Risk Preference Types, Limited Consideration, and Welfare

Abstract

We provide sufficient conditions for semi-nonparametric point identification of a mixture model of decision making under risk, when agents make choices in multiple lines of insurance coverage (contexts) by purchasing a bundle. As a first departure from the related literature, the model allows for two preference types. In the first one, agents behave according to standard expected utility theory with CARA Bernoulli utility function, with an agent-specific coefficient of absolute risk aversion whose distribution is left completely unspecified. In the other, agents behave according to the dual theory of choice under risk combined with a one-parameter family distortion function, where the parameter is agent-specific and is drawn from a distribution that is left completely unspecified. Within each preference type, the model allows for unobserved heterogeneity in consideration sets, where the latter form at the bundle level—a second departure from the related literature. Our point identification result rests on observing sufficient variation in covariates across contexts, without requiring any independent variation across alternatives within a single context. We estimate the model on data on households’ deductible choices in two lines of property insurance, and use the results to assess the welfare implications of a hypothetical market intervention where the two lines of insurance are combined into a single one. We study the role of limited consideration in mediating the welfare effects of such intervention.

KEYWORDS:

• (Non-)expected utility
• Risk preferences
• Semi-nonparametric identification
• Unobserved consideration sets

Acknowledgments

We thank the editor, Ivan Canay, two anonymous reviewers, Matias Cattaneo, Cristina Gualdani, Elisabeth Honka, Xinwei Ma, Yusufcan Masatlioglu, Julie Mortimer, Deborah Doukas, Roberta Olivieri, and conference participants at FUR22 and at the JBES session at the ESWM23 for helpful comments.

Disclosure Statement

The authors report there are no competing interests to declare.

Notes

1 This assumption is sometimes viewed as an aspect of rationality (e.g., Kahneman 2003), and is credible in our empirical study of demand in very similar contexts (collision and comprehensive deductible insurance).

2 Within a single insurance company, typically in a given context if an agent faces a larger price than another agent for one alternative, the first agent faces a (proportionally) larger price for all other alternatives.

3 See Barseghyan, Molinari, and Thirkettle (2021b) for a formal discussion and Section 4.3 for further details.

4 See Section 4.2 for additional information on the data.

5 The multiplicative factors $\{g^{\mathcal{l}j}:\mathcal{l}\in {\mathcal{D}}^j\}$ are known as the deductible factors and δ^j is a small markup known as the expense fee.

6 Multiple preference types are a focus of the literature that estimates risk preferences using experimental data (e.g., Bruhin, Fehr-Duda, and Epper 2010; Harrison, Humphrey, and Verschoor 2010; Conte, Hey, and Moffatt 2011), although preferences are homogeneous within each type, at most conditioning on some observed demographic characteristics.

7 Other preferences that are characterized by a scalar parameter include ones exhibiting constant relative risk aversion (CRRA), or negligible third derivative (NTD; see, e.g., Cohen and Einav 2007; Barseghyan et al. 2013). Under CRRA, it is required that agents’ initial wealth is known to the researcher.

8 Recall that our analysis conditions on ${\mu }_i^j$, hence, the distribution of preferences may depend on it.

9 All papers that estimate risk preferences in the field as reviewed in Barseghyan et al. (2018) impose it.

10 The SCP is satisfied in many contexts, ranging from single agent models with goods that can be unambiguously ordered based on quality, to multiple agents models (e.g., Athey 2001).

11 We assume that while ν and ω have bounded support, the utility functions in ${\mathcal{U}}^1$ and ${\mathcal{U}}^0$ are well defined for any real valued ν and ω, respectively.

12 For a discussion of possible failures of SCP, see Apesteguia and Ballester (2018).

13 Recall that our analysis conditions on ${\mu }_i^j$, hence, the distribution of consideration sets may depend on it.

14 For example, a $500 deductible at price ${\mathbf{x}}^{\mathtt{I}}$ in collision insurance and a $500 deductible at price ${\mathbf{x}}^{\mathtt{II}}$ in comprehensive insurance would enter the consideration set independently.

15 The results extend easily to more than two contexts, at the cost of heavier notation.

16 See Figure 3.1 and its discussion below.

17 In our empirical model described in Section 4, this intersection point corresponds to ν = 0 and ω = 1, that is, respectively, no risk aversion and no probability distortions.

18 Recall that these assumptions, jointly, imply that any agent who draws $\nu <{\mathcal{V}}_{2,1}^{1,1}({\mathbf{x}}^{\mathtt{I}\prime })<{\mathcal{V}}_{1,2}^{1,1}({\mathbf{x}}^{\mathtt{II}\prime })$ unambiguously prefers alternative ${\mathcal{l}}^{1\mathtt{I}}$ to all other alternatives in ${\mathcal{D}}^{\mathtt{I}}$, unambiguously prefers alternative ${\mathcal{l}}^{1\mathtt{II}}$ to all other alternatives in ${\mathcal{D}}^{\mathtt{II}}$, and therefore unambiguously prefers bundle ${\mathcal{I}}_{1,1}$ to any other bundle in $\mathcal{D}$.

