Welfare with Imperfect Foreknowledge: The Case of Risk Aversion

1. Introduction

In the context of recreational resources, it is not uncommon that the status of the amenities users enjoy, or conversely, of the disamenities they wish to avoid, is unknown to individuals before they visit a site. This may be less important when agents travel to the sites regularly and therefore form reasonable expectations about the amenities. But when the attributes have changed substantially, past information may be of little value. For example, fecal coliform densities may change quickly from increases in water temperature. Oil spills may contaminate a beach tomorrow even if it was clear of oil today. Likewise, the availability of target species for hunting or fishing is unlikely to be known ahead of time. In these circumstances, agents must select which site to visit with incomplete information (Leggett 2002; Allcott 2013; Schmeiser 2014).

When agents make site choices without full knowledge of amenities’ levels, the classical analytical expressions for assessing welfare impacts associated with changes in attributes (Small and Rosen 1981) may no longer hold. Train (2015) and Glasgow and Train (2018) have analyzed such a situation when agents are risk neutral, that is, when their preferences can be characterized by linear utility. Specifically, these authors assume that agents hold beliefs on the attributes’ levels at the different locations, but allow for the possibility that those beliefs may be biased (that is, they may over- or underestimate the true state of the amenities/disamenities). Thus, individuals may be better or worse off ex post than they expected to be, and welfare measures need to be adjusted accordingly. There are other settings, however, where agents’ preferences may exhibit declining marginal values for additional amenity levels and consequently may not be properly represented by linear utility (Foster and Just 1989; Holzer and McConnell 2017). Faced with the uncertainty associated with incomplete information on attributes, users would behave as risk-averse agents. Consider the case when the attribute is water quality, as given by the concentration of fecal coliform. One could envision a large marginal utility for reductions in fecal coliform at high concentrations that declines to zero for low levels (as the likely health impacts of fecal coliform depend on the level of exposure). Similarly, in rock climbing, the marginal utility associated with the technical difficulty of the climbs is likely to decline as the rock configurations become progressively more challenging and the risk of injury higher. In the same vein, in so-called meat fisheries (as opposed to catch- and-release sport fisheries) in which individuals prefer to bring fish home in a cooler, the utility associated with an additional kept fish is likely to be decreasing in the catch. In such cases, it would make sense to consider the impact of nonlinear utility on welfare estimates for changes in attribute levels.

In this paper we investigate the impact of risk aversion in attributes on the determination of welfare measures when users make choices under biased beliefs. Using a model with additively separable and concave (in attributes) utility, we show that there are two critical components in welfare calculations: (1) a risk component, defined by the willingness to pay to eliminate uncertainty in attributes; and (2) a bias component, the ex post correction in utility due to the agent’s mistaken beliefs about the attribute. The bias component in (2) coincides with the correction used by Train (2015) and Glasgow and Train (2018) to reconcile welfare measures based on beliefs with ex post realization of utility (based on the actual status of attributes). However, our analysis reveals that when agents’ utility is concave in attributes, in order to compute ex post welfare corrections, we must initially determine the agent’s disutility from uncertainty in the quality of amenities. We illustrate our results with a Monte Carlo analysis based on risk preferences estimated from a set of choice experiments conducted among anglers on the East Coast of the United States. By comparing the cases of linear utility with constant absolute risk aversion (CARA) utility, we show that ex post corrections of welfare losses are attenuated by greater dispersion in beliefs about amenities. Additionally, we show conditions under which the partial disclosure of information—information on the true status of a subset of recreational sites—may reduce ex post welfare for risk averse individuals.

2 Welfare Measures

In this section, we introduce the analytical framework that will then be used in the empirical application. Assume that an individual chooses among a set of discrete alternatives indexed by j=1,…,J. The utility the person expects from alternative j depends on the vector of observable characteristics xj and environmental quality Q_j ∈ [Q,Q]. Suppose first that perceptions of quality are correct. Upon selecting alternative j, indirect utility is given by [1] where y is the income available for the choice occasion, and c_j is the cost of alternative j. We assume v_j is strictly increasing in all its arguments. Under perfect information, equation [1] represents both anticipated and experienced utility as there is no discrepancy between the two. In this paper, we are concerned with situations where the individual does not know the level of environmental quality before visiting the site. Instead, individuals hold beliefs regarding the quality they will encounter at the site.¹ These heterogeneous beliefs reflect the information available to each agent at the time the decision is made. We take in-dividuals’ beliefs as a primitive of the model and let the vector of parameters ∂_j summarize the consumer’s prior on site j’s environmental quality q̃ (Leggett 2002). Thus, we can write the consumer’s expected (anticipated) indirect utility conditional on choosing site j as [2]

