The cultural transmission of trust and trustworthiness

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The cultural transmission of trust and trustworthinessR Akira Okada Professor Emeritus, Hitotsubashi University, Japan

a r t i c l e

i n f o

Article history: Received 1 March 2019 Revised 19 October 2019 Accepted 31 October 2019 Available online xxx JEL classification: C72 D02 D64 D91

a b s t r a c t We consider the cultural transmission of trust and trustworthiness in a trust game with spatial matching. Players are assumed to enjoy psychological benefits from good conduct. The game has a unique equilibrium with different behavioral modes, depending on social distance between players. Parents rationally choose efforts to transmit their own values to their children. Trust and trustworthiness coevolve, and the transmission dynamics has a globally stable stationary point if the increasing rate of educational marginal costs is high. While “local” enforcement applied to transactions between closer partners deteriorates the diffusion of trust and trustworthiness, they are influenced by institutions in different ways. Better “intermediate” enforcement crowds out trust and crowds in trustworthiness. © 2019 Elsevier B.V. All rights reserved.

Keywords: Crowding effect Cultural transmission Random matching game Social distance Trust Trustworthiness

1. Introduction Trust and trustworthiness underlie almost all human relationships. During the last two decades, trust research has grown rapidly in economics beyond traditional approach of rational choice under selfish preference. Many empirical studies show that trust plays important roles in economic performance: economic growth (Knack and Keefer, 1997), government effectiveness (La Porta et al., 1997), financial development (Guiso et al., 2004), and international trade and investment (Guiso et al., 2009). The trust game (Berg et al., 1995) is employed as a standard framework for the study of trust and trustworthiness as the prisoner’s dilemma is so for that of cooperation. Given that trust and trustworthiness are fundamental for better economic outcomes, important questions are how the values of trust and trustworthiness are inherited between generations,1 and how one can increase them in a society. The aim of this paper is to answer these questions. Namely, we consider cultural transmission of trust and trustworthiness and their relationship to institutions with external enforcement.

R I would like to thank two anonymous reviewers for their very useful comments and suggestions. I am also grateful to Nozomu Muto, Takeshi Nishimura, Yasuhiro Shirata, and seminar participants at Waseda University and Game Theory Workshop 2019 for their valuable comments. E-mail address: [email protected] 1 Guiso et al. (2008) consider intergenerational transmission of beliefs about the trustworthiness of others.

https://doi.org/10.1016/j.jebo.2019.10.025 0167-2681/© 2019 Elsevier B.V. All rights reserved.

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Recent empirical studies indicate the relevancy of cultural transmission of trust in economics. Dohmen et al. (2012) provide empirical supports for intergenerational transmission of risk and trust attitudes. Based on the data from the 2003 and 2004 waves of the German Socio-Economic Panel Study, they find that these attitudes are strongly positively correlated between parents and children, after controlling personal and environmental characteristics. They also find a positive impact of prevailing attitudes in local environments on child attitudes. Their empirical findings indicate that socialization plays an important role in the transmission process beyond a genetic process.2 Guiso et al. (2006, p. 23) define culture as “those customary beliefs and values that ethnic, religious, and social groups transmit fairly unchanged from generation to generation,” and document a strong effect of ethnic origin on trust which an American has toward others. Using data on bilateral trust between European countries, Guiso et al. (2009) document that bilateral trust is affected by cultural factors such as the history of conflicts and religious similarities, and that lower bilateral trust leads to less trade and less portfolio and direct investment. Tabellini (2008a) documents that the value of generalized morality has widely diffused in societies that were ruled by non-despotic political institutions in the distant past. Bisin and Verdier (2001) present a model of intergenerational cultural transmission which incorporates two channels of transmission, “direct vertical” socialization inside family and “oblique” socialization by society. They show that if the two types of transmissions are “culturally substitute” so that parents have less incentives to educate their children the more widely dominant are their types in the population, then the dynamics of cultural transmission has a globally stable stationary point with heterogeneous types. Focusing on the vertical socialization, Tabellini (2008b) explicitly formulates parents’ educational efforts to transmit the good value of cooperation to their children. In the prisoner’s dilemma with spatial random matching, he proves that “good” parents with a higher level of generalized morality make positive efforts to educate their children and “bad” parents with a lower level of it do not. Based on the parents’ optimal educational choice, he further proves that better “local” enforcement which applies to transactions between socially close individuals in families, groups and communities hurts the diffusion of cooperation, while better “legal” enforcement applying to transactions between strangers enhances it. Adapting the framework of Tabellini (2008b), we consider the cultural transmission of trust and trustworthiness in a spatial random matching model of the trust game. We assume that players are intrinsically motivated to these good values inherited from earlier generations. Economic incentives are distinguished from individual values of the good conduct. Preferences which determine individual behavior are influenced by incentives and values. The model captures the distinction of general trust (trust in others in general) and knowledge-based trust stressed by social psychologists (Yamagishi et al., 1998; Yamagishi and Yamagishi, 1994). While general trust can be applied to everyone over a large range of social distance, knowledge-based trust is limited to close partners such as families and group members with committed relationships. This paper is distinguished from the work of Tabellini (2008b) in two critical ways. First, trust and trustworthiness are different concepts as many empirical studies show (Glaeser et al., 20 0 0; Buchan et al., 20 02; Cox, 20 04; Ashraf et al., 20 06; Bohnet and Baytelman, 2007; Kiyonari et al., 2006; Ben-Ner and Halldorsson, 2010, for example). They are not identical to the value of cooperation. Trust and trustworthiness interrelate. In contrast to Tabellini (2008b), we consider the coevolution of these two cultural traits. Second, trust and trustworthiness involve different incentive structures. A trustor is willing to trust a trustee if he or she is expected to be trustworthy. On the other hand, the trustee is economically motivated to cheat the partner’s trust. This implies that the trust game does not possess a strategic complementarity between values and behavior, which plays an important role in the analysis of Tabellini (2008b). By this reason, the result of Tabellini (2008b) cannot be directly related to a large volume of literature of empirical trust studies employing the trust game framework. We summarize the results of the paper. First, we characterize an equilibrium of a trust game with spacial matching. An equilibrium configuration is complex, compared to that in the prisoner’s dilemma. Depending on social distance between players, a unique equilibrium of the trust game has nine behavioral modes in general, owing to different incentives of trust and trustworthiness. Two thresholds of social distance for trust and trustworthiness play a critical role in the analysis. Second, we consider the cultural transmission of trust and trustworthiness. Trust and trustworthiness coevolve endogenously over time. We show that the transmission dynamics has a globally stable stationary point if the increasing rate of educational marginal costs is high. Third, we consider how institutions which externally enforce economic transactions affect the diffusion of trust and trustworthiness. Extending the result of Tabellini (2008b), we show that the strengthening of local and legal enforcement crowds out and crowds in trust and trustworthiness, respectively, and moreover that different institutions affect the diffusion of the two good values in different ways. Better “intermediate” enforcement crowds out trust but crowds in trustworthiness. The remainder of the paper is organized as follows. Section 2 presents the trust game with spatial matching. Section 3 characterizes an equilibrium of the game. Section 4 examines the cultural transmission of trust and trustworthiness. Section 5 considers the effects of institutions on them. Section 6 discusses some empirical works consistent with our results. All proofs are given in Appendix.

2 There exists a considerable volume of the literature on genetic transmission of trust. For example, Kosfeld et al. (2005) provide neurobiological evidence of trust using oxytocin. Cesarini et al. (2008) find that genetic differences play a significant role for behavior in the trust game experiment played by monozygotic and dizygotic twins.

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Table 1 Material payoffs in the trust game.

C N

R

N

βγ ω, (1 − β )γ ω ω, 0

0, γ ω ω, 0

Table 2 Trust game with psychological payoffs. R C N

βγ ω + d1 e ω , d 2 e −θ2 y

N −θ1 y

, ( 1 − β )γ ω + d2 e

−θ2 y

d 1 e −θ1 y , γ ω ω, 0

2. The model We consider a random matching model of the trust game (Berg et al., 1995).3 There are two types of players called an investor (player 1) and a receiver (player 2). Let I = {1, 2}. A continuum of each type of players is uniformly distributed on the circumference of a circle of size 2S. The maximum distance of two players is S. A player is randomly matched with the other player at distance y with a probability density function g(y) > 0. The probability that a pair of an investor and a S receiver is matched between 0 and S is equal to one, that is, 0 g(y )dy = 1.4 Distance y could be not only geographical one but also socioeconomic differences. Examples include kinship, ethnicity, language, religion, and class. A larger circle with higher S implies a more heterogeneous society in terms of social distance. Two matched players observe their distance y and play the trust game as follows. An investor (trustor) decides whether to send the endowment ω > 0 to a receiver (trustee) or not. The investment gives the receiver a return γ ω where γ > 1. The receiver decides whether to repay a fixed portion, β ∈ (0, 1), of γ ω to the investor, or not. If repayment is made, then the investor obtains material payoff βγ ω and the receiver obtains (1 − β )γ ω. If not, the receiver exploits the whole return γ ω. If the investor does not send the endowment, then the investor’s payoff is ω and that of the receiver is zero. The material payoffs of two players are summarized in Table 1.5 In Table 1, an investor (a row player) has two actions C and N which mean “invest” and “not invest”, respectively, and a receiver (a column player) has two actions R and N which mean “repay” and “not repay”, respectively. It is natural to interpret the investment (C) as the trust that an investor places in a receiver, and the repayment (R) as a trustworthy action of the receiver (Glaeser et al., 20 0 0).6 In what follows, parameters ω, γ and β are fixed. The return rate to an investor is either βγ or 0, depending on whether the receiver repays. We assume that βγ > 1 so that an investor is better off by repaid investment than by no investment. The trust game in Table 1 has only the Nash equilibria where the investor chooses N and the receiver may choose N with any probability not less than 1 − 1/(βγ ). Thus, the maximization of material payoffs cannot explain positive investment and repayment which have been observed in most experiments of one-shot trust games. Recent experimental studies show that social preferences play a key role in trusting behavior (Fehr, 2009). Sapienza et al. (2013) argue that the quantity of money sent in the trust game is a combination of two components: preferences (risk aversion, inequality aversion, altruism) and beliefs in receivers’ trustworthiness. By cross-cultural experiments in Russia, South Africa and the United States, Ashraf et al. (2006) show that trust is based on beliefs on trustworthiness and on unconditional kindness, and that trustworthiness is related to unconditional kindness and reciprocity. In view of the experimental results mentioned above, we assume that individuals are motivated not only by material payoffs but also by the value of good conduct. Specifically, following Tabellini (2008b), we assume that besides the material payoffs, each player i ∈ I enjoys a noneconomic (psychological) benefit di when he or she plays the good conduct (C and R), irrespective of an opponent’s action. The noneconomic benefit of player i decays with distance at exponential rate θ i > 0. Player i enjoys noneconomic benefit di e−θi y by the good conduct in a match with distance y. The material and psychological payoffs of two players are summarized in Table 2.

