Simon Fraser University
Spring 2016
Econ 302 Assignment 6
Due: in Lecture, Tuesday March 22, 2016.
Show all your work.
1. (2 points) Assume that w > 0. Answer the following questions using second derivatives.
Is u(w) = 1 − e−αw a risk-averse utility function for α > 0? Is u(w) = ln(w) a risk-averse
utility function? For what values of σ is the utility function u(w) = wσ risk averse? For
what values of σ is the utility function u(w) = wσ risk loving?
2. (2 points) You have utility over wealth u(w) = ln(w) and start with $10000. Consider a
risky project with probability 0.5 of gaining $5000, and probability 0.5 of losing $3500. Find
the expected value of this project. Find your expected utility, certainty equivalent, and risk
premium for the this project. Would you take the project?
3. (3 points) You have utility over wealth u(m) = ln(m) and have initial wealth of w > 0
dollars. Consider the following gamble: if you bet x dollars, then with probability p you win
b × x (in addition to getting back your wager of x), and with probability 1 − p you lose your
wager of x. (b is called the gambling odds; b = 1 means 1-to-1 odds.) Assume that b and p
are known constants. Find your optimal bet.1
4. (3 points) Consider the model in the Moral Hazard Lecture (slide 5). Assume now that
the agent is risk neutral: u(w) = w. And assume that pV > 1. Solve for the principal’s first
best and second best outcomes.
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The solution to this problem is known as the Kelly criterion or the Kelly strategy, and its variants are
reportedly used by Warren Buffett and Bill Gross. For more information, see Poundstone, William (2005),
Fortune’s Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall
Street, New York: Hill and Wang, ISBN 0-8090-4637-7.
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