A graphical exposition of the profit maximizing two-part tariff
This note presents a simple graphical approach for deriving the profit maximizing two-part tariff when there are two types of consumers in the market. The exposition covers both the case in which a monopoly can offer only one two-part tariff plan and the case in which the monopoly can offer two different plans. The American Economist (forthcoming).
(With M. 't Sas-Rolfes) We present an economic model of a market for wildlife products. We use it to study theoretically the potential impact on the poaching of wildlife animals if a legal market for synthetic wildlife products is created. We show that allowing for a legal trade in synthetic substitutes in general has two opposing effects on poaching level: a price effect that reduces poaching by lowering the revenue generated from poaching; and a laundering effect that encourages poaching by making it easier for poachers to sell their products. When demand for wildlife goods is inelastic, the price effect is bigger, while the laundering effect is smaller; hence, establishing a legal market for synthetic substitutes is more likely to reduce poaching when demand is less sensitive to price changes. Measures that make it more difficult for poachers to launder their products reduce the laundering effect and enhance the conservation value of producing synthetic substitutes. Ecological Economics 180 (2021), 106873.
Why discuss fake rhino horns? (With F. Vollrath) SWARA: The Voice of Conservation in East Africa, January-March, 2020, 48-53.
Leland and Meyer showed through an example that when the distribution of consumer types is nondegenerate, there exist two-block pricing plans that yield strictly higher producer surplus than the profit maximizing two-part tariff scheme. This led them to pose the following question: does weak dominance (in the sense of profit) rather than strict dominance of two-block pricing over two-part tariff hold only when the distribution of buyer-types is degenerate? This note shows that the answer to this question is, No: even when the distribution of consumer types is continuous, it is possible that the best two-block pricing scheme performs no better than the best two-part tariff. The American Economist 65 (2020), 204 –213 .
A simple dynamic model of vaccination is presented and analyzed to study how the amount of vaccines available affects people’s vaccination decisions. In addition, the model is used to examine how the level of vaccination in equilibrium compares to the efficient or socially optimal level. It is shown that, when the stock of vaccines is large so that a shortage could never arise, an equilibrium is generically unique, and there is too little vaccination compared to the social optimum. When the stock of vaccines is small so that not everyone in a population could get vaccinated, a shortage could occur in equilibrium. Moreover, the occurrence of a shortage could be self-fulfilling: when agents expect a shortage to develop, they have a greater incentive to demand the vaccine right away, which increases the likelihood that a shortage will result; and when agents do not expect a shortage to arise, they are more willing to delay their vaccination decision, which reduces the likelihood of a shortage developing. This leads to the possible co-existence of multiple equilibria that differ in the level of vaccination. Multiple equilibria can arise, however, only if agents are uncertain about the cost of being infected – if agents are sufficiently certain about the cost of infection, then an equilibrium is unique. Furthermore, when the stock of vaccines is small, an equilibrium level of vaccination may be too high relative to the socially optimal level. Increasing the vaccine stock could have ambiguous effects on the level of vaccination in equilibrium but unambiguously increases social welfare. Journal of Theoretical Biology 446 (2018), 19-32.
(With R. Stevens*) Seasonal influenza imposes an enormous burden on society every year, yet many people refuse to obtain flu shots due to misconceptions of the flu vaccine. We argue that recent research in psychology and behavioral economics may provide the answers to why people hold mistaken beliefs about flu shots, how we can correct these misconceptions, and what policy-makers can do to increase flu vaccination rates. Health Promotion International 32 (2017), 1067-1073.
(With R. Regan*) We study the incentives that museums face in determining how much resources to invest in the protection of their artwork from theft. We present and analyze a game-theoretic model of art heists that accounts for the strategic interactions between museums’ and art thieves’ decisions and that incorporates several key features of the black market for stolen art. We find that the equilibrium level of security museums choose need not be monotonic in the true market value or the black market value of artwork, i.e., increasing the value of an art piece—whether it is the true market value or the black market value—does not necessarily lead museums to invest more in protecting their artwork. The effects of parameter changes in the model that reflect a shift of public policy depend critically on what type of policy change is considered. For instance, an increase in the penalty imposed for committing art theft cannot raise the amount of theft in equilibrium and could in fact lead museums to increase their level of security. On the other hand, investing more resources on law enforcement agencies so that they are better able to solve art crimes can actually increase the amount of theft in equilibrium by causing museums to spend less on security. Journal of Cultural Economics 41 (2017), 283-307.
