What is the profit-maximizing number of sheep to own

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Reference no: EM13183726

A village has 10 farmers. Each summer, all the farmers graze their sheep on the village green. The cost of buying and caring for a sheep is $1, independent of how many sheep a farmer owns. The value to a farmer of grazing a sheep on the green when a total of F sheep are grazing is v(F) per sheep:

F v(F) (price of a sheep)

1 $108

2 $105

3 $100

4 $93

5 $84

6 $73

7 $60

8 $45

9 $18

10 $9

11 $0

12+ $0

For example, if one farmer owns 2 sheep and no other farmer owns any sheep. Then, the farmer who owns 2 sheep reaps 2x$105-2x$1=$208. Similarly, when each farmer owns 1 sheep, each farmer gets 1x$9 - 1x$1 = $8.

During the spring, farmers simultaneously choose how many sheep to own.

Suppose you are one of the 10 farmers in the game.

Q1. What is your optimal choice if the other 9 farmers choose to own 0 sheep?

Q2. What is your optimal choice if each of the other 9 farmers chooses to own 1 sheep?

Q3. What is your optimal choice if the other 9 farmers altogether choose to own 10 sheep?

Q4. Is the situation in which each farmer (including yourself) chooses to own 1 sheep a Nash equilibrium outcome?

Q5. Is the situation in which you choose to own 10 sheep while the other 9 farmers choose to own 0 a Nash equilibrium?

Q6. What is the profit-maximizing number of sheep to own for the 10 farmers?

Q7. What is your optimal choice if the other 9 farmers choose to own 5 sheep?

Q8. What is your optimal choice if the other 9 farmers choose to own 6 sheep?

Q9. What is your optimal choice if the other 9 farmers choose to own 7 sheep?

Q10. Is any Nash equilibrium for this game efficient (profit-maximizing for the 10 farmers)? If not, can you think of a way for them to reach the efficient outcome?

Reference no: EM13183726

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