Proficiency,Attitude,and Conventions in Minority Languages

Abstract

In this article, we study a simple mathematical model of a bilingual community in which all agents are fluent in the majority language but only a fraction of the population has some degree of proficiency in the minority language. We investigate how different distributions of proficiency, combined with the speakers’ attitudes toward or against the minority language, may influence its use in pair conversations.

Keywords

social choice bilingualism minority languages linguistics mathematical model

Introduction

The European Union has a long tradition of linguistic diversity within its member states. According to the 2006 Eurobarometer, (European Commission 2006), only 9¹ out of the 28 states can be considered as linguistically homogeneous, that is, with at least 97 percent of the population reporting a common mother tongue. Twelve states² report figures of between 90 percent and 96 percent and a further six³ between 70 percent and 89 percent. In Belgium, 56 percent of the population report Dutch as their mother tongue and 38 percent report French. With the exceptions of Belgium, Estonia, and Luxembourg, there is always a majority language spoken by the whole population.

The European Charter for Regional or Minority Languages, which entered into force in 1998, seeks to assure respect for protection and promotion of regional and minority languages in Europe. It defines regional or minority languages as languages that are “traditionally used within a given territory of a State by nationals of that State who form a group numerically smaller than the rest of the State’s population; and different from the official language(s) of that State.” A minority language is not necessarily a language at risk of extinction though many of them are classified as either vulnerable or endangered according to the Atlas of the World Languages in Danger published by the United Nations.

The use of regional or minority languages varies greatly from one language and one territory to another. In Catalonia (Spain), more than 60 percent of the population are able to speak and write in Catalan, and 36 percent report that they use Catalan exclusively on a daily basis, while another further 12 percent use both Catalan and Spanish on a daily basis (Generalitat de Catalunya 2009). In the Basque Autonomous Community (Spain), the reported knowledge of Basque is around 35 percent, while the use is around 13 percent (Gabinete de Prospección Sociológica, Eusko Jaurlaritza-Gobierno Vasco 2011). In Ireland, the percentage of the population who can speak Irish is 40.6 percent and 2 percent speak Irish on a daily basis outside the education system (Central Statistics Office, Government of Ireland 2012). The relationship between knowledge and use of languages is thus far from linear.

Abrams and Strogatz (2003) propose a macroscopic model of language competition which predicts that two languages cannot coexist stably—one will eventually drive the other to extinction. These authors arrive at the conclusion that increasing language status is crucial in order to prevent the decline of minority languages. Their model however assumes that individuals are monolingual. Minett and Wang (2008) incorporate bilingualism and language transmission, but their model yields the same extinction conclusion.

The objective of this article is to study language competition at a microscopic level: We seek to understand the mechanism that determines which language is chosen for conversation when various languages coexist. One fundamental distinction between languages in competition is that the majority language is spoken by all, while minority languages are spoken by only a fraction of the population. The model is kept as simple as possible: We concentrate on bilingual conversations and start with a static model (i.e., we disregard from learning concerns). Each speaker is characterized by two features considered by Fishman (1991:49) as being of overriding importance: attitude and competence (or proficiency). We follow van Parijs (2007) and consider that communication exchanges yield value for speakers through these two different dimensions: efficiency and expressive value. The proficiency of the speakers participating in the conversation affects the efficiency in exchanging information. The attitude or motivation of the speakers participating in the conversation gives value for expressive reasons. We take the view that proficiency and attitude in generating worth or value are equally important. Note that the efficiency of a conversation is determined by the speaker with the lower proficiency (the “maximin law of communication,” van Parijs 2007). By contrast, the expressive value of a conversation depends on the speaker’s attitude toward the language. Once the value of a conversation is fixed for each speaker, a social convention selects the language to be spoken. That social convention is accepted by the whole of society as a social norm and therefore has some reasonable properties. We focus on conventions that are anonymous, unanimous, and monotone.⁴

The main question is whether the mechanism above described is neutral in the sense introduced by May (1952). In our context, the mechanism is neutral if it does not favor either language. Individual preferences are neutral: If the names of the languages are changed, the preferred language is not changed. In a conversation, the proficiency of the partner matters when determining the value of a conversation. Indeed, the value of a conversation depends on the proficiency of both speakers. This introduces a bias in favor of the majority language: Preferences in conversations are no longer neutral. This bias is due exclusively to the fact that the value of the information exchange suffers from a (negative) externality given by a lack of proficiency in the minority language. It is shown that no convention can offset this bias in favor of the majority language, that is, even the convention that is most favorable to the minority language cannot offset the bias in favor of the majority language that is generated by the interaction of the proficiency of speakers in a conversation.

