中級微觀經(jīng)濟(jì)學(xué)習(xí)題及答案

1、ANSWERSThe MarketSuppose 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. It would be constant at $500 for 25 apartments and then drop to $200.In the above example, what would the equilibrium pr

2、ice be if there were 24 apartments to rent What if there were 26 apartments to rent What if there were 25 apartments to rent. 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.If people have dierent reservation pric

3、es, why does the market demand curve slope down. 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.In the text we assumed that the condominium purchasers came from the inner

4、-ring people people who were already renting apartments. What would happen to the price of inner-ring apartments if all of thecondominium purchasers were outer-ring people the people who were not currently renting apartments in the inner ring. The price of apartments in the inner ring would go up si

5、nce demand for apartments would not change but supply would decrease.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. The price of apartments in the inner ring would ri

6、se.What do you suppose the eect of a tax would be on the number of apartments that would be built in the long run. A tax would undoubtedly reduce the number of apartments supplied in the long run.Suppose the demand curve is D(p) = 1002p. What price would the monopolist set if he had 60 apartments Ho

7、w many would he rent What price would he set if he had 40 apartments How many would he rent. 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 would be given by the solution to D(p) = 1002p = 40, whi

8、ch is p = 30.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.Everyone who had a reservation price higher than the equilibrium price in the competitive market, so that the nal outcome would be

9、 Pareto ecient. (Of course in the long run there would probably be fewer new apartments built, which would lead to another kind of ineciency.)Budget ConstraintOriginally 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,

10、and income becomes 4 times larger. Write down an equation for the new budget line in terms of the original prices and income.The new budget line is given by 2p1x1 +8p2x2 =4m.What happens to the budget line if the price of good 2 increases, butthe price of good 1 and income remain constant.The vertic

11、alintercept ( ,axis) decreasesand the horizontal interceptC axis) stays the same. Thus the budget line becomes atter.If the price of good 1 doubles and the price of good 2 triples, does the budget line become atter or steeper.Flatter. The slope is 2 /3| _What is the denition of a numeraire good.A go

12、od whose price has been set to 1; all other goodsprices are measured relative to the numeraire good s price.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 of7 cents a gallon. What net tax is this combination equiv

13、alent to.A tax of 8 cents a gallon.Suppose that a budget equation is given by p、i + _ 1=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.(pi+ t) X +(p：s)X2 = mu.If the income of

14、the consumer increases and one of the prices decreases at the same time, will the consumer necessarily be at least as well-o.Yes, since all of the bundles the consumer could aord before are aordable at the new prices and income.PreferencesIf we observe a consumer choosing (m , x?) when 勺 i, y：) is a

15、vailable one time, are we justied in concluding that g,工2) ( i ,”).No. It might be that the consumer was indierent between the two bundles. All we are justied in concluding is that (1 ,x2) ( i，丫2).Consider a group of people A, B, C and the relation“ at least as tas, “as in As at least as tall as B.

16、Is this relation transitive Is it complete.Yes to both.Take the same group of people and consider the relation “strictly taller than. Is this relation transitive Is it reexive Is it complete .It is transitive, but it is not complete two people might be the same height. It is not reexive since it is

17、false that a person is strictly taller than himself.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.It is transitive, but not complete. What if A were bigger but slower thanB W

18、hich one would he preferCan an indierence curve cross itself For example, could Figure depict a single indierence curve. Yes. An indierence curve can cross itself, it just can ctross another distinct indierence curve.Could Figure be a single indierence curve if preferences are monotonic. No, because

19、 there are bundles on the indierence curve that have strictly more of both goods than other bundles on the (alleged) indierence curve.If both pepperoni and anchovies are bads, will the indierence curve have a positive or a negative slope. A negative slope. If you give the consumer more anchovies, yo

20、u ve made him worse o, so you have to take away some pepperoni to get him back on his indierence curve. In this case the direction of increasing utility is toward the origin.Explain why convex preferences means that“ averages are preferredto extremes. ”. Because the consumer weakly prefers the weigh

21、ted average of twobundles to either bundle.What is your marginal rate of substitution of $1 bills for $5 bills. 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 axi

22、s.If good 1 is a“ neutral, ” what is its marginal rate of substitution forgood 2. Zero if you take away some of good 1, the consumer needs zero unitsof good 2 to compensate him for his loss.ANSWERS A13Think of some other goods for which your preferences might be concave. Anchovies and peanut butter,

23、 scotch and Kool Aid, and other similar repulsive combinations.UtilityThe 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 casef(u)=uA2.).The function f(u)=uA2 is a mono