Next, denote μ_j = E[q̃_j], the expected environmental quality at site j under the agent’s current beliefs, where we allow for the possibility that μ_j≠ Q_j, the experienced quality. There are various reasons why agents may hold biased beliefs (e.g., Kahneman and Tversky 1973; Rabin and Schrag 1999; DellaVigna 2009). Instead, what we have in mind is variation in the quality of the initial information readily accessible to Bayesian agents. Hence, after enough observations, dispersion in an agent’s prior beliefs would disappear and beliefs would converge to the truth (Blackwell and Dubins 1962; Millner and Ollivier 2016). Define the individual’s willingness to pay (ρ_j) to remove uncertainty from the assessment on environmental quality and face instead a quality level μ. as [3]

This is a true willingness to pay since we are assuming individuals make choices exclusively based on these subjective priors. From equation [3], parameter ρ_j can be interpreted as an added cost of selecting site j. Indeed, this cost can be avoided only by not choosing the uncertain alternative. In other words, once choice j has been selected, the monetized disutility associated with the uncertain prospect (ρ_j) is experienced regardless of the true ex post environmental quality Q_j at that location. As such, this disutility will need to be accounted for in the ex post welfare calculations. Furthermore, if the agent faces un-certainty in the assessment of environmental quality at each of the sites in his choice set, his only option to avoid risk altogether is to give up the recreational trip. Thus, we write ex post (experienced) utility as v_j(y – c_j – ρ_j,x_j,Q_j), and the difference between experienced and anticipated utility as d_j = v_j(y – c_j – ρ_j,x_j,Q_j) – E_∂j [v_j(y - c_j, x_j, q̃_j)]. Using expression [3] above, this wedge in utilities can be rewritten as d_j = v_j(y – cj – ρj,xj,Q_j) –v_j(y – c_j – ρ_j,xj,μ_j), which collapses to the case studied by Train (2015) and Glasgow and Train (2018) when v_j is linear in attributes, that is, when ρ_j = 0.

Further insights come from the widely used quasi-linear utility function. Let v_j = φ(x_j,Q_j) + α(y – c_j), where φ(⋅) is strictly increasing and concave in all its arguments and α is a constant. Thus, while individuals are risk neutral in income, they exhibit disutility from variability in site attributes. In the standard consumer choice model, risk aversion stems from changes in the marginal utility of income. However, in the discrete choice case where uncertainty pertains to site attributes, risk aversion can be specified with quasi-linear preferences (Holzer and McConnell 2017). We can rewrite [3] as [4]

Rearranging terms, we have [5] where σ_j corresponds to the certainty equivalent environmental quality level defined by φ(xj,σ_j) = E$. [φ(xj,q̃_j)], with σ_j < μ_j by Jensen’s inequality. Under the assumption of quasi-linear utility, the difference between actual and anticipated utility can be written as [6]

Thus, d_j can be calculated with Q_j, experienced quality, and μ_j, the mean of the agent’s subjective distribution of q̃_j. Determination of μ_j from changes in visitation to site j, however, requires knowledge of the agent’s subjective distribution of q̃_j or, equivalently, P_j. This illustrates one of the principal findings of the paper: the welfare corrections when experienced environmental quality differs from expected environmental quality cannot be calculated without accounting for the agent’s risk aversion.

