Josh is playing blackjack for real money. He has reference-dependent preferences over money: if his earnings are m and his reference point is r, then his utility is v(m - r), where the value function v satisfy es v(x) = ln(x + 1) for x >= 0, and v(x) = -2 ln(-x + 1) for x =< 0.
(a) Graph Josh's utility function as a function of m - r.
(b) Does Josh's utility function satisfy loss aversion? Does it satisfy diminishing sensitivity?
Suppose that Josh has linear probability weights (that is, he does NOT have prospect theory's non-linear probability weighting function). Hence, if he has a fifty- fifty chance of getting amounts m and m', and his reference point is r, his expected utility is 1/2v(m- r) + 1/2v(m' - r). For parts (c), (d), and (e), assume that Josh's reference point is $0 (that is, no wins or losses) and for the given situation, answer the following questions: (i) What is the g for which Josh would be indifferent between taking a fifty- fifty win $g or lose $5 gamble? (ii) Does this reflect risk loving or risk averse behavior? (iii) What feature of Josh's reference-dependent preferences is driving this choice?
(c) This is the fi rst round and Josh has not won or lost any money yet.
(d) Josh is $10 down.
(e) Josh is $10 ahead.
(f) What does the process of gambling|i.e. winning or losing|do to Josh's risk attitudes? Explain the intuition.