24、tonic transformation for positive u, but not for negative u.Which of the following are monotonic transformationsu =2 v13; (2) u =八2；(3)u =1/丫A2；(4)u = In v; (5)u = eAv; (6)u=vA2; (7) u = v、2 for v0;(8) u = v、2 for vy or yx, which means that one of the bundles has more of both goods. But if preferenc

25、es are monotonic, then one of the bundles would have to be preferred to the other.What kind of preferences are represented by a utility function of the form u(x1,x2)=、l 二 What about the utility function v(x1,x2)= 13x1 +13x2 .Both represent perfect substitutes.What kind of preferences are represented

26、 by a utility function of the form u(x1,x2)=x1 + ；x：/Is the utility function v(x1,x2)=x2 1 +2x1 7 +x2 a monotonic transformation of u(x1,x2).Quasilinear preferences. Yes.Consider the utility function u(x1,x2)= ： . What kind of preferences does it represent Is the function v ( , _) = _ _a monotonic t

27、ransformation of u(*,x) Is the function wg,i) =xx；f a monotonic transformation of u (Xi, x?).The utility function represents Cobb-Douglas preferences. No. Yes.Can you explain why taking a monotonic transformation of a utility function doesn t change the marginal rate of substitution.Because the MRS

28、is measured along an indierence curve, and utility remains constant along an indierence curve.ChoiceIf two goods are perfect substitutes, what is the demand function for good 2.X2=0 whenPOi,工工=m/良 when SP】，and anything between 0 and m/p2 when Pi = P2.Suppose that indierence curves are described by s

29、traight lines with a slope of b. Given arbitrary prices and money income p1, p2, and m, what will the consumer s optimal choices look like.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.Suppose that a consumer always consumes 2 spoons of

30、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.Let z be the number of cups of coee the consumer buys. Then we know that 2z is the number of t

31、easpoons of sugar he or she buys. We must satisfy the budget constraint2 iz + 卜），z = m.Solving for z we havemz =m 卜 p/Suppose that you have highly nonconvex preferences for ice cream and olives, like those given in the text, and that you face prices p1, p2 and have m dollars to spend. List the choic

32、es for the optimal consumption bundles.We know that you ll either consume all ice cream or all olives. Thus the two choices for the optimal consumption bundles will be x1 = m/pj, x2 = 0, or x1 = 0, x2 = m/.If a consumer has a utility function u(x1,x2)=x1x4 2, what fraction of her income will she spe

33、nd on good 2.This is a Cobb-Douglas utility function, so she will spend 4/(1 + 4) = 4/5 of her income on good 2.For what kind of preferences will the consumer be just as well-o facing a quantity tax as an income tax.For kinked preferences, such as perfect complements, where the change in price doesn

34、 t induce any chdegeiind.DemandIf the consumer is consuming exactly two goods, and she is always spending all of her money, can both of them be inferior goods.No. If her income increases, and she spends it all, she must be purchasing more of at least one good.Show that perfect substitutes are an exa

35、mple of homothetic preferences.The utility function for perfect substitutes is u( if u(x,x：) u (yi,v2), we have xi + x2yi + It follows that t xi + t 2 tyj + &工 so that u(txi,txt) u (tyi, ty?).Show that Cobb-Douglas preferences are homothetic preferences.The Cobb-Douglas utility function has the prop

36、erty thatu(t X ,t K2)=(tx垃= tatljME1 x21-a 2 = txx-t = t*u(x1, xz). Thus if u(xi, x：) u (yi, y?), we know that u(tM ,tx)u (t 力，so that Cobb-Douglas preferences are indeed homothetic.The income oer curve is to the Engel curve as the price oer curve is to .The demand curve.If the preferences are conca

37、ve will the consumer ever consume both of the goods together.No. Concave preferences can only give rise to optimal consumption bundles that involve zero consumption of one of the goods.Are hamburgers and buns complements or substitutes.Normally they would be complements, at least for non-vegetarians

38、.What is the form of the inverse demand function for good 1 in thecase of perfect complements . We know that x1 = m/(p1 + p2). Solving for p1 as a function of the other variables, we have p1 = m x1 p2.True or false If the demand function is x1 = p1, then the inverse demand function is x = 1/p1. Fals

39、e.Revealed PreferenceWhen prices are (p1,p2) = (1 ,2) a consumer 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. No. This consumer violates the Weak Axiom of Revealed Preference

40、since when he bought (x1,x2) he could have bought (y1,y2) and vice versa. In symbols:plxl + p2x2 =1 X1+2X2=5 4=1 X2+2Xl=p1y1 + p2y2 andqlyl + q2y2 =2 乂 2+1 乂 1=5 4=2 x 1+1 乂 2=q1x1 + q2x2.When prices are (p1,p2) = (2 ,1) a consumer demands (x1,x2) = (1 ,2), and when prices are ( q1,q2) = (1 ,2) the

