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Economic Theory

by: Dr. Janiya Bernier

Economic Theory ECON 101A

Dr. Janiya Bernier

GPA 3.77


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This 77 page Class Notes was uploaded by Dr. Janiya Bernier on Thursday October 22, 2015. The Class Notes belongs to ECON 101A at University of California - Berkeley taught by Staff in Fall. Since its upload, it has received 36 views. For similar materials see /class/226720/econ-101a-university-of-california-berkeley in Economcs at University of California - Berkeley.


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Date Created: 10/22/15
Economics 101A Lecture 10 Stefano DellaVigna October 3 2006 Outline 1 Application 2 Intertemporal choice 2 Application 3 Altruism and Charitable Donations 1 Intertemporal choice 0 Nicholson Ch 17 pp 502 506 OLD Ch 23 pp 628 632 0 So far we assumed people live for one period only 0 Now assume that people live for two periods t O people are young t 1 people are old 0 t 0 income M0 consumption CO at price p0 1 o t 1 income M1 gt MO consumption c1 at price P1 1 0 Credit market available can lend or borrow at inter est rate 7 0 Budget constraint in period 1 0 Sources of income M1 M0 c0 gtlt 1 7 this can be negative 0 Budget constraint 013M1M0 0017 or C0 C1 Mo M1 1 1 Utility function Assume 1 U U uCo C1 00 1 6 01 U gt 0 U lt 0 6 is the discount rate Higher 6 means higher impatience Elicitation of 6 through hypothetical questions Person is indifferent between 1 hour of TV today and 1 6 hours of TV next period o Maximization problem max Uc0 1 6Ucl 875 c c ltM M 0 1 7 1 0 1 7 1 o Lagrangean 0 First order conditions 0 Ratio of focs Uco 1 7 UCl 16 o Caser6 gtllt gtllt CO cl Substitute into budget constraint using c3 cizc k 27 1 M M 1TC 01 l r 1 or 1 8 7 M1 M 2 02 l r We solved problem virtually without any assump tion on U Notice M0 lt c lt M1 0 Casergt6 gtllt gtllt CO cl 0 Comparative statics with respect to income M0 0 Rewrite ratio of focs as 1 U Co 1 gU m 0 0 Substitute c1 in using c1 M1M0 c0 1 r to get 17 U Co 16 UMl M0 Co 1 7 0 0 Apply implicit function theorem 803 7 M iZ U 01 1 7 3M0 U Co i iEU c1 1 7 Denominator is always negative Numerator is positive 8c 7 M 8M0 gt 0 consumption at time 0 is a normal good Can also show 8c 7 M 8M1 gt 0 0 Comparative statics with respect to interest rate 7 0 Apply implicit function theorem 803 7 M 1 1H5UCl 37 U Co Uquotc1 1 7 1 Ul01Mo Co U Co i iEU Kcl lt1 m o Denominator is always negative 0 Numerator First term negative substitution eff o Numerator Second term income effect positive if MO gt CO negative if M0 lt co 2 Altruism and Charitable Dona tions 0 Maximize utility satisfy self interest o No not necessarily o 2 person economy Mark has income MM and consumes CM Wendy has income MW and consumes CW 0 One good c with price p 1 Utility function with u gt O u lt 0 Wendy is altruistic she maximizes ucWau CM with 04 gt 0 Mark simply maximizes ucM Wendy can give a donation of income D to Mark Wendy computes the utility of Mark as a function of the donation D Mark maximizes r9an uCM 8i CM Solution cj w 2 MM D Wendy maximizes max ucW one MM D CMD 8i CW S MW D o Rewrite as mgxuMW D one MM D 0 First order condition u MW D au MM D 0 0 Second order conditions u MW D om MM D lt 0 0 Assume 04 1 Solution uMW D u MM D Airy D lt 44iDgtllt Ot Dgtlt 2 MW MM 2 Transfer money so as to equate incomes Careful D lt 0 negative donation if MM gt MW 0 Corrected maximization mgxuMW D one MM D 8tD 2 0 0 Solution 04 1 D iwa MMgtO 0 otherwise Assume interior solution D gt 0 Comparative statics 1 altruism 8D u MM D 804 u MW D om MM D gt0 Comparative statics 2 income of donor 8D u MW D gt 0 BMW u MW D om MM D Comparative statics 3 income of recipient 8D om MM D 0 MM u MW 0 om MM 0 lt 3 Next Lectures 0 Introduction to Probability 0 Risk Aversion o Coefficient of risk aversion Applications Insurance Portfolio choice Department of Economics Microeconomic Theory