ANSWERS
1 The Market
1. Suppose that there were 25 people who had a reservation price of $500, and the 26th person had a reservation price of $200. What would the demand curve look like?
1.1. It would be constant at $500 for 25 apartments and then drop to $200.
2. In the above example, what would the equilibrium price be if there were 24 apartments to rent? What if there were 26 apartments to rent? What if there were 25 apartments to rent?
1.2. In the ?rst case, $500, and in the second case, $200. In the third case, the equilibrium price would be any price between $200 and $500.
3. If people have di?erent reservation prices, why does the market demand curve slope down?
1.3. Because if we want to rent one more apartment, we have to o?er a lower price. The number of people who have reservation prices greater than p must always increase as p decreases.
4. In the text we assumed that the condominium purchasers came from the inner-ring people—people who were already renting apartments.
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What would happen to the price of inner-ring apartments if all of the condominium purchasers were outer-ring people—the people who were not currently renting apartments in the inner ring?
1.4. The price of apartments in the inner ring would go up since demand for apartments would not change but supply would decrease.
5. Suppose now that the condominium purchasers were all inner-ring people, but that each condominium was constructed from two apartments. What would happen to the price of apartments? 1.5. The price of apartments in the inner ring would rise.
6. What do you suppose the e?ect of a tax would be on the number of apartments that would be built in the long run?
1.6. A tax would undoubtedly reduce the number of apartments supplied in the long run.
7. Suppose the demand curve is D(p) = 100?2p. What price would the monopolist set if he had 60 apartments? How many would he rent? What price would he set if he had 40 apartments? How many would he rent?
1.7. He would set a price of 25 and rent 50 apartments. In the second case he would rent all 40 apartments at the maximum price the market
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would bear. This would be given by the solution to D(p) = 100?2p = 40, which is p? = 30.
8. If our model of rent control allowed for unrestricted subletting, who would end up getting apartments in the inner circle? Would the outcome be Pareto e?cient?
1.8. Everyone who had a reservation price higher than the equilibrium price in the competitive market, so that the ?nal outcome would be Pareto e?cient. (Of course in the long run there would probably be fewer new apartments built, which would lead to another kind of ine?ciency.)
2 Budget Constraint
1. Originally the consumer faces the budget line p1x1 + p2x2 = m. Then the price of good 1 doubles, the price of good 2 becomes 8 times larger, and income becomes 4 times larger. Write down an equation for the new budget line in terms of the original prices and income. 2.1. The new budget line is given by 2p1x1 +8p2x2 =4m.
2. What happens to the budget line if the price of good 2 increases, but the price of good 1 and income remain constant?
2.2. The vertical intercept (???? axis) decreases and the horizontal intercept (???? axis) stays the same. Thus the budget line becomes ?atter.
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3. If the price of good 1 doubles and the price of good 2 triples, does the budget line become ?atter or steeper? 2.3. Flatter. The slope is ?2????/3????.
4. What is the de?nition of a numeraire good?
2.4. A good whose price has been set to 1; all other goods’ prices are measured relative to the numeraire good’s price.
5. Suppose that the government puts a tax of 15 cents a gallon on gasoline and then later decides to put a subsidy on gasoline at a rate of 7 cents a gallon. What net tax is this combination equivalent to? 2.5. A tax of 8 cents a gallon.
6. Suppose that a budget equation is given by
???????? +???????? = m. The government decides to impose a lump-sum tax of u, a quantity tax on good 1 of t, and a quantity subsidy on good 2 of s. What is the formula for the new budget line? 2.6. (????+ t) ???? +(?????s) ???? = m?u.
7. If the income of the consumer increases and one of the prices decreases at the same time, will the consumer necessarily be at least
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as well-o??
2.7. Yes, since all of the bundles the consumer could a?ord before are a?ordable at the new prices and income.
3 Preferences
1. If we observe a consumer choosing (????, ????) when (????, ????) is available one time, are we justi?ed in concluding that (????, ????) >(????, ????)? 3.1. No. It might be that the consumer was indi?erent between the two bundles. All we are justi?ed in concluding is that (????, ????)> (????, ????).
2. Consider a group of people A, B, C and the relation “at least as tall as,” as in “A is at least as tall as B.” Is this relation transitive? Is it complete? 3.2. Yes to both.
3. Take the same group of people and consider the relation “strictly taller than.” Is this relation transitive? Is it re?exive? Is it complete? 3.3. It is transitive, but it is not complete—two people might be the same height. It is not re?exive since it is false that a person is strictly taller than himself.
4. A college football coach says that given any two linemen A and B, he always prefers the one who is bigger and faster. Is this preference relation transitive? Is it complete?
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