The rest of the article is organized as follows. In the second section , we study the individual preferences. The third section shows the bias that arises in conversations. The fourth section reviews the convention and the fifth section shows that there is a bias in the choice of language for any convention. The sixth section concludes with some discussion. The Appendix contains the proofs.

Individual Preference for a Language

Consider a population of speakers that can make use of two languages A and B in (for the sake of simplicity bilateral) communication exchanges. The majority language A is spoken by the whole population, while not all individuals are able to maintain a conversation in the minority language B. In this section, we describe the relative (positive or negative) preference for language B with respect to language A.

Speakers positively value a communication exchange in a given language for two reasons: information exchange and expressive reasons. The quantity and quality of information exchanged is critically determined by the proficiency of the speaker: The higher the proficiency, the more information is exchanged and the better its quality is.⁵ Languages are also used as carriers of cultural identity and of expressive meanings beyond the pure informative statements.⁶

The importance of the language as an expressive meaning is captured by the attitude of the speakers toward the language. Hence, the speaker’s preference for one language over the other results from the interaction of these two parameters, proficiency and attitude.

For the sake of simplicity, we assume that all individuals have the same level of proficiency in language A, and the same attitude toward language A and choose language A as a benchmark. A speaker can then be characterized by $\vec{x}$ = (m, p), where m denotes the individual’s relative attitude toward language B and p denotes her or his relative proficiency in language B. We assume −1 ≤ p ≤ 1 and −1 < m < 1 with the following interpretation: Monolingual individuals (who only speak language A) have p = −1 and bilingual individuals have p > −1. An individual with −1 ≤ p < 0 is more proficient in language A, and an individual with 0 < p ≤ 1 is more proficient in language B. An individual equally proficient in both languages has p = 0. Similarly, the interpretation of m = 0 is that the attitude toward both languages is identical. The quantity m is strictly positive (0 < m < 1) if the individual has a more positive attitude toward language B, while 0 > m > −1 reflects a more positive attitude toward language A. Any speaker can be identified by her or his characteristics, $\vec{x}$ = (m, p). The space of speakers is referred to as X = ]−1,1[ × [−1,1].

Both characteristics have positive effects on the preference for the minority language compared to the majority language: The more proficient or the more positive the attitude, the more preferred the minority language is relative to the majority language. We consider that the effects of proficiency and attitude on language preference are separable additively, hence allowing for cases where proficiency and attitude do not go hand in hand: A monolingual individual may have a positive attitude toward the use of the minority language or an individual with higher proficiency in the minority language may have a negative attitude toward the minority language. We take the view that proficiency and attitude are at least partially substitutes: The preference for a language can be maintained if one characteristic increases and the other decreases. For instance, a more positive attitude toward the minority language can offset a lower level of proficiency. For the sake of simplicity, we consider that the two characteristics are perfect substitutes. The individual utility of the minority language relatively to the majority language (or individual utility for short) is the sum of the relative attitude and relative proficiency. Denoting by u_ii the individual utility of speaker ${\vec{x}}_{i}$ = (m_i , p_i ), the following is obtained:

u_{i i} = m_{i} + p_{i} .

This is interpreted as ${\vec{x}}_{i}$ ’s relative preference for language B over language A. If u_ii < 0, speaker ${\vec{x}}_{i}$ prefers language A, she or he is indifferent if u_ii = 0, and prefers language B if u_ii > 0. The absolute value of u_ii measures the intensity of the (positive or negative) preference. Note that a monolingual individual (i.e., if p_i = −1) always prefers language A (u_ii < 0, given that m_i < 1).

