中级微观经济学习题及答案
中级微观经济学习题答案
第一部分 消费者理论1. 当11xx 时,加数量税t,画出预算集并写出预算线预算集:).....(. (1)12211x x m x p x p).........(..........)(1112211x x x t m x p x t p2. 如果同样多的钱可以买(4,6)或(12,2),写出预算线。
mx p x p 2211 则有mp p 2164,mp p 21212不妨假设12 p ,则可解得:8,211 m p 。
预算线为82121 x x 3.(1)0.4100x y(2)0.2100.............300.4106.............30x y if x x y if x(3)0.4106x y4. 证明:设两条无差异曲线对应的效用分别为21,uu ,由曲线的单调性假设,若21uu ,则实为一条曲线。
若21uu ,假设两曲线相交,设交点为x,则21)(,)(ux u u x u ,可推出21uu ,存在矛盾,不可能相交。
5. -5(把一元纸币放在纵轴上)或者-1/5(把一元纸币放在横轴上),6. 中性商品是指消费者不关心它的多少有无的商品商品2 如果也是中性商品那么该题就无所谓无差异曲线,也无所谓边际替代率了. 商品2如果不是中性商品:边际替代率是0(把中性商品放在横轴上)或者 (把中性商品放在纵轴上)7. (1)x1 is indefinitely the substitution of x2, and five units of x1 can bring the same utility as that one unit of x2 can do. With the most simple form of the utilityfunction,125u x x x , and assume that the prices of those two goods are p1 and p2 respectively and the total wealth of the consumer is m, the problem can be writtenas121112max ,..u x xst p x p x m③ Because 5p1=p2, any bundle 12,x x which satisfies the budget constraint, is thesolution of such problem.(2) A cup of coffee is absolutely the complement of two spoons of sugar. Let x1 and x2 represent these two kinds of goods, then we can write the utility function as12121,min ,2u x x x xThe problem of the consumer is121112max ,..u x xst p x p x mAny solution should satisfies the rule that 1212x x , and the budget constraint.So replace x1 with (1/2)(x2) in the budget constraint and we can get 1122mx p p,and 21222mx p p.8. (1) Because the preference is Cobb-Douglas utility, we can simplify thecomputation by the formula that the standardized parameter of one commodity means its share of total expenditure.So directly, the answer is 1123m x p , 213mx p.(详细方法见8(2)). (2)库恩-塔克定理。
中级微观经济学48题及答案【考研考博专用,必备48题!!】
第一部分 消费者选择理论1.有两种商品,x1和x2,价格分别为p1和p2,收入为m 。
当11x x ≥时,政府加数量税t,画出预算集并写出预算线2. 消费者消费两种商品(x1,x2),如果花同样多的钱可以买(4,6)或(12,2),写出预算线的表达式。
3.重新描述中国粮价改革(1)假设没有任何市场干预,中国的粮价为每斤0。
4元,每人收入为100元。
把粮食消费量计为x ,在其它商品上的开支为y ,写出预算线,并画图。
(2)假设每人得到30斤粮票,可以凭票以0。
2元的价格买粮食,再写预算约束,画图。
(3)假设取消粮票,补贴每人6元钱,写预算约束并画图。
4. 证两条无差异曲线不能相交5. 一元纸币(x1)和五元纸币(x2)的边际替代率是多少?6.若商品1为中性商品,则它对商品2的边际替代率? 7. 写出下列情形的效用函数,画出无差异曲线,并在给定价格(p 1,p 2)和收入(m )的情形下求最优解。
(1)x 1=一元纸币,x 2=五元纸币。
(2)x 1=一杯咖啡,x 2=一勺糖, 消费者喜欢在每杯咖啡加两勺糖。
8. 解最优选择(1) 21212(,)u x x x x =⋅(2)2u x =9. 对下列效用函数推导对商品1的需求函数,反需求函数,恩格尔曲线;在图上大致画出价格提供曲线,收入提供曲线;说明商品一是否正常品、劣质品、一般商品、吉芬商品,商品二与商品一是替代还是互补关系。
(1)212x x u +=(2)()212,m in x x u =(3)b a x x u 21⋅=(4) 12ln u x x =+,10.当偏好为完全替代时,计算当价格变化时的收入效用和替代效用(注意分情况讨论)。
11. 给定效用函数 (,)x y xy =,p x =3,p y =4,m=60,求当p y 降为3时价格变化引起的替代效应和收入效应。
12. 用显示偏好的弱公理说明为什么Slutsky 替代效应为负。
中级微观经济学45道题(含答案)
中级微观经济学45道题(含答案)中级微观经济学期末考试复习题(版权归13企业管理班所有,翻版必究,哈哈!)1.实现委托代理最优合约设计的两个约束条件是什么、答:⼀种是代理⼈的个⼈理性约束,即委托⼈得保证让代理⼈不跳槽,安于经理岗位。
另⼀种是对代理⼈的激励相容约束,即让代理⼈⾃⼰去选择⾏动值a,使其期望的边际效⽤值达到最⼤。
2、为何需求的价格弹性⼤于1时,降价能增加收益,⽽需求的价格弹性⼩于1时,涨价能增加收益,请给出数学证明。
答:需求的价格弹性公式为:由公式可知,当|e|>1,即富于弹性时,MR<0,边际收益为负,即提⾼价格,收益降低,相反,降低价格则收益升⾼。
当|e|<1,即缺乏弹性时,MR>0,边际收益为正,即提⾼价格,收益升⾼,相反,降低价格,收益变少。
3.简述公共产品与私⼈产品的差异。
(微观经济学⼗⼋讲P352)答:公共品是指由公共部门提供⽤来满⾜社会公共需要的商品和服务。
公共品具有不可分割性、⾮竞争性和⾮排他性。
但是必须明确并不是全部的公共品都应由公共部门提供。
私⼈品是指那些具有效⽤上的可分割性,消费上的竞争性和受益上的排他性的产品。
公共品和私⼈品的区别在于,公共品是可以让⼀群⼈同时消费的物品,⽽私⼈品在任何时候只能为⼀个使⽤者提供效⽤。
%4、毕加索油画的供给价格弹性是多少,为什么答:弹性0,因为供给的价格弹性反映价格变动对供给数量变动的影响。
毕加索的油画是唯⼀的,因此,不管价格如何变动,供给为1,即供给不随价格变动⽽变动,弹性为0。
5、完全竞争市场条件下,为什么⾏业中所有⼚商的经济利润在长期均衡时都会为零这是否意味着⼚商的⽣产变得没有意义西⽅经济学中所谓长期均衡时利润为零,是指经济利润为零,并不是会计利润为零。
所谓经济利润,通常也叫超额利润,就是⼀个⼚商赚取了较之⼀般利润⽔平更⾼的利润。
之所以如此,这是因为,在西⽅经济学理论上,会计利润被计⼊⼚商投⼊⾃有要素所应获得的报酬,是产品的隐含成本。
中级微观经济学复习题及答案
一、简述题1.如果我们看到在(1y ,2y )可以同时得到的情况下,消费者却选择了(1x ,2x ),那么,(1x ,2x ) (1y ,2y )的结论是否正确?(第二章,题1) 答:不正确,因为也可能是消费者恰好在这两个消费束之间无差异。
也就是说,根据题目的已知条件我们只能断定(1x ,2x ) (1y ,2y ),这是弱偏好。
对本题加上什么样的假设前提,题目中的断定就是正确的?如果加上消费者的偏好是严格凸的这一限制条件,断定(1x ,2x ) (1y ,2y )就是正确的。
因为严格凸性条件下,最优解若存在则只有一个。
2.若某个消费者的偏好可以由效用函数22121122(,)10(2)50u x x x x x x =++-来描述,那么对消费者而言,商品1和商品2是完全替代的吗?为什么?(第二章,题5)答:两种商品完全替代即它们的边际替代率为常数。
边际替代率是在消费者保证效用相等的条件下,用一种商品替代另一种商品的比率。
因此有: 商品1的边际效用为MU 1=du /dx 1=10(2x 1 +2 x 2)商品2的边际 效用为MU 2= du /dx 2=10(2x 1 +2 x 2)商品1对商品2的边际替代率MRS 12= MU 1 / MU 2 =1。
满足完全替代品的效用函数特征,因此这个说法是正确的。
3.假定消费者购买x 和y 两种商品,起初,x x y yMU P MU P =,若x P 下降,y P 保持不变,再假定x 的需求价格弹性大于1,则y 的购买量会不会发生变化?(第三章,题3)答:原来消费处于均衡状态。
设消费者花在x 商品上的支出为1m ,则1x m p x =。
对该式求x p 的导数有,11x x x x x p dm dx dx p x x dp dp dp x ⎡⎤=+=+⎢⎥⎣⎦,因x 的需求价格弹性大于1(绝对值),所以有10x dm dp <,即随着价格下降,消费者花在x 商品上的支出会增加。
中级微观经济学习题及部分答案
中级微观经济学习题及部分答案一、单项选择题(每题0.5分,共15分,将正确答案填入答题纸上)1.有下列因素除哪一种外都会使需求曲线移动?(B)a消费者收入变化;b商品价格变化;c消费者偏好变化;d其他相关商品价格变化。
2.假定某耐用消费品的需求函数Qd=400-5P时的均衡价格为50,当需求函数变为Qd=600-5P时,(供给不变)均衡价格将(B)a低于50b高于50c等于50d不变3.下列命题中哪个是规范经济学的命题?(A)a.征税对中等收入家庭是不公平的c.1982年8月政府把贴现率降到10%b.1981年失业率超过9%d.社会保险税的课税依据现已超过30000美元4.当供求力量自由作用时,一次谷物歉收通过什么显示在市场上?(B)a.政府规定的购买量限制b.谷物价格上涨c.劝说人们减少购买量的广告d.谷物贸易量的增加5.某月内,X商品的替代品的价格上升和互补品的价格上升,分别引起X商品的需求变动量为50单位和80单位,则在它们共同作用下该月X商品需求数量:(B)。
a.增加30单位b.减少30单位c.增加130单位d.减少130单位6.在完全竞争市场中,(C)。
