Uncertainty and Information
Chapter 18 & 19
1.
Emma grows mushrooms on a small farm near Kansas City.
Her daily cost function is
TC=219+0.85q+0.003q
2
, where q is the number of one pound containers of mushrooms
produced per day.
Emma sells her mushrooms at a farmer's market in Kansas City.
Business
at the market is brisk on sunny days, but the market has few shoppers on rainy days.
Emma
is able to sell a pound of fresh mushrooms for $5.25 in sunny weather, but she needs to
reduce her price to $4.35 on rainy days.
The weather in Kansas City is sunny 50 percent of
the time.
Assume that the production decision must be made at the start of the day before
Emma knows whether is will rain in town.
a)
Find Emma's supply curve.
b)
How much should Emma produce to maximize expected profit?
c)
Suppose that Emma maximizes her expected utility instead of her expected profit.
Her utility function is U=1e
0.02W
.
Find her Pratt measure of risk aversion.
Discuss
(no math required) how we would expect her to adjust her output compared to part b.
d)
Now suppose Emma maximizes expected profit and discovers an Internet website
that accurately predicts weather in town.
Of course, Emma has no affect on the
weather, but she can make production decisions each day after she learns what the
weather will be.
What will be the firm's expected profits be in this case?
e)
If the Emma is maximizing expected profits, what is the maximum amount that the
she will be willing to pay for access to the weather prediction?
Answer:
a)
Her supply curve is MC above AVC.
MC=0.85+0.006q. P=MC, so P=0.85+0.006q is
the inverse supply curve.
q=(1/.006)P(0.85/.006) or simplified q=166.67P141.67.
b)
E
π
=0.5(5.25qTC(q))+0.5(4.35qTC(q))=4.8qTC(q).
P=MC, so 4.8=0.85+0.006q,
and q*=658.33.
c)
The Pratt measure is r(W)=U''(W)/U'(W).
U'=(0.02)e
0.02W
=0.02e
0.02W
>0 and
U''=0.0004e
0.02W
<0.
Then, r(W)=0.02.
The Pratt measure being positive means that
Emma is risk adverse.
Since she is risk adverse, she weights the prospects of bad
outcomes more heavily that the symmetric prospects of good outcomes, so she will
reduce output.
d)
She can now pick the profit maximizing output given each price and does not need to
produce for the "average" price.
We can read off the output levels from the supply
equation in part a.
If it will be rainy, she should produce 583.33 and earn profits of
801.83.
If it will be sunny, she should produce 733.33 and earn profits of 1394.33.
Expected profits are 1098.083.
e)
Compare the expected profits from producing 685.33 in part b (1081.208) with the
expected profits in part d (1098.083).
Emma is willing
to pay up to 16.875 for the
information.
2.
Rachel is a young television producer, and she is deciding whether to produce cowboy
movies or holiday specials.
Holiday specials do well if the weather is really cold around
Christmas, because then everybody stays home and watches the specials.
In a cold year, she
can earn 70 million dollars from holiday specials as compared with only 25 million in a warm
year.
Cowboy movies attract a more stable following irrespective of the weather, so they will
always earn 35 million in cold years and 45 million in warm years.
Suppose the probability
of a cold year is 0.4, and Rachel’s utility is U(Y)=ln Y, where Y is measured in millions of
dollars.
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 Winter '08
 Buddin
 Microeconomics, Tim, Expected Profit, Pratt measure

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