老王阅读经典之： 经济数学讲义 著者：Mobius 层次：提高 阅读理由: 很好的经济数学即数理经济学梳理 还能加强英语
Fall 2002 Economics 2030: Mathematics for Economists Markus M. Mobius Instructor: Markus M. Mobius, Littauer 327, mobius@. Office hours: Monday, 10-12pm or appointment. Teaching Fellow: TBA. Grades problem sets and organizes recitations. Lectures: Tu-Th, 1:00-2:30pm, Littauer M15. Textbooks: Carl P. Simon and Lawrence Blume, Mathematics for Economists, Norton, 1994. (available at the COOP or ) will be used for most of the lectures. Intriligator, M. Mathematical Optimization and Economic Theory, Prentice-Hall, 1971 will be used for Dynamic Optimization. Math Prerequisites: Calculus of one variable and basic linear algebra. Functions, graphs, maxima and minima, vectors, matrices. The following Chapters of Simon and Blume review the basic material: 1, 2, , , , 4, 5, 6, 8. Math 20 is an undergraduate course at Harvard (fall semester) that covers the basics math needed for economics. Math 20 is less advanced than Ec 2030. Assignments: -There are six problem sets. Count for 60% of the final grade. Students are encouraged to form groups to solve problem sets. th- Final exam scheduled on January 21. Length: 1h30. Counts for 40% of the final grade. - Drop-off and pick-up location: box outside my office.
GOALS OF THE CLASS: The first goal of the class is to teach the key mathematical tools that are useful to follow the Econ PhD classes. Most emphasis is put on math tools used in the first year Micro sequence, especially Microeconomics (Green) and General Equilibrium (Calvet). However, linear algebra should be useful to Econometrics as well and many of the maximization tools and differential equations are commonly used in the Macro sequence. The second goal of the class is to show real (but relatively simple) mathematical proofs so that you can get familiar with mathematical reasoning. This should be helpful to understand proof arguments in micro or econometric classes. Some of the material taught in this class might not appear at first sight to be useful to understand economics. This is because math is a logical construction and must be presented in the right order. The lectures have been built so that you can get exposure to the key tools over the semester. More advanced students may skip the first 6 lectures. Students less interested in macro can skip the last 6 lectures. COURSE OUTLINE: Section 1: Review of Basic Concepts (5 lectures) Introduction, Sets, Numbers, and Proofs SB 1, A1 One-variable Calculus: Sequences, Limits, Continuity. SB 2, 3, 12 One-variable Calculus-Applications: Chain Rule, Taylor Expansions, SB 4, 29, One-variable Calculus, Concavity, Convexity, SB 29 Linear Algebra: vector, matrices, SB 6, 7, 8. Linear Algebra: subspaces, rank, determinants SB 9, 10, 11. Section 2: Calculus of Several Variables (4 lectures) Basic topology in vector spaces, SB 12, 29. Calculus of Several Variables, SB 13, 14, 15 Convex and Concave functions of Several Variables, SB 21 Implicit functions and Derivatives, SB 15, 30 Section 3: Optimization (4 lectures) Unconstrained optimization SB 17 Constrained Optimization SB 18 Constrained Optimization SB 19 Homogeneous functions, Quasiconcavity SB 20, 21 Economic Applications (consumer theory) SB 22 Section 4: Linear Algebra and Differential Equations (4 lectures)
Matrices, Eigenvalues and Eigenvectors SB 23 Subspaces and Applications SB 28 Symmetric Matrices, Quadratic Forms SB 23 Ordinary Differential Equations SB 24 Ordinary Differential Equations: systems SB 25 Section 5: Dynamic Optimization (4 lectures) Linear Programming, Intriligator 5 Control Problem, Intriligator 11 Dynamic Programming, Intriligator 13 Maximum Principle, Intriligator 14 Section 6: Probability Theory (3 lectures) Probability Spaces, Grimmett and Stirzaker Random Walks, Grimmett and Stirzaker Markov Chains, Grimmett and Stirzaker Supplementary Reading List Dixit, Avinash, Optimization in Economic Theory, Oxford University Press, 1990. Extensive analysis of most optimization problems in economics. Develops many useful economic intuitions. Requires a fairly good math background. Grimmett, Geoffrey and David Stirzaker, Probability and Random Processes, Oxford University Press,1992. This is a great introduction to probability theory, random processes and martingales. It has nothing on mathematical statistics, however – so it won’t be useful for econometrics. Intriligator, M. Mathematical Optimization and Economic Theory, Prentice-Hall, 1971. Classical reference on optimization and economic applications. Very well written and easy to follow. Janich, Klaus. Linear Algebra, Springer-Verlag, 1994. Nice and short book on linear algebra. Starts slowly and covers all that is needed in linear algebra for economics and econometrics. The lecture notes borrow heavily from it. Kelley, ., General Topology, Van Nostrand, 1955. Classical reference on Topology. Useful if you want to learn more topology in a very rigorous way. Lang, Serge, Undergraduate Analysis, Springer-Verlag, 1995.
Excellent Analysis textbook. Very rigorous, more advanced than Blume and Simon. The lectures notes borrow heavily from it. Lucas, Robert and Nancy Stokey, Recursive Methods in Economics Dynamics, Harvard University Press, 1989. Best reference on dynamic optimization. This is a must if you plan to do Real Business Cycle macro simulations. Nice mathematical appendix. Rudin, Walter, Principles of Mathematical Analysis, McGraw-Hill, 1964. Short and very concentrated. Contains virtually all the math results that are useful for Ph. D. students. Rudin, Walter, Real and Complex Analysis, McGraw Hill, 1987. Advanced excellent textbook on measure theory, integration, and functional analysis. Simon, Carl and Lawrence Blume, Mathematics for Economists, Norton, 1994. Key textbook for the class. I find it too long and chatty relative to the material it covers. The lectures are more condensed and cover more topics. Solow, Daniel, How to Read and Do Proofs: An Introduction to Mathematical Thought Processes, Jon Wiley and Sons, 1990. Very simple book that teaches the basics of mathematical proofs. Useful for students that have very little experience with math proofs. rdSpivak, Michael, Calculus, 3 edition, 1994. Classical undergraduate textbook in calculus. Goes very slowly. Sydsaeter, Knut and Peter Hammond, Mathematics for Economic Analysis, Prentice-Hall, 1995. A big textbook very similar to Simon and Blume Takayama, A. Analytical Methods in Economics, University of Michigan Press, 1993. Advanced textbook that provides an elaborate mathematical analysis of micro-economics and general equilibrium.
IntroductiontoEcon2030September17,20021TheRoleofMathematicsinEconomicsEconomicsandsociologyusedtobedefinedintermsoftheirobjectofanalysis:,growth,,aneweconomicmethodologywasdefinedaroundmathematics:•economicandsocialsystemsconsistofindividualagents(consumers,firms,govern-ments,tradeunionsetc.)•eachagenthasstablepreferenceoverasetofoutcomesandaccesstoanumberofresources•eachagenttakesactionswhichmaximizehisobjectivefunction•agentsactionscaninfluenceeachother;wesaythataneconomyisinequilibriumifagents’perceptionsofothertheactionsofothersequaltheactualactionsofthoseagents1
(CarlMenger),,sociologywiththeexplanationofthoseends(LionelRobbinsandTalcottParsons).However,duringthelast50yearseconomics,orbettereconomicmethodology,,-styled’Economicimperialists’,ledbyGaryBeckerandothers,proclaimthat’theeconomicapproachisacomprehensiveonethatisapplicabletoallhumanbehavior—allhumanbehaviorcanbeviewedasinvolvingparticipantswhomaximizetheirutilityfromastablesetofpreferencesandaccumulateanoptimalamountofinformation’(Becker,TheEconomicApproachtoHumanBehavior).2NumbersandSetsThebasicbuildingblockofallnumbersarethenaturalnumberswhichwedenotewithℵ:ℵ={1,2,3,4,..}(1)
AsubsetAofsomesetBiswrittenA⊂,ifwewanttowritethesubsetofallnaturalnumbersbetween1and100wecouldwrite:A={1,2,..,100}={x∈ℵ|1≤x≤100}(2):n{}mQ=x=|m,n∈Zandn =0(3),:,forexample,thefollowinglemma:n(n+1):[drawfigure]Let’sdenotes=1+2+..+n(4)nFromthefigurewecanseethat:22∗s−n=n(5)
,,assumethatyouwrotedownthesumofthefirstnintegersforn=1,2,3andgotthesequence1,3,6,10...Youplotitonacomputer,anditlookslikeaquadraticn(n+1)
-sically,youassumetheoppositeoftheclaimyouwanttoproveandshowthatitleadstoacontradiction.√2Lemma3Thereisnorationalnumberxsuchthatx=2().mAssumeitwouldbeafractionsuchthatm,:n22m=2∗n(7),mhastobeanevennumberandwecanwritem=2m˜.Wethenobtain:222∗m˜=n(8)√
LectureII:,|a−b|.∈≤bforalla∈S.[biscalledanlowerboundforSiffb≤aforalla∈S.]Definition2IfbisanupperboundforSandnoelementsmallerthanbisanupperboundforS,:•SetS={1,2,},.•S={,,,....},1/3=..
,,,,:S={x∈Q|x<2}hasnoupper√:•LetS={a|i∈I}.Ifa≤Mforanyi∈Ithensupa≤•Ifa<Mforanyi∈Ithensupa≤{a|i∈I,j∈J}beaboundedsetofrealnumbers,then:ijsupinfa≤infsupaijijjjii∗Proof:,∗infa≤a(1)-handsidedominatestheleft-handsidewemusthave(check!:supinfa≤∗supa(2)ijiijji2
∗:supinfa≤infsupa∗(3)ijiij∗:N→(u)nn∈NExamplesare:•u=nn•u=1/nnKeydefinitionoflimits:Definition6Wesaythatasequenceofrealnumbers(u)hasalimitl∈Riff:nn∈N•forany >0,thereissomenumberNsuchthatn>N=⇒|u−l|< n•Wesaythatasequenceofrealnumbers(u)hasforlimit+∞iff:nn∈N∀A>0,existsNsuchthatn>N=⇒u>A(4)nInformally,[drawpicture]Notation:3
•Becarefulwiththeuseofquantifiers.•Weindicatealimitofasequenceeitherbyu→lorlimu=→∞n11Example7Thefollowingholds:→0ifα>0,→+∞ifα<0ααnnαProof: > which1/αcanberewrittenasn>(1/ )=N. nExample8u=(−1),thatis(u)=(1,−1,1,−1,1,−1,...).:Supposeitdoesconvergetosomel,andtake =1/|u−l|<1/4,andtherefore|1−l|<|−1−l|=|1+l|<(triangleinequality).:′′•(Uniqueness)Ifu→landu→lthenl=′′′•Ifu→landv→lthenwehaveu+v→l+landuv→′′l>0,u/v→l/•(SandwichTheorem)If|v|<uandu→0,thenv→φ:N→Nisincreasing,y=xdefines(y)asanφ(n)nsubsequenceof(x).(u)nkkorevenmoresimple(u′).nProposition11If(x)hasforlimitl,
Examples:(x),(x2)2nnVeryimportantproperty:Proposition12Thesequenceudoesnotconvergetoliffthereissome andsomen0subsequenceuthatremainsoutsidetheinterval(l− ,l+ ).φ(n)00⇐Thisiseasy.⇒ >0suchthatforanyNthereisn0somen>Nsuchthat|u−l|≥ .=1,thenthereissomen=φ(1)>1suchthat|u−l|≥ .phi(1)0TakeN=φ(1)+1,thenissomen=φ(2)≥N>φ(1)suchthat|u−l|≥ .φ(1)0SoIcanconstructasubsequenceuoutsidetheinterval(l− ,l+ ).φ(n)(u)isincreasingifu≤uforallm≤:Ifasequence(u)isincreasingitconvergesneithertoafinitelimitorto+∞.Weareinthefirstcaseiffthesequenceisboundedfromabove,.∃M/u<::BoundedLetl=:nnnthenthereisan >0sothatthereisasubsequenceuoutside(l− ,l+ ).0φ(n)00Infact,thesubsequenceisbelowl− becauselisanupperboundof(u).0nTakeanyn,thenn≤φ(n)andsou≤u<l− butthismeansthatl− isnφ(n)
CaseII:UnboundedForanyA>0,thereissomeNsuchthatu>AbutasN(u)isincreasing,infactforalln≥N,u≥u>-Weierstrass:Ifasequence(u)isbounded(thereissomeMnsuchthat|u|<Mforalln)then(u)(moreadvanced):(u)hasamonotonoussubsequence(eitherincreasingorndecreasing).Suppose(u),lookforn1thefirstelementu>u,thentakethefirstu>u,,,u≤uforanyn>=φ(1).-1φ(1)+(u)isCauchyifforall >0thereexistssomeNsuchthatnforalln,m>Nwehave|u−u|< .:
Proof(moreadvanced):(1)FirstweshowthataCauchysequenceisbounded.(2)Itthenhasaconvergentsubsequence.(3):•openinterval(a,b)={x∈R|a<x<b}•closedinterval[a,b]={x∈R|a≤x≤b}•semi-openand-closedintervals:(a,b],[a,b)•intervalsinvolvinginfinity:(−∞,b],(a,+∞):X→Riscontinuousinx∈X0iff:Forall >0thereexistsη>0suchthatforallx∈X,|x−x|<η=⇒|f(x)−f(x)|< ,∈Xiffforanysequence(x)inX,(x)→x0nn0implies(f(x))→f(x)n0Proof:=⇒iseasy⇐=
>0suchthatforany00h=1/n,Icanfindxin(x−h,x+h)suchthatf(x)outside(f(x)− ,f(x)+n00n000 ).SoIhaveconstructedasequencexconvergingtoxsuchthatf(x)doesnot0n0nconvergetof(x).0So,:Proposition20iffandgarecontinuous,f+g,fg,f/g,:Sum,product,(x),thenf◦giscontin-00uousatx::Supposex→xtheng(x)→g(x)becausefcontinuousatxandthenn0n00f(g(x))→f(g(x))becausefcontinuousatg(x).Sof◦gcontinuousatx:n000Anotherwayofrewritingthat:limg(f(x))=g(limf(x))x→xx→:ln,exp,cos,sin,
-MaxTheoremLetfbeacontinuousfunctiononaclosedinterval[a,b].Thenthereissomecin[a,b]wherefreachesitsmaximumandsomedin[a,b]:First,:thenthereexistssomesequencef(x)≥-Weierstrassxhasaconvergentsubsequencex′.=sup{f(x),x∈[a,b]}andxsequencesuchthat|f(x)−M|<1/nforanynnn>′convergestoanumbercandf(x′)convergestof(c),f(c)=’no-gap’[a,b].Thenforanymintheinterval[f(a),f(b)]or([f(b),f(a)]),thereissomec∈[a,b]suchthatf(c)=:Supposef(a)<f(b).LetS={x∈[a,b]|f(x)≤m},=,ofcoursef(x)≤msof(c)≤>(x)>msothelimitf(c)≥(c)=[a,b].f([a,b])=[m,M].