3

While Berg et al. (1995) give the name of an investment game, we call it a trust game, following the literature. This model of random matching over a circle is originally considered by Dixit (2004) in the context of the repeated prisoner’s dilemma. 5 In the experiment of Berg et al. (1995), parameters are such that ω = 10 and γ = 3. Choice variables are continuous. An investor chooses any investment amount x with 0 ≤ x ≤ ω and a receiver does any amount y with 0 ≤ y ≤ γ ω for return. A large volume of experimental studies show that, on average, investors send about 50 percent and receivers repay about 95 percent of what has been invested (Camerer, 2003). Besides the original version of the trust game, binary-choice versions of it are frequently used in experimental studies (Bohnet and Zeckhauser, 2004; Ermisch and Gambetta, 2010, for example). The result of this paper is not affected in any critical manner if an investor has an option to invest a fixed portion of the endowment ω, or nothing. 6 There, however, is no consensus on the proper definition of trust in the literature (Fehr, 2009). Recent literature examines whether a positive investment in the trust game appropriately measures the trust of an investor, and whether a positive repayment does the trustworthiness of a receiver. See Bohnet and Zeckhauser (20 04), Cox (20 04), Ashraf et al. (2006), Kiyonari et al. (2006), Fehr (2009), Ben-Ner and Halldorsson (2010) and Sapienza et al. (2013) among others. 4

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The formulation of individuals’ preference above reflects the view that the intrinsic values of trust and trustworthiness depend on social connection. Within a circle of more familiar individuals, the norm of trust and trustworthiness can be applied with stronger force, and it becomes weaker among less familiar individuals. Coleman (1990) discusses that familiarity breeds trust. Individuals trust others more familiar to themselves than those unfamiliar. There exists a large body of evidence consistent with the assumption that the values of good conduct are strengthened by social connectedness. Glaeser et al. (20 0 0) present experimental evidence that social connectedness rises both the amount sent and returned in the trust game. Based on the General Social Survey (GSS) of the United States in the years 1974-94, Alesina and La Ferrara (2002) find that interpersonal trust is lower in more racially heterogeneous communities and in those with higher income inequality. They argue that the effect of heterogeneity on trust is largely due to the fact that individuals trust those more similar to themselves. Hoffman et al. (1996) find that decreasing social distance increases other-regarding behavior in the dictator game. For parameter values, we assume:

d1 > ω, d2 > βγ ω > ω.

(A1)

This assumption means that when two individuals play the trust game at zero distance (y = 0 ), C is the dominant action of an investor and R is that of a receiver, and thus the trust game has a unique Nash equilibrium (C, R). The introduction of psychological payoffs at a positive distance y > 0 changes the strategic property of the trust game in Table 1 so that the receiver prefers R to N for any value of y if the investor chooses N. This means that all Nash equilibria under material payoffs (the investor chooses N for sure and the receiver may choose N with positive probability not less than 1 − 1/βγ ) disappear. In every Nash equilibrium under psychological payoffs, an investor may choose C with a positive probability because the investor prefers C to N if the receiver chooses R. We assume that there are two types of each player, a “good” type (g) and a “bad” type (b). Let J = {g, b}. Both types enjoy the same psychological benefit d of choosing good conduct, but they differ in their rates at which the benefit decays with j g distance. For each i ∈ I and j ∈ J, let θi be the decaying rate for player i of type j. We assume that 0 < θi < θib . For any positive distance, a good type values the good conduct more than a bad type. The two types of an individual are introduced to capture the distinction of general trust and knowledge-based trust. According to Yamagishi and Yamagishi, 1994, knowledge-based trust is limited to particular objects (people or organizations). Individuals may have a strong belief in the benevolence of their partners based on the knowledge obtained in long-term and/or committed relationships such as family, friends and groups. Knowledge-based trust can be interpreted as assurance to be a perception of the partner’s incentive leading to cooperative actions. In contrast, general trust is a belief in the benevolence of human nature in general and is not limited to particular objects. In the model, good types are assumed to have a higher degree of general trust than bad ones. We are mainly concerned with the cultural transmission of general trust and its relationship with institutions. In the population of each player i ∈ I, let ni ∈ (0, 1) denote the fraction of good types at any point in the circle. When individuals play the trust game, they observe distance y but not types of their partners. In the next section, we will characterize a Nash equilibrium of the trust game with random matching when the fractions ni are fixed for all i ∈ I. 3. Equilibrium In the trust game with random matching, a mixed strategy for player i ∈ I of type j ∈ J is a function σi from [0, S] to j

[0, 1] where σi (y ) is the probability that he or she may choose the good conduct in a match with distance y. For a mixed j

strategy profile (σi )(i∈I, j∈J ) , the average probability π i (y) that an individual in the population of player i chooses the good conduct is given by j

πi (y ) = ni σig (y ) + (1 − ni )σib (y ).

(1)

Since players observe distance y, a Nash equilibrium of the random matching game is composed of those of the trust game in Table 2, given y. j

We first consider the optimal behavior of an investor. An investor of type j ∈ J obtains payoff βγ ωπ2 (y ) + d1 e−θ1 y by playing C, and obtains payoff ω, otherwise. Thus, an investor of type j optimally plays C if

βγ ωπ2 (y ) + d1 e−θ1 y > ω. j

(2)

(2) is rewritten as7

y < T1j (π2 (y )) :=

1

θ1j

ln

d1

ω (1 − βγ π2 (y ))

.

(3)

Since θ1 < θ1b , (3) holds for a good type whenever it holds for a bad type. This means that whenever a bad type chooses C with positive probability, a good type chooses C with probability one.8 Thus, in a pure-strategy equilibrium, the investor’s g

If 1 − βγ π2 (y ) < 0, we set T1j (π2 (y )) = +∞ for every j. In this case, an investor of every type chooses C against a receiver at any distance. If the equality holds in (3), then an investor is indifferent between playing C and N. In this case, he or she may choose C with a positive probability in equilibrium. 7 8

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5

Fig. 1. Trust and trustworthiness threshold functions.

average probability π 1 (y) of good conduct may take three values, 0, n1 , and 1, which correspond to the three cases: (i) no types invest, (ii) only the good types invest, and (iii) all types invest, respectively. In a mixed-strategy equilibrium, there are two possible cases, 0 < π 1 (y) < n1 and n1 < π 1 (y) < 1. In the first case, only good types may invest with positive probability and bad types do not. In the second case, good types invest with probability one and bad types may invest with j positive probability. Given the equilibrium probability π 1 (y), the mixed strategies σ1 (y ) for types j are determined by (1). j

We next consider the optimal behavior of a receiver. A receiver of type j ∈ J obtains payoff (1 − β )γ ωπ1 (y ) + d2 e−θ2 y by playing R and obtains payoff γ ωπ 1 (y), otherwise. Thus, a receiver of type j optimally plays R if

(1 − β )γ ωπ1 (y ) + d2 e−θ2 y > γ ωπ1 (y ). j

(4) is rewritten

(4)

as9

y < T2j (π1 (y )) :=

1

θ

j 2

ln

d2

βγ ωπ1 (y )

.