a.消费者是价格接受者,而企业不是b.消费者和企业都不是价格接受者c.消费者和企业都是价格接受者d.企业是价格接受者,而消费者不是/doc/3914130.html,C曲线(A)。
a.当LMC<LAC时下降,而当LMC>LAC时上升b.随LMC曲线下降而下降c.随LMC曲线上升而上升 d.通过LMC曲线的最低点8.一个企业在以下哪种情况下应该关闭?(A)a.P<A VCb.P<SACc.发生亏损时d.SMC>MR9.寡头垄断和垄断竞争之间的主要区别是(C)。
a.厂商的广告开支不同b.非价格竞争的种类不同c.厂商之间相互影响的程度不同d.以上都不对10.短期内,完全竞争厂商只能通过对(D)调整来实现最大利润。
a.生产规模b.价格c.全部生产要素d.产量11.不完全竞争市场中出现低效率的资源配置是因为产品价格(A)边际成本a.大于b.小于c.等于d.大于或等于12.当市场处于长期均衡状态时,向右下方倾斜的需求曲线相切于长期平均成本的最低点的左边,产品的价格等于生产的平均成本,产品产量较高但存在多余的生产能力,则该市场属于下列哪一类(B)。
中级微观经济学习题答案
中级微观经济学习题答案第⼀部分消费者理论1. 当11x x ≥时,加数量税t,画出预算集并写出预算线预算集:).....(..........112211x x m x p x p ≤≤+).........(..........)(1112211x x x t m x p x t p ≥+≤++ 2. 如果同样多的钱可以买(4,6)或(12,2),写出预算线。
m x p x p≤+2211 则有 m p p =+2164,m p p =+21212不妨假设12=p ,则可解得:8,211==m p 。
预算线为82121=+x x3.(1)0.4100x y +=(2)0.2100.............300.4106.............30x y if x x y if x +=≤??+=>?(3)0.4106x y +=4. 证明:设两条⽆差异曲线对应的效⽤分别为21,u u ,由曲线的单调性假设,若21u u =,则实为⼀条曲线。
若21u u ≠,假设两曲线相交,设交点为x ,则21)(,)(u x u u x u ==,可推出21u u =,存在⽭盾,不可能相交。
5. -5(把⼀元纸币放在纵轴上)或者-1/5(把⼀元纸币放在横轴上),6. 中性商品是指消费者不关⼼它的多少有⽆的商品商品2 如果也是中性商品那么该题就⽆所谓⽆差异曲线,也⽆所谓边际替代率了. 商品2如果不是中性商品:边际替代率是0(把中性商品放在横轴上)或者∞(把中性商品放在纵轴上)7. (1)x1 is indefinitely the substitution of x2, and five units of x1 can bring the same utility as that one unit of x2 can do. With the most simple form of the utility function, ()125u x x x =+, and assume that the prices of those two goods are p1 and p2 respectively and the total wealth of the consumer is m, the problem can be written as()121112max ,..u x x s t p x p x m+≤③ Because 5p1=p2, any bundle ()12,x x which satisfies the budget constraint, is the solution of such problem.(2) A cup of coffee is absolutely the complement of two spoons of sugar. Let x1 and x2 represent these two kinds of goods, then we can write the utility function as()12121,min ,2u x x x x ??=The problem of the consumer is()121112max ,..u x x s t p x p x m+≤Any solution should satisfies the rule that 1212, and the budget constraint. So replace x1 with (1/2)(x2) in the budget constraint and we can get 1122mx p p =+,and 21222mx p p =+.8. (1) Because the preference is Cobb-Douglas utility, we can simplify thecomputation by the formula that the standardized parameter of one commodity means its share of total expenditure. So directly, the answer is 1123m x p =, 213mx p =.(详细⽅法见8(2)) . (2)库恩-塔克定理。
中级微观经济学习题及答案
中级微观经济学习题及答案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 equilibrium 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?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 o?er 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 purchasers came from the inner-ring people—people who were already rentingapartments. What would happen to the price of inner-ring apartments if all of the condominium purchasers were outer-ring people—the people who were not currently renting apartments in the inner ring? 1.4. The price of apartments 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 of apartments in the inner ring would rise.6. What do you suppose the e?ect 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) = 100?2p. What price would 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 marketwould bear. This would be given by the solution to D(p) = 100?2p = 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 e?cient?1.8. Everyone who had a reservation price higher than the equilibrium price in the competitive market, so that the ?nal outcome would be Pareto e?cient.(Of course in the long run there would probably be fewer new apartments built, which would lead to another kind of ine?ciency.)2 Budget Constraint1. Originally 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.2.1. The new budget line is given by 2p1x1 +8p2x2 =4m.2. What happens to the budget line if the price of good 2 increases, but the price of good 1 and income remain constant? 2.2. The vertical intercept (axis) decreases and the horizontalx2intercept (axis) stays the same. Thus the budget line becomes ?atter.x13. If the price of good 1 doubles and the price of good 2 triples, does the budget line become ?atter or steeper?2.3. Flatter. The slope is ?2/3.p1p24. What is the de?nition 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 good’s price.5. 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 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. Thep1x1p2x2 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?2.6. (+ t)+(?s)= m?u.p1x1p2x27. If the income of the consumer increases and one of the prices decreases at the same time, will the consumer necessarily be at leastas well-o??2.7. Yes, since all of the bundles the consumer could a?ord before are a?ordable at the new prices and income.3 Preferences1. If we observe a consumer choosing (,) when (,) is availablex1x2y1y2one time, are we justi?ed in concluding that (,)>(,)?x1x2y1y23.1. No. It might be that the consumer was indi?erent between the two bundles. All we are justi?ed in concluding is that (,)> (,).x1x2y1y22. 