- Email: [email protected]
Risk preferences over simple and compound public lotteries
Economics Letters 170 (2018) 85–87
Contents lists available at ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
Risk preferences over simple and compound public lotteries David Scrogin Department of Economics, University of Central Florida, Orlando, FL 32816-1400, USA
highlights • Risk preferences are estimated over multi-attribute public lotteries. • Simple and compound lottery choice models are estimated under CRRA preferences. • Significant risk aversion is found in male and female applicant preferences.
article
info
Article history: Received 27 March 2018 Received in revised form 9 May 2018 Accepted 10 May 2018 Available online 31 May 2018
a b s t r a c t Lottery choice experiments with monetary payoffs have a long tradition for eliciting risk preferences. This note demonstrates how risk preferences over multi-attribute public lotteries may be estimated. Consistent with experimental findings, results from simple and compound lottery choice models indicate significant differences in the degree of risk aversion between male and female applicants. © 2018 Elsevier B.V. All rights reserved.
JEL classification: D81 C93 Keywords: Risk preferences Simple lotteries Compound lotteries
1. Introduction An expansive literature has employed lottery choice experiments with monetary payoffs to elicit risk preferences and test for differences in risk preferences between individuals (see Harrison and Rutström, 2008 and Croson and Gneezy, 2009). Outside of the lab, lotteries are commonly used to allocate access to natural resources and recreational privileges on public lands and waterways, and the choice data that is collected remains under-explored (Yoder et al., 2014). This note demonstrates how risk preferences may be estimated with choice data from public lotteries. Simple and compound lottery choice models are evaluated with data from New Mexico’s lotteries for allocating elk hunting licenses. Consistent with experimental findings, individuals are found to have risk averse preferences over the multi-attribute public lotteries considered here, with male applicants on average exhibiting significantly less risk aversion than their female counterparts.
E-mail address: [email protected]. https://doi.org/10.1016/j.econlet.2018.05.012 0165-1765/© 2018 Elsevier B.V. All rights reserved.
2. Lottery frameworks 2.1. Simple lotteries In the simple lottery, participants are drawn with probability p(d), and those who are drawn receive a non-monetary ‘payoff’ characterized by a vector of deterministic attributes x. The lottery is comprised of direct and indirect costs, which include a fee F collected by the administrator and transactions costs TC incurred in the marketplace conditional upon being drawn, such as travel costs required to realize the payoff x. The administrator may require F to be pre-paid in the choice period t = 0, with a portion α ∈ [0, 1] refunded in period t = 1 to participants who were not drawn, or instead payment of F may only be required in period t = 1 by participants who were drawn. For consistency with the empirical application the latter, post-payment pricing arrangement is considered here. Denoting individual income in period t by It , and assuming intertemporal utility is additively time-separable, the expected utility of the simple lottery under post-payment pricing is E(U) = U(I0 ) + p(d)U(I1 − F − TC , x) + (1 − p(d))U(I1 )
(1)
86
D. Scrogin / Economics Letters 170 (2018) 85–87
As the elements of U(.) in (1) are non-stochastic, p(d) is the sole source of randomness in the simple lottery.
3. Analysis of lottery choices 3.1. Data
2.2. Compound lotteries
To proceed towards estimation of risk preferences, U(.) is assumed additively separable between the lottery and other goods, and risk preferences are assumed separable between the lottery and other goods. Denoting the marginal utility of income by δ , (1) may be written
The choice data was obtained from the New Mexico Department of Game and Fish lotteries for big-game hunting licenses (see Scrogin, 2005). Resident choices over the J = 136 lotteries for elk licenses are evaluated here. The data contains 23,324 applicants who were at least eighteen years of age and from households that submitted only one application. The average applicant age was 41.9 years, and 90.9% of the applicants were male. Applicants who were drawn received a license to harvest an elk within a specific geographic area and time period, and the license fees (F in (1)–(6)) ranged from $37 to $60. Lottery attributes in the models include the license fee (F ) and a travel cost (TC ) proxy, hunt time and location, license type, and the number of licenses to be awarded. In addition, the probability of being drawn (p(d)) and the probability of harvesting an elk (p(s)) must be specified. As neither is known ex ante, each is proxied from information that was available to applicants and reported in the application book.2 For each lottery, p(d) is defined as the number of resident licenses to be awarded divided by the number of prior year resident applicants, and p(s) is defined as the prior year harvest rate. p(d) ranged from 0.09 to 1 (average = 0.56), and p(s) ranged from 0 to 1 (average = 0.36).3 In the simple lottery model, p(s) enters E(U) as an attribute in x, and in the compound model p(s) enters E(U) as seen in (6).
