1.   (30 pts) Danielle is a risk-averse, expected utility maximizer. Lottery A = ($0, 0.25;  $40, 0.25;  $80, 0.25;  $120, 0.25) and lottery B = ($0, 0.25;  $20, 0.25; $100, 0.25;  $120, 0.25).

a.   Use the concept of a mean preserving spread to determine her preferences between lotteries A and B. Clearly demonstrate your logic.

E  A   0.25  0  0.25  40  0.25  80  0.25  120   $60  E  B   0.25  0  0.25  20  0.25  100  0.25  120   $60

Starting from A’s probability mass function we can shift the 0.25 units of mass on $40 to the left to $20 and shift 0.25 units of the mass on $80 to   the right to $100 and we will end up with B’s exact probability mass        function. (Illustrating this on a graph would be helpful.)

B is a mean preserving spread of A so Sarah will prefer A over B.

b.   Use the concept of zero conditional mean noise to determine her          preferences between lotteries A and B. Clearly demonstrate your logic.

$0                                  if A  $0

$20, p;  $60,1 p     if A  $40

   $60, q;  $20,1 q      if A  $80

$0                                ifA  $120

E   | A   0 :

 20p  60  1  p   0               p  

 60q  20  1  q   0               q  

p  $0 | A     0.25  p  $0 | B 

p  $20 | A     0.25   0.25   0.25  p  $20 | B     p  $100 | A     0.25   0.25   0.25  p  $100 | B 

p  $120 | A     0.25  p  $120 | B 

B  A   so she prefers A over B.

(Note: It is enough to check any 3 of the 4 probabilities.)

2.   (25 pts) Lottery C = ($60, p;  $120, 1 – p) where 0 < p < 1 and lottery D = ($20, 0.2;  $80, 0.8).

a.   Find the full range ofp such that lottery C second order stochastically   dominates (SOSD) lottery D. Clearly demonstrate your logic. If no such range exists, clearly demonstrate why.

p

1

FD              +

p

F  C

–

0.2

0       20                  60         80            140     x

80

 FD  s  FC  s   ds   60  20  0.2   80  60  p  0.2   0

20

p  

If x FD  s  FC  s   ds  0 for x = $80, it will also be greater than or

20

equal to 0 for all x.

b.   Find the full range ofp such that lottery D second order stochastically   dominates (SOSD) lottery C. Clearly demonstrate your logic. If no such

range exists, clearly demonstrate why.

No such range exists because FD starts above FC.

3.   (45 pts) One version of the Allais Paradox is as follows. A = $30, B =

$0,  ;  $30,  ;  $50,  , C = $0,  ;  $30,   and D = $0,  ;  $30,  ;  $50,  .

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.

C          p1 = p(x1)

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: 

0     ifp  0

  p        if 0  p   ?

 p    if   p  1

Demonstrate algebraically. Assume v($0) = 0 and v($50) = 1.