University of California Berkeley Economics 101A March 12 2007 Spring 2007 Economics 101A Microeconomic Theory Section Notes for Week 8 1 Nash Equilibrium For simplicity consider two rms Firm 17s pro ts 7T11112 depend upon its own actions 11 and the actions of rm 2 12 Similarly rm 27s pro ts are denoted 7T21112 Examples of possible actions could be setting a price level of output or determining whether to enter or exit an market 11 De nition A set of actions ailing constitute a Nash Equilibrium if and only if 7T1aiva v 2 7T111 a vWal and N N N 1011702 2 7T1 11702W 11 So a rm7s Nash equilibrium action is the action that gives it the highest pro t given that the other rm is playing their Nash equilibrium action 12 Solving for Nash Equilibria To nd a Nash equilibrium solution rm 1 must solve max 7T11112 a1 while rm 2 solves max 7T21112 a2 Assuming the rms7 pro t functions are continuously di erentiable concave and a1 12 gt 0 then the rst order necessary conditions for a maximum are 5W1alv7a v 6 FOO1 0 01 N N F002 67T2g a 0 2 which yield a system of two equations and two unknowns The second order conditions for a a v to be a global maximum of 7T1 and 7T2 respectively are 62711011 12 SOC 0 1 3a 627T211 12 SOC 7 0 2 ag 13 Best Response Functions Each FOC implicitly de nes a rm7s best response function Firm 17s best response function7 a1 7a2 gives the the most pro table action for rm 1 given that rm 2 is taking action 12 Similarly rm 27s best response function can be written as 12 2011 A Nash equilibrium is any points where these function cross each other7 so that all 7a v and 19 waV Substituting7 gives ail w aV which shows that the Nash equilibrium de nes a xed point7 and provides a way to solve for ail 14 Cartel Monopoly Equilibrium If the rms can coordinate their actions then it is said that they are operating as a cartel and will achieve the same level of output as a monopolist Their problem is then max 7T111712 7T2a1 a2 l741l742 the rst order condiditions from this problem are 67110117 12 67Tg11712 0 601 601 37Tz01702 5W101702 0 602 602 Note7 W in the rst equation represents the externality that rm 17s actions have on rm 27s pro ts W in the second equation has a similar interpretation 2 Duopoly Examples 21 Cournot Competition Example Consider the following setup 2 rms Competition in quantities Market deman function is Dp 12 7 p There are no costs so MC 0 The inverse demand function is D 1q1q2 12 7 qL qz p Firm 17s pro t is thus 7T1Q17Q2 10th 0 12 i 11 1211 The rst order condition is 5W1ltQ17 12 1272 7 0 aql 11 Q2 solving for Q1 gives the best response function for rm 1 1 11 101012 6 12 By symmetry rm 27s best response function is 1 12 102011 6 EQI solving for the Nash equilibrium qiv 1 2in 1 1 N N 7 67 677 ql 2lt 2q1 11V 46 by symmetry We could also use the symmetry directly 1 Q 10415 6 7 5qu 3 N 7 6 2q5 qiv 7 4in 1 The price under Cournot competition is found by reading from the demand function p 12 7 qiv 7 q v 4 Pro ts are then 7T1 pq1 16 and 7T2 pq2 16 Finally7 note that 6701 1 7 7 0 6q2 2 lt 6152 1 7 7 0 aql 2 lt Both best responses are decreasing in each others actions in this example In this case actions qhqz are called strategic substitutes 22 Hotelling Competition Example Consider a different setup 2 rms Competition in prices due to differentiated products demand for product 1 is D1p1p2 12 7 p1 p2 demand for product 2 is D2p1p2 12 101 7102 There are no costs so MC 0 The pro t functions of the rms are 7T1017102 10111 10112 101 102 7T1017102 10292 10212 101 102 the FOC7s are F0011272pivp v 0 FOCZ1272p VpV 0 the best response functions are l Piv 1011 6 5103 by symmetry 1 10 6 1N 7 i 6 2p5 piv 12plvp v Note 6 1 7 gt 0 6q2 2 6 l gt 0 aql 2 best responses are increasing in each others7 actions In this case actions 101102 are called strategic complements Economics 101A Lecture 23 Stefano DellaVigna April 22 2008 Outline 1 Second price Auction 2 Auctions eBay Evidence 3 Dynamic Games 4 Oligopoly Stackelberg 