In expression [6] d_j can be interpreted as coming from bias in beliefs. Note that, unlike previous work (Allcott 2013; Schmeiser 2014; Train 2015; Glasgow and Train 2018), in this setup the disutility associated with imperfect foreknowledge comprises not only d_j but also the term αρ_j=φ(x_j,μ_j) -φ(x_j,σ_j) representing the disutility associated with risky outcomes. Using the definitions above, we can write the ex post utility associated with each site as a function of ex ante utility and a term stemming from bias in beliefs, v_j= Ev_j+ d_j, or, in the case of quasi-linear utility, φ(X_j,Q_j)+α(y – c_Γρ_j) = [φ(x_j,u_j)d_j + α(y-C_j-P_j). Note that, under unbiased beliefs, μ_j = Q_j, expression [6] vanishes, and ex ante and ex post utility coincide and equal φ(x_j,Q_j) + α(y -C_j - P_j). As an example, imagine environmental quality refers to the presence of oil on the beach. Very low levels of oil, represented by a high Q, might be just background levels. In contrast, low Q could indicate oil concentrations that severely impact utility. Assume that Q_j > μ_j, reflecting a relatively oil-free, benign condition relative to the agent’s beliefs. In this case, [6] refers to the additional utility the individual experiences once at the site when oil contamination is less than anticipated, Q_j - μ_j> 0. Even though the agent expects a moderate oil contamination level of μ_j, he experiences disutility associated with the risk, represented by αP_j, that an unlucky draw may result in a heavily oiled site. This disutility is likely to be especially salient when site attributes result in physical discomfort, such as oil at the beach, or health hazards, such as swimming in water contaminated with fecal coliform. Note that while αP_j ≥ 0 for concave φ(·), it holds that d_j > 0 only if the alternative is actually better than the individual expected, and d_j < 0 otherwise. To continue with our example, imagine the individual has the option to visit either site j or an alternative, site m. Out of these two alternatives, the agent selects the one with the higher expected (ex ante) utility. Depending on the structure of beliefs, however, this choice may prove suboptimal ex post. To see that this may indeed be the case, assume that the agent’s expected utility is higher for site m, φ(x_j,μ_j)+α(y - C_j-p_j < φ(x_m,μ_m) + α(y-c_m-p_m) so he decides to visit that location. Furthermore, imagine that upon arrival he confirms that the level of oil pollution on that beach is as expected, that is, Q_m = μm and d_m = 0. Under these circumstances, the individual experiences utility ν_m = φ(x_m,μm) + α(y - c_m-p_m). Had he visited site j instead, he would have been pleasantly surprised, as indicated earlier, to observe less oil on the beach than expected, that is, Q_j - μ_j > 0 and d_j > 0. In fact, for a large enough d_j > [φ(x_m,μm) – φ(x_j,μ_j)] + α(C_j + P_j - C_m - P_m), he would have experienced higher utility and been better off at destination j, ν_j > ν_m.

For empirical implementation that facilitates the measurement of these terms, we adopt the standard logit specification [7] where ε_j are iid and follow an extreme value type I distribution. In what follows we assume that the ε_j’s are independent of the d_j’s and the q̃_j’s. The choice probabilities, based on anticipated utility and beliefs ϑ = {ϑ₁,…,ϑ_J, are given by [8]

Each individual chooses among the alternatives based on anticipated utility but obtains actual utility from the selected alternative. The optimal choice based on ex ante information is i, for which Ev_i > Ev_j, ∀j ≠i. However, actual utility may differ from anticipated utility by d_i, as shown in equation [6]. Define consumer surplus as the maximum willingness to pay for access to the set of sites, conditional on choosing at least one of them. Then the (actual) consumer surplus from the ex ante optimal choice i for a given individual, based on current beliefs (Schmeiser 2014), is [9] which, using equation [4], can be rewritten as [10]

Next, denote the alternative that provides the highest actual (ex post) utility as k, that is, v_k>v_j, ∀j≠k. Following Train (2015), we write the ex post loss due to imperfect foreknowledge as the difference in ex post utility between the site actually chosen with the ex ante information and the alternative that would have been selected under complete information. This loss is E[v_i]/α-E[v_k]/α, where the second term is the expected maximum utility if the individual had correctly anticipated the actual environmental quality at each site. This ex post loss for the agent can be rewritten as [11]

△CS comprises the change in log-sum terms from the anticipated attributes to actual attributes, plus the average difference between actual and anticipated utility (Train 2015). The loss from imperfect foreknowledge is also the value of obtaining full information about the actual environmental attributes. Note that equation [11] is analogous to Train’s (2015) equation [2]. However, in order to isolate the j and arrive at [11] it has been necessary to discompose the effect of imperfect information according to equation [4], that is, between bias in beliefs and the disutility associated with risky outcomes. In fact, even with unbiased beliefs, there is a reduction in consumer surplus compared to the perfect foreknowledge scenario. Equation [11] collapses to the expression derived by Train (2015) only either (1) in the absence of uncertainty, and/or (2) when individuals’ preferences are linear in attributes.