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 taller than.” Is this relation transitive? Is it re?exive? Is it complete? 3.3. It is transitive, but it is not complete—two people might be the same height. It is not re?exive 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, he always prefers the one who is bigger and faster. Is this preferencerelation 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 indi?erence curve cross itself? For example, could Figure 3.2 depict a single indi?erence curve?3.5. Yes. An indi?erence curve can cross itself, it just can’t cross another distinct indi?erence curve.6. Could Figure 3.2 be a single indi?erence curve if preferences are monotonic?3.6. No, because there are bundles on the indi?erence curve that have strictly more of both goods than other bundles on the (alleged) indi?erence curve.7. If both pepperoni and anchovies are bads, will the indi?erence curve have a positive or a negative slope?3.7. A negative slope. If you give the consumer more anchovies, you’ve made him worse o?, so you have to take away some pepperoni to get him back on his indi?erence 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 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 or?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 for good 2?3.10. Zero—if 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 combinations.4 Utility1. The text said that raising a number to an odd power was amonotonic transformation. What about raising a number to an even power? Is this a monotonic transformation? (Hint: consider the case f(u)=u^2.)4.1. The function f(u)=u^2 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 = ?1/v^2;(3)u =1/v^2; (4)u = ln v; (5)u = ?e^?v; (6)u = v^2; (7) u = v^2 for v>0; (8) u = v^2 for v<0.4.2. (1) Yes. (2) No (works for v positive). (3) No (works for v negative).(4) Yes (only de?ned for v positive). (5) Yes. (6) No. (7) Yes. (8) No.3. We claimed in the text that if preferences were monotonic, then a diagonal line through the origin would intersect each indi?erence curve exactly once. Can you prove this rigorously? (Hint: what would happen if it intersected some indi?erence curve twice?)4.3. Suppose that the diagonal intersected a given indi?erence curve at two points, say (x,x) and (y,y). Then either x>y or y>x, which means that one of the bundles has more of both goods. But if preferences are monotonic, then one of the bundles would have to be preferred to the other.4. What kind of preferences are represented by a utility function of thex1+x2form u(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 utility function of thex2x2 form u(x1,x2)=x1 +? Is the utility function v(x1,x2)=x2 1 +2x1+x2 a monotonic transformation of u(x1,x2)?4.5. Quasilinear preferences. Yes.x1x26. Consider the utility function u(x1,x2)=. What kind of pref- erences does it represent? Is the function v(,)= ax1x2x12x22 monotonic transformation of u(,)? Is the function w(,) =x1x2x1x2x12a monotonic transformation of u(,)?x22x1x24.6. The utility function represents Cobb-Douglas preferences. No. Yes.7. Can you explain why taking a monotonic transformation of a utility function doesn’t change the marginal rate of substitution?4.7. Because the MRS is measured along an indi?erence curve, and utility remains constant along an indi?erence curve.5 Choice1. If two goods are perfect substitutes, what is the demand function forgood 2?5.1.=0 when>, = m/when <, and anything betweenx2 p2p1x2p2p2p10 and m/p2 when = .p1p22. Suppose that indi?erence curves are described by straight lines witha slope of ?b. Given arbitrary prices and money income p1, p2, and m, what will the consumer’s 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. Suppose that a consumer always consumes 2 spoons of sugar with each cup of co?ee. If the price of sugar is p1 per spoonful and the price of co?ee is p2 per cup and the consumer has m dollars to spend on co?ee and sugar, how much will he or she want to purchase?5.3. Let z be the number of cups of co?ee the consumer buys. Then we know that 2z is the number of teaspoons of sugar he or she buys. We must satisfy the budget constraint2z + z = m.p1p2Solving 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 face prices p1, p2 and have m dollars to spend. List the choices for the optimal consumption bundles.5.4. 