E(U) = p(d)(U(x) − δ (F + TC )) + δ (I0 + I1 )
3.2. Estimation results
In the compound lottery, participants are also drawn with probability p(d). However, the attribute vector x contains both deterministic and random elements. Assuming the latter is comprised of a single, binary (0–1) random variable, and denoting the conditional probability of a ‘success’ (i.e., a 1) by p(s) and the deterministic elements of x by z, the expected utility of the compound lottery with post-payment pricing is E(U) = U(I0 ) + p(d)(p(s)U(I1 − TC − F , z , 1)
+ (1 − p(s))U(I1 − TC − F , z , 0)) + (1 − p(d))U(I1 )
(2)
The compound lottery is therefore a simple lottery within a simple lottery. 2.3. Preference structures over lotteries
(3)
Similarly, (2) may be written E(U) = p(d)(p(s)U(z , 1) + (1 − p(s))U(z , 0) − δ (F + TC ))
+ δ (I0 + I1 )
(4)
To allow for risk aversion over lotteries, U(.) in (3) and (4) is specified according to constant relative risk aversion (CRRA). With the simple lottery U(x) = (x′ β)1−r /(1 − r), and with the compound lottery U(z , 0) = (z ′ γ )1−r /(1 − r) and U(z , 1) = (z ′ γ + γ0 )1−r /(1 − r). The term r is the coefficient of constant relative risk aversion, β and γ are parameter vectors, and γ0 is a constant. Concavity of U(.) and risk aversion holds for r > 0, risk neutrality holds for r = 0, and risk seeking behavior holds for r < 0. Substituting for U(x) in (3), E(U) in the simple lottery is
( E(U) = p(d)
(x′ β)1−r 1−r
) − δ (TC + F ) + δ (I0 + I1 )
(5)
Substituting for U(z , 0) and U(z , 1) in (4), E(U) in the compound lottery is
( E(U) = p(d) p(s)
(z′ γ + γ0 )1−r 1−r
+ (1 − p(s))
(z′ γ )1−r 1−r
− δ (TC + F )
) (6)
+ δ (I0 + I1 ) Given a set of lotteries from which expected utility maximizing individuals jointly select a lottery and consumption of other goods, the parameters in (5) and (6) may be estimated by discrete choice methods, and McFadden’s conditional logit model is employed here.1 The data and estimation results are discussed below. Of interest is the risk aversion coefficient r, which is conditioned upon individual characteristics in order to test for differences in risk preferences across lottery participants. 1 The derivations of the log-likelihood functions are available in an appendix, and the term δ (I0 + I1 ) in (5) and (6) drops from estimation as it exhibits no withinindividual variation between lotteries.
Estimates of the coefficients of the risk function (r = r0 + r1 Age + r2 Male) in the simple and compound lottery choice models are reported in Table 1. Applicant age has a significant positive effect on the degree of risk aversion (p < 0.05), and female applicants are significantly more risk averse than male applicants on average (p < 0.01). The estimated age and gender coefficients are highly similar between the simple and compound lottery choice models, while the estimated constant is marginally smaller in the former, indicating the simple lottery will yield more conservative estimates of the degree of risk aversion for applicants of a given age and gender.4 Estimates of the CRRA parameter (r) are reported in Table 2 for the full sample and the male and female sub-samples. Overall, the results indicate a significant but moderate degree of risk aversion for the average aged lottery applicant, and consistent with Table 1 the estimated r from the simple lottery model is less than the compound model (0.061 versus 0.074). Decomposing the sample by applicant gender, the estimate of r for a female applicant is more than three times larger than that of a male applicant on average in the compound lottery model (0.061 versus 0.209), and from the simple model the difference is estimated to be more than four times larger for female versus male applicants (0.047 versus 0.194). 2 An implicit assumption therefore is that the proxies for p(d) and p(s) were the applicants’ subjective probabilities. While this may not be unreasonable, the lottery choice data from field-like experiments such as those considered here could be augmented with data from belief elicitation tasks in order to estimate applicant subjective probabilities together with risk preferences over lotteries (see e.g., Andersen et al., 2014 and Harrison et al., 2017 for discussion and experimental implementation of belief elicitation tasks via linear and quadratic scoring rules). 3 For comparison over the lotteries, the proxy for p(d) employed here differs from the true probability (calculated with the actual number of applicants) by 0.054 on average, and the absolute value of the difference between the probabilities is less than 0.1 in about forty-five percent of the cases. 4 The estimated coefficients of the variables contained in the attribute vectors (x and z) are also highly similar. The full set of estimation results is available in an appendix.