3 Information Disclosure

Assume next that information on the true environmental quality at locations in the subset I is made available to the agents before they choose which site to visit. This could occur, for example, if the sites are located across multiple jurisdictions, with authorities in different jurisdictions disclosing environmental quality data at various (uncoordinated) times (e.g., beach advisories by different municipalities in the Gulf of Mexico). In this case, individuals’ ex ante and ex post indirect utility from visiting sites j∈I coincide and are equal to φ(x_j,Q_j)+α(y-c_j). According to equation [6] it follows that , where a denotes after disclosure of information. We are interested in specifying conditions under which this partial release of information is valuable to agents. Using equation [9] we write the value of information as [12] where the probabilities and are defined as in equation [8] evaluated, respectively, at v_j=φ(x_j,Q_j)+α(y-c_j) and . Using the fact that and , [12] can be rewritten as [13] which, using equation [4], becomes [14] where, as indicated earlier, P_j = (φ(x_j,μ_j) - φ(x_j,σ_j)) / α is positive for strictly concave φ(⋅). Letting O_j,k=Π_j/ Π_k denote the relative odds of choosing site j over k, it follows that for all j∉I and k∈I. Thus, provided αρ_k+d_k>0 for all k∈I, disclosure of the true condition of some sites shifts probability mass away from locations of uncertain environmental quality and toward those with full information. The riskier the sites k∈ I and/or the more pronounced the underestimation of their quality before the release of information, the larger this shift in probabilities.

In particular, note from equation [14] that for voi < 0 it must be the case that [15]

This condition will never hold in the context of unbiased beliefs, that is, d_j’s = 0. Indeed, in this case the left-hand side of [15] vanishes while the right-hand side is always positive provided that, before the release of information, ρ_j > 0 for some j∈I. It is only by changing the ex ante ranking of sites and inducing a different location choice that will prove sub-optimal ex post, that the new information may reduce agents’ welfare. This can occur only in the presence of bias in beliefs regarding environmental quality in at least one of the alternatives. In applications that involve many choices such as recreation, however, it seems plausible that individuals may hold biased beliefs regarding the environmental quality of at least some of the sites.

As an example, assume the agent holds unbiased beliefs on environmental quality at sites j∈I. Under this assumption, the disclosure of information on j∈ I simply eliminates the disutility associated with risky outcomes at those sites. Equation [15] then becomes [16]

The right-hand side represents the increase in ex ante utility due to the reduction in risk at sites j∈ I. The left-hand side is the foregone ex post utility due to the selection of suboptimal sites. If completely eliminating the risk from j∈ I increases the probability of visiting (ex post) suboptimal alternatives, then [16] will hold and agents will be hurt by the additional information. The rationale for the result is as follows: by reducing uncertainty in the assessment of attributes at sites j∈I, the manager makes those sites ex ante more attractive than they were before the release of information. This, in turn, increases the probability of visiting the sites at the expense of alternatives j∉I. If, ex post, sites j∉I turn out to be higher quality sites than agents anticipated (due to biased beliefs), the reduction in the probability of selecting these sites may result in lower welfare. This is a somewhat surprising result since completely eliminating risk from a subset of sites may make risk averse individuals worse off. As stated earlier, however, this could occur only ex post and in the presence of biased beliefs regarding the attributes at some sites.

4 Implications of Concavity: Monte Carlo Experiments

Our interest in the implications of concavity (of utility) on site attributes for the computation of welfare under imperfect information is primarily conceptual but has been stimulated by the work of Train (2015) and Glasgow and Train (2018). These authors assumed linear preferences, as in earlier papers by Leggett (2002), Allcott (2013), and Schmeiser (2014), but dealt implicitly with uncertainty over amenities. We know from equation [6] in Section 2 that uncertainty must be purged before the site correction terms (i.e., the d_j’s originating from bias in beliefs) needed for determining welfare losses can be determined. However, this result is derived from the theoretical assumption of concavity in attributes, which was not grounded empirically. It could be that the utility function is concave but the actual deviation from linearity is too small to matter. Our Monte Carlo analysis is based on preferences estimated in a plausible setting. It demonstrates that concavity of utility matters and can alter welfare assessments in settings of incomplete information.