To examine the potential impact of synthetic horns to reduce rhino poaching, a formal model of the rhino horn market in which there exist firms with the capability to produce high quality synthetic horns is presented and studied. The analysis shows that whether the availability of synthetic horns would decrease the equilibrium supply of wild horns—and how much the reduction would be—depends on market structure—i.e., how competitive the synthetic horn production sector is—and on how substitutable the synthetic horns are for wild horns. The implications of these results for conservation policies are derived and discussed. Synthetic horn producers would benefit more by promoting their products as being superior to wild horns, but this could increase horn prices and lead to more rhino poaching. For conservation purposes, it may be beneficial to incentivize firms to produce inferior fakes—synthetic horns that are engineered to be undesirable in some respect but difficult for buyers to distinguish from wild horns. The analysis also shows that promoting competition in the production of synthetic horns in general is desirable from a conservation standpoint as synthetic horn producers may prefer to keep prices at a high enough level that could still encourage significant amount of poaching. Ecological Economics 141 (2017), 180-189.
Habitat loss, climate change, poaching, and illegal wildlife trafficking are threatening a large number of animal populations. Recent research suggests that people place higher value on wildlife or wildlife‐derived goods when they are considered rare or uncommon. Although supply‐side antipoaching policies—such as strengthening the enforcement of regulations against illegal hunting—decrease people's incentives to engage in poaching, one significant consequence of the demand for rarity is that it weakens the impact of these policies. While the demand for rarity cannot completely offset the effects of supply‐side antipoaching policies, the reduction in the amount of poaching that can be achieved using these policies may not be large when people are willing to pay more for rare wildlife goods. This calls into question whether actions targeting the supply side of the market for poaching are the most effective ones for protecting endangered or threatened species affected by poaching. Conservation Letters 9 (2016), 65-69.
(With F. Toxværd) The market for vaccinations is widely believed to be characterized by market failures, because individuals do not internalize the positive externalities that their vaccination decisions may confer on other individuals. Francis (1997) provided a set of assumptions under which the equilibrium vaccination pattern is socially optimal. We show that his conditions are not necessary for the welfare theorem to hold but that in general, the market yields inefficiently low vaccination uptake. Equilibrium non-optimality may obtain if (i) agents can recover from infection, (ii) vaccines are imperfect, (iii) individuals are ex ante heterogeneous, (iv) vaccination timing is inflexible or (v) the planning horizon is finite. Apart from the case with heterogeneity, inefficiencies result from the presence of strategic interaction. Journal of Theoretical Biology 363 (2014), 105-117.
The economic approach to modeling self-protective behavior in epidemiology
In P. Manfredi and A. d'Onofrio (eds.), Modeling the Interplay Between Human Behavior and the Spread of Infectious Diseases. New York: Springer, 2013.
(With J. Heckelman) We develop a lottery procedure for selecting multiple winners that is strategy proof. The rule assigns points to each candidate based on any standard scoring rule method, and then uses one draw to select a single winning set of candidates in proportion to their collective score. In addition to being strategy proof, the lottery rule is also shown to have several other attractive normative properties. Violations of some other important normative properties are noted as well. Journal of Public Economic Theory 15 (2013), 103-123.
(With J. Raley*) In this paper, we advocate the writing of mathematical fiction (i) as an aid for students in learning mathematics, and (ii) to engage students in doing mathematics. Journal of Humanistic Mathematics 3(2) (2013), article 7.
(with A. Griffith, A. Cottrell, and Y.-L. Wong) We report the results of a study we conducted using a simple multiplayer online game that simulates the spread of an infectious disease through a population composed of the players. We use our virtual epidemics game to examine how people respond to epidemics. The analysis shows that people's behavior is responsive to the cost of self-protection, the reported prevalence of disease, and their experiences earlier in the epidemic. Specifically, decreasing the cost of self-protection increases the rate of safe behavior. Higher reported prevalence also raises the likelihood that individuals would engage in self-protection, where the magnitude of this effect depends on how much time has elapsed in the epidemic. Individuals' experiences in terms of how often an infection was acquired when they did not engage in self-protection are another factor that determines whether they will invest in preventive measures later on. All else being equal, individuals who were infected at a higher rate are more likely to engage in self-protective behavior compared to those with a lower rate of infection. Lastly, fixing everything else, people's willingness to engage in safe behavior waxes or wanes over time, depending on the severity of an epidemic: when prevalence is high, people are more likely to adopt self-protective measures as time goes by; when prevalence is low, a ‘self-protection fatigue’ effect sets in whereby individuals are less willing to engage in safe behavior over time. PLOS ONE 8(1) (2013), e52814.
The decision of individuals to engage in public avoidance during epidemics is modeled and studied using game theory. The analysis reveals that the set of Nash equilibria of the model, as well as how the equilibria compare to the social optimum, depend on the contact function that governs the rate at which encounters occur in public. If the contact ratio – defined to be the ratio of the contact rate to the number of people out in public – is increasing with the number of people out in public, then there exists a unique Nash equilibrium. Moreover, in equilibrium, the amount of public avoidance is too low with respect to social welfare. On the other hand, if the contact ratio is decreasing in the number of people out in public, then there can be multiple Nash equilibria, none of which is in general socially optimal. Furthermore, the amount of public avoidance in equilibrium with a decreasing contact ratio is too high in that social welfare can be increased if more susceptible individuals choose to go out in public. In the special case where the contact ratio does not vary with the number of people out in public, there is a unique Nash equilibrium, and it is also the socially optimal outcome. Journal of Theoretical Biology 302 (2012), 18-28.