19 Equivalently, bundle ${\mathcal{I}}_{\mathcal{l},q}$ is chosen if and only if it is the first best among the ones considered:

$\mathrm{Pr}({\mathcal{I}}^{*}={\mathcal{I}}_{\mathcal{l},q}|\mathbf{x})=\alpha \sum _{{\mathcal{I}}_{\mathcal{l},q}\in \mathcal{K}}{\mathcal{Q}}_1(\mathcal{K})\int \mathbf{1}({\text{CE}}_{\nu }({\mathcal{I}}_{k,r},\mathbf{x})\le {\text{CE}}_{\nu }({\mathcal{I}}_{\mathcal{l},q},\mathbf{x}) \forall {\mathcal{I}}_{k,r}$

$\in \mathcal{K}|\mathbf{x};\nu )\text{dF}+(1-\alpha )\sum _{{\mathcal{I}}_{\mathcal{l},q}\in \mathcal{K}}{\mathcal{Q}}_0(\mathcal{K})\int \mathbf{1}({\text{CE}}_{\omega }({\mathcal{I}}_{k,r},\mathbf{x})$

$\le {\text{CE}}_{\omega }({\mathcal{I}}_{\mathcal{l},q},\mathbf{x}) \forall {\mathcal{I}}_{k,r}\in \mathcal{K}|\mathbf{x};\omega )dG.$

20 For $\frac{\partial \mathrm{Pr}({\mathcal{I}}^{*}={\mathcal{I}}_{1,1}|\mathbf{x})}{\partial {\mathbf{x}}^{\mathtt{II}}}$, the right-hand side of (3.9) remains as is, with $\partial {\mathbf{x}}^{\mathtt{II}}$ replacing $\partial {\mathbf{x}}^{\mathtt{I}}$.

21 Alternatively, ${\mathcal{O}}_1(\{{\mathcal{I}}_{1,1},{\mathcal{I}}_{1,2}\};\varnothing )={\mathcal{O}}_1(\{{\mathcal{I}}_{1,1},{\mathcal{I}}_{1,2}\};\{{\mathcal{I}}_{2,2},{\mathcal{I}}_{2,1}\})$ can replace the last condition in Assumption 3.5-(II). In our application this alternative restriction is satisfied because bundle ${\mathcal{I}}_{1,2}$ (which is the deductible bundle $\{\$1000,\$500\}$ ) is chosen with probability zero, and hence both probabilities are zero.

22 These conditions are available from the authors upon request, and require that $\partial {\mathcal{V}}_{1,2}^{1,1}(\mathbf{x})/\partial {\mathbf{x}}^{\mathtt{II}}$ does not equal a specific linear function of $\partial {\mathcal{V}}_{2,1}^{1,1}(\mathbf{x})/\partial {\mathbf{x}}^{\mathtt{I}}$.

23 If one had variation in ${\mathbf{x}}^j$ across alternatives and unbounded support, letting the observed covariate (say, price) for a given alternative go to infinity would be akin to assuming that one observes agents repeated choices in context j while facing feasible sets that include/exclude each single alternative.

24 For example, if for type t_i = 1 alternative ${\mathcal{I}}_{\mathcal{l},k}$ dominates alternative ${\mathcal{I}}_{q,r}$, ${\mathcal{Q}}_1(\{{\mathcal{I}}_{\mathcal{l},k},{\mathcal{I}}_{q,r}\})$ cannot be separately identified from ${\mathcal{Q}}_1(\{{\mathcal{I}}_{\mathcal{l},k}\})$.

25 For example, Barseghyan, Molinari, and Thirkettle (2021b) require that whenever ${\mathcal{l}}^{1\mathtt{I}}$ is considered, ${\mathcal{l}}^{2\mathtt{I}}$ is also considered. They do so because there is not a one-to-one mapping between $\partial \mathrm{Pr}({\mathcal{I}}^{*}={\mathcal{I}}_1|\mathbf{x})/\partial {\mathbf{x}}^{\mathtt{I}}$ and the (up-to-scale) density function evaluated at a single point. Rather, $\partial \mathrm{Pr}({\mathcal{I}}^{*}={\mathcal{I}}_1|\mathbf{x})/\partial {\mathbf{x}}^{\mathtt{I}}$ maps into a linear combination of the density function evaluated at cutoffs ${\mathcal{V}}_k^1({\mathbf{x}}^{\mathtt{I}}),k>1$. In contrast, here by properly using variation in ${\mathbf{x}}^{\mathtt{II}}$ we are able to create such a mapping even though there can be multiple preference types.

26 Probability distortions are featured also in, for example, prospect theory (Kahneman and Tversky 1979; Tversky and Kahneman 1992), rank-dependent expected utility theory (Quiggin 1982), Gul (1991) disappointment aversion theory, and Kőszegi and Rabin (2006, 2007) reference-dependent utility theory.

