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1、ANSWERS1 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 eq
2、uilibrium 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 die
3、rent 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 pu
4、rchasers came from the inner-ring peoplepeople who were already renting apartments. 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 i
5、n 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
6、 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 wou
7、ld 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 would bear. This w
8、ould 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 equilibrium p
9、rice 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 the
10、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 g
11、ood 2 increases, but the price of good 1 and income remain constant?2.2. The vertical intercept (x2 axis) decreases and the horizontal intercept (x1 axis) stays the same. Thus the budget line becomes atter.3. If the price of good 1 doubles and the price of good 2 triples, does the budget line become
12、 atter or steeper?2.3. Flatter. The slope is 2p1/3p2. 4. 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 then l
13、ater 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 p1x1 +p2x2 = m. The government decides to impose a lump-sum tax of u, a quantity tax on good 1 of t, a
14、nd a quantity subsidy on good 2 of s. What is the formula for the new budget line?2.6. (p1+ t) x1 +(p2s) x2 = mu. 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 as well-o?2.7. Yes, since all of the bundles the co
15、nsumer could aord before are aordable at the new prices and income.3 Preferences1. If we observe a consumer choosing (x1, x2) when (y1, y2) is available one time, are we justied in concluding that (x1, x2) (y1, y2)?3.1. No. It might be that the consumer was indierent between the two bundles. All we
16、are justied in concluding is that (x1, x2) (y1, y2). 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 talle
17、r 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 and B,
18、he always prefers the one who is bigger and faster. Is this preference relation 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 singl
19、e 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 than
20、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 back o
21、n his indierence curve. In this case the direction of increasing utility is toward the origin.8. 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 marginal rate o
22、f 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 of substi
23、tution 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 repulsive
24、 combinations.4 Utility1. The text said that raising a number to an odd power was a monotonic 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 positive u, bu
25、t 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, then one
26、of the bundles would have to be preferred to the other.4. What kind of preferences are represented by a utility function of the form u(x1,x2)=x1 + x2? What about the utility function v(x1,x2)= 13x1 + 13x2?4.4. Both represent perfect substitutes.5. What kind of preferences are represented by a utilit
27、y function of the form u(x1,x2)=x1 +x2? Is the utility function v(x1,x2)=x2 1 +2x1x2 +x2 a monotonic transformation of u(x1,x2)?4.5. Quasilinear preferences. Yes.6. Consider the utility function u(x1,x2)= x1 x2 . What kind of pref- erences does it represent? Is the function v(x1, x2) = x12x22a monot
28、onic transformation of u(x1, x2)? Is the function w(x1, x2) = x12x22 a monotonic transformation of u(x1, x2)?4.6. 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 rate of substi
29、tution?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 good 2?5.1. x2=0 when p2p1, x2 = m/p2 when p2p1, and anything between 0 and m/p2 when p1 = p2.2.
30、Suppose that indierence curves are described by straight lines with a slope of b. Given arbitrary prices and money income p1, p2, and m, what will the consumers optimal choices look like?5.2. The optimal choices will be x1 = m/p1 and x2 = 0 ifp1/p2 b, and any amount on the budget line if p1/p2 = b.3
31、. 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
32、 the consumer buys. Then we know that 2z is the number of teaspoons of sugar he or she buys. We must satisfy the budget constraint2p1z + p2z = m.Solving for z we havez =m2p1+ p2 4. Suppose that you have highly nonconvex preferences for ice cream and olives, like those given in the text, and that you
33、 face prices p1, p2 and have m dollars to spend. List the choices for the optimal consumption bundles.5.4. We know that youll either consume all ice cream or all olives. Thus the two choices for the optimal consumption bundles will be x1 = m/p1, x2 = 0, or x1 = 0, x2 = m/p2.5. If a consumer has a ut
34、ility function u(x1,x2)=x1x4 2, what fraction of her income will she spend on good 2?5.5. This is a Cobb-Douglas utility function, so she will spend 4/(1 + 4) = 4/5 of her income on good 2.6. For what kind of preferences will the consumer be just as well-o facing a quantity tax as an income tax?5.6.