1 Secondprice Auction 0 Nicholson Ch 18 pp 659 66 Not in old book 0 Sealed bid auction Highest bidder wins object 0 Price paid is second highest price 0 Two individuals I 2 0 Strategy 87 is bid bi 0 Each individual knows value 127 o Payoff for individual 73 is 07 bi if by gt bi U7qub 7L w 5 02 if bi 5 7 0 if bz lt b7 0 Show weakly dominant to set I 2 vi 0 To show 164117554 2 16457554 for all bi for all b7 and for 73 12 1 Assume b7 gt 127 0 164117554 0 WU 5 7 for any bi lt 5 7 0 1645 7 5 7 1173 5 7 2 lt 0 O uibi bi 07 bi lt 0 for any 37 gt bi 2 Assume now b7 2 vi 3 Assume now b7 lt 127 2 Auctions Evidence from eBay 0 In second price auction optimal strategy is to bid one s own value o Is this true 0 eBay has proxy system If you have highest bid you pay bid of second highest bidder 0 eBay is essentially a second price auction 0 Two deviations 1 People bid multiple times they should not in this theory 2 People may overbid An example eBay Bidding for a Board Game Bidding environment with clear boundary for rational willingness to pay buyit now price Empirical environment unaffected by commonvalue arguments presumably bidding for private use in addition buyitnow price Still nonnegligible amount 100200 9 Is there evidence of overbidding 9 If so can we detect determinants of overbidding The Object in miken Ynur Financial Genius The Data Cash ow 101 board game with the purpose of nanceaccounting education Retail price 195 plus shipping cost 1075 from manufacturer Two ways to purchase Cash ow 101 on eBay Auction quasisecond price proxy bidding Buyitnow Handcollected data of all auctions and Buyit now transactions of Cash ow 101 on eBay from 2192004 to 962004 Sample Listings 206 by individuals 187 auctions only 19 auctions with buyitnow option 493 by two retailers only buyitnow Remove nonUS terminated unsold items and items Without simultaneous professional buyitnow listing 9 169 auctions Buyit now offers of the two retailers Continuously present for all but siX days Often individual buyit now offers present as well they are often lower 100 and 999 positive feedback scores Same prices 12995 until 07312004 13995 since 08012004 Shipping cost 995 other retailer 1095 New items with bonus tapesvideo Listing Example 02122004 Rmh Dad s Cash ow Quadrant Rmh dad 9 Rmh Dad s Cash ow Quadrant by Robert T Real Estate Investment Cash ow Software 0001 aquot CASHFLOW 101202 Robert K1 osalu Best Pak W rav 1T ToDAv WITH Aesorurrw NO arsx CASHIFLOW 101 Robert nyosalu Plus Bonuses1 W 3 bank Vent sstrstsmen rs euaaamre 100 MJNT Czsh nw 101 Ruben Kilnsaki Game NR 126 Bvand New stt11sea1ed as easy a be m eash ow Hard Money Fundng 101 real estate 126 BRANDN39EW RICHZDAD CASHFLOW FOR KIDS Er GAME 9 CASHIFLOW 101 Robert nyosalu Plus Bonuses1 W bank Vent sstrstsmen rs euaaamre 100 CASHIFLOW 101 202 Robert nyosalu Best Pak W rav 1T ToDAv WITH Aesorurrw NO R15 3 to 0129 95 14000 14 99 2000 0129 95 0207 96 4 a swim EEuyMW swim mow 3914le 1d 00h 14m 1d 00h 43m 1d 04h 36m 1d 06h 47m 1d 08h 02m 1d 08h 04m 1d 09h 28m 1d13h54rn 1d 14h 17m 1d15h47rn Listing Example Magni ed ASHFLOW 1m 20 Robert 11 vsak Ben Pal2 x 90 207 92 WWW Pricing TRY KTTODAY WITH AasoLurEm NO RIS Pricing 4 mun1m 14000 B1dd1ng hlstogy of an Item Aim F1 Edt v1w Favu tas Van s Hub VbBazk v n V 331 Search avnnlas Nema g u I 1117921 3 ma 11119113 any 31 smm 1 g 1 zasbucm 31111111 150mm 1 1amm1a OAS LOW1EI1 Euavd Gama Rmh Bad Paar Dad T1m312 Anewquot has Ended Leam On1acma1b1ds mum abuu mun UserlD Bid Amnum Dale nvhm beezaehugs a t Aug11EI4EI9512 PDT 11131331111171 Usmuun Aug11m EIE 39 53 PDT baggga ugs g1 US 140 an AugrDErDA 1216 D5 PDT a mm 1 US mm 11 Augr rm 23 49 D2 PDT successbmkey mm US11DDD Augr rm 1955 26 PDT successhmkey g has U551D5DD Augr rm 17 15239 PDT um i uswzan AugDam 17 11 339 PDT successbmkey inure uswunu Augr rDA 1541 AU PDT M L US 599 ED AugrDEVUA 17 1D 43 PDT um 1 Us WEED AugDEDA 17 1D239 PDT 12mm 1 iv US ma El Augr rm D313 am PDT mm L i US MEED Augr rm 114733 