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Economics 171: Decisions Under Uncertainty

Final Exam Solutions (Wednesday Exam): Fall 2021

1.   (8 pts)

a.   When eliciting someone’s preferences state two advantages of using       hypothetical decision problems instead of paying the person the result of his or her chosen lotteries.

    Large payouts.

    No wealth effects.

b.   When eliciting someone’s preferences state two disadvantages of using  hypothetical decision problems instead of paying the person the result of his or her chosen lotteries.

    Lack of effort by the decision maker.

     People may behave in a less risk averse way in hypothetical

situations.

     Subject does not have financial incentive to choose optimally.

2.   (36 pts) One version of the Constant Proportion Paradox is as follows. A = $0,  ;  $80,  , B = $0,  ;  $160,  , C = $0,  ;  $80,   and D =

$0, ;  $160,  . A paradox occurs when A is preferred to B and when D is

preferred to C.

a.   Use a probability triangle to demonstrate that these preferences violate  Expected Utility Theory. It may be helpful to make your graph to scale. Briefly explain your reasoning.

0         A                  p1 = p(x1)     C       1

It is impossible to explain the pair of preferences with parallel, linear indifference curves.

b.   Is possible these preferences are consistent with Prospect Theory with:

 1  p           0  p   1

  p    2                               3 ?

5 p   1      1  p  1

Demonstrate algebraically. Assume v($0) = 0 and v($160) = 1.

V  A   V  B 

  v  $0     v  $80     v  $0     v  $160  0      v  $80   0      1

3 v  $80    1

v  $80   

 

V  D   V  C 

  v  $0     v  $160     v  $0     v  $80  0    1    v  $80 

1    1  v  $80

v  $80   

It is possible if     v  $900      .

c.   Briefly explain why the weighting function in (b) could or could not           resolve these preferences. (Hint: Thinking about the inequality involving   the fixed ratio of probabilities may help. We used this inequality to resolve the Constant Proportion Paradox.)

It underweights high probabilities.

We get the inequality,    , to hold by decreasing the

weight on pq, π(2/5) = 1/4, relative to the weight on p, π(4/5) = 3/4. This is from the negative intercept that only applies to high probabilities (1/3 < p  ≤ 1).

3.   (32 pts) Andy’s preferences are consistent with Prospect Theory. His value function is v(x) = x3 . Lottery A = (–$100, 0.5;  $100, 0.5) and lottery B = $0.

a.   Does Andy’s value function have the same curvature that K&T assume in the gain region? Does Andy’s value function have the same curvature that K&T assume in the loss region? Clearly demonstrate your logic and         provide a brief explanation.

No and no.

v ' x   3x2 ,   v '' x   6x  

Andys value function is convex in gains and concave in losses.

b.   Does Andys value function have the property that losses loom larger than

gains? Clearly demonstrate your logic and provide a brief explanation.

No.

Let   0.

v '   32   v '   2   3    2

The slopes in the loss region are the same as the slopes in the gain region.

c.   Suppose his weighting function has the property that π(p) + π(1 – p) = 1     for 0 < p < 1. How do we know π(0.5) = 0.5? What are Andy’s preferences between lotteries A and B? Illustrate using a graph.

  0.5    1  0.5   1

  0.5   0.5

100

A  B . V(A) is read off the line segment at

  A     0.5  100    0.5 100   0 . V(B) is read off v($0).

d.   Suppose his weighting function has the property that π(p) + π(1 – p) < 1 for 0 < p < 1. What are Andy’s preferences between lotteries A and B?  Briefly explain. You may refer to your graph in (c).

A  B . His previous V(A) is now multiplied by a number that is less than 1. Since his previous V(A) was 0, the new V(A) will also be 0.

4.   (24 pts) Danielle’s preferences are consistent with Prospect Theory. Her value      function is convex in losses. Lottery A = –$400. Lottery B = (–$400, 0.25; –$800, 0.75) and lottery C = (–$400, 0.75; –$800, 0.25). Her weighting function is:

 0        p    1  p

 1

ifp  0

if 0  p  1

ifp  1

a.   Use the graph of a convex value function to determine whether or not her preferences between lotteries A and B are consistent with first order         stochastic dominance (FOSD). Clearly demonstrate your logic. Provide a brief, intuitive explanation.

-800    -500       

v(x)

  B    1  0.25 400   1  0.75 800    500

A  B  which is consistent with FOSD.

Since π(p) + π(1 – p) = 1, V(B) will be on the line segment connecting v(–$400) and v(–$800) which must be below v(–$400).

b.   Use the graph of a convex value function to determine whether or not her preferences between lotteries B and C are consistent with first order         stochastic dominance (FOSD). Clearly demonstrate your logic. Provide a brief, intuitive explanation.

 

  C    1  0.75 400   1  0.25 800    700

B  C  which violates FOSD.

π(p) is downward sloping which means more weight is placed on less        likely outcomes. Lotteries B and C have the same two outcomes. B’s has a higher probability of the worst but the weighting function puts lower          weight on this outcome.