北大光华微观经济学作业09 Problem Set-3

合集下载

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。

2011年北京大学光华管理学院869经济学(微观经济学部分)考研真题及详解【圣才出品】

2011年北京大学光华管理学院869经济学（微观经济学部分）考研真题1．假设有两种商品X和Y，某消费者的效用函数具有以下形式：U（X，Y）＝ln（X ＋3）＋ln（Y－2），其中，X≥0，Y＞2。

商品X的价格为p，Y的价格为q，消费者的收入为I。

（1）求出消费者关于X和Y的最优消费量，并说明I≥3p＋2q是使得X和Y存在有效需求量的必备条件。

（2）求出消费者对X和Y的需求收入弹性，X和Y是否属于奢侈品（luxury good）？（3）X和Y是否有劣质品（inferior goods）或吉芬商品（Giffen goods）的情形？请给出严格的证明。

2．一个垄断厂商面临两种类型的消费者。

第一类消费者的需求函数为p＝6－0.8q，第二类消费者的需求函数为p＝12－q。

某市场上共有第一类消费者10人，第二类消费者20人。

该厂商的边际成本始终为3。

（1）若厂商实行三级价格歧视，则对于两类消费者分别确定的价格和产量为多少？（2）若厂商对于首次进入市场的消费者一次性收取固定费用F，对于消费者按价格p 收取费用。

若厂商需要保证两类消费者都能消费，那么最优的F和p是多少？若厂商只需要保证一类消费者能够消费，那么最优的F和p是多少？厂商会做出何种选择？3．假设某企业为价格接受者，其成本函数为C（q1）＝（α＋βq1）2，其中α＞0，β＞0。

（1）求出该企业的供给函数；（2）如果有两个企业，每个企业的成本函数都是C（q1）＝（α＋βq1）2，那么这两个企业的平均供给与价格是什么关系？如果企业数目为4，企业的平均供给与价格的关系如何？如果N趋于无穷大呢？（3）假定市场需求曲线为p＝a－bQ，其中Q为市场总的需求量，a和b为正的常数，如果整个市场只有上述一个企业提供产品（假定企业仍然为价格接受者）。