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/,p1 x2 = 0, or x1 = 0, x2 = m/.p25. If a consumer has a utility 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. For kinked preferences, such as perfect complements, where the change in price doesn’t 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 spends it all, she must be purchasing more of at least one good.2. Show that perfect substitutes are an example of homotheticpreferences.6.2. The utility function for perfect substitutes is u(,)=+ .x1 x2 x1 x2Thus if u(,) >u (,), we have + >+ . It follows x1 x2 y1 y2 x1 x2 y1y2that t+ t> t+ t, so that u(t,t) >u (t, t).x1 x2 y1y2 x1 x2 y1y23. Show that Cobb-Douglas preferences are homothetic preferences.6.3. The Cobb-Douglas utility function has the property that u(t,t)=x1 x2= 2 = t 2 = t*u(x1,). ( t x1)a( t x2)1?a t a t1?a x1a x21?a x1a x21?a x2Thus if u(,) >u (,), we know that u(t,t) >u (t,), so x1 x2 y1 y2 x1 x2 y1t y2that Cobb-Douglas preferences are indeed homothetic.4. The income o?er curve is to the Engel curve as the price o?er 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.。
《中级微观经济学》试题与答案
《中级微观经济学》试题与答案一、名词解释(5道题)1. 消费者剩余-解释:消费者愿意支付的最高价格与实际支付价格之间的差额,是消费者获得的净收益。
2. 边际替代率-解释:消费者在维持同一效用水平的情况下,愿意用一种商品替代另一种商品的比率。
3. 规模经济-解释:当企业的生产规模扩大时,平均成本随着产量的增加而下降的现象。
4. 纳什均衡-解释:在博弈论中,每个参与者在给定其他参与者策略的情况下,所选择的最佳策略组合。
5. 帕累托最优-解释:资源配置的一种状态,无法在不使任何人变得更差的情况下,使某些人变得更好。
二、填空题(5道题)1. 在短期生产函数中,边际产量递减规律是指(在其他投入固定时,增加一个单位可变投入,产量的增加量递减)。
2. 供给弹性大于1表示(供给是弹性的)。
3. 无差异曲线上的点表示(消费者获得相同效用的不同商品组合)。
4. 在完全竞争市场中,长期均衡时,企业的经济利润是(零)。
5. 价格上限政策可能导致的直接结果是(商品短缺)。
三、单项选择题(5道题)1. 下列哪一项不是完全竞争市场的特征?()。
- A. 大量的买者和卖者- B. 同质产品- C. 自由进入和退出市场- D. 厂商具有定价权-答案:D2. 在长期,完全竞争市场中的企业会选择生产在()。
- A. 平均成本最低的产量- B. 边际成本最低的产量- C. 平均总成本等于价格的产量- D. 边际成本等于价格的产量-答案:D3. 如果一种商品的需求是价格无弹性的,那么价格上升10%将导致需求量()。
- A. 增加10%- B. 减少10%- C. 减少少于10%- D. 减少多于10%-答案:C4. 在垄断市场中,垄断者的利润最大化产量是()。
- A. 边际成本等于价格- B. 边际收益等于价格- C. 边际收益等于边际成本- D. 平均成本等于边际成本-答案:C5. 在短期内,完全竞争企业的供给曲线是()。
- A. 平均总成本曲线- B. 平均可变成本曲线- C. 边际成本曲线- D. 边际成本曲线位于平均可变成本曲线之上的部分-答案:D四、多项选择题(5道题)1. 影响需求的主要因素有()。
中级微观经济学复习题与答案
一、简述题1.如果我们看到在(1y ,2y )可以同时得到的情况下,消费者却选择了(1x ,2x ),那么,(1x ,2x )(1y ,2y )的结论是否正确?(第二章,题1)答:不正确,因为也可能是消费者恰好在这两个消费束之间无差异。
也就是说,根据题目的已知条件我们只能断定(1x ,2x )(1y ,2y ),这是弱偏好。
对本题加上什么样的假设前提,题目中的断定就是正确的?如果加上消费者的偏好是严格凸的这一限制条件,断定(1x ,2x )(1y ,2y )就是正确的。
因为严格凸性条件下,最优解若存在则只有一个。
2.若某个消费者的偏好可以由效用函数22121122(,)10(2)50u x x x x x x =++-来描述,那么对消费者而言,商品1和商品2是完全替代的吗?为什么?(第二章,题5)答:两种商品完全替代即它们的边际替代率为常数。
边际替代率是在消费者保证效用相等的条件下,用一种商品替代另一种商品的比率。
因此有:商品1的边际效用为MU 1=du /dx 1=10(2x 1 +2 x 2)商品2的边际 效用为MU 2= du /dx 2=10(2x 1 +2 x 2)商品1对商品2的边际替代率MRS 12= MU 1 / MU 2 =1。
满足完全替代品的效用函数特征,因此这个说法是正确的。
3.假定消费者购买x 和y 两种商品,起初,x x y yMU P MU P =,若x P 下降,y P 保持不变,再假定x 的需求价格弹性大于1,则y 的购买量会不会发生变化?(第三章,题3)答:原来消费处于均衡状态。
设消费者花在x 商品上的支出为1m ,则1x m p x =。
对该式求x p 的导数有,11x x x x x p dm dx dx p x x dp dp dp x ⎡⎤=+=+⎢⎥⎣⎦,因x 的需求价格弹性大于1(绝对值),所以有10xdm dp <,即随着价格下降,消费者花在x 商品上的支出会增加。
中级微观经济学习题答案
第一部分 消费者理论1. 当11x x ≥时,加数量税t,画出预算集并写出预算线 预算集:).....(..........112211x x m x p x p ≤≤+).........(..........)(1112211x x x t m x p x t p ≥+≤++ 2. 如果同样多的钱可以买(4,6)或(12,2),写出预算线。
m x p x p ≤+2211 则有 m p p =+2164,m p p =+21212不妨假设12=p ,则可解得:8,211==m p 。
预算线为82121=+x x3.(1)0.4100x y +=(2)0.2100.............300.4106.............30x y if x x y if x +=≤⎧⎨+=>⎩(3)0.4106x y +=4. 证明:设两条无差异曲线对应的效用分别为21,u u ,由曲线的单调性假设,若21u u =,则实为一条曲线。
若21u u ≠,假设两曲线相交,设交点为x ,则21)(,)(u x u u x u ==,可推出21u u =,存在矛盾,不可能相交。
5. -5(把一元纸币放在纵轴上)或者-1/5(把一元纸币放在横轴上),6. 中性商品是指消费者不关心它的多少有无的商品商品2 如果也是中性商品那么该题就无所谓无差异曲线,也无所谓边际替代率了. 商品2如果不是中性商品:边际替代率是0(把中性商品放在横轴上)或者∞(把中性商品放在纵轴上)7. (1)x1 is indefinitely the substitution of x2, and five units of x1 can bring the same utility as that one unit of x2 can do. With the most simple form of the utility function, ()125u x x x =+, and assume that the prices of those two goods are p1 and p2 respectively and the total wealth of the consumer is m, the problem can be written as()121112max ,..u x x s t p x p x m+≤③ Because 5p1=p2, any bundle ()12,x x which satisfies the budget constraint, is the solution of such problem.(2) A cup of coffee is absolutely the complement of two spoons of sugar. Let x1 and x2 represent these two kinds of goods, then we can write the utility function as()12121,min ,2u x x x x ⎧⎫=⎨⎬⎩⎭The problem of the consumer is()121112max ,..u x x s t p x p x m+≤Any solution should satisfies the rule that 1212x x =, and the budget constraint. So replace x1 with (1/2)(x2) in the budget constraint and we can get 1122mx p p =+,and 21222mx p p =+.8. (1) Because the preference is Cobb-Douglas utility, we can simplify thecomputation by the formula that the standardized parameter of one commodity means its share of total expenditure. So directly, the answer is 1123m x p =, 213mx p =.(详细方法见8(2)) . (2)库恩-塔克定理。
中级微观经济学习题与答案
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 efficient? 1.8. Everyone who had a reservation price higher than the equilibrium price in the competitive market, so that the final outcome would be Pareto efficient. (Of course in the long run there would probably be fewer new apartments built, which would lead to another kind of inefficiency.)
