## Problems for Section 51

2. If she bought wheat at 5 bushels of wheat per book she would get before taxes 5 bushels for each book given up. But after the government took its cut in taxes she would get only 2j bushels, not enough to compensate her for the sacrificed book (she demands 4 bushels per book). And a market price of 3 bushels for one book would be even worse. Only if the before tax price is more than 8 bushels per book will she care to sell books (that is, buy wheat). If she entered the market to buy books, on the other hand, she could get at most j of a book for a single bushel (or worse at the other price = | of a book). Since she demands J of a book to compensate for the sacrifice of 1 bushel, the deal looks good. But again the government takes a cut, half of the | of a book, leaving her only g of a book—not enough to make the deal worth her while. Only if the 50% tax left her with more than j of a book would she want to sell wheat (that is, buy books). Fifty percent of ^ is J, so any number of books more than \ book per bushel of wheat would do the trick. In short, at a price of books in terms of wheat less than 2 bushels per book she will buy books; at a price of books more than 8 bushels per book (the earlier result) she will sell books. In between she does nothing: taxes, like transport costs, hinder trade.

5. a. Haines and McQuaid alone would clearly make one transaction; Haines, McQuaid, and Menard, three. Four people would give six transactions, as you can see by arranging four dots in a rough circle and counting how many lines it takes to hitch every dot with every other. By the same method, five gives 10 transactions, and six gives 15.

b. The total number of links for a society of N people is (N — 1) + (N — 2) + (N — 3) + • • • + N - (N - 1): for 5, it is 4 + 3 + 2 + 1 = 10; for 6, it is 5 + 4 + 3 + 2 + 1 = 15. It can be shown that for very large N (say, 10,000 = a very small city), the number of links is approximately N2 (100 million for N = 10,000). Those who have had calculus will be able to see the argument: (N - V + (N - 2) + ■ • ■+N-(N-1)=N + N + N + N+ - ■ ■ - 1 - 2 - 3 - ■ • • - (N - 1)

= (N — 1) (N) — 2 N. The limit of the last expression as N gets very large is N2.

N=l c. There are now N transactions, one for each person with the single General Store. That is, for a society of six people, six transactions with Gordon as against 15 without him. For a society of 10,000 people, 10,000 transactions as against 100,000,000 without him. Hurrah for Gordon! I don't know about you, but I feel just great about middlemen now.