PDT 11mm L 3 US MED Augr rm 1145 439 PDT 11 DE 1113 iv US WED AugDam 1u45 EIE PDT 12231571121111le 3 US31ED Aug DA DE 4915PDT m 1 US 30 El Aug EIJ D4 15 45 at PDT ha 1 1 US WEED AugEIEEIA DEAD 29 PDT 1 vsnbuHsEZ 3 US 25ED Augr rm U648 D39 PDT Wynn and mm mdumpmama sums 111 amuum the aar11arb1d1akaspv1umy E aszanHMQ m 165111511421 2 wmsa 61mm r m 1 In w Ian Hypotheses Given the information on the listing website H1 An auction should never end at a price above the concurrently available purchase price H2 Mentioning of higher outside prices should not affect bidding behavior Figure 1 Starting Price startprz ce 45 below 20 mean46 SD4388 only 6 auctions with rst m not price above buyitnow Frequency 8 1 DEED llll 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 Starting Price Figure 2 Final Price nalprice 41 are above buyit now mean 132 SD 1683 OEEH DD 90 100 110 120 130 140 150 160 170 180 FinalPIice Figure 4 Total Price incl shipping cost 51 are above buyit now plus its shipping cost mean14420 SD1500 120 130 140 150 160 170 180 190 Total PIice The Other Lesson Some unsolicited eBay advice Can make money by selling Cash ow 101 to those who aspire to become nancially smart and overpay for the board game Sellers add exaggerated retail price pay 20 cents extra now 40 cents for 10 day listing Buyers check out the buyitnow price before you bid 3 Dynamic Games 0 Nicholson Ch 8 pp 255 266 better than Ch 15 pp 449 454 9th 0 Dynamic games one player plays after the other 0 Decision trees Decision nodes Strategy is a plan of action at each decision node 0 Example battle of the sexes game She He Ballet Football Ballet 21 00 Football 00 12 0 Dynamic version she plays first o Subgame perfect equilibrium At each node of the tree the player chooses the strategy with the highest payoff given the other players strategy 0 Backward induction Find optimal action in last pe riod and then work backward 0 Solution 0 Example 2 Entry Game 1 2 Enter Do not Enter Enter 1 1 100 Do not Enter 05 00 0 Exercise Dynamic version 0 Coordination games solved if one player plays first 0 Can use this to study finiter repeated games 0 Suppose we play the prisoner s dilemma game ten times 1 2 D ND D 4 4 1 5 ND 5 1 2 2 o What is the subgame perfect equilibrium o The result differs if infinite repetition with a proba bility of terminating 0 Can have cooperation 0 Strategy of repeated game Cooperate ND as long as opponent always co operate Defect D forever after first defection 0 Theory of repeated games Econ 104 4 Oligopoly Stackelberg o Nicholson Ch 15 pp 543 545 better than Ch 14 pp 423 424 9th 0 Setting as in problem set 0 2 Firms 0 Cost Cy 2 cy with c gt O 0 Demand pY a bY with a gt c gt O and bgt0 0 Difference Firm 1 makes the quantity decision first 0 Use subgame perfect equilibrium Solution Solve first for Firm 2 decision as function of Firm 1 decision rnygx a by2 byiquot y2 cy2 Foca 2by 2quot by lk c O Firm 2 best response function a c 341 2b 239 y2 o Firm 1 takes this response into account in the max im ization my xw by1 bka y1y1 Cyl or b W D max a c yl 91 2b 2 91 91 oFoc a 2by1 by1 CO or a c and 0 Total production c a Y5y1ltygt2k3T Price equals blt3a cgt 1 3 a a c p 4 b 4 4 0 Compare to monopoly a c yM 2b and ac PM 2 0 Compare to Cournot ygtk gtlt 2265C D y1y2 3b and 1 2 a c 131 3 3 Compare with Cournot outcome Firm 2 best response function 24 2 2b Firm 1 best response function a c 2b y1 Intersection gives Cou rnot a c yl 2 y 2 2 Stackelberg Equilibrium is point on Best Response of Firm 2 that maximizes profits of Firm 1 Plot iso profit curve of Firm 1 l a C 741 by1y2 574 Solve for yg along iso profit a c 1 y2 b yl byl Iso profit curve is flat for dyg dyl 5041 or 91 Figure 5 Next lecture 0 General Equilibrium o Edgeworth Box Department of Economics Microeconomic Theory University of California7 Berkeley Economics 101A April 117 2007 Spring 2007 Economics 101A Microeconomic Theory Section Notes for Week 12 1 Ef cient Insurance We will