D. Scrogin / Economics Letters 170 (2018) 85–87 Table 1 Estimates of the coefficients of the risk aversion function.
4. Conclusions
Risk function (r = x′ r)
Simple lottery
Compound lottery
Age
0.011** (0.006) [−0.003, 0.026] −0.146*** (0.027) [−0.217, −0.076] 0.146*** (0.036) [0.053, 0.239]
0.012** (0.006) [−0.003, 0.026] −0.149*** (0.028) [−0.220, −0.077] 0.160*** (0.037) [0.066, 0.255]
−102,280
−102,277
Male
Constant
Log-likelihood
NOTE: The full set of estimation results is reported in an appendix. Standard errors are in parentheses and 99% confidence intervals are in brackets. *** Significance at the 1% level **
Significance at the 5% level
Table 2 Estimates of the risk aversion coefficient. Risk aversion coefficient (r)
Simple lottery
Compound lottery
Full sample
0.061*** (0.017) [0.017, 0.104] 0.047*** (0.018) [0.002, 0.093] 0.194*** (0.028) [0.121, 0.266]
0.074*** (0.018) [0.027, 0.121] 0.061*** (0.019) [0.012, 0.109] 0.209*** (0.028) [0.136, 0.282]
Male
Female
87
NOTE: Standard errors are in parentheses and 99% confidence intervals are in brackets. *** Significance at the 1% level
Similar gender differences in risk preferences over monetary lotteries are reported in the survey of experimental findings by Croson and Gneezy (2009), and as CRRA is a standard specification of utility in the experimental literature, the results reported here can be compared to those from lottery choice experiments.
Consistent with experimental findings from choices over monetary lotteries, this study finds significant risk aversion in individual choices over multi-attribute public lotteries for big-game hunting licenses. Estimation results are robust across simple and compound lottery choice models, with male applicants exhibiting significantly less aversion to risk than female applicants under post-payment lottery pricing rules. Acknowledgments I am grateful to a referee for detailed comments and suggestions that improved the paper. Appreciation is also extended to the New Mexico Department of Game and Fish for providing the data used in the analysis. Any errors and the interpretations of the results are my own. Appendix A. Supplementary data Supplementary material related to this article can be found online at https://doi.org/10.1016/j.econlet.2018.05.012. References Andersen, S., Fountain, J., Harrison, G.W., Rutström, E.E., 2014. Estimating subjective probabilities. J. Risk Uncertain. 48 (3), 207–229. Croson, R., Gneezy, U., 2009. Gender differences in preferences. J. Econ. Lit. 47 (2), 1–27. Harrison, G.W., Rutström, E.E., 2008. Risk aversion in the laboratory. In: Cox, J.C., Harrison, G.W. (Eds.), Risk Aversion in Experiments. In: Research in Experimental Economics, vol. 12, Emerald Group Publishing Limited, pp. 41–196. Harrison, G.W., Martinez-Correa, J., Swarthout, J.T., Ulm, E.R., 2017. Scoring rules of subjective probability distributions. J. Econ. Behav. Organ. 134, 430–448. Scrogin, D., 2005. Lottery-rationed public access under alternative tariff arrangements: Changes in quality, quantity, and expected utility. J. Environ. Econ. Manag. 50 (1), 189–211. Yoder, J., Ohler, A., Chouinard, H.H., 2014. What floats your boat? Preference revelation from lotteries over complex goods. J. Environ. Econ. Manag. 67 (3), 412– 430.