Proposition26Takef:R→R,(x)=+∞andlimf(x)=x→+∞x→−∞−∞,thenf(R)=:ExistenceofanequilibriumpriceIfSupply(0)=Demand(+∞)=0andSupply(+∞)=Demand(0)=+∞thenthereexistssomepricepsuchthatSupply(p)=Demand(p).(5)[a,b]thenf([a,b])=[f(a),f(b)]andfisaone-to-onefunctionfrom[a,b]onto[f(a),f(b)].−1Theinversefisaone-to-onefunctionfrom[f(a),f(b)]onto[a,b].−:Assumethatlimf(x)=,(x),too.−
LectureIII:,20021DifferentiationUptonowweconsideredlimitsofsequencesandlookedatn→∞.(x)=f(a).x→aIfthefunctionisnotdefinedatawecanlookatleftandrightlimits(seeclass).f(x+h)−f(x)Letusintroducethefunctionφ(h)=.(x+h)−f(x)′Definition1Iflimexists,wenoteitf(x)=df/dx(x)thederivativeoffh→0hinx,f(1/n)−f(0)Example2x −→|x|iscontinuousin0butnotderivable:comparethelimitsofand1/nf(−1/n)−f(0)−1/nAnotherwaytowritethederivativeis:′′f(x+h)=f(x)+hf(x)+h (h)=f(x)+hf(x)+o(h)(1)1
o(h)isaquantitysuchthath (h)=o(h)where (h)→0ash→0o(h)o(h)=h→0h0.•Thisisausefulnotationtosaveonspace.•Economistsalsooftenusephysicists’notationwheredxrepresentsasmallchangearoundx.′′f(x+dx)=f(x)+f(x)dx+o(dx)=f(x)+f(x)dx(2)Proposition3•derivable⇒continuous′′ffg−gf′′′′′′′•(f+g)=f+g,(fg)=fg+gf,()=2gg′′′•(f◦g)(x)=f(g(x))g(x)′′n′n−1•(ln)(x)=1/x,(exp)=exp,(x)=nx,(sin)’=cos,(cos)’=-sinα′′α−1•(f)=αff′−11•f(x)=′−1f(f(x))Proof:Let’:′f(x+h)=f(x)+hf(x)+o(h(3)′g(x+h)=g(x)+hg(x)+o(h4Note,thatwethesamesameo(h)eventhoughthetwoo’:′′f(x+h)g(x+h)=[f(x)+hf(x)+o(h)][g(x)+hg(x)+o(h)]=(5)′′f(x)g(x)+hf(x)g(x)+hg(x)f(x)+o(h)(6)
(x)Whenapplyingthechainrulewatchoutthaty=g(x)isintheinteriorofthedomainoff(y).Forinversefunctions,notethat−1f(f(x))=x(7)=’’strueforn+:y=f(u)andu=g(x)then,dy/dx=dy/du·du/(a,b)andletc∈(a,b)beamaxforf:f(c)≥f(x)foranya<x<b.′Thenf(c)=0Proof:f(c+h)≤f(c)so(f(c+h)−f(c))/h≤0forh>0sorightlimitislessorequal′tozero,sof(c)≤0.′Forh<0,getf(c)≥−1Note,
,thefunctionf(x)=|x|:Letfbecontinuouson[a,b]anddifferentiableon(a,b)such′thatf(a)=f(b)thenthereisc∈(a,b)suchthatf(c)=:,-Maxtheoremithasaninterior′extremumsuchthatf(c)=[a,b]andbedifferentiableon(a,b).Thenthereisc∈(a,b)′suchthatf(b)−f(a)=f(c)(b−a).Proof:Takeg(x)=f(x)−(f(b)−f(a))(x−a)/(b−a),.′Corollary8Letfbecontinuouson[a,b]andbedifferentiableon(a,b).Assumef(x)>0forallxthenfisstrictlyincreasing.′(equivalentresult:f(x)<0impliesfstrictlydecreasing).→[a,b]andbedifferentiableon(a,b).′1)f(x)≥0forallxifffisweaklyincreasing.′2)f(x)≤0forallxifffisweaklydecreasing.′3)f(x)=
:′′′′(2)(n)(f)=f=fandsoon,fforthen−(I).Setofrealfunctionsthathavenderivativesandwhosen−thderivativeiscontinuousnisdenotedbyC(I).∞C(I)forfunctionsthatareinfinitelyderivable(setofsmoothfunctions).Properties:nnf,g∈C(I)thenfg,f+g,λfinCand(n)Leibnizformula(fg)=...(n)nn+1(1/x)=(−1)n!/xk(n)k−n(x)=k(k−1)...(k−n+1)xfork≥nand0fork<,:x01•exp(x)=esuchthate=1,e=e=...,x+yxy•e=eex•e≥0.•Slope1at0.•exp’=exp•Naturallogarithm,logorlnistheinverseofexp′•log(x)=1/xdefinedon(0,∞).5
2BasicIntegrationAfunctionon[a,b]boundedintervalispiecewisecontinuousifitiscontinuouseverywhereexceptonafinitenumberofpointsinIandthatateverypointwhereitisnotcontinuousitadmitsleftandrightlimits.(drawpictures).Definition10RiemannIntegrationForpiecewisecontinuousfunctionsonintervalI,wedefineforanyaandbinI(a<b):∫nb∑b−akf(x)dx=limf(a+(b−a))(8)n→+∞annk=1∫∫abBydefinition,Isay:f(x)dx=−f(x),:∫∫∫bbb•(f+g)=f+gaaa∫∫•λf(x)dx=λf(x)dx∫∫∫bcb•f(x)dx=f(x)dx+f(x)dxaacIfwefixsomeawecancalculatethefollowingfunctionF(x):∫xF(x)=f(x)dx(9)a6
ByvaryingthevalueawegetaclassoffunctionsF(x)whichwecalltheindefiniteintegraloffandwriteas∫F(x)=f(x)dx(10)Theorem13TheindefiniteintegralF(x)isdifferentiableiffiscontinuousandsatis-fies:′F(x)=f(x)(11)Proof:,thatF(x):∫x+hF(x+h)−F(x)f(x)dxx=(12)hhItiseasytoseeinapicture,thattheareaofthissmallintegralisapproximatelyf(x)×
LectureIII:,20021BasicIntegrationAfunctionon[a,b]boundedintervalispiecewisecontinuousifitiscontinuouseverywhereexceptonafinitenumberofpointsinIandthatateverypointwhereitisnotcontinuousitadmitsleftandrightlimits.(drawpictures).Definition1RiemannIntegrationForpiecewisecontinuousfunctionsonintervalI,wedefineforanyaandbinI(a<b):∫nb∑b−akf(x)dx=limf(a+(b−a))(1)n→+∞annk=1∫∫abBydefinition,Isay:f(x)dx=−f(x),
Proposition3Usefulpropertiesoftheintegral:∫∫∫bbb•(f+g)=f+gaaa∫∫•λf(x)dx=λf(x)dx∫∫∫bcb•f(x)dx=f(x)dx+f(x)dxaacIfwefixsomeawecancalculatethefollowingfunctionF(x):∫xF(x)=f(y)dy(2)aTheorem4FundamentalTheoremofCalculusThefunctionF(x)isdifferentiableiffiscontinuousandsatisfies:′F(x)=f(x(3)Proof:,thatF(x):∫x+hF(x+h)−F(x)f(x)dxx=(4)hhItiseasytoseeinapicture,thattheareaofthissmallintegralisapproximatelyf(x)×,thattherearemanypossibleF(x),allF(x)differonlybyaconstant(easytoshow).Itseemsthatintegrationisthe’opposite’(x)a′primitiveifF(x).Thefundamentaltheoremshowsthattheintegralfunctionisalwaysa∫,thatanyprimitiveG(x)differsfromF(x)=f(y)[a,b]andthatthereexists′′functionsFandGsuchthatF(x)=G(x)=f(x).ThenFandGdifferonlybyaconstant,(x)=G(x)+
′Proof:DefineD(x)=F(x)−G(x).WeknowthatD(x)=(a,b)andcontinuousover[a,b].Henceitisconstant(useMVT).Combiningallresultswehaveaneasywaytocalculateanintegralofsomecontinuousfunctionf:Theorem6AssumeGisaprimitiveoffoveraninterval[a,b].Thenthefollowingholds:∫bf(y)dy=G(b)−G(a(5)[a,b]suchthatf(x)≤g(x)forallx∈[a,b]then∫∫bbf(x)dx≤g(x)dxaaProof:Lookatthedifferenceg−∫∫bb|f(x)dx|≤|f(x)|dxaa∫b|f(x)dx|≤sup|f(x)|(b−a)a∫∫bb|f(x)g(x)dx|≤sup|g(x)||f(x)|dxaaProof:Takeg(x)=|f(x)|≥0continuousonsomeinterval[a,b]andf(c)>0forc∈[a,b]thenthe∫bintegralf(x)
3UsefulIntegralsandTricks∫b•(1/x)dx=ln(b)−ln(a)a∫b•exp(x)dx=exp(b)−exp(a)a∫α+1α+1bαb−a•tdx=aα+1•Theindefiniteintegral(orprimitive)oflog(x)isxlog(x)−•Theprimitiveof1/(1+x)isarctan(x):LetJandJbeintervals(withmorethanonepoint):12′Letf:J→Jandg:J→,binJ,1∫∫bf(b)′g(f(x))f(x)=g(u)duaf(a)′Proof:◦fistheprimitiveofx→g(f(x))f(x)(chainrule).BothtermsareequaltoG(f(b))=G(f(a)).∫2221xxxExample11ConsidertheintegralI==eanddu=∫ewecanwriteI=du=e−:∫∫bb+αf(t+α)dt=f(u)du(6)aa+α∫∫bαbf(u)f(αt)dt=du(7)aαaα4
Proposition12Integrationbypart:′′SupposeFandGaredifferentiableon[a,b].SupposeF=fandG=:∫∫bbf(t)G(t)dt=[F(b)G(b)−F(a)G(a)]−F(t)g(t)dt(8)aa′Youcanrederivethisformulaquicklybynoticingthat[FG]=fG+∫ba∫b′d[(x)f(t)dt]/dx=b(x)f(x)a∫b′d[(x)f(t)dt]/dx=−a(x)f(x):[a,b]×I→R(t,x)→f(t,x)1)fiscontinuousin[a,b]×I2)∂f/∂xexistsandiscontinuousin[a,b]×I∫b1Thenx→F(x)=f(t,x)dtisCin[a,b]and:a∫b′F(x)=∂f/∂x(t,x)(x+h)−F(x)Sketchofaproof::h∫bF(x+h)−F(x)f(t,x+h)−f(t,x)=dt(9)
:Proposition14Leibnitzrule:∫∫b(t)b(t)d∂f(x,t)′′f(x,t)dx=dx+b(t)f(b(t),t)−a(t)f(a(t),t)(10)dta(t)a(t)∂t∂FProof:Let’slookataparticularprimitiveF(x,t)off(x,t)suchthat=∂xthefundamentaltheoremofcalculuswecandeducethat∫b(t)f(x,t)dx=F(b(t),t)−F(a(t),t)(11)a(t)Wecanderivethisfunctionwithrespecttotusingthechainruleandfind:∫b(t)d∂F(b(t),t)∂F(b(t),t)∂F(a(t),t)∂F(a(t),t)′′f(x,t)dx=b(t)+−a(t)−dta(t)∂x∂t∂x∂t(12)4ImproperIntegrals∫(t)dtexists,A→+∞a∫+∞wenoteitf(t),sometimesafunctionf(x)isunboundedoveranintervala∫b[a,b]butthelimitlimf(x)dxexists. →0a+ ∫−rt−ra+∞−rte+∞eExample15edt=[−]=aarr√∫∫+∞+∞−rt−tExample16Let’scomputetedt(useanintegrationbypart)andedt(usea0√thechangeofvariableu=t)αExample17Animportantfunctionineconomicmodelsisf(x)=1/-ticularlyinterestedwhenthefunctionisintegrableat0and∞.6
-tionfunctionF(x)isdefinedas:∫xF(x)=f(t)dt(13)−∞TheexpectationE(X)is∫∞E(X)=xf(t)dt(14)−∞andthemthmomentis∫∞mE(X)=xf(t)dt.(15)2−∞2ThevarianceisdefinedasVar(X)=E(X)−E(X).’sLawTheoneregularitywhichpopsupinlotsofdatasourcesinsocialsciencesisZipf’sLaworPareto’(1902-1950)foundthatifoneranksEnglishwordsinatextbyfrequencyofoccurrencethenthefrequencyXis−broughlyproportionaltoanegativepowerofitsrankS,∼’slawholdsfamouslyforcities(seeappendixofthesenotes).Gabaix(QJE,1998)hasshownthatZipf’slawwithunitycoefficientcanbeexplainedbyastochasticgrowthprocessknownasGibrat’,theParetodistributionhassupport[x,∞]anddensityminaxminf(x)=a(16)a+1xfora>:()axminF(x)=1−(17)xTherankSofobservationxissimplyS=1−F(x).7
5TaylorExpansionsIdeaistogeneralizetheexpansionoffunctionsaroundapointx:′f(x+h)=f(x)+f(x)h+o(h).+:I→RbeafunctionofclassC,Forxandx+hinI,wehave,′′(n)f(x)f(x)′2nf(x+h)=f(x)+f(x)h+h+..+h+Rn2n!where∫n1(1−t)n+1(n+1)R=hf(x+ht)dtn0n!Proof:=0,wewanttoshowthat∫1′f(x+h)=f(x)+hf(x+ht)dt(18)0Thisiseasybyusingachangeofvariablesu=x++1byintegrationbyparts:∫n1(1−t)n+1(n+1)R=hf(x+ht)dtn0n![]′∫n+11(1−t)n+1(n+1)=h−f(x+ht)dt0(n+1)![]1n+1(n+1)=hf(x)+R(19)n+1(n+1)!QEDnWecangeneralizerthenotationo(h)whichweintroducedearlieranddefineo(h)=nnnh (h)with (h)→0oro(h)/h→:8
+:I→RbeafunctionofclassC,ForxintheinteriorofI,wehave,′′(n)f(x)f(x)′2nnf(x+h)=f(x)+f(x)h+h+..+h+o(h)2n!(n+1)(n+1)Proof:LetM=supf(x)andm=inff(x)[a,b][a,b]∫n1(1−t)n+1n+1R≤hMdt=hM/(n+1)!n0n!∫n1(1−t)n+1n+1R≥hmdt=hm/(n+1)!n0n!n+1som≤R/(h/(n+1)!)≤Mn(n+1)n+1Sothereisc∈[a,b]suchthatR=f(c)h/(n+1)!.nn+1Itisclearthat|R|≤:xnn•e=1+x+..+x/n!+o(x)2n+1n•log(1+x)=x−x/2+...+(−1)x/n+...α2n•(1+x)=1+αx+α(α−1)x/2+..+α..(α−n+1)x/n!+...Remark21Note,:222222222222log(1+x+x)=(x+x)−(x+x)/2+o((x+x))=x+x−x/2+o(x)=x+x/2+o(x)(20)9
6LocalMaximaandMinimaDefinition23Afunctionfhasalocalmaximum(minimum)atcifthereexistssome neighborhoodsuchthatfreachesamaximum(minimum)over[c− ,c+ ].′Iffisdifferentiablewealreadyknowthatf(c)=′theinteriorofI,thenf(x)=-Younghelpsustocharacterizelocalextremabyusingthefollowingrelationship:′′′22f(x+h)=f(x)+f(x)h+f(x)h/2+o(h)(21)2′(x)=0forxintheinteriorofI:′′•Iff(x)>0thenitisalocalminimum.′′•Iff(x)<0thenitisalocalmaximum.′′•Iff(x)=0wecan’′′′′ofIthenf(x)≤(x)≥
Zipf in graphs, AmericaCity properUrban = + = + = = = + = = + = = + = = + =
= + = = + = = + = + = = = + = + = =
enough = + = enough datapointsIndexUnited = + = + = = = + = + = =
EmpiricalDistributionsofFirmSalesFirm Sales Rank (by SIC)17275 -- Commercial Printing305 -- Rubber Products100025050101371 -- Motor Vehicles283 -- Drugs100025050101525100100020000525100100020000Sales (Million US $).
LectureIV:,20021ConvexSetsDefinition1AsubsetSofvectorspaceVisconvexiffforanyxandyisS,andany0≤λ≤1realnumber,wehaveλx+(1−λ)y∈:•λx+(1−λ)•Convexsetscanbeflat(segmentinR)•:I−→<λ<1,f(λx+(1−λ)y)≤λf(x)+(1−λ)f(y)fisconcaveiffforanyxandyinIandany0<λ<1,f(λx+(1−λ)y)≥λf(x)+(1−λ)f(y)1
Note,(x,f(x)0and(y,f(y)).2•AnequivalentdefinitionwouldbethattheRsetdefinedby{(x,y)|y≥f(x)}isconvex:theuppergraphisconvex.•Linearfunctionsf(x)=ax+:I−→(x =y)inIandany0<λ<1,f(λx+(1−λ)y)<λf(x)+(1−λ)f(y)fisconcaveiffforanyxandyinI(x =y)andany0<λ<1,f(λx+(1−λ)y)>λf(x)+(1−λ)f(y)•Strictconvex(concave)functionsarestrictlybelow(above)thesegment[f(x),f(y)].Linearfunctionareneitherstrictlyconcavenorstrictlyconvex.•fconcaveiff−fconvex(idemforstrictly)•minofconcavefunctionsisconcave.•maxofconvexfunctionsisconvex(drawpictures).•sumofconvexfunctionsisconvex.•:I−→,foranyz<z<z1232
f(z)−f(z)f(z)−f(z)f(z)−f(z)213132≥≥z−zz−zz−z213132Proof:zliesinthesegment[z,z]thusthereissome0<λ<1suchthatz=2132λz+(1−λ)λ=(z−z)/(z−z))and1−λ=(z−z)/(z−z).32312131f(z)=f(λz+(1−λ)z)≥λf(z)+(1−λ)f(z)impliesf(z)−f(z)≥(1−2131321λ)(f(z)−f(z))(z)−f(z)≥λ(f(z)−f(z),thefunctiony→[f(y)−f(x)]/(y−x):I−→.′′′′Moreover,f(x)≥f(x)andf(x)andf(x):φ(h)=[f(x+h)−f(x)]/hforh>,weknowthatφ(h),,[f(x)−f(y)]/(x−y)≥φ(h)foranyhso[f(x)−f(y)]/(x−y)≥φ(h)decreasingash<′′′cannotbeminusinfinitebecauseitislargerthanf(x).Sof(x)≥f(x).rlrNote,
′′Whyaref(x)andf(x)weaklydecreasing?lrTakex<yandh>0smallsuchthat:x<x+h<y<y+hinI:Useinequalitiestoget:f(x+h)−f(x)f(y+h)−f(y)≥hh′′Takelimitsash→0fromabove:f(x)≥f(y).rr′′Adapttheproofwithh<0toshowthatf(x)≥f(y).QEDll′′Ifitisdifferentiableatanypointsuchthatf(x)=f(x).lr′′Functionhasakinkwhenf(x)>f(x):thiscanonlyhappenatmostacountablelrsetofpoints(whichisofmeasurezero).Therefore,aconcave(orconvex)′′f(x)≥a≥f(x),wehavef(y)−f(x)≤a(y−x)
