Old MacDonald has a farm. On her farm she can grow either tomatoes or cacti (with a
squish squish here and a squish squish there, e-i-e-i-o). Cacti are drought resistant, while
tomatoes are not. A crop of cacti will produce a profit of $18,000 regardless of the
weather, while a crop of tomatoes will produce a profit of $30,000 if there is no drought
and $10,000 if there is a drought. The probability of a drought is ½.
a. Draw Old MacDonald's decision tree.
b. If Old MacDonald is an expected value maximizer (value is profit, in
this case), what crop should she grow? What would her expected
profit be?
The expected profit of tomatoes is:
0.5x$10,000 + 0.5x$30,000 = $20,000
The expected profit of cactuses is $18,000.
Since the question told you to assume that Old MacDonald is an expected profit
maximizer (i.e. that she is risk neutral), she should grow tomatoes for an expected
profit of $20,000. This is the solution to the decision tree in Figure 1.
c. You find out that Old MacDonald chose to grow cacti. What can you
conclude about Old MacDonald?
If Old Mac decided to grow cactuses, she must not be maximizing expected
profits (or she made a mistake). She probably decided to grow cactuses because
there is lower risk. In fact, the profit from growing cactuses is known with
certainty. If this is the reason, then Old Mac is risk averse.
d. Now market prices for tomatoes and cacti change. A crop of cacti will
now produce a profit of $12,600 regardless of the weather, while a
crop of tomatoes will produce a profit of $21,000 if there is no drought
and $7,000 if there is a drought. The probability of a drought is still
½. What will Old Mac do? What would she do if she were risk
neutral and just wished to maximize expected profit?
The ratio of expected profits from the two crops is still 10:9 ($14,000 versus
$12,600), but if Old MacDonald was growing cactuses because she was risk
averse, she might now grow tomatoes. The risk involved is smaller since both
expected values are smaller. Since people tend to be less risk averse over small
gambles than over large ones, Old Mac might be willing to take this gamble (and
grow tomatoes) now that the risk is smaller. If she were risk neutral, she would
grow tomatoes, since the expected value of growing tomatoes ($14,000) is greater
than the expected value of growing cactuses ($12,600).
Your birthday is this weekend. You are trying to decide between going picnicking on
Angel Island and going to a nice restaurant that just opened in San Francisco. Your
happiness in either case can be measured by “utils”, units of utility. You check the
weather forecast and it appears that there is a 40% chance of precipitation. You value
going picnicking on Angel Island at 10 utils if it rains and 200 utils if it doesn’t rain. On
the other hand, you are not sure about the quality of the new restaurant, having never
eaten there before. You think that there’s about a 50% chance you will value eating there
at 100 utils and a 50% chance you will value eating there at 160 utils.
a) Draw a decision tree that represents the situation you face.
Good Food [p=0.5]
Restaurant
160
EV=130
Bad Food [p=0. 5]
100
Good Weather [p=0.6]
200
EV=130
EV=124
Angel
Island
Bad Weather [p=0.4]
10
b) You end up going to the restaurant. What does that say about your risk
preferences? Do we know whether you risk-neutral, risk-loving, or risk-averse?
You could be risk-neutral, as you chose the option with the highest EV. You could also
be risk-averse, since you are choosing the relatively safer option. You could even be
slightly risk-loving, since your love for risk would make you value choosing Angel Island
but your love for utils would make you value choosing the restaurant, and it is possible
that this latter love wins out even though it’s less risky.
Note: Risk neutral people are always EV maximizers, but it is not true that if someone
chooses an action consistent with EV maximization they are necessarily risk neutral.
c) You are good friends with a chef in France who knows the chef of the new
restaurant. You consider calling your friend long-distance to learn whether you
will enjoy eating there. However, that would cost some money, and you value
each dollar you have at 1 util. Assuming you are risk-neutral, what is the most
you would be willing to pay for the long-distance telephone call to get the
information?
If you know beforehand what the food will be like, it will only change your decision to
go eat there if your friend tells you the food will be bad. In that case, you will choose
Angel Island, as a risk-neutral EV-maximizer. So you stand to gain 124 - 100 utils 50%
of the time, or 12 utils. With the 1:1 utils:money conversion assumed in the question,
you’d be willing to pay up to $12 for the long-distance call.
Nancy currently earns $100,000 per year. She is told that if she stays within her current
firm, there is a 80% chance she will be promoted to a position that would pay her a salary
of $150,000. If she’s not promoted, her salary will remain at $100,000. She is also
considering an offer from a new firm that would pay her an annual salary of $115,000.
Assume that Nancy is considering her situation for the next year only and not taking into
account her options thereafter. If she decides to stay at her current firm, what can you
conclude about her risk preferences? In other words, is her behaviour consistent with risk
aversion only? With risk neutrality only? With risk seeking only? Or what combination
of the three?
Answer: Nancy faces two options: 1) earn 115,000 for sure or 2) earn 100,000 with 20%
probability and 150,000 with 80% probability. The expected value of option 2 is 140,000.
Because she rejected option 1, we know that Nancy’s CE for the gamble is greater than
115,000. This could mean that she is risk averse, risk neutral, or risk seeking.