请求出市场均衡的价格，并说明存在惟一均衡的条件。

4．衡量行业集中度的一个重要指标是赫芬达尔指数，其表达式为，其中αi为各企业的市场份额。

北大光华本科微观作业micro-chapter3

pter 3 Chapter 3p Consumer Behavior Consumer BehaviorThere are three steps involved in the ystudy of consumer behavior1.Consumer Preferences◦T o describe how and why people prefer one good to another22.Budget Constraints◦People have limited incomesConsumer Behavior3Given preferences and limited incomes 3.Given preferences and limited incomes,what amount and type of goods will bepurchased?◦What combination of goods will consumersgbuy to maximize their satisfaction?Budget Constrainty Describe budget constraint◦AlgebraG h◦Graphy Describe changes in budget constraint y Government programs and budget constraintsy Non-linear budget linesConsumption Bundley A consumption bundle containing x1units of commodity 1 xcommodity 1, x2units of commodity 2 and so on up to x n units of commodity n is denoted by the ect r ()vector (x1, x2, … , x n).Physical ConstraintsNon negative:y Non-negative:Consumption set:X={ (x1, … , x n) | x1≥0, … , x n≥ 0 }y A consumption set is the collection of all physically possible consumption bundles hto the consumery Y ou only have 24 hours a dayy yy Subsistence needEy Etc.Budget ConstraintsCommodity prices are p p py1, p2, … , p n.y Q: When is a bundle (x1, … , x n) affordable at prices p1, … , p n?y A: Whenp1x1+ … + p n x n≤mh i th’(di bl) where m is the consumer’s (disposable) income.Budget ConstraintsThe bundles that are only just affordable yform the consumer’s budget constraint. This is the set{ (x1,…,x n) | x1 ≥0, …, x n≥ 0and+ … + p=p1x1 … p n x n m }.x2Budget constraint is +p =mm /p 2p 1x 1+ p 2x 2= m. x 1m /p 1x2Budget constraint is +p =mm /p 2p 1x 1+ p 2x 2= m.x 1m /p 1Budget Set and Constraint for T woCommoditiesx2Budget constraint is +p =mm /p 2p 1x 1+ p 2x 2= m.Just affordablex 1m /p 1Budget Set and Constraint for T woCommoditiesx2Budget constraint is +p =mm /p 2p 1x 1+ p 2x 2= m.Not affordableJust affordablex 1m /p 1x2Budget constraint is +p =mm /p 2p 1x 1+ p 2x 2= m.Not affordableAff d bl Just affordable Affordablex 1m /p 1x2Budget constraint is +p =mm /p 2p 1x 1+ p 2x 2= m.th ll tithe collection of all affordable bundles.Budget x 1m /p 1SetBudget Set and Constraint for T wo Commoditiesx2p 1x 1+ p 2x 2= m is =(p +m /p 2x 2= -(p 1/p 2)x 1+ m /p 2so slope is -p 1/p 2.Budget x 1m /p 1SetBudget Constraintsx 2Slope is -p 1/p 2-p +1p 1/p 2x 1Budget ConstraintsxOpp cost of an extra unit of2Opp. cost of an extra unit ofcommodity 1 is p1/p2unitsf f dit2-pforegone of commodity 2.+1p1/p2x1Budget ConstraintsxOpp cost of an extra unit of2Opp. cost of an extra unit ofcommodity 1 is p1/p2unitsf f dit2foregone of commodity 2.Opp. cost of an extraunit of commodity 2 is1units foregone-p/p+1p2/p1 units foregoneof commodity 1.p2p1x1Budget Sets & Constraints; Income and Price ChangesThe budget constraint and budget setydepend upon prices and income. What happens as prices or income change?How do the budget set and budgetg g constraint change as income m increases? x2Originalbudget setgx1Higher income gives more choiceNew affordable consumption x New affordable consumption choices2Original and Original and new budget constraints are constraints are parallel (same l )Originalbudget set slope).gx 1How do the budget set and budget constraint ow o t e bu get set a bu get co st a tchange as income m decreases?x 2Original budget set gx 1How do the budget set and budget constraint ow o t e bu get set a bu get co st a tchange as income m decreases?x 2Consumption bundles th t l that are no longer affordable.Old and newNew, smaller budget set constraints are parallel.x 1budget setBudget Constraints Budget Constraints --Income ChangesyIncreases in income m shift the constraint outward in a parallel manner thereby outward in a parallel manner, thereby enlarging the budget set and improving h ichoice.y Decreases in income m shift the constraint inward in a parallel manner, thereby shrinking the budget set and reducing choice.Budget ConstraintsBudget Constraints --Income ChangesNo original choice is lost and new choices yare added when income increases, so higher income cannot make a consumer worse off.y An income decrease may (typically will) make the consumer worse offmake the consumer worse off.Budget ConstraintsBudget Constraints --Price Changes What happens if just one price decreases? yy Suppose p1decreases.How do the budget set and budget constraint change as p1decreases from p1’to p1”?x2m/p2p-p1’/p2Originalbudget setgx1m/p1’m/p1”How do the budget set and budget constraint change as p1decreases from p1’to p1”?x2m/p2New affordable choicesp-p1’/p2Originalbudget setgx1m/p1’m/p1”Budget ConstraintsBudget Constraints --Price Changes Reducing the price of one commodityypivots the constraint outward. No old choice is lost and new choices are added, so reducing one price cannot make theg pconsumer worse offSimilarlyy Similarly, increasing one price pivots the constraint inwards, reduces choice and(i ll ill) k hmay (typically will) make the consumer worse off.Uniform Ad Valorem Sales T axesAny ad valorem sales tax （从价营业税）leviedat a rate of 5% increases all prices by 5%, from pto (1+005)p = 105pto (1+0.05)p = 1.05p.y An ad valorem sales tax levied at a rate of ti ll i b f t(1+)increases all prices by t p from p to (1+t)p.y A uniform sales tax is applied uniformly to allcommodities.Uniform Ad Valorem Sales T axes A uniform sales tax levied at ratey t changes the constraint fromp1x1+ p2x2= mto(1+t)p1x1+ (1+t)p2x2= m Uniform Ad Valorem Sales T axes A uniform sales tax levied at ratey t changes the constraint fromp1x1+ p2x2= mto(1+t)p1x1+ (1+t)p2x2= mi.e.p1x1+ p2x2= m/(1+t).Uniform Ad Valorem Sales T axesx 2m p 1x 1+ p 2x 2= m p 2p 1x 1+ p 2x 2= m/(1+t )m t p ()12+x m m 1p 1t p ()11+Uniform Ad Valorem Sales T axesx 2m Equivalent income loss m p 2q ism t t p ()12+m t tm−+=+11x m m 1t p ()11+p 1Uniform Ad Valorem Sales T axesx 2A uniform ad valorem l t l i d t t m sales tax levied at rate t is equivalent to an incomem p 2tax levied at ratet .t p ()12+t1+x m m 1t p ()11+p 1The Food Stamp ProgramFood stamps are coupons that can beylegally exchanged only for food.y How does a commodity-specific gift suchas a food stamp alter a family’s budgetp y g constraint?The Food Stamp ProgramSuppose m = $100 p ySuppose m = $100, p F = $1 and the price of “other goods” is p G = $1y“Other goods” is a composite good◦It simplifies the analysis to a 2-good modelyThe budget constraint is thenF +G =100.F G 100.The Food Stamp ProgramG F +G 100b f tF +G = 100: before stamps.100F100The Food Stamp ProgramF +G 100b f tGF +G = 100: before stamps.B d t t ft 40f d Budget set after 40 food stamps issued.(F-40) + G = 100 for F ≥40G=100 for F<40The Food Stamp ProgramG F +G 100b f tF +G = 100: before stamps.B d t t ft 40f d100Budget set after 40 food stamps issued.The family’s budget set is enlarged set is enlarged.F 10014040(F 40)+G 100f F (F-40) + G = 100 for F ≥40G=100 for F<40The Food Stamp ProgramWhat if food stamps can be traded on a yblack market for $0.50 each?y F+G=100+0.5×(40-F) for F<40(F-40)+G=100 for F y (F-40)+G 100 for F ≥40The Food Stamp ProgramG F +G 100b f t F + G = 100: before stamps.Budget constraint after 40120100gfood stamps issued.Budget constraint with Budget constraint with black market trading.F10014040F+G=100+05(40F)for F<40 F+G=100+0.5×(40-F) for F<40 (F-40)+G=100 for F ≥40The Food Stamp ProgramG F +G 100b f t F + G = 100: before stamps.Budget constraint after 40120100gfood stamps issued.Black market trading Black market trading makes the budget set larger again.F10014040TA Contact Info王娟juanwangpku@gmail com yjuanwangpku@ y 刘诗颖liushiying@ y 宫晴gongqing@ ripple wrz@gmail com y 王融璋ripple.wrz@ y 章天乐ztl@y TA Section: T uesday evening (7-9pm) at MBA Club (in the basement)下载课件作业和材料的路径下载课件、作业和材料的路径“公共课程”y公共课程/周黎安/中级微观2010 y ftp://162.105.15.109y username: u1y password: u11Shapes of Budget Constraints Q:y What makes a budget constraint a straight line?y A: A straight line has a constant slope and the constraint isp1x1+ … + p n x n= mso if prices are constants then a constraint is a straight line.Shapes of Budget Constraints But what if prices are not constants? yy E.g.bulk buying discounts, or price penalties for buying “too much”. Then constraints will be curvedy Then constraints will be curved.Shapes of Budget Constraints Shapes of Budget Constraints --Quantity DiscountsSuppose py2is constant at $1 but that p1=$2 for 0 ≤x1≤20 and p1=$1 for x1>20.Shapes of Budget Constraints Shapes of Budget Constraints --Quantity DiscountsySuppose p 2is constant at $1 but that p 1=$2 =$1 for x >20 for 0 ≤x 1≤20 and p 1$1 for x 1>20. y Then the constraint’s slope is2 f 0 2, for 0≤x 1≤20p 1/p 2=1, for x 1> 20Shapes of Budget Constraints with a Quantity Discount=$100x m = $100100Slope = -2 / 1 = -2(21)2(p 1=2, p 2=1)Slope = -1/ 1 = -1=1p (p 1=1, p 2=1)502080x 1Shapes of Budget Constraints with aQuantity Discount=$100x m = $100100Slope = -2 / 1 = -2(21)2(p 1=2, p 2=1)Slope = -1/ 1 = -1=1p (p 1=1, p 2=1)502080x 1Budget Constraints with a QuantityDiscountThe constraint isy2x 1+x 2=m for 0 ≤x 1≤202×20+(x 1-20)+x 2=m for x 1> 20Shapes of Budget Constraints with aQuantity Discount=$100x m= $1001002Budget ConstraintBudget Set502080x1Shapes of Budget Constraints with a Quantity Penaltyx2BudgetBudgetConstraintBudget Setx1Consumer PreferencesDescribe preferencesyy Indifference curves (无差异曲线)y Well-behaved preferencesy Marginal rate of substitution (边际替代率)Rationality in EconomicsBehavioral Postulatey:A decisionmaker always chooses its most preferred alternative from its set of available alternatives.y So to model choice we must model decision makers’ preferencesdecision-makers preferences.Preference RelationsyComparing two different consumptionbundles x and y:bundles, x and y: ◦strict preference (严格偏好): x is more preferred th than y◦Indifference (无差异): x is exactly as preferred as yf( ◦weak preference (弱偏好): x is as at least as preferred as is yAssumptions about Preference RelationsCompleteness (y完备性): For any two bundles x and y it is always possible to make the comparison between x and y y Reflexivity (反身性): Any bundle x is always at least as preferred as itselfAssumptions about Preference RelationsyT ransitivity (传递性: Ify ()x is at least as preferred as y, and y is at least as preferred as z, then x is at least as preferred as z .Indifference CurvesT ake a reference bundle x’ yake a reference bundle x. The set of all bundles equally preferred to x’ is theindifference curve containing x’; the set of all bundles y ∼x’.yIndifference Curvesx 2x’ ∼x” ∼x”’x’x”x”’x 1Indifference Curvesx 2z xypp xzyx 1Indifference CurvesAll bundles in I x 2x All bundles in I 1are strictly preferred to ll i I I 1all in I 2.zAll b dl i I I 2yAll bundles in I 2arestrictly preferred tox all in I 3.I 31Weakly Preferred Set (弱偏好集)x 2WP(x), the set of x (),bundles weaklypreferred to x.I(’)pI(x’)I(x)x 1Weakly Preferred Set (弱偏好集)x 2WP(x), the set of (),bundles weaklypreferred to x.x pWP(x)includes I(x).I(x)x 1Strictly Preferred Set (严格偏好集)x 2SP(x), the set of (),bundles strictlypreferred to x,x p ,does notincludeI(x).I(x)x 1Indifference Curves Cannot Intersectx 2I 1I 2xyzx 1Slopes of Indifference CurvesWhen more of a commodity is always ypreferred, the commodity is a good .y If every commodity is a good thenindifference curves are negatively sloped.g y pSlopes of Indifference Curves Good2Good 2Two goodsa negatively slopedindifference curve.indifference curve.Good 1Slopes of Indifference CurvesIf less of a commodity is always preferred ythen the commodity is a bad.Slopes of Indifference Curves Good2Good 2One good and onebad apositively slopedpositively slopedindifference curve.Bad 1ExamplesPerfect substitutes (y完全替代)y Perfect complements (完全互补) y Satiation (餍足)Extreme Cases of Indifference Curves: Perfect Curves: Perfect Substitutes Substitutesx 2Slopes are constant at Slopes are constant at --1.15I 2Bundles in I 2all have a totalof 15 unitsx 115Extreme Cases of Indifference Curves: Perfect Curves: Perfect Complements Complementsx 245oEach of (5,5), (5,9)and (95)and (9,5)contains 5 pairs so each is ll f d9equally preferred.I 15x 159Indifference Curves ExhibitingSatiationx 2S ti ti Satiation (bliss)rpointe t t e x 1B Well Well--Behaved PreferencesConvexity y: Mixtures of bundles are (at least weakly) preferred to the bundles themselves. E.g., the 50-50 mixture of thebundles x and y isy z = (0.5)x + (0.5)y.z is at least as preferred as x or y.Well Well--Behaved Preferences Behaved Preferences ----Convexity.x 2+xx+y is strictly preferred x 2+y 22z =x y 2is strictly preferred to both x and y.y y 2x 1y 1x 1+y12Well Well--Behaved Preferences Behaved Preferences ----Convexity.x 2xz =(tx +(1-t)y , tx +(1-t)y )1122is preferred to x and y for all 0 < t < 1.y o a 0t y 2x 1y 1Well Well--Behaved Preferences Behaved Preferences ----Convexity.Preferences are strictly convexwhen all mixtures z x 2xare strictlyto theirz preferred to their component bundles x and y ybundles x and y.y 2x 1y 1Well Well--Behaved Preferences Behaved Preferences ----Weak Convexity.x’z’Preferences areweakly convexif at z y least one mixture z is equally preferred is equally preferred to a component bundle xzy’bundle.yyNonNon--Convex Preferencesx2The mixture zz is less preferredthan x or y.t a o yy2x1y1More NonMore Non--Convex Preferencesx2The mixture zz is less preferredthan x or y.t a o yy2x1y1Marginal Rate of Substitution x2MRS at x’ islim-{Δx2/Δx1}Δx10= = --dx2/dx1at x’Δx2x’Δx1x1Marginal Rate of Substitution=MRS*dx so,at x’,MRS is x2dx2MRS dx1so, at x, MRS isthe rate at which the consumer isonly just willing to exchangeonlyonly just willingjust willing to exchangecommodity 2 for a small amountof commodity1.dx2of commodity 1.Marginal willingness to pay.x’dx1x1Marginal Rate of SubstitutionClothing 16AMRS = 6CMRS Δ−=1214-6FΔ810B1-4= 26DE14-2MRS 2Food24G11-123451MRS & Ind Curve PropertiesMRS & Ind. Curve Properties x 2MRS = 0.5MRS decreases(becomes more negative)(becomes more negative)as x 1increasesnonconvex preferences nonconvex preferencesMRS = 5x 1MRS & Ind Curve PropertiesMRS & Ind. Curve Properties MRS is not always increasing as x 2MRS is not always increasing asx 1increases nonconvexMRS MRS = 1preferences.= 0.5MRS = 2MRS 2x 1MRS and Utility Function_21),(Ux x U =2121_0),(dx Udx U x x dU U d =∂+∂==221U U dx x x ∂∂=−=∂∂211/|_x x dx MRS u u ∂∂=Utility Function and PreferencesA consumer’s preferences can be yA consumer s preferences can be represented by a utility functiony The “perfect substitutes” preference canbe represented by U(x, y)=x+yp y (,y)y y The “perfect complements” preferences:U( ) { }U(x, y)= min {x, y}Is the utility function representation unique y p q for a given preference?Utility Function and PreferencesThe utility representation is not unique yy A positive monotonic transformation of a utility function represents the same preference as the original utility function p g yy As long as the utility functions give the d i f th ti b dl same ordering of the consumption bundles, we say they represent the same preferencey The utility is ordinal , rather than cardinalPositive monotonic transformation)]2121//)('/0)(')],,([),(x u x u u f x v u f x x u f x x V ∂∂∂∂∂∂>=212121//)('/x u x u u f x v MRS ∂∂=∂∂=∂∂=Consumer ChoiceRational constrained choice yy Computing ordinary demands◦Interior solution （内在解）◦Corner solution （角点解）Economic RationalityThe principal behavioral postulate is that ya decision-maker chooses its mostpreferred alternative from those available to it.Th il bl h i i h y The available choices constitute the choice set.y How is the most preferred bundle in the choice set located?Rational Constrained Choicex 2More preferred bundlesAffordable bundlesx 1Rational Constrained Choicex 2(x *,x *)is the most 1,x 2) is the most preferred affordable bundle.x 2**x 1x 1Rational Constrained ChoiceThe most preferred affordable bundle is ycalled the consumer’s ORDINARYDEMAND （or DEMAND ）at the given prices and budget.p gy Ordinary demands will be denoted by p m) and x p m)x 1*(p 1,p 2,m) and x 2*(p 1,p 2,m).Rational Constrained ChoiceWhen x y1* > 0 and x 2* > 0 the demanded bundle is INTERIOR .y If buying (x 1*,x 2*) costs $m then thebudget is exhausted.g Rational Constrained Choicex 2*,x *)is interior.(x 1,x 2) is interior.(x 1*,x 2*) exhausts thebudget.x 2**x 1x 1Rational Constrained Choicex 2(x *,x *)is interior.1,x 2) is interior.(a) (x 1*,x 2*) exhausts the budget;p *+p *=m.budget; p 1x 1 p 2x 2 m.x 2**x 1x 1Rational Constrained Choicex 2(x *,x *)is interior .1,x 2) is interior .(b) The slope of the indiff.curve at (x *,x *)equals curve at (x 1,x 2) equals the slope of the budget x 2*constraint.*x 1x 1Rational Constrained Choicey(x *,x *) satisfies 1,x 2) satisfies two conditions :y (a) the budget is exhausted;* + *p 1x 1* + p 2x 2* = m y (b) tangency: the slope of the budget ()g y p g constraint, -p 1/p 2, and the slope of the *,x *) are indifference curve containing (x 1,x 2) are equal at (x 1*,x 2*).Meaning of the T angency ConditionyConsumer’s marginal willingness to pay equals the g g p y q market exchange rate.y Suppose at a consumption bundle (x 1, x 2),MRS 2 P 1MRS= 2, P 1/P 2=1◦The consumer is willing to give up 2 unit of x 2to exchange for an additional unit of x 1◦The market allows her to give up only 1 unit of x 2to obtain an additional x 1y (x 1, x 2) is not optimal choicey She can be better off increasing her consumption of x 1.x 2x 1x 1Computing Ordinary DemandsSolve for 2 simultaneous equations ySolve for 2 simultaneous equations.◦T angency◦Budget constraintyThe conditions may be obtained by using the Lagrangian multiplier method, i.e., constrained optimization in calculusconstrained optimization in calculus.Computing Ordinary DemandsHow can this information be used toylocate (x 1*,x 2*) for given p 1, p 2and m?Computing Ordinary Demands Computing Ordinary Demands --a Cobb Cobb--Douglas Example.Suppose that the consumer has Cobb ySuppose that the consumer has Cobb-Douglas preferences.βα2121),(x x x x U =Computing Ordinary Demands Computing Ordinary Demands --a Cobb Cobb--Douglas Example.y At (x 1*,x 2*), MRS = p 1/p 2so the tangency diti (MRS = /) icondition (MRS = p 1/p 2) is yα12p p x x MRS β==121p β=)1(122x p x αComputing Ordinary Demands Computing Ordinary Demands --a Cobb Cobb--Douglas Example.y(x 1*,x 2*) also exhausts the budget so2mx x =+)(2211p pComputing Ordinary Demands Computing Ordinary Demands --a Cobb Cobb--Douglas Example.The solution to the simultaneous yequations (1) and (2) is:m 1*1p x βαα+=*2mx ββ=2p α+Lagrange Multipliers..),(221121=+m x p x p t s x x U Max )(),(221121=−∂=∂−−+=UL x p x p m x x U L λ0111∂∂∂∂UL p x x λ0222∂=−∂=∂p x x λ02211=−−=∂x p x p m LλEqual Marginal Principlex U x U ∂∂=∂∂=//21λn x x x U of case the In p p ∂),,...,,(2121n p x U x U x U ∂==∂∂=∂∂=/...//21λnp p 21Understanding Understanding lamda lamda∂∂+∂∂=dx U dx U dU 21∂=∂==UU dm x dm x dm 21λ∂∂+dxdx dU p x p x dx p dx p dm Since //,22112211λλλ+=+=dm dx p dx p dm p dm p dm /)(22112211λ=⇒dmdUHow to Allocate Time Efficiently?s s s U Max ni in ...11=++=∑=t f t f t f s t s ni i i i i i i 0)(,0)(),()1(..'''<>=Tt i i )2(1≤∑=t t f t t f t t f n n n /)(..../)(/)(222111λ=∂∂==∂∂=∂∂⇒timeof price shadow :λRational Constrained Choice: SummaryWhen x * > 0 and x * > 0 y1 > 0 and x2 > 0and (x 1*,x 2*) exhausts the budget,and indifference curves have no‘kinks’, the ordinary demands are obtained by solving:y (a) p 1x 1* + p 2x 2* = y (b) h l f h b d i / d f y(b) the slopes of the budget constraint, -p 1/p 2, and of the indifference curve containing (x 1*,x 2*) are equal t (**)at (x 1*,x 2*).Rational Constrained ChoiceBut what if x y1* = 0?y Or if x 2* = 0?y If either x 1* = 0 or x 2* = 0 then the *x *) is at a ordinary demand (x 1,x 2) is at a corner solution (角点解) to the problem of i i i tilit bj t t b d t maximizing utility subject to a budget constraint.Examples of Corner Solutions Examples of Corner Solutions ----the Perfect Substitutes Casex 2MRS = -1x 1Examples of Corner Solutions Examples of Corner Solutions ----the Perfect Substitutes Casex 2MRS = -1Slope =-p with p >p Slope = -p 1/p 2with p 1> p 2.x 1Examples of Corner Solutions Examples of Corner Solutions ----the Perfect Substitutes Casex 2MRS = -1Slope =-p with p >p Slope = -p 1/p 2with p 1> p 2.x 1Examples of Corner Solutions Examples of Corner Solutions ----the Perfect Substitutes Casex 2MRS = -1x y p 22*=Slope =-p with p >p Slope = -p 1/p 2with p 1> p 2.x 1x *=10Examples of Corner Solutions Examples of Corner Solutions ----the Perfect Substitutes Casex 2MRS = -1Slope =-p with p <p *Slope = -p 1/p 2with p 1< p 2.x 1y*x 20=x p 11=Examples of Corner Solutions Examples of Corner Solutions ----the Perfect Substitutes Casex2MRS = -1Slope = -p1/p2with p1= p2.yp2x1yp1Is T angency Condition Sufficient? Ty angency condition is sufficient and necessary if(1) Preferences are convex(2) Solutions are interior。