Our Monte Carlo illustration is conducted in the context of recreational fishing. Agents’ preferences were estimated from choice experiments conducted among recreational anglers from the East Coast of the United States (Holzer and McConnell 2017). One attribute, the number of fish kept per trip, is defined in these choice experiments as a random variable in the form of a range that is varied experimentally. From the these data we have estimated linear and CARA specifications for anglers’ utility (Table 1). Thus, these preferences are based on realistic choices, and not simply assumed as in many Monte Carlo experiments.

Our analysis begins with a population of 50,000 anglers, characterized by demographic variables drawn from the sample of respondents to the original choice experiment survey. We assume each angler faces a single choice occasion, in which he decides among four sites, labeled A, B, C, and D, as well as an opt-out alternative. As depicted in Table 1, the utility function is specified either as linear or as quasi-linear. In the latter case, we have v_j=φ(x_j,q_j)+α(y-C_j), with where x _j is a vector of nonrandom site characteristics, q_j denotes the keep of the target species at the site, and r is the Arrow-Pratt measure of absolute risk aversion (for details see Holzer and McConnell 2017). We adopt the standard logit specification for the discrete choices. Using these linear and CARA preferences, we first establish the baseline number of trips for the 50,000 anglers, computed as the sum of the probabilities of visiting each site.

Next, we introduce a disruptive event with the potential to negatively impact the catch at site A. We label this event “algal bloom,” but it could alternatively be an oil spill or some other form of impermanent contamination. The “algal bloom” label is simply used as a device to convey our ideas and is not related to any effort to model this particular type of event realistically. The contamination associated with the event creates uncertainty about the number of fish kept through its potential impact on the fish stock and the catch, but by assumption does not otherwise affect utility. In turn, this uncertainty about the keep influences decisions on whether to visit site A. The end result is that total visitation to this site decreases. We use an approach similar to that of Glasgow and Train (2018) to construct the various scenarios. We start by defining—for each scenario—a quantity of lost trips to site A after the “algal bloom” takes place. We then proceed to infer the corresponding reduction in utility that would be implied by these lost trips. Lastly, we decompose that reduction in utility into its risk component (ρ_A) and its bias component (d_A), as discussed in the previous section. We then compute the correction associated with bias in beliefs to determine welfare losses in each case.

Hence, in our experiments the entire impact of water contamination is channeled through a single attribute, the number of kept fish, which is perceived as random by anglers. Agents must make a decision before knowing if the contamination associated with the “algal bloom” persists. Those who visit the site will attain utility according to the true state of the site, that is, according to whether there is “algal bloom” present. When the “algal bloom” has impacted site A, the utility lost from the contamination comes from the reduction in the keep rate at that site. On the other hand, when “algal bloom” is absent, there is no such reduction in keep, and welfare losses stem exclusively from the reduction in visitation to a high-quality site (i.e., due to anglers’ biased beliefs that mistakenly assumed that the fishing experience at the site would be impaired by pollution).

In our Monte Carlo analysis, we begin with a baseline (prior to the contamination event) of 24,995 trips to site A for the case of CARA preferences, and 28,151 trips to site A for linear preferences. Changing the assumption on preferences naturally results in a different baseline number of trips. We then analyze the welfare impacts, under CARA and linear preferences, of nine alternative scenarios associated with the possibility of an “algal bloom.” These scenarios are composed of three levels of lost trips at site A (4,000, 2,500, and 1,000 trips) and, for each of them, three alternative specifications for agents’ beliefs about keep at the site. Throughout the simulations we assume all agents hold identical beliefs. Beliefs are characterized by a uniform distribution, ¢A - U(μA - ł,μA + ℓ), where μA = E[¢A] varies by scenario, and where the assumed dispersions are given by ℓ = 0 (no uncertainty), ℓ = 2, and ^{ℓ= 3}. This increase in dispersion is naturally equivalent to an increase in risk. The nine scenarios are summarized in Table 2 (columns 3 through 7 for CARA preferences; columns 8 through 11 for linear preferences). Specifically, each scenario is described by a pair of rows in which the top row corresponds to the case of unbiased beliefs, that is, the case where “algal bloom” is indeed present at site A and negatively impacts its keep as anticipated by anglers. The bottom row, on the other hand, depicts the case in which the “algal bloom” is actually absent and the keep remains unchanged at the baseline level of 7 fish. In this latter case, agents would have been mistaken and their beliefs biased. The degree of bias for each scenario is shown in columns (7) (CARA preferences) and (11) (linear preferences).