′Proof:(y)−f(x)≤f(x)(y−x).Applyittoxandty+(1−t)x:′′f(x)−f(ty+(1−t)x)≤f(ty+(1−t)x)[x−(ty+(1−t)x)]=−tf(ty+(1−t)x)(y−x)(1)′Similarly,f(y)−f(ty+(1−t)x)≤(1−t)f(ty+(1−t)x)(y−x)Multiplyby(1−t)andtandsumtoobtain:(1−t)f(x)+tf(y)≤f(ty+(1−t)x).Importantconsequence:Property10LetIbeanintervalinRandf:I−→RarealfunctiononIdifferentiable.′fisconvexandf(x)=0thenxisglobalminimumoffonI.′fisconcaveandf(x)=:I−→.′′fisconvexifff(x)≥0foranyxinI′′fisconcaveifff(x)≤[a,b],oneofthefollowingistrue′(x)=”interiorsolution”[a,b]sothatthemaximumisf(b).(”cornersolution”).[a,b]sothatthemaximumisf(a).(”cornersolution”).5
3RiskaversionandJenseninequalityLemma13LetIbeaninterval,takeu:I→R,concaveandann-tuple(p,...,p)of1npositivenumberssuchthat:p+...+p=1(2)1nThen∀x...,x∈I:1,nu(px+...px)≥pu(x)+...+pu(x)(3)11nn11nnProof:Thiscanbeprovedbyinductiononn:px+...+px22nnu(px+...px)=u[px+(p+...+p)()](4)11nn112np+...+p2nThenexttheoremisextremelyusefulandimportant:Theorem14Jensen’sInequalityIfuconcaveandfisacontinuousfunctiononIthenthefollowingholds:∫∫u(xf(x)dx)≥u(x)f(x)dx(5)IIProof:’(suchindependenceofirrelevantalternatives,continuityetc.)allowsustorepresentpreferencesoverlotterieswithavon-Neumann-Morgensternexpectedutilityfunctionwhichisuniqueuptoapositivelineartransformation,(x,..,x)canberepresentedbyp=(p,..,p)
vNMutilityfunctionsrepresentingthesamepreferencesthenu=αv+βforsomeα,βandα>:∫∞u(f)=u(x)f(x)dx(6)∞Iftherandomvariabledefinedbyfcanonlytakefinitelymanyvalueswithprobabilityp,...pexpectedutilityiscorrespondinglydefinedas:1n∑u(f)=pu(x)(7)iiiTheinterpretationofJensen’: withprob1/2and− withprob1/:Eu=u(W+ )+u(W− )22WeknowalreadyfromJensenthatEu<u(W).Buthowfarisit:canweestimateEu−u(W)?WearegoingtodosobyusingsecondorderTaylorexpansions:Supposeuistwicederivable:2 ′′′2u(W+ )=u(W)+ u(W)+u(W)+o( )22 ′′′2u(W− )=u(W)− u(W)+u(W)+o( )27
So,byaddingthetwo:2 ′′2Eu=u(W)+u(W)+o( (8) small,:2e′′′u(W)−u(W)p=Eu=u(W)+u(W)2So:′′1u(W)2p=− (9)′2u(W)′′′−u/uiscalledtheabsolutedegreeofriskaversion.−γxConstantandequaltoγinthecaseofCARAfunctionsu(x)=−:relativeriskaversioncoefficient:′′′η=−xu(x)/u(x)Functionswithconstantrelativeriskaversiontaketheform:1−ηu(x)=x/(1−η).lossaversion:,whichiscontinuousbutnotderivable(thereisakinkinW):u(x)=W+(x−W)=xifx>WW−2(W−x)ifx<WTheideaisthatWisthe”referencepoint”
WenowhaveEu=u(W)−,
LectureV:,:elements(called’vectors’)canbeadded,:Definition1Atriple(V,+,·)consistingofasetV,amap(addition),+:V×V→V(x,y)→x+yandamap(calledmultiplication)·:R×V→V(λ,x)→λxiscalledarealvectorspaceifthefollowing8conditionshold:(1)(x+y)+z=x+(y+z)forallx,y,zinV(2)x+y=y+xforallx,yinV(3)Thereisanelement0∈V(calledthezerovector)suchthatx+0=xforanyx.(4)Foreachelementx∈V,thereisanelement−x∈Vsuchthatx+(−x)=
(5)λ(µx)=(λµ)xforallλ,µ∈R,x∈V.(6)1x=xforallxinV(7)λ(x+y)=λx+λyforallλinR,x,yinV.(8)(λ+µ)x=λx+µxforallλ,µ∈R,x∈:′′′1)0isunique(axiom2and30=0+0=0+0=0)2)Opposite−xisunique3)0·x=0NotationalConvention:x+(−y)=x−yBecarefultodistinguishbetween0∈Vand0∈,.(notamuchuseineconomics).∈Risdenotedasx=(x,..,x).”Definition2Letx=(x,..,x)andy=(y,..,y):x+y=(x,..,x)+(y,..,y)=(x+y,..,x+y)1n1n11nn2
Definition3Letx=(x,..,x)beavectorandλarealnumber1nDefine:λ·x=λ·(x,..,x)=(λx,..,λx)[0,1].Definef+gsuchthat(f+g)(x)=f(x)+g(x)andλfsuchthat(λf)(x)=λf(x).Themostusefulfunctionvectorspacessatisfycertainpropertiesthatareconservedbyadditionandmultiplication:•SetofcontinuousfunctionsfromIsubsetofRintoRdenotedC(I,R),orsetofderivablefunctions,...n•C(I,R)⊂Viscalledavectorsubspace(orsubspace)ofVifUnonemptyandforanyx,y∈Uandλ∈R,x+y∈Uandλx∈:•,−xalsois.•IfUandUaresubspaceofVthenU∩UisalsoasubspaceofV(provethis!).1212•TheNullspace{0}
,letv,..,vbevectorsandλ,..λ=1r1rλv+..+λviscalledalinearcombinationofv,..,,..,:1rSpan(v,..,v)={λv+..+λv|λ∈R}⊂V1r11rriofalllinearcombinationsofv,..,viscalledthelinearhullof(v,..,v).Byassump-1r1rtionL(∅)={0}.Lemma6ThesetSpan(v,..,v),..,-tuple(v,..,v)ofvectorsissaidtobe1rlinearlyindependentifffromλv+..+λv=0itfollowsnecessarilythatλ=..=11rr1λ=-tuple(v,..,v):,ifthetupleisnotindependent,thenthereexistssomeλsuchi∑rthatλv=0andλ>=-tuple(v,..,v)ofvectorsinViscalleda1nbasisofViff1)(v,..,v)arelinearlyindependent1n2)Span(v,..,v)=V(spanthespace)1n4
Proposition10If(v,..,v)isabasisiffforeachvinVthereexistsexactlyone1nn(λ,..,λ)∈Rsuchthatv=λv+..+λ:=⇒Thereisatleastone,supposetherearetwodecomp,theyareidenticalbecause(v,..,v)∑n⇐=(v,..,v)spansthespaceandisabasisbecauseλv=0hasaunique1niii=1solutionandλ=:e=(0,..,1,..0)(1):,..,v,w,..,,..,varelinearlyindependentandSpan(v,..,w)=V1r1sthenbysuitablychosenvectorsfromw,..,w,onecanextendv,..,:Considerw,eitheritisinspan(v,..,v),takeinonboard,11r(v,..,v,w),...1r12Whenfinishedyouhave(v,..,v,w,..,w),otherwise,(v,..,w)=V,itisalsothecasethatspan(v,..,v,w,..,w)=:Suppose(v,..,v)and(w,..,w)arebasesofV,then1n1mforeachv,thereissomewsuchthat(v,..,v,w,v,..,v)−1ji+1nProof:ConsiderSpan((v,..,v,v,..,v):thissetisnotequaltoVbecausevisnot1i−1i+(v,..,v,w,v,..,v)j1i−1ji+
Now,as(v,..,v,w,v,..,v,v)spansV,−1ji+1nieither(v,..,v,w,v,..,v)or(v,..,v,w,v,..,v,v)−1ji+1n1i−1ji+(v,..,v)and(w,..,w)arebasesofVthenn=:Supposenotsuchthatn>,andv,..,v12n′:Definition14If(v,..,v),{0},..,varer>:=(v,..,v)thenitiscalledaninfinitedimensional1nspaceandwenotedimV=∞.0Example:C([0,1])(provethis!.Definition18U+V={u+v,u∈U,v∈V}+
Theorem20dim(U∩U)+dim(U+U)=dimU+dimU121212Proof:StartwithabasisofU∩U(rvectors)andthencompleteittobasisofU(s121morevectors)andtobasisofU(tmorevectors).Theunionofthesetwobasisis2abasisofU+++∩:ifU∩U={0}thenwenoteU+U=U⊕U121212Thenextcorollaryisaveryimportantspecialcaseofthepreviousresult:Corollary21dim(U⊕U)=dimU+=R·∈/HsuchthatH⊕R·v=,HhyperplaneiffdimH=n−
LectureVI:LinearAlgebraII-EuclideanSpaces,,20021InnerProductsandNormsAninnerproductdefinesageometryonavectorspace:wecannowtalkaboutconceptssuchas’angles’betweenvectors(inparticularorthogonality),×:<x,y>=x·y(1)Theinnerproductis•linearinbotharguments•commutative•satisfies<x,x>≥0and<x,x>=0iffx=0Aninnerproductallowsustodefineanormonavectorspace.+AnormofavectorspaceisafunctionV→Rwhichsatisfiesthefollowingproperties:1.||x+y||≤||x||+||y||(triangleinequality)2.||x||≥0and||x||=0iffx=0(obvious)1
3.||λx||=|λ|·||x||(obvious)Ifwehaveaninnerproductwecandefineanorm||x||asfollows:√||x||=<x,x>(2)Tocheckthatthisisanormweneedthefollowingimportanttheoremwhichwestatewithoutproof:Theorem1Cauchy-SchwarzIfx,y∈VandVhasaninnerproductthen|<x,y>|≤||x||||y||(3)Remark2ItiseasytoproveCauchySchwartzusingstandardresultsonquadraticequa-tionsandthebinomialexpansionof<x+θy,x+θy>forsomex,yandanyrealθ.∑<x,y>=xy(4)iii=1Thenormbecomes:√√22||x||=<x|x>=x+..+x(5)
,..,visorthonormaliff||v||=1(normality)and1ni<v,v>=0fori =j(orthogonality).,:,..,fbasisofF(whichweknowexists).1k′=f/||f||,111∗′′′′∗∗=f−<f,f>=f/||f||.2221112223....∗′′′′′∗∗=f−[<f,f>f+..+<f,f>f].Normalize,f=f/||f||.kkkk11k−1k−⊥nDefinition6F={x∈R|<f,x>=0foranyf∈Fn⊥⊥Proposition7R=F⊕(F)=n−dim(F)⊥Proof:ItiseasytoseethatF∩F={0}.ThentakeanyorthonormalbasisofF,expandittoacompleteorthonormalbasisf,f,..,∈Vwehave:n∑∑=<x,f>f(6)iixi=1perpIfx∈Fthefirstm=,..fisam+1n⊥
⊥⊥Remark8(F)=Fn⊥nTheorem9Hyperplanes1)ifxisanon-zerovectorofR,(R·x))={y∈R|<x,y>=0})Conversely,anysubspaceofdimensionn-1isoftheform{y∈R|<x,y>=0}:1).⊥2)SupposethatHisofdimensionn−-dimensional.⊥Letx∈’=F⊕G,forallvectorx,thereisauniquedecompositionx=x+x,whereFGx∈Fandx∈•x=P(x)isthelinearprojectionofxonF,•P(λx+µy)=λP(x)+µP(y)∀λ,µ∈R(linearity).FFF•P(P(x))=P(x)(thispropertyiscalledidempotence)FFF2Infact,anylineartransformationsuchthatP=:orthogonalprojectiononasubspaceFn⊥⊥nUsingR=F⊕F,wecandefinetheprojectiononF,=F⊕G,forallvectorx,thereisauniquedecompositionx=x+x,wherex∈Fandx∈⊥•Notethatx−P(x)∈FisorthogonaltoP(x)∈FFF222•||x||=||P(x)||+||x−P(x)||FF4
2222•∀f∈F,||x−f||=||x−P(x)+P(x)−f||=||x−P(x)||+||P(x)−f||≥FFFF2||x−P(x)||Thisinequalityisanequalityifandonlyiff=P(x).(x)=ArgMin||x−f||(7)Ff∈FThatis,×nmatrixAoverRisanarrayofmrowsandncolumnsofrealnumbers:describedasfollowsA=(a)ijaarethecoefficients,ij(a..a)arethemrowsi1in(a,..,a)arethencolumns1jmjSetofallmatricesofsizem×nisdenotedbyM(R).m,×nmatricesA=(a)andB=(b).ijijThesumofmatricesA+Bisthem×nmatrix(a+b).ijijTheproductofAwiththerealnumberλisdefinedasλA=(λa).·,nijhaszeroentrieseverywhereexceptforthecoefficientati,jwhichis1:∑A=aE(8)ijijij5
Definition12Theproductofanmtimesnmatrixwithan×1columnvectorxisamvectorinRdefinedasfollows:n∑(Ax)=ax(9)jijjj=1nmAmatrixcanbeinterpretedasalineartransformationTwhichmapsRintoRandisdefinedasT(x)=:(x+y)=Tx+(λx)=λTxNote,[e,...,e]∈M(R),B∈M(R).ThenC=AB∈M(R)definedby:m,nn,pm,p∑n(C)=ABi,ji,kk,jk=(AB)C=A(BC),(A+B)C=AC+BC,C(A+B)=CA+,B∈M(R).Theyarenotequaln,(R).Whentheyareequal,wesaythatAandBn,,:AI=IA=A(10)6
′Definition15A∈M(R),thenthetransposeA∈M(R)isdefinedby:m,nn,m′(A)=A(11)i,jj,iProposition16′′′′′(A+B)=A+B,(λA)=λA,∀λ∈R′′′(AB)=BA′′(A)=A∑nDefinition17A∈M(R),thenthetracetr(A)=An,niii=1Proposition18A,B∈M(R)n,ntr(A+B)=tr(A)+tr(B)tr(λA)=λtr(A),∀λ∈Rtr(AB)=tr(BA)′tr(A)=tr(A)∑∑′2tr(AA)=ai∈[1,n]j∈[1,n]i,j2′n||X||=XXforanyvectorX∈={Ax,x∈R}=A(R)={x∈R,Ax=0}[e,...,e]isthecanonicalbasisofRand1nAe=C(12)jjImA=span(C,..,C)1n7
mProposition20AissurjectiveiffImA=RAisinjectiveiffkerA={0}Theorem21IfAisanm×nmatrix,thenn=dimImA+:Takev,..,,..,=1rr++iIclaimthatw,..,−r∑∑ItspansImAbecauseanyAx=A(tv)=∑∑∑Itislinearlyindependenttw=0soA(tv)=0sotv∈kerAsoiiir+iir+i∑∑tv=pvimpliesthatalltarezerobecause(v)+ijjiinmSincelineartransformationsfromRtoRarethesamethingasmatricesM(R)m,
LectureVII:,(typicallyRandfunctionspaces).||.||isafunctionV→Rsuchthat:1)||x||≥0and||x||=0iffx=02)||x+y||≤||x||+||y||3)||λx||=|λ|·||x||Recall,thatinnerproductspaceshaveanaturalnormdefinedby||x||=<x,x>.:•Thesup-norm||x||=sup|x|∑n•Theabsolutenorm||x||=|x|=11
:Proposition2If||.||and||.||aretwonormsonafinite-dimensionalvectorspaceV12thenthereexistconstantsaandbsuchthata||x||≤||x||≤b||x||(1)121forallx∈[a,b].:d(x,y)=||x−y||(2):(x,y)=d(y,x)(x,y)≥0andd(x,y)=0iffx=(x,z)≤d(x,y)+d(y,z)AnyfunctiononV×∈nC[a,b]wedefine||f||=sup|f(x)|(3)x∈[a,b]Note,
||x−x||convergeskkto0(inR).Equivalentnorms(suchasallnormsinfinitedimensionalmetricspaces)(wherewecanfocusWLOGontheEuclideannormsinceallnormsareequivalent).nProposition5IntheEuclideanspaceRthesequencex=(x,..,x)convergestokk1knx=(x,..,x)iffforeachi∈{1,..,n},:Ifallcoordinatesconvergethenlim||x−x||=,notethat:k→∞k|x−x|≤||x−x||(4)kiiknHence,,(ageneralizationoftheleastupperboundpropertyoftherealline)andcompactness(ageneralizationofclosed,boundedsetsontherealline).ThesetwopropertiestogetherwiththetriangleinequalityofthemetricspaceallowustotransplantallcrucialresultssuchasBolzano-Weierstrasstheorem,3
theintermediatevaluetheorem(moregenerally,fixedpointtheorems)andthemin-maxnntheoremfromthereallineintometricspacessuchasRandC[a,b].∈ isB(x)={y∈V|d(y,x)< }. Onthereallineopenballsaresimplyopenintervals(x− ,x+ ).⊂∈S,thereissome >0suchthatB(x)⊂S. ⋃Proof:ConsideropensetsVwithindexseti∈IandtheunionV=∈Vii∈iIthenx∈Vforsomei∈IandhenceB(x)⊂ iiimpliesthatB(x)⊂. Nowconsiderx∈V∩(x)⊂VandsomeballB(x)⊂ 1 212Therefore,take =min( , ).TheopenballB(x)isthenbothasubsetofVand12 1V(hereweusethetriangleinequality-why?)andhenceasubsetofV∩⋂exampleV=(−1/n,1/n)suchthatV=V={0}∈NiDefinition11LetS⊂−
Forexample,[a,b]isclosedandthesingletonset{a}⊂Risclosediffforanyconvergingsequencex∈S,:⇒→x∈/∈∈/SimpliesB(x)⊂R−Sbutx∈B(x). n n⇐,V=R−,thereisx∈Vsuchthatforany >0,B(x)∩Snonempty. Take =1/nandconstructasequencex∈:WeuseDeMorgan’slaw:nnR−∪S=∩(R−S)iinnR−∩S=∪(R−S)⊂={x∈S|thereis >0suchthatB(x)⊂S}. :{a}=∅[0,1]=(0,1)=intA5
nDefinition15LetS⊂R.¯TheclosureofSisS=clSissmallestclosedsetcontainingS⋂clS=FS⊂FIntuition::clS=R−int(R−S)Theorem16Forany >0andx∈clS,B(x)∩Snonempty. Proof:SupposenotandB(x)∩S=∅−B(x)isclosedandcontains ⊂FbutxisnotinFsox∈/:(a,b)=[a,b]{1,1/2,1/3,...}={0,1,1/2,1/3,...}=(x)=={y∈R|||y−x||≤ } nDefinition17LetS⊂∂S=clS−intSExamples:1.∂[0,1]=[0,1](0,1)={0,1}2.∂(0,1)=[0,1](0,1)={0,1}n3.∂B(x)={y∈R|||y−x||= } n4.∂RisemptyIntuition:⊂>0andsomey∈Vsuchthatd(x,y)≤Mforallx∈:1.[a,b]
,{V}(whichisacollectionofopensets)ifi⋃X⊂∈iIForexample,anopencoverof[0,1]wouldbe(i− ,i+ )wherei∈[0,1].Definition20CompactnessAsetXiscompactifeveryopencoverofXhasafinite⋃msubcover,{i,i,..,i}suchthatX⊂=:ConsiderthecompactsetXandasequence(x)⊂Xwhichconvergeston∗∗∈/:V=n{}∗1y|d(x,y)>.⊂∗’[a,b]×[a,b]×..×[a,b]