show that if actuarially fair insurance is offered7 then it will be optimal for consumers to fully insure Actuarially fair insurance is an insurance contract where the expected cost to the consumer and the expected pro t of the rm selling the contract are both zero The model set up is The consumer begins with income of yo The consumer has probability7 p gt 0 of getting into an accident which will cause the consumer to incur a loss of income of L With probability 1 7p there won7t be an accident An insurance contract is available which has a payout of 7WO if no accident occurs and a payout of C 7 7T0 if an accident happens We assume that the consumer is an expected utility maximizer and thus picks the level of coverage 0 to solve mnglu7Jl 7 mgxpuyo 7 L O 7 7TC17 puyo 7 7TC where yquot is a random variable representing the consumers income We can think of the amount of income in the accident state7 51 and the no accident state7 52 as state contingent goods If 51 occurs then the consumer has income 1 yo 7 L C 7 7T0 and if 52 occurs then the consumer has income x2 yo 7 7T0 To solve the consumers problem we nd the rst order condition p17 ityo 7 L C 7 7T0 7 1 7p7Tuy0 7 7T0 0 use 7 we 7 e u y07LC77TC This relationship provides the same intuition that we saw graphically in lecture yesterday 1 If rm7s offer actuarially fair insurance then 7139 p and the odds ratio on the right hand side equals one Since gt 0 and u lt 0 everywhere marginal utility is always decreasing in income7 thus u x2 u x1 implies that 1 2 which means that income is the same no matter which state of the world occurs The consumer is fully insured 2 If less than actuarially fair insurance is offered7 then 7139 gt p hence the right hand side will be less than one This implies that u x2 lt u x1 thus 2 gt 1 meaning that income is higher if no accident occurs or that the consumer is less than fully insured 9 Similarly7 if more than actuarially fair insurance is offered7 then 7139 lt p hence the right hand side will be greater than one This implies that u x2 gt u x1 thus 2 lt 1 meaning that income is higher if the accident occurs or that the consumer is more than fully insured 2 Adverse Selection The setup for adverse selection is the same as the previous model7 except now there are two types of individuals with different probabilities of having an accident We assume that the individuals have the exact same expected utility function which implies that they have the same degree of risk aversion The low type has an accident with probability pL and the high type has an accident with probaility pH where pL lt pH 21 E icient Solution Perfect Monitoring First we consider the case where insurance rms can tell which type of consumer they are facing this assumption is sometimes referred to as perfect monitoring In this case the insurance agency will set a premium for the low type 7TL pL and a premium for the high type 7TH pH Note that this implies that for the low type uyo LCV 13m 1 Uyo 7 L O 7 WLO 1er which means that 1 2 for the low type they insure fully Similarly for the high type u yo 7 WHO m 1 Uyo 7 L O 7 WHO 13H which implies that the high type also fully insures Note7 however7 that income in both states is lower for the high type than it is for the low type since WHO gt 7TLO 22 Failure of Ef ciency Next we consider the case where rms can7t tell whether they are dealing with a consumer who has a low probability of getting into an accident or a high probability of getting into an accident The model is the same except now we assume that A is the proportion of the population that is low risk and the 1 7 A is the proportion of the population that is high risk The ef cient contract where the rm offers two premiums 7TL pL and 7TH pH will no longer work To see this7 note that all high risk type consumers will pretend to be low risk since in order to get the lower premium7 thus the rms pro ts will be Elprofz ts ApL7TL 71C 17pL7TLC17 ApH7TL 71C 17pH7TLC since 7TL