We illustrate the analysis by discussing the scenario that begins with a reduction in visitation to site A of 4,000 trips. The linear preference case is straightforward. We solve for the change in keep that would reduce trips by 4,000. To this end, we first find the reduction in site A’s utility (d_A) that brings the total number of trips from 28,151 to 24,151: [17] where d_k = 0 for all k ≠ A, i indexes anglers, and s denotes the opt-out option, which is assumed to depend on the vector of individual characteristics z_i. Once d_A is determined from the equation above, we compute the expected keep that would induce the same reduction in trips as μA = E [q̃_A] = qA + dA / Λ where γ is the coefficient on keep for the linear case from Table 1. As shown in Table 2, a reduction in linear utility of d_A = -0.33, which is equivalent to an expected keep of μ_A = 5.06, would result in a total number of visits to site A equal to 24,151. At this point we are ready to proceed with the welfare calculation in equation [11], which as discussed earlier will depend on what actually happened at the site.

If the “algal bloom” has impacted site A (scenario 1, algal bloom, columns 8-11), then the new reduced keep will effectively be 5.06 and angler’s expectations will be realized ex post (i.e., 0% bias). In these circumstances, we can use the standard expression of difference in log-sums (Small and Rosen 1981) to compute welfare losses, which are composed of the losses associated with the reduced visitation to A, plus the losses resulting from a lowered number of kept fish for those who visited the site. In Table 2, those losses equal $1,795,678. Alternatively, and contrary to anglers’ expectations, site A may remain uncontaminated. In this “no algal bloom” case (scenario 1, no algal bloom, columns 8-11), individuals that travel to the site expect to keep 5.06 fish but are pleasantly surprised when they are able to keep 7 fish instead. In other words, anglers hold biased beliefs regarding the keep rate at the site (equivalent to an underestimation bias of 100× (1 - 5.06 / 7) or 27.7%). Accordingly, the anticipated losses due to a lower keep are not realized ex post. The only losses that remain stem from the reduction in trips to site A, a site that turns out to be of high quality for those who visit it. These losses amount to $137,653 and are calculated using equation [11] for linear utility when ρj = 0 ∀j, d_A = -0.33, and dB= dC= dD= 0. In this case, equation [11] naturally collapses to the expression used by Train (2015) and Glasgow and Train (2018) in their analysis.

For the analysis of the various scenarios under anglers’ CARA preferences, we follow the implications of equation [3]. That is, in order to calculate the d_A that allows for the determination of the welfare losses under “no algal bloom”—the losses in the case in which there is disparity between ex ante and ex post utility—we must first remove the impact of uncertainty from the assessment of the site’s amenity. From equation [6] we can write [18] where q_A is the ex post, realized keep at site A when unaffected by the “no algal bloom” (7 fish). The parameter μ_A is determined starting with the dispersion in beliefs, as defined by the width of the support of the uniform distribution, 2ℓ, and finding the μ_A that results in the desired reduction in total trips to site A. For example, in the scenario of 4,000 lost trips and ℓ = 3, beliefs are described by U(μ_A-3,μ_A+3), and consequently μ_A is found by equating the sum of anglers’ probabilities of visiting site A, as given in equation [8], to 20,995. Once μ_A is identified, d_A is readily determined as in equation [18], and welfare changes are computed using equation [11]. As depicted in Table 2, when the number of lost trips to site A equals 4,000, then μ_A = 3.26 fish kept and losses equal $479,937 (scenario 7, no algal bloom, columns 3-7). The percentage bias is computed as 100 × (μ_A -7)/7 = -53.5%. As in the linear case, d_A represents the wedge between anticipated and realized utility and is driven by the agent’s bias in beliefs. Indeed, absent bias, d_A = 0 and equation [11] collapse to the difference in log-sums from Small and Rosen (1981). This is the case labeled as “algal bloom” (scenario 7, algal bloom, columns 3-7), where the total loss equals $1,608,019. This figure is significantly larger than the previous $479,937 because it not only includes losses from lost trips to site A, but also reduced utility associated with visits to a contaminated site with expected keep of 3.26 rather than the original 7 fish. Importantly, in each of the risky scenarios under CARA preferences, whether “algal bloom” or “no algal bloom,” the welfare loss includes the monetized disutility due to uncertainty in the keep, given by ρ_A. Thus, when uncertainty stems from the mere possibility of the disruptive event impacting site A, it becomes part of the welfare costs of environmental deterioration even in the absence of algal bloom.