Neutrality requires that both languages be treated equally, that is, if languages were labeled differently, the results would remain identical. If we had chosen language B as the benchmark instead of language A, the relative attitude and proficiency levels of individuals would just have the opposite sign: A speaker would be characterized by − ${\vec{x}}_{i}$ = (−m_i , −p_i ) instead of ${\vec{x}}_{i}$ = (m_i , p_i ). The space of characteristics is neutral, given that ${\vec{x}}_{i}$ ∈ X ⇔ − ${\vec{x}}_{i}$ ∈ X. One implication of neutrality is that in a uniformly distributed population (i.e., same density of individuals in X), both languages are preferred by the same number of speakers.

Now consider the neutrality of preferences. If language B were taken as reference, the preference for the majority language would be reflected by a positive utility and negative utility would mean a preference for the minority language. Preferences are neutral if the following equivalence holds: The utility of any speaker ${\vec{x}}_{i}$ is strictly positive if and only if that of speakers − ${\vec{x}}_{i}$ is strictly negative. For any speaker ${\vec{x}}_{i}$ , let ${\bar{u}}_{i i}$ denote the utility of − ${\vec{x}}_{i}$ . Given that − ${\vec{x}}_{i}$ = (−m_i , −p_i ), it emerges that ${\bar{u}}_{i i}$ = −m_i − p_i and neutrality is trivially satisfied: For any ${\vec{x}}_{i}$ ∈ X, it holds that u_ii > 0 ⇔ ${\bar{u}}_{i i}$ < 0.

Biased Preferences in a Conversation

The high proficiency of a speaker is worth nothing in a conversation if the other speaker is monolingual or has a poor knowledge of the language. The exchange of information is limited by the speaker with the lower proficiency. Formally, in a conversation between speakers ${\vec{x}}_{i}$ = (m_i , p_i ) and ${\vec{x}}_{j}$ = (m_j , p_j ), the exchange is constrained in terms of proficiency by min{p_i , p_j }. By contrast, the expressive value of the conversation is not constrained by the other speaker. Depending on the individual’s degree of altruism, the attitude in a conversation can take any value between m_i and m_j . Here we consider that individuals are purely self-interested: Their valuation is intrinsic to the individual and is not affected by the characteristics of the other participant. The attitude of ${\vec{x}}_{i}$ is simply m_i . The relative preference of individual ${\vec{x}}_{i}$ for a language in conversation with ${\vec{x}}_{j}$ is determined by attitude of ${\vec{x}}_{i}$ and the proficiency of the conversation min{p_i , p_j }. In a conversation between ${\vec{x}}_{i}$ and ${\vec{x}}_{j}$ , let u_ij denote speaker ${\vec{x}}_{i}$ ’s utility. We have the following definition:

u_{i j} \overset{def}{=} m_{i} + min {p_{i}, p_{j}} .

If u_ij < 0, speaker ${\vec{x}}_{i}$ prefers language A in a conversation with ${\vec{x}}_{j}$ , she or he is indifferent if u_ij = 0, and she or he prefers language B if u_ij > 0. The absolute value of u_ij measures the intensity of the (positive or negative) preference.

The utilities obtained by speakers ${\vec{x}}_{i}$ and ${\vec{x}}_{j}$ in a conversation generally differ: u_ij ≠ u_ji when m_i ≠ m_j . If u_ij ≠ u_ji , both speakers share the same preference for a language if u_ij ·u_ji > 0, or one speaker prefers the minority language while the other prefers the majority language if u_ij ·u_ji < 0. Similarly, speaker ${\vec{x}}_{i}$ may have different preferences for languages in conversation with different partners ${\vec{x}}_{j}$ and ${\vec{x}}_{k}$ . That is, we may have u_ij ·u_ik < 0.

For any conversation between ${\vec{x}}_{i}$ and ${\vec{x}}_{j}$ , the values of u_ij and u_ji vary in the interval ]−2,2[. All individuals are assumed to have the same utility function u:

u : X^{2} \to] - 2, 2 [, ({\vec{x}}_{i}, {\vec{x}}_{j}) \mapsto u ({\vec{x}}_{i}, {\vec{x}}_{j}) = u_{i j} .

