Download Problem Set 3 1) Go to court or settle? Assume Alan and Anna are

Problem Set 3 1) Go to court or settle? Assume Alan and Anna are involved in a patent dispute, where Alan is allegedly infringed on Anna’s patented product idea. Both Alan and Anna start with wealth of $100,000. Scenario A: There is a 5 percent chance that Alan will be found guilty and have to pay $5,000. The alternative is to settle out of court for a fair amount of $250. Scenario B: There is a 50 percent chance you will win the case and get $5,000. The alternative is to settle out of court for a fair amount $2,500 to settle out of court. Assume that Alan and Anna have the utility function: where x is final wealth and r is the reference point. The natural reference point here is a not-­‐guilty verdict, meaning r = $100,000. Assume Anna and Alan both use rank-­‐
dependent expected utility functions V =
n
X
i=1
2 0
1
i
X
U (xi ) 4⇡ @
P r(j)A
j=1
0
13
i 1
X
⇡@
P r(j)A5 j=1
p
Anna’s decision weights are given by ⇡(p) =
with γ = 0.61. 1
(p
+
(1
p)
)
p
Alan’s decision weights are given by ⇡(p) =
with δ = 0.69, 1
(p + (1 p) )
a) Consider Scenario A: Compute Alan’s rank-­‐dependent expected utility function if they go to court and his utility for the proposed settlement. Which one does Alan prefer? b) Consider Scenario A: Compute Anna’s rank-­‐dependent expected utility function if they go to court and her utility for the proposed settlement. Which one does Anna prefer? Would you expect a settlement to occur? c) (Optional) Repeat part (a) and (b) for Scenario B. 2) Modeling Disappointment A way to capture disappointment and its consequences was proposed by Loomes and Sugden (1986). The basic idea is to measure payoff relative to some reference prior expectation of what the payoff will be. If outcome is below the expected level, then the consumer experiences disappointment. If the outcome is above the expected level, then the consumer experiences elation. Elation and disappointment are captured by some function, D. So, if the outcome payoff is m his utility will be u(m) = m + D(m prior)
The prior expectation can be context specific but a natural candidate is the expected payoff of the gamble. Consider a consumer with the following disappointment/elation function: ⇢
✓z 2 for z 0
D(z) =
✓z 2 for z < 0
where θ = 0.0002 measures the importance of disappointment. Consider the following gambles from the Allais Paradox. Table 3.10 details the utility of each prospect for different values of θ. When θ = 0 we have a standard expected utility model and Alan would choose prospect A over B, and C over D. If disappointment is given weight θ = 0.0002, we see that prospect B becomes preferred to prospect A. Why? The expected Calculate the expected utility of each gamble. What are his choices between Gamble A and B and between Gamble C and D? Interpret your findings in relation to the Allais Paradox. 3) Your current wealth consists of a collection of valuable artwork worth $400,000 and $100,000 cash. Suppose that your preferences are given by rank-­‐dependent expected utility function with ⇢
p
(x r)0.9 if x r
⇡(p)
=
u(x) =
1
2.25(r x)0.9 if x < r
(p + (1 p) )
where x is your wealth, r is your reference point and with δ = 0.69. You are considering an insurance policy that will pay you the value of your collection should anything destroy it. Suppose that the probability of your artwork being damaged is 0.03. Considering that the current value of your wealth as your reference point (r=500,000) , what is the most you would be willing to pay for the coverage? 4) Representation of Gambles on Marschak-Machina triangle
Consider the following gamble with 3 outcomes that are a priori fixed: 0, 100, and 25 and with varying probabilities. 0 with probability x 100 with probability y 25 With probability 1-­‐x-­‐y The consumer’s utility is given by 𝑢 𝑚 = 𝑚 a) Draw the corresponding Marschak-­‐Machina triangle, carefully labeling the axes. b) Draw the indifference curves that passes through the following gambles: (i) x=0.5 y=0.5 (ii) x=0.25 y=0.25 (iii)
getting 100 as the sure outcome (iv)
getting 25 as the sure outcome c) Find all the set of all the gambles that the consumer prefers to getting 25 as the sure outcome. 5) Consider the utility function given by u(m) = ln m and the set of gambles with the possible outcomes $10, $20 and $30. a) Graph the indifference curves implied by expected utility in the Marschak–
Machina triangle. What is the slope of the indifference curves? b) Alternatively, consider the reference dependent utility function given by ⇢
(m r)2 if m r
u(m)
=
(m r)2 if m < r
Plot the implied indifference curves in the Marschak–Machina triangle, assuming the reference outcome of r=$19.