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The gross income

A potential victim is characterized by his income or wealth, which we denote by y. Let a be the perceived probability of this person being attacked. A victim can seek protection against attack; think of this as “defense” d. While not directly affecting a itself ðthough in equilibrium a will be endogenousÞ, an individual’s investment in defense lowers the probability that the attack will be effective. Write this probability as p 5 pðdÞ, with p continuous and decreasing in d. While we regard d somewhat abstractly here, it has several interpretations to which we return below. For now, we simply view a potential victim with income y as picking d to maximize ð1 2 aÞy 1 afpðdÞð1 2 mÞy 1 ½1 2 pðdÞð1 2 bÞyg 2 cðdÞ; where cðd Þ is the direct or opportunity cost of defense, assumed continuous and increasing in d; m is the fraction of gross income lost by the victim in the event of a successful attack; and b ðpresumably smaller than mÞ is the fraction lost if an attack occurs and turns out to be unsuccessful, where the word “successful” is used from the aggressor’s point of view.9 This specification incorporates the fact that an attack, successful or not, may still be costly to the victim: 0 ≤ b < m ≤ 1. This problem is equivalent to the one of choosing d to minimize aðm 2 bÞpðdÞ 1 ½cðdÞ=y; 8 Of the Moradabad riots in 1980, Rajgopal ð1987, 75Þ observes that “the incident was sparked off by the entry of a pig towards the Namazis ðMuslims offering prayersÞ.” A more common list includes “encroachment on places of worship,” “music before mosques,” “teasing of girls belonging to the other community,” and “provocative articles in magazines” ð87Þ. 9 One could just as easily write this out in a more sequenced way. For instance, there could be some explicit prior stage at which defense resources are chosen, followed by a second stage in which attacks possibly happen. Our results are also robust to the use of a constant-elasticity utility function defined on net income. The online appendix, which the reader is encouraged to read, contains a precise formulation of these and other issues. 726 journal of political economy This content downloaded from 128.122.149.145 on Tue, 9 Sep 2014 10:18:07 AM All use subject to JSTOR Terms and Conditions where the first term details the extra loss that will accrue from a successful attack, and the second term is the cost of lowering the success probability. Under our assumptions, there is always a solution to the maximization problem. As we track the choices of d ðand so p 5 pðdÞÞ over different values of a, we obtain a best response mapping in ða; pÞ space, which we call the protection function.