Contents lists available at ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
Risk preferences over simple and compound public lotteries David Scrogin Department of Economics, University of Central Florida, Orlando, FL 32816-1400, USA
highlights • Risk preferences are estimated over multi-attribute public lotteries. • Simple and compound lottery choice models are estimated under CRRA preferences. • Significant risk aversion is found in male and female applicant preferences.
article
info
Article history: Received 27 March 2018 Received in revised form 9 May 2018 Accepted 10 May 2018 Available online 31 May 2018
a b s t r a c t Lottery choice experiments with monetary payoffs have a long tradition for eliciting risk preferences. This note demonstrates how risk preferences over multi-attribute public lotteries may be estimated. Consistent with experimental findings, results from simple and compound lottery choice models indicate significant differences in the degree of risk aversion between male and female applicants. © 2018 Elsevier B.V. All rights reserved.
JEL classification: D81 C93 Keywords: Risk preferences Simple lotteries Compound lotteries
1. Introduction An expansive literature has employed lottery choice experiments with monetary payoffs to elicit risk preferences and test for differences in risk preferences between individuals (see Harrison and Rutström, 2008 and Croson and Gneezy, 2009). Outside of the lab, lotteries are commonly used to allocate access to natural resources and recreational privileges on public lands and waterways, and the choice data that is collected remains under-explored (Yoder et al., 2014). This note demonstrates how risk preferences may be estimated with choice data from public lotteries. Simple and compound lottery choice models are evaluated with data from New Mexico’s lotteries for allocating elk hunting licenses. Consistent with experimental findings, individuals are found to have risk averse preferences over the multi-attribute public lotteries considered here, with male applicants on average exhibiting significantly less risk aversion than their female counterparts.
E-mail address: [email protected]. https://doi.org/10.1016/j.econlet.2018.05.012 0165-1765/© 2018 Elsevier B.V. All rights reserved.
2. Lottery frameworks 2.1. Simple lotteries In the simple lottery, participants are drawn with probability p(d), and those who are drawn receive a non-monetary ‘payoff’ characterized by a vector of deterministic attributes x. The lottery is comprised of direct and indirect costs, which include a fee F collected by the administrator and transactions costs TC incurred in the marketplace conditional upon being drawn, such as travel costs required to realize the payoff x. The administrator may require F to be pre-paid in the choice period t = 0, with a portion α ∈ [0, 1] refunded in period t = 1 to participants who were not drawn, or instead payment of F may only be required in period t = 1 by participants who were drawn. For consistency with the empirical application the latter, post-payment pricing arrangement is considered here. Denoting individual income in period t by It , and assuming intertemporal utility is additively time-separable, the expected utility of the simple lottery under post-payment pricing is E(U) = U(I0 ) + p(d)U(I1 − F − TC , x) + (1 − p(d))U(I1 )
(1)
86
D. Scrogin / Economics Letters 170 (2018) 85–87
As the elements of U(.) in (1) are non-stochastic, p(d) is the sole source of randomness in the simple lottery.
3. Analysis of lottery choices 3.1. Data
2.2. Compound lotteries
To proceed towards estimation of risk preferences, U(.) is assumed additively separable between the lottery and other goods, and risk preferences are assumed separable between the lottery and other goods. Denoting the marginal utility of income by δ , (1) may be written
The choice data was obtained from the New Mexico Department of Game and Fish lotteries for big-game hunting licenses (see Scrogin, 2005). Resident choices over the J = 136 lotteries for elk licenses are evaluated here. The data contains 23,324 applicants who were at least eighteen years of age and from households that submitted only one application. The average applicant age was 41.9 years, and 90.9% of the applicants were male. Applicants who were drawn received a license to harvest an elk within a specific geographic area and time period, and the license fees (F in (1)–(6)) ranged from $37 to $60. Lottery attributes in the models include the license fee (F ) and a travel cost (TC ) proxy, hunt time and location, license type, and the number of licenses to be awarded. In addition, the probability of being drawn (p(d)) and the probability of harvesting an elk (p(s)) must be specified. As neither is known ex ante, each is proxied from information that was available to applicants and reported in the application book.2 For each lottery, p(d) is defined as the number of resident licenses to be awarded divided by the number of prior year resident applicants, and p(s) is defined as the prior year harvest rate. p(d) ranged from 0.09 to 1 (average = 0.56), and p(s) ranged from 0 to 1 (average = 0.36).3 In the simple lottery model, p(s) enters E(U) as an attribute in x, and in the compound model p(s) enters E(U) as seen in (6).