The results for all cases are summarized in Table 3. The three panels—each analyzing scenarios for the loss of 4,000, 2,500, and 1,000 trips, under both “algal bloom” and “no algal bloom”—depict rising risk, as represented by increasing, ℓ, from ℓ, = 0, to ℓ = 2, and ℓ = 3 fish kept (corresponding, respectively, to standard deviations of , and ). First, note that the results for the linear utility case are the same for each panel, because only the first moment matters in this context. In other words, since ρ_A = 0, the shift in utility (d_A) required for a given reduction in visitation is the same regardless of dispersion. With CARA preferences, we notice that the welfare correction due to bias in beliefs for the “no algal bloom” case declines as uncertainty increases. For example, the adjust-ment required to properly assess the ex post losses associated with the 4,000 trip reduction in visitation to site A, computed as the difference between welfare losses with and without “algal bloom,” goes from $1,469,262 to $1,305,527 to $1,128,082 as changes from ℓ = 0 to ℓ = 2 to ℓ = 3. Table 3 recapitulates the results in percentage terms. The correction, that is, the percentage reduction of the ex ante welfare losses due to bias in beliefs, goes, respectively, from 91% (computed as 100×(1,608,635-139,373)/1,608,635) to 81% to 70%. Similar attenuation holds for the 2,500 and 1,000 trip loss cases. As increases, the less important is the correction. The reason for this pattern is as follows. An observed reduction in the number of visits to site A (e.g., 4,000 trips lost) must be driven by a decrease in the ex ante utility associated with that site. In turn, this lowered utility must stem from the additional cost of making a choice under uncertainty (ρ_A) and/or from bias in beliefs (d_A). As dispersion in the assessment of environmental quality increases, ρ_A accounts for a progressively larger share of the reduction in trips, in detriment of the bias component. Since it is d_A that determines the size of the correction in equation [11], this correction becomes naturally smaller as ρ_A increases. The systematic effect is that uncertainty translates into a smaller correction of losses.

5 Conclusion

In this paper we focus on the assessment of welfare changes in discrete choice models when there may be a disparity between anticipated and experienced (ex post) utility. Unlike previous work on the topic (Leggett 2002; Allcott 2013; Schmeiser 2014; Train 2015; Glasgow and Train 2018), which assumes linear utility, we explore the case of preferences characterized by decreasing marginal utility of the environmental amenity. The proper representation of preferences remains an empirical question, but nonlinear utility may apply, for example, to recreational settings involving the potential of water or beach contamination, or to the number of kept fish in so-called meat fisheries. We have demonstrated that the analyses of Train (2015) and Glasgow and Train (2018) carry over to this context, but with two caveats. First, computation of ex post welfare losses with quasi-linear preferences requires previous decomposition of ex ante utility into a term stemming from bias in beliefs and a component representing willingness to pay for eliminating uncertainty in attributes. Without this decomposition, proper assessment of ex post welfare losses (e.g., as in an environmental disaster) is infeasible. Second, unlike the case of linear utility, increases in uncertainty translate into progressively smaller corrections of ex ante welfare losses. We illustrated our results with Monte Carlo simulations that rely on preferences (CARA) estimated from a choice experiment survey conducted among anglers on the East Coast of the United States.