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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 wouldthe 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 p
2、rice be if therewere 24 apartments to rent?What if there were 26 apartments torent?What if there were 25 apartments to rent?1.2.In the first 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 different reservation pri
3、ces,why does the marketdemand curve slope down?1.3.Because if we want to rent one more apartment,we have to offer alower price.The number of people who have reservation prices greaterthan p must always increase as p decreases.4.In the text we assumed that the condominium purchasers came fromthe inne
4、r-ring peoplepeople who were already renting apartments.What would happen to the price of inner-ring apartments if all of thecondominium purchasers were outer-ring peoplethe people whowere not currently renting apartments in the inner ring?1.4.The price of apartments in the inner ring would go up si
5、nce demandfor apartments would not change but supply would decrease.5.Suppose now that the condominium purchasers were all inner-ringpeople,but that each condominium was constructed from twoapartments.What would happen to the price of apartments?1.5.The price of apartments in the inner ring would ri
6、se.6.What do you suppose the effect of a tax would be on the number ofapartments that would be built in the long run?1.6.A tax would undoubtedly reduce the number of apartments suppliedin the long run.7.Suppose the demand curve is D(p)=100-2p.What price would themonopolist set if he had 60 apartment
7、s?How many would he rent?What price would he set if he had 40 apartments?How many would herent?1.7.He would set a price of 25 and rent 50 apartments.In the secondcase he would rent all 40 apartments at the maximum price the marketwould bear.This would be given by the solution to D(p)=100-2p=40,which
8、 is p*=30.8.If our model of rent control allowed for unrestricted subletting,whowould end up getting apartments in the inner circle?Would theoutcome be Pareto efficient?1.8.Everyone who had a reservation price higher than the equilibriumprice in the competitive market,so that the final outcome would
9、 bePareto efficient.(Of course in the long run there would probably befewer new apartments built,which would lead to another kind ofinefficiency.)2 Budget Constraint1.Originally the consumer faces the budget line plxl+p2x2=m.Thenthe 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 thenew budget line in terms of the original prices and income.2.1.The new budget line is given by 2plxl+8p2x2=4m.2.What happens to the budget line if the price of good 2 increases,butthe price of good 1 and income remain constant?2.2.The v
11、ertical intercept(x2 axis)decreases and the horizontalintercept(x】axis)stays the same.Thus the budget line becomes flatter.3.If the price of good 1 doubles and the price of good 2 triples,doesthe budget line become flatter or steeper?2.3.Flatter.The slope is-2p1/3p2.4.What is the definition of a num
12、eraire good?2.4.A good whose price has been set to 1;all other goods/prices aremeasured relative to the numeraire good/s price.5.Suppose that the government puts a tax of 15 cents a gallon ongasoline and then later decides to put a subsidy on gasoline at a rate of7 cents a gallon.What net tax is thi
13、s combination equivalent to?2.5.A tax of 8 cents a gallon.6.Suppose that a budget equation is given byPiXi+p2x2=m.The government decides to impose a lump-sum taxof u,a quantity tax on good 1 of t,and a quantity subsidy on good 2 ofs.What is the formula for the new budget line?2.6.(pi+t)xi+(p2-s)x2=m
14、-u.7.If the income of the consumer increases and one of the pricesdecreases at the same time,will the consumer necessarily be at leastas well-off?2.7.Yes,since all of the bundles the consumer could afford before areaffordable at the new prices and income.3 Preferences1.If we observe a consumer choos
15、ing 的 甩)when(y 2)is availableone time,are we justified in concluding that(x1,x2)(yi，yz)?3.1.No.It might be that the consumer was indifferent between the twobundles.All we are justified in concluding is that(x1,x2)(y1/y2).2.Consider a group of people A,B,C and the relation Mat least as tall as,”as in
16、 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“strictlytaller than/*Is this relation transitive?Is it reflexive?Is it complete?3.3.It is transitive,but it is not completetwo people might be the sameheig
17、ht.It is not reflexive since it is false that a person is strictly taller thanhimself.4.A college football coach says that given any two linemen A and B,healways prefers the one who is bigger and faster.Is this preferencerelation transitive?Is it complete?3.4.It is transitive,but not complete.What i