The value u_ii is the utility obtained by ${\vec{x}}_{i}$ in a conversation with another speaker with identical characteristics (m_i , p_i ). It is also the individual utility of ${\vec{x}}_{i}$ as defined in equation (1). Obvious properties can be derived from equations (1) and (2).

The highest utility of ${\vec{x}}_{i}$ is reached when conversing with a partner with identical characteristics: u_ij ≤ u_ii for any ${\vec{x}}_{j}$ ∈ X or

u_{i i} = \max_{{\vec{x}}_{j} \in X} {u_{i j}} .

If ${\vec{x}}_{i}$ individually prefers language A (i.e., if u_ii < 0), then this preference is kept in any conversation: u_ij < 0 for any ${\vec{x}}_{j}$ ∈ X.

Speaker ${\vec{x}}_{i}$ prefers language A in a conversation with ${\vec{x}}_{j}$ if either ${\vec{x}}_{i}$ or ${\vec{x}}_{j}$ does not have sufficient proficiency: u_ij < 0 if p_j < −m_i or p_i < −m_i .

Speaker ${\vec{x}}_{i}$ individually prefers language B but prefers language A in a conversation with ${\vec{x}}_{j}$ if ${\vec{x}}_{j}$ has a lower proficiency than ${\vec{x}}_{i}$ (p_j < p_i ) and the attitude of ${\vec{x}}_{i}$ toward the minority language is not sufficiently large to offset this lower proficiency: p_j < −m_i < p_i .

Speaker ${\vec{x}}_{i}$ prefers language B in a conversation with ${\vec{x}}_{j}$ only if both ${\vec{x}}_{i}$ and ${\vec{x}}_{j}$ have sufficient proficiency: u_ij > 0 if p_j > −m_i and p_i > −m_i .

If ${\vec{x}}_{i}$ prefers language B in a conversation with any ${\vec{x}}_{j}$ , then ${\vec{x}}_{i}$ individually prefers language B: u_ij > 0 ⇒ u_ii > 0.

Thus, two conditions must be satisfied for a speaker to prefer the minority language in a conversation: Her or his attitude has to offset her or his own lack of proficiency and that of the other participant; by contrast, failure to meet just one condition is enough for the speaker to prefer the majority language. Preferring the majority language in a conversation with a speaker of the same level is a sufficient condition to prefer the majority language in any conversation, while preferring the minority language in a conversation with a speaker of the same level is not a sufficient condition to prefer the minority language in any conversation.

As a result, although the preferences of individuals are neutral, neutrality is lost for preferences in conversations. Neutrality would require the signs of the utilities in a conversation between − ${\vec{x}}_{i}$ and − ${\vec{x}}_{j}$ to be the opposites of the signs of the utilities in the conversation between ${\vec{x}}_{i}$ and ${\vec{x}}_{j}$ . Using the following notation:

{\bar{u}}_{i j} \overset{notation}{=} u (- {\vec{x}}_{i}, - {\vec{x}}_{j}) .

The preferences of ${\vec{x}}_{i}$ would be neutral if u_ij · ${\bar{u}}_{i j}$ < 0 or u_ij = ${\bar{u}}_{i j}$ = 0 for any ${\vec{x}}_{j}$ . It is shown below that preferences are not neutral as there are some ( ${\vec{x}}_{i}$ , ${\vec{x}}_{j}$ ) ∈ X ² such that u_ij · ${\bar{u}}_{i j}$ > 0. Nevertheless whenever u_ij > 0 it holds that ${\bar{u}}_{i j}$ < 0. This means that there is a bias in favor of the majority language: Whenever ${\vec{x}}_{i}$ prefers language B in a conversation with ${\vec{x}}_{j}$ , it results that − ${\vec{x}}_{i}$ prefers language A in a conversation with − ${\vec{x}}_{j}$ . By contrast, it may be that ${\vec{x}}_{i}$ prefers language A in a conversation with ${\vec{x}}_{j}$ and − ${\vec{x}}_{i}$ also prefers language A in a conversation with − ${\vec{x}}_{j}$ .

