Homework 6
1. In the game of
Life
the first choice that a player has to make is whether or not he wants to go to
college or if he wants to begin his career immediately.
If he begins his career immediately then
he will earn $36,000 for sure.
If he chooses to go to college then he must first pay $20,000 in
tuition and fees.
After paying for college he will begin his career.
However the salary that he
receives is uncertain.
There is a 50% chance that he will get a job that pays $100,000 and there is
a 50% chance that he will get a job that pays only $40,000.
Assume that the player is risk neutral
.
a)
Assume that the player does not know which salary he will receive after college before
making his decision whether or not to go to college. He does know the probabilities of
those salaries given above.
Illustrate his decision problem in a decision tree.
Which
action
“go to college”
or
“start career”
is
ex ante
optimal?
What is his expected income
(net of any tuition) from his optimal action?
Is his choice always
ex post
optimal?
According to the rules of the game the player is not supposed to know which salary he will
receive.
However, players often cheat.
A player can cheat in this game by simply looking at the
cards.
If he looks at the cards then he will know for sure which salary he will receive.
b) Assume that there is no
possibility that the other players will catch him cheating.
Illustrate his decision problem with the new option of cheating in a decision tree. Which
action
“go to college”
,
“start a career”
, or
“cheat”
is
ex ante
optimal?
What is his
expected income (net of any tuition) from his optimal action?
c)
Assume that the probability of being caught cheating is 1. How big must a fine have to be
in order to (just) discourage the player from cheating?
Answer:
a) If player starts career, he will get 36,000 for sure. If player goes to college, he gets
100,000 – 20,000 = 80,000 if he gets a good job, and 40,000 – 20,000 = 20,000 if he gets
a worse job. The expected income will be
0.5*80,000 + 0.5* 20,000 = $50,000
Since the player is risk neutral and he can get higher expected income after college
education, it’s
ex ante
optimal to do so, but not always
ex post
optimal. If he gets the
$40,000 job, he would have been better off had he started his career right away for he
would have received $36,000 instead of $20,000.
b) The optimal action is to cheat, since expected income will now be