The market for high-quality caviar is dependent on the weather. If the weather is good, there are many fancy parties and caviar sells for 300 SEK per kg. In bad weather it sells for only 200 SEK per kg. Caviar produced one week will not keep until the next week, i.e. it has to be sold and consumed the same week it is produced. A small caviar producer has a cost function given by C=5q2 - 50q + 1000 where q is the weekly caviar production. Production decisions must be made before the weather (and hence the price of caviar) is known, but it is known that good weather and bad weather each occur with a probability of 0.5. a) How much caviar should the firm produce per week if it wishes to maximise its expected profit? b) Suppose the owner of the firm has a utility function of the form U(π)=√π where π is weekly profits. What is the expected utility associated with the strategy in (a)? c) Suggest a weekly production level that would give the owner higher expected utility than the answer you found in (b). Show that this production level gives higher expected utility, and explain why. d) Suppose that the firm owner has inside information on the week’s weather forecast and hence can predict (but not influence) the price. What production strategy would maximise expected profits in this case? What would expected profits be?
The market for high-quality caviar is dependent on the weather. If the weather is good, there are many fancy parties and caviar sells for 300 SEK per kg. In bad weather it sells for only 200 SEK per kg. Caviar produced one week will not keep until the next week, i.e. it has to be sold and consumed the same week it is produced. A small caviar producer has a cost function given by
C=5q2 - 50q + 1000
where q is the weekly caviar production. Production decisions must be made before the weather (and hence the price of caviar) is known, but it is known that good weather and bad weather each occur with a probability of 0.5.
a) How much caviar should the firm produce per week if it wishes to maximise its expected profit?
b) Suppose the owner of the firm has a utility function of the form
U(π)=√π
where π is weekly profits. What is the expected utility associated with the strategy in (a)?
c) Suggest a weekly production level that would give the owner higher expected utility than the answer you found in (b). Show that this production level gives higher expected utility, and explain why.
d) Suppose that the firm owner has inside information on the week’s weather
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