Peer effects in risk taking: Envy or conformity?

Abstract

We examine two explanations for peer effects in risk taking: relative payoff concerns and preferences that depend on peer choices. We vary experimentally whether individuals can condition a simple lottery choice on the lottery choice or the lottery allocation of a peer. We find that peer effects increase significantly, almost double, when peers make choices, relative to when they are allocated a lottery. In both situations, imitation is the most frequent form of peer effect. Hence, peer effects in our environment are explained by a combination of relative payoff concerns and preferences that depend on peer choices. Comparative statics analyses and structural estimation results suggest that a norm to conform to the peer may explain why peer choices matter. Our results suggest that peer choices are important in generating peer effects and hence have important implications for modeling as well as for policy.

Appendix A: Theoretical framework

1.1 A.1 A model of relative payoff concerns

Assume the utility in state j ( j ∈ {g, b}) of having chosen lottery i ( i ∈ {A, B}) and earning ${m^{j}_{i}}$, to be given by the sum of two terms: a consumption utility, which is solely determined by individual risk preferences, plus a social utility term, which depends on payoff differences. This implies $v^{j}_{i,k}=u\left ({m^{j}_{i}}\right ) + R\left ({m^{j}_{i}}-{m^{j}_{k}}\right ),$ where k ∈ {A, B} is the lottery of the peer, and R(⋅) is a function of payoff differences and defined as follows:

$$ R(x) = \left\{\begin{array}{ll} x & \text{if }x\geq0, \\ \lambda x & \text{if }x<0. \end{array}\right. $$

The parameter λ captures how large losses with respect to the peer loom relative to gains. An individual’s expected utility from choosing lottery i is

$$V_{i,k}=U_{i} + \sum\limits_{j} p_{j}R\left({m^{j}_{i}}-{m^{j}_{k}}\right), $$

where U _i is the expected consumption utility of lottery i. If the peer holds a lottery that yields a lower consumption utility, the individual may nevertheless choose it, if he experiences a strong disutility from falling behind the peer, i.e. if λ is large enough. Let us define an individual’s strategy space as $\mathcal {S}=\{\text {imitate} = (i; AA,BB),\, \text {deviate} = (i; BA,AB),\, \text {stay} = (i; iA,iB), \,\text {change} = (i; -iA,-iB); \,\text {for } i\in \{A,B\}\}$. Here i ( −i) denotes his (opposite) choice in Part I, and the tuple ik describes the choice of lottery i in Part II given that his peer has lottery k. Then, the cutoffs are given by the following proposition.

Proposition 1

Define ${\Delta }\equiv \frac {U_{B}-U_{A}}{p\delta }+\frac {(1-p)(c-\delta )}{p\delta }$ and ${\Theta }\equiv \frac {U_{A}-U_{B}}{(1-p)(c-\delta )}+\frac {p\delta }{(1-p)(c-\delta )}$ . An individual imitates if λ > max{Δ, Θ}. An individual deviates if λ< min{Θ, Δ}. An individual stays with his Part I choice otherwise.

Note that whether Δ is smaller or greater than Θ is determined by the individual’s choice in Part I, i.e. by his expected consumption utility U _A and U _B .

Proof

An individual imitates if V _{A, A} > V _{B, A} and V _{B, B} > V _{A, B} . V _{B, B} > V _{A, B} is equivalent to

$$\begin{array}{@{}rcl@{}} \lambda(1-p)(c-\delta)> U_{A}-U_{B}+p\delta\quad \Leftrightarrow \quad \lambda>{\Delta}\equiv\frac{U_{A}-U_{B}}{(1-p)(c-\delta)}+\frac{p\delta}{(1-p)(c-\delta)}. \end{array} $$

V _{A, A} > V _{B, A} is equivalent to

$$\begin{array}{@{}rcl@{}} \lambda p\delta&>U_{B}-U_{A}+(1-p)(c-\delta)\quad \Leftrightarrow \quad \lambda>{\Theta}\equiv\frac{U_{B}-U_{A}}{p\delta}+\frac{(1-p)(c-\delta)}{p\delta}. \end{array} $$

Hence, for an individual to imitate it must hold that λ > max{Δ, Θ}.

Similarly, an individual deviates if V _{A, A} < V _{B, A} and V _{B, B} < V _{A, B} . It follows directly from above that this is satisfied if λ < min{Δ, Θ}. □

1.2 A.2 A model based on social comparison theory

Consider a model in which the closer the individual risky choice is to the social anchor, the more utility the individual derives. In a setting with only two options, this can be captured by an additional utility γ when the option chosen coincides with the social anchor. In particular, the expected utility of lottery i given the anchor k is

$$V_{i,k}=U_{i} +\gamma \cdot\mathbf{1}(i=k), $$

where 1(⋅) is the indicator function. (Cooper and Rege, 2011, also assume this form of utility when examining conformity.) Based on the argument above, we would expect γ to differ across treatments and γ _C , in Choice, to be larger than γ _R , in Random. This would generate an increase in imitation in Choice. Further, since the effect of γ is independent of lottery characteristics, we would expect the change in imitation across treatments to be symmetric with respect to the two available options, A or B.