,,forthecaseofC[a,b]-AscoliAsubsetXofC[a,b] >0thereexistsaδ>0suchthat|x−y|<δimplies|f(x)−f(y)|< forallf∈:[a,b],ithastobeshownthatfiscontinuousandhenceanelementofC[a,b].8
6Bolzano-WeierstrassFinally,(x)⊂,thateachcoordi-natesequence(x)oftheboundedsequence(x)(x)
LectureVIII:CalculusofSeveralVariables-Continuity,,,,TheBellmanoperatormapscontinuousboundedvaluefunctionsintocontinuousboundedfunctions,:C[a,b]→C[a,b].Definition1Afunctionf:X→Yiscontinuousinx∈Xiffforall >0thereexists0η>0suchthatforallx∈X||x−x||<η=⇒||f(x)−f(x)||< (1)∈Xiffforanysequence(x)inX,(x)→x0nn0implies(f(x))→f(x)n0Continuityispreservedbyusualtransformations:Proposition3iffandgcontinuous,f+g,fg,f/g,f◦
Theproofsofbothpropositionareexactlythesameasforrealfunctions(seelectures2-3).Note,thatwemakeuseofthetriangleinequalitywhichisacrucialpropertyofmetricspaces!Thefollowingcharacterizationofcontinuousfunctionsisextremelyuseful::X→Yiscontinuousifffor−1−1anyVopeninYf(V)isopeninXiffforanyFclosedinYf(F))Iff:K→Riscontinuousthenf(K))Iff:K→Riscontinuousthenfhasamaximizer(andaminimizer).Proof:1)letf(x)beasequenceinf(K).NowxisasequenceinKsoithasacon-nn′vergingsubsequencexconvergingtoxinKandthereforebecausefiscontinuousnf(x)convergestof(x)whichisinf(K).n2)f(K)∗intheclosureoff(K)thusinf(K).Soithasamaxf(x))Iff:X→Riscontinuousthenf(X),iff(x)<f(y)forx,y∈X,thenanyvalueinbetweenf(x)andf(y)
:U→(x)=(f(x,..,x),..,f(x,..,x)(2)(wedon’tworryaboutcornerssinceit’sanopenser).=j−j(x,..,x,x,..,x).Wecannowconsiderforeach1≤i≤pthefunction1j−1j+1nf(x)=f(x,x)(3)ijij−,wenotethederivativef(x+h,x)−f(x,x)ij−jij−j∂f/∂x=lim(4)ikh→0hThesetofderivatives∂f/∂x,∂f/∂x..formavector:1k2k()∂fip:U→R(5)∂xknWecancollectallthesenvectors(foreachcoordinateinR)inasinglematrixcalledtheJacobianmatrix:J=∂f/∂x(x)(6):U→R(=1)anditspartialderivatives∂f/∂x:U→,denotethesehigher-orderpartialderivativeswith∂f/∂x∂,theijorderofdifferentiationmatters,
2Theorem7SchwarzLetf:U→∂f/(∂x∂x)ij2and∂f/(∂x∂x):Fixx∈U⊂Randdefine:A(h,l)=f(x+h,x+l)−f(x+h,x)−f(x,x+l)+f(x,x)12121212B(h)=f(x+h,x+l)−f(x+h,x(71212Wecanseethatforsome0<θ<11∂f∂f′A(h,l)=B(h)−B(0)=hB(θh)=h[(x+θh,x+l)−(x+θh,x)](8)1112112∂x∂-tivegivesus:2∂fA(h,l)=hl(x+θh,x+θl)(9)1122∂x∂x12Similarly,wecandefineC(l)=f(x+h,x+l)−f(x+h,x)andwrite12122∂f′A(h,l)=C(l)−C(0)=hC(θh)=hl(x+θh,x+θl)(10)3i3j4∂x∂x21Thisallowsustoinferthat22∂f∂f(x+θh,x+θl)=(x+θh,x+θl)(11)i1j2i3j4∂x∂x∂x∂x1221Note,,asweleth→0andl→0(sequentially)weget:22∂f∂f(x,x)=(x,x)(12)ijij∂x∂x∂x∂x1221Note,-orderpartialderivatives:()2∂fH=(13)∂x∂xiji,j4
,:U→Rafunction:f(x)=[f(x,..,x),..,f(x,..,x)](14)(n=p=1),fdifferentiableinxifff(x+h)=f(x)+′f(x)h+o(h).Wecangeneralizethisdefinitiontofunctionsofmanyvariables:npDefinition9LinearApproximationThelinearfunctionAfromRintoR(orequivalentlythep×nmatrixA)istangenttofatxiff(x+h)−f(x)=Ah+o(h)(15)o(h)meaningthato(h)/||h||→0ash→=:D=A−A,Dh=o(h)soD=,assumethatD ==esatisfies|Dh|/||h||→aforsomea>-example:f(x,y)=xy(x−y)/(x+y)and2f(0,0)=-orderderivativesarenotcontinuousinbothxandyand∂f/(∂x∂y)(0,0)=12but∂f/(∂y∂x)(0,0)=−
pDefinition11Iff:U→:1Ae=lim([f(x+te)−f(x)])=J(16)111t→:U→RisofclassCiff′1)foranyx∈U,f(x)exists′2)x→f(x)(x)=AxisCandthederivativeDf=:U→RisinC(U)(continuouslydifferentiable)-valuetheoreminthesamewaywhenweprovedSchwarz’:U⊂R→Rbedifferentiableatx∈Uandletebeavectorinn′(x)(x).eProposition16Thederivativeinthedirectionofesatisfies1′f(x)e=lim([f(x+te)−f(x)])(17)+t→0t′′Thatis,f(x)e=Φ(0)whereΦ(t)=f(x+te):f(x,y)=y/xifx =0andf(0,y)=
′alternativenotationf(x)=Df(x)whichimpliesthatf(x+h)=f(x)+Df(x)h+o(h).′Remark18Sometimeswewrite:f(x+dx)−f(x)=df=f(x):real-valuedfunctions′Letf:U→(x)isa1×nvectorcalledthegradientoffsometimesdenoted∇forDf:xxf(x+h)=f(x)+<∇f,h>+o(h)=f(x)+∇f·h+o(h)(18)xxImportantexamples::x→||x||=x+..+xwehave∇f=,orbydirectlyexpanding||x+h||.√:x→||x||=x+..+xisdifferentiableatanypointx =0and∇g=x/||x||.,:′′′(f+g)(x)=f(x)+g(x),′′(λf)(x)=λf(x)(19)Thenextresultisthekeytocomputingmanyusefulderivatives:7
npmTheorem20Chainrulef:U⊂R→V⊂Randg:V→(x)theng◦fisdifferentiableatxand′′′(g◦f)(x)=g[f(x)]f(x)(m×n=m×p·p×n):Writedownthingscarefully:notethatf(x+h)=f(x)+k(h),wherek(h)=f′(x)h+o(h).(g◦f)(x+h)=g[f(x)+k(h)]′=(g◦f)(x)+g[f(x)]k(h)+o[k(h)]′=(g◦f)(x)+g[f(x)]f′(x)h+o(h).,sometimesthechainruleiswrittenoutexplicitly:p∑∂(g◦f)∂g∂fiik=[f(x)](x).(20)∂x∂y∂xjkjk=′n′Considerthereal-valuedfunctionf:U→:U→R,x→f(x).′Youcanaskwhetherfisinturndifferentiable.′′Thenf(x)isan×′′′′Definition21fisofclassC(U)ifff(x)existsforanyx∈Uandx→f(x)
:U→RisofclassCthen∫1′′′′f(x+h)=f(x)+f(x)h+(1−t)hf(x+th)hdt(21)02Proof:Let’sdefineΦ(t)=f(x+th)suchthatΦ:[0,1]→:′′Φ(t)=f(x+th)h′′′′′Φ(t)=hf(x+th)hQEDTheorem23Taylor-Youngexpansion:2Iff:U→RisofclassCandletx∈I:′′′′2f(x+h)=f(x)+f(x)h+hf(x)h/2+o(h)∑∑∑22f(x+h)=f(x)+∂f/∂xh+∂f/(∂x∂x)hh/2+ (h)hiiijijii,jjj∫1′′′′′′′Proof:DefineD=f(x+h)−f(x)−f(x)h−hf(x)h/2=(1−t)h[f(x+th)−0′′f(x)]hdt′2RememberhAh≤||A||·||h||∫1′′′′22So|D|≤sup||f(x+th)−f(x)||(1−t)||h||dt= (h)||h||/2t∈[0,1]03LocalExtremeLetAbeamatrixm×:R→(usethesequencecharacterizationofacontinuousfunction).ThusasclosedunitballBiscompact,Icandefine:|A|=sup|Ax|.(22)x∈BThen|Ax|≤|A|·|x|.Youcancheckthat|A|definesanormonthevectorspaceM(R).n,mAlsonote,thatforM(R)ifAisinvertiblethereisa>0suchthat|Ax|≥a|x|(becausen,n−1n|Ay|≤b|y|foranyy∈R,takey=Axanda=1/b).9
∈Missymmetriciffa=an,nijji∑′′ThenxAy=yAx=(x,Ay)=(Ax,y)=axyijijij′Definition24Matrixissemi-definitepositiveiffAissymmetricandxAx≥0foralln′nx∈≥0forallx∈,wecandefinenegativesemi-defineness.′2WehavexAx≤|x|·|Ax|≤|A||x|′2Lemma25IfAdefinite-positivethenthereisa>0suchthatxAx≥a|x|.′Proof:xAx=0impliesthatAx=0(wewillseethatlateron)andthusx=0.′x→xAxiscontinuoussoreachesaminoncompactunitsphereequaltoa>(strictandnonstrict)extremumforf:U→RforUopensubsetofR.′Defineacriticalpointf(x)=:U→RforUopensubsetofRisCandhasalocalextremum′atxthenf(x)=0(necessarycondition).Proof:UseTaylorexpansionandapplyh=(0,..,h,..0).i3Againthereciprocalstatementisnottrueasthecounterexamplex→′Theorem27LetU→RbeCandassumethatthereisx∈Usuchthatf(x)=0.′′A)Iff(x)invertible(nonsingular)′′1)Ifthematrixf(x)
′′2)Ifthematrixf(x)isdefinitenegativethenxisstrictlocalmaximum.′′3)Iff(x)isnondefinite(neitherweaklypositiveornegative)thennolocalextremum(saddlepoint).′′B)Iff(x)issingular:1)Mixedsignthensaddlepoint2)Weaklypositiveornegative,:2Theorem28LetU→RbeCandassumethatthereisx∈Ulocalextremumthen′f(x)=0.′′1)Ifxisalocalmaxthenf(x)≤0.′′2)Ifxisalocalminthenf(x)≥ =−c>0anda>0anddefinitenegative22iffab−c>0anda<,ab−c>0anda>0impliesstrictlocalminimumand22ab−c>0anda<−c<∈Cwehaveλx∈Cforallλ>.αDefinition30fC:→Rishomogeneousofdegreeαifff(λx)=λf(x).11
:1Proposition31IffishomogeneouswithdegreeαandisCthen∂fishomogeneousiofdegreeα−1.αProof:Derivef(λx)=λf(x):C→α.′(x)x=αf(x)foranyx∈:⇒Derivewrttoλandtakeλ=1⇐Fixxanddefineg:λ→f(λx)′λg(λ)=αg(λ)αSolve:g(λ)=λf(x)-valuedfunctionsonR:nDefinition33LetVbeaconvexsubsetofR:Letf:V→
fisconvexiffforanyxandyinVandany0<λ<1,f(λx+(1−λ)y)≤λf(x)+(1−λ)f(y)fisconcaveiffforanyxandyinVandany0<λ<1,f(λx+(1−λ)y)≥λf(x)+(1−λ)f(y)nEquivalentdefinition:fisconvexifftheuppergraph{(x,y)∈R×R|y≥f(x)}isn+:V−→(x =y)inVandany0<λ<1,f(λx+(1−λ)y)<λf(x)+(1−λ)f(y)fisconcaveiffforanyxandyinV(x =y)andany0<λ<1,f(λx+(1−λ)y)>λf(x)+(1−λ)f(y)Somebasicpropertiesofconcaveandconvexfunctions:−fconvex(idemforstrictly)(drawpictures).
:V−→.∑′fisconcaveifff(y)−f(x)≤f(x)·(y−x)=∂f(x)(y−x)foranyxandyinViiii′Iffisconvexifff(y)−f(x)≥f(x)·(y−x):nCorollary37LetVbeaconvexsetinRandf:V−→RarealfunctiononVdifferentiable.′(x)=0thenxisglobalminimumoffonV.′(x)=:V−→.′′(x)≥0(positivesemi-definite)foranyxinV′′(x)≤0(negative-semi-definite)foranyxinVnProof:FixxinVandletbeh∈=x+ht∈→g(t)=f(ty+(1−t)x).fconcaveimpliesthatgisconcave.(checkthat)14
′′′2thusg(t)=(y−x)Df(ty+(1−t)x)(y−x)≤0′22soatthelimit:hDf(x)h≤0impliesthatDf(x)(x)≤0foranyx∈V.∫′′′2UseTaylorformula:f(y)=f(x)+(y−x)f(x)+(1−t)f(x+t(y−x))(y−x)dt≤′f(x)+(y−x)f(x):nLetX=(X..X)∈RvectorofthedependentvariableX∈M(R).npWewanttofindthelinearcombinationoftheX’β=Xβ+...+Xβ||Y−Xβ||2Considerthefunctionβ→f(β)=||Y−Xβ||Composedoftwofunctions:′′′nf(β)=2(Y−Xβ)·X=2(YX−XXβ)vectorinR.′′′f(β)=2XX≥0(showit).’sassumethat.′¯¯βsuchthatf(β)=0isglobalmin.′′¯2(YX−XXβ)=0′−1′¯β=(XX)
LectureVIII-2:,(avectorwithnelements).Utilityisrepresentedbyacontinuousanddifferentiableutilityfunctionu(x).(p,u):e(p,u)=minpxsuchthatu(x)≥u(1)Note,thatthesetAofallvectorsxsuchthatu(x)≥uisbothconvexandclosed(bycontinuity).Thisisimportanttoshowthattheexpenditurefunctioniswelldefined().Theparticularvectorxwhichminimizesexpenditure(itisuniquebecauseindifferencesetsarestrictlyconvex)iscalledHicksiandemandh(p,u).Note,thatbydefinition:e(p,u)=h(p,u)p(2)1
:i∂e(p,u)=h(p,u)(3)i∂piThisisasimpleconsequenceoftheenvelopetheorem(whichwewilldoinafewlectures).Toseethisfromfirstprincipleslet’sfirstusethechainrule:∂e(p,u)∂h=h(p,u)+p(4)∂p∂p∑∂h(p,u)iWewaredoneifwecanshowthatp=∂pchoosetheoptimalHicksiandemandh(p,u)tominimizeexpenditurebutinsteadh(p˜,u)iwithp˜:∂h(p˜,u)i|p=0(5)p˜=p∂p˜Otherwise,:(p,u)andMarshalliandemandx(p,m)arecloselyrelatedthroughthefollowingequation:x(p,e(p,u))=h(p,u(6)Wecanusethechainruleagainonthisequationandget:∂h∂x∂x∂e∂x∂xiiiii=+=+h(p,u)(7)i∂p∂p∂m∂p∂p∂:∂h∂x∂x∂e∂x∂x=+=+h(p,u)(8)∂p∂p∂m∂p∂p∂m2
Note,thattheverylasttermisan×1columnvectorstimesa1×nrowvectorandhenceann×nmatrix!∂e(p,u)′Sinceh==e(p)(ifwefixu)wecanwritetheSlutskyequationinmatrixform∂pas:∂x∂x′′S=e(p,u)=+h(p,u)(9)∂p∂(fixingu).’=h(p,u)andx˜=h(p˜,u).Nowlookatpricespˆ=λp+(1−λ)p˜andtheoptimalHicksiandemandxˆ.Thenwehave:e(pˆ,u)=λpxˆ+(1−λ)p˜xˆ≥λe(p,u)+(1−λ)e(p˜,u)(10)Thelastinequalityfollowsbecauseatpricesptheexpenditurepxˆ’sall...goodluckwithJerry’
LectureX:,20021ImplicitFunctionTheoremLet’slookatthefollowingfunction:22f(x,y)=x+y−1(1)Supposewelookattheequationf(x,y)=0andazeroofthefunctionat(x¯,y¯).Theequationf(x,y)=0definesundercertaincircumstancesanimplicitfunctiony=φ(x).Clearly,wehavey¯=φ(x¯).Itwillbeveryusefultounderstandthebehaviorofthisimplicitfunction(ifitexists)aroundthepoint(x¯,y¯).Moreprecisely,wewanttofindopenintervalsUcontainingx¯andVcontainingy¯andφ:U→Vsuchthatφ(x¯)=y¯andforx∈U,andy∈V,φ(x)=yifff(x,y)=φ[x,φ(x)]=0forallx∈U,,wegetusingthechainrule:′f(x,φ(x))+f(x,φ(x))φ(x)=0(2)xy′Soφ(x)=−f/ =,thereisxyyaproblemifwewanttofindtheimplicitfunctionaroundx=1,y=
+mparameters:x=(x,..,x),y=(y,..,y)1n1mf(x,y)=0,f(x,y)=0,..,f(x,y)=012mf=(f,..,f)(3)1mSupposethatthevectors(x¯,y¯)areasolutionofthesystemf(x,y)=(x),..,y(x)?Note,+mparameters:f(x,y)=0(4)nm1LetΩbeanopensetofR×Randassumethatf∈C[Ω].Choose(x¯,y¯)∈Ωsuchthatf(x¯,y¯)=×msquarematrixDf(x¯,y¯)=(∂f/∂y)isoffullrankm()yijthenthefollowingholds:⊂Rcontainingx¯andanopensetV⊂Rcontainingy¯suchthatU×V⊂Ω.Thereisafunctionφ:U→Vsuchthatforany(x,y)∈U×V,f(x,y)=0⇐⇒y=φ(x)(5)12.φisC(U,V)and′−1φ(x)=−(Df(x,φ(x)))Df(x,φ(x))(6)yxwhereDf=(∂f/∂x).xijProof:Part1ishardtoprovefullyformally,butit’(x,y)=0anditszeroat2
(x¯,y¯).Usingthedefinitionofdifferentiabilityofamulti-variatefunctionweget:′f(x,y)=f(x¯,y¯)+f(x¯,y¯)(x−x¯,y−y¯)+o(h)=f(x¯,y¯)+Df(x¯,y¯)(x−x¯)+Df(x¯,y¯)(y−y¯)+o(h)(7)xywhereh=(x−x¯,y−y¯).Nownote,thatf(x¯,y¯)=(h)wegetalinearequationsystemwhichapproximatelydefinestheimplicitfunctionφ:Df(x¯,y¯)(x−x¯)+Df(x¯,y¯)(y−y¯)=0(8)xyThequestioniswhencanweexpressyintermsofx?Well,wecanjustsolvetheaboveequation:Df(x¯,y¯)(y−y¯)=−Df(x¯,y¯)(x−x¯)yx−1y=y¯+(Df(x¯,y¯))Df(x¯,y¯)(x−x¯)(9)yxForthistobewelldefinedweneedDtobeinvertible-sinceitisanm×mmatrixyithastohavefullrank.′Thesecondpartfollowsfromthisargument-wecanjustreadoffthederivativeφ.However,,wecanthinkofyasthesolutiontosomeequilibriumwithf(x,y)=:nmaxf:R→Rsuchthatg(x)=0(10)3