2. In the above example, what would the equilibrium 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 first case, $500, and in the second case, $200. In the third case, the equilibrium price would be any price between $200 and $500.
中级微观经济学考试试题答案完整版
一.简答题1.机会成本相关答:是指为了得到某种东西而所要放弃另一些东西的最大价值。
在稀缺性的世界中选择一种东西意味着放弃其他东西。
一项选择的机会成本,也就是所放弃的物品或劳务的价值。
机会成本是指在资源有限条件下,当把一定资源用于某种产品生产时所放弃的用于其他可能得到的最大收益。
2.劣等品和低档品的逻辑推导(P78)答:低档品或劣等品(inferior goods):指消费量随收入的上升而下降的物品。
对于正常商品,替代效应与价格呈反方向变动,收入效应也与价格成反方向变动,所以总效应与价格成反方向变动,因此正常商品的需求曲线向右下方倾斜。
对于低档品来说,替代效应与价格成反方向变动,收入效应与价格同向变动,但替代效应大于收入效应的作用,总效应与价格反方向变动,需求曲线向右下方倾斜。
吉芬商品,替代效应与价格反方向变动,收入效应与价格通向变动,收入效应极大地大于替代效应,使总效应与价格同向变动,其需求线向右上方倾斜。
二.计算题1.生产理论:已知企业的生产函数要素价格需求函数企业生产量求:市场长期均衡的产品价格和企业数量EG A: 已知生产函数为Q=KL-0.5L2—0.32K2,其中Q表示产量,K表示资本,L表示劳动,若K=10,求:(1)写出劳动的平均产量函数和边际产量函数。
(2)分别计算出当总产量、平均产量和边际产量达到极大值时,厂商雇佣的劳动量。
(3)证明当APL达到最大值时,APL=MPL=2AN A:(1)TP=Q=10L-0.5L^2-30 把K=10带进去边际产量(MPL )函数 就是上式对L 求导。
MPL=10-L平均产量(APL )函数 就是总产量除以投入的劳动。
APL=TP/L=10-0.5L-30/L(2)当TP 最大时,MPL=0。
令MPL=10-L=0 ,解得L=10,所以当劳动投入量L=10时,劳动的总产量TP 达到极大值。
当APL 最大时,是APL 与MPL 相交的时候。
令APL 的导数=0,解得L=2倍根号15(负值舍去),所以当劳动投入量L=2倍根号15 时,劳动的平均产量达到极大值。
中级微观经济学例题及答案
0. 5
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时的消费者剩余。
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=(180-160)/(100-80)×(80+100)/(160+180)=0.53
3、 假定下表是供给函数 Qs=-3+2P 在一定价格范围内 的供给表。 某商品的供给表 价格 (元) 供给量 性。 (2) 根据给出的供给函数,求 P=4 是的供给的价格 点弹性。 (3) 根据该供给函数或供给表作出相应的几何图 形,利用几何方法求出 P=4 时的供给的价格点 弹性。它与(2)的结果相同吗? 解:(1) (2)
7
(3)若某工会愿意接纳张某为会员,会费为 100 元,但张某可以 50%的价格购买 X,则张某是否应该 加入该工会? 解:(1)由效用函数 U=X2Y2 可得 MUX=2XY2,MUY =2YX2 消费者均衡条件为 MUX/MUY =2XY2/2YX2 =Y/X=Px/Py =2/5 500=2·X+5·Y 可得 X=125 者均衡。 (2)消费者可以原价格的 50%购买 X,意味着商品 X 的价格发生变动,预算约束线随之变动。消费者均 衡条件成为: Y/X=1/5 500=1·X+5·Y 可得 X=250 Y=50 张某将消费 250 单位 X,50 单位 Y。 (3)张某收入发生变动,预算约束线也发生变动。 消费者均衡条件成为:
中级微观经济学期末考试及答案
中级微观经济学期末考试及答案一、选择题(每题2分,共20分)1、当某消费者对商品W的消费达到饱和点时,则边际效用MUₓ为( C )A、正值B、负值C、零D、不确定2、当吉芬商品的价格上升时,应该有( B )A、替代效应为正值,收入效应为负值,且前者的作用大于后者B、替代效应为负值,收入效应为正值,且前者的作用小于后者C、替代效应为负值,收入效应为正值,且前者的作用大于后者D、替代效应为正值,收入效应为负值,且前者的作用小于后者3、对于生产函数Q=F(L,K)和成本方程C=WL+RK来说,在最优的生产要素组合点上应该有( ABC )A、等产量曲线和成本线相切B、 MRTSᴋʟ=W/RC、MP ʟ/W=MP ᴋ/RD、MP ʟ=MP ᴋ4、在短期成本曲线图形中,从原点出发的射线与TC曲线相切的点上,必有( BCD )A、MC曲线处于下降阶段B、AC最小C、AC=MCD、MC曲线处于上升阶段5、对于完全竞争厂商,如果在厂商的短期产量是,AR小于SAC但大于AVC,则厂商( B )A、亏损,立即停产B、亏损,但继续生产C、亏损,生产或不生产都行D、获得正常利润,继续生产6、垄断厂商实现利润最大化的均衡条件是( C )A、MR=ACB、AR=ACC、MR=MCD、AR=MC7、在垄断厂商的长期均衡点上,不可能出现的情形是( B )A、价格P大于LACB、价格P等于最小的LACC、价格P小于LACD、AR大于MR8、市场失灵的表现是( ABCD )A、信息不完全B、存在垄断C、公共物品生产D、外部性9、在非完全竞争的产品市场,企业的边际收益线( A )A.向右下方倾斜,在AR的下方B.向右上方倾斜,在AR的下方C.平行于横轴,在AR的上方D.不能确定10、当边际产量大于平均产量时,平均产量( C )A.递减 B不变 C.递增 D先增后减二、判断题(每题1分,共10分)1、西方经济学所说的吉芬商品是一种劣等品。