2015年北京大学光华管理学院869经济学(微观经济学部分)考研真题及详解【圣才出品】

2015年北京大学光华管理学院869经济学（微观经济学部分）考研真题1．一个人有250000元的资产，他从中拿出200000元用来买车，车出事故的概率为5%，出事故后，车子的价值降为40000元。

已知这个人的冯·诺依曼-摩根斯坦效用函数为U（W）＝W0.5。

（1）你认为这个人是风险爱好、风险中性还是风险厌恶，并说明原因。

（2）求消费者为补偿所有损失的完全保险愿意支付的最高价格，并结合数学等式与图形加以说明。

（3）求补偿所有损失的公平保险价格，并结合数学等式与图形加以说明。

（4）结合（2）与（3），判断保险市场是否存在交易的可能，并说明原因。

2．已知某产品的生产函数为f（x1，x2）＝[min（x1，x2）]α，α＞0，x1，x2为生产要素，生产要素的价格为w1＝10，w2＝20，p＝50，p为产品价格。

（1）求规模报酬递增、规模报酬不变、规模报酬递减情况下的取值范围。

（2）求利润最大化时x1，x2的要素需求函数。

（3）求成本函数。

（4）前面的（2）、（3）问如何依赖于α的取值。

3．有两个合伙人1和2，他们共同创办了一家企业，两个人的努力程度分别为e1，e2（e i＞0，i＝1，2）各自的努力成本函数为C（e i）＝e i2/2，合伙公司的收入函数为R（e1，e2）＝e1＋e2＋e1e2/2。

（1）若两人采取合作共赢策略，此时π＝R（e1，e2）－C（e1）－C（e2），求利润最大化时，双方的努力程度e1，e2。

（2）若两人平分收入，每个人只考虑自己的努力程度，求纯策略纳什均衡。

（3）在（2）中，若两人提前签订了契约：若公司收入大于等于6，两人就平分收入；若小于6，公司的收入捐献给慈善机构。

证明在此情况下（1）为纯策略纳什均衡。

4．小李与小王博弈，小李首先开始行动，他可以选择H或者选择L。

小王无法观测到小李的行为，但是他可以获得信号h与l，并有如下分布：Pr（h|H）＝p，Pr（l|H）＝1－p，Pr（h|L）＝q，Pr（l|L）＝1－q，p＞0.5＞q。

2005年北京大学光华管理学院869经济学(微观经济学部分)考研真题及详解【圣才出品】

2005年北京大学光华管理学院869经济学（微观经济学部分）考研真题1．简答题（1）假设消费者只消费两种商品，而且他总是花光全部货币，那么消费者消费的能否都是奢侈品？（2）为什么效用可以度量人的偏好？（3）什么是无差异曲线？为什么无差异曲线不能相交？（4）什么是风险规避？（5）微观经济学中的效用和保险中的期望效用有什么区别？2．已知需求函数为Q d＝100－20p，供给函数为Q s＝20＋20p。

（1）计算出均衡价格与均衡数量。

（2）假如存在数量税为每单位商品0.5元，那么均衡价格和均衡数量变为多少？（3）计算税收的无谓损失。

（4）如果存在两种征税方式，一种是对生产者征税，一种是对消费者征税，分别计量均衡数量、均衡价格和无谓损失。

3．下表为丈夫（H）和妻子（W）的博弈的支付矩阵：请问：什么是纳什均衡？求出该博弈所有可能的纳什均衡，利用图形说明求出的纳什均衡的意义。

表1 丈夫和妻子的博弈的支付矩阵4．消费者的效用函数为U（x1，x2）＝Ax1αx21－α，其中x1和x2分别为第一期和第二期的消费商品量。

消费者初始禀赋为m。

假设消费者只消费一种商品，该种商品当前价格为p1。

已知市场利率为r。

（1）试述效用函数中参数α的经济学意义。

（2）若当前商品的价格变为p1′，试求出价格变化引起的替代效应和收入效应。

2005年北京大学光华管理学院869经济学（微观经济学部分）考研真题及详解1．简答题（1）假设消费者只消费两种商品，而且他总是花光全部货币，那么消费者消费的能否都是奢侈品？（2）为什么效用可以度量人的偏好？（3）什么是无差异曲线？为什么无差异曲线不能相交？（4）什么是风险规避？（5）微观经济学中的效用和保险中的期望效用有什么区别？答：（1）消费者消费的不可能都是奢侈品。

原因如下：若商品x为奢侈品，则其收入弹性e i＝（dx i/dm）·（m/x i）＞1，其中m为收入。

设商品x1和x2的价格分别为p1和p2，则有：p1x1＋p2x2＝m①①式两边对m求导得：p1（dx1/dm）＋p2（dx2/dm）＝1；可以整理为：p1·（dx1/dm）·（m/x1）·x1＋p2·（dx2/dm）·（m/x2）·x2＝m＝p1x1＋p2x2；即：p1x1e1＋p2x2e2＝p1x1＋p2x2；即：p1x1（e1－1）＋p2x2（e2－1）＝0；所以，上式中，（e1－1）与（e2－1）异号或者同时为0。

2003年北京大学光华管理学院869经济学(微观经济学部分)考研真题【圣才出品】

2003年北京大学光华管理学院869经济学（微观经济学部分）考研真题及详解1．某垄断企业由两个工厂构成，工厂Ⅰ的生产函数为y1＝x1αx21－α，工厂Ⅱ的生产函数为y2＝x1βx21－β，其中x1和x2为两种要素的投入数量，α与β为常数。

如果要素市场为完全竞争市场，r1和r2为两种要素的价格，则该企业的成本函数如何？解：（1）首先求两个工厂各自的成本函数。

它们的成本最小化问题分别为：对工厂Ⅰ，建立拉格朗日函数：L1（x1，x2，λ1）＝r1x1＋r2x2－λ1（x1αx21－α－y1）；一阶条件为：∂L1/∂x1＝r1－λ1αx1α－1x21－α＝0∂L1/∂x2＝r2－λ1（1－α）x1αx2－α＝0∂L1/∂λ1＝－x1αx21－α＋y1＝0联立以上三式可解得：因此，厂商Ⅰ的成本函数为：同理可得厂商Ⅱ的成本函数为：（2）再求厂商总成本函数。

令：则厂商总成本为：C（y1，y2）＝Ay1＋By2。

因此，可建立垄断厂商成本最小化的拉格朗日函数为：由Kuhn-Tucker定理可知，存在u i≥0，i＝1，2，使一阶条件∂L3/∂y1＝A－λ－μ1＝0和∂L3/∂y2＝B－λ－μ2＝0成立，且满足互补松弛条件：若y1＝0，则μ1≥0。

此时y2＝y，则μ2＝0，代入上式中有：A＝λ＋μ1≥λ，B＝λ，因此A≥B；若y2＝0，则μ2≥0。

此时y1＝y，则μ1＝0，代入上式中有：B＝λ＋μ2≥λ，A＝λ，因此A≤B。

因此，当A＞B时，工厂Ⅰ不生产，工厂Ⅱ产量为y，总成本函数为C（y）＝By；当A ＜B时，工厂Ⅰ产量为y，工厂Ⅱ不生产，总成本函数为C（y）＝Ay；当A＝B时，工厂Ⅰ、Ⅱ任意分配产量，总成本函数为C（y）＝Ay。

2．已知某企业的生产函数为f（x1，x2）＝[min{x1，x2}]1/α，x1和x2为两种投入要素的数量，α＞0为常数。

已知产品价格为p，要素价格分别为ω1和ω2。

请求出利润最大化的要素需求函数、供给函数和利润函数。

北京大学光华管理学院经济学专业真题

北京⼤学光华管理学院经济学专业真题2008年光华管理学院研究⽣⼊学考试微观经济学考题⼀、10分，每题5分1、如果⼀个投资者在有风险的股票和⽆风险的国债之间选择了股票，是否可以说他是风险爱好者？为什么？2、股市有⼀种追涨现象，即有⼈喜欢买价格上涨了的股票。

这是否说明对这些⼈来说，股票事劣等品？奢侈品？吉芬品？为什么？⼆、20分假如⼀个经济中有100个消费者，⼤家的偏好是完全⼀样的。

他们只消费两种商品，x和y，并且他们的偏好可以⽤效⽤函数U（x，y）=x+2.94ln (y)来表⽰。

商品x是由国际市场提供，且价格为1。

商品y只由该经济⾃⼰⽣产。

该经济中有48个技术完全⼀样的企业可以⽣产y。

它们的⽣产函数都是y= 。

⽣产要素x1和x2的价格分别是4元和1元，短期内⽣产要素x2是固定在1的⽔平，请问：在短期，市场的均衡价格是多少？每个⼚商⽣产多少？每个⼚商的利润是多少？在长期，市场的均衡价格是多少？市场的总产量是多少？有多少个⼚商进⾏⽣产？三、15分在⼀个两种投⼊品z1和z2和两种产出品q A和q B的经济中，他们的技术分别是,(1)假设z1和z2的价格分别是r1和r2，在均衡时r1和r2=(1,1/4)，且两种商品都⽣产，求出均衡的产出⽐率P A/P B(2)如果经济的禀赋是(z1,z2)=(100,900)，那么投⼊品价格⽐率改变的可能区间是什么？同时求出可能的价格产出⽐率的变化区间。

四、15分经济中存在⼀种商品，经济中有两种状态，产⽣状态1的概率是3/4，Alex是风险中性的，Bev是风险爱好者，他的效⽤函数V(c)=ln(1+c)。

经济的禀赋(ω1,ω2)=(100,200)(1)最⼤的均衡状态权益价格⽐率（state claims price ratio）是什么？(2)对于什么样的禀赋，使得风险中性者承担所有均衡风险？五、15分⼀个卖者有不可分割的单位某种物品出售，有两个潜在的买家。

第⼀个买家对出售物的评价是V1，第⼆个买家对出售物的评价是V2，且这两个评价对他们来说是共同知识。

北大光华微观经济学练习

北大光华微观经济学练习Intermediate MicroeconomicsProblem Set 2(Due Date: Nov. 14)1． Jane owns a house that is worth $100,000. She cares only about her wealth, whichconsists entirely of the house. In any given year, there is a 20% chance that the house will burn down. If it does, its scrap value will be $22,500. Jane’s von Neumann-Morgenstern utility function is 1/2()u W W =.a) Is Jane risk averse, risk neutral or a risk lover? Explain.b) W hat is the expected monetary value of Jane’s uncertainty?c) What is the maximum that Jane is willing to pay to completely insure her house against being destroyed by the fire?d) Say that Homer is the president of an insurance company. He is risk neutral and has a utility function of the following type ()u W W =, where W is his wealth. What is the lowest price at which he is willing to provide a fair insurance contract to completely insure Jane’s risk?2．Suppose that the process of producing lightweight p arkas by Polly’s Parkas isdescribed by the function Q = 10K 0.8(L-40)0.2, where Q is the number of parkas produced, K is the number of machine hours, and L is the number of person-hours of labors.e) Derive the cost-minimizing demands for K and L as a function of Q, wage rates (w), and rental rates on machines (r). Use these to derive the total cost function. f) This process requires skilled workers, who earn $32 per hour. The rental rate is$64 per hour. At these factor prices, what are total costs as a function of Q? Does this technology exhibit decreasing, constant, or increasing returns to scale?g) Polly’s Parkas plans to produce 2000 parkas per week. At the factor prices given above, how many workers should they hire (at 40 hours per week) and how many machines should they rent (at 40 machine-hours per week)? What are themarginal and average costs at this level of production?3. The short-run cost functions of two firms are given by211100y C +=and 2222816y y C ++=.a) Suppose the two firms are plants of the same firm in a competitive market. If the firm wants to produce 24 units of output, how much should it produce in each plant? Why not produce this output simply in one plant? Briefly explain.b) Suppose the two firms act independently. What is the short-run supply curve of the two firms?c) If the price is 6, what is the number of firms active in the market in the short run? Explain why.4． A firm produces a product with labor and capital and its production function is described by Q = LK, where L denotes labor and K denotes capital. Suppose that the price of labor (w) equals 2 and the price of capital (r) equals 1.a)Suppose that in the short run, capital level is fixed at K=10. Write downmathematical expressions for the short run total average cost curve, averagevariable cost curve, average fixed cost curve, and marginal cost curve.b)Derive the long-run total cost curve, average cost curveand marginal costcurve.5. Suppose the market for widgets can be described by the following equationsDemand: P = 10 – Q; Supply: P = Q – 4.where Q is the quantity in thousands of units and P is the price in dollars per unit.a)What is the equilibrium price and quantity?b)Suppose the government imposes a tax of $1 per unit to reduce widgetconsumption and raise government revenues. What will the new equilibriumquantity? What price will the buyer buy? What amount per unit will the sellerreceive?c)Suppose the government has changed its mind and decide to remove the taxand grant a subsidy of $1 per unit to widget producers. What will equilibriumquantity be? What amount per unit (including the subsidy) will the sellerreceive? What will be the total cost to the government?。

北京大学微观经济学习题及答案

北京大学微观经济学习题及答案2008年春微观经济学习题注意：作业严禁抄袭！作业分数取决于写作业的认真程度，与是否正确无关，如有发现作业雷同者，作业成绩全部为0分。

1、（第一章）假定某消费者关于某种商品的消费数量Q 与收入M 之间的函数关系为M=100Q 2。

求：当收入M=2500时的需求的收入点弹性。

2、（第一章）假定需求函数为N MP Q -=,其中M 表示收入，P 表示商品价格，N(N>0)为常数。

求：需求的价格点弹性和需求的收入点弹性。

3、（第一章）利用图阐述需求的价格弹性的大小与厂商的销售收入之间的关系，并举例加以说明。

4、（第二章）假定某商品市场上只有A 、B 两个消费者，他们的需求函数各自为P Q d A 420-=和P Q d B 530-=。

计算市场需求函数。

5、（第二章）、假设某消费者的效用函数为x x U 5.025.01=，两商品的价格分别为P 1,P 2,消费者的收入为M 。

分别求该消费者关于商品1和商品2的需求函数。

6、（第二章）用图说明序数效用论者对消费者均衡条件的分析，以及在此基础上对需求曲线的推导。

7、（第三章）已知某企业的生产函数K L Q 3132=，劳动的价格2=ω，资本的价格r ＝1。

求：1) 当成本C ＝3000时，企业实现最大产量时的L ，K ，Q 值。

2) 当产量Q ＝800时，企业实现最小成本时的L ，K ，C 值。

8、（第四章）已知某企业的短期总成本函数是5108.004.0)(23++-=Q Q Q Q STC ，求最小的平均可变成本值。

9、（第五章）已知某完全竞争行业中的单个厂商的短期成本函数101521.023++-=Q Q Q STC 。

试求：（1）当市场上产品的价格为P ＝55时，厂商的短期产量和利润；（2）当市场价格下降到多少时，厂商必需停产；（3）厂商的短期供给函数。

10、（第五章）已知某完全竞争市场的需求函数为D ＝6400－400P ，短期市场供给函数为SS’=3000+1500P ；单个企业在LAC 曲线最低点的价格为6，产量为50；单个企业的成本规模不变。

范里安微观作业(姚澜6次集合版)

Problem SetⅠPartⅠ(True or False)1.There are3goods.The price of good1is-1,the price of good2is+1,and the price of good3is+2.It is physically possible for a consumer to consume any commodity bundle with nonnegative amounts of each good.A consumer who has an income of10could afford to consume some commodity bundles that include5units of good1and6units of good2.2.If all prices are doubled and money income is left the same,the budget set does not change because relative prices do not change.3.If there are two goods and the prices of both goods rise,then the budget line must become steeper.4.If someone has the utility function U=1,000+2min{x,y},then x and y are perfect complements for that person.5.If there are two goods,if a consumer prefers more of each good to less,and if she has a diminishing marginal rate of substitution,then her preferences are convex.6.If preferences are convex,then for any commodity bundle x,the set of commodity bundles that are worse than x is a convex set.7.Bill Katz prefers more of good1to less and he prefers less of good2to more. Bill has convex preferences.If we draw his indifference curves with good1on the horizontal axis and good2on the vertical axis,then his indifference curves have positive slope but get steeper as they rise.8.A consumer who is unable to detect small differences in the amount of water in her beer could have a transitive strict preference relation but is unlikely to have a transitive indifference relation.9.If Melody has more classical records than rock and roll records,she is willing to exchange exactly1classical record for2rock and roll records,but if she has more rock and roll records than classical records,then she is willing to exchange exactly1 rock and roll record for2classical records.Melody has convex preferences.10.With quasilinear preferences,the slope of indifference curves is constant along all rays through the origin.11.Wanda Lott has the utility function U(x,y)=max{x,y}.Wanda’s preferences are convex.12.Mr.Surly consumes only two goods and hates them both.His utility function is U(x,y)=-max{x,y}.Mr.Surly has(weakly)convex preferences.13.Alice’s utility function is U(x,y)=x2y.Steve’s utility function is U(x,y)=x2y+ 2x.Alice and Steve have the same preferences since Steve’s utility function is a monotonic transformation of Alice’s.14.A consumer has preferences represented by the utility function U(x1,x2)= 10(x21+2x1x2+x22)-50.For this consumer,goods1and2are perfect substitutes. 15.A person with the utility function U(x1,x2)=min{x1+2x2,2x1+x2}has convex but not strictly convex preferences.16.Max Gross has the utility function U(x,y)=max{x,y}.If the price of x is the same as the price of y,Max will buy equal amounts of x and y.17.If a consumer does not have convex preferences,then a point of tangency between her indifference curve and her budget line must be an optimal consumption point.18.Ambrose’s utility function is U(x,y)=x+4y1/2.The price of x is$1and the price of y is$2.If his income rises from$100to$150,his consumption of y increases by more than10%but less than50%.19.Mary Granola’s utility function is U(x,y)=min{x+2y,y+2x}.Mary maximizes her utility subject to a budget constraint.If she chooses the bundle(5,6), then the price of x must be exactly twice the price of y.PartⅡ(Single choice)1.Billy Bob wants to gain some weight so that he can play football.Billy consumes only milk shakes and k shakes cost him$1each and spinach costs$2per serving.A milk shake has850calories and a serving of spinach has200calories.Billy Bob never spends more than$20 a day on food and he always consumes at least8,000calories per day.Which of the following is necessarily true?a.Billy Bob consumes at least9milk shakes a day.b.Billy Bob never consumes more than6servings of spinach a day.c.Billy Bob never consumes positive amounts of both goods.d.Billy Bob consumes only milk shakes.e.None of the above.2.If you have an income of$40to spend,commodity1costs$4per unit,and commodity2 costs$8per unit,then the equation for your budget line can be writtena.x1/4+x2/8=40.b.(x1+x2)/12=40.c.x1+2x2=10.d.5x1+9x2=41.e.12(x1+x2)=40.3.If you could exactly afford either5units of x and21units of y,or9units of x and5 units of y,then if you spent all of your income on y,how many units of y could you buy?a.57b.14c.25d.41e.None of the above.4.Murphy used to consume100units of X and50units of Y when the price of X was$2and the price of Y was$4.If the price of X rose to$4and the price of Y rose to$9,how much would Murphy’s income have to rise so that he could still afford his original bundle?a.$700b.$450c.$350d.$1,050e.None of the above.5.Suppose there are two goods,the prices of both goods are positive,and a consumer’s income is also positive.If the consumer’s income doubles and the price of both goods triple,a.the consumer’s budget line gets steeper and shifts inward.b.the slope of the consumer’s budget line does not change but the budget line shifts outward away from the origin.c.the consumer’s budget line gets steeper and shifts outward.d.the slope of the consumer’s budget line does not change but the budget line shifts inward toward the origin.e.the consumer’s budget line gets flatter and shifts inward.6.If there are only two goods,if more of good1is always preferred to less,and if less of good 2is always preferred to more,then indifference curvesa.slope downward.b.slope upward.c.may cross.d.could take the form of ellipses.e.None of the above.7.Charlie’s indifference curves have the equation x B=constant/x A,where larger constants denote better indifference curves.Charlie strictly prefers the bundle(10,17)toa.the bundle(11,16).b.the bundle(17,10).c.the bundle(12,15).d.more than one of these bundles.e.none of these bundles.8.Recall that Tommy Twit’s mother measures the departure of any bundle from her favorite bundle for Tommy by the sum of the absolute values of the differences.Her favorite bundle for Tommy is(2,7),that is,2cookies and7glasses of milk.Tommy’s mother’s indifference curve that passes through the point(c,m)=(4,5)also passes througha.the point(6,3).b.the points(2,3),(6,7),and(4,9).c.the point(2,7).d.the points(4,7),(2,5),and(2,9).e.None of the above.9.I prefer6apples and1orange to5apples and2oranges.My preferencesa.are transitive.b.are complete.c.are convex.d.obey the Law of Demand.e.None of the above.10.Ike’s utility function is U(x,y)=25xy.He has12units of good x and8units of good y.Ben’s utility function for the same two goods is U(x,y)=4x+4y.Ben has9units of x and13units of y.a.Ike prefers Ben’s bundle to his own bundle,but Ben prefers his own bundle to Ike’s.b.Ben prefers Ike’s bundle to his own,but Ike prefers his own bundle to Ben’s.c.Each prefers the other’s bundle to his own.d.Neither prefers the other’s bundle to his own.e.Since they have different preferences,there is not enough information to determine who envies whom.11.Tim has preferences represented by the utility function U(x,y)=min{6x+y,x+2y}.If x is on the horizontal axis and y is on the vertical axis,what is the slope of his indifference curve at the point(8,9)?a.-6b.-2/6c.-1/2d.-1/6e.-8/912.Mac’s utility function is U(x,y)=max{2x-y,2y-x}.a.Mac’s preferences are quasilinear.b.If Mac has more x than y,any increase in his consumption of y would lower his utility.c.If Mac has more x than y,a decrease in his consumption of y would raise his utility.d.Mac always prefers more of each good to less.e.Goods x and y are perfect substitutes.13.Raymond’s preferences are represented by the utility function U(x,y)=x/y if y>0and U(x,y)=0if y=0.a.Raymond has indifference curves that are rectangular hyperbolas.b.Raymond prefers more of each good to less.c.Raymond has quasilinear preferences.d.Raymond has a bliss point.e.Raymond has indifference curves that are upward-sloping straight lines if y>0.14.Joe Bob’s cousin Peter consumes goods1and2.Peter thinks that4units of good1is always a perfect substitute for2units of good2.Which of the following utility functions is the only one that would not represent Peter’s preferences?a.U(x1,x2)=min{2x1,4x2}.b.U(x1,x2)=20x1+40x2-10,000.c.U(x1,x2)=2x1+4x2+1,000.d.U(x1,x2)=4x21+16x1x2+16x22.e.More than one of the above does not represent Peter’s preferences.15.Wanda Littlemore’s utility function is U(x,y)=x+46y-2y2.Her income is$135.If the price of x is$1and the price of y is$18,how many units of good x will Wanda demand?a.5b.12c.16d.0e.916.Janet consumes two commodities x and y.Her utility function is min{x+2y,y+2x}.She chooses to buy10units of good x and20units of good y.The price of good x is$1.Janet’s income isa.$40.b.$50.c.$30.d.$20.e.There is not enough information in the problem to determine her income because we are not told the price of good y.17.Howard has the utility function U(x,y)=x-(1/y).His income is$30.a.Howard does not like good y.b.Howard has a bliss point.c.If the price of good x is$4and the price of good y is$1,Howard will buy2units of y.d.Howard will buy good y only if it is cheaper than good x.e.None of the above.18.Elmer’s utility function is U(x,y)=min{x,y2}.If the price of x is$25and the price of y is$15 and if Elmer chooses to consume7units of y,what must his income be?a.$2,660b.$280c.$1,430d.$1,330e.There is not enough information to determine his income.PartⅢ(Problems)1.Brenda likes hot dogs and Coca-Cola.Hot dogs cost$1each and Cokes cost$.50per bottle. There is a special promotion for Coke that will last for one month.If Brenda sends in the bottle tops from the Cokes she drinks during the next month,she will get a refund of$.20for every bottlecap beyond the first12that she returns.For example,if she returns25bottle caps she will get back$2.60=$.20(25-12).Brenda has$40to spend on hot dogs and Coke during the next month.Draw her budget line with Coke on the horizontal axis and hot dogs on the vertical axis. Find the points where the budget line hits the axes and the point where it has a kink.At each of these three points write down the quantities of each good consumed.2.Coach Steroid likes his players to be big,fast,and obedient.If player A is better than player B in two of these three characteristics,Steroid will prefer A to B.Three players try out for quarterback.Wilbur Westinghouse weighs320pounds,runs very slowly,and is quite obedient. Harold Hotpoint weighs240pounds,runs extremely fast,and is extremely disobedient.Jerry Jacuzzi weighs150pounds,runs at average speed,and is extremely obedient.Does Coach Steroid have transitive preferences?Explain your answer.3.Max has the utility function U(x,y)=x(y+1).The price of x is$2and the price of y is$1. Income is$10.How much x does Max demand?How much y?If his income doubles and prices stay unchanged,will Max’s demand for both goods double?e separate graphs to draw indifference curves for each of the following utility functions:a.U(x,y)=min{2x+y,2y+x}.b.U(x,y)=max{2x+y,2y+x}.c.U(x,y)=x+min{x,y}.In which of these cases are preferences convex?5.Les has the utility function U(x,y)=(x+1)(y+4).The price of y is$1.Les spends all of his income to buy6units of y and no x.What must the price of x be?Explain your answer and draw a diagram to illustrate it.Answer to problem set1PartⅠ1.true2.false3.false4.true5.true6.false7.false8.true9.false10.false11.false12.true13.false14.true15.true16.false17.false18.false19.truePartⅡ1.b2.c3.d4.b5.d6.b7.e8.b9.e10.a11.c12.c13.e14.a15.e16.d17.c18.dPartⅢ1.The budget line runs from(0,40)on the vertical axis to a kink point(12,34)and from(12,34) to about(125.3,0).2.No.Steroid prefers W to H because W is heavier and more obedient.He prefers H to J becauseH is heavier and faster.But he prefers J to W because J is more obedient and faster than W.Since his preferences have a cycle,they cannot be transitive.3.To set his MRS equal to the price ratio,Max sets(y+1)/x=2.His budget constraint is2x+y= 10.Solve these two equations to find that x=11/4and y=9/2.If his income doubles and prices stay unchanged,his demand for both goods does not double.A quick way to see this is to note that if quantities of both goods doubled,the MRS would not stay the same and hence would not equal the price ratio,which has stayed constant.4.If you take a point on the line x=y and draw two lines through it,one with a slope of-1/2and the other with a slope of-2,the outer envelope of these lines will be an indifference curve for(a) and the inner envelope will be an indifference curve for(b).The indifference curves for(c) passing through a point on the line x=y consist of a line segment going down and to the right with slope-1and a line segment going up and to the left with slope-2.Cases(a)and(c)display convex preferences,and case(b)does not.5.The price of x must be at least$10.His marginal rate of substitution at the bundle(6,0)is10.If the price of x is$10or greater,he will choose that corner.Problem SetⅡPartⅠ(True or False)1.If consumers spend all of their income,it is impossible for all goods to be inferior goods.2.If the demand curve is a downward-sloping straight line,then the price elasticity of demand is constant all along the demand curve.3.When other variables are held fixed,the demand for a Giffen good rises when income increases.4.Quasilinear preferences are homothetic when the optimal amount of one of the goods is not affordable.5.The strong axiom of revealed preference requires that if a consumer chooses x when he can afford y and chooses y when he can afford z,then he will not choose z when he can afford x.6.If a consumer maximizes a utility function subject to a budget constraint and has strictly convex preferences,then his behavior will necessarily satisfy the weak axiom of revealed preference and the strong axiom of revealed preference.7.Prudence was maximizing her utility subject to her budget constraint.Then prices changed. After the change,she is better off.Therefore the new bundle costs more at the old prices than the old bundle did.8.Patience was maximizing her utility subject to her budget constraint.Prices changed and Patience was less well off than before.Therefore,at the old prices her new bundle must cost less than her old bundle.9.In the case of homothetic preferences,the entire change in demand from a price change is due to the substitution effect.10.John purchases two goods,x and y.Good x is an inferior good for some range of income.There must be another range of income for which good x is a normal good.11.A consumer has the utility function U(x,y)=x+2y1/2.The price of good x is2and the price ofgood y is1.The consumer’s income is20.If the price of good y rises to2,then entire change in demand for y is due to the substitution effect.12.Suppose a consumer has strictly convex preferences and her Engel curve for a good is avertical line for some range of income.In that same income range,her demand curve for the good slopes down.13.If somebody is buying10units of x and the price of x falls by$3,then that person’s netconsumer’s surplus must increase by at least$30.14.Bernice has the utility function U(x,y)=min{x,y}.The price of x used to be3but rose to4.The price of y remained at1.Her income is12.The price increase was as bad for her as a loss of$3in income.15.If there is a price increase for a good that Marilyn consumes,her compensating variation is the change in her income that allows her to purchase her new optimal bundle at the original prices.16.With quasilinear preferences,the equivalent variation and the compensating variation inincome due to a tax are the same.17.In general,aggregate demand depends only on prices and total income and not on incomedistribution.18.If the demand function is q=3m/p,where m is income and p is price,then the absolute valueof the price elasticity of demand decreases as price increases.19.If a rational consumer must consume either zero or one unit of a good,then an increase in theprice of that good with no change in income or in other prices can never lead to an increase in the consumer’s demand for it.20.If the demand curve were plotted on graph paper with logarithmic scales on both axes,then itsslope would be the elasticity of demand.PartⅡ(Single Choice)1.Mike consumes two commodities,x and y,and his utility function is min{x+2y,y+2x}.He chooses to buy8units of good x and16units of good y.The price of good y is$.50.What is his income?a.$32b.$40c.$24d.$16e.Mike’s income cannot be found unless the price of x is given too.2.For m>p2,the demand functions for goods1and2are given by the equations,x1=m/p1-1and x2=p1/p2,where m is income and p1and p2are prices.Let the horizontal axis represent the quantity of good1.Let p1=1and p2=2.Then for m>2,the income offer curve isa.a vertical line.b.a horizontal line.c.a straight line with slope2.d.a straight line with a slope of1/2.e.None of the above.3.Ambrose’s brother Augustine has an income of$105and a utility function U(x1,x2)=40x1/21+ x2.The price of good1(nuts)is$5and the price of good2(berries)is$1.How many units of nuts will Augustine demand?a.12b.16c.14d.26e.304.Daisy received a tape recorder as a birthday gift and is not able to return it.Her utility function is U(x,y,z)=x+z1/2f(y),where z is the number of tapes she buys,y is the number of tape recorders she has,and x is the amount of money she has left to spend.f(y)=0if y<1and f(y)=24 if y is1or greater.The price of tapes is$4and she can easily afford to buy dozens of tapes.How many tapes will she buy?a.9b.11c.7d.13e.We need to know the price of tape recorders to solve this problem.5.When prices are($4,$2),Tomoko chooses the bundle(9,18),and when prices are($1,$2), she chooses the bundle(8,14).a.The bundle(8,14)is revealed preferred to the bundle(9,18)and she does not violate WARP.b.She violates SARP but not WARP.c.The bundle(9,18)is revealed preferred to the bundle(8,14)and she does not violate WARP.d.She violates WARP.e.None of the above.6.Stan Ford currently spends$100a week on entertainment.A rich uncle offers him a choice between a$50a week allowance and the opportunity to buy all of his entertainment at half price. Stan has no kinks in his difference curves.Stan woulda.prefer the$50allowance.b.prefer the half-price subsidy.c.be indifferent between the allowance and the subsidy.d.prefer the subsidy if entertainment is a normal good and be indifferent otherwise.e.prefer the allowance if entertainment is an inferior good and prefer the subsidy otherwise.7.A student spends all of her income on pizza and books.When pizzas cost$3each and bookscost$10each,she consumed30pizzas and3books per month.The price of pizzas fell to$2.90 each while the price of books rose to$11each.The price changea.made her worse off.b.left her exactly as well off as before.c.left her at least as well off as before and possibly helped her.d.might have helped her,might have harmed her.We can’t tell which unless we observe what she consumed after the price change.e.had the same effect as a$3increase in her income.8.When the prices were($5,$1),Vanessa chose the bundle(x,y)=(6,3).Now at the new prices, (p x,p y),she chooses the bundle(x,y)=(5,7).For Vanessa’s behavior to be consistent with the weak axiom of revealed preference,it must be thata.4p y<p x.b.p x<4p y.c.5p y<p x.d.p y=5p x.e.None of the above.9.Cindy consumes goods x and y.Her demand for x is given by x(p x,m)=0.05m—5.15p x.Now her income is$419,the price of x is$3,and the price of y is$1.If the price of x rises to$4and if we denote the income effect on her demand for x by DI and the substitution effect on her demand for x by DS,thena.DI=-0.28and DS=-0.52.b.DI=-0.28and DS=-4.88.c.DI=-0.52and DS=-0.52.d.DI=0and DS=-2.00.e.None of the above10.Ernest’s income elasticity of demand for natural gas is.4.His price elasticity of demand for natural gas is-.3,and he spends10%of his income on natural gas.What is his substitution price elasticity?a.-.26b.-.34c..20d.-.12e.None of the above.11.In2000,Bruce spent his income on two goods,x and y.Between2000and2001,the price of good x rose by8%and the price of good y rose by8%.In2001,Bruce bought the same amount of x as he bought in2000,but he bought more of good y than he had bought in2000.a.y is a normal good.b.y is an inferior good.c.x is an inferior good.d.Nothing can be said about inferiority or superiority,since we don’t know what happenedto income.e.Bruce is acting irrationally,since the relative prices of x and y did not change.12.Rob consumes two goods,x and y.He has an allowance of$50per week and is not endowed with either of the goods.If the price of good x increases and his substitution and income effects change demand in opposite directions,a.good x must be a Giffen good.b.good x must be an inferior good.c.WARP is violated. d.good x must be a normal good.e.There is not enough information to judge whether good x is a normal or inferior good.13.Poindexter’s utility function is U(x,y)=min{x+2y,3x+y},where x is butter and y is guns.If the price of butter is$4and the price of guns is$5,what would it cost Poindexter to buy the cheapest bundle that he likes as well as4units of butter and3units of guns?a.$31b.$32c.$29d.$28e.None of the above.14.Albin has quasilinear preferences and he loves pretzels.His inverse demand function forpretzels is p(x)=49-6x,where x is the number of pretzels that he consumes.He is currently consuming8pretzels at a price of$1per pretzel.If the price of pretzels rises to$7per pretzel,the change in Albin’s consumer surplus isa.-$90.b.-$56.c.-$42.d.-$45.e.-$42.15.If Bernice(whose utility function is min{x,y},where x is her consumption of earrings and y is money left for other stuff)had an income of$15and was paying a price of$8for a pair of earrings, then if the price of earrings went up to$13,the equivalent variation of the price change would bea.$5.36.b.$8.33.c.$16.67.d.$2.68.e.$6.85.16.Lolita,the Holstein cow,has a utility function is U(x,y)=x-x2/2+y,where x is her consumption of cow feed and y is her consumption of hay.If the price of cow feed is$.30,the price of hay is$1,and her income is$4,and if Lolita chooses the combination of hay and cow feed that she likes best from among those combinations she can afford,her utility will bea. 4.25.b.3.70.c.0.25.d.6.25.e. 2.25.17.If the demand function is q=m-2(ln p)over some range of values of p,then at all such values of p,the absolute value of the price elasticity of demanda.increases as p increases.b.decreases as p increases.c.is constant as p changes.d.increases with p at small values and decreases with p at large values.e.decreases with p at large values and increases with p at small values.18.The inverse demand function for grapes is described by the equation p=518-5q,where p is the price in dollars per crate and q is the number of crates of grapes demanded per week.When p =$38per crate,what is the price elasticity of demand for grapes?a.-190/96b.-5/518c.-5/96d.-96/38e.-38/48019.A person with a quasilinear utility function willa.have a price elasticity of demand equal to zero for some goods.b.have an income elasticity of demand equal to one for some goods.c.necessarily consume zero quantity of some good.d.necessarily consume positive amounts of every good.e.None of the above.20.The only quantities of good1that Tomoko can buy are1unit or0units.For all positive values of x2,Tomoko’s preferences are represented by the utility function(x1+14)(x2+16).If her income is$20and the price of good2is$1,then Tomoko’s reservation price for good1isa.$.88.b.$8.50.c.$2.40.d.$4.80.e.$1.04.PartⅢ(Problems)1.An indifference curve map is homothetic if each line,which passes through the origin cuts the indifference curve at points,where all the indifference curves have the same slopes.a)Please prove such an indifference curve map is homothetic;Now assume that the consumer has a homothetic indifference curve map.b)Which is the general type of utility function,which represents this type of preference?Give some examples.c)Represent the income-consumption curve;d)Show that the Engle curves are lines;e)Prove that neither of the goods is Giffen.2.Marie’s utility function is U(x,y)=min{3x+2y,2x+5y},where x is the number of units of sugar she consumes and y is the number of units of spice she consumes.She is currently consuming12units of sugar and40units of spice and she is spending all of her income.a.Draw a graph showing her indifference curve through this point.b.The price of spice is$1.In order for this to be her consumption bundle,what must the price of sugar be and what must her income be?3.The indirect utility function for a consumer with a utility function U(x1,x2)is defined to bea function V(p1,p2,m)such that V(p1,p2,m)is the maximum of U(x1,x2)subject to the constraint that the consumer can afford(x1,x2)at the prices(p1,p2)with income m.a.Find the indirect utility function for someone with the utility function U(x,y)=2x+y.b.Find the indirect utility function for someone with the utility function U(x,y)=min{2x,y}. Explain how you got your answers.4.Suppose that the inverse demand function for wool is p=A/q for some constant A.Suppose that 1/4of the world’s wool is produced in Australia.a.If Australian wool production increases by1%and the rest of the world holds its output constant,what will be the effect on the world price of wool?b.How does the marginal revenue to Australia from an extra unit of wool relate to the price of wool?5.The demand for Craftmatic Adjustable Beds is described by Qc=P-1.40c I-0.60P0.20m A0.25,where Qc is the number of Craftmatic Adjustable Beds demanded,Pc is the price of a Craftmatic Adjustable Bed,I is per capita income,Pm is the price of a battery-powered massage pillow,and A is the advertising budget.a.If the marginal cost of producing a Craftmatic Adjustable Bed is$200,what is the profit-maximizing price?b.Per capita income in the United States is forecast to rise by3%next year.How will this impact Craftmatic’s sales?c.The price of battery-powered massage pillows suddenly fell by10%.How will this impace Craftmatic’s sales?Answer to problem set2PartⅠ1.T2.F3.F4.T5.T6.T7.T8.F9.F10.T11.T12.T13.T14.F15.F16.T17.F18.F19.T20.TPartⅡ1.d2.b3.b4.a5.d6.e7.c8.a9.b10.a11.a12.b13.d14.d15.a16.a17.a18.e19.e20.cPartⅢ1.a)Suppose the vector of the prices of goods consumed is P,and the consumer’s income is m. Since the indifference curve is differentiable everywhere,without any loss,we can suppose bundle x is preferred to y at price p and income m where x and y are affordable,then for any t>0,tx and ty are also affordable at price p and income tm because the indifference curve is homothetic and tx must be preferred to ty.Otherwise,if ty is strictly preferred to tx,let t=1,then y is strictly preferred to x,which is a contradiction.b)The general utility function is of homogeneity of one and the typical ones are Cobb-Douglas, complete substitution and complete comple-mentary utility functions.They represent Cobb-Douglas,complete substitution and complete complementary preference.c)See textbooks.d)When keep the prices unchanged,for any bundle x and income m,since the preference is homothetic,when x is consumed under m,then tx must be consumed under tm for any t>0.As (x,m)and(tx,tm)are on the same line,the Engel line of homothetic preference is a line.e)It is sufficient to prove neither is an inferior good.Without any loss,we can suppose good1’s price decreases,then the total income will increase.If the demand for good1decreases,then the demands for any other goods must decrease because the preference is homothetic which is impossible because at this time the income constraint is not binding.2.a.Her indifference curve is a broken line consisting of the outer envelope of the two lines3x+ 2y=116and2x+5y=116.The point(12,40)is on the line3x+2y=116.b.The price of sugar must be$1.50and her income must be$58.3.a.m/(min{p1/2,p2}).Consumer spends entire income on x or y depending upon which gives more utility.b.2m/(p1+2p2).To maximize utility,consumer spends income in increments of one x and 2y’s.4.a.Price will fall by about0.25%.b.Marginal revenue is75%of price.5. a.$700.b.Demand will decrease by1.8%.c.Demand will decrease by2%.。

北大光华微观经济学作业ProblemSet-1

北大光华微观经济学作业ProblemSet-1Problem Set 11. Draw the indifference curves for the following individuals’ preferences for two goods: hamburger and beer.a) Al likes beer but hates hamburger. He always prefers more beer no matter how many hamburgers he has.b) Chris eats one hamburger and washes it down with one beer. He will not consume an additional unit of one item without an additional unit of the other.c) Doreen loves beer but is allergic to beef. Every time she eats a hamburger shebreaks out in hives.2. Connie Consumer has a monthly income of $200, which she allocates between two goods: meat and potatoes.a) Suppose meat costs $4 per pound and potatoes cost $2 per pound. Draw her budget constraint.b) Suppose also that her utility function is given by the equation U(M, P)=2M+P. What combination of meat and potatoes should she buy to maximize her utility? c) Connie’s supermarket has a special promotion. If she buys 20 pounds of potatoes (at $2 per pound), she gets the next 10 pounds for free. This offer applies only to the first 20 pounds she buys. All potatoes in excess of the first 20 pounds(excluding bonus potatoes) are still $2 per pound. Draw her budget constraint. d) An outbreak of potato rot raises the price of potatoes to $4 per pound. Thesupermarket ends its promotion. What does her budget constraint look like now? What combination of meat and potatoes maximizes her utility?4．Assume that a utility function is given by Min{X, Y}, where Min is the minimum of the two values of X and Y. What is the Slutsky equation that decomposes the change in the demand for X in response to a change in its price? What is the income effect? What is the substitution effect?5．Using Slutsky equation shows that Giffen goods must be inferior goods, but inferior goods need not be Giffen goods.6．Consider a consumer who purchases two goods, x and . The consumer’s utility function is . The prices are given by y 2(,)U x y xy =x P and , and the consumer’sincome is y P I .1) Set up the Lagrangian expression for solving the consumer’s utility maximization problem (subject to budget constraint) and write down the first order conditions.2) Derive the demand functions for goods x and y .Assume initially 1$30I =,$x P =, an $1d y P =. Now assume the price of y increases to y P $8′=.3) Compute the initial optimal bundle A before the price change, and the newoptimal bundle C after the price change.4) Identify the substitution effect and income effect. What is the direction of the substitution effect? Is good y a normal, inferior, or Giffen good? Explain.。

北京大学光华管理学院经济学参考书目及真题分析

北京大学光华管理学院经济学参考书目及真题分析一、参考书目近年来，北京大学一直不指定参考书目。

根据往年的招生简章以及考题难度和考生心得，北京大学光华管理学院“经济学”考试科目“微观经济学部分”推荐使用以下参考书目：1．范里安《微观经济学：现代观点》，格致出版社2．朱善利《微观经济学》，北京大学出版社3．平新乔《微观经济学十八讲》，北京大学出版社4．尼科尔森《微观经济学：基本原理与拓展》，北京大学出版社5．范里安《微观经济学：高级教程》，经济科学出版社6．蒋殿春《高级微观经济学》，北京大学出版社7．张维迎《博弈论与信息经济学》，格致出版社需要提醒考生注意的是，上述推荐参考书目中，有些书是必读的，如范里安《微观经济学：现代观点》；有些书是选读的，如平新乔《微观经济学十八讲》和尼科尔森《微观经济学：基本原理与拓展》属于同一级别，可全看或选其一。

二、真题分析通过对近年来北京大学光华管理学院“经济学”考研真题的研究分析，可以看出，呈现如下命题规律：1．难度系数非常大，偏重计算和推导查看北京大学光华管理学院“经济学”近几年考研真题，发现微观经济学部分是重点，且很有难度。

需要提醒考生注意的是，考生复习过程中，特别是强化、冲刺阶段，除了要看北京大学光华管理学院“经济学”之前年份的考研真题，还要看北京大学国家发展研究院历年考研真题，很有参考价值。

2．各本书的复习方法存在差异考虑到微观经济学部分知识点就那么多，推荐考生复习过程中采取题海战术，多做考题。

推荐参考书目，建议考生前期多花时间学习范里安《微观经济学：现代观点》，需要反复多看几遍；平新乔《微观经济学十八讲》重点以习题为重，建议把课后习题做两到三遍。

后期，建议多看看蒋殿春《高级微观经济学》和张维迎《博弈论与信息经济学》，对知识点理解很有帮助。

北京大学光华管理学院经济学（微观经济学部分）考研真题及详解1．一个纳税人，效用函数为lnw，w为其财富，是固定值。

国家按照固定税率t（0＜t＜1）对纳税人上报的收入征税，但此人可以少报收入，即报的收入为x（0＜x＜w）。

(完整版)北京大学微观经济学期末试题及答案.docx

2014~2015 学年北京大学光华管理学院微观经济学期末试题（普本/B 卷）一、单选题（每题 2 分，共 20 分）1、经济学研究的基本问题是（）A、怎样生产B、生产什么，生产多少C、为谁生产D、以上都是2、下列各项哪一项会导致一国生产可能性曲线向外移动（）A、失业B、有用性资源增加或技术进步C、通货膨胀D、经济能力增长惟一决定于劳动力数量3、保持所有其它因素不变，某种商品的价格下降将导致（）A、失业B、有用性资源增加或技术进步C、通货膨胀D、经济能力增长惟一决定于劳动力数量4、若某行业中许多生产者生产一种标准化产品，我们可以估计到其中任何一个生产者的需求将是：（）A、毫无弹性B、有单元弹性C、缺乏弹性D、富有弹性5、如果人们收入水平提高，食物在总支出中的比重将（）A 、大大增加B、稍有增加C、下降D、不变6、若无差异曲线上任一点的斜率为：－1/2，这意味着消费者有更多x 的时，他更愿意放弃（）单位x而获得一单位y。

A、1/2B、2C、1D、1.57、恩格尔曲线从（）导出A 、价格——消费曲线B、收入——消费曲线C、需求曲线D、无差异曲线8、使用自有资金也应计算利息收入，这种利息从成本角度看是（）A、固定成本B、隐含成本C、会计成本D、生产成本9、长期成本曲线呈“ U”型的原因与（）有关A 、规模报酬B、外部经济与不经济C、要素的边际生产D、固定成本与可变成本所占比重10、在拐折需求曲线模型中，拐点左右两边的需求弹性是（）A 、左边大，右边小B、左边小，右边大C、左右一样大D、、以上均不对二、判断是非题（每题 1 分，共 10 分）（注：对的打“√” ，错的打“×”）1、实证经济学回答的问题是：应该是什么（）2、对于一个正常品来说，它的替代品价格上涨时，它的需求曲线回向右上方移动（）3、市场上某产品存在超额需求是由于该产品供不应求（）4、降低价格一定会使供给量下降（）5、提高价格可以使总收益增加（）6、 SAC 决不会小于 LAC （）7、消费者均衡点处于无差异曲线与预算线相切处（）8、劳动的供给会随价格的上升而增加（）9、完全竞争厂商不愿做广告（）10、什均衡属于优势策略均衡（占优均衡）（）三、简答题（每题 5 分，共 25 分）1、简述实证分析和规范分析的差异。

光华历年真题答案(微观96-09)

赤焰永明（刘明）光华人，梦想起飞的舞台
2、解：根据资本资产定价模型
R1 R f ( Rm R f )
R1 ( Rm R f ) R f 1.2 (0.2 0.1) 0.1 0.22
p 20 16.4 1 0.22
3、解：寡头 2 视寡头 1 的产量给定，由
Q q1 q2 a1 a2 (b1 b2 ) p
( pQ) 1 a a c p [ 1 2 c] Q 2 b1 b2
Q ''
a1 a2 b1 b2 c 2 2
Q ' Q ''
Email: liuming1204@
f (tx1 ,tx2 ,,txn )

r (x )
i 1 i i
f ( x1 , x2 ,, xn )
LAC
故长期平均成本曲线应为一条水平线。 2、答：不正确。除了政府干预外，只要交易成本为零，明碀所有权也可以达到帕累托有效率条件，政府干预的方法有：一、对污染企业按污染程度征收排污费；二、对给企业分配排污标准；三、关闭污染企业；四、兼并污染企业。科思定理：只要法定权利可以自由交换，且交易成本为零，那么法定权利的最初配置对资源的配置效率而言无关紧要。 3、答：即使不存在逆向选择的情况下，如果提供全额赔偿，会导致严重败德行为，从而破坏市场的运作，不能达到市场的有效性。二、计算题 1、解：根据市场的分配法则，各市场的边际收益相等，且等于边际成本。
t qt / Q 代表企业 t 的市场份额。
勒纳指数：用于衡量垄断厂商对于市场的垄断程度，即 ( p MC ) / p
1 。 Ed
其中，

北大光华历年真题(微观96-09)

MR1 A aT ， MR 2 B bT ， MR3 C cT 。其中，T是公共电视播放的时间，A、
B、C、a、b、c都是常数，假定公共电视服务是纯公共产品，提供该公共产品的边际成本等于常数，即每小时M元，问：（1）公共电视有效播放时间是多少？（2）如果由竞争的私人市场提供公共电视服务，提供的时间应该是多少？ 2、请给出斯拉茨基方程(Slutsky Equation)微分形式的全部推导过程，并说明它的经济学含义。 3、王老汉和张老汉是自由市场上仅有的两个出售西红柿的人。西红柿的市场需求函数为
X [ x1 , x2 xn ] ,要素对应的价格向量为： R [r1 , r2 rn ] ，厂商规模收益不变。
请推导出厂商超额利润与垄断力的关系(15分) 三、ry)和信息经济学(Information Economics)的区别和联系。 2、委托—代理理论对中国国有企业改革有什么借鉴意义？(15 分)
Email: liuming1204@
赤焰永明（刘明）光华人，梦想起飞的舞台
北大光华1998年研究生入学考试试题(微观经济学)
一、解释并比较下列经济学基本概念(每题6分，共计30分) 1 私人成本(Private Cost)与社会成本(Social Cost) 2 边际替代率(Marginal Rate of Substitution)与边际技术替代率(Marginal Rate of T echnical Substitution) 3 产业集中指数(Industrial Concentration Index),勒纳指数(Lerner’s Index)与贝恩指数(Bain’s Index) 4 价格歧视(Price Discrimination)与二重价(Two-part Tariff) 5 占优均衡(Dominant Equilibrium)与纳什均衡(Nash Equilibrium) 二、简答下列各题(每题5分，共计30分) 1、有些专家论证说，北京市居民大量丢失自行车，原因是自行车盗窃者把盗窃来的自行车转手在旧车交易市场上卖掉，因此建议取缔北京市所有旧车交易市场。请给出一个支持该观点的论据，给出一个反对该观点的论据。 2、某君在研究生入学考试复习期间，从经济学中悟出一个道理：应该使投入每门课程最后一单位时间的收益都相等。换句话说，与其投入大量的时间使数学成绩从90分提高到95分，不如把这些时间投入到微观经济学中，使其成绩从60分提高到80分。请给出一个支持该观点的论据，给出一个反对该观点的论据。 3、你怎样理解新制度经济学关于制度安排与交易成本的关系？请举例说明。 4、垄断存在着效率损失，请说明是怎样产生的？ 5、一般认为，竞争性市场是有效率的，但为什么有时会出现市场失灵？解决市场失灵的方法有哪些？既然市场会出现失灵，为什么我们还要建立市场经济体制？ 6 、委托代理问题产生的原因是什么？三、计算题(每题8分，共计24分) 1、某竞争行业所有厂商的规模都相等，都是在产量达到500单位时达到长期平均成本的最低点4元，当用最优的企业规模生产600单位产量时，每一个企业的短期平均成本为4.5元，市场需求函数为Q=70000-5000P，供给函数为Q=40000+2500P，求解下列问题：（1）市场均衡价格是多少？该行业处于短期均衡还是长期均衡？当处于长期均衡时，该行业有多少厂商？（2）如果市场需求变化为Q=100000-5000P，求行业与厂商新的短期均衡价格与产量，在新的均衡点，厂商盈利还是亏损？ 2、假定有n家相同的厂商处于古尔诺均衡(Cournot Equilibrium)情况下，求证：市场需求曲线的弹性一定大于1/n。 3、家用电气市场上有两个厂商，各自都可以选择生产空调和彩电，彼此的利润如下列收益矩阵所示：

北京大学光华管理学院考研微观经济学部分历年真题汇编

北大、人大、中财、北外、中传教授创办
集训营、一对一保分、视频、小班
二、2009 年微观经济学（一）厂商成本函数为 C（Q）=F+0.5a Q2 1．求厂商的规模报酬区间 2．厂商的供给函数 3．若市场需求函数为 P=A-b Q（A,b 均为正数），市场由该厂商唯一供给，且厂商是价格接受者，在什么条件下存在唯一的市场均衡？
府先确定s，两国再同时确定产量，政府最大化本国福利（国家1 为生产国，故其福利为利润与政府税收之
和或是利润与政府补贴之差）
ⅰ）求最优的政府贸易政策s，两国的产量，利润。与情形2）比较，两国的产量与利润有何变化；国家1 的
福利
ⅱ）解释最优贸易政策的作用
（四）还是在上一题的框架下
1）求出国家3 对产品1 和2 的直接需求函数
。
生产要素x1 和x2 的价格分别是4 元和1 元，短期内生产要素x2 是固定在1 的水平，请问：
侈品？吉芬品？为什么？
（二）假如一个经济中有100 个消费者，大家的偏好是完全一样的。他们只消费两种商品，x 和y，并且他
们的偏好可以用效用函数U（x，y）=x+2.94ln (y)来表示。商品x 是由国际市场提供，且价格为1。商品y 只
由该经济自己生产。该经济中有48 个技术完全一样的企业可以生产y。它们的生产函数都是y=
北京大学光华管理学院考研历年真题（微观经济学部分）
一、2010 年微观经济学（一）农民效果函数为 alogX1+（1-a）log（X2-K）,k>0,X1 为大米，X2 为其他商品组合，农民禀赋为（R1,R2），禀赋固定不变。 1.求农民什么时候出售大米，出售的大米量与大米价格正相关吗？ 2.若农民消费其他商品有配额限制，对农民出售大米的意愿有影响吗？（二）一个企业有三个车间，各自的成本函数为 C1=4X1+X12，C2=3X2+2X22，C3=6X3 问：当这个个企业要生产 8 单位时，应如何分配产量，使成本最小。（三）市场上有 A、B 两种消费者，各占 50%。A 的效用函数为 5x - 0.5x2+y，B 为 6x- 0.5x2+y 垄断厂商的成本为 C(x)=X, x 为垄断厂商生产的商品，y 为其他商品组合，价格为 1 1.求 A,B 各自的需求曲线 2.若垄断厂商在 AB 中选一种出售商品，那么你认为应该选择哪个？垄断厂商采用 2 部分定价，那么边际价格和一次性付费是多少？ 3.若同时在 AB 中出售，并且采用 2 部分定价，问厂商以什么边际价格和一次性付费为多少能够达到利润最大化。（四）市场需求曲线 P=a – bQ，企业 1 和企业 2 的博弈过程如下：第一阶段：企业 1 决定边际成为 C 和固定成本 F，但若企业 2 也进入市场的话，将与企业 1 一样，边际成本为 C 和固定成本为 F 第二阶段：企业 2 决定是否进入市场第三阶段：若企业 2 不进入市场，企业 1 将是唯一的垄断厂商，但企业 2 若进入，两企业将达到古诺均衡。问：1.对于任意给定的 C，F 至少为多少才能使企业 2 不进入市场。 2．企业 1 会选择什么样的 F 使企业 2 不进入 3．选择让企业 2 不进入是企业 1 最优选择吗？（五）两支军队争夺一个岛屿，一开始军队 2 占领岛屿，但是军队 1 可以进攻岛屿，每次进攻方和防守方都损失一个营并且由进攻略占领岛屿，军队 1 有 K 个营，军队 2 有 L 个营，两支军队的统领可以选择进攻还是放弃进攻岛屿，都认为得到岛屿的价值高于一个营而低于两个营，但若一次进攻后，两支军队都没有剩余的营了，得到的价值将为 0。问：根据子博弈纳什均衡，谁将占领这个岛屿？