(With M. Jiang, S. Rabidoux*, and S. Robinson) In this paper, we present a mathematical model of infectious disease transmission in which people can engage in public avoidance behavior to minimize the likelihood of acquiring an infection. The framework employs the economist's theory of utility maximization to model people's decision regarding their level of public avoidance. We derive the reproductive number of a disease which determines whether an endemic equilibrium exists or not. We show that when the contact function exhibits saturation, an endemic equilibrium must be unique. Otherwise, multiple endemic equilibria that differ in disease prevalence can coexist, and which one the population gets to depends on initial conditions. Even when a unique endemic equilibrium exists, people's preferences and the initial conditions may determine whether the disease will eventually die out or become endemic. Public health policies that increase the recovery rate or encourage self-quarantine by infected people can be beneficial to the community by lowering disease prevalence. However, it is also possible for these policies to worsen the situation and cause prevalence to rise since these measures give people less incentive to engage in public avoidance behavior. We also show that implementing policies that result in a higher level of public avoidance behavior in equilibrium does not necessarily lower prevalence and can result in more infections. Journal of Theoretical Biology 278 (2011), 107-119.
The author presents a simple geometric method to graphically illustrate the expected utility from a gamble with more than two possible outcomes. This geometric result gives economics students a simple visual aid for studying expected utility theory and enables them to analyze a richer set of decision problems under uncertainty compared to what they typically see in an intermediate-level microeconomics course. Journal of Economic Education 41 (2010), 63-70.
(With A. Kays*) In this note, we use game theory to study the incentives that college students have for purchasing season tickets to their schools’ football games if the schools implement a lottery system whereby students can try to win free tickets by entering an online drawing. Mathematics Magazine 82 (2009), 283-289.
How the quality of information about the prevalence of an infectious disease affects individuals’ incentives to adopt self-protective actions to reduce the risk of infection is studied using an economic/game-theoretic model of epidemics. In the model, agents make inferences regarding the current prevalence of a disease by observing the health status of a subset of the population. Therefore, the higher the number of agents whose infection status can be observed, the better one’s information about the current prevalence is. In particular, it is assumed that an agent’s estimate of the current prevalence depends on observations of the current health status of other agents and on the agent’s estimate of past prevalence, and that the agent places more weight on the current observations in forming an estimate of the prevailing prevalence when the number of observations increases. It is shown that the likelihood of eradicating an infectious disease through behavioral changes depends critically on the amount of information that individuals have access to, which also determines whether prevalence will be relatively stable or will exhibit cyclical patterns over time. Increasing the amount of information that individuals possess may lower the likelihood of eradication. Mathematical Biosciences 217 (2009), 125-133.
(With A. Cottrell) The dynamics of an epidemic model with voluntary vaccinations are studied. Individual vaccination decisions are modelled using an economic/game-theoretic approach: agents in the model decide whether to vaccinate or not by weighing the cost and benefit of vaccination and choose the action that maximizes their net benefit. It is shown that, when vaccine efficacy is low, there are parameter values for which multiple steady-state equilibria and periodic equilibria coexist. When multiplicity of steady states is obtained, which one the population reaches in some cases depends entirely on agents’ expectations concerning the future course of an epidemic and not on the initial conditions of the model. Journal of Biological Dynamics 3 (2009), 357-375.
Rebated losses revisited
(With J. Grosjean) In S. N. Ethier and W. R. Eadington (eds.), Optimal Play: Mathematical Studies of Games and Gambling. Reno, NV: Institute for the Study of Gambling and Commerical Gaming, 2007.
An issue about more than disease. Winston-Salem Journal, June 23, 2007.
The impact that the introduction of vaginal microbicides and subsequent changes in condom usage have on HIV prevalence in men and women depends critically on the efficacy of microbicides in blocking transmission to women as well as their ability to reduce the infectiousness of infected women. Neglecting to consider how differences in microbicide efficacy by direction of transmission can affect HIV prevalence over time will result in underestimating the effect of condom replacement. AIDS 20 (2006), 1551-1553.
An susceptible-infected epidemic model with endogenous behavioral changes is presented to analyze the impact of a prophylactic vaccine on disease prevalence. It is shown that, with voluntary vaccination, whether an endemic equilibrium exists or not does not depend on vaccine efficacy or the distribution of agent-types. Although an endemic equilibrium is unique in the absence of a vaccine, the availability of a vaccine can lead to multiple endemic equilibria that differ in disease prevalence and vaccine coverage. Depending on the distribution of agent-types, the introduction of a vaccine or, if one is available, a subsidy for vaccination can increase disease prevalence by inducing more risky behavior. Journal of Mathematical Biology 53 (2006), 253-272.