whereg(x)=0consistsofmseparateconditionsg(x)=0,g(x)=0,..,g(x)=¯satisfiesg(x¯)=0andf(x)≤f(x¯)foranyx∈C(thatis,anyxsuchthatg(x)=0).Definition4Thelocalmaximizerx¯satisfiesg(x¯)=0andthereisanopensetUcon-tainingx¯suchf(x)≤f(x¯)foranyx∈C∩(x,x)=x+yandg(x,y)=x+2y−√√(1/2,1/2).Howcanwefinditthroughsomealgorithm?Supposethat(x¯,y¯)(x¯,y¯) =0-thenwecanuseimplicitfunctiontheoremtoreplaceconstrainty√2locallybyy=φ(x)=1−¯maximizesf(x,φ(x))′f(x¯,φ(x¯))+f(x¯,φ(x¯))φ(x)=0(11)xyAlsowehave′g+gφ(x)=0(12)xyCombiningthesetworesultsweobtain:fgxx′φ(x)=−=−(13)fgyyTheorem6Supposethatx¯isalocalconstrainedmaxoff(x)stg(x)=0andm<n.′Supposethatthematrixg(x¯)ofsizen×misofrankm.(Constraintqualification:constraintsmustbeindependentatx¯).4
mThenthereisavectorλ=(λ,..,λ)∈R(calledthevectorofLagrangemultipliers)1msuchthatm∑∂f∂gk(x)=λ(x)(14)k∂x∂xiik=1i=1,..,norequivalently:m∑∇f(x¯)=λ∇g(x¯)(15)kkk=,:L(x,λ)=f(x)−λ·g(x)(16)Ifx¯isasolutionthen,assumingtheconstraintqualificationholds,itmustsatisfythefollowingnecessaryconditions:∂L∂L=0and=0(17)∂x∂λikThatis,alocalmaxisacriticalpointfortheLagrangianfunction.(byexpandingthenumberofvariables,wehavetosolveanonconstrainedproblem).2Remark7The(rarelyused)second-orderconditionoftheLagrangianisDL≤:f(x,y)=x+ystx+y=15
LectureXI:,20021InequalityConstraintsIneconomics,weoftenfindmaximizationproblemswithinequalityconstraintsratherthanequalityconstraints:maxf(x)suchthatnx∈Rg(x)=b,..,g(x)=band11mmh(x)≤c,..,h(x)≤c(1)11ppWeassumen>m+p(otherwisethesystemisoverdetermined).,..,{∇g(x),..,∇g(x),∇h(x),..,∇h(x)}λ,...,λ,µ,...,µsuchthat:1m1pmp∑∑∂f∂g∂hkl(x)=λ(x)+µ(x)kl∂x∂x∂xiiik=1l=11
,µ≥0foralll=1,..,p,andlµ[h(x)−c]=0lllforalll=1,..,pRemarks:•concisenotationintermofgradients:mp∑∑∇f=λ∇g+µ∇h(2)kkllk=1l=1•Thelastconditionsimplymeansthatµ=0foranyconstraintthatdoesn’•
′′Proof:f(x+h)=f(x)+f(x)h+o(h).Iff(x+h)=f(x)thenf(x)·h=0.′f(x)pointsinthedirectionwherefincreases.′Takehgoingthesamedirectionoff(x)thenf(x+h)>f(x).∑∑mpL(x,λ,µ)=f(x)−λg(x)−µh(x)kkllk=1l=1WriteFOCasmp∑∑∂L∂f∂g∂hkl(x,λ,µ)=(x)−λ(x)−µ(x)=0kl∂x∂x∂x∂xiiiik=1l=,,∇f(x¯)=µ∇h(x¯)+µ∇h(x¯)(x,x)stpx+px≤I,x≥0,x≥012112212withU(x,x)=U(x,x)−λ(px+px−I)+µx+µx1211221122Threepossibilities:1)µ=µ=:U=λpU=λp1211222)µ>0,µ==0,x=I/pandU=λp−µ.)µ=0,µ<
22Maximizef(x,y)=x+x+4yst2x+2y≤1andx≥0andy≥=x+x+4y−λ(2x+2y−1)+λx+λy123FOC:∂L=2x+1−2λ+λ12∂x∂L=8y−2λ+λ13∂yWeseethatf(x,y)isincreasinginxandy,thereforetheconstraint2x+2y=λ>(x,y)=(0,1/2)andf(0,1/2)=λ>(x,y)=(1/2,0)andf(1/2,0)=3/λ=λ=+1=8y,implyingthaty=(2x+1)/+(2x+1)/4=1orequivalentlyx=3/10,y=1/(3/10,1/5)=11/’ −→f(x,a).1Supposethatforeacha,thereisasolutionx(a)(a)=f[x(a),an]=Maxf(x,a)x∈RThen:∂f′V(a)=[x(a),a](3)∂a4
∂fProof:Firstordercondition:[x(a),a]=,∂x∂f∂f∂f′′V(a)=[x(a),a]+x(a)[x(a),a]=[x(a),a].QED∂a∂x∂aThefunctionV(a):n1EnvelopeTheorem(ConstrainedCase).Letf,g,...,g:R×R→(a)denotethesolutionofproblemMaxnf(x,a)x∈Rstg(x,a)=0,..,g(x,a)=(Someofthesecanbeinequalityconstraints)1Supposethatthesolutionx(a)andtheLagrangemultipliersλ(a),..,λ(a),∂L′V(a)=[x(a),λ(a),..,λ(a)](4)1m∂am∑∂f∂gk=[x(a),a]−λ(a)[x(a),a]5k∂a∂ak=1Proof:Attheoptimumm∑V(a)≡L[x(a),λ(a),a]≡f[x(a),a]−λ(a)g[x(a),a]kkk=1sinceeitherg[x(a),a]orλ(a)=:nm∑∑∂fdx∂fi′′V(a)=[x(a),a]+[x(a),a]−λ(a)g[x(a),a]kk∂xda∂aii=1k=1{}mn∑∑∂gdx∂gkik−λ(a)[x(a),a]+[x(a),a]k∂xda∂aik=1i=15
andthusm∑∂L′′V(a)=[x(a),λ(a),..,λ(a)]−λ(a)g[x(a),a]1mkk∂ak=1{}nm∑∑dx∂f∂gik+[x(a),a]−λ(a)[x(a),a]kda∂x∂xiii=1k=1WenotethattheFOCsimplythatm∑∂f∂gk[x(a),a]−λ(a)[x(a),a]=∂x∂xiik=1′Wethenshowthatλ(a)g[x(a),a]=[x(a),a]=0,′Org[x(a),a]<λ(a)=0onaneighborhoodofaandλ(a)=(a,...,a).1qApplication:g(x,a)=h(x)−=f(x)−λ[h(x)−a]1′ThenV(a)=∂L/∂a=λ.Example:utilitymaximizationproblemλ,
LectureXII:,20021SecondOrderConditionsWeasktwofurtherquestions:(1)(Howcanweknowifalocalextremumisamaximizerorminimizer?Thisraisesthequestionofsecondorderconditions.(2)Howcanweensurethatalocalmaximizer/minimizerisaglobalmaximizer/minimizer?•Intheunconstrainedcase,wesimplylookattheHessianmatrixoftheobjectivefunction.•Intheconstrainedcase,,∇g(x¯),k=k1,..,′WesaythatamatrixAisnegativedefiniteonasubspaceVofRiffvAv≤0forallv∈(SecondOrderCondition)Letf,g,..,-1msidertheproblemofmaximizingf(x)ontheconstrainedsetnC={x∈R:g(x)=b,...,g(x)=b}.(1)11mm1
Considerx¯∈Candsupposethat:(a)Thequalificationconstraintholds.∑(b)TheFOCholds:DL(x¯,λ)=∇f(x¯)−λ∇g(x¯)=0xkkk2(c)TheHessianmatrixH=DL(x¯,λ)isnegativedefiniteonthetangentsubspacexxnV={v∈R:∇g(x¯).v=0forallk}.kThenx¯:ConsideraCpathx(t)ontheconstrainedset,x(0)=x¯,andthefunctionϕ(t)≡f[x(t)].Letv=x˙(0).Weknowthatn∑∂f′ϕ(t)≡∇f[x(t)].x˙(t)=[x(t)]x˙(t)i∂xii=1andnn2∑∑d∂f∂f′′ϕ(t)≡∇f[x(t)].x˙(t)=[x(t)]x¨(t)+x˙(t)[x(t)]x˙(t)iijdt∂x∂x∂xijii=1j=1T2=∇f[x(t)].x¨(t)+x˙(t)Df[x(t)]x˙(t).Sincex¯isalocalmaximumofthefunctionfonC,weknowthat′ϕ(0)=∇f(x¯).v=0and′′T2ϕ(0)=∇f(x¯).x¨(0)+vDf(x¯)v≤0.∑[Sincethefirstequalityholdsforeveryv,weinferthat∇f(x¯)=λ∇g(x¯)underkkktheconstraintqualification.]Wecansaymoreaboutx¨(0).Sinceg[x(t)]≡0,thesecondderivativeofg[x(t)]iskkalsoidenticallyequaltozero:T2∇g[x(t)].x¨(t)+x˙(t)Dg[x(t)]x˙(t)≡
Inparticular,T2∇g(x¯).x¨(0)=−vDg(x¯)∑∇f(x¯).x¨(0)=λ∇g(x¯).x¨(0)kkk∑T2=−λ[vDg(x¯)v]kkkTheFOCisthus∑′′T2T2ϕ(0)=vDf(x¯)v−λ[vDg(x¯)v]≤∑2m2ThematrixH=Df(x¯)−λDg(x¯)isthereforenegativesemi-definiteonkkk=-TuckercasetheSOCwithinequalityconstraintsbecomes:Necessarycondition:Ifx¯isaconstrainedlocalmaximum,theHessianmatrixH=2nDL(x¯,λ)isnegativesemi-definiteonthesubspace{v∈R:∇g(x¯).v=0forallkxxbindingconstraintsk}..′′Example2Supposefisastrictlyconcavefunctionsuchthatf(x)(x)·x=I′FOC:f(x)=λp′′SOC:H=f(x)nItisnegativedefiniteonR,andthereforeonthespacep·v=0.′SoifIcanfindλ>0andf(x)=λ
2ConcaveProgramingWithconcavefunctions,⊂:U→,..,gareconvexfunctionsU→(x)stg(x)≤0.(notethatconstrainedsetisconvex).¯Supposeyoucanfindx¯∈Uandλst:∑m¯∇f(x¯)=λ∇g(x¯)kkk=1¯¯whereλ≥0,g(x¯)≤0andλg(x¯)=¯¯isastrictglobalconstrainedmaximumoff(andthusuniquemax).Proof:Letx∈Ubeanelementoftheconstrainedset:g(x)≤(x¯)=0binds,kthen∇g(x¯).(x−x¯)≤g(x)−g(x¯)=g(x)≤0kkkk¯Ifnonbindingthenλ=0sok¯λ∇g(x¯).(x−x¯)≤0forallkkkThussummingoveri,∇f(x¯).(x−x¯)≤∇f(x¯).(x−x¯)≥f(x)−f(x¯)asfconcaveimpliesthatf(x)≤f(x¯).QEDNotethatiffisstrictlyconcavethenx¯:4
Theorem5SupposethattheconstraintsetCisconvex(forexamplebecausethegareiquasi-convex)andtheobjectivefunctionf(.).∏nExample6Considermaximizingthefunctionf(x,x,..,x)=i=1xsubjectto12ni∑nx≤1andx≥=1∑nfunctioncanbewrittenaslnf=ln(x).Eachofthelogfunctionsisconcave-henceii= beapreferencerelationdefined+nonS=:+:foranyxandyinS,x yory :foranyx,x :x yandy zimpliesx ≥yandx
=yimpliesx y(goodsaregoods):+Foranyx,S(x)={y∈S|y x}(thesetofbundlesbetterorequalthanx)and−S(x)={y∈S|y x}(thesetofbundlesworseorequalthanx) zandxclosetoz(ineuclidiannorm)thenx
nTheorem7RepresentationTheoremLetS= +satisfiesaxioms1to5then ,thereissomecontinuousu:S→Rsuchthatu(x)≥u(y)iffx :Lete=(1,..,1)∈∈S,letu(x)beanumbersuchthatx∼u(x)e.+Wehavetoshowthatu(x)={t∈R|te x}andW={t∈R|te x}.,bothsetsareclosedandW∪B==infB:=supW=(x)suchthatpx≤m(2),weknowthatpx= isstrictlyconvexiffy x,andz x,andz
=yimpliesthatαy+(1−α)z xforany0<α<
Ifpreferencesatisfiesaxioms1-5andisstrictlyconvex,thenuisstrictlyquasiconcave(veryclosetobeingstrictlyconcave).,(inparticu-larindependenceofirrelevantalternatives)(seeJensen’sinequality).(x)definedoverconvexsetV(ingeneralRnorR).Assumethatuisstrictlyincreasing.+•goodsvectorx=(x,..,x)1n•pricevectorp=(p,..,p).p>•Wealthm.•Budgetconstraintp·x≤,thattheconstraintg(x)=p·x−m≤0islinear(thusconvex).(x)suchthatg(x)≤0,x≥0.(3)-Tuckerandknowthattheglobalmaximizersatisfies:thereis′λ≥0suchthatu(x¯)=λ,becauseustrictlyincreasing,λ>¯
(p,m)(p,m).TheLagrangianmultipliersoftheproblemaredenotedwithλ(p,m).Proposition12Propertiesoftheindirectutilityfunction:1)v(p,m))v(p,m)ishomogenousofdegree0in(p,m).n3)v(p,m)isquasi-convexinp.{p∈R|v(p,m)≤k}isconvex.+nProof:,p∈=tp+(1−t)+nnDefineB={x∈R|px≤m}andB={x∈R|px≤m}.ii++ThenB⊂B∪B(assumenotandgetcontradiction).12Sov(p,m)=maxu(x)≤maxu(x)≤∈Bx∈B∪B12FurtherusefulProperties1)px(p,m)=mProof:)x(tp,tm)=x(p,m)fort>0homogeneityofdegree0Proof:samebudgetconstraint,:usingEulerequation:∑3)p∂x(p,m)/∂p+m∂x(p,m)/∂m=0iijijDerive1)withrespecttoR:∑4)p∂x/∂m=1(marginalconsumption)iiDerive1)withrespecttop:i∑5)x+p∂x/∂p=0ijjij8
Proposition13Theindirectutilityfunctionsatisfies∂v/∂m=λ∂v/∂p=−λp=−p∂v/∂m(Roy’sidentity)(4)iiiProof:Wejustapplytheenvelopetheoremonthevaluefunctionv(p,m)=u(x(p,m))=u(x(p,m))−λ(p,m)(px(p,m)−m):∂v/∂m=λ∂v/∂p=−λp=−p∂v/∂m(5)·xsuchthatu(x)≥u,x≥:max−p·xsuchthat−u(x)+u≤0(6)-tiontotheaboveproblemdefinestheexpenditurefunctione(p,u)andtheHicksianorcompensateddemandfunctionsh(p,u).Proposition14Importantproperties:•,wehave:u(h(p,u))=u.(7)•e(p,u)•e(p,u)
Proof:Let(p,x)and(p,x)=tp+(1−t)(p,u)=p·x=tp·x+(1−t)p··x≥e(p,u)and1211p·x≥e(p,u).Soe(p,u)≥te(p,u)+(1−t)e(p,u).2212Infact,onecanshowthate(.,u):(p,u)=p·h(p,u)(tp,u)=te(p,u)(tp,u)=h(p,u)4.∂e/∂p=h(p,u)(sincetheconstraintsarecompactsets).’same’(.)iscontinuous,preferencesarestronglymonotonic,(thusthetwoproblemshavesolutions)then:∗∗∗=u(x)thenxalsosolvestheexpenditureminimizationproblem.∗∗∗=p·
′′Proof:1)Supposenot,letxsolvetheexpenditureminimizationproblem,hencepx<∗′∗′′′′′′∗pxandu(x)≥u(x).Takex>>xclosetoxsuchthatpx<px=Rthen′′∗u(x)>u(x)whichisimpossible.′′∗2)Supposenot,letxsolvetheutilitymaximizationproblem,sothatu(x)>u(x)′∗′′′′′∗andpx=px==(1− )xwith >0smallsothatu(x)>u(x)(by′′continuityofu)andpx≤:e(p,v(p,m))=m(8)Inwords:theminimalexpenditurenecessarytoreachutilityv(p,m)ism.∗Ifinsteadwestartwithexpenditureminimization,thenxsolvesutilitymaximizationsuchthat:v(p,e(p,u))=u(9)Themaximumutilitywhichcanbederivedfromincomee(p,u)-encesthenwehavethefurtheridentities:3)x(p,m)=h(p,v(p,m))4)h(p,u)=x(p,e(p,u))Animportantconsequenceofidentity(3)istheSlutskyequation:Theorem16SlutskyEquationAssumestrictlymonotonic,=(∂h/∂p)issymmetricandnegativedefinitejiandcanbeexpressedas:∂h/∂p=∂x/∂p+x∂x/∂m(10)jijiijDrawpictureindimension2:
Proof:Rememberfromtheenvelopetheoremthat∂e(p,u)/∂p=h(p,u).TheHessianiiHofp→e(p,u)issymmetricandthusbySchwarz’theorem:∂h/∂p=∂h/∂p(11)ijjiSincetheexpenditurefunctionisstrictlyconcaveinptheHessian()(3),thatthederivative∂x/∂misnotnecessarilypositive(Giffengoods!):butthiscaseishardtofindinpracticeisinceingeneralincomeeffectsgointheotherdirection(mostgoodsarenormalgoods).,:(p)=maxp·ysuchthaty∈Y(productionpossibilityset)(12)Remark17Note,thatwealwaysassumedthatconsumersdonothavepricingpower().,
LectureXIII:,:(intersectionofallconvexsetscontainingB).TheconvexhullcontainsalllinesconnectingpointsinsidethesetB:∑J∑co(B)={pjxj,xj∈B,pj≥0,pj=1}(1)j=1jRecallthatahyperplaneinRnisann−:Definition1Anaffinehyperplaneisgeneratedbysomep∈Rnnonzeroandc∈RsuchthatHp,c={z∈Rn,p·z=c}(2)Wecanalsodefinethehalfspaceaboveandbelowthehyperplane:thehalfspaceaboveisthesetofzsuchthatp·z≥c,andthehalfspacebelowisthesetofzsuchthatp·z≤∈/∈Rnnonzeroandc∈Rsuchthatp·x>c>p·bforanyb∈
Proof:Considerf(b)=||b−x2||forb∈:B→Riscontinuousthusreachesitsminimumatsomepointb∗∈,graphically,b∗=x−b∗andc′=p·b∗.p·x−c′=p·(x−b∗)=||x−b∗2||>0Foranyb∈B,p·b−c′=p·(b−b∗)=(x−b∗)·(b−x+x−b∗)=||x−b∗2||−(x−b∗)·(x−b)≤=c′+ where =||x−b∗||/∈/∈Rnnonzerosuchthatp·x≥p·bforanyb∈¯∈B¯().Inthatcasetakexm→xandxm∈/B¯.∈Rn,notidenticallyzero,andascalarcsuchthatp·a≤c≤p·bforalla∈Aandb∈,continuousandmonotonicpreferences(-concavecontinuousutilityfunctionrepresentingt∑heirpreferences)andinitialendowmentseisuchthatthe’socialendowment’ise=ni=
Theorem5SecondWelfareTheoremConsideraPareto-optimalallocationxiofthesocialendowmenttoeachconsumeri().Assumethatthisallocationisstrictlypositive(xik>0foreachgoodk).Then(xo)istheallocationpartofaWalrasianequilibrium,∑of:Weknowthatxi=eˆ≤{xcanbeallocatedamongthenconsumers}inawaythatstrictlyX∗=x∈X|Paretodominates(xi)(3)ItiseasytoseethatX∗∗asthesocial’better-than’ˆ{=x∈RK}|x≤e(4)ˆandX∗·x≤cforallx∈Xˆandp·x≥cforallx∈X∗.Thiscanbeinterpretedasthesocialbudgetconstraintoftheplanner:givenincomecandpricespthesocialplannerwouldchoosetheallocationeˆ.Thepricevectorpsatisfiesseveralproperties:≥ˆwhichisa’box’.1Thesecondwelfaretheoremiscloselyrelatedtothethedebateabout’equalityofoutcomes’versus’equalityofopportunities’.Basically,’staticefficiency’:,itisgenerallydynamicallyinefficient,
