中级微观经济学习题及答案

1、ANSWERS IThe Market 1 Suppose that there were 25 peop I e who had a reservat i on pr i ce of $500, and the 26th person had a reservation price of $200. What would the demand curve look I ike It would be constant at $500 for 25 apartments and then drop to $200. 2. In the above example, what would the

2、 equi librium price be if there were 24 apartments to rent What if there were 26 apartmerrts to rent What if there were 25 apartments to rent In the fi rst case, $500, and in the second case, $200 In the thi rd case, the equiIibr ium pr ice wouId be any pr ice between $200 and $500. 3. If peopIe hav

3、e diff erent reservation prices, why does the market demand curve slope down Because i f we want to rent one more apartment, we have to off er a Iower pr ice The number of peop I e who have reservation pr ices greater than p must a I ways increase as p decreases. 4. In the text we assumed that the c

4、ondominium purchasers came from the inner-ring peopIepeopIe who were already renting apartmerrts. What would happen to the price of inner-ring apartmerrts if al I of the condominium purchasers were outer-ring peopIehe peopIe who were not currently renting apartmerrts in the inner ring The pr ice of

5、apartments in the inner r ing wouId go up sinee demand for apartments would not change but suppIy would decrease 5. Suppose now that the condominium purchasers were al I inner-ring people, but that each condominium was construeted from two apartments. What would happen to the price of apar tmerrts T

6、he pr ice of apar tments i n the in ner ring wou I d r i se. 6. What do you suppose the eff ect of a tax would be on the number of apartments that would be built in the long run A tax wouId undoubtedly reduce the number of apartments suppIied in the long run. 7. Suppose the demand curve is D(p) =100

7、- 2p What price would the monopoI ist 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 He wou Id set a pr ice of 25 and rent 50 apartments. I n the second case he wouId rent a I I 40 apartments at the maximum pr ice the marke t

8、 would bear. Th i s wou I d be given by the solution to D (p) =100- 2p 二 40, which is p* 二 30. 8. If our model of ren t control a I I owed for unrestr ic ted sub I et ti ng, who would end up gett i ng apar tmerrts in the inner circle Would the outcome be Pareto efficient Everyone who had a reservati

9、on pr ice higher than the equi I ibr ium price i n the compe titive marke t, so that the fi na I out come wou I d be Pare to effi c i en t(Of course in the long run there wouId probabIy be fewer new apartments bui It, which would Iead to another kind of ineffi ciency.) 2 Budget Constraint 1. Origina

10、lly 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. The new budget line is given by 2p1x1 +8p2

11、x2 二4m. 2. What happens to the budget I ine if the price of good 2 increases, but the price of good 1 and income remain constarrt The ver ti ca I int ercep t(X2 ax i s) decreases a nd t he hor i z ontal int ercep t (x】ax i s) st ays the same. Thus the budge t I in e becomes fl atter. 3. If the pr ic

12、e of good 1 doubIes and the price of good 2 tr iples, does the budget Iine become fl atter or steeper Flatter. The s I ope i s - 2pi/3P2. 4. What is the defi nit ion of a numeraire good A good whose pr i ce has been set to 1; a I I other goods pr i ces are measured re I ative to the numera i re good

13、 s pr i ce 5. Suppose that the governmen t puts a tax of 15 cen ts a gal I on on gasol ine and then later decides to put a subsidy on gasol ine at a rate of 7 cen ts a gallon. Wha t net tax is this comb i nat i on equivalerrt to A tax of 8 cents a gaI Ion. 6. Suppose that a budget equation is given

14、by P1X1 +P2X2 = m The government decides to impose a Iump-sum tax of u, a quant ity tax on good 1 of t, and a quantity subsidy on good 2 of s. What is the formula for the new budget line （P1+ t） Xi +（P2- s） X2 二 m- u. 7. I f the i ncome of the consumer i ncreases and one of the pr ices decreases at

15、the same time, wi I I the consumer necessari ly be at least as we 11-off Yes, s i nee a I I of the bund I es the consumer cou I d aff ord before are aff ordable at the new pr ices and income 3 Preferences L If we observe a consumer choosing (x, X2) when (yi, Y2) is ava i I ab I e one time, are we ju

16、s ti 丘 ed in cone I ud i ng that (X1, X2)(yi, Y2) No. It mi gh t be that the con sumer was in diff ere nt bet wee n the two bun d I es Al I we are jus tifi ed in cone I ud i ng i s that (xi, x2) (yi, Y2). 2. Consider a group of peop I e A, B, C and the re I ation least as tal I as, as in A is at lea