中级微观经济学题库及答案
中级微观经济学题库及答案一、名词辨析1.规范分析与实证分析;实证分析方法:实证分析不涉及价值判断,只讲经济运行的内在规律,也就是回答“是什么”的问题;实证分析方法的一个基本特征它提出的问题是可以测试真伪的,如果一个命题站得住脚,它必须符合逻辑并能接受经验数据的检验。
规范分析方法:规范分析以价值判断为基础,以某种标准为分析处理经济问题的出发点,回答经济现象“应当是什么”以及“应当如何解决”的问题,其命题无法证明真伪。
2.无差异曲线与等产量线;无差异曲线是指这样一条曲线,尽管它上面的每一点所代表的商品组合是不同的,但消费者从中得到的满足程度却是相同的。
等产量线表示曲线上每一点的产量都相等,是由生产出同一产量的不同投入品组合所形成的曲线。
3.希克斯补偿与斯卢茨基补偿;希克斯补偿是指当商品价格发生变化后,要维持原有效用水平所需补偿的货币数量。
斯卢茨基补偿是指当商品价格发生变化后,要维用所需要的生产要素的实际支出;隐性成本:是指厂商本身所拥有的且被用于该企业生产过程的那些生产要素的总价格。
7.机会成本与会计成本;会计成本:是指企业财务人员记载的、企业运行过程中所发生的每一笔成本和费用;机会成本:是指生产者所放弃的使用相同要素在其他生产用途上所能得到的最高收入;8.生产者剩余与消费者剩余;生产者剩余:所有生产单位商品的市场价格和边际成本之间的差额。
消费者剩余:指消费者愿意为某一物品支付的最高价格与它的市场价格之间的差额。
9.规模经济与规模报酬;规模经济指的是企业在生产过程中能够以低于双倍的成本获得双倍产出。
规模报酬是指在其他条件不变的情况下,企业内部各种生产要素按相同比例变化时所带来的产量变化。
规模经济与规模报酬的区别在于,规模报酬是指所有投入品增加一倍,产出增加量超过一倍;规模经济所指是成本增加一倍,产出增加超过一倍。
也就是说,规模经济不一定是所有投入同比例增加,但它包括了所有投入品同比例增加一倍的情形。
中级微观经济学作业及答案(可编辑修改word版)
中级微观经济学第一次作业答案1、假设政府对一个每月收入400美元的贫困家庭进行补贴。
有三种方案:第一,允许该家庭购买400美元的食品券,单位美元食品券的价格为0.5;第二,政府直接发给该家庭200美元的食品券补贴;第三,政府直接发给该家庭200美元的货币补贴。
画出三种方案下该家庭的预算线,解释该家庭的最优选择,并分析三种方案的优劣。
解:如上图所示,横轴表示花费在食品上的货币数量,纵轴表示花费在其他商品上的货币量,初始预算线为CD。
第一种补贴方案下,该家庭可以用200美元购买400美元的食品券,因此预算线变为折线CE1B,最优选择为E1点,效用水平为U1;第二种补贴方案下,政府直接发放给该家庭200美元食品券补贴,因此预算线变为CE2B,最优选择为E2点,效用水平为U2;第三种补贴方案下,政府直接发放给该家庭200美元的货币补贴,因此预算线直接平移到AB,最优选择为E3点,效用水平为U3。
综上所述,因为U3>U2>U1,所以对于该家庭而言,第三种方案最好,第二种方案次之,第一种方案最差。
2、请画出以下各位消费者对两种商品(咖啡和热茶)的无差异曲线。
(1)消费者A喜欢喝咖啡,对喝热茶无所谓;(2)消费者B喜欢1杯热茶和1杯咖啡一起喝;(3)消费者C认为,在任何情况下,1杯热茶和2杯咖啡是无差异的;(4)消费者D喜欢喝咖啡,讨厌喝热茶。
3、写出下列情形的效用函数,画出无差异曲线,并在给定价格(p1,p2)和收入(m )的情形下求最优解。
(1)x1=一元纸币,x2=五元纸币。
U (x )=x 1+5x 2(2)x1=一杯咖啡,x2=一勺糖, 消费者喜欢在每杯咖啡加两勺糖。
()12121,min ,2u x x x x ⎧⎫=⎨⎬⎩⎭,1122m x p p =+,21222m x p p =+ 解:(1)当p1/p2>0.2时,x1=0, x2=m/p2;当p1/p2=0.2时,(x 1,x 2)={(x 1,x 2)|p 1x 1+p 2x 2=m,x 1>0,x 2>0} 当p1/p2<0.2时,x1=m/p1, x2=0(2){x 1=12x 2=mp 1x 1+p 2x 2=m解得:x 1=m p 1+2p 2, x 2=2mp 1+2p 24、假设某消费者的效用函数为:U (x 1,x 2)=lnx 1+x 2试问:给定商品1和商品 2 的价格为p 1和p 2,如果该消费者的收入I 足够高,则收入的变化是否会导致该消费者对商品1的消费,并解释原因。
中级微观经济学考题及答案
中级微观经济学考题及答案
1. 何为微观经济学?