E(U) = p(d)(U(x) − δ (F + TC )) + δ (I0 + I1 )
3.2. Estimation results
In the compound lottery, participants are also drawn with probability p(d). However, the attribute vector x contains both deterministic and random elements. Assuming the latter is comprised of a single, binary (0–1) random variable, and denoting the conditional probability of a ‘success’ (i.e., a 1) by p(s) and the deterministic elements of x by z, the expected utility of the compound lottery with post-payment pricing is E(U) = U(I0 ) + p(d)(p(s)U(I1 − TC − F , z , 1)
+ (1 − p(s))U(I1 − TC − F , z , 0)) + (1 − p(d))U(I1 )
(2)
The compound lottery is therefore a simple lottery within a simple lottery. 2.3. Preference structures over lotteries
(3)
Similarly, (2) may be written E(U) = p(d)(p(s)U(z , 1) + (1 − p(s))U(z , 0) − δ (F + TC ))
+ δ (I0 + I1 )
(4)
To allow for risk aversion over lotteries, U(.) in (3) and (4) is specified according to constant relative risk aversion (CRRA). With the simple lottery U(x) = (x′ β)1−r /(1 − r), and with the compound lottery U(z , 0) = (z ′ γ )1−r /(1 − r) and U(z , 1) = (z ′ γ + γ0 )1−r /(1 − r). The term r is the coefficient of constant relative risk aversion, β and γ are parameter vectors, and γ0 is a constant. Concavity of U(.) and risk aversion holds for r > 0, risk neutrality holds for r = 0, and risk seeking behavior holds for r < 0. Substituting for U(x) in (3), E(U) in the simple lottery is
( E(U) = p(d)
(x′ β)1−r 1−r
) − δ (TC + F ) + δ (I0 + I1 )
(5)
Substituting for U(z , 0) and U(z , 1) in (4), E(U) in the compound lottery is
( E(U) = p(d) p(s)
(z′ γ + γ0 )1−r 1−r
+ (1 − p(s))
(z′ γ )1−r 1−r
− δ (TC + F )
) (6)
+ δ (I0 + I1 ) Given a set of lotteries from which expected utility maximizing individuals jointly select a lottery and consumption of other goods, the parameters in (5) and (6) may be estimated by discrete choice methods, and McFadden’s conditional logit model is employed here.1 The data and estimation results are discussed below. Of interest is the risk aversion coefficient r, which is conditioned upon individual characteristics in order to test for differences in risk preferences across lottery participants. 1 The derivations of the log-likelihood functions are available in an appendix, and the term δ (I0 + I1 ) in (5) and (6) drops from estimation as it exhibits no withinindividual variation between lotteries.
Estimates of the coefficients of the risk function (r = r0 + r1 Age + r2 Male) in the simple and compound lottery choice models are reported in Table 1. Applicant age has a significant positive effect on the degree of risk aversion (p < 0.05), and female applicants are significantly more risk averse than male applicants on average (p < 0.01). The estimated age and gender coefficients are highly similar between the simple and compound lottery choice models, while the estimated constant is marginally smaller in the former, indicating the simple lottery will yield more conservative estimates of the degree of risk aversion for applicants of a given age and gender.4 Estimates of the CRRA parameter (r) are reported in Table 2 for the full sample and the male and female sub-samples. Overall, the results indicate a significant but moderate degree of risk aversion for the average aged lottery applicant, and consistent with Table 1 the estimated r from the simple lottery model is less than the compound model (0.061 versus 0.074). Decomposing the sample by applicant gender, the estimate of r for a female applicant is more than three times larger than that of a male applicant on average in the compound lottery model (0.061 versus 0.209), and from the simple model the difference is estimated to be more than four times larger for female versus male applicants (0.047 versus 0.194). 2 An implicit assumption therefore is that the proxies for p(d) and p(s) were the applicants’ subjective probabilities. While this may not be unreasonable, the lottery choice data from field-like experiments such as those considered here could be augmented with data from belief elicitation tasks in order to estimate applicant subjective probabilities together with risk preferences over lotteries (see e.g., Andersen et al., 2014 and Harrison et al., 2017 for discussion and experimental implementation of belief elicitation tasks via linear and quadratic scoring rules). 3 For comparison over the lotteries, the proxy for p(d) employed here differs from the true probability (calculated with the actual number of applicants) by 0.054 on average, and the absolute value of the difference between the probabilities is less than 0.1 in about forty-five percent of the cases. 4 The estimated coefficients of the variables contained in the attribute vectors (x and z) are also highly similar. The full set of estimation results is available in an appendix.