pL the rst term equals zero Thus Elprofz ts 17 ApH7TL 71C 1 pH7TLO C1 MpHWL PH 7TL 10H7TLl 01 Ml PH 7TLl 01 MlpL le lt 0 meaning that the rms will choose to go out of business hence the ef cient contracts outcome that we found under perfect monitoring is not an equilibrium 23 Other Pooling Equilibria Are there any coverage level and premium combinations WP OP that an insurance company could offer to both types and still make a pro t in a competitive industry No For the rm to be able to earn zero pro ts and stay in business it must be able to subsidize its losses from the high risk types with gains from the low risk types However pH gt pL implies that the MRS of the high types is always greater than the MRS of the low types Thus for any possible pooling contract a rival rm can o fer another contract that is better for the low types but worse for the high types This contract will pull low type business away from the rst rm thus causing them to earn negative pro ts and go out of business This argument can be made for every possible pooling contract thus there are no equliibirum pooling contracts Draw Graph 24 Separating Equilibria Even though pooling equilibria do not exist there are separating equilibria where one pre mium coverage combination is chosen by the low risk types 7T5 CE and a different combi nation is chosen by the high risk types Wfl For a separating equilibrium to be stable it must be the case that the high risk types have no incentive to select the contract intended for the low risk types and the rm o ereing the contracts must earn expected pro ts 2 0 so it will stay in business Competition among rms for the high risk consumers will ensure that they are able to buy their ef cient contract thus W Cf1 pHL To ensure that the high risk types don7t want to switch to the contract intended for the low risk types we must make sure that the expected utility of the high risk types is lower when they choose MEGS than it is when they choose 105111 the contract intended for them Thus the boundary for a feasible separating contract for the low risk type is implicitly de ned by the equation pHuyo L 05 W505 1 PHuyo 7Tfo pHuyo 7 L L pHL 1 pHuyo pHL yoii DHId The other restriction comes from the fact that rms must be making at least zero expected pro ts The rms expected pro ts are Eipmfz39ts Aipmrf 1Oflt1 e pmfofi 0 AOflPLWf 10L f i mel AOlef PLl Z 0 Thus zero expected pro ts occurs when TIE pL making insurance actuarially fair Substi tuting back into out condition for the separating equilibrium to be feasible we get pHuyo L Of pLCLS 1 pHuyo pLOf uyo pHL an equation which implicitly de nes Cf in terms of known parameters Economics 101A Lecture 4 Stefano DellaVigna January 29 2009 Outline 1 Convexity and concavity 2 Constrained Maximization 3 Envelope Theorem 4 Preferences 1 Convexity and concavity 0 Alternative characterization of convexity o A function f twice differentiable is concave if and only if for all 90 the subdeterminants of the Hessian matrix have the property H1 S O H2 Z 0 H3 S O and so on o For the univariate case this reduces to f S O for all 90 o For the bivariate case this reduces to fag 96 S O and 2 3 fyy lt 2 0 o A twice differentiable function is strictly concave if the same property holds with strict inequalities 0 Examples 1 For which values of a b and c is f ax3 13902 cm d is the function concave over R Strictly concave Convex 2 Is fxy x2 y2 concave o For Example 2 compute the Hessian matrix fa 7fy 3330 7 79 fl 9513 7 gay Hessian matrix H I H lt 773733 ITy y 0 Compute H1l 3233 and lH2l mm 99 lt gt2 339 Why are convexity and concavity important Theorem Consider a twice differentiable concave convex function over C C R If the point X0 satisfies the fist order conditions it is a global max imum minimum For the proof we need to check that the second order conditions are satisfied These conditions are satisfied by definition of con cavity We have only proved that it is a local maximum 2 