(5)

By the same arguments as in the case of an investor, it can be shown that the receiver’s average probability π 2 (y) of good conduct may take three values, 0, n2 , and 1 in a pure-strategy equilibrium, and may take two values, 0 < π 2 (y) < n2 and n2 < π 2 (y) < 1, in a mixed-strategy equilibrium. j j We call T1 (π2 ) and T2 (π1 ) in (3) and (5) the trust threshold function of an investor and the trustworthiness threshold function of a receiver, respectively. These functions describe the upper limit of social distance so that players behave in the good manner, depending on the average probability that a partner behaves in the good manner. If a distance is smaller than the threshold, then players are willing to behave in the good manner. If the threshold is larger than the size S of a society, then players choose the good conduct, regardless of a distance. The two threshold functions play a critical role in the analysis. Their graphs are depicted in Fig. 1. The trust threshold j function T1 (π2 ) of an investor is monotonically increasing in the average probability π 2 that a receiver may reciprocate. If

π 2 is larger than

1 βγ , T1 (π2 ) becomes the infinity. In contrast, the trustworthiness threshold function T2 (π1 ) of a receiver is j

j

monotonically decreasing in the average probability π 1 that an investor may trust him or her. These functions are affected by the change of the return rate βγ to an investor in the opposite way. The functions T1 and T2 shift upward and downward, respectively, as βγ rises. The opposite monotone properties of the threshold functions for trust and trustworthiness shown in Fig. 1 can be explained as follows. An investor bears the risk of trusting a (possibly untrustworthy) receiver. The investor’s net material payoff, ω − βγ ωπ2 , from not trusting is monotonically decreasing in the average probability, π 2 , that a receiver may be j

trustworthy. An investor trusts a receiver if the psychological benefit, d1 e−θ1 y , from trusting exceeds it. It implies that an investor trusts a receiver in greater distance as π 2 increases. On the other hand, a receiver is economically motivated to cheat an investor. The receiver’s net material payoff, βγ ωπ 1 , from cheating is monotonically increasing in the average probability,

9

If π1 (y ) = 0, we set T2j (π2 (y )) = +∞ for every j. In this case, a receiver of every type chooses R against an investor at any distance.

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A. Okada / Journal of Economic Behavior and Organization xxx (xxxx) xxx Table 3 Equilibrium actions where “m” stands for playing a mixed strategy.

good investor bad investor good receiver bad receiver

1

2

3

4

5

6

7

8

9

C C R R

C C R N

C C N N

C m m N

C m R m

C N R N

C N N N

m N m N

m N R m

j

π 1 , that investment may be made. A receiver behaves in a trustworthy manner if the psychological benefit, d2 e−θ2 y , from the good conduct exceeds it. Thus, a receiver reciprocates an investor in shorter distance as π 1 increases. Corresponding to the three possible values, 0, ni and 1, of the equilibrium average probability π i (y) of the good conduct by player i, the trust and trustworthiness thresholds are given as follows.10 For every j ∈ J,

T1j (0 ) =

1

θ1j

ln

T2j (0 ) = +∞

d1

ω

T1j (n2 ) =

T2j (n1 ) =

1

θ2j

1

θ1j

ln

ln

d1

T1j (1 ) = +∞

ω (1 − βγ n2 )

d2

βγ ωn1

T2j (1 ) =

1

θ2j

ln

(6)

d2

(7)

βγ ω

The following theorem characterizes a Nash equilibrium in the trust game with random matching. Theorem 1. There exists a unique equilibrium in the trust game with random matching. The equilibrium probabilities of the good conduct by two players are characterized as follows.11

1.

π1 ( y ) = 1 , π2 ( y ) = 1

for

0 ≤ y ≤ T2b (1 )

2.

π1 ( y ) = 1 , π2 ( y ) = n 2

for

T2b (1 ) < y ≤ (T1b (n2 ), T2g (1 ))−

3.

π1 ( y ) = 1 , π2 ( y ) = 0

for

4.

n 1 < π1 ( y ) < 1 , 0 < π 2 ( y ) < n 2

for

5.

n 1 < π1 ( y ) < 1 , n 2 < π2 ( y ) < 1

for

(T1b (n2 ), T2b (1 ))+ < y < T2b (n1 )

6.

π1 ( y ) = n 1 , π2 ( y ) = n 2

for

(T1b (n2 ), T2b (n1 ))+ < y ≤ (T1g (n2 ), T2g (n1 ))−

7.

for

8.

π1 ( y ) = n 1 , π2 ( y ) = 0 0 < π1 ( y ) < n 1 , 0 < π2 ( y ) < n 2

for

(T1b (0 ), T2g (n1 ))+ < y ≤ T1g (0 ) (T1g (0 ), T2g (n1 ))+ < y < T1g (n2 )

9.

0 < π1 ( y ) < n 1 , n 2 < π2 ( y ) < 1

for

(T1g (n2 ), T2b (n1 ))+ < y ≤ S.

T2g (1 ) < y ≤ T1b (0 )

(T1b (0 )

, T2g

(1 ))+ < y < (T1b (n2 ), T2g (n1 ))−

The theorem shows that there exist nine cases regarding the equilibrium probabilities that investors and receivers behave in the good manner, depending on social distance between them. The equilibrium configuration is summarized in Table 3. When social distance is sufficiently small, investors trust receivers, who reciprocate their trust. This property is the direct result of the assumption that individuals internalize the norm of good conduct more strongly for partners in closer distances. The trust game exhibits a complex equilibrium pattern compared to the prisoner’s dilemma.12 A mixed-strategy equilibrium possibly prevails as distance becomes far. In particular, when distance is very far, a good investor may trust a receiver only with a positive probability. The risk of being cheated causes the investor to behave cautiously. A detailed explanation of each case is given below. The nine cases are divided into four phases. Phase 1 (cases 1, 2, and 3): A pure-strategy equilibrium prevails. Social distance between an investor and a receiver is close. Every type of the investor trusts the receiver. The receiver behaves in the trustworthy manner with decreasing probabilities, 1, n2 , and 0, as distance increases. The intuition for this behavioral mode can be explained as follows. Since the distance y is sufficiently close (0 < y ≤ T1b (0 )), every type of the investor chooses C owing to the psychological payoff, even if the receiver does not repay (π2 (y ) = 0 ). Since the same thing holds if the receiver repays, C is the dominant action of the investor, namely, he or she is better off by choosing C rather than N, regardless of how the receiver behaves. Given that the investor chooses g C, all types of the receiver choose R if y ≤ T2b (1 ) (case 1), only good types choose R if T2b (1 ) < y ≤ T2 (1 ) (case 2), and no g types choose R if T2 (1 ) < y (case 3). Social distances in phase 1 correspond to those in families, working places, and local communities where individuals strongly internalize the norm of good conduct. If 1 − βγ n2 < 0, then we set T1j (n2 ) = +∞. See footnote 7. We use notations (a, b)+ = max{a, b} and (a, b)− = min{a, b} for two real numbers a and b. 12 Tabellini (2008b) shows that the (Pareto superior) equilibrium of the prisoner’s dilemma with random matching has three cases. With two thresholds, Y0 and Y1 , for a bad type and for a good type, both types cooperate in a match with distance y if 0 ≤ y ≤ Y0 , and only good types cooperate if Y0 < y ≤ Y1 and no types cooperate if Y1 < y ≤ S. 10 11

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Phase 2 (cases 4 and 5): A mixed-strategy equilibrium prevails. The reason for this can be explained as follows. Distance y is so large to exceed the trustworthiness threshold T2b (1 ) for a bad receiver when investment is made. The receiver is motivated to cheat the investor’s trust. This means that the action pair (C, R) is not an equilibrium in the trust game in Table 2. Similarly, a bad investor is motivated not to trust an untrustworthy receiver since the distance exceeds the trust threshold T1b (0 ). Accordingly, the action pair (C, N) is not an equilibrium, either. Thus, there exists no pure-strategy equilibrium in the trust game. g g The equilibrium probability of the good conduct is determined as follows. Consider first case 4. Because T2 (1 ) < y < T2 (n1 ) g and the trustworthiness threshold function T2 (π1 ) is monotonically decreasing, there exists some π 1 ∈ (n1 , 1) satisfying g y = T2 (π1 ). The average probability that an investor may choose C is given by π 1 . Since π 1 > n1 , a good investor chooses C for sure, and a bad investor plays a mixed strategy. Similarly, because T1b (0 ) < y < T1b (n2 ) and the trust threshold function T1b (π2 ) is monotonically increasing, there exists some π 2 ∈ (0, n2 ) satisfying y = T1b (π2 ). The average probability that a receiver may choose R is given by π 2 . Since π 2 < n2 , a good receiver plays a mixed strategy, and no bad receiver chooses R. Case 5 is analysed in the same way as in case 4. Since T2b (1 ) < y < T2b (n1 ), there exists some π 1 ∈ (n1 , 1) such that y = T2b (π1 ). The average probability that an investor may choose C is given by π 1 . A good investor chooses C for sure, and a bad investor plays a mixed strategy. Similarly, because T1b (n2 ) < y, there exists some π 2 ∈ (n2 , 1) such that y = T1b (π2 ). The average probability that a receiver may choose R is π 2 . A good receiver chooses R for sure, and a bad receiver plays a mixed strategy. Phase 3 (cases 6 and 7): A pure-strategy equilibrium prevails. The equilibrium probability of the investor’s trusting decreases to n1 . In case 6, g since T1b (n2 ) < y ≤ T1 (n2 ), only good investors choose C if the receiver may choose R with probability n2 . Further, since g b T2 (n1 ) < y ≤ T2 (n1 ), only good receivers choose R if the investor may choose C with probability n1 . Thus, case 6 has the equilibrium probability pair (n1 , n2 ) where only good types of both players follow the norm of the good conduct. Similarly, in g g case 7, since T1b (0 ) < y ≤ T1 (0 ), only good investors choose C if no receiver chooses R. Further, since T2 (n1 ) < y, no receiver chooses R if the investor may choose C with probability n1 . Thus, case 7 has the equilibrium probability pair (n1 , 0) where only good investors follow the norm of the good conduct. Phase 4 (cases 8 and 9): A mixed-strategy equilibrium prevails. Social distance is far between an investor and a receiver, and thus the norm of g g good conduct is weak. In case 8, it holds that T2 (1 ) < y since T2 (n1 ) < y. Every type of the receiver is motivated to cheat the g investor’s trust. In turn, since T1 (0 ) < y, every type of the investor is motivated not to trust such an untrustworthy receiver. There exists a cycle of best responses in the trust game in Table 2: (C, R) → (C, N) → (N, N) → (N, R) → (C, R). Therefore, the trust game has no pure-strategy equilibrium. The equilibrium probability of the good conduct is determined in the same g g way as in phase 2. Consider first case 8. Since T2 (n1 ) < y, there exists some π 1 ∈ (0, n1 ) satisfying y = T2 (π1 ). On average, an investor may trust a receiver with probability π 1 . A good investor plays a mixed strategy, and a bad investor does not g g g trust a partner. Since T1 (0 ) < y < T1 (n2 ), there exists some π 2 ∈ (0, n2 ) satisfying y = T1 (π2 ). A receiver may repay with probability π 2 . A good receiver plays a mixed strategy, and a bad receiver does not repay. Case 9 is analysed in the same way as in case 8. Since T2b (n1 ) < y, there exists some π 1 ∈ (0, n1 ) satisfying y = T2b (π1 ). An investor may choose C with g probability π 1 . A good investor plays a mixed strategy, and a bad investor does not trust a receiver. Since T1 (n2 ) < y, there g exists some π 2 ∈ (n2 , 1) satisfying y = T1 (π2 ). A receiver may choose R with probability π 2 . A good receiver repays for sure, and a bad receiver plays a mixed strategy. A large number of experimental studies support the prediction of Theorem 1 that trust and trustworthiness can be maintained more likely among socially closer partners. Glaeser et al. (20 0 0) report experimental evidence that when individuals are socially closer (in the number of friends in common and the duration of their acquaintanceship), both levels of trust and trustworthiness rise. Specifically, trustworthiness declines when partners are of different races or nationality, even after controlling social connection. In their experiments, eleven out of the twelve times with no money returned, partners were of different races. Employing an economically relevant measure of social distance, Etang et al., 2011 conduct trust game experiments in two villages in rural Cameroon. They found that significantly more money was sent when players lived in the same village, although levels of trust and trustworthiness towards the other village were still high. Gender, education and membership of rotating credit groups also influenced transfers. Many experiment studies in social psychology show that even the ad hoc categorization (minimal-group condition) of individuals affect their cooperative behavior, in general. In cross-national experiments of the trust game, Buchan et al. (2002) show that the level of trust is higher among “neighbors” than among “strangers” where the group identity of neighborhood is experimentally manipulated.