微观经济学是研究个体之间的经济行为的学科,旨在了解单个人、家庭、公司和其他组织如何做出经济决定,以及如何受到政府政策和
其他因素的影响。
该学科利用统计技术和模型,以及涉及决策行为的
理论分析,来分析经济活动。
它也研究如何应用经济政策来影响经济
绩效和人民生活。
2. 市场均衡与价格敏感
市场均衡是指市场中购买者和销售者都在一定价格下能获得最大
利益的一种经济状态。
价格敏感性是指消费者在价格的变动性的程度,也就是说,当价格上升时,消费者购买的量会减少。
市场均衡和价格
敏感性之间存在直接关系,当价格敏感性增大时,消费者对价格变化
会有密切关注,进而影响市场均衡。
3. 动态效用和信息不对称
动态效用指的是由于消费者向高效用的货物和服务偏爱的行为,
使其短期的满意度增加而长期的满意度降低的状态。
信息不对称指的
是消费者在购买时无法获得完整的可靠信息,导致其在决策过程中受到巨大的影响的状况。
动态效用和信息不对称之间存在联系,当消费者在购买中缺乏信息时,他们会关注动态效用,从而做出更糟糕的决策。
4. 最优消费艺术和最优生产艺术
最优消费艺术是指消费者以最大限度满足他们欲望的最有效价格分配货物和服务的方式,可以以最少成本实现最大收益。
最优生产艺术指的是采取最有效的生产方式和技术,以最小的成本获得最大的收益。
最优消费艺术和最优生产艺术的实现都受到道德和法律的限制,二者之间也存在着联系,一方面,最优生产需要有效的资源配置,来满足消费需求;另一方面,最优消费除了遵循经济模型外,还需考虑在效用函数中考虑善恶和道德行为等因素。
中级微观经济学作业及标准答案
中级微观经济学第一次作业答案1、假设政府对一个每月收入400美元的贫困家庭进行补贴。
有三种方案:第一,允许该家庭购买400美元的食品券,单位美元食品券的价格为0.5;第二,政府直接发给该家庭200美元的食品券补贴;第三,政府直接发给该家庭200美元的货币补贴。
画出三种方案下该家庭的预算线,解释该家庭的最优选择,并分析三种方案的优劣。
解:如上图所示,横轴表示花费在食品上的货币数量,纵轴表示花费在其他商品上的货币量,初始预算线为CD。
第一种补贴方案下,该家庭可以用200美元购买400美元的食品券,因此预算线变为折线CE1B,最优选择为E1点,效用水平为U1;第二种补贴方案下,政府直接发放给该家庭200美元食品券补贴,因此预算线变为CE2B,最优选择为E2点,效用水平为U2;第三种补贴方案下,政府直接发放给该家庭200美元的货币补贴,因此预算线直接平移到AB,最优选择为E3点,效用水平为U3。
综上所述,因为U3>U2>U1,所以对于该家庭而言,第三种方案最好,第二种方案次之,第一种方案最差。
2、请画出以下各位消费者对两种商品(咖啡和热茶)的无差异曲线。
(1)消费者A喜欢喝咖啡,对喝热茶无所谓;(2)消费者B喜欢1杯热茶和1杯咖啡一起喝;(3)消费者C认为,在任何情况下,1杯热茶和2杯咖啡是无差异的;(4)消费者D喜欢喝咖啡,讨厌喝热茶。
3、写出下列情形的效用函数,画出无差异曲线,并在给定价格(p1,p2)和收入(m )的情形下求最优解。
(1)x1=一元纸币,x2=五元纸币。
U (x )=x 1+5x 2(2)x1=一杯咖啡,x2=一勺糖, 消费者喜欢在每杯咖啡加两勺糖。
()12121,min ,2u x x x x ⎧⎫=⎨⎬⎩⎭,1122m x p p =+,21222m x p p =+ 解:(1)当p1/p2>0.2时,x1=0, x2=m/p2;当p1/p2=0.2时,(x 1,x 2)={(x 1,x 2)|p 1x 1+p 2x 2=m,x 1>0,x 2>0} 当p1/p2<0.2时,x1=m/p1, x2=0(2){x 1=12x 2=mp 1x 1+p 2x 2=m解得:x 1=m p 1+2p 2, x 2=2mp 1+2p 24、假设某消费者的效用函数为:U (x 1,x 2)=lnx 1+x 2试问:给定商品1和商品 2 的价格为p 1和p 2,如果该消费者的收入I 足够高,则收入的变化是否会导致该消费者对商品1的消费,并解释原因。
中级微观经济学(1-4章习题及答案)
(1) 将 1代入 CES 函数,立即得到 y A1x1 A 2x 2 ,这是线性函数。
(2)当 0 时,CES 函数成为不定式,为求其极限,对 CES 函数取对数。
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币;假设他现在处于均衡状态,即是说目前的市价下他不想买进也不想卖出。证明:无论钱
币市场上钱币的价格上涨还是下跌,这个人的效用水平总会增加。(该题是关于显示偏好的
内容,参照显示偏好内容进行理解)
提示:假设现在收藏家在( M* , X* )处达到均衡,其中 M 指钱币数量, X 是所有其他消费
品(的支出)。在图中,预算线与一条无差异曲线I* 相切。如果 pM 上升, pM/pX 增大,预 算约束线较以前陡峭,但它必然还通过( M* ,X* )点,因为这点的坐标满足预算线方程[注
ln(
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(3)对 CES 函数取对数,求极限,利用洛必达法则
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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. It would be constant at $500 for 25 apartments and then drop to $200.2. In the above example, what would the equilibrium 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 . 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 prices, why does the market demand curve slope down. Because if we want to rent one more apartment, we have to offer 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 purchasers camefrom the inner-ring people—people who were already renting apartments. What would happen to the price of inner-ring apartments if all of the condominium 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 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. The price of apartments in the inner ring would rise.,6. What do you suppose the effect 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.7. Suppose the demand curve is D(p) = 100−2p. What price would the monopolist set if he had 60 apartments How many would herent 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) = 100−2p = 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 efficient'. Everyone who had a reservation price higher than the equilibrium price in the competitive market, so that the final outcome would be Pareto efficient. (Of course in the long run there would probably be fewer new apartments built, which would lead to another kind of inefficiency.)2 Budget Constraint1. Originally 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 +8p2x2 =4m.2. What happens to the budget line if the price of good 2 increases, but the price of good 1 and income remain constant . The vertical intercept (x x axis) decreases and the horizontal intercept (x 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, does the budget line become flatter or steeper. Flatter. The slope is −2x x/3x x.4. What is the definition of a numeraire good. A good whose price has been set to 1; all other goods’ prices are measured relative to the numeraire good’s price.5. 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 of 7 cents a gallon. What net tax is this combination equivalent to. A tax of 8 cents a gallon.:6. Suppose that a budget equation is given byx x x x +x x x x = 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. (x x+ t)x x +(x x−s)x x = m−u.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-off. Yes, since all of the bundles the consumer could afford before are affordable at the new prices and income.3 Preferences1. If we observe a consumer choosing (x x,x x)when (x x,x x) is available one time, are we justified in concluding that (x x,x x) >(x x,x x)—. No. It might be that the consumer was indifferent between the two bundles. All we are justified in concluding is that (x x,x x)> (x x,x x).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. Yes to both.3. Take the same group of people and consider the relation “strictly taller than.” Is this relation transitive Is it reflexive Is it complete. It is transitive, but it is not complete—two people might be the same height. It is not reflexive since it is false that a person is strictly taller than himself.:4. A college football coach says that given any two linemenA 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 than B Which one would he prefer5. Can an indifference curve cross itself For example, could Figure depict a single indifference curve. Yes. An indifference curve can cross itself, it just can’t cross another distinct indifference curve.6. Could Figure be a single indifference curve if preferences are monotonic. No, because there are bundles on the indifference curve that have strictly more of both goods than other bundles on the (alleged) indifference curve.:7. If both pepperoni and anchovies are bads, will the indifference curve have a positive or a negative slope. A negative slope. If you give the consumer more anchovies, you’ve made him worse off, so you have to take away some pepperoni to get him back on his indifference 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.”. Because the consumer weakly prefers the weighted average of two bundles to either bundle.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 need to compensate you Five $1 bills will do nicely. Hence the answer is −5 or−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 for good 2. Zero—if 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.. Anchovies and peanut butter, scotch and Kool Aid, and other similar repulsive 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)=u^2.). The function f(u)=u^2 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 = −1/v^2; (3)u =1/v^2; (4)u = ln v;(5)u = −e^−v; (6)u = v^2; (7) u = v^2 for v>0; (8) u =v^2 for v<0.. (1) Yes. (2) No (works for v positive). (3) No (works for v negative). (4) Yes (only defined for v positive). (5) Yes. (6) No. (7) Yes. (8) No.