1、1ANSWERS1 The Market1. 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 e
2、quilibrium 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
3、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 oer 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 p
4、urchasers came from the inner-ring peoplepeople who were already renting 2apartments. What would happen to the price of inner-ring apartments if all of the condominium purchasers were outer-ring peoplethe people who were not currently renting apartments in the inner ring?1.4. The price of apartments
5、 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
6、of apartments in the inner ring would rise.6. What do you suppose the eect 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) = 1002p. What price w
7、ould 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 3would bear. Thi
8、s would be given by the solution to D(p) = 1002p = 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 ecient?1.8. Everyone who had a reservation price higher than the equilibriu
9、m price in the competitive market, so that the nal outcome would be Pareto ecient. (Of course in the long run there would probably be fewer new apartments built, which would lead to another kind of ineciency.)2 Budget Constraint1. Originally the consumer faces the budget line p1x1 + p2x2 = m. Then t
10、he 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 o
11、f good 2 increases, but the price of good 1 and income remain constant?2.2. The vertical intercept ( axis) decreases and the horizontal 2intercept ( axis) stays the same. Thus the budget line becomes atter.143. If the price of good 1 doubles and the price of good 2 triples, does the budget line beco
12、me atter or steeper?2.3. Flatter. The slope is 2 /3 . 1 24. What is the denition 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 goods price.5. Suppose that the government puts a tax of 15 cents a gallon on gasoline and the
13、n 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 11 22government decides to impose a lump-sum tax of u, a quantity tax on good 1 of t, an
14、d a quantity subsidy on good 2 of s. What is the formula for the new budget line?2.6. ( + t) +( s) = mu. 1 1 2 27. If the income of the consumer increases and one of the prices decreases at the same time, will the consumer necessarily be at least 5as well-o?2.7. Yes, since all of the bundles the con
15、sumer could aord before are aordable at the new prices and income.3 Preferences1. If we observe a consumer choosing ( , ) when ( , ) is available 1 2 1 2one time, are we justied in concluding that ( , ) ( , )?1 2 1 23.1. No. It might be that the consumer was indierent between the two bundles. All we
16、 are justied in concluding is that ( , ) ( , ). 1 2 1 22. 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 t
17、aller than.” Is this relation transitive? Is it reexive? Is it complete?3.3. It is transitive, but it is not completetwo people might be the same height. It is not reexive since it is false that a person is strictly taller than himself.4. A college football coach says that given any two linemen A an
18、d B, he always prefers the one who is bigger and faster. Is this preference 6relation transitive? Is it complete?3.4. It is transitive, but not complete. What if A were bigger but slower than B? Which one would he prefer?5. Can an indierence curve cross itself? For example, could Figure 3.2 depict a
19、 single indierence curve?3.5. Yes. An indierence curve can cross itself, it just cant cross another distinct indierence curve.6. Could Figure 3.2 be a single indierence curve if preferences are monotonic?3.6. No, because there are bundles on the indierence curve that have strictly more of both goods
20、 than other bundles on the (alleged) indierence curve.7. If both pepperoni and anchovies are bads, will the indierence curve have a positive or a negative slope?3.7. A negative slope. If you give the consumer more anchovies, youve made him worse o, so you have to take away some pepperoni to get him
21、back on his indierence curve. In this case the direction of increasing utility is toward the origin.78. Explain why convex preferences means that “averages are preferred to extremes.”3.8. Because the consumer weakly prefers the weighted average of two bundles to either bundle.9. What is your margina
22、l rate of substitution of $1 bills for $5 bills?3.9. If you give up one $5 bill, how many $1 bills do you need to compensate you? Five $1 bills will do nicely. Hence the answer is 5 or1/5, depending on which good you put on the horizontal axis.10. If good 1 is a “neutral,” what is its marginal rate
23、of substitution for good 2?3.10. Zeroif you take away some of good 1, the consumer needs zero units of good 2 to compensate him for his loss.ANSWERS A1311. Think of some other goods for which your preferences might be concave.3.11. Anchovies and peanut butter, scotch and Kool Aid, and other similar
24、repulsive combinations.4 Utility1. The text said that raising a number to an odd power was a 8monotonic transformation. What about raising a number to an even power? Is this a monotonic transformation? (Hint: consider the case f(u)=u2.)4.1. The function f(u)=u2 is a monotonic transformation for posi
25、tive u, but not for negative u.2. Which of the following are monotonic transformations? (1) u =2 v13; (2) u = 1/v2; (3)u =1/v2; (4)u = ln v; (5)u = ev; (6)u = v2; (7) u = v2 for v0; (8) u = v2 for vy or yx, which means that one of the bundles has more of both goods. But if preferences are monotonic,
26、 then one of the bundles would have to be preferred to the other.94. What kind of preferences are represented by a utility function of the form u(x1,x2)= ? What about the utility function v(x1,x2)= x1 + x213x1 + 13x2?4.4. Both represent perfect substitutes.5. What kind of preferences are represented
27、 by a utility function of the form u(x1,x2)=x1 + ? Is the utility function v(x1,x2)=x2 1 +2x1 x2 x2 +x2 a monotonic transformation of u(x1,x2)?4.5. Quasilinear preferences. Yes.6. Consider the utility function u(x1,x2)= . What kind of pref- 1 2 erences does it represent? Is the function v( , ) = a 1
28、 2 1222monotonic transformation of u( , )? Is the function w( , ) =1 2 1 2a monotonic transformation of u( , )? 1222 1 24.6. The utility function represents Cobb-Douglas preferences. No. Yes.7. Can you explain why taking a monotonic transformation of a utility function doesnt change the marginal rat
29、e of substitution?4.7. Because the MRS is measured along an indierence curve, and utility remains constant along an indierence curve.5 Choice1. If two goods are perfect substitutes, what is the demand function for 10good 2?5.1. =0 when , = m/ when b, and any amount on the budget line if p1/p2 = b.3.
30、 Suppose that a consumer always consumes 2 spoons of sugar with each cup of coee. If the price of sugar is p1 per spoonful and the price of coee is p2 per cup and the consumer has m dollars to spend on coee and sugar, how much will he or she want to purchase?5.3. Let z be the number of cups of coee the consumer buys. Then we know that 2z is the number of teaspoons of sugar he or she buys. We must satisfy the budget constraint2 z + z = m.1 2Solving for z we havez =21+ 2 .