4. Cultural transmission In this section, we consider how the values of trust and trustworthiness evolve through cultural transmission from parents to children. Parents rationally choose values to transmit to their children by a form of paternalistic altruism called “imperfect empathy”, which means that parents care material payoffs of their children but evaluate the expected welfare of the children with their own values (Bisin and Verdier, 2001). Parents attempt to socialize their children to their own Please cite this article as: A. Okada, The cultural transmission of trust and trustworthiness, Journal of Economic Behavior and Organization, https://doi.org/10.1016/j.jebo.2019.10.025

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preference trait. Dohmen et al. (2012) provide empirical evidence for intergenerational transmission of risk and trust attitudes based on the German Socio-Economic Panel Study. The trust game with random matching is repeatedly played over infinitely many periods. Each period has two generations, children and parents, in two populations of investors and receivers. In the first period of their life, children are educated by their parents. The education increases the probability that a child becomes a good type. Thereafter, children play the trust game, observing their own types. In the second period of their life, each player becomes a parent of a single child and devotes effort to education. Education is costly. We analyse how the proportions of types in the two populations are evolved through parents’ education. Given educational efforts ei ≥ 0 by parents in population i ∈ I, children will become a good type with probability δi + ei where 0 < δi + ei < 1. Parameter δ i > 0 reflects the natural growth rate of a good type child under no education. For each j j ∈ J, let ei denote efforts by parents of type j. Then, by the Law of Large Numbers, the proportion of a good type child, ni,t , in generation t will be changed to

ni,t+1 = (δi + egi )ni,t + (δi + ebi )(1 − ni,t ).

(8)

In this formulation, we rule out the possibility that bad type parents exert effort to transmit their own “bad” value to children. We will later prove (Proposition 1) that only good parents devote positive effort to transmit the good value to children, and thus that bad parents exert no effort. A child of a bad parent grows at the natural rate δ to become a good type. At the end of this section, we will discuss how the result changes if we allow bad parents to exert effort to transmit their own value to children. To analyse the dynamics of cultural transmission (8), we first consider the parents’ optimal educational choice where their children behave according to the equilibrium of Theorem 1. In the population of investors (player 1), let σ1i (y ) be the probability that a child of type i ∈ J conducts in the good

manner in a match with distance y, and π 1 (y) be the average one that a child conducts in the good manner. Let V1 denote ij

the type i parent’s evaluation of having a type j child. Under the assumption of imperfect empathy,

V1i j =



S 0

ij V1

is given by

{u1j (σ1j (y ), π2 (y )) + d1 σ1j (y )e−θ1 y }g(y )dy i

(9)

where u1 (σ1 (y ), π2 (y )) is the material payoff that a child of type j receives in a match with distance y when a partner may j

j

conduct in the trustworthy manner with probability π 2 (y). The parent’s evaluation V1 implies that the parent cares about the child’s material payoff in the same way as a child, but that parents value the good conduct (trust) in their own way, ij possibly different from the children’s. For the population of receivers (player 2), the type i parent’s evaluation, V2 , of having a type j child is defined in the same way as (9). The rule of cultural transmission implies that parents are motivated to educate their children if they evaluate a good child higher than a bad one. The next proposition shows the parent’s educational choice. ij

Proposition 1. In every population i ∈ I, a “good” parent devotes a positive effort to educating a child. A “bad” parent exerts no effort. gg

The intuition for this result can be explained with help of Table 3. While the proposition shows that Vi Vibb

bg Vi ,

gb

> Vi

and

> namely, that parents evaluate children who are of the same types as theirs higher than those of the different types in terms of the expected evaluations over the whole range [0, S] of social distance, it actually shows that parents weakly prefer having children of the same types as theirs in all matches in [0, S], and strictly do so in some matches. Let us evaluate the equilibrium outcome of the trust game played by children in each match in Table 3 as follows. Consider first the population of investors. In cases 1, 2 and 3, good and bad children behave in the same way, playing C. Thus, parents are indifferent to a type of their children. In cases 4 and 5, a good child chooses C and a bad child plays a mixed strategy. This implies that a good child prefers playing C rather than playing the mixed strategy. Since good parents have preferences aligned with good children, they prefer having a good child who plays C rather than a bad child who plays the mixed strategy. On the other hand, bad children are indifferent between playing C and N since they play a mixed strategy in equilibrium. This further implies that bad children are indifferent between playing C and the mixed strategy. Since bad parents have preferences aligned with bad children, they are indifferent between having a bad child who plays the mixed strategy and having a good child who plays C. In cases 6 and 7, good children choose C and bad children do N. By the same reasoning as in cases 4 and 5, good parents prefer having good children, and bad parents have the opposite preference. In cases 8 and 9, good parents are indifferent to a type of their children, and bad parents prefer having bad children. Consider next the population of receivers. The same reasoning as in the case of investor parents can be applied. In cases 1, 3, and 7, parents are indifferent to types of their children because good and bad children behave in the same way in equilibrium. In cases 2 and 6, a good child chooses R and a bad one does N. Thus, good parents prefer having good children, and bad parents have the opposite preference. In cases 4 and 8, a good child plays a mixed strategy, and thus good parents are indifferent to types of their children. In contrast, a bad child chooses N, and thus bad parents prefer having bad children. Finally, in cases 5 and 9, a good child chooses R and a bad one plays a mixed strategy. Thus, good parents prefer having good children, and bad parents are indifferent to types of their children. Please cite this article as: A. Okada, The cultural transmission of trust and trustworthiness, Journal of Economic Behavior and Organization, https://doi.org/10.1016/j.jebo.2019.10.025

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9

We now consider the optimal effort level for education. We assume that a good parent incurs guadratic costs re2 where e > 0 is educational effort. Cost parameter r > 0 captures the increasing rate of marginal costs with respect to effort e. In each population i ∈ I, good parents choose effort ei to maximize the expected evaluation to educate their children,

(δi + ei )Vigg + (1 − δi − ei )Vigb − re2i ,

(10)

given the effort chosen by all other parents. The first-order optimality condition is given by

2rei = Vigg − Vigb > 0.

(11)

Let ei = ei (n1,t , n2,t ) denote the optimal educational effort of a good parent in population i ∈ I in generation t. In general, the effort level ei in each population depends on proportions of good type individuals, n1t and n2t , in the two populations. A child turns out to be a good type with probability δi + ei if cost parameter r is sufficiently small so that 0 < δi + ei < 1, while a child of a bad parent becomes a good type only with probability δ i . The dynamics (8) of the fraction of good children in population i ∈ I is given as

ni,t+1 = (δi + ei (n1,t+1 , n2,t+1 ))ni,t + δi (1 − ni,t ) = δi + ei (n1,t+1 , n2,t+1 )ni,t .

(12)

According to (12), the good values of trust and trustworthiness coevolve through strategic interactions in the process of cultural transmission. The next theorem shows that the dynamics (12) has a globally stable stationary point if the increasing rate r of marginal educational costs is so large that the parent’s optimal effort ei (n1,t , n2,t ) is small for every n1,t and n2,t . Theorem 2. For sufficiently large r, the cultural transmission dynamics (12) in population i ∈ I has a unique stationary point n∗i which is globally stable. The stationary point satisfies n∗i =

δi

1−ei (n∗1 ,n∗2 )

.