\3. We claimed in the text that if preferences were monotonic, then a diagonal line through the origin would intersect each indifference curve exactly once. Can you prove this rigorously (Hint: what would happen if it intersected some indifference curve twice). Suppose that the diagonal intersected a given indifference curve at two points, say (x,x) and (y,y). Then either x>y or y>x, which means that one of the bundles has more of both goods. But if preferences are monotonic, then one 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)=√ What about the utility function v(x1,x2)= 13x1 + 13x2. Both represent perfect substitutes.5. What kind of preferences are represented by a utilityfunction of the form u(x1,x2)=x1 +√xx Is the utility function v(x1,x2)=x2 1 +2x1√xx +x2 a monotonic transformation of u(x1,x2). Quasilinear preferences. Yes.(6. Consider the utility function u(x1,x2)= √x x x x. What kind of pref- erences does it represent Is the function v(x x,x x)=x x x x x x a monotonic transformation of u(x x,x x)Is the function w(x x,x x)=x x x x x x a monotonic transformation of u(x x,x x). The utility function represents Cobb-Douglas preferences. No. Yes.7. Can you explain why taking a monotonic transformation ofa utility function doesn’t change the marginal rate of substitution. Because the MRS is measured along an indifference curve, and utility remains constant along an indifference curve.5 Choice1. If two goods are perfect substitutes, what is the demand function for good 2@.x x=0 when p2>px, x x = m/pxwhen px<px, and anythingbetween 0 and m/p2 when px = px.2. Suppose that indifference curves are described by straight 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 , x1 = 0 andx2 = m/p2 if p1/p2 >b, and any amount on the budget line if p1/p2 = b.3. Suppose that a consumer always consumes 2 spoons of sugar with each cup of coffee. If the price of sugar is p1 per spoonful and the price of coffee is p2 per cup and the consumer has m dollars to spend on coffee and sugar, how much will he or she want to purchase. Let z be the number of cups of coffee the consumer buys. Then we know that 2z is the number of teaspoons of sugar he or she buys. We must satisfy the budget constraint2px z + pxz = m.$Solving for z we have z =m2p x+ p x.4. 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 choices for the optimal consumption bundles.. We know that you’ll either cons ume all ice cream or all olives. Thus the two choices for the optimal consumption bundles willbe x1 = m/px , x2 = 0, or x1 = 0, x2 = m/px.5. If a consumer has a utility function u(x1,x2)=x1x4 2, what fraction of her income will she spend 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.&6. For what kind of preferences will the consumer be just as well-off facing a quantity tax as an income tax. For kinked preferences, such as perfect complements, where the change in price doesn’t 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. No. If her income increases, and she spends it all, she mustbe purchasing more of at least one good.2. Show that perfect substitutes are an example of homothetic preferences.—. The utility function for perfect substitutes isu(x x,x x)=x x + x x. Thus if u(x x,x x) >u (yx ,yx), wehave x x + x x>yx + yx. It follows that t x x + t x x > t yx+t yx , so that u(t x x,t x x) >u (t yx, t yx).3. Show that Cobb-Douglas preferences are homothetic preferences.. The Cobb-Douglas utility function has the property that u(t x x,t x x)=( t x x)a( t x x)1−a = t a t1−a x x a x x1−a 2 = t x x a x x1−a2 = t*u(x1,x x). Thus if u(x x,x x) >u (y x,y x),we know that u(t x x,t x x) >u (t yx ,t yx), so that Cobb-Douglaspreferences are indeed homothetic.4. The income offer curve is to the Engel curve as the price offer curve is to .... The demand curve.|5. If the preferences are concave 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.6. Are hamburgers and buns complements or substitutes. 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 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./8. True or false If the demand function is x1 = −p1, then the inverse demand function is x = −1/p1.. False.7 Revealed Preference1. 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 themodel of maximizing behavior. No. This consumer violates the Weak Axiom of Revealed Preference since when he bought (x1,x2) he could have bought (y1,y2) and vice versa. In symbols:p1x1 + p2x2 =1×1+2×2=5> 4=1×2+2×1=p1y1 + p2y2and,q1y1 + q2y2 =2×2+1×1=5> 4=2×1+1×2=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 model of maximizing behavior. Yes. No violations of WARP are present, since the y-bundle is not affordable 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. 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.—4. We saw that the Social Security adjustment for changing prices would typically make recipients at least as well-offas they were at the base year. What kind of price changes would leave them just as well-off, 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.5. In the same framework as the above question, what kind of preferences would leave the consumer just as well-off as he was in the base year, for all price changes. 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 effect . 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 consume 25 blue pencils, a change which is entirely due to the income、2. Suppose that preferences are concave. Is it still the case that the substitution effect is negative. Yes.3. In the case of the gasoline tax, what would happen if the rebate to the consumers were based on their original consumption of gasoline, x, rather than on their final consumption of gasoline, x’. Then the income effect would cancel out. All that would be left would be the pure substitution effect, which would automatically be negative.4. In the case described in the preceding question, would the government be paying out more or less than it received in tax revenues/. They are receiving tx’in revenues and paying out tx, so they are losing money.5. In this case would the consumers be better offor worse offif the tax with rebate based on original consumption were in. Since their old consumption is affordable, the consumers would have to be at least as well-off. This happens because the government is giving them back more money than they are losing due to the higher price of gasoline.9 Buying and Selling1. 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 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 (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 and purchase a bundle that she would 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,q2) = (2 ,4). Could the consumer be better off 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.4. The . currently imports about half of the petroleum 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 off. Yes, but only if the . switched to being a net exporter of oil.。