Definition 1: Preferences in conversations are

A-biased if u_ij > 0 ⇒ ${\bar{u}}_{i j}$ < 0 for any ( ${\vec{x}}_{i}$ , ${\vec{x}}_{j}$ ) ∈ X ²,

B-biased if u_ij < 0 ⇒ ${\bar{u}}_{i j}$ > 0 for any ( ${\vec{x}}_{i}$ , ${\vec{x}}_{j}$ ) ∈ X ².

Preferences in conversations are neutral if they are both A-biased and B-biased. The bias is systematic if preferences are biased toward one language and not toward the other. The following proposition sums up what is stated above (proof is given in the Appendix).

Proposition 1: Preferences in conversations are systematically A-biased.

Neutrality holds for individual preferences but ceases to hold for preferences in conversation: There is a systematic bias in favor of the majority language. The implication is that although both languages are individually preferred in a uniformly distributed population by the same number of speakers (i.e., there are as many speakers with u_ii < 0 as u_ii > 0), more individuals prefer the majority language in conversations. Indeed, the speakers who individually prefer the majority language keep their preferences in all conversations, while those who individually weakly prefer the minority language switch their preference if the interlocutor lacks sufficient proficiency. The scale of the bias depends on how speakers are distributed within X. In particular, it depends on the speaker’s individual preference and the interlocutor’s proficiency as shown in the following proposition (proof of which is given in the Appendix).

Proposition 2: Given a speaker ${\vec{x}}_{i}$ ∈ X,

If u_ii < 0, then u_ij < 0 for any ${\vec{x}}_{j}$ ∈ X.

If u_ii = 0, then u_ij ≤ 0 for any ${\vec{x}}_{j}$ ∈ X. The equality u_ij = 0 holds if p_j ≥ −m_i .

If u_ii > 0, then (a) u_ij < 0 for any ${\vec{x}}_{j}$ such that p_j < −m_i , (b) u_ij > 0 for any ${\vec{x}}_{j}$ such that p_j > −m_i , and (c) u_ij = 0, whenever p_j = −m_i .

Linguistic Conventions

So far, we have explored the speakers’ preferences for one language or the other. In a conversation between ${\vec{x}}_{i}$ and ${\vec{x}}_{j}$ , the respective preferences of the speakers are given by u_ij and u_ji . The utilities space is denoted by U = ]−2,2[ × ]−2,2[. An element of the space (u_ij , u_ji ) ∈ U represents a pair of utilities obtained in a conversation. A linguistic convention C associates either languages A or B with any pair (u_ij , u_ji ):

C : U \to {A, B}, (u_{i j}, u_{j i}) \mapsto C (u_{i j}, u_{j i}) = A o r B .

We thus assume that there is a linguistic convention which is accepted by the whole society as a social norm. In all conversations, the same rule is applied to select the language. This rule is exclusively based on the respective utilities of both speakers in the conversation. A convention satisfies certain reasonable properties. In order to be accepted as a convention, a rule should satisfy (at least) three properties:

Anonymity: ∀(u_ij , u_ji ) ∈ U: C(u_ij , u_ji ) = C(u_ji , u_ij ).

Unanimity: ∀(u_ij , u_ji ) ∈ U such that (u_ij , u_ji ) ≠ (0, 0):

If u_ij ≥ 0 and u_ji ≥ 0 then C(u_ij , u_ji ) = B,

If u_ij ≤ 0 and u_ji ≤ 0 then C(u_ij , u_ji ) = A.

Monotonicity: ∀(u_ij , u_ji ), (v_ij , v_ji ) ∈ U:

If C(u_ij , u_ji ) = A, then (u_ij ≥ v_ij and u_ji ≥ v_ji ) ⇒ C(v_ij , v_ji ) = A,

If C(u_ij , u_ji ) = B, then (u_ij ≤ v_ij and u_ji ≤ v_ji ) ⇒ C(v_ij , v_ji ) = B.

Anonymity means that the identities of the speakers do not matter. This is a reasonable property as long as individuals have no role in the conversation.⁷ Unanimity requires that if both speakers share the same preference for a language then the convention selects the preferred language, and if one speaker is indifferent and the other strictly prefers one language, then the convention selects the language preferred by the speaker with strict preference. In our context, this is equivalent to the Pareto efficiency condition: If the choice of the minority (majority) makes at least one speaker better-off and does not make the other speaker worse off, then the minority (majority) language is chosen. Finally, monotonicity states that if a convention selects one language for two given speakers, then it must select the same language for two other speakers with the same or stronger preferences for that language.