The proof of the theorem uses the fact that the dynamic system (12) defines the contraction mapping from [0, 1] × [0, 1] to itself for sufficiently large r. The two values of trust and trustworthiness coevolve through the education of parents who rationally choose their efforts, anticipating interactions of their children in future. These values globally converge to the long run equilibrium where good and bad individuals coexist. Starting from an initial point where there are few good individuals in the populations of investors and receivers, the fractions of good types monotonically increase. The upward (downward) shift of the parents’ optimal educational effort in (11) rises (falls) the long run level of good values.13 To conclude the section, we discuss how the analysis can be extended to the case that bad parents attempt to transmit their bad value to children. By the same argument as that of the optimal effort level of good parents given in (11), the optimal effort level fib of bad parents in population i ∈ I is given by

2r fib = Vibb − Vibg > 0.

(13)

Then, the dynamics (12) of cultural transmission is modified to

ni,t+1 = (δi + egi )ni,t + (δi − fib )(1 − ni,t ) = δi − fib + (egi + fib )ni,t .

(14)

(14) shows how the fraction of good individuals changes, depending on the efforts by different types of parents. If f ib = 0, g g g then (14) is reduced to (12). Since the absolute values of ei , fib , ∂ ei /∂ ni,t+1 , ∂ ei /∂ n j,t+1 , ∂ fib /∂ ni,t+1 and ∂ fib /∂ n j,t+1 are sufficiently small for large r > 0, it can be shown that (14) has the same dynamic property as (12). Theorem 2 holds in the situation that parents attempt to transmit their own values to children. The long run equilibrium satisfies n∗i = g

δi − fi∗ 1−e∗i − fi∗

where e∗i = ei (n∗1 , n∗2 ) and fi∗ = fib (n∗1 , n∗2 ). The long run equilibrium of the fraction of good individuals, n∗i , in population i may or may not exceed the natural growth rate δ i , depending on the magnitudes of good and bad parents’ efforts e∗i and fi∗ . It exceeds δ i if the effort ratio e∗i / fi∗ is greater than (1 − δi )/δi . 5. Trust and institution Tabellini (2008b) investigates whether good values of cooperation are crowded in or out by better external enforcement, in the context of the prisoner’s dilemma. The quality of external enforcement is formulated as the probability of cheating being detected. Comparing two different, local and legal, types of enforcement, which are applied to neighboring and distant matches, respectively, he argues that better legal enforcement contributes the diffusion of good values, but that better local enforcement destroys good values. The reason for this result is that good players cooperate and bad players do not over a larger range of matches under better legal enforcement, and thus parents devote more efforts to education. In contrast, when local enforcement is strengthened, the threshold of cooperation for bad players increases, and thus the range of matches where bad and good types behave differently shrinks. As a result, parents put less efforts to education. In this section, we examine whether or not the insight of Tabellini (2008b) can be extended to the values of trust and trustworthiness. 13 Replacing imperfect empathy with full altruism, Tabellini (2008b) considers the cultural transmission by parents who care about children’s welfare evaluated by the children themselves. He shows that all (good and bad) parents devote positive efforts to educating their children since the expected utility of a good child is always higher than that of a bad child. It can be shown that the same result holds in the trust game.

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A. Okada / Journal of Economic Behavior and Organization xxx (xxxx) xxx Table 4 Equilibrium actions under (A2) and (A3) where “m” stands for playing a mixed strategy.

good investor bad investor good receiver bad receiver

1

2

3

4

C C R R

C C R N

C m m N

m N m N

Recently, many empirical studies of trust show that trust is composed of social preferences and beliefs in reciprocal behavior (Ashraf et al., 20 06; Fehr, 20 09; Ben-Ner and Halldorsson, 2010; Sapienza et al., 2013, for example). Better enforcement positively affects people’s belief in others’ reciprocal behavior. But, how about social preferences? If external intervention through monetary rewards or punishments undermines intrinsic motivation as the motivation crowing theory suggests,14 better enforcement may hurt trust and trustworthiness, and may have a long run effect on social, economic, and political outcomes through cultural transmission. Parents choose their educational effort levels with the expectation of how their children play in the trust game with random matching. Since Theorem 1 shows that there are nine cases regarding the equilibrium configuration, the relationship between parents’ education and institution is naturally complicated. To obtain a clear insight, we place some restrictions on g g model parameters to reduce the number of cases. We assume T1 (0 ) < T2 (1 ), which is rewritten as

ln d1 − ln ω < ln d2 − ln βγ ω

θ1g . θ2g

(A2)

Under (A2), the smallest threshold of trustworthiness for good types is larger than that of trust. See Fig. 1. We further j assume that the long run equilibrium level n∗2 of trustworthiness satisfies T1 (n∗2 ) > S for every j ∈ J, which is rewritten as

n∗2 >

1

βγ

 1−

d1

ω

 e − θ1 S . j

(A3)

Under (A3), given that good type receivers reciprocate trust, the trust threshold of every type of investors is larger than the size S of the society. That is, every type of investors trust their partners, regardless of a distance, if they expect that good receivers reciprocate their trust. Note, however, that good receivers may not reciprocate in equilibrium, depending on the distance. Given S, (A3) means that the fraction n∗2 of good receivers is larger than a certain threshold (depicted by level t(S) 1 1 in Fig. 1). The threshold is an increasing function of S, being bounded from above by βγ . If n∗2 is larger than βγ , (A3) holds for all S. Since the long run equilibrium is globally stable (Theorem 1), (A3) holds after some number of generations, starting even at an initial point without it. In the following analysis, we assume (A2) and (A3). Later, we will discuss how the result can be extended without these assumptions. Under (A2) and (A3), the equilibrium configurations of the trust game are reduced to four cases. (A2) rules out cases 3 and 7 and (A3) does cases 5, 6 and 9 in Theorem 1. Proposition 2. Under (A2)and (A3), a unique equilibrium of the trust game with random matching is characterized as follows.

1.

π1 ( y ) = 1 , π2 ( y ) = 1

for

0 ≤ y ≤ T2b (1 )

2.

π1 ( y ) = 1 , π2 ( y ) = n 2 n 1 < π1 ( y ) < 1 , 0 < π 2 ( y ) < n 2 0 < π1 ( y ) < n 1 , 0 < π2 ( y ) < n 2

for

T2b (1 ) < y ≤ T2g (1 )

for

T2g (1 ) < y < T2g (n1 )

for

T2g (n1 ) < y ≤ S

3. 4.

Proposition 2 is a corollary of Theorem 1. The equilibrium has a simple structure. When distance is small (case 1), all types of individuals conduct in the good manner. When distance is very far (case 4), good individuals play mixed strategies, and bad ones do not conduct in the good way. In between (cases 2 and 3), good investors trust their partners, but bad receivers do not reciprocate. Bad investors change their behavior from trusting to the mixed strategy, and good receivers do from the trustworthy action to the mixed strategy. The equilibrium configurations are summarized in Table 4.

14 Frey and Jegen (2001) provide an excellent survey on empirical evidence supporting the existence of motivation crowding effects in a wide variety of social, economic and political situations.

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Table 5 Material payoffs under enforcement.

C N

R

N

βγ ω, (1 − β )γ ω ω, 0

βγ pω, (1 − β p)γ ω ω, 0

With help of Table 4, we derive an explicit formula of the parents’ optimal effort given by (11). For investors, it follows from Proposition 2 that

 2re1 =

T2g (n1 ) T2g

(1 )



= d1

{βγ ωπ2 (z ) + d1 e−θ1 z − ω}g(z )dz g

T2g (n1 )

T2g

(1 )

(e−θ1 z − e−θ1 z )g(z )dz. g

b

(15)

By the same argument as in Proposition 1, it can be shown that parents devote a positive effort only in case 3 of Proposition 2. The second equality is derived by the equilibrium condition for a bad child’s mixed strategy that βγ ωπ2 (z ) −

d1 e−θ1 z = ω. (15) shows that the optimal effort e1 (n1 ) is decreasing in n1 . The investor parents are less willing to devote effort to education as the fraction of good individuals increases in their population. Intuitively, as n1 increases, the trustworthiness g threshold T2 (n1 ) of receivers decreases, and thus receivers reciprocate less likely. This causes the investor parents to evaluate less good type children. As a result, they devote less effort to education. Similarly, for receivers, we have b

 2re2 =  =

T2g (1 ) T2b (1 ) T2g (1 ) T2b (1 )

{(1 − β )γ ω + d2 e−θ2 z − γ ω}g(z )dz g

(d2 e−θ2 z − βγ ω )g(z )dz. g

(16)

The parents devote a positive effort only in case 2 of Proposition 2. We now formalize external enforcement in the trust game as follows. If investors send money and receivers cheat them in a match of distance y, then cheating may be detected with probability p(y). In the case of detection, a receiver is enforced to implement a contract, that is, to repay the β portion of return γ ω to an investor. If the cheating is not detected, the two players receive the same payoffs as in Table 1. Note that enforcement is relevant only for a receiver. The quality of external enforcement is captured by the detection probability p(y). For an action pair (C, N), an investor obtains expected payoff βγ p(y)ω, and a receiver does (1 − β p(y ))γ ω. The payoff matrix under enforcement is given in Table 5 where p = p(y ). We assume that βγ p < 1 so that the trust games in Tables 1 and 5 have the same strategic properties. An investor of type j is indifferent between playing C and N if

βγ ω{π2 (y ) + (1 − π2 (y )) p(y )} + d1 e−θ1 y = ω j

(17)

where π 2 (y) is the average probability that a receiver may choose R. A receiver of type j is indifferent between playing R and N if

(1 − β )γ ωπ1 (y ) + d2 e−θ2 y = (1 − β p(y ))γ ωπ1 (y ) j

(18)

where π 1 (y) is the average probability that an investor may choose C. From (17) and (18), the trust and trustworthiness thresholds (6) and (7) are modified under enforcement p = p(y ) as follows.