17、st as tall as B” this re I ation transitive Is it comp Iete Yes to both. 3. Take the same group of peopIe and consider the relation strictly taller than. Is this relation transitive Is it refl exive Is it comp Iete It is transitive, but it is not comp Ietetwo peopIe mi ght be the same height. It is

18、not refl exive sinee it is false that a person i s str i ctly ta I I er than himseIf. 4. A co I Iege footba 11 coach says that given any two Ii nemen A and B, he a Iways prefers the one who is bigger and faster. Is this preference relation transitive Is it comp Iete It is transiti ve, but not comp I

19、ete. What if A were bigger but sIower than B Which one wouId he prefer 5. Can an ind iff erence curve cross itself For example, could Figure depict a single ind iff erence curve Yes An in diff ere nee curve can cross it self, it jus t can t cross another distinct indiff erenee curve. 6. Could Figure

20、 be a s i ng I e i nd iff erence curve if preferences are monotonic case the d i rec tion of increasing utility is to ward the origin. No, because there are bund I es on the i nd i ff erence curve that have str ictly more of both goods than other bundies on the (a I I eged) i nd iff erence curve. 7.

21、 If both pepperoni and anchov i es are bads. wi 11 the i nd iff erence curve have a pos i t i ve or a negative A negative sI ope. If you give the consumer more anchov i es, you， ve made him worse off , so you have to take away some pepperon to get him back on h i s i nd iff ere nee curve. n this 8.

22、Exp lain why convex preferences means that averages are preferred to extrernes. Because the consumer weak Iy prefers the we i ghted average of two bundies to either bundIe. 9. 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 nee

23、d to compe nsate you Five $1 b i I I s wi I I do ni ce I y. Hence the an swer is - 5 or- 1/5, depending on which good you put on the hor i zontaI ax i s. 10. If good 1 is a neutral, what is its marginal rate of substitution for good 2 Zero一i f you take away some of good 1, the consumer needs zero un

24、its of good 2 to compensate him for his Ioss ANSWERS A13 11 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 mono tonic transforma tion (Hin t: consider the case f (u)二lT2.) i s a monoton i c transformation for

25、 pos itive u, but not for negative u. 2. Which of the following are monotonic transformations (1) u =2 v- 13;(2) u = - 1/v2； (3)u =1/v2;u 二 In v; 口 = - e- v;u = v2; u = v2 for v0;(8) u = 2 for vy or yx, which means that one of the bund I es has more of both goods. But if preferences are monotonic, t

26、hen one of the bundIes wouId have to be preferred to the other 4. What kind of preferences are represented by a uti I ity funct i on of the form u (x1, x2) = x x1 + x2 What about the uti I ity function v(x1,x2)= 13x1 + 13x2 Both represent perfect substitutes. 5. What kind of preferences are represen

27、ted by a utility function of the form u(x1, x2) =x1 +、x2 Is the uti I ity function v(x1,x2) =x2 1 +2x1 x2 +x2 a monotonic transformation of u(x1,x2) Quas iIinear preferences Yes. 6. Consider the uti I ity function u(x1, x2)=、X2 What kind of pref- erences does it represent Is the function v(xX2) =Xi

28、X2 a mono tonic t ransforma tion of u(xi, X2) Is the func tion w(xi, X2) = X X2 a monotonic transformation of u(xi, X2) The utility function represe nts Cobb-Douglas prefere nces No. Yes. 7. Can you exp lain why taking a monotonic transformation of a utility function doesn t change the marginal rate

29、 of substitution Because the MRS is measured along an i nd iff erence curve, and utility remains constant along an ind iff erence curve. 5 Oho i ce 1 If two goods are perfect substitutes, what is the demand funct i on for good 2 X2=0 whe n P2P1, X2 二 m/P2 whe n P2P1, and anything bet wee n 0 and m/p

30、2 when Pi = P2. 2. Suppose tha t i nd iff erence curves are described by straigh t I ines with a slope of - b. Given arbitrary prices and money income p1, p2, and m, what wi 11 the consumer s opt i ma I choices Iook Iike The opt ima I choices wi I I be x1 = m/p1 and x2 = 0 ifp1/p2 b, and any amount