Constrained Maximization Ch 2 pp 36 42 38 44 9th Ed 0 So far unconstrained maximization on R or open subsets o What if there are constraints to be satisfied 0 Example 1 maxyac y subject to 3x y 5 0 Substitute it in maxing a gtlt 5 396 0 Solution 90 0 Example 2 maxing my subject to ac expyy expx 5 0 Solution Graphical intuition on general solution Example 3 maxing fxy a gtlt y st hxy 2 y2 1 0 Draw 0 hxy 2 y2 1 Drawxgt1ltyKwith KgtOVaryK Where is optimum Where dydm along curve my 2 K equals dydm along curve 902 y2 1 0 Write down these slopes Idea Use implicit function theorem 0 Heuristic solution of system rggyx fy st hxy 0 0 Assume continuity and differentiability of h hy7 0orh7 0 o Implicit function Theorem Express y as a function of 90 or 90 as function of y 0 Write system as maxg fxgx fmgx Wm 51 o What is 839 63 0 Substitute in and get O or f 9 hamp79 fg 9 h 79 o Lagrange Multiplier Theorem necessary condi tion Consider a problem of the type mlrpfgn f 961 962 m an p hl 1 2 an p 0 h2 17 27 397 71 0 hm 1 2 an p 0 with n gt m Let X Xp be a local solution to this problem 0 Assume f and h differentiable at 90 the following Jacobian matrix at X has maximal rank ah ah Tm X a711 X J ah ah Tm X a X 0 Then there exists a vector A A1 gtm such that X A maximize the Lagrangean function Lx A fx p ikg h x P j o Casen2m1 0 First order conditions are afXpgt3hXp0 8967 8907 fori12 o Rewrite as fml hml 1 12 h mQ o Constrained Maximization Sufficient condition for the case n 2m 1 o If X satisfies the Lagrangean condition and the de terminant of the bordered Hessian ah 8h 0 X X 8h 8 L a L ah 82L 82L 8 2X 83318332 X 83028332 X is positive then X is a constrained maximum 0 If it is negative then X is a constrained minimum 0 Why This isjust the Hessian of the Lagrangean L with respect to A 1 and 2 Example 4 maxm x2 yy2 st x2y2 p 0 maxmy 2 96y 242 M962 242 p Foc with respect to ac Foc with respect to y Foc with respect to A Candidates to solution Maxima and minima 3 Envelope Theorem II o Envelope Theorem Ch 2 pp 42 43 44 9th Ed 0 Envelope Theorem for Constrained Maximiza tion In problem above consider Fp E fxp p We are interested in dFpdp We can neglect in direct effects 3fXpp m A8hjXPJP dpi 8197 320 7 3197 Example 4 continued maxing 2 my y2 st df 9619 yp dp Envelope Theorem 4 Preferences 0 Part 1 of our journey in microeconomics Consumer Theory 0 Choice of consumption bundle 1 Consumption today or tomorrow 2 work study and leisure 3 choice of government policy 0 Starting point preferences 1 1 egg today gt 1 chicken tomorrow 2 1 hour doing problem set gt 1 hour in class gt gt 1 hour out with friends 3 War on Iraq gt Sanctions on Iraq 5 Next Class 0 Properties of Preferences 0 From Preferences to Utility 0 Common Utility Functions Department of Economics Microeconomic Theory University of California7 Berkeley Economics 101A March 147 2007 Spring 2007 Economics 101A Microeconomic Theory Section Notes for Week 9 1 Spence s 1973 Job Market Signalling Game This example is taken from the book Game Theory for Applied Economists77 by Robert Gibbons It is an example of a dynamic game with incomplete information Although I will not formally de ne the equilibrium7 the sketch I provide is based upon an equilibrium concept called Perfect Bayesian Equilibrium 11 Setup There are two possible types of workers High productivity workers denoted H and low productivity workers denoted L 1 Nature determines whether the productive ability of the worker7 77 77 H with proba bility q and 77 L with probability 1 7 q note 0 lt q lt 1 2 The worker learns his or her type and chooses how many years of schooling to complete S 2 0 3 The labor market77 observes the workers level of schooling but not the workers type and makes a wage o er equal to his or her expected productivity 4 The worker accepts the o er Note the worker will always accept the o er since w 7 C777 S gt 0 7 C78 The payo to the worker will be w 7 C78 The term C77S represents the cost both mental and