A decision-maker with initial wealth w> 0 is choosing how many tickets to buy for a raffle. Tickets cost 1 dollar each, and are perfectly divisible; thus the decision-maker can buy any number y € [0, w] of tickets (where y need not be an integer). If he buys y tickets, he has a probability 52 of winning R dollars, in which case his final wealth is w − y + R. With the remaining probability 22, he wins nothing, giving a final wealth of w - y. y+2 (a) For what values of R is the decision-maker's expected wealth decreasing in y? (b) Suppose the decision-maker is an expected utility maximizer with von Neumann-Morgenstern utility u(x) = e over his final level of wealth. Find his optimal number y* of raffle tick-

A decision-maker with initial wealth w> 0 is choosing how many tickets to buy for a raffle. Tickets cost 1 dollar each, and are perfectly divisible; thus the decision-maker can buy any number y € [0, w] of tickets (where y need not be an integer). If he buys y tickets, he has a probability 52 of winning R dollars, in which case his final wealth is w − y + R. With the remaining probability 22, he wins nothing, giving a final wealth of w - y. y+2 (a) For what values of R is the decision-maker's expected wealth decreasing in y? (b) Suppose the decision-maker is an expected utility maximizer with von Neumann-Morgenstern utility u(x) = e over his final level of wealth. Find his optimal number y* of raffle tick-

ENGR.ECONOMIC ANALYSIS

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ISBN:9780190931919

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Chapter1: Making Economics Decisions

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A decision-maker with initial wealth w> 0 is choosing how many tickets to buy for a raffle.
Tickets cost 1 dollar each, and are perfectly divisible; thus the decision-maker can buy any
number y = [0, w] of tickets (where y need not be an integer). If he buys y tickets, he has a
probability of winning R dollars, in which case his final wealth is w - y + R. With the
y+2
remaining probability 72, he wins nothing, giving a final wealth of w - y.
Y
y+2¹
(a) For what values of R is the decision-maker's expected wealth decreasing in y?
(b) Suppose the decision-maker is an expected utility maximizer with von Neumann-Morgenstern
utility u(x) = e over his final level of wealth. Find his optimal number y* of raffle tick-
ets.
(c) For the decision-maker in part (b), compare the certainty equivalent associated with the
optimal number y* of raffle tickets to that associated with some other number y = [0, w].
Can you say which of these certainty equivalents is higher? Note that you do not need
to solve for the certainty equivalents.
(d) Now suppose the decision-maker has prospect theory preferences with a reference point
of 0 final wealth. If 1 < R < 4, is it possible to say whether the decision-maker will buy
a positive (non-zero) number of tickets? Explain carefully.

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