An epidemiological model of HIV transmission with endogenously determined risk behavior is presented, and its predictions regarding the impact of improving access to antiretroviral therapy (ART) are derived. It is shown that, if antiretroviral treatment decreases the transmission probability by a higher percentage than the treatment-induced reduction in AIDS mortality rate, then HIV incidence may decrease by expanding ART coverage. Otherwise, increasing access to treatment will worsen the epidemic. Although expanding ART coverage can result in disease eradication if the transmission probability is lowered suﬃciently, the conditions under which eradication obtains are more stringent than those predicted by conventional epidemiological models with exogenously speciﬁed individual behavior. The Mathematical Scientist 31 (2006), 11-20.
An epidemic model of HIV transmission with self-protective behavior and preferred mixing is presented. Individuals in the model are assumed to choose their levels of risk behavior by comparing the costs and benefits of self-protective actions. Unlike in models which treat individual risk behavior as exogenously given and fixed, the condition under which an endemic steady state equilibrium exists does not depend on the extent of assortative mixing in the population. Specifically, a unique endemic equilibrium exists when the basic reproductive number of the disease, which is given in the model by the expected number of secondary infections caused by an infected individual in the absence of any self-protection, is strictly greater than one. Otherwise, the disease-free equilibrium is the only steady state equilibrium. With respect to changes in contact patterns, it is shown that, if the degree of preferred mixing is increased, the disease prevalence can decrease in the high-risk subpopulation consisting of individuals who are more likely to engage in unsafe practices. The situation is reversed for the low-risk subpopulation, which is composed of individuals who are less willing to engage in risky practices, so that increasing the likelihood of mixing with members of one’s own group may increase the prevalence level within the low-risk subpopulation. Mathematical Biosciences 199 (2006), 141-159.
(With J. Heckelman) The pairwise lottery system is a multiple round voting procedure which chooses by lot a winner from a pair of alternatives to advance to the next round where in each round the odds of selection are based on each alternative’s majority rule votes. We develop a framework for determining the asymptotic relative likelihood of the lottery selecting in the final round the Borda winner, Condorcet winner, and Condorcet loser for the three alternative case. We also show the procedure is equivalent to a Borda lottery when only a single round of voting is conducted. Finally, we present an alternative voting rule which yields the same winning probabilities as the pairwise lottery in the limiting case as the number of rounds of the pairwise lottery becomes large. Economic Theory 26 (2005), 607-617.
It is well-known from Becker’s [Becker, G., 1973. A theory of marriage, Part I. Journal of Political Economy 81, 813–846] analysis of the neoclassical marriage market that matching is positive assortative if agent-types are complements in household production. Shimer and Smith [Shimer, R., Smith, L., 2000. Assortative matching and search. Econometrica 68, 343–369] have shown that this result does not generally extend to markets with search frictions, and they provide additional conditions on the production function that guarantee assortative matching in these settings. Here, the relationship between the Shimer–Smith restrictions and equilibrium matching pattern is considered, and alternative proofs of their results on assortative matching are provided. Journal of Mathematical Economics 41 (2005), 705-721.
The susceptible-infected (SI) model is extended by allowing for individual optimal choices of self-protective actions against infection, where agents differ with respect to preferences and costs of self-protection. It is shown that a unique endemic equilibrium prevalence exists when the basic reproductive number of a STD is strictly greater than unity, and that the disease-free equilibrium is the unique steady state equilibrium when the basic reproductive number is less than or equal to one. Unlike in models that take individual behavior as given and fixed, the endemic equilibrium prevalence need not vary monotonically with respect to the basic reproductive number. Specifically, with endogenously determined self-protective behavior, a reduction in the basic reproductive number may in fact increase the endemic equilibrium prevalence. The global stability of the endemic steady state is established for the case of a homogeneous population by showing that, for any non-zero initial disease prevalence, there exists an equilibrium path which converges to the endemic steady state. Theoretical Population Biology 66 (2004), 307-316.
Unraveling in a dynamic matching market with Nash bargaining
Equilibrium sorting in a finite−horizon, two−sided matching market with heterogeneous agents is considered. It is shown that, if the match production function is additively separable in agent−types and if the division of match output is determined by the Nash bargaining solution, then an unraveling of the market obtains as the unique equilibrium in which all matches are formed in the first period. Economics Bulletin 3 (2003), 1-6.
Equilibrium sorting in marriage markets with search frictions and transferable utility is considered. In particular, a sufficient condition on payoffs for the formation of matches between agents of the same types is provided. Economics Letters 78 (2003), 317-321.
* student or former student