T1jp (0 ) =

1

θ1j

ln

d1

ω (1 − βγ p)

, T1jp (n2 ) =

1

θ1j

ln

d1

ω (1 − βγ n2p )

, T1jp (1 ) = ∞

(19)

p

where n2 = n2 + (1 − n2 ) p,15 and

T2jp (0 ) = ∞, T2jp (n1 ) =

1

θ

j 2

ln

d2

β ( 1 − p )γ ω n1

, T2jp (1 ) =

1

θ

j 2

ln

d2

β ( 1 − p )γ ω

.

(20)

The introduction of enforcement increases all thresholds of trust and trustworthiness. This is a critical fact for the analysis of trust and institution. 15

If 1 − βγ n2p < 0, then we set T1jp (n2 ) = +∞.

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A. Okada / Journal of Economic Behavior and Organization xxx (xxxx) xxx Table 6 The long run effect of institution on trust and trustworthiness.

Trust Trustworthiness

Local

Intermediate

Formal

0 -

+

+ 0

The equilibrium analysis in Section 3 can be applied to the case of enforcement, replacing the thresholds in (6) and (7) with those in (19) and (20). In what follows, we consider the same types of equilibrium as in Proposition 2. (A2) is p easily modified. (A3) holds under enforcement since n2 < n2 . We consider three types of enforcement. Let y1 belong to a right-hand neighborhood of T2b (1 ), and y2 to a right-hand g neighborhood of T2 (1 ). Three enforcement levels p0 , p1 , and p2 are applied to matches in the ranges [0, y1 ], (y1 , y2 ], and 2 (y , S], respectively. The parameter p0 corresponds to local enforcement which is relevant within families, small groups, and communities. The parameter p2 corresponds to legal or formal enforcement that is more relevant beyond communities where local enforcement is ineffective. The parameter p1 captures intermediate enforcement. The following proposition shows how the optimal efforts of parents are affected by different types of enforcement institutions in the short run. Proposition 3. The educational effort of an investor parent is constant, monotonically decreases, and monotonically increases as local, intermediate, and legal enforcement increase, respectively. That of a receiver parent monotonically decreases, increases, and is constant as local, intermediate, and legal enforcement increase, respectively. The intuition for the proposition is as follows. The introduction of enforcement increases all trustworthiness thresholds g g for receivers, T2b (1 ), T2 (1 ), T2 (n1 ), which determine the equilibrium behavior of children. By the definition of the three g types of enforcement, the smallest threshold T2b (1, p0 ) depends on p0 , and the intermediate one T2 (1, p1 ) does on p1 , and g 2 2 1 2 the largest one T2 (n1 , p ) does on p (assuming that y and y are not very large). All the thresholds are monotonically increasing functions of the corresponding levels of enforcement. How does the increase of enforcement affect the education g of investor parents? First, a parent devotes no educational effort over the range [0, T2 (1, p1 )] where good and bad children g 0 behave in the same way. The increase of local enforcement p does not affect the range [0, T2 (1, p1 )], while it increases the b 0 threshold T2 (1, p ), which is in the range. This means that the educational effort of the parent remains unchanged as the g g local enforcement p0 increases. Second, a good child trusts a partner over the range [T2 (1, p1 ), T2 (n1 , p2 )], and a bad child g does not. Thus, a good parent evaluates a good child more than a bad child over this range. Since the threshold T2 (1, p1 ) 1 1 1 is increasing in p , the range shrinks if p increases. As a result, if the intermediate enforcement p is strengthened, then the expected evaluation of a parent for having a good child decreases and thus, the parent is less willing to educate a g g child. Finally, the increase of legal enforcement p2 expands the range [T2 (1, p1 ), T2 (n1 , p2 )] where good children trust their partners, and thus parents become more willing to educate their children. The same logic as in the case of an investor parent can be applied to the case of a receiver parent, although the enforcement level enhancing the receiver parent’s educational effort is different from that for an investor parent. For example, the increase of intermediate enforcement p1 increases the g effort of the parent because it expands the range [T2b (1, p0 ), T2 (1, p1 )] where good children behave in a trustworthy manner. While Proposition 3 is restricted to the four cases of an equilibrium configuration under (A2) and (A3), the basic insight of it can be applied to the nine cases of an equilibrium in Theorem 1 without the two assumptions. In Theorem 1, good parents of investors devote zero effort to education over the range [0, T1b (0 )] (in cases 1–3) and devote positive effort over g g g [(T1b (0 ), T2 (1 ))+ , T1 (0 ))] (in cases 4–7). See Table 3. Local enforcement applying to closer matches below the threshold T2 (1 ) does not affect the first range, and thus the parent’s effort remains constant by the increase of it. Intermediate enforcement shrinking the second range decreases the parent’s effort, and formal enforcement expanding the second range increases it. The same analysis can be applied to the education of a receiver parent, while the effect of enforcement on the education is not very clear compared to the case of an investor parent. The short run effect of enforcement on parents’ educational effort changes the long run equilibrium of cultural transmission. If the parents’ effort increases, then the fraction of good individuals increases in the short run according to the dynamics (12). Since the institution has the property of inertia, that is, it tends to remain once it is introduced, the short δ run effect is accelerated in the process of cultural transmission. The formula of the long run equilibrium, n∗i = 1−e (ni∗ ,n∗ ) , i

1

2

shows that the increase of the effort ei enhances the long run level of good values. The effect of institution on the long run equilibrium is summarized in Table 6. From the results above, we conclude that the insight of Tabellini (2008b) into the relationship between institution and the value of cooperation in the prisoner’s dilemma can be basically extended to the values of trust and trustworthiness. Enforcement which is applied to transactions between socially closer partners is more likely to crowd out trust and trustworthiness, and the one which is applied to more distant matches is more likely to crowd in these values. A new finding of our analysis is that different institutions influence trust and trustworthiness in different ways. For example, intermediate institution deteriorates the diffusion of trust, but it enhances that of trustworthiness. This institutional difference in trust and trustworthiness is caused by a difference in individuals’ incentives to the two values. As the payoff Please cite this article as: A. Okada, The cultural transmission of trust and trustworthiness, Journal of Economic Behavior and Organization, https://doi.org/10.1016/j.jebo.2019.10.025

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matrix of the trust game in Table 1 shows, investors are willing to trust their partners who behave in a trustworthy manner, but receivers have an incentive to cheat their partners’ trust. Owing to this difference in strategic incentives between investors and receivers, there exists some range of transactions (case 3 in Table 4) where only good investors trust their partners, but all receivers cheat their trust (at least with a positive probability). The improvement of intermediate institution which externally enforces such transactions shrinks the range, and change individuals’ behavior for some matches in the region such that all investors trust the partners and good receivers reciprocate trust. Consequently, better intermediate institution crowds out trust and crowds in trustworthiness. Finally, Bohnet and Baytelman (2007) provide some empirical evidence supporting the crowding-out results discussed above. With subjects of senior executives, they compare the amounts of money sent and returned in anonymous one-shot trust games with those under four institutional constraints (pre-play communication, post-play communication, post-play punishment, and repetition). They find that compared to the one-shot trust game, all settings with tighter institutional constraints decrease intrinsically motivated trust. In contrast, the institutional constraints do not affect trustworthiness. Their experimental result supports our theoretical one that trust and trustworthiness are crowded out under different types of institution. Frey and Oberholzer-Gee (1997) present a case study of the siting of a noxious facility (repositories for nuclear wastes) in Switzerland, and show that financial compensation reduces the willingness to host the facility due to motivation-crowding out. 6. Discussion We have considered the cultural transmission of trust and trustworthiness when players are intrinsically motivated by these good values. Cultural transmission takes place through parents’ education. Parents with imperfect empathy evaluate the welfare of their children by their own preferences. As a result, only “good” parents with high motivation make positive efforts to educate their children. Trust and trustworthiness coevolve in the dynamic process of transmission, and the ratio of good players in each of two populations globally converges to a stationary point if the increasing rate of educational marginal costs is high. The diffusion of trust and trustworthiness is affected by institutions which externally enforce economic transactions. The strengthening of “local” enforcement which applies to transactions between socially closer partners deteriorates the diffusion of trust and trustworthiness, and that of “formal” enforcement which applies to transactions between individuals with far distance enhances them. Different institutions, however, influence trust and trustworthiness in different ways. Better “intermediate” enforcement crowds out trust, and crowds in trustworthiness. Institutions have a long-run effect on the levels of trust and trustworthiness in a society. We discuss some empirical works which are consistent with theoretical results of this paper. Guiso et al. (2016) study the long-term effect of a historical event on civic capital, the beliefs and values that help people overcome the free-rider problem of cooperation. At the beginning of the second millennium, a number of independent citystates emerged in the North part of Italy, caused by the demise of the Holy Roman Empire. To examine the long-term persistence of culture in development, Guiso et al. (2016) test the conjecture of Putnam et al. (1993) that the regional differences in civic capital in Italy are due to the free city-state experience. The number of nonprofit organizations per capita is employed as the major measure of civic capital, with two additional measures such as the existence of an organ donation organization and the frequency of cheating in a national mathematics exam taken by eighth-graders. They provide the empirical evidence that Northern cities that experienced self-government in the Middle Ages have a higher level of civic capital today than similar cities in the same area that did not, in all three measures. Their work shows the long-term persistence of a positive historical shock in individual behavior. To explain an underlying mechanism, they conjecture that a self-efficacy belief, a belief in one’s own ability to complete tasks and reach goals, is the cultural trait fostered by the free-state experience, that this belief is transmitted through education and socialization, and that the positive attitude of individuals enhances civic capital (cooperation) today. They find that pupils in Northern cities with the free-state experience have a higher degree of self-efficacy than those in Northern cities without the experience. While we focus the intrinsic values of trust and trustworthiness rather than beliefs, Theorem 2 is consistent with the empirical works of Guiso et al. (2016) in showing that the upward shift of the parents’ optimal educational effort rises the long run level of civic capital (good values). The positive historical experience of cooperation during the free city-state period might strengthen the level of education and socialization in regions. Studying cross-national differences of trust, Yamagishi and his associates (Yamagishi et al., 1998; Yamagishi and Yamagishi, 1994) provide the empirical evidence that American people are more trusting other people in general than Japanese people. Yamagishi and Yamagishi, 1994 conduct a cross-national questionnaire survey with 1136 Japanese respondents and 501 American respondents, and find that the mean of the General Trust Scale16 is significantly greater among American respondents than Japanese respondent. They argue that this cross-national difference prevails because networks of committed relations play a more prominent role in Japanese society than in American society. Yamagishi et al., 1998 (Yamagishi et al., 1998, p.166) write: “strong and stable relations (such as family ties and group ties) promote a sense of security within such

