Reference no: EM132208485

Question: We are going to consider a common pool problem. This is a particular type of externality in which a specific resource has no clear property rights and hence it ends up being overexploited. This is known as the “Tragedy of the Commons” and a classic example of this is marine fisheries. Because nobody owns the fish in the sea, fishermen do not face the true social marginal costs of their fishing, which include the increasing difficulty of catches for other fishermen and reduced stocks in the future. As a consequence, there is pervasive overfishing. In the picturesque village of Hillbillia families typically supplement their earnings with the proceeds from fattening cows. The villagers collectively own the lush meadows around the village and they typically leave their cows free to roam. The problem is that the more cows are on the grasslands, the less grass can each of them eat, and hence they are not putting on much weight. To capture this notion, assume that if there are x cows wandering the meadows, each cow will weigh fx = 100 (10x âˆ' x^2) kilograms at the end of the year.

a. Note that if there are few cows, f(x) is actually increasing! This captures the fact that eating grass is a much better pastime if you have someone with whom to chat about the weather. Given this, what is the number of cows so that their average weight is maximised?

b. Now assume that cows sell at $10 per kg. Assume that buying a cow costs $2000 and there are already 7 cows in the meadows. Your family has to decide whether to get a cow this year. Would it be profitable for your family to buy a cow if there are already 7 cows on the meadows?

c. If you add an 8th cow, how much are you reducing the other villagers' profits? (hint: calculate their individual profits when there are 7 cows and when there are 8 cows. The difference is how much each of them loses when you get a cow. If you multiply this by 7 you get the total externality that you impose on the families that own the first 7 cows.)

d. How many cows will be in the fields if every villager can buy a cow and let it roam? (hint: find the largest number of cows for which profits are still non-negative). What are aggregate profits for the whole village in this case?

e. Show that total profits in the village are maximized at 7 cows (hint: instead of taking first order conditions, show this by calculating total profits at 6, 7 and 8 cows and showing that the maximum is reached at 7). This is the efficient number of cows! Explain intuitively why the market does not reach this number. What is the externality here?

f. To solve this externality, the village council decides to charge a tax on people that buy cows. How big has the tax to be so that the efficient number of cows are roaming the meadows in equilibrium? How tax revenue would this tax generate? How could you distribute the proceeds of the tax such that this is a Pareto improvement (i.e. everyone is better off than in the outcome you found in part d).