31、on the budget Iine if p1/p2 = b. 3 Suppose that a consumer a I ways consumes 2 spoons of sugar with each cup of coff ee. If the pr ice of sugar is p1 per spoonful and the price of coff ee is p2 per cup and the consumer has m dol lars to spend on coff ee and sugar, how much wi 11 he or she want to pu

32、rchase Let z be the number of cups of coff ee the consumer buys Then we know that 2z i s the number of teaspoons of sugar he or she buys We must satisfy the budget constra i nt 2piz + P2Z 二 m. Solving for z we have m z P2 4. Suppose that you have highly nonconvex preferences for ice cream and ol ive

33、s, I ike those given in the text, and that you face pr i ces p1, p2 and have m dol I ars to spend. Li st the cho ices for the opt i ma I consumpt i on bundles. We know that you I I ei ther consume a I I ice cream or a I I ol i ves. Thus the two choices for the optima I consumption bund Ies wi I I be

34、 x1 = m/Pi, x2 二 0, or x1 = 0, x2 = m/P2. 5. If a consumer has a utility function u (x1, x2) =x1x4 2, wha t fraction of her income wiI I she spend on good 2 This is a Cobb-Doug I as utility function, so she wi11 spend 4/(1 + 4) = 4/5 of her income on good 2. 6. For what kind of preferences wi 11 the

35、 consumer be just as we 11 -off f ac i ng a quantity tax as an income tax For k i nked preferences, such as perfect comp I ements, where the change in price doesrf t induce any change in demand. 6 Demand 1- If the consumer is consuming exactly two goods, and she is always spending al I of her money,

36、 can both of them be inferior goods No. If her i ncome in creases, and she spends it all, she mus t be purchasing more of at least one good. 2. Show that perfect substitutes are an examp I e of homothetic preferences. The utility function for perfect substitutes i s u ( xi, X2)二 xi + X2. Thus if u (

37、 xi, X2)u ( yi, Y2), we have X1 + X2 V1 + Y2. It fol lows that t X1 + t X2 t yi + so that u(t X1, t X2)u (t yi, ty2). 3. Show that Cobb-Douglas preferences are homothetic preferences. The Cobb-Douglas utility function has the property that u (t X1 , t X2 )= ( t xj a ( t X2) 1 _ 0 = tat1a X21 - a 2 =

38、 t XX21 a2 = t*U (x1, X2). Thus if u( X1, X2)U ( yi, Y2), we know that u(t x】，t X2) u (t yi,t y2), so that Cobb-Doug I as preferences are indeed homothetic. 4. The income off er curve is to the Engel curve as the price off er curve is to The demand curve. 5. If the preferences are con cave wi 11 the

39、 consumer ever consume both of the goods together No. Concave preferences can only give r ise to optima I con sump tion bund I es that involve zero consump tion of one of the goods. Are hamburgers and buns comp I ements or substitutes Norma I Iy they would be comp I ements, at least for nonvegetar i

40、 ans. 7. What is the form of the inverse demand function for good 1 in the case of perfect comp Iements We know that x1 二 m/ (p1 + p2). Solving for p1 as a function of the other var iables, we have p1 = m x1 - p2. 8. True or false If the demand function is x1 = - p1, then the inverse demand function

41、 is x = - 1/p1. 7 ReveaIed Preference 1- When 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 Re

42、veaIed Preference since when he bought (x1, x2) he could have bought (y1,y2) and vice versa. In symboIs: p1x1 + p2x2 =1X1+2X2=5 4=1 X2+2X 1=p1y1 + p2y2 and q1y1 + q2y2 =2X2+1 X1=5 4=2X1+1 X2=q1x1 + q2x2. 2. When prices are (p1, p2) = (2 11) a consumer demands (x1, x2) =(1 ,2), and when prices are (

43、q1, q2) = (1 ,2) the consumer demands (y1,y2) = (2,1). Is this behavior consistent with the model of maximizing behavior Yes. No violations of WARP are present, since the y-bundIe is not aff ordable when the x-bundIe was purchased and vice versa 3. In the precedi ng exercise, which bundle is preferr

44、ed by the consumer, the x-bundIe or the y-bundIe Since the y-bund Ie was more expensive than the x-bund Ie when the x-bundIe was purchased and vice versa, there is no way to teI I which bund Ie i s preferred 4. We saw that the Socia I Security adjustment for changing prices would typ i ca11y make re