ˆisontheboundaryofX∗aswellasontheboundaryofX∗,wehaveeˆ·p=caswellase·p=·ei=p·(p,(xi)):ˆisuchthatpxˆi≤α<1wehavethatαxˆiisstillstrictlypreferredtoxiandp·αxˆi<˜i=αxˆ’budgetsurplus’topurchasemoregoodsforconsumersingroupIsuchthatthereisastrictlyParetodominatingallocationx˜suchthatx˜∈X∗andp·x˜<-bothinstaticmodels()anddynamicmodels(avaluefunctionsolvingaBellmanequationtypicallyexists).,andishenceusefultostateagain:Theorem6Assumeacontinuousfunctionf:[a,b]→[a,b].Thenthereexistssomex∈[a,b]suchthatf(x)=:Weconsiderg(x)=f(x)−(a)≥0andg(b)≤∈[a,b]
Itisusefultopointoutthecrucialassumptionswhicharenecessaryforthefixedpointresult:Continuity:-ness:fmapseveryelementofitsdomainintoanotherelementofthedomain(here[a,b]).Compactness:Thedomainhastobecompact()Convexity:’sandSchauder’sFixedPointTheoremsBrower’sfixedpointtheoremgeneralizedtheabovecorollarytofunctionsofmanyvari-ables,andSchauder’stheoremmoregenerallytocontinuousfunctionsongeneralvectorspaces(suchasC[a,b]withthesupnorm).Theorem7BrowerFixedPointLetAbeanon-emptyconvexandcompactsubsetofRmandfisacontinuousmapf:A→∈Asuchthatf(x)=,’-dimensionalspacessuchasC[a,b]thereisSchauder’-emptyconvexandcompactsubsetofacompletevectorspaceXandletfbeacontinuousmapf:A→∈Asuchthatf(x)=:Corollary9LetAbeanon-empty,convexandboundedsubsetofC[a,b]:A→∈Asuchthatf(x)=
Thisfollowsimmediatelyfrom(a)firstapplyingArzela-AscoliandshowingthatAiscompactand(b)usingSchauder’’sandSchauder’,someintuitioncanbederivedfromthesimpleapplicationoftheIVTabove(whichbythewayisaspecialcaseofBrower’smoregeneraltheorem).:A→,y∈Awehave:||f(x)−f(y)||≤C||x−y||(5)forsomeC<:Proposition10Afunctionf∈C1(A)forsomecompactA⊂Risacontractionmap-pingif|f′(x)|<1forallx∈:Thisisanimmediateconsequenceofthemeanvaluetheorem:f(x)−f(y)=f′(ζ)(6)x−yforsomeζ∈[x,y].Butf′<1suchthat|f′(x)|≤,<,thatbythemeanvaluetheo-remanycontinuousfunctiononRnfulfilltheinequality||f(x)−f(y)||≤C||x−y||forsomeC>
:Thisfollowsimmediatelyfromthedefinition:takeysufficientlyclosetoxandwecansurelyfulfill||f(y)−f(x)||≤ Wenextdefineasequence(xn)⊂Astartingfromsomearbitraryx0∈A:xn+1=f(xn)(7)Lemma13Thesequence(xn)isCauchyandconvergestosomex∗.Proof:Notethat||xn+1−xn||=||f(xn)−f(xn−1)||≤C||xn−xn−1||≤Cn||x1−x0||(8),forsomem>nwecanapplythetriangleinequalityandfind:||xm−xn||≤||xm−xm−1||+||xm−1−xm−1||+..+||xn+1−xn||≤Cm||x1−x0||+Cm−1||x1−x0||+..+Cn||x1−x0||[]=Cn||x1−x0||Cm−n+Cm−n−1+..+1≤Cn11−C||x1−x0||(9)(xn)∗.QEDLemma14Thelimitpointx∗:Simplyconsiderthedefinitionxn+1=f(xn),∗.7
. not (x)x1 x2Figure1:Illustrationofupperandlowerhemi-continuityofacorrespondenceF(x)Proof:Assumethattherearetwofixedpointsx∗andxˆ.Thenwehave||x∗−xˆ||=||f(x∗)−f(xˆ)||<C||x∗−xˆ||.Thiscanonlybetrueif||x∗−xˆ||=0whichimpliesx∗=xˆ.,⊂RmandY⊂-valuedmappingF:x→F(x)⊂=(x,y)|y∈F(x).8
-continuous,.,atx0ifF(x0)isnonemptyandforeverysequencexn→x0andeveryy0∈F(xo)thereexistsasequence(yn)suchthatyn∈F(xn)andyn→-graphpropertyAcorrespondenceisupperhemi-continuous,.,ifforallsequences(xn,yn)suchthatxn→xandyn→yandyn∈F(xn)wehavey∈F(x).:X→XwithX⊂Rmhasafixedpointx∈Xsuchthatx∈F(x),(x)(x)’sfixedpointtheoremisaspecialcaseofKakutani’:’
’(strategicform)={1,2,..,I}.,eachoftheplayershastheoptiontogotheEmpireStatebuildingormeetattheoldoaktreeinCentralPark(whereeverthatis...).SothestrategysetsofbothplayersareS1=S2={E,C}.’strategyprofile’,inourgametherearefourpossibleoutcomes-bothplayersmeetattheEmpirestatebuilding(E,E),theymiscoordinate,(E,C)and(C,E),ortheymeetinCentralPark(C,C).Mathemat-ically,thesetofstrategyprofiles(oroutcomesofthegame)isdefinedasS=S1×S2Inourcase,,andplayer2cantake10possibleactions,’,too,fine!IftheychooseEbuttheotherplayerchoosesC,,notactions(ofcoursetheiractionsinfluencetheoutcome-butforeachactiontheremightbemanypossibleoutcomes-inourexampletherearetwopossibleoutcomesperaction).Recall,thatwecanrepresentpreferencesover10
Figure2:Exampleofa2by2game-NewYorkGameECE1,10,0C0,01,,preferencesoveroutcomesaredefinedas:ui:S→RInourexample,ui=1ifbothagentschoosethesameaction,:Amoreformaldefinitionofagameisgivenbelow:Definition25Anormal(strategic)formgameGconsistsof•AfinitesetofagentsD={1,2,..,I}.•StrategysetsS1,S2,..,SI•Payofffunctionsui:S1×S2×..SI→R(i=1,2,..,n)We’llwriteS=S1×S2×..×SIandwecalls∈Sastrategyprofile(s=(s1,s2,..,sI)).Wedenotethestrategychoicesofallplayersexceptplayeriwiths−ifor(s1,s2,..,si−1,si+1,..sI).-StrategyNashEquilibriaWriteΣi(also∆(Si))ΣforΣ1×..×Σσ∈ΣisanI-tuple(σ1,..,σI)withσi∈Σ
Wewriteui(σi,σ−i)forplayeri’sexpectedpayoffwhenheusesmixedstrategyσiandallotherplayersplayasinσ−i.∑ui(σi,σ−i)=(10)si,s−iui(si,s−i)σi(si)σ−i(si)(strategyprofiles)σ∗∈Σsuchthat()()uiσ∗,σ∗≥u∗i−iiσi,σ−iforalliandallσi∈Σ-ResponseCorrespondenceThemixedstrategybestresponsecorrespondenceforplayeriBRi:Σ−i→ΣiisdefinedbyBRi(σ−i)=argmaxui(σi,σ−i)σi∈Σi()Proposition29σ∗isaNashequilibriumifandonly∗ifσ∗∈BRiiσ−
LRU1,10,4D0,22,1Figure3:2by2gameInthe2by2gameoffigure3wehave:Uifβ>23BR1(βL+(1−β)R)={αU+(1−α)D|α∈[0,1]}ifβ=23Difβ<23Lifα<14BR2(αU+(1−α)D)={βL+(1−β)R|β∈[0,1]}ifα=14Rifα>’best-responsecorrespondencesinasinglecorrespondencewecandefineBR:Σ→Σsuchthatforamixedstrategyprofileσ∈Σwehave:BR(σ)=(BR1(σ−1),BR2(σ−2),..,BRI(σ−I))ThenextlemmajustfollowsdirectlyfromthedefinitionofmixedNashequilibrium:Lemma30Amixedstrategyprofileσ∗isaNashequilibriumifandonlyifitisafixedpointoftheBRcorrespondence,.σ∗∈BR(σ∗).13
’:TheindividualmixedstrategysetsΣ,thatwecandenoteamixedstrategythroughthevector(p1,p2,..,pn)σm=(pm1,..,pmn)whichconvergesto(p∗1,..,p∗n).Thenthislimitpointisagainamixedstrategyprofilebecausetheprobabilitiessumupto1(thelimitofthesumisthesumofthelimits!).Therefore,thesetsΣ:Foranyplayeriandanyprofileσ−iweknowthatthelinear(andhencecontinuous)utilityfunctionuimapsthecompactsetΣiintoacompactsetontherealnumbersdefinedasA={ui(σi,σ−i)|σi∈Σi}.-responsetoσ:Assumethatσ∗andσ∗∗arebothBRofplayeritoσ,:Assumethatσnisasequenceofstrategyprofilesandσ˜n∈BR(σn).Bothsequencesconvergetoσ∗andσ˜∗,()uiσ˜n)(,σn≥u,σni−iiσ′i−
forallσ′∈Σ,,wecantakethelimitonbothsideswhilepreservingtheinequalitysign:()()uiσ˜∗,σ∗≥ui−iiσ′,σ∗i−i()forallσ′∈Σ∈σ˜∗iiσ∗−andthereforeσ˜∗∈BR(σ∗).
LectureXIV:,(1989)∑∞supβtF(xt,xt+1)+1∈Γ(xt)andx0∈Xgiven(1)(xt+1)∞t=0t=0Γ(xt)isacorrespondencewhichdefinesanequationofmotion:eachstatevariablextismappedintoanewstatevariablext+1takenfromthesetΓ(xt).
=f(xt)andconsumesct=yt−xt+1whichgiveshimper-periodutilityF(xt,xt+1)=u(ct)=u(f(xt)−xt+1).Note,thatΓ(xt)=[0,f(xt)].Definition2Givenx0∈X,thesetΠ(x0)ofallfeasibleplansisgivenbyΠ(x0)={(x∞t)t=0|xt+1∈Γ(xt)}(2)(notnecessarilybounded).Γ(x)isnon-empty,compact-valuedandcontinuousforallx∈×,or,alternatively,thestatevariablesliveinaboundedset(andhenceboundedinitsdomain).InourRobinson-example,theutilityfunctionisnotnecessarilybounded,butthemaximumoutputlevelisboundedfromabovebyybecausetheareaofRobinson’∗isgivenbyv∗(x0)=supU(x)(3)x∈Π(x0)Note,∗satisfiestheBellmanequa-tionv(x)=max{F(x,y)+βv(y)}(4)y∈Γ(x)Conversely,everysolutionoftheBellmanequationsolvesthesupremumproblemifβnv(xn)→:seeStokeyandLucas,,
3TheoremoftheMaximumWecanshow,Γ:Theorem6Underassumption3thesupremumfunctionv∗(x):Considerv(x+h)(xˆt)withxˆ0=(x+h)≤F(x+h,xˆ1)+βv(xˆ1)=F(x+h,xˆ1)−F(x,xˆ1)+v(x)v(x+h)−v(x)≤|F(x+h,xˆ1)−F(x,xˆ1)|(5)assumingthatxˆ1∈Γ(x)andΓ(x+h).Otherwisewecanfindsomex′1closetoxˆ1bythecontinuityofthecorrespondenceΓ.Bysymmetrywecanrepeattheaboveargumentandfind:v(x)−v(x+h)≤|F(x,x˜1)−F(x+h,x˜1)|(6)Inbothcases,,∈C(X).Recall,:C(X)→C(X):(Tv)(x)=max[F(x,y)+βv(y)](7)y∈Γ(x)Note,thatthismaximumisalwayswell-definedbecauseΓ
solvingthefollowingproblem-theright-handsideisacontinuousfunctionH(x,y)=F(x,y)+βv(y)inxandyandwetrytofind:h(x)=maxH(x,y)(8)y∈Γ(x)Themaximizercorrespondencey(x)isthesetofy-valueswhichmaximizeH(x,y)Γ(x)iscontinuousinxandthemaximizercorrespondencey(x):,however(drawgraphs).’sTheoremandConstructionoftheValueFunctionThefinalelementinouranalysisisBlackwell’,w∈C(X)theBellmanoperatorsatisfies||T(v)−T(w)||≤β||v−w||.Proof:Assumethatthemaximizerofvisyv(x)andofwisyw(x).Thenwecanwrite:Tv(x)=F(x,yv(x))+v(yv(x))=F(x,yv(x))+w(yv(x))−w(yv(x))+v(yv(x)))≤Tw(x)+||v−w|||Tv(x)−Tw(x)|≤||v−w||||Tv−Tw||≤||v−w||(9)4
QEDSinceTisacontractionmappingthereisauniquefixedpointwhichcanbeconstructedrecursivelybystartingfromsomearbitraryguessforavaluefunctionv0andtheniteratingsuchthatvn+1=,(x):Assumption10Foreachy,F(.,y),Γismonotoneinthesensethatx≤x′impliesΓ(x)subsetΓ(x′):.Γ,(x)isacontinuous,:’(themaximizer)
7PolicyConvergenceConsiderasequenceofvaluefunctions(vn),becauseoftenwecanshowsomepropertyPholdsforeachgn(suchasdifferentiabilityorconcavityofthepolicyfunction).Ifconvergenceholds,thenthelimitpolicyfunctionwillbedifferentiable/,(x)=g(x).IfXiscompactthenconvergenceisalsouniform,.||gn−g||→-OrderConditionsAssumption15FiscontinuouslydifferentiableontheinteriorofX×,15,10and3thevaluefunctioniscontinuouslydifferentiableforeachx0∈-orderconditionsfortheBellmanmaximizationproblemaswellasapplytheenvelopetheorem:∂F(x,y)+βV′(y)=0∂yV′(x)=∂F(x,y)(10)∂xWecancombinetheseequationstoobtaintheEulerequation:∂F∂F(xt,xt+1)+β(xt+1,xt+2)=0(11)∂y∂xThisistheequationwhichwewouldobtainfrom’naively’optimizingtheinfinitesumdefiningthesupremumfunctionv∗.6
9SufficiencyofEulerequationsNote,thattheEulerequationallowsustocalculatext+2givenxtandxt+’forwardpath’.→inftyβtFx(xt,xt+1)=,,wecaninterpretFx·:
LectureXV:,:-tially,whichmeansit’,:#include<>#include<>#include<>/*Definesizeofthestatespace-onedimensiononlyinthisexample;epsilondescribesthemaximumacceptabledifferencebetweenV-T(V)whereTistheBellmanoperator*/1
#definen#
}}}/*calculatediff=||v-v_new||*/diff=0;for(k=0;k<n;++k){diff=diff+abs(v[k]-v_new[k]);}v=v_new;}writedata();}/*Thisroutineisoptional:itoutputsyourdataintoatextfilesoyoucanreaditintoMATLABorExcelanddrawthepolicyandvaluefunctions*/intwritedata(){charname[10];FILE*fp;externdoublev[n],g[n];intk;printf("NameofData-File:");scanf("%7s",&name);strcat(name,".dat");fp=fopen(name,"w");/*Nowwritevaluefunctionandpolicyfunction*/for(k=0;k<n;++k){fprintf(fp,"%,%\n",v[k],g[k]);}fclose(fp);}3
•.•:typicallyyouhavetosearchoverallpossiblex∈Γ(x)):ifxisone-dimensionalt+,
(k)=max[u(c)+E(V(f(k)−c+η)](1)ηcwhereη-otherwisetheprocesstendstosettledownveryquicklyaroundthesteadystateandwedon’tlearnmuchaboutthevaluefunctionexceptitssteadystatevalue.•’tknowthevaluefunctioneverywhere-jyoujusthaveestimatesofthefunctionatstatesyouvisitedbefore.∗•,sincetransitionstothosestatesarerare(otherwiseyouwouldhavegoodestimates)youdon’thavetoworrytoomuchaboutthem.∗•FortheoptimalctherewillbepossiblestochastictransitionstoseveralstatesK=f(k)−c+η.Chooseoneparticulartransitiontosomestatek∈+1j+1j+1•Calculateanewestimatedvaluefunctionwasfollows:j+1w(k)ik =kjj+1w(k)=(2)j+1∗w(k)+α[u(c)+w(k)−w(k)]ifk=kjjj+1jj+1Theparameterαispositiveanddeterminesthespeedoflearning:’backward’
α,thatthevaluefunctionisnotacompletearray-it’,:LinearizationAroundSteadyStateProblemset3showedyouthatyoucanusethefollowingingredientstosolvefortheapproximatepolicyfunctionaroundthesteadystate:•+1equationswherenisthenumberofstatevariables.•Solveforthesteadystatewherex=+1t•Linearizethen+1conditionsyouobtainedabovearoundthesteadystate.′′∗•YouhavepartialderivativesoftheformV(x)-inatealltheseunwantedconstantsfromyourn+:InspiredGuessesSometimesyoucanguessthepolicyfunctionorthevaluefunctionbylookingatitforashort(orverylong)’=1000Dollarsandcanbuyartichokesand/
(a,b,a,b,..).-Douglasoftheform(ab)andhisdiscountfactorisδ===,:[](y,p)=max(ab)+(((y−a−pb),)+((y−a−pb),))a,b(3)=kyandb=ky/−(y,p)=,oncewepluginthepolicyandvaluefunctionsonbothsideswecandeletetandpandgetanequationinKandk:[]−−=k++(1−2k)(4)-handsideas:[]−−−(ab)++(y−a−pb)p(5)2︸︷︷︸CThetwoFOCarenow:11−−−=KC(y−a−pb)−−=KC(y−a−pb)p(6)42b1Weimmediatelycandeducethat=
Wealsogetbyreplacinga=pb:()−−=2KC−2bp1yb=(7)222(1+2CK)py1Henceb=kjustasrequiredandk=.22p2(1+2CK)Sowehavetwoequationsintwounknownswhichhasasolution(checkthis!).