A convention C can be graphically represented by dividing the U-space into the subspaces U_A (C) and U_B (C), where A and B are chosen, respectively. Anonymity requires U_A (C) and U_B (C) to be symmetric with respect to the first diagonal. Unanimity requires the first quadrant of U to be included in U_B (C) and the third quadrant U to be included in U_A (C). Monotonicity requires U_A (C) and U_B (C) to be connected. Graphic examples of conventions are shown in Figure 1.

Figure 1.

Examples of linguistic conventions.

Obviously, the larger U_B (C) (resp. U_A (C)) is, the more favorable the convention is toward the minority (resp. majority) language. Convention C ₁ is more favorable to B than convention C ₂ (or convention C ₂ is more favorable to A than convention C ₁) if U_B (C ₂) ⊂ U_B (C ₁).

The convention that is most favorable to B is denoted by C_B . It selects the minority language unless both speakers share the same preference for the majority language:

C_{B} (u_{i j}, u_{j i}) = {\begin{cases} A if u_{i j} \leq 0, u_{j i} \leq 0 and (u_{i j}, u_{j i}) \neq (0, 0) \\ B otherwise \end{cases} .

Similarly C_A can be defined as the convention that is most favorable to the majority language.

C_{A} (u_{i j}, u_{j i}) = {\begin{cases} B if u_{i j} \geq 0, u_{j i} \geq 0 and (u_{i j}, u_{j i}) \neq (0, 0) \\ A otherwise \end{cases} .

We now focus on the condition of neutrality and the space of utilities when the minority language is taken as the benchmark instead of the majority one. If (u_ij , u_ji ) is the pair of utilities when A is taken as benchmark, then the pair of utilities if B were taken as benchmark would be given by (−u_ij , −u_ji ). The space of utilities is neutral, given that (u_ij , u_ji ) ∈ U ⇔ (−u_ij , −u_ji ) ∈ U. A convention is neutral if the choice of language for (−u_ij , −u_ji ) is the opposite of the choice for (u_ij , u_ji ). That is, if, for all (u_ij , u_ji ) ∈ U,

C (u_{i j}, u_{j i}) = A \Leftrightarrow C (- u_{i j}, - u_{j i}) = B .

This condition reduces to C(u_ij , −u_ij ) ≠ C(−u_ij , u_ij ), whenever u_ij + u_ji = 0. This is incompatible with anonymity, which requires C(u_ij , u_ji ) = C(u_ji , u_ij ). In order to avoid this incompatibility, we only require equivalence for (u_ij , u_ji ) ∈ U such that u_ij + u_ji ≠ 0.

If only one part of the equivalence (6) holds for all (u_ij , u_ji ) ∈ U, the convention is systematically biased toward one language or the other.

Definition 2: If for any pair (u_ij , u_ji ) ∈ U such that u_ij + u_ji ≠ 0,

C (u_{i j}, u_{j i}) = B \Rightarrow C (- u_{i j}, - u_{j i}) = A then convention C is A-biased,

C (u_{i j}, u_{j i}) = A \Rightarrow C (- u_{i j}, - u_{j i}) = B then convention C is B-biased .

A convention is neutral if it is both A-biased and B-biased. It is clear that C_A is systematically A-biased, while C_B is systematically B-biased. Other examples are given in Figure 1 along with an example of a neutral convention and an example of a nonbiased convention, that is, a convention such that there are (u_ij , u_ji ), (v_ij , v_ji ) ∈ U with C(u_ij , u_ji ) = C(−u_ij , −u_ji ) = A and C(v_ij , v_ji ) = C(−v_ij , −v_ji ) = B.

A neutral convention can be characterized up to pairs (u_ij , u_ji ) such that u_ij + u_ji = 0 as shown by the following proposition (see the Appendix for proof).