41、consumer demands (y1,y2) = (2 ,1). Is this behavior consistent with the model of maximizingbehavior . Yes. No violations of WARP are present, since the y-bundle is not aordable when the x-bundle was purchased and vice versa.In the preceding exercise, which bundle is preferred by the consumer, the x-

42、bundle or the y-bundle. Since the y-bundle was more expensive than the x-bundle when the x-bundle was purchased and vice versa, there is no way to tell which bundle is preferred.We saw that the Social Security adjustment for changing prices would typically make recipients at least as well-o as they

43、were at the base year. What kind of price changes would leave them just as well-o, no matter what kind of preferences they had. If both prices changed by the same amount. Then the base-year bundle would still be optimal.In the same framework as the above question, what kind of preferences would leav

44、e the consumer just as well-o as he was in the base year, for all price changes . Perfect complements.Slutsky EquationSuppose 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. Yes.

45、 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

46、nsume 25 blue pencils, a change which is entirely due to the income eect.Suppose that preferences are concave. Is it still the case that the substitution eect is negative. Yes.In the case of the gasoline tax, what would happen if the rebate to the consumers were based on their original consumption o

47、f gasoline, x, rather than on their nal consumption of gasoline, x . Then the income eect would cancel out. All that would be left would be the pure substitution eect, which would automatically be negative.In the case described in the preceding question, would the government be paying out more or le

48、ss than it received in tax revenues. They are receiving tx in revenues and paying out tx, so they are losing money.In this case would the consumers be better o or worse o if the tax with rebate based on original consumption were in eect. Since their old consumption is aordable, the consumers would h

49、ave to be at least as well-o. This happens because the government is giving them back more money than they are losing due to the higher price of gasoline.Buying and SellingIf a consumer s net demands are (5,3) and her endowment is (4,4), what are her gross demands. Her gross demands are (9,1).The pr

50、ices are (p1,p2) = (2 ,3), and the consumer is currently consuming (x1,x2) = (4 ,4). There is a perfect market for the two goods in which they can be bought and sold costlessly. Will the consumer necessarily prefer consuming the bundle (y1,y2) = (3 ,5) Will she necessarily prefer having the bundle (

51、y1,y2). The bundle (y1,y2) = (3 ,5) costs more than the bundle (4,4) at the current prices. The consumer will not necessarily prefer consuming this bundle, but would certainly prefer to own it, since she could sell it andpurchase a bundle that she would prefer.The prices are (p1,p2) = (2 ,3), and th

52、e consumer is currently consuming (x1,x2) = (4 ,4). Now the prices change to (q1,q2) = (2 ,4). Could the consumer be better o under these new prices. Sure. It depends on whether she was a net buyer or a net seller of the good that became more expensive.The . currently imports about half of the petro

53、leum that it uses. The rest of its needs are met by domestic production. Could the price of oil rise so much that the . would be made better o. Yes, but only if the . switched to being a net exporter of oil.Suppose that by some miracle the number of hours in the day increased from 24 to 30 hours (wi

54、th luck this would happen shortly before exam week). How would this aect the budget constraint. The new budget line would shift outward and remain parallel to the old one, since the increase in the number of hours in the day is a pure endowment eect.If leisure is an inferior good, what can you say a

55、bout the slope of the labor supply curve. The slope will be positive.Intertemporal ChoiceHow much is $1 million to be delivered 20 years in the future worth today if the interest rate is 20 percent. According to Table , $1 20 years from now is worth 3 cents today at a 20percent interest rate. Thus $

56、1 million is worth .03x 1,000,000 = $30,today.As the interest rate rises, does the intertemporal budget constraint be- come steeper or atter. The slope of the intertemporal budget constraint is equal to (1+r). Thus as r increases the slope becomes more negative (steeper).Would the assumption that go

57、ods are perfect substitutes be valid ina study of intertemporal food purchases. If goods are perfect substitutes, then consumers will only purchase the cheaper good. In the case of intertemporal food purchases, this implies that consumers only buy food in one period, which may not be very realistic.

58、A consumer, who is initially a lender, remains a lender even after a decline in interest rates. Is this consumer better o or worse o after thechange in interest rates If the consumer becomes a borrower after the change is he better o or worse o. In order to remain a lender after the change in intere

59、st rates, the consumer must be choosing a point that he could have chosen under the old interest rates, but decided not to. Thus the consumer must be worse o. If the consumer becomes a borrower after the change, then he is choosing a previously unavailable point that cannot be compared to the initia

60、l point (since the initial point is no longer available under the new budget constraint), and therefore the change in the consumer is unknown.What is the present value of $100 one year from now if the interest rate is 10% What is the present value if the interest rate is 5%. At an interest rate of 1