��

"

��

�ln A(Z��

)#ln �(Z�)�. (B.2)

In equilibrium a bilingual parent must be indi!erent between the two possiblechoices. Setting the di!erence equal to 0 in the "nal line produces the result inthe paper. Observe that the productivity advantage of educating any type ofchild by bilingual parents is crucial to the result that (15) contains the dynamicsof the bilingual share of the M population.

What remains to be shown is that the convergence is monotonic. Given thefunctional forms for A and �, the equal utility condition reduces to

�[#Z��(1!)]"#Z�

��(1!). (B.3)

Solving for Z��

produces the nonlinear di!erence equation:

Z��

"[A#BZ��]� �,

0(A"

�!1!

(1,

0(B"

�(1!)1!

(1. (B.4)


The restriction on A comes from the assumption 1'�'. The restriction onB comes from the assumptions: ' and 0(�(1. The slope of the di!erenceequation is given by

�Z��

�Z�

"

BZ��

[A#BZ��]��

(B.5)

By assumption �51. When �"1, the nonlinear di!erence equation becomesa linear di!erence equation. The coe$cient on Z

�is B(1, which implies

stability. When �'1, notice that

[A#BZ��]�� 'B�� Z��

�'BZ��

�. (B.6)

The "rst inequality follows from the fact that A'0, and that �'1. The secondinequality comes from the fact that 1'B'0 and �'1. Thus the slope of thedi!erence equation is less than 1. Therefore the steady state is globally stable.

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Documents

Translators: Market makers in merging markets