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Question 1: The ﬁgure above shows the growing number of maritime piracy incidents in South-East Asian waters during the period 1993–2007. The economic costs of maritime piracy globally is estimated to be between 7 and 12 billion U.S. dollars. A feature of maritime piracy in South-East Asia is that many pirates are not professional criminals but are instead ﬁshermen who use piracy to supplement their legal income. Using this fact, consider the following model of occupational choice with risk. Suppose that an individual (individual B) has the option of choosing between two occupations. He can become a ﬁsherman. If he does so, he receives an annual payoﬀ of 40 with certainty. Instead, if he chooses to become a pirate, he receives an annual payoﬀ of 80 if he is not caught. If he is caught, he still retains his illicit payoﬀ of 80 but must pay a penalty of 75. You can think of this penalty as the cost, in monetary units, of being jailed. Let the probability that this person is caught be 0.4.
Suppose that this individual’s preference towards risk can be characterized by the following utility function:
W1−γ 1−γ where W ≥ 0 represents the individual’s payoﬀ and 0 ≤ γ < 1 is a constant.
(a) [5 marks] To measure this individual’s risk aversion we will use a concept called relative risk aversion. Let this individual’s relative risk aversion, R, be deﬁned as
R = −W
where U0(W) = dU/dW is individual B’s marginal utility and U00(W) = dU0/dW is his change in marginal utility with respect to his payoﬀ. Show that individual B’s relative risk aversion is equal to γ. Is R an increasing or decreasing function of his risk aversion? Use an appropriate diagram to explain.
(b) [3 marks] Suppose γ = 0.75. Given the information above, will this individual choose to be a ﬁsherman or a pirate? Fully explain your answer. You can assume that the individual must choose one occupation and cannot divide his time between both.
Suppose that a second individual (individual C) comes along and proposes that both B and C get together and form a gang. As part of being in a gang, both individuals will undertake piracy activities independently but will face the same risk of capture. Their individual payoﬀs will also be identical, but B will receive a fraction α of their total payoﬀ, where 0 ≤ α ≤ 1.
(c) [6 marks] For what values of α will B choose to form a gang? Why does forming a gang make sense for B for these values of α? Explain.
(d) [3 marks] Now suppose that γ = 0.5. For what