18、f A were bigger but slowerthan B?Which one would he prefer?5.Can an indifference curve cross itself?For example,could Figure 3.2depict a single indifference curve?3.5.Yes.An indifference curve can cross itself,it just cant cross anotherdistinct indifference curve.6.Could Figure 3.2 be a single indif
19、ference curve if preferences aremonotonic?3.6.No,because there are bundles on the indifference curve that havestrictly more of both goods than other bundles on the(alleged)indifference curve.7.If both pepperoni and anchovies are bads,will the indifference curvehave a positive or a negative slope?3.7
20、.A negative slope.If you give the consumer more anchovies,you/vemade him worse off,so you have to take away some pepperoni to gethim back on his indifference curve.In this case the direction ofincreasing utility is toward the origin.8.Explain why convex preferences means that“averages are preferredt
21、o extremes?/3.8.Because the consumer weakly prefers the weighted average of twobundles to either bundle.9.What is your marginal 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 tocompensate you?Five$1 bills will do nicely.Hence the answer is-5or-
22、1/5,depending on which good you put on the horizontal axis.10.If good 1 is a neutral,“what is its marginal rate of substitution forgood 2?3.10.Zeroif you take away some of good 1,the consumer needs zerounits of good 2 to compensate him for his loss.ANSWERS A1311.Think of some other goods for which y
23、our preferences might beconcave.3.11.Anchovies and peanut butter,scotch and Kool Aid,and othersimilar repulsive combinations.4 Utility1.The text said that raising a number to an odd power was amonotonic transformation.What about raising a number to an evenpower?Is this a monotonic transformation?(Hi
24、nt:consider the casef(u)=uA2.)4.1.The function f(u)=uA2 is a monotonic transformation for positive u,but not for negative u.2.Which of the following are monotonic transformations?(1)u=2 v-13;(2)u=-l/vA2;(3)u=l/vA2;(4)u=In v;(5)u=-eA-v;(6)u=vA2;(7)u=vA2 for v0;(8)u=vA2 for vyor yx,which means thatone
25、 of the bundles has more of both goods.But if preferences aremonotonic,then one of the bundles would have to be preferred to theother.4.What kind of preferences are represented by a utility function of theform u(xl,x2)=x 4-x2?What about the utility function v(xlzx2)=13x1+13x2?4.4.Both represent perf
26、ect substitutes.5.What kind of preferences are represented by a utility function of theform u(xlzx2)=xl+Vx2?Is the utility function v(xl,x2)=x2 1+2xlVx2+x2 a monotonic transformation of u(xl,x2)?4.5.Quasilinear preferences.Yes.6.Consider the utility function u(xl/x2)=A/x1x2.What kind of preferences
27、does it represent?Is the function v(x1/x2)=x12x22a monotonictransformation of u(x17x2)?Is the function w(xi,x?Tx/x22 amonotonic transformation of u(xlzx2)?4.6.The utility function represents Cobb-Douglas preferences.No.Yes.7.Can you explain why taking a monotonic transformation of a utilityfunction
28、doesnt change the marginal rate of substitution?4.7.Because the MRS is measured along an indifference curve,andutility remains constant along an indifference curve.5 Choice1.If two goods are perfect substitutes,what is the demand function forgood 2?5.1.X2=0 whenp2Pi，x2=m/p2 when p2Pi，and anything be
29、tween0 andm/p2 when=p2.2.Suppose that indifference curves are described by straight lines witha slope of-b.Given arbitrary prices and money income pl,p2,andm,what will the consumers optimal choices look like?5.2.The optimal choices will be xl=m/pl and x2=0 ifpl/p2 b,and any amount on the budget line
30、 if pl/p2=b.3.Suppose that a consumer always consumes 2 spoons of sugar witheach cup of coffee.If the price of sugar is pl per spoonful and the priceof coffee is p2 per cup and the consumer has m dollars to spend oncoffee and sugar;how much will he or she want to purchase?5.3.Let z be the number of
31、cups of coffee the consumer buys.Then weknow that 2z is the number of teaspoons of sugar he or she buys.Wemust satisfy the budget constraint2Plz+p2z=m.Solving for z we havemZ-2P1+P24.Suppose that you have highly nonconvex preferences for ice creamand olives,like those given in the text,and that you
32、face prices pl,p2and have m dollars to spend.List the choices for the optimalconsumption bundles.5.4.We know that you/ll either consume all ice cream or all olives.Thusthe two choices for the optimal consumption bundles will be xl=m/plzx2=0,orxl=0,x2=m/p2.5.If a consumer has a utility function u(xl,
33、x2)=xlx4 2,what fraction ofher 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-offfacing a quantity tax as an income tax?5.6.For kinked preferences,such
34、as perfect complements,where thechange in price doesnt induce any change in demand.6 Demand1.If the consumer is consuming exactly two goods,and she is alwaysspending all of her money,can both of them be inferior goods?6.1.No.If her income increases,and she spends it all,she must bepurchasing more of
