Suppose there are 10 farmers near in the town of New Paltz, New York, each farmer has 10 cows. Farmers utilize a large open grazing area which is maintained by the village government, who allow the farmers to graze their cows for free. Farmers must pay $100 per cow per month in veterinary costs regardless of whether the cow grazes. The following equation describes the relationship between lbs. of grass eaten/month per cow (g) and gallons of milk produced/month per cow (m):

m = 0.5g

Further, suppose that the amount of grass each cow consumes per month is a function of the total number of cows (C) sent to graze:

g = 800 − 2C

Lastly, suppose that the market for milk is such that price is completely determined by the level of supply:

P _gallon = 10 − 0.0001M

Where M is the total number of gallons of milk brought by all farmers to the market for sale. You may assume throughout that farmers have no other possible source of income.

[i.] Each month they all simultaneously decide how many of their cows to send out to graze. That is, they do not observe other farmers' decisions before sending cows out. How many cows are sent out and what are profit for each farmer? HINT [to find profits, you must determine the market price of a gallon of milk]

[ii.] The farmers begin to notice that portions of the grazing area are not growing due to increased drought from climate change and not allowed to recover due to too many grazing cows. Suppose that it is determined that the grass eaten per cow equation is altered to reflect this new fact:

where ^P3_t C_t is the number of cows sent to graze in the previous three months combined.

Assume your answer in part 1 is month 1's outcome. What happens in months 2 4? If this process of cow grazing continues, what occurs to both the market for milk in the village and the quality of the grazing area?

[iii.] Suppose instead that farmers collectively decide how to proceed after month 4, including the pooling of profits. Assuming no one cheats, how many cows are sent out to maximize profits in the long run?