27 Vuong tests comparing the various models confirm the good fit of our preferred specification.

28 Except when both degenerate into net present value calculations with ${\nu }_i=0$ and ${\omega }_i=1$.

29 Independence results from the assumption that claims follow a Poisson distribution, which is imposed in estimating the probability of a claim (see Barseghyan et al. 2013; Barseghyan, Teitelbaum, and Xu 2018).

30 Inspection of (A.2)–(A.4) in the Appendix shows that under Assumption 4.4, $f(·)$ and $g(·)$ are identified, provided the intervals $[{\nu }^{*},{\nu }^{**}]$ and $[{\omega }^{*},{\omega }^{**}]$ in Assumption 3.4 are not singletons.

31 Alternatively, we could assume that if the realized consideration set is empty, agents choose one of the alternatives in $\mathcal{D}$ uniformly at random. Our estimation results are robust to this modeling assumption.

32 As explained in Barseghyan, Molinari, and Thirkettle (2021b), the dataset is an updated version of the one used in Barseghyan et al. (2013). It contains information for an additional year of data and puts stricter restrictions on the timing of purchases across different lines. These restrictions are meant to minimize potential biases stemming from nonactive choices, such as policy renewals, and temporal changes in socioeconomic conditions.

33 An analogous fact can be established even if an iid, type-specific, noise term were added to the utility function in (4.3) at the coverage level or, more broadly, for any model that abides a notion of generalized dominance formally defined in Barseghyan, Molinari, and Thirkettle (2021b).

34 We use subsampling because the parameter vector is on the boundary of the parameter space.

35 Given the choice patterns in the data discussed in Section 4.3, this is not surprising, as MLE sets the consideration probability of never-chosen bundles to zero.

36 Recall that we assume that $(\$1000,\$1000)$ is considered with probability one.

37 Under full consideration, the likelihood of observing nonzero shares of never-the-first-best alternatives is zero. Due to this, in estimation we set the consideration probability of each bundle to 0.99 instead of 1.00.

38 These results are sensitive to the choice of the simple lottery to benchmark willingness to pay. Changing the stakes will induce a nonlinear response by the EU types but a linear one by the DT types. Changing the loss probability will induce a nonlinear response by the DT types but a linear one by the EU types.

39 The narrow consideration model implies a rank correlation of 0.42 while in the data and under the broad consideration model this coefficient equals 0.61 and 0.62, respectively. In comparison, in the Mixed Logit model with full consideration this correlation is 0.45, while with lower triangular consideration it is 0.65.

40 See, for example, the model with imperfect information in (Gualdani and Sinha 2023, Example 2).

41 As we allow for multiple preference types, our analysis extends that of Barseghyan, Molinari, and Thirkettle (2021b) even in the simplified framework where consideration is independent across contexts.

42 Many important papers in the theory literature—including papers on revealed preference analysis under limited attention, limited consideration, rational inattention, and other forms of bounded rationality that manifest in unobserved heterogeneity in consideration sets—also grapple with the identification problem (e.g., Masatlioglu, Nakajima, and Ozbay 2012; Manzini and Mariotti 2014; Caplin and Dean 2015; Lleras et al. 2017; Cattaneo et al. 2020). However, these papers generally assume rich datasets—for example, observed choices from every possible subset of the feasible set—that often are not available in applied work, especially outside of the laboratory.

43 Examples for the first approach include De los Santos, Hortaçsu, and Wildenbeest (2012), Conlon and Mortimer (2013), Honka, Hortaçsu, and Vitorino (2017), Honka and Chintagunta (2017); for the second, Goeree (2008), van Nierop et al. (2010), Gaynor, Propper, and Seiler (2016); Heiss et al. (2021). Recent examples for the third approach include Abaluck and Adams (2020), Crawford, Griffith, and Iaria (2021), Lu (2022).

44 These derivations are based on repeated use of facts such as

${\mathcal{O}}_1(\{{\mathcal{I}}_{1,1},{\mathcal{I}}_{2,2}\};\varnothing )={\mathcal{O}}_1(\{{\mathcal{I}}_{1,1},{\mathcal{I}}_{2,2},{\mathcal{I}}_{2,1}\};\varnothing )+{\mathcal{O}}_1(\{{\mathcal{I}}_{1,1},{\mathcal{I}}_{2,2}\};\{{\mathtt{I}}_{2,1}\})$

$={\mathcal{O}}_1(\{{\mathcal{I}}_{1,1},{\mathcal{I}}_{2,2},{\mathcal{I}}_{1,2}\};\varnothing )+{\mathcal{O}}_1(\{{\mathcal{I}}_{1,1},{\mathcal{I}}_{2,2}\};\{{\mathtt{I}}_{1,2}\})$

Additional information

Funding

Financial support from NSF grants SES-1824448 and SES-2149374 is gratefully acknowledged.