LectureXVI:,20021IntroductionAdifferentialequationisarelationshipbetweenafunctiony(t)(t)wheret∈I⊂Rthatsatisfiesadifferentialequationoftheform′′′(n)F[t,y(t),y(t),y(t),..,y(t)]=0.(1)(k),..,yandt∈Isuchthat:y(t)=−100k′Example1y(t)+y(t)−t=(t)=(n)(n)F(t,y,..,y)=a(t)y+..+a(t)y+b(t)=0(2)0n(n)islineariny,..,
′Ahomogenouslinearfirst-orderdifferentialequationhastheformy(t)=a(t)y(t)wherea(t)isagivenfunction.∫t′•Solutionisy/y=a(t)fromwhichwederivey(t)=Cexp(a(s)ds).0•Ifyouspecifyy(0)=y,thesolutionisunique,=-HomogenouslinearequationsNowconsiderequationsoftheform′y(t)=a(t)y(t)+b(t(3)wherea(t)andb(t)aretwogivenfunctions.∫tAssumethaty(t)’ssubstituteanewfunctionC(t)=y(t)exp(−a(s)ds).0∫tSoy(t)=C(t)exp(a(s)ds).That’stheformofthehomogenousequation,butyouno0longerassumethatC(t)isconstant.∫∫∫ttt′′′•Theny(t)=C(t)exp(a(s)ds)+C(t)a(t)exp(a(s)ds)=C(t)exp(a(s)ds)+000a(t)y(t)=a(t)y(t)+b(t)∫t′•SoC(t)=b(t)exp(−a(s)ds)0∫∫tr•SoC(t)=C+b(r)exp(−a(s)ds)drandthus00[]∫t∫∫rt−a(s)ds)a(s)ds00y(t)=C+b(r)edre0isthegeneralformofthesolution.•Onceagainuniqueifyouspecifyy(takeC=y).002
′f(y)y(t)=g(t(4)∫xLetF(x)=f(s)dsdenotetheprimitiveoffsuchthat0′′F[y(t)]y(t)=g(t(5)Sointegratingdelivers:∫tF[y(t)]=c+g(s)ds.(6)0Sometimesitispossibletoobtainanexplicity(t)-′1tialvalueproblem:y(t)=f[t,y(t)]andy(t)=-valued002functiondefinedonaopensetUofRandthat(t,y)∈,theproblemhasa00uniquemaximalsolution,:Ifweknowy(t)wecanintegratebothsidesandfind:∫xy(x)=y(x)+f(t,y(t))dt(7)0x0WecanthendefineamapT:C(U)→C(U)suchthat:∫xTy(x)=y(x)+f(t,y(t))dt(8)0x0Therefore,’stheorem,wehavetofocusonacompactsubsetofC(U),
Thepreciseproofofthetheoremisquitehard,buttheintuitionisnotsodifficult:essentially,,mayloose′2/,theproblemy(t)=3y,y(0)=0hastwosolutions:y(t)=013andy(t)==0].,consider′222y=1+y,y(0)=(t,y)=1+ydefinedoverRbutsolutiony(t)=tan(t)isdefinedonlyover(−pi/2,pi/2).(p)(p−1)y(t)=f[y(t),...,y(t),t](9)(p−1)andgivenvaluesfory(0),...,y(0).(t)isavector:y(t)∈,.′′′y(t)+by(t)+cy(t)=0(10)4
withb,crealnumbers.λtλt′λtWe’guess’(t)=e,y(t)=λe,′′2λty(t)=λ,λhastosatisfythefollowingquadraticequation:2λt(λ+bλ+c)e=0(11)2andthusλ+bλ+c=:2P(λ)=λ+bλ+c(12)whichhastworootsλandλ.122Therearethreecasestodistinguishdependingon∆=b−∆>0:Rootsarerealanddistinct.λtλt12Thegeneralsolutionisnowy(t)=ce+′forallinitialconditionsy(0)andy(0)wehavetoplugthemintothecandidatesolution:c+c=y(0),12′λc+λc=y(0)(13)1122orequivalently11cy(0)1=.(14)′λλcy(0)∈∆=0:λt1y(t)=(c+tc)e(15)
∆<0:RootsarecomplexconjugatesThezerosofthecharacteristicpolynomialareλ=−b/2+iw,λ=−b/2−−bt/2y(t)=e[ccos(wt)+csin(wt)](16)′′′y+by+cy=f(t)(17)wheref(t)isgivenandisgenerallynon-zero.∗∗Thestrategyistofindaparticularsolutiony(t).Ifyisalsoasolution,thenz=y−y′′′satisfiesz+bz+cz=,weobtainthegeneralsolution:∗y=y+generalsolutionofthehomogeneousequation.(18)•Itisgenerallytrickytofindaparticularsolution.•Iff(t)=fconstant,thenyisaconstant.•Iffisapolynomial,lookforapolynomialsolution•(t)livesinRandthecurvex(t):′x˙=x(t)=Ax(t)(19)6
whereA∈M(R)andinitialconditionx(0)=()willdefinetheoptimalconsumption/,,itisextremelyusefultounderstandthepropertiesofitsunique(!).′Corollary6Thelinearsystemx(t)=Ax(t)hasauniquesolutionwhenx(0)=∈R.′LetV={x:x(t)=Ax(t)}.(t),..,s(t)is∑iabasisofsolutions,thegeneralsolutioniscs(t),wherec∈∑iAnyparticularsolutionisobtainedbyfindingtheconstantsc,..,csuchthatcs(0)=1niix(0).:n=1Inthiscasethereisauniquesolutionwhenx(0)isspecifiedoftheformx(t)=x(0)exp(at)(20)Case:MatrixAisDiagonal′AssumeA=diag(λ,..,λ)thenx(t)=λx(t)andtheuniquesolutionisx(t)=1niiiix(0)exp(λt).iiAbasisforthesetofsolutionsisexp(λt)
-zerovectorsuchthatAv=λvforsomeλ∈λiscalledtheeigenvalue.λt′Theneatfactabouteigenvectorsisthatt→evisasolutionofx=,..,=(v..v).Pisinvertiblebecauseitscolumnvectorsarelinearlyindependent1n−1andA=PDPwhereD=diag(λ,..,λ).Theeasiestwaytothinkaboutidentityis1nthefollowing:•.−1•Pmapseacheigenvectorintoacanonicalvector.•Thisvectoristhenstretchedbyλ.•Finally,thestretchedvectorismappedbackintooneoftheeigenvectors.−1IfAisdiagonalizablewecandefineachangeinvariablesy(t)=Px(t).Then′−1′−1y(t)=Px(t)=PAx(t)=Dy(t).′y(t)=λy(t)(21)iiiλtiSoy(t)=ceandii∑λtix(t)=Py(t)=cev(22),..,vbeabasisofeigenvec-1n∑n′λ=Axisoftheform:x(t)=cevwherec∈λti(thesetofsolutionsisavectorspacespannedbyev).i8
Example11Consider−13A=−24Theeigenvaluesareλ=1,vectorv=(3,2)andλ=2,vectorv=(1,1).Thegeneral1122solutionis∑λtt2tix(t)=cev=cev+cev.(23)ii1122it2tt2tThatis,x(t)=3ce+ceandx(t)=2ce+:x(0)=1andx(0)=:123c+c=1,122c+c=0(24)12soc=1andc=−
LectureXVII:,:whenandhowcanwediagonalizeamatrix?1ComplexNumbers2Acomplexnumberisoftheformz=x+iysuchthati=−.•Differentiatingacomplexfunctioninvolvesdifferentiatingtherealandimaginary′′termseparately:z′=x+iy.•Theconjugatez¯ofacomplexnumberzhasthesamerealpartbutnegativeimagi-narypartofz:z¯=x−iy(1)•Theabsolutevalueofacomplexnumberis:222|z|=zz¯=x+y(2)iθ•Itisalsousefultodefinee=cosθ+isinθ.Thisfollowsimmediatelyfromthedefinitionofthetrigonometricfunctionsandtheexponentialfunctionasseriesex-1
pansions:∞∑izexp(z)=i!i=0∞∑2izisin(z)=(−1)(2i)!i=0∞∑2i+1zicos(z)=(−1)(3)(2i+1)!i=0•Anyrealnumberzcanberepresentedasz=|z|exp(iθ)forsomeθ.(),v,..,:12nP=(v,v,..,v)(4)(check!):iiPe=v(5)ii−×,weassumethecanonical2
,usingtheabovematrixPitiseasytoswitchbasisandcalculatethecorrespondingnewAmatrixwhichinducesthesamelineartransformation:−1B=PAP(6)Inwords:−1•Pchangesthebasisfromvtothecanonicalbasis.•Ainducesthelineartransformationinthecanonicalbasis.•,,,B∈M(R).WesaythatAandBaresimilariffthereisPn−1invertiblesuchthatA=(C)m,nwhichconsistsofallm×∈M(C).IfAx=λxandx =0,wesaythatλ∈Cisaneigenvaluen,nofAandxaneigenvectorassociatedtoλ.Therefore,λisaneigenvalueiffdim[Ker(A−λI)]≥(A−λI)iscalledtheeigenspaceofAfortheeigenvalueλ.dim[Ker(A−λI)]iscalledthegeometricmultiplicityoftheeigenvalueλ.Lemma4Ifx,..,xareeigenvectorsforλ,..,λdistincteigenvaluesthenx,..,
Proof:=−1andlet’sworkwithcaser:r∑∑µx=0impliesµλx=0(7)iiiiii=1i∑r−1Takedifference,thisimplies(λ−λ)µxwhichinreturnimpliesbythein-riiii=1ductionassumptionthat(λ−λ)µ=µ=0fori=1,..,r− =0wealsohaveµ=,∈M(C),:’:•det(A)=det(B)•tr(A)=tr(B)•
Proposition8ThematrixA∈M(R)isdiagonalizableiffthereexistsaninvertiblenmatrixP∈M(R)suchthatn,nλ1−1A=PP.λn−1Proof:TheproofisimmediateifyouthinkofPandPasmapsfromcanonicalbasisintothebasisofeigenvectors,=(v,v,..,v).QED12nNotethatifAdiagonalizable,•det(A)=λ...λ,1n•tr(A)=λ+...+λ1nkλ1k−1•A=PPkλnk•λ∈CisaeigenvalueforAiffKer(A−λI) ={0}iffdet(A−λI)=∈M(C)isdefinedasnP(X)=det(A−IX).AProposition10LetAbeamatrixinM(C).ThecomplexnumberλisaneigenvaluenforAiffP(λ)=:nrr1kP(X)=(−1)(X−λ)..(X−λ)(8)A1kwhereλ,..,λ∈C,rareintegers≥1andr+..r=
Theorem11Cayley-HamiltonThematrixAsatisfiesthematrixequationP(A)=⇐⇒dim[Ker(A)]≥1⇔Aisnotinvertible⇔det(A)=×,”arandomlychosenmatrix(foraprobabilitymeasureequivalenttoLebesgue’s)isalmostsurelydiagonalizable”.νofλisequaltothemultiplicityroftherootofP(X).≤ν≤=ν:,thenyouaredone!StepII:!FindthebasisbysolvingthecorrespondinglinearequationsAv=λvforeachrootλ.6
StepIII::IfstepIIIgivesyoulessthanneigenvectorsthematrixisNOTdiagonalizable.−13Example14A=−24∣∣∣∣∣−∣1−λ3∣∣det(A−λI)==(λ+1)(λ−4)+6∣∣∣∣−24−λ2andthusP(λ)=λ−3λ+λ=1,λ=:v =0suchthat(A−I)v=−23A−I=andv=(3,2)−23Noteofcoursethatifviseigenvectorthentvisalsoeigenvectorforanyt∈−33A−2I=andv=(1,1)works2−2232SoP=(v,v)=12211−2−12−1P=−=−232−310−1CheckthatA=PP02nWhenrootsarecomplex,thenyouneedtolookforeigenvectorsinCandthematrixPwillbeinM(C)λ=a+ibisacomplexeigenvalue¯associatedwithcomplexvectorv=u+λ=a−ibisalsoaneigenvaluewitheigenvectorv¯=u−
¯Proof:SinceAv=λv,weinferthatAv=Av=λ′Definition16AsquarematrixAissymmetriciffA=A′′Proposition17AissymmetriciffforallvectorsX,Y:XAY=.<X,AY>=<AX,Y>.′′′′′′Proof:IfthematrixAissymmetric,weknowthatXAY=(XAY)=YAX=YAX.′′Conversely,letX=eandY==eAe=eAe=′Example18AAissymmetric.′Proposition19ForanymatrixArk(AA)=rk(A).′2′′′Proof:AAX=0implies||AX||=XAAX=0impliesAX=0,soKer(AA)⊂∑n′′Exercise20ShowthatAA==1usedineconometrics.′Definition21AmatrixΩ∈M(R)iscalledorthogonalifΩΩ=ΩformanorthonormalbasisofR.′Proof:LetC,...,CdenotethecolumnvectorsofΩ.Wecheckthat(ΩΩ)=<C,C>.1ni,jijQED•Orthonormaltransformationspreservetheinnerproduct:′′′′<ΩX,ΩY>=(ΩX)ΩY=XΩΩY=XY=<X,Y>ThisimpliesthatΩ
•Italsoleavesdistancesunchanged:<ΩX,ΩX>=<X,X>.Forthisreason,Ω∈M(R)suchthat:nλ1−1A=ΩΩ(9)λn′andΩΩ=::AssumeAv=λv,Av=λvandλ =λ.Then11122212<Av,v]=λ<v,v>=<v,Av>=λ<v,v>(10)1211212212Therefore<v,v>=012⊥Proposition25FactII:Ifvisaneigenvector,thenH=R·v)isinvariantunderA,⊂:<Ah,v>=<h,Av>=λ<h,v>=:
Proof:λ=a+ibsuchthatAz=λzwithnz∈=x+iywithx,y∈R(11)Nowseparaterealandimaginarypartsandfind:Ax=ax−byAy=ay+bx(12)Asymmetricimplies<Ax,y>=<x,Ay>,therefore<ax−by,y>=<x,ay+bx>(13)22Thisgivesusb(||x||+||y||)=0sob=,λ∈Rsothereisarealneigenvectorx∈:•Theoremisobviouslytrueindimension1.•∗⊥considerA:H→HwhichistherestrictionofAonsubspacetheH=∗isofdimensionn−′Definition27AsymmetricmatrixissaidtobepositivedefiniteiffXAX>
,,,definitepositive,ifandonlyifthereexistsaninvertible2symmetricmatrixPsuchthatA=P−1Proof:A=Ωdiag(λ,...,λ)Ω1n√√√√−1−1=[Ωdiag(λ,...,λ)Ω][Ωdiag(λ,...,λ)Ω]1n1n11
LectureXVIII:,20021LinearSystems-ComplexRootsWecontinuetolookatthedifferentialequationsystemoftheformx˙=Ax(1)WerealizedthatwehavetodiagonalizethematrixwhichtypicallyinvolvessolvingthecharacteristicequationP(λ).,ifAisrealthenyouknowthatifλ=a+ibisacomplexeigenvalueassociated¯withcomplexvectorv=u+iwthenλ=a−ibisalsoaneigenvaluewitheigenvectorv¯=u−iw.¯λtλtThenev,ev¯,(t)=x(t)+ix(t)isacomplexsolutionthenx(t)andx(t)arereal1212solutionstothelinearequationsystemwitharealmatrixA.′′′′′Proof:x=Aximpliesx+ix=Ax+=Axandx=
Therefore,togettherealsolutions,taketherealandpureimaginarypartsofλtλtatev=e(u+iw)=e[cos(bt)+isin(bt)](u+iw),(2)andthusλtatatev=e[cos(bt)u−sin(bt)w]+ie[sin(bt)u+cos(bt)w].(3)atatSoe[cos(bt)u−sin(bt)w]ande[sin(bt)u+cos(bt)w]arerealsolutions.¯λtNotethatwegetthesamerealsolutionswithev¯.Practicalrecipe:DiagonalizeAinC,×2realmatrixwithcomplexeigenvaluesa±ibandcorresponding′eigenvectorsu±=Axisatatx(t)=ce[cos(bt)u−sin(bt)w]+ce[sin(bt)u+cos(bt)w]’sconsiderthe2simplecaseofAofsize2×2andonesingleeigenvalueλ,P(X)=(X−λ),λ∈R,(v,v)=(v,v),wehave12λα−1PAP=B=.0β2SinceP(X)=P(X)=(X−λ),weinferthatβ=λ.DA′Changeofvariabley=Pxleadstoy=By.′λty=λysoy(t)=′λty=λy+αce1212
λt′Tryy=c(t)e,getc(t)=αcsoc(t)=αct+c1221λtλtWeconcludethaty(t)=(c+αct)e,y(t)=λtλtx=Py=yv+yv=(c+αct)ev+cev112212122λtλtx=cev+ce(αtv+v)11212Example321A=−102FindthatP(λ)=(λ−1).Thereisasingleeigenvectorv=(1,−1)suchthatAv==(1,0)andwegetAv=(2,−1)=v+:′nnny=f(y)wherey:R→Randf:R→R.(4).∗∗∗Ifysatisfiesf(y)=0,theconstantfunctiony(t)=yisasolutionofthesystem.∗∗(y)=0iscalledasteadystateorstationarysolution.∗n∗yiscalledasymptoticallystableifthereexistsanopensetU⊂Rcontainingysuch∗thatforeveryy(0)∈U,limy(t)=y.[Everysolutiony(t)thatstartssufficientlyt→+∞∗∗closetoyconvergestoy].∗∗yiscalledgloballyasymptoticallystableifanysolutionconvergestoy.[Regardlessofy(0)]∗yiscalledneutrallystableifitisnotasymptoticallystableandeverysolutiony(t)that∗∗startsclosetoystaysclosetoyforallt.∗
′Wecalculatethesolutionsofx=×2matrixA.∑2λtiEitherwehavetworealrootsλandλ.=1Asymptoticallystableifλ,λ<λ,andλ−→−→Orwehavecomplexrootsa±ibwithcorrespondingeigenvectorsu+−→−→at−→−→x(t)=ecos(bt)(cu−cw)−esin(bt)(cu+cw)1221Itisasymptoticallystableifa<=0(periodicsolutions)Theorem4Thesteadystatex=,∗∗∗Supposethatf(y)==y−y′′∗∗y=f(y)=f(y)(y−y)+o(x)=Ax+o(x)′∗whereA=f(y).Theoremofstabilitycarriesover.∗′Theorem5Supposethatyisasteadystateofy=f(y).′∗LetA=f(y)∗IfeveryeigenvalueofAhasanegativerealpart,theny=
