北大光华微观经济学作业09 Problem Set-3
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?。
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Problem Set 3
1． Why is there a social cost to monopoly power? If the gain to producers from
monopoly power could be redistributed to consumers, would the social cost of monopoly power be eliminated? Explain briefly.
2．A monopolistic firm faces the following average revenue (demand) curve: P = 100 – 0.01Q where Q is weekly production and P is price, measured in cents per unit. The firm’s cost function is given by C = 50Q+30000.
A) What is the level of production, price, and total profit per week?
B) The government decides to levy a tax of 10 cents per unit on this product. What will the new level of production, price, and profit be as a result?
3．A monopolistic firm has two factories, for which costs are given by:
Factory #1: C 1(Q 1) = 10Q 12
Factory #2: C 2(Q 2) = 20Q 22
The firm faces the following demand curve: P = 700 – 5Q, where Q is total output (Q=Q 1+Q 2).
A) Calculate the values of Q 1, Q 2, Q, and P that maximize profits. Also draw a diagram to illustrate the optimal choices of this firm.
B) Suppose labor costs increase in Factory 1 but not in Factory 2. How should the firm adjust Q 1, Q 2, Q, and P?
4. Suppose that a monopoly can produce any level of output it wishes at a constant marginal cost $5 per unit. Assume that the monopoly sells its goods in two different markets that are separated by some distance. The demand curve in the first market is given by Q 1=55-P 1, and the demand curve in the second market is given by Q 2=70-2P 2.
1) If the monopolist can maintain the separation between the two markets, what level of output should be produced in each market, and what price will prevail in each market? What are total profits in this situation?
2) How would your answer change if it only costs demanders $5 to transport goods between the two markets? What would be the monopolist’s new profits in this situation?
3) How would your answer change if transportation costs were zero and the firm was forced to follow s single-price policy?
4) Suppose the firm could adopt a linear two-part tariff under which marginal prices must be equal in the two markets but lump-sum entry fees might vary. What pricing policy should the firm follow?
5. A monopoly supplies its products in two segmented markets. The first market’s demand is , and the second is 1110P Q −=22P A Q −=. The firm’s cost function is , where , and Q is the output.
Q C 5=2010≤<A
a) Suppose the monopoly has to set the same price in these two markets, what is it? Why should we assume that ? What if this condition does not hold?
20≤A
b) Suppose the monopoly is able to implement the third-degree discrimination. Derive the profit-maximizing prices in the two markets.
c) Compare social welfares under a) and b). Does the third-degree discrimination improve social welfare? Why? Please give an intuitive interpretation.。