Proposition 3: A convention C is neutral if and only if it meets the following conditions:

\begin{array}{l} (i) u_{i j} + u_{j i} < 0 \Rightarrow C (u_{i j}, u_{j i}) = A, \\ (i i) u_{i j} + u_{j i} > 0 \Rightarrow C (u_{i j}, u_{j i}) = B . \end{array}

Interestingly, a neutral convention can be seen as a utilitarian convention because it chooses the language that maximizes the sum of utilities. Furthermore, a neutral convention also satisfies the Rawlsian criterion because it selects the language in such a way that it benefits the speaker with the highest degree of preference (the one who would be worse off if her or his less preferred language were chosen).

Biased Choice of Language

Given a pair of speakers ( ${\vec{x}}_{i}$ , ${\vec{x}}_{j}$ ), the preferences in the conversation are determined by u_ij = u( ${\vec{x}}_{i}$ , ${\vec{x}}_{j}$ ) and u_ji = u( ${\vec{x}}_{j}$ , ${\vec{x}}_{i}$ ) as defined in equation (3). The language is chosen by applying a convention to (u_ij , u_ji ). The choice of language under a convention can thus be seen as a mechanism that associates the choice of a language with any pair of characteristics ( ${\vec{x}}_{j}$ , ${\vec{x}}_{i}$ ) ∈ X ².

The question of neutrality in the choice of the language can be addressed. The X ²-space is obviously neutral. The relevant question is whether the pair (− ${\vec{x}}_{i}$ , − ${\vec{x}}_{j}$ ) will always result in a different choice of language than the pair ( ${\vec{x}}_{i}$ , ${\vec{x}}_{j}$ ). The answer depends on the convention. It seems clear that the convention that is the most favorable to the majority language will result in a choice in favor of the majority language: Indeed, the preferences in conversation are A-biased. Can the convention in favor of the minority language offset the bias in preferences and result in a neutral choice of language?

To answer that question, a formal definition of neutral choice of language is needed. That definition is the natural extension of the definitions above: The language choice is neutral under a convention if it is A-biased and B-biased. Biases are defined using notation (5).

Definition 3: The choice of a language under convention C is

A-biased if for any pair ( ${\vec{x}}_{i}$ , ${\vec{x}}_{j}$ ) such that u_ij + u_ji ≠ 0 the following holds: C(u_ij , u_ji ) = B ⇒ C( ${\bar{u}}_{i j}$ , ${\bar{u}}_{j i}$ ) = A.

B-biased if for any pair ( ${\vec{x}}_{i}$ , ${\vec{x}}_{j}$ ) such that u_ij + u_ji ≠ 0 the following holds: C(u_ij , u_ji ) = A ⇒ C( ${\bar{u}}_{i j}$ , ${\bar{u}}_{j i}$ ) = B.

As shown in the following proposition, the answer to the question of whether a neutral choice of language can be obtained is no.

Proposition 4: Whatever the convention C, the choice of language under convention C is never B-biased.

The proof in the Appendix shows that the choice of language is never B-biased, as there are ( ${\vec{x}}_{i}$ , ${\vec{x}}_{j}$ ) with C(u_ij , u_ji ) = C( ${\bar{u}}_{i j}$ , ${\bar{u}}_{j i}$ ) = A. Note that a stronger property could be proved: For any ${\vec{x}}_{i}$ ∈ X, there is ${\vec{x}}_{j}$ ∈ X with C(u_ij , u_ji ) = C( ${\bar{u}}_{i j}$ , ${\bar{u}}_{j i}$ ) = A. That is, whatever the convention and whoever the speaker ${\vec{x}}_{i}$ , there is ${\vec{x}}_{j}$ such that the choice of language is the majority language in their conversation and in the conversation between − ${\vec{x}}_{i}$ and − ${\vec{x}}_{j}$ .

Preferences in conversations are A-biased, even if individual preferences are neutral. This bias cannot be offset by the convention, not even by the convention that is most favorable to the minority language, C_B . The choice of language under C_B is not B-biased and thus not neutral. By contrast, the choice of language under C_A is systematically A-biased. The following proposition (proof of which is in the Appendix) shows a stronger result: Other conventions result in an A-biased choice of language.

Proposition 5: The choice of language under any A-biased convention is A-biased.

One direct consequence of this proposition is that the choice of language under any neutral convention is systematically A-biased.