35、 at least one good.2.Show that perfect substitutes are an example of homotheticpreferences.6.2.The utility function for perfect substitutes is u(xlz x2)=+x2.Thus if u(xlz x2)u(y i，y2),we have+x2 yi+y2.It follows thattx i+1 x2 t yi+ty2,so thatu(t xlzt x2)u(t y i，ty2).3.Show that Cobb-Douglas preferen
36、ces are homothetic preferences.6.3.The Cobb-Douglas utility function has the property thatu(t xlzt x2)=(t x j)a(tx2)1-a=tat1-a xa x21-a 2=t X ja x21-a2=t*u(xl,x2).Thus if u(X,x2)u(y17 y2),we know that u(t xlzt x2)u(t ylzt y2),so that Cobb-Douglas preferences are indeed homothetic.4.The income offer
37、curve is to the Engel curve as the price offer curveis to.?6.4.The demand curve.5.If the preferences are concave will the consumer ever consume bothof the goods together?6.5.No.Concave preferences can only give rise to optimal consumptionbundles that involve zero consumption of one of the goods.6.Ar
38、e 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 thecase of perfect complements?6.7.We know that xl=m/(pl+p2).Solving for pl as a function of theother variables,we have
39、 pl=m xl-p2.8.True or false?If the demand function is xl=-pl,then the inversedemand function is x=-1/pl.6.8.False.7 Revealed Preference1.When prices are(pl,p2)=(1,2)a consumer demands(xl,x2)=(1,2),and when prices are(ql,q2)=(2,1)the consumer demands(yl,y2)=(2,1).Is this behavior consistent with the
40、model of maximizingbehavior?7.1.No.This consumer violates the Weak Axiom of RevealedPreference since when he bought(xl,x2)he could have bought(yl,y2)and vice versa.In symbols:plxl+p2x2=lxl+2x2=5 4=lx2+2xl=plyl+p2y2andqlyl+q2y2=2x2+lxl=5 4=2xl+lx2=qlxl+q2x2.2.When prices are(pl,p2)=(2,1)a consumer de
41、mands(xl,x2)=(1,2),and when prices are(ql,q2)=(1,2)the consumer demands(yl,y2)=(2,1).Is this behavior consistent with the model of maximizingbehavior?7.2.Yes.No violations of WARP are present,since the y-bundle is notaffordable when the x-bundle was purchased and vice versa.3.In the preceding exerci
42、se,which bundle is preferred by the consumer,the x-bundle or the y-bundle?7.3.Since the y-bundle was more expensive than the x-bundle when thex-bundle was purchased and vice versa,there is no way to tell whichbundle is preferred.4.We saw that the Social Security adjustment for changing priceswould t
43、ypically make recipients at least as well-off as they were at thebase year.What kind of price changes would leave them just as well-off,no matter what kind of preferences they had?7.4.If both prices changed by the same amount.Then the base-yearbundle would still be optimal.5.In the same framework as
44、 the above question,what kind ofpreferences would leave the consumer just as well-off as he was in thebase year,for all price changes?7.5.Perfect complements.8 SlutskyEquation1.Suppose a consumer has preferences between two goods that areperfect substitutes.Can you change prices in such a way that t
45、he entiredemand response is due to the income effect?8.1.Yes.To see this,use our favorite example of red pencils and bluepencils.Suppose red pencils cost 10 cents a piece,and blue pencils cost5 cents a piece,and the consumer spends$1 on pencils.She would thenconsume 20 blue pencils.If the price of b
46、lue pencils falls to 4 cents apiece,she would consume 25 blue pencils,a change which is entirelydue to the income effect.2.Suppose that preferences are concave.Is it still the case that thesubstitution effect is negative?8.2.Yes.3.In the case of the gasoline tax,what would happen if the rebate tothe
47、 consumers were based on their original consumption of gasoline,x,rather than on their final consumption of gasoline,x?8.3.Then the income effect would cancel out.All that would be Ie代would be the pure substitution effect,which would automatically benegative.4.In the case described in the preceding
48、question,would thegovernment be paying out more or less than it received in tax revenues?8.4.They are receiving tx in revenues and paying out tx,so they arelosing money.5.In this case would the consumers be better off or worse off if the taxwith rebate based on original consumption were in effect?8.
49、5.Since their old consumption is affordable,the consumers wouldhave to be at least as well-off.This happens because the government isgiving them back more money than they are losing due to the higherprice of gasoline.9 Buying and Selling1.If a consumers net demands are(5,-3)and her endowment is(4,4)
50、,what are her gross demands?9.1.Her gross demands are(9,1).2.The prices are(pl,p2)=(2,3),and the consumer is currentlyconsuming(xl,x2)=(4,4).There is a perfect market for the two goodsin which they can be bought and sold costlessly.Will the consumernecessarily prefer consuming the bundle(yl,y2)=(3,5