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A visitor of the casino has initial wealth W and a vN-M utility function u(w)...

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Accounting

A visitor of the casino has initial wealth W and a vN-M utility function u(w) for wealth satisfying u(w) > 0 for all w R. For a xed payment to the casino, he can play an electronic pinball machine with the following payos: with probability 12 he wins an amount z > 0, and with probability 12 he will not win anything. Assume that the visitor does not get utility from gambling itself, but only from potential monetary rewards. 1 (a) Let s denote the maximum amount of money the visitor will be willing to pay to play the pinball machine. Explain why s satises the equation: u ( W ) = 12 u ( W s ) + 12 u ( W + z s ) . (b) Provide an argument as to why s must be strictly less than z. (c) Now assume that u(w) < 0. Prove that s must be strictly less than z2 . (d) In general, the maximum amount of money, s, will be a function s(z), i.e., it will depend upon the amount z he can potentially win. Calculate ds/dz and nd its range. What is the economic interpretation of this derivative? What's the range of ds/dz when the visitor is risk-aversive and risk-loving, respectively? (hint: You need to take total dierentiation of the equation in (a).)

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