D. Scrogin / Economics Letters 170 (2018) 85–87 Table 1 Estimates of the coefficients of the risk aversion function.
4. Conclusions
Risk function (r = x′ r)
Simple lottery
Compound lottery
Age
0.011** (0.006) [−0.003, 0.026] −0.146*** (0.027) [−0.217, −0.076] 0.146*** (0.036) [0.053, 0.239]
0.012** (0.006) [−0.003, 0.026] −0.149*** (0.028) [−0.220, −0.077] 0.160*** (0.037) [0.066, 0.255]
−102,280
−102,277
Male
Constant
Log-likelihood
NOTE: The full set of estimation results is reported in an appendix. Standard errors are in parentheses and 99% confidence intervals are in brackets. *** Significance at the 1% level **
Significance at the 5% level
Table 2 Estimates of the risk aversion coefficient. Risk aversion coefficient (r)
Simple lottery
Compound lottery
Full sample
0.061*** (0.017) [0.017, 0.104] 0.047*** (0.018) [0.002, 0.093] 0.194*** (0.028) [0.121, 0.266]
0.074*** (0.018) [0.027, 0.121] 0.061*** (0.019) [0.012, 0.109] 0.209*** (0.028) [0.136, 0.282]
Male
Female
87
NOTE: Standard errors are in parentheses and 99% confidence intervals are in brackets. *** Significance at the 1% level
Similar gender differences in risk preferences over monetary lotteries are reported in the survey of experimental findings by Croson and Gneezy (2009), and as CRRA is a standard specification of utility in the experimental literature, the results reported here can be compared to those from lottery choice experiments.
Consistent with experimental findings from choices over monetary lotteries, this study finds significant risk aversion in individual choices over multi-attribute public lotteries for big-game hunting licenses. Estimation results are robust across simple and compound lottery choice models, with male applicants exhibiting significantly less aversion to risk than female applicants under post-payment lottery pricing rules. Acknowledgments I am grateful to a referee for detailed comments and suggestions that improved the paper. Appreciation is also extended to the New Mexico Department of Game and Fish for providing the data used in the analysis. Any errors and the interpretations of the results are my own. Appendix A. Supplementary data Supplementary material related to this article can be found online at https://doi.org/10.1016/j.econlet.2018.05.012. References Andersen, S., Fountain, J., Harrison, G.W., Rutström, E.E., 2014. Estimating subjective probabilities. J. Risk Uncertain. 48 (3), 207–229. Croson, R., Gneezy, U., 2009. Gender differences in preferences. J. Econ. Lit. 47 (2), 1–27. Harrison, G.W., Rutström, E.E., 2008. Risk aversion in the laboratory. In: Cox, J.C., Harrison, G.W. (Eds.), Risk Aversion in Experiments. In: Research in Experimental Economics, vol. 12, Emerald Group Publishing Limited, pp. 41–196. Harrison, G.W., Martinez-Correa, J., Swarthout, J.T., Ulm, E.R., 2017. Scoring rules of subjective probability distributions. J. Econ. Behav. Organ. 134, 430–448. Scrogin, D., 2005. Lottery-rationed public access under alternative tariff arrangements: Changes in quality, quantity, and expected utility. J. Environ. Econ. Manag. 50 (1), 189–211. Yoder, J., Ohler, A., Chouinard, H.H., 2014. What floats your boat? Preference revelation from lotteries over complex goods. J. Environ. Econ. Manag. 67 (3), 412– 430.