Microeconomics, Module 19, “Common Property and Public Goods” (Chapter 14)

Homework Assignment

(The attached PDF file has better formatting.)

Suppose many people want to catch fish in a well-stocked lake.

●    As the number of fishermen increases, the cost of hiring boats and equipment rises; that is, the marginal cost of fishing rises with the number of fishermen.
●    As the number of fishermen increases, the number of fish caught by each person decreases; the marginal value of fishing decreases with the number of fishermen.

The table below shows the marginal cost and value of fishing:

Fishermen    Marginal Cost of
Fishing (Per Person)    Total Value of
Fish Caught    Value per
Fisherman    Social Marginal
Benefit
0    –    $0    –    –
1    $5    $20        
2    $6    $36        
3    $7    $48        
4    $8    $56        
5    $9     $60        
6    $10    $60        
7    $11    $56        
8    $12    $48        


A.    Complete the last two columns of this table. The “value per fisherman” is the average value per fisherman, assuming all fishermen have equal ability to catch fish. For example, the value per fisherman for two fishermen is $36 / 2 = $18.
B.    The social marginal benefit is the additional social value of the last fisherman. For example, the social marginal benefit of the second fisherman is $36 – $20 = $16.
C.    If the lake is common property, how many fishermen will use it? (Choose the row where the marginal cost to the fisherman in the second column equals the average value of fishing to the fisherman in the fourth column.)
D.    How much social gain is created by fishing in the lake? (For the row determined above, use the value of the fish caught minus the product of the number of fishermen and the marginal cost of each fisherman.) You should get an answer of $0, since all gains from common property are dissipated.

Question: The marginal cost includes the value of the fisherman’s time. This example says that the net present value of fishing is zero. That is true for all work; if we include the value of one’s time, the net present value of any occupation should be zero (in a competitive labor market). What is different about common property?

Answer: The cost includes the value of the fisherman’s time but not the opportunity cost of the lake. The lake is a valuable asset. If there were only one fisherman, he would pay $20 – $5 = $15 to use the lake. The lake produces a daily income of $15, which might be worth $50,000 (depending on the opportunity cost of capital, the growth rate of the daily income, and the lifetime of the daily income).

If there were only two fishermen, they would pay $36 – 2 × $6 = $24 to use the lake. The lake produces a daily income of $24, which might be worth $80,000, depending on the opportunity cost of capital, the growth rate of the daily income, and the lifetime of the daily income. These variables depends on the daily depletion of fish from the lake.

Our concern is not the value of fishing as an occupation. If fishing is competitive, fishermen earn a normal wage for their efforts. Our concern is the value of the lake itself. Although the lake is clearly a valuable asset, all its value is dissipated by overuse.

Did anyone look at the 7 dwarfs question in the book?  Can you tell me how to get started thinking about it?  The only answer they have is that the queen would charge 16 for cave A and 18 for cave B.  But that would mean that some of the dwarfs would want to be in cave A, and I can't see how that would be the case since there is more gold in B.  Any insights?

It is confusing because you don't know if the entry fee is per person or 18 a day for the mine.  So I thought about it both ways

18 / 16 per person

If she charges 18 per person for B at most five will work in in B.  If five work there than she gets all their gold, and the two A take home 2 nuggets a piece.  If only 4 work in B they take home 3 a piece, and the three guys in A give all their gold to the queen.  So she has priced the entry fee to maximize the gold she will recieve.  She will either get all the gold in A or all the gold in B.  This assumes that everyone goes to work. 

 

18/16 per person

So she charges 18 for mine B, regardless of how many workers and 16 for mine A, again regardless of workers.  This would force workers to divide 3 in A and 4 in B Eventually all would go to B, and get 9.48 nuggets a day.  If one left he would only recieve 4.  If three left the would earn 10.667 in A and the others in B would earn 16.5.  If they were out for themselves they would go to be and try to run the others out, but I would bet they would all be in B waiting for the others to leave.  I don't know how they came up with 18 and 16 though.  I would have figured the tax so that the miners would earn the same in each mine. 

that second paragraph should be titled 18 / 16 per mine

Thanks for the explanation! It makes so much more sense now. I think they mean pieces of gold per miner, because it makes more sense that way and you also get whole numbers. I got through the problem, and this time I got the answer that the social optimum is when 3 work in A for 16 nuggets apiece and 4 work in B for 21 nuggets apiece for a total social gain of 132 nuggets. It seems, then, about the queen question that she would want to charge 16 for A and 21 for B. Doesn't it? That way, 3 would still choose to work in A and 4 would choose to work in B (because any other way, somebody would be losing money and would choose to work in the other place). And if the entrance fee is the same as the amount gained, the queen gets all the gold. Does this make sense? Am I still thinking about it wrong, or did the book make a mistake?

I think the reason the owner is described as "wicked" is to point out a Principal-Agent problem.  If the fee collector is more concerned with maximizing fees, then only 5 dwarves work: 2 in Mine A and 3 in Mine B.  This produces 2*16+3*18=86 nuggets which is greater than 7*12=84 nuggets. 

Responding to the posts from newjenl, jercox, and sleepyinseattle above, with regards to Question 2 on page 494 in the textbook (the seven dwarves problem):

I agree with all of these candidates that the book's solution to part d must be wrong, although I don't exactly agree with any of the candidates' postings either. According to the textbook, the wicked queen will charge each miner 16 nuggets per day to enter Mine A, and 18 nuggets per day to enter Mine B. Clearly, under this fee structure, of the first six dwarves, 2 would choose to enter Mine A and 4 would choose to enter Mine B. The seventh dwarf would then face the choice of paying a fee of 16 nuggets to enter Mine A (which would yield 16 nuggets) or a fee of 18 nuggets to enter Mine B (which would yield 18 nuggets) or to stay home and earn nothing. Since the seventh dwarf has nothing to gain by entering either Mine, I think he would stay home and the wicked queen would only collect six entrance fees totaling 104 nuggets (2 * 16 + 4 * 18). In the best-case scenario for the queen, the seventh dwarf would be Dopey and would choose to pay the 18-nugget fee to work all day collecting 18 nuggets in Mine B, rather than staying home, meaning the queen would collect 104 + 18 = 122 nuggets. That is probably what the book believed would happen. Under the book's solution, the queen gets, at most, 122 nuggets.

A better admission fee structure for the queen would be to charge 15 nuggets to enter Mine A and 20 nuggets to enter Mine B. Under this scenario, all seven dwarves would have some incentive to mine, because they would each harvest more nuggets than their admission fee. Three of them would choose to enter Mine A, paying 15 nuggets each to harvest 16 nuggets each. Four of them would choose to enter Mine B, paying 20 nuggets each to harvest 21 nuggets each. Each dwarf goes home at the end of the day with one nugget in his pocket, getting at least some reward for going off to work for a day. Here, the queen collects seven entrance fees totaling 125 nuggets (3 * 15 + 4 * 20). The book's solution clearly does not maximize the wicked queen's profit.

I think the solutions to all of question 2 are as follows:    a) If entry to the mines is free, 2 miners work in A and 5 miners work in B.    b) At social optimum, 3 miners work in A and 4 miners work in B.    c) One way to bring about that social optimum would be to charge anywhere between 3 and 6 nuggets to work in Mine B, while still letting access to Mine A be free of charge.    d) The wicked queen would charge 15 nuggets to enter Mine A and 20 nuggets to enter Mine B.