45、cipients at least as we 11-off as they were at the base year. What kind of price changes would leave them jus t as we 11 -off , no matter wha t k i nd of preferences they had I f both pr i ces changed by the same amount Then the base-year bund Ie wouId still be optima I. 5. In the same framework as

46、the above question, what kind of preferences would leave the consumer just as we 11 -off as he was in the base year, for al I price changes Perfect comp I ernents. 8 SIutsky Equat i on L Suppose a consumer has preferences between two goods that are perfect substitutes. Can you change prices in such

47、a way that the entire demand response is due to the income eff ect Yes. To see t hi s, use our favor i te examp I e of red pen ci I s and b I ue penc i I s. Suppose red pen oils cos t 10 cents a piece, and bIue penci Is cost 5 cents a piece, and the consumer spends $1 on penci Is. She wouId then con

48、sume 20 bIue penci I s If the pr ice of b I ue penci I s fa I I s to 4 cents a piece, she wou I d con sume 25 bIue penoils, a change which is entirely due to the income eff ect. 2. Suppose that preferences are concave. Is it still the case that the substitution eff ect is negative Yes. 3. In the cas

49、e of the gasol ine tax, what would happen if the rebate to the consumers were based on their original consumption of gasoline, x, rather than on their nal consumption of gasoline, x Then the income eff ect wouId canceI out. Al I that wouId be I ef t wou I d be the pure subs ti tut ion eff ec t, whic

50、h would automaticaI Iy be negative. 4. In the case descr ibed in the preceding question, wouId the governme nt be paying out more or I ess t han it rece i ved i n tax revenues They are receiving tx in revenues and paying out tx, so they are losing money. 5. In t his case would the consumers be bet t

51、er off or worse off if the tax with rebate based on original consumption were in eff ect Since thei r oId consumption is aff ordable, the consumers wou I d have to be at leas t as we I I-off . Th i s happens because the government is giving them back more money than they are losing due to the higher

52、 pr ice of gasol ine 9 Buying and Selling 1. If 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). 2. The prices are (p1, p2) = (2 ,3), and the consumer is currently consuming (x1, x2) = (4,4). There is a perfect market for the t

53、wo goods in which they can be bought and sold cost I ess I y. Wi 11 the consumer necessar i ly prefer consuming the bund Ie (y1, y2) = (3,5) Wi 11 she necessar i ly prefer having the bund Ie (y1, y2) The bund I e (y1, y2) = (3 , 5) costs more than the bund I e (4, 4) at the current pr ices. The cons

54、umer wi I I not necessar i Iy prefer con sum i ng t his bund I e, but would cer tainly prefer to own it, since she couId sei I it and purchase a bundIe that she wouId prefer. 3. The prices are (p1, p2) = (2 ,3), and the consumer is currently consuming (x1,x2) = (4 ,4). Now the prices change to (q1,

55、q2) = (2 ,4). Could the consumer be better off under these new prices Sure It depe nds on whe ther she was a net buyer or a net se I I er of the good that became more expensive. 4. The currently imports about half of the petroleum that it uses. The rest of its needs are met by domes tic production.

56、Could the price of oil rise so much that the would be made better off Yes, but only if the . switched to being a net exporter of oil. 5. Suppose that by some miracle the number of hours in the day increased from 24 to 30 hours (with luck this would happen shortly before exam week). How would this af

57、f ect the budget constraint The new budget I ine wouId shift outward and rema in para I lei to the oId one, since the increase in the number of hours in the day i s a pure endowment eff ect. 6. If leisure is an inferior good, what can you say about the slope of the labor supply curve The sI ope wiI

58、I be pos itive. 10 IntertemporaI Choice ! How much is $1 mi 11 ion to be del ivered 20 years in the future worth today if the interest rate is 20 percent According to Tab Ie , $1 20 years from now is worth 3 cents to day at a 20 perce nt int eres t rate. Thus $1 mi I I ion is worth 03X 1,000, 000 =

59、$30, 000 today. intertemporal budget 2. As the irrterest rate rises, does the constrairrt be- come steeper or fl atter The s I ope of the intert empora I budge t const ra i nt is equa I to -(1+r). Thus as r in creases the s I ope becomes more n ega tive (steeper). 3. Would the assumption that goods

60、are perfect substitutes be valid in a study of irrtertemporaI food purchases If goods are perfect substitutes, then consumers wiI I only purchase the cheaper good In the case of intertemporaI food purchases, th i s imp Ii es that consumers only buy food i n one per iod, which may not be very reaIist