′Indimension1:y=y(2−y)∗∗′Twoequilibria:y=0andy=(y)=2(1−y)′f(0)=2unstable′f(2)=−:′y=y(4−y−y)1121′y=y(6−y−3y)(0,0),(0,6),(4,0),and(1,3).[Ify=0,theny(6−y)=0,whichimpliesthaty∈{0,6}1222Ory =0,andtheny+y=:Ify=0,theny=4,otherwisethe11221steadystatesolvesy+y=4,3y+y=6,implyingy,y=(1,3).]1212124−2y−y−y121′f(y)=−3y6−2y−3y22140′At(0,0),f(y)=(0,6)eigenvalues(−2−6)(4,0)(1,3)
′Inthecasen=2,wecandepictsolutionsofx=f(x).′x=f(x,x)1121′x=f(x,x)2122Ateachpoint(x,x)drawthevectorequalto[f(x,x),f(x,x)](t).′′Example:uncoupledsystem:x=2xandx=−2tx=x(0)e11−2tx=x(0)e22Notethatxxisaconstant(hyperbola)12LinearSystems′x=AxAsolutionisavectorx(t)=[x(t),x(t)](t))Twodistinctrealrootsλ,λ.12λtλt12Solutionisx(t)=cev+cev1122Saddlepathwhenλ<0<λ(importantcase)122)(λ)=0,,:6
′y=f(y,y)1121′y=f(y,y)2122Drawcurvesf(y,y)=0andf(y,y)=0(linesinthecaseoflinearsystems).112212Intersectiongivestheequilibria.′∗∗Studythematrixf(y),
LectureXIX:,-EulerequationSupposeyouwanttofindadifferentablefunctionk:[0,T]→Rthatmaximizesanintegraloftheform∫T′J(k)=u[k(t),k(t),t]dt0′subjecttotheconstraintk(0)=kandk(T)=kwhereu(k,k,t).∗Supposethatk(t)(t)beafunctionsuchthatg(0)=g(T)=ε∈R.ε∗Thenk(t)=k(t)+εg(t)isadmissible.ε∗SoI(ε)=J(k)≤J(k)=I(0)′SoI(ε)ismaximizedatε=0andthereforeI(0)=0.∫T∗′′I( )=u[k(t)+εg(t),k(t)+εg(t),t]dt01
∫T′′So′I(0)=[ug(t)+ug(t)]dt=0kk0∫∫TTdu′′TkWeintegratebyparts:u′′g(t)dt=[u()g(t)]−g(t)′that[u()g(t)]=0k0∫T′ThereforeI(0)=[u−du′/dt]g(t)dt=(t)soitmustthecasethat:du′k∗′∗∗′∗u[k(t),k(t),t]=[k(t),k(t),t]kdtTakeforexampleg(t)=u−du′/∗Theorem1IfkmaximizesJstk(0)=kandk(T)=kthenEulerequationholdsat0Teacht∈(0,T).Example:StandardRamseymodelofmacroeconomicswithoneconsumer:′Capitalaccumulation:k(t)=f[k(t)]−c(t)Objectivefunction:∫T−θtJ=v[c(t)]edt0∫T′−θtJ=v[f(k)−k(t)]′′−θtNotethatu(k,k,t)=v[f(k)−k]e′′′−θtu=f(k)v[f(k)−k]ek′′−θtu′=−v[f(k)−k]ekEulerequation′′−θt′−θtv(c)f(k)e=−d[v(c)e]/dtttt′−θt′′′′−θtSod[v(c)e]/dt=[v(c)c(t)−θv(c)]ettt′′−θt′′′′′−θtHencev(c)f(k)e=v(c)[θ−v(c)c(t)/v(c)]ettttt′′′′′f(k)=θ−v(c)c(t)/v(c)′′′′′andthereforec(t)=−[v(c)/v(c)][θ−f(k)]2
Systemoftwodifferentialequations:′k=f(k)−c′′′′′c=−[v(c)/v(c)][f(k)−θ]Equilibrium:′∗′f(k)=θ(modifiedgoldenrule,c=0)∗∗′f(k)=c(curvek=0)Linearizationaroundtheequilibrium:′∗∗k=θ(k−k)−(c−c)′∗c=−β(k−k)′′∗′∗′′∗whereβ=f(k)v(c)/v(c)>0Bysubstitution:′′′∗k−θk−β(k−k)=:,(T)==∞.:Choosefunctionc:[0,T]→Rcalledthecontrolfunction,tomaximize∫TW=u[k(t),c(t),t]dt,0′wherek(t)isthestatevariabledeterminedasthesolutionofk(t)=Q[k(t),c(t),t]withtheboundaryconditionk(0)=
Theideaistochoosec(t).Givenk,k(t)isuniquelydefinedbythedifferentialequation0′k=Q(k,c,t).≥:H=u(k,c,t)+µ(t)Q(k,c,t)FOC:′µ(t)=−H(co-stateequation)kH=0(standardFOC)cTheorem2Pontryagin’sMaximumprinciple:∗Supposethatc(t)µ(t)such∗′thatateacht,c(t)maximizestheHamiltonian(H=0),µ(t)=−H,andµ(T)=0ck(transversalitycondition).4
LectureXX:,’sproblem:∫J=maxu(c)exp(−rt)dtc˙k=f(k)−xk(0)=k(1)’shadow’’()cthereforehastosatisfy:cmaximizesH=u(c)exp(−rt)+µ(t)(f(k)−c(2)IftheHamiltonianisconcaveincwecansimplyfindthemaximumthroughtheFOCH=
:Robinsonhastobeindifferentbetweenholdingonhiscapitalbetweentimetandt+∆µ(t)+∆thesellshisadditionalunitkofcapitalatpriceµ(t+∆t).Hencethetotalprofitshemakes(theycanbenegative!)are:−µ(t)+H∆t+µ(t+∆t(3)∆t→∆tandthentakethelimityouget:µ˙=−H(4),(0)2
’tcareabouthowmuchcapitalhehasleftintheendthenµ(T)=,(sinceitdoesn’texist...).,asinthecaseoftheinsectproblem,µ(T)hastoequalthemarginalreturntocapitalattimeT,,µ(t)k(t)=0(5)t→∞3
γ.Thedeathrateofresearchersandeducatorsisδ.Thegovernmentcaninfluencetheproportionofscientistsα:˙E=αγE−δE˙R=(1−α)γE−δR0≤α≤(6)∗∗α(t)suchthatthefollowingismaximized:∫TJ=−1dt(7)0TheHamiltonianistherefore:H=−1+λ(αγE−δE)+µ((1−α)γE−δR)(8)Theoptimalcontrolisofthebang-bangtypeagain:1ifλ−µ>0α=(9)0ifλ−µ<0Thetwoarbitrageequationsare:˙−λ=λ(αγ−δ)+µ(1−α)γ−µ˙=−δµ(10)δtFromthelastequationwefindthatµ(t)=
Assumethatλ(T)>µ(T).Inthiscaseα(T)=λλ(T)<µ(T).Thisimpliesthatα(T)=:δt˙λ=−γCe(11)γCδtThisgivesusλ=B−δ∗∗whereλ(t)=µ(t)definedas:γC∗∗δtδtB+e=Ce(12)δAssoonasthispointisreachedthedifferentialequationdescribingλbecomes:˙λ=−λ(γ−δ)(13)Thissolutionhastheformλ=Dexp(−(γ−δ)t)whereDisafunctionofBandCsinceweknowthatλ∗:T,,
Economics200A–Fall,2003MeasureTheorymostrecentrevision:November25,2003IntroductionMeasureTheoryisabranchofmathematicsthatcomesfromhistoricalattemptstoformalizelength,area,,:;’characteristics;.
AbstractMeasureSpaceDefinition:Aσ-algebrainthenon-emptysetΩisacollectionΣofsubsetsofΩsuchthat•Ω∈Σ•IfA∈Σ,thenthecomplementAc∈Σ.•IfA1,A2,...isasequenceofsetswithAn∈Σforeachn=1,2,...,then∪∞i=nAi∈Σ.Ameasureon(Ω,Σ)isafunctionµsuchthat•µ:Σ→[0,+∞]•µ(∅)=0•µ(∪∞n=1An)=∑∞n=1µ(An)foranysequence(An)∞n=1ofmutuallydisjointsetsinΣAmeasurespaceisatriple(Ω,Σ,µ)whereΩisanon-emptyset,Σisaσ-algebraofsubsetsofΩ,andµisameasureon(Ω,Σ).ProbabilisticInterpretationRecallthataprobabilityspaceisatriple(S,E,P)wherethesamplespaceSrepresentsthesetofallpossibleoutcomesofachanceexperiment,thecollectionofsubsetsEaretheeventsthatmightoccur,andP(E),theprobabilityspace(S,E,P)isameasurespacewiththespecialpropertythatP(S)=
MoreTerminologyIntheestablishedliteratureonmeasuretheorythereareseveraldifferenttermsforthesameconcept,severaldifferentusesforthesameterm,-emptyset(sometimescalledΩorXor...),acollectionofsubsets(sometimescalledSorAor...)couldbeanynon-emptycollectionwithlessstructurethanthatofaσ-algebra,andafunction(sometimescalledµorνorαor...):LetΩbeanon-emptysetandSacollectionofsubsetsofΩcontainingatleasttheemptyset.•Callµasetfunctionon(Ω,S)ifµ:S→[0,∞]andµ(∅)=0.•Asetfunctionµon(Ω,S)isfinitelyadditiveifµ(A∪B)=µ(A)+µ(B)forallA,B∈SsuchthatA∩B=∅andA∪B∈S•Asetfunctionµon(Ω,S)iscountablyadditiveif∑∞µ(∪∞n=1An)=µ(An)forallA1,A2,...inSwith(An)mutuallydisjointand∪∞n=1∈Sn=1Inparticular,ameasureisacountablyadditivesetfunctiondefinedonaσ,sincethereisverylittlestructureassumedaboutthecollectionofsubsetsonwhichthesetfunctionisdefined,:GivenacollectionSofsubsetsofΩ,theσ-algebrageneratedbySisdefinedtobethesmallestσ-algebrainΩσ-algebrasinΩiseasilyseentobeaσ-algebrainΩ,andsincethereisalwaysatleastoneσ-algebrainΩcontainingS(Ω),itfollowsthattheintersectionofalltheσ-algebrascontainingSistheuniquesmallestσσ-algebraisσ(S).σ
Definition:LetΩΩisasemiringprovided•∅∈S•IfA,B∈SthenA∩B∈S.•IfA,B∈SthenA\B=∪ni=1Ciforsomecollectionofpairwisedisjointsets(Ci):•Asetfunctionνismonotoneifν(A)≤ν(B)wheneverA⊆B.•Asetfunctionνissubadditiveifν(A∪B)≤ν(A)+ν(B).•Anoutermeasureνisamonotone,subadditivesetfunctiondefinedonallsubsetsofΩ.4
ExamplesWegivesomebasicexamplesofmeasures;moresophisticatedexampleswillbeseenbelowafterthesectiononCarathe´Ω{#(A)forAfinµ(A)=ite+∞otherwisewhereforanyfinitesetA,onedenotesby#(A)(Ω,P(Ω),µ)Ω∈Ωdefinethesetfunction{1ifx0∈Aδx0(A)=0otherwiseItisclearthat(Ω,P(Ω),δx0)<p<1andn≥1aninteger,thebinomialproba-bilitymeasureis∑nµ=C−i(n,i)pi(1−pn)δii=λ>0,thePoissonprobabilitymeasureis∑∞iµ=exp−λλδii!i=05
Carathe´odoryExtensionThegoalistoconstructmeasuresonreasonablylargeσ´:•Sasemiringofsubsets(theelementarysubsets)ofanon-emptysetΩ•µ:S→[0,+∞]acountablyadditivesetfunctiononthesemiringSStep1(Extend):Defineµ∗(A):P(Ω)→[0,+∞]by{∑∞}µ∗(A)=infµ(An):An∈S,A⊆∪∞n=1Ann=1Theresultingsetfunctionµ∗isanoutermeasuredefinedonallsubsetsofΩwiththepropertythatµ∗:P(Ω)→[0,∞]isanextensionofµ:S→[0,∞],µ∗(A)=µ(A)forallA∈µ∗(A)iscomputedbycloselycoveringAwithsetsfromSandaddinguptheµµ∗neednotbeadditive,muchlesscountablyadditive,(Restrict):DefineasetA⊆Ωtobeµ-measurableifµ∗(S)=µ∗(S∩A)+µ∗(S∩Ac)forallS⊆ΩLetΣµbethecollectionofallµ:•ThecollectionΣµofµ-measurablesetsisaσ-algebraofsubsetsofΩ.•Thesetfunctionµ∗restrictedtotheσ-algebraΣµµ.•S⊆σ(S)⊆Σµ•Thesetsintheσ-algebrageneratedbytheelementarysetsarecloseapproxima-tionstotheµ-measurablesets,∈Σµ,thenthereexistsasetB∈σ(S)suchthatA⊆Bandµ(B)=µ(A).6
LebesgueMeasureThemostcelebratedmeasureconstructiblebytheCarathe´-openintervalsasthesemiringofelementarysets;:•S={[a,b):a≤binR}•λ([a,b))=b−aObservethatλ´odoryextensionprocedure,thereisanextension,againcalledλ,whichisacountablyadditivesetfunctionontheσ-algebraΣλofλ,(R,Σλ,λ),itcanbeshownthattheextensionisunique,,itisevidentthattheσ-algebragener-atedbythecollectionofhalf-openintervalsisthesmallestσσ-algebraofR,⊂B⊂Σλ⊂P(R)-dimensionalLebesguemeasureλ=2wemeasuretheareaofsetsinR2;and,forn=3,:•S={[a1,b1)×···×[an,bn):ai≤biinRforeachi=1,...,n}•λn([a1,b1)×···×[an,bn))=∏ni=1(bi−ai)7
ProbabilityMeasuresProbabilitymeasurescanberigorouslyconstructedusingtheCarathe´,supposetherealreadywereaprobabilityspace(Ω,Σ,P).(x)=P[X≤x]forallx∈RThefunctionF:R→Rhasfourproperties:•0≤F(x)≤1forallx∈R.•Fisnon-decreasing.•Fisright-continuous.•F(−∞)=0andF(+∞)=,weusethenotationF(+∞)=limx→+∞F(x)andF(−∞)=limx→−∞F(x).Finally,notethatthecumulativedistributionfunctionsatisfiesF(b)−F(a)=P[a<X≤b]Conversely,ifweweretostartwithanydistributionfunctionF,:R→Rsatisfyingthefourpropertiesabove,thenwecanconstructaprobabilitymeasureµ:={(a,b]:a≤binR}2.µF((a,b])=F(b)−F(a)Notethatµ´odoryextensionprocedurethereisameasure,againcalledµF,definedontheσ-algebraΣµFofµ,thetotalmassisµF(R)=1;.µ:(i)viathesamplepoints(theexperimentaloutcomes)byfocusingonthespace(Ω,Σ,P);(ii)viathevaluesoftherandomvariablebyfocusingonthespace(R,Σ,µ).Inparticular,theexpectationE(X)ofarandomvariableXcanbecomputedby∫E(X)=XdPΩ8
orby∫+∞∫E(X)=xdF(x)=xdµF−∞RHere,thefirstintegralistheLebesgueintegralperformedin(Ω,Σ,P);thethirdintegralistheLebesgueintegralperformedin(R,Σ,µ);,-ityspace(Ω,Σ,P)isthespaceofthe36outcomes{(o1,o2):oi=1,2,...,6foreachi=1,2}∫E(X)=XdPΩ=X((1,1))P({(1,1)})+X((1,2))P({(1,2)})+···+X((6,6))P({(6,6)})=12∗+13∗+···+112∗363636=7Ontheotherhand,µFisadiscretemeasureonthe11valuesinthesubset2,3,...,12ofRwhichassignsmass136tothevalue2,236tothevalue3,1...,∫E(X)=xdµFR=112∗+23∗+···+2∗363636=7/
Economics200A–Fall,2003Probabilitylastrevision:October31,2003Consideranexperiment(arealphysicalexperimentorathoughtexperiment)inwhichthereisuncertaintyabouttheoutcomeoftheexperimentbeforetheexperimentisperformed,:tossingacoin,tossingacoinonehundredtimes,throwingtwodice,countingthenumberofmisprintsinamanuscript,choosingoneperson”atrandom”fromapopulationof500people,whetherornotitwillraintomorrow,whetherajob-seekerwillfindemploymenttomorrow,andobservingtherateofreturnofafinancialinvestmentinagiven(future)∈;:E→Rsuchthat0≤P(E)≤1forallEP(S)=1⋃∞∑∞P(Ei)=P(Ei)forpairwisedisjointeventsEii=1i=={(1,1),(1,2),...,(6,6)}.Ifweassumethateachofthe6facesofadieareequallylikelyandthatthereisnoinfluenceofonedieontheother,theneachoftheoutcomeshasequalprobability1/;={(1,5),(2,4),(3,3),(4,2),(5,1)}.ThenP(E)=5/,wearenotreallyconcernedwiththeoutcomesdirectly,,,wewouldnotbeinterestedinthenamesofthechosenresidents;instead,wewouldbeinterestedintheannualincomeofthoseresidentschosen;
,,,infinancethisapproachmodels(amongotherthings)thesituationofone-periodinvestment:,=thesumofthenumberofspotsonthetwoupfacesofthediceHerethepossiblevaluesaretheintegers2,3,4,5,6,7,8,9,10,11,,:•ForarandomvariableweuseuppercaseletterslikeX,Y,....•Forpossiblevaluesofanrvweuselowercaseletterslikex,y,....•ForprobabilitystatementsaboutanrvweuseP[statementaboutXanditsvalues][X=6]=536Therearetwokindsofrandomvariables:•Adiscreterandomvariablehasafiniteorcountablenumberofpossiblevalues.•AcontinuousrandomvariableXhasthepropertythatP[X=x]=0foreachvaluex∈R2
,,x2,...thenthedensityfunctionofXisdefinedtobef(xi)=P[X=xi]Wetakef(x)=:•f(x)≥0forallx∑•f(xii)=,thenweinterpretf(x)asf(x)dx=P[x≤X≤x+dx].Rigorously,fisafunctionsuchthat∫bf(x)dx=P[a<X≤b]forall−∞≤a≤b≤+∞aAcontinuousdensityfunctionhastwocharacteristicproperties:•f(x)≥0forallx∫+∞•−∞f(x)dx=13
,...,,{1f(x)=nforx=1,...,-imentsuchthat(i)theoutcomeofatrialdoesnotinfluencetheoutcomeofanyothertrial,(ii)therearetwopossibleoutcomesforeachtrial,SandF,and(iii).(Forexample,.)Thedensityfunctionforthisdiscreterandomvariableis{(n)xforx=0,...,nfