Cuestionario “ad hoc”

In this model, we assume a DM can incur a perceived utility cost γi ≥ 0 in order to choose an option that is not the default. Here we assume that the utility cost is neoclassical, such as an administrative fee for opting into an alternative retirement plan. Although we initially focus on the case in which γi reflects a real cost (i.e., the perceived cost equals the true cost), this formal model also captures “as if” transaction costs that nonetheless shape decision-makers’

behavior, as explored in Appendix C.3 below. In independent work, Bernheim, Fradkin and Popov (2015) develop a model like this one, with a little more structure on the distribution of uiand γi, to estimate the distribution of privately optimal contribution rates to a 401(k) plan and opt-out costs. Because we are primarily focused on the relationship between positive models and the identification strategies described in the body of our paper, however, our discussion of the implications of these models is largely orthogonal to their work,

A defining feature of this class of model is that the decision-maker knows the potential utility gain from choosing the non-default option and selects it if and only if the benefit from doing so exceeds the perceived opt-out cost. DM’s choice is thus given by

yi(d) = arg max

y∈X ui(y) − γi1{ci 6= d}

When di = d0, the solution to this problem is given by yi0 = 1 ⇐⇒ ¯ui > γ_i. When d= d1,the solution is given by yi1 = 1 ⇐⇒ −¯ui < γ_i. We can summarize the three distinct possibilities for the choices of individual i as follows:

(yi0, yi1) =













(0, 0) if − ¯ui > γi

(0, 1) if -¯ui < γi, ¯ui < γi

(1, 0) if ¯ui > γi

(26)

From (26), it is straightforward to verify that the consistency principle and frame mono-tonicity will hold. The two statistics studied in our paper will be given in this model by

y_i^∗ = 1 ⇐⇒ ¯ui >0 ci = 1 ⇐⇒ |¯uⁱ| > γⁱ

When transaction costs are homogenous, γi = γ ∀i, the average (ordinal) preferences of the consistent decision-makers is given by: E[y^∗_i|ci = 1] = P (ui >0 | ui ∈ (−∞, −γi] ∪ [γi,∞)),

or E[y_i^∗|cⁱ = 1] = 1 − F(γ)

1 − F(γ) + F(−γ)

Similarly, for the inconsistent decision-makers we have E[y^∗_i|ci = 0] = P (ui >0 | uⁱ ∈ (−γ, γ)),

or E[y_i^∗|cⁱ = 0] = F(γ) − F(0)

F(γ) − F(−γ)

Note that heterogeneity in decision-makers’ consistency in this model is driven by het-erogeneity in the intensity of their preferences as well as the size of their transaction costs.

Consequently, decision-quality independence will not generally be satisfied:¹¹ cov(y_i^∗, ci) = p(¯ui > γi) − p(¯u > 0)p(¯ui <−γi or ¯ui > γi)

which will equal zero if and only if the distribution of preferences happens to satisfy p(ui >

γi|ui > 0) = p(ui < −γi|ui < 0). That decision-quality independence usually fails here is not surprising: whether an individual is consistent in this model depends strongly on her preferences.

Nevertheless, additional structure can make the statistics on ordinal preferences studied in the body of the paper sufficient for optimal policy. Consider a utilitarian social planner choosing d ∈ {d0, d₁} to maximize

W(d) = ˆ

i

ui(y(d)) − ργⁱ1{yⁱ(d) 6= d}di.

where ρ ∈ [0, 1] governs the normative relevance of “as-if” opt out costs. One can show that when 1) ¯ui and γi are independent and 2) the distribution of ¯ui is single-peaked and symmetric, that the optimal default for a utilitarian social planner is d = 1 if and only if

11cov(yi^∗, ci) = E[yi^∗cii] − E[y^∗i]E[cii] = p(y^∗i = cii = 1) − p(y^∗i = 1) p(cii= 1).

Figure 6: Sometimes Consistent Choosers in the Costly Opt-out Model the optimal policy obtain regardless of whether γ is a real utility cost that enters the planner’s objective or merely a normatively irrelevant “as-if” cost, i.e. it holds for any ρ ∈ [0, 1].

Decision-quality instruments are immensely useful in this model, both for accounting for selection into the consistent subgroup and for identifying preference intensity. In this model changes in the cost of opting out constitute valid decision-quality instruments. Reductions in these costs could be obtained, for example, by easing the administrative requirements (such as paperwork) for choosing the non-default option. Suppose that transactions costs change from γi to γ_i⁰ ≤ γi,with γ_i⁰ < γ_i for some i. Then variation in transactions costs will meet the criteria for being a decision-quality instrument and we will have:

(yi0(γ), yi0(γ⁰), yi1(γ⁰), yi1(γ)) =

The second and fourth cases correspond to the sometimes-consistent decision-makers whose ordinal preferences are captured by the statistic YS in Section 3.3. Figure 6 depicts the different cases in Equation (27), given two values of a decision-quality instrument.

Because of the two-sided, symmetric nature of selection into the consistent subgroup in this model, we can identify the cardinal utility parameters governing the distribution of ¯ui

and γi. With sufficient (observable) variation in γi and/or functional form assumptions on

12If the conditional distributions of γi and uihave these properties when conditioning on given observable characteristics, using our matching on observables approach to tailor defaults by individual characteristics, as discussed in Section 4, would also lead to optimal policies.

the joint distribution of γi and ¯ui, one can back out the underlying structural parameters using maximum-likelihood estimation or semi-parametric techniques. The setup of these estimation strategies is similar to that of the one in Section 3.3.2, except that selection is two-sided here and one-sided in Section 3.3.2.

Here we will focus on the situation where γ is homogeneous and known.¹³ Figure 7 illustrates the recovery of preference intensity given arbitrarily rich (exogenous) variation in γ. The top panel depicts Equation (26) for some γ. From this panel we can see that Equation (26) implies that given some γ, E[yi0(γ)] tells us about the fraction of consistent choosers with ¯ui > γ, while E[yi1(γ)] tells us about the fraction of consistent choosers with

¯ui <−γ. The bottom panel shows how given rich variation in γ, so that we know E[yi0(γ)]

and E[yi1(γ)] as a function of γ ∈ [0, ∞), we can recover the full cumulative distribution function of ¯ui, where the units of ¯ui are measured in the same units as γ.