35、 For kinked preferences, such as perfect complements, where the change in price doesnt induce any change in demand.6 Demand1. If the consumer is consuming exactly two goods, and she is always spending all of her money, can both of them be inferior goods?6.1. No. If her income increases, and she spen
36、ds it all, she must be purchasing more of at least one good.2. Show that perfect substitutes are an example of homothetic preferences.6.2. The utility function for perfect substitutes is u( x1, x2)= x1 + x2. Thus if u( x1, x2) u ( y1, y2), we have x1 + x2 y1 + y2. It follows that t x1 + t x2 t y1 +
37、ty2, so that u(t x1,t x2) u (t y1, ty2).3. Show that Cobb-Douglas preferences are homothetic preferences.6.3. The Cobb-Douglas utility function has the property that u(t x1,t x2)=( t x1)a( t x2)1-a = tat1-a x1a x21-a 2 = t x1a x21-a2 = t*u(x1, x2). Thus if u( x1, x2) u ( y1, y2), we know that u(t x1
38、,t x2) u (t y1,t y2), so that Cobb-Douglas preferences are indeed homothetic.4. The income oer curve is to the Engel curve as the price oer curve is to .?6.4. The demand curve.5. If the preferences are concave will the consumer ever consume both of the goods together?6.5. No. Concave preferences can
39、 only give rise to optimal consumption bundles that involve zero consumption of one of the goods.6. Are hamburgers and buns complements or substitutes?6.6. Normally they would be complements, at least for non-vegetarians.7. What is the form of the inverse demand function for good 1 in the case of pe
40、rfect complements?6.7. We know that x1 = m/(p1 + p2). Solving for p1 as a function of the other variables, we have p1 = m x1 p2.8. True or false? If the demand function is x1 = p1, then the inverse demand function is x = 1/p1.6.8. False.7 Revealed Preference1. When prices are (p1,p2) = (1 ,2) a cons
41、umer demands (x1,x2) = (1 ,2), and when prices are ( q1,q2) = (2 ,1) the consumer demands (y1,y2) = (2 ,1). Is this behavior consistent with the model of maximizing behavior?7.1. No. This consumer violates the Weak Axiom of Revealed Preference since when he bought (x1,x2) he could have bought (y1,y2
42、) and vice versa. In symbols: p1x1 + p2x2 =11+22=5 4=12+21=p1y1 + p2y2andq1y1 + q2y2 =22+11=5 4=21+12=q1x1 + q2x2.2. When prices are (p1,p2) = (2 ,1) a consumer demands (x1,x2) = (1 ,2), and when prices are ( q1,q2) = (1 ,2) the consumer demands (y1,y2) = (2 ,1). Is this behavior consistent with the
43、 model of maximizing behavior?7.2. Yes. No violations of WARP are present, since the y-bundle is not aordable when the x-bundle was purchased and vice versa.3. In the preceding exercise, which bundle is preferred by the consumer, the x-bundle or the y-bundle?7.3. Since the y-bundle was more expensiv
44、e than the x-bundle when the x-bundle was purchased and vice versa, there is no way to tell which bundle is preferred.4. We saw that the Social Security adjustment for changing prices would typically make recipients at least as well-o as they were at the base year. What kind of price changes would l
45、eave them just as well-o, no matter what kind of preferences they had?7.4. If both prices changed by the same amount. Then the base-year bundle would still be optimal.5. In the same framework as the above question, what kind of preferences would leave the consumer just as well-o as he was in the bas
46、e year, for all price changes?7.5. Perfect complements.8 Slutsky Equation1. Suppose a consumer has preferences between two goods that are perfect substitutes. Can you change prices in such a way that the entire demand response is due to the income eect?8.1. Yes. To see this, use our favorite example of red pencils and blue pencils. Suppose red pencils cost 10 cents a piece, and blue pencils cost 5 cents a piece, and the consumer spends $1 on pencils. She would then consume 20 blue pencils. If the price of blue pencils falls to 4 cents a piece, she would co