In a uniformly distributed population, both languages are individually preferred by the same number of speakers, but more individuals prefer the majority language in conversations. If conversations between pairs of individuals are random, all conventions select the majority language more often. The frequency depends on the distribution of the population within X and on the convention. We have this final proposition.

Proposition 6: For speakers ${\vec{x}}_{i}$ ≠ ${\vec{x}}_{j}$ ∈ X such that (u_ij , u_ji ) ≠ (0, 0), the following is found: for any convention C,

if min{m_i , m_j } + min{p_i , p_j } ≥ 0 then C(u_ij , u_ji ) = B,

if max{m_i , m_j } + min{p_i , p_j } < 0 then C(u_ij , u_ji ) = A.

The proof is omitted, as it is a direct consequence of Lemma 1 (see Appendix). Note that if −max{m_i , m_j } ≤ min{p_i , p_j } ≤ −min{m_i , m_j }, then u_ij ·u_ji ≤ 0 and the choice of the language depends on the convention.

Discussion

Our model shows that a minority language in contact with a majority language suffers from a disadvantage from a static point of view. This disadvantage comes from the externality originated by the interlocutor’s lack of proficiency as compared to one’s own attitude for the minority language. The speaker’s preference in a conversation crucially depends on her or his attitude for the minority language. The larger a speaker’s attitude is, the lower the proficiency that she or he may accept from a partner in a conversation. By contrast, a speaker with an individual preference for the majority language always prefers the majority language in any conversation.

The importance of attitude in minority language is well known among linguists. As put by Grenoble (2013:797), “Clearly more positive attitudes toward the language tend to strengthen its usage; and more negative attitudes to weaken it.” Our research provides an understanding of the mechanism that yields this effect. It also shows that a positive attitude is more important for those who have high proficiency levels (i.e., those who individually prefer the minority language) than for those who have low proficiency levels. In particular, it shows that the phenomenon depends on how the population is located in the spectrum of proficiency and attitude. Our research also suggests that the key element is preferences in conversations. Once these are formed and present a bias, no convention can offset this bias not even the convention that selects the minority language unless both individuals prefer the majority language. This result would still hold with more altruistic utility functions, in particular, if individuals take into account the partner’s attitude in their own utility function.⁸

Further work includes more sophisticated preferences and an application to case studies. Proficiency and attitude are not necessarily equally important nor perfect substitutes. Collecting data is not without difficulties as “there is usually no practical alternative to either collecting self report data about them via ‘scales’ or questionnaires” (see Fishman 1991:49). Our model does not preclude the possibility of introducing dynamics. For example, an individual may decide to learn the language or not and if the language is not used competence may decrease. We guess that the long-term effect of these dynamic elements will compound the (static) bias in favor of the majority language that we have identified here in each period of the dynamic setting. Nevertheless, a dynamic model might help suggest possible avenues for policy intervention. For example, as already suggested, a government could try to promote the language in order to increase motivation.

Footnotes

Appendix

This Appendix is devoted to proofs.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The paper was inspired by a working group at the department of Economics I of the University of the Basque Country. The authors benefitted from these discussions. Comments from Sergio Faria, Durk Gorter and Jose Ramon Urriarte are gratefully acknowledged. We also thank the suggestions made by an anonymous referee. The first author acknowledges the support of the Spanish Ministerio de Economía y Competitividad under project ECO2015-67519-P, and of the Departamento de Educación, Política Lingüística y Cultura from the Basque Government (Research Group IT568-13). The second author acknowledges financial support from IKERBASQUE, Basque Foundation for Science, through a Research Fellowship, and is grateful to Fundamentos de Análisis Económico I department at UPV/EHU for their hospitality and great research atmosphere. The third author acknowledges the support of the Advanced Grant NUMERIWAVES/FP7-246775 of the European Research Council Executive Agency, the BERC 2014-2017 program of the Basque Government, the FA9550-15-1-0027 of AFOSR, the MTM2014-52347 and MTM2011-29306 Grants and the Severo Ochoa program SEV-2013-0323 of the MINECO, and a Humboldt Award at the University of Erlangen-Nürnberg.

Notes

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