monetary that a worker of type 77 must incur to complete S years of schooling The payo to the rm will be y777 S 7 w if the rm hires the worker or 0 if the rm does not hire the worker 12 Assumptions For a Signalling Equilibrium The critical assumption is that the marginal cost of schooling is greater for the low type of worker than for the high type This condition can be expressed as 60 L S 6 H S lt gt gt clt gtVS 65 65 Graphically7 the workers indi erence curves can be represented in 7718 space The above condition means that for particular level of schooling a type L worker will always have a steeper indifference curve than a type H worker The implication is that type L workers will need a larger rise in wages to compensate them for an increase schooling level from S0 to S than is required by type H workers Draw Graph A second assumption is that competition among rms drive their expected pro ts to zero which is why the worker is o fered a wage equal to his or her expected productivity Let MnlS denote labor markets7s beliefs about whether the the worker is type 77 conditional on observing that he or she has gone to school for S years Speci cally the market believes that a worker with S years of schooling is type H with probability MHlS Under this assumption the expected wage conditional upon years of schooling can be written as wS MHl5yH7 5 i 1 MHl5yL75 13 Complete Information Case To analyze the equilibria let7s rst think about how the game would play out under complete information that is to say if the market knew the type of worker it was offering a wage to In this case MS 077 5 facing this wage the worker would choose S to solve msaxym 5 i 0077 5 let S77 denote the optimal level of schooling for a worker of type 77 Then w77 y77 S77 This solution can be represented graphically Draw graph 14 Incomplete Information Case Now we will revert back to the initial setup with incomplete information In this case the market knows that the initial probability of a type H worker is q Then based upon the level of schooling that the worker chooses the market will update its assessment of the probability that the worker is type H It is assumed that the market uses Bayes7 Rules to update its belief uHlS about the probability that the worker is type H There are several possible types of equilibria in this model pooling separating or sig nalling and hybrid We will focus on the separating equilibrium but rst mention the pooling equilibrium In the pooling equilibrium the signal years of schooling S does not reveal any new information to the market thus the markets belief remains uHlS Q In the separating equilibrium the signal is fully revealing meaning that after observing the level of schooling that at worker has chosen the market will know the workers type with certainty Two cases are possible The rst case called the no envy77 case occurs when it is too expensive for the type L workers to aquire schooling level SH even though doing so would allow them to masquerade as a type H worker and earn wH This occurs when7 wL CL7 STD gt wH CL7 5H Draw graph The second case is more interesting In this case7 the type H worker must invest in extra schooling relative to both the perfect inforrnation case and the no envy case in order to dissuade the type L worker from masquerading as a type H worker We will denote this new equilibrium choice of education for the type H worker as 5 gt SH Draw graph One speci cation for the markets belief that supports this equilibrium is that the worker is type H if S 2 SS and type L if S lt 55 The markets beliefs are thus 7 0 ifSltSS Hl5 1 ifSZSS the markets strategy will thus be to offer wage w 7 yLS ifS lt55 7 yHS ifS 2 SS Finally7 we can brie y mention the hybrid equilibria In these equilibria7 one type chooses one level of schooling with certainty while the other randornizes between pooling with the rst type and separating from the rst type


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