16 The General Trust Scale was constituted by six items in the questionnaire: “Most people are basically honest,” “Most people are trustworthy,” “Most people are basically good and kind,” “Most people are trustful of others,” “I am trustful,” and “Most people will respond in kind when they are trusted by others.”

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relations but endanger trust that extends beyond these relations.” General trust is irrelevant under commitment and assurance. Proposition 3 provides a theoretical support to their arguments. It is conceivable that Japanese society has a higher level of local enforcement which is applied to closely connected individuals such as families, firms, and communities. Thus, caused by the crowding-out effect shown in Proposition 3, the value of trust has been diffused less in Japanese society than in American society. Ermisch and Gambetta (2010) provide the direct evidence that people with strong family ties have a lower level of trust in strangers, based on experimental data from a trust game combined with panel survey data, both from a sample of the British population. We close by noting a possible avenue for future work. Institution is exogenously given in our analysis. We have focused on a one-way relation from institution to behavior, showing that institution affects the intrinsic motivation of trust and trustworthiness through education. Tabellini (2008b) extends his analysis to an issue of endogenous institution, considering the opposite relation from behavior to institution. Viewing that the quality of institution is determined by a political process, he argues that politics and culture interact with mutually reinforcing effects. Further analysis of interactions between trust and politics is left to future work.

Declaration of Competing Interest None.

Appendix Proof of Theorem 1. For i ∈ I and j ∈ J, let σi (y ) be the equilibrium probability that player i of type j conducts in the good manner in a match at distance y. Also, let π i (y) be the average probability that player i conducts in the good manner at distance y, defined by (1). As we have shown in Section 3, there are three cases, πi (y ) = 0, ni , 1, in a pure-strategy equilibrium, and two cases, 0 < π i (y) < ni , ni < π i (y) < 1, in a mixed-strategy equilibrium. We will check equilibrium conditions in all 13 (= 3 × 3 + 2 × 2 ) cases. The following case numbers correspond to those in the theorem. j

Case 1. π1 (y ) = 1, π2 (y ) = 1 : If π2 (y ) = 1, then σ1 (y ) = 1 for every y and every j ∈ J since T1 (1 ) = +∞, and thus π1 (y ) = 1. Given that π1 (y ) = 1, it j

σ2j (y )

j

holds that = 1 if y ≤ Thus, (π1 (y ) = 1, π2 (y ) = 1 ) is an equilibrium if y ≤ T2b (1 ). Case 1’. (π1 (y ) = n1 , π2 (y ) = 1 ) and (π1 (y ) = 0, π2 (y ) = 1 ) : Because π2 (y ) = 1 implies π1 (y ) = 1, neither (π1 (y ) = n1 , π2 (y ) = 1 ) nor (π1 (y ) = 0, π2 (y ) = 1 ) is an equilibrium. j T2 (1 ).

Case 2. π1 (y ) = 1, π2 (y ) = n2 : Given that π2 (y ) = n2 , it holds that σ1 (y ) = 1 if y ≤ T1 (n2 ). Given that π1 (y ) = 1, σ2 (y ) = 1 and σ2b (y ) = 0 if T2b (1 ) < g g y ≤ T2 (1 ). Thus, (π1 (y ) = 1, π2 (y ) = n2 ) is an equilibrium if T2b (1 ) < y ≤ (T1b (n2 ), T2 (1 ))− . j

j

g

Case 3. π1 (y ) = 1, π2 (y ) = 0 : Given that π2 (y ) = 0, it holds that σ1 (y ) = 1 if y ≤ T1 (0 ). Given that π1 (y ) = 1, it holds that σ2 (y ) = 0 if T2 (1 ) < y. g Thus, (π1 (y ) = 1, π2 (y ) = 0 ) is an equilibrium if T2 (1 ) < y ≤ T1b (0 ). j

j

j

j

Case 4. n1 < π 1 (y) < 1, 0 < π 2 (y) < n2 : It holds that 0 < σ1b (y ) < 1 and 0 < σ2 (y ) < 1. Then, it must be that a bad type investor is indifferent between C and N, and thus y = T1b (π2 (y )). Since the trust threshold function T1b is a monotonically increasing function, 0 < π 2 (y) < n2 g g is equivalent to T1b (0 ) < y < T1b (n2 ). Similarly, we have y = T2 (π1 (y )). Since the trustworthiness threshold function T2 g g is a monotonically decreasing function, n1 < π 1 (y) < 1 is equivalent to T2 (1 ) < y < T2 (n1 ). Thus, (n1 < π 1 (y) < 1, g g 0 < π 2 (y) < n2 ) is a mixed-strategy equilibrium if (T1b (0 ), T2 (1 ))+ < y < (T1b (n2 ), T2 (n1 ))− . g

Case 5. n1 < π 1 (y) < 1, n2 < π 2 (y) < 1: It holds that y = T1b (π2 (y )) and y = T2b (π1 (y )). By the same argument as in case 4, it can be shown that (n1 < π 1 (y) < 1, n2 < π 2 (y) < 1) is a mixed-strategy equilibrium if (T1b (n2 ), T2b (1 ))+ < y < T2b (n1 ), noting that T1b (1 ) = +∞. Case 6. π1 (y ) = n1 , π2 (y ) = n2 : Given that π2 (y ) = n2 , it holds that π1 (y ) = n1 if T1b (n2 ) < y ≤ T1 (n2 ). Given that π1 (y ) = n1 , it holds that π2 (y ) = n2 if g g g T2b (n1 ) < y ≤ T2 (n1 ). Thus, (π1 (y ) = n1 , π2 (y ) = n2 ) is an equilibrium if (T1b (n2 ), T2b (n1 ))+ < y ≤ (T1 (n2 ), T2 (n1 ))− . Case 6’. π1 (y ) = 0, π2 (y ) = n2 and π1 (y ) = 0, π2 (y ) = 0 : g

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Given that π1 (y ) = 0, it holds that π2 (y ) = 1 since T2 (0 ) = +∞ for every j ∈ J. A contradiction. Thus, neither (π1 (y ) = 0, π2 (y ) = n2 ) nor (π1 (y ) = 0, π2 (y ) = 0 ) is an equilibrium. j

Case 7. π1 (y ) = n1 , π2 (y ) = 0 : Given that π2 (y ) = 0, it holds that σ1 (y ) = 1 and σ1b (y ) = 0 if T1b (0 ) < y ≤ T1 (0 ). Given that π1 (y ) = n1 , it holds that g

σ2j (y )

= 0 if

g

< y. Thus, (π1 (y ) = n1 , π2 (y ) = 0 ) is an equilibrium if

j T2 (n1 )

(T1b (0 ), T2g (n1 ))+

g

< y ≤ T1 (0 ).

Case 8. 0 < π 1 (y) < n1 , 0 < π 2 (y) < n2 : It holds that y = T1 (π2 (y )) and y = T2 (π1 (y )). By the same argument as in case 4, it can be shown that (0 < π 1 (y) < n1 , g g g g 0 < π 2 (y) < n2 ) is a mixed equilibrium if (T1 (0 ), T2 (n1 ))+ < y < T1 (n2 ), noting that T2 (0 ) = +∞. g

g

Case 9. 0 < π 1 (y) < n1 , n2 < π 2 (y) < 1: It holds that y = T1 (π2 (y )) and y = T2b (π1 (y )). By the same argument as in case 4, it can be shown that (0 < π 1 (y) < n1 , g g n2 < π 2 (y) < 1) is a mixed-strategy equilibrium if (T1 (n2 ), T2b (n1 ))+ < y ≤ S, noting that T1 (1 ) = +∞ and T2b (0 ) = +∞.  g

Proof of Proposition 1. Consider the population of investors. For distance y, let σ1 (y ) be the equilibrium probability that children of type j ∈ J trust their partners, and π 2 (y) the average equilibrium probability that their trust is reciprocated. Let vi1j (σ1j (y ), π2 (y )) be the expected utility that a type j child receives where psychological benefit from trusting is evaluated by the value of a parent of type i ∈ J. Namely, we have j

vi1j (1, π2 (y )) = βγ ωπ2 (y ) + d1 e−θ1 y , i

Let

ij V1

vi1j (0, π2 (y )) = ω.

(A.1)

be the evaluation of a type i parent for having a type j child. By (9), it holds that

V1i j =



S 0

vi1j (σ1j (y ), π2 (y ))g(y )dy. gg

gb

We will show that V1 > V1

vii1 (σ1i (y ), π2 (y ))

≥

(A.2) bg

and V1bb > V1 . By (A.2), it suffices us to show that for every i, j ∈ J with i = j,

vi1j (σ1j (y ), π2 (y ))

in all cases in Theorem 1, and the strict inequality holds in some cases.

Cases 1, 2, and 3: Since σ1 (y ) = σ1b (y ) = 1, it holds that v1 (1, π2 (y )) = vib (1, π2 (y )) for every i ∈ J. 1 g

ig

Cases 4 and 5: Since σ1 (y ) = 1, it holds that v1 (1, π2 (y )) > v1 (0, π2 (y )) = ω. Since a bad child plays a mixed strategy 0 < σ1b (y ) < 1, it holds that g

gg

gg

vgb (σ1b (y ), π2 (y )) = σ1b (y )vgg (1, π2 (y )) + (1 − σ1b (y ))ω 1 1 gg < v1 (1, π2 (y )). A bad child is indifferent between choosing 0(N) and 1(R), playing a mixed strategy 0 < σ1b (y ) < 1. This im-

plies that vbb (σ1b (y ), π2 (y )) = vbb (1, π2 (y )) = vbb (0, π2 (y )) = ω. By definition, we have vbb (1, π2 (y )) = v1 (1, π2 (y )). Thus, 1 1 1 1 bg

vbb (σ1b (y ), π2 (y )) 1

=

vbg (1, π2 (y )). 1

Cases 6 and 7: g Since σ1 (y ) = 1, it holds bb v1 (0, π2 (y )) > vbg (1, π2 (y )). 1

that v1 (1, π2 (y )) > v1 (0, π2 (y )) = v1 (0, π2 (y )). Similarly, since σ1b (y ) = 0, it holds that gg

gg

gb

Cases 8 and 9: Since a good child plays a mixed strategy 0 < σ1 (y ) < 1, it holds that v1 (σ1 (y ), π2 (y )) = v1 (1, π2 (y )) = v1 (0, π2 (y )) = g

vgg (σ1g (y ), π2 (y )) 1

ω. Thus, vbg (σ1g (y ), π2 (y )). 1

=

vgb (0, π2 (y )). 1

Since

σ1b (y )

gg

= 0, it holds that

g

gg

vbb (0, π2 (y )) 1

vbb (1, π2 (y )). 1

>

gg

Thus,

vbb (0, π2 (y )) 1

>

By the same arguments as in the case of investors, it can be shown that v2 (π1 (y ), σ2 (y )) = v2 (π1 (y ), σ2b (y )) in gg

g

gb

cases 1, 3, 4, 7, and 8, and v2 (π1 (y ), σ2 (y )) > v2 (π1 (y ), σ2b (y )) in cases 2, 5, 6, and 9, and that vbb (π1 (y ), σ2b (y )) = 2 gg

vbg (π1 (y ), σ2g (y )) 2

g

in cases 1, 3, 5, 7, and 9, and

gb

vbb (π1 (y ), σ2b (y )) 2

> v2 (π1 (y ), σ2 (y )) in cases 2, 4, 6, and 8. bg

g



Proof of Theorem 2. For sufficiently large r > 0, it follows from (11) that

0 < δi + ei (n1 , n2 ) < 1

for all 0 < n1 , n2 < 1.

(A.3)

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Suppose that (12) can be described as the system

ni,t+1 = fi (n1,t , n2,t )

for i = 1, 2

(A.4) R2 .

where fi is a differentiable function on By (12) and (A.3), fi is a mapping from X = [0, 1] × [0, 1] to itself. We introduce the distance d(x, y) on R2 such that d (x, y ) = maxi=1,2 (|xi − yi | ). It is well-known that this distance is (topologically) equivalent to the Euclidean distance. For any initial point n0 ∈ X, let {nt } be the sequence in X generated by (A.4). Without loss of generality, we can assume that ni,t−1 < ni,t for every i. By the mean value theorem, there exists some ai,t ∈ (ni,t−1 , ni,t ) for any t such that

ni,t+1 − ni,t = fi (n1,t , n2,t ) − fi (n1,t−1 , n2,t−1 ) = fi (n1,t , n2,t ) − fi (n1,t−1 , n2,t ) + fi (n1,t−1 , n2,t ) − fi (n1,t−1 , n2,t−1 )

∂ fi (a1,t , n2,t ) ∂ f (n ,a ) (n1,t − n1,t−1 ) + i 1,t−1 2,t (n2,t − n2,t−1 ). ∂ n1 ∂ n2

=

(A.5)

By (12), we have for i = j,

  ∂ ni,t+1 ∂ ei ∂ n j,t+1 ∂ ei ∂ ni,t+1 = ei + + ni,t ∂ ni,t ∂ ni,t+1 ∂ ni,t ∂ n j,t+1 ∂ ni,t 

∂ ni,t+1 = ∂ n j,t

(A.6)

 ∂ ei ∂ n j,t+1 ∂ ei ∂ ni,t+1 + ni,t ∂ ni,t+1 ∂ n j,t ∂ n j,t+1 ∂ n j,t ∂ ei

where ei = ei (n1,t+1 , n2,t+1 ). By (11), |ei |, | ∂ n

i,t+1

(A.7)

|, and | ∂ n∂ ei | are sufficiently small for large r. Then, by (A.6) and (A.7),

there exists some 0 < k < 1 such that for every i, j ∈ J,

j,t+1

     ∂ ni,t+1   ∂ fi (n1,t , n2,t )  k  = <  ∂ n j,t    2 ∂ n j,t

(A.8)

for every (n1,t , n2,t ). It follows from (A.5) and (A.8) that d (nt+1 , nt ) < kd (nt , nt−1 ). Thus, the dynamic system (A.4) defines the contraction mapping from X to itself. It is well-known that the system has a unique fixed point which is globally stable.  Proof of Proposition 3. Consider the equilibrium in Proposition 2 under enforcement. Players’ trust and trustworthiness j thresholds Tip (πk ) are given by (19) and (20) where πk = 0, nk , 1 is the average probability that a partner may behave in the good manner. Emphasizing that Tip (πk ) is a function of enforcement level p, we denote it as Ti (πk , p). It can be shown j

j

j that ∂∂p Ti (πk , p) > 0 for every i and j. The three enforcement levels p0 , p1 , and p2 are applied to matches in the ranges [0,

y1 ], (y1 , y2 ], and (y2 , S], respectively, where y1 belongs to a right-hand neighborhood of T2b (1 ) and y2 does to a right-hand g neighborhood of T2 (1 ). By (15), the optimal effort of an investor parent is given by

 2re1 =

T2g (n1 ,p2 )

T2g

(

1,p1



= d1

)

{βγ ω{π2 (z ) + (1 − π2 (z )) p} − ω + d1 e−θ1 z }g(z )dz g

T2g (n1 ,p2 ) T2g

(

1,p1

)

(e−θ1 z − e−θ1 z )g(z )dz. g

b

g

g

Note that we take values p1 and p2 so that T2 (1, p1 ) < y2 < T2 (n1 , p2 ). The investor parent’s effort does not depend on p0 .

With

θ1g

<

θ1b ,

∂e ∂e g g it can be shown that ∂ p12 > 0 > ∂ p11 since T2 (1, p1 ) and T2 (n1 , p2 ) are increasing functions of p1 and p2 ,

respectively. This proves the first part of the proposition. By (16), the optimal effort of a receiver parent is given by

 2re2 =

T2b (1,p0 )

 =

T2g (1,p1 )

T2g (1,p1 )

T2b (1,p0 )

{(1 − β )γ ω + d2 e−θ2 z − (1 − β p)γ ω}g(z )dz g

{d2 e−θ2 z − β (1 − p(z ))γ ω}g(z )dz. g

g

We take values p0 and p1 so that T2b (1, p0 ) < y1 < T2 (1, p1 ) < y2 . The receiver parent’s effort e2 does not depend on p2 . g Since T2b (1, p0 ) and T2 (1, p1 ) are increasing functions of p0 and p1 , respectively, it can be shown without much difficulty ∂e

∂e

that ∂ p21 > 0 > ∂ p20 .17 This proves the second part of the proposition.



17 Note that the integrand is increasing in p(z) where p(z ) = p0 on [T2b (1, p0 ), y1 ] and p(z ) = p1 on [y1 , T2g (1, p1 )]. We use the derivative formula of a a F (a ) = 0 f (x, a )dx for a differential function f(x, a): F (a ) = f (a, a ) + 0 ∂∂a f (x, a )dx.

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Please cite this article as: A. Okada, The cultural transmission of trust and trustworthiness, Journal of Economic Behavior and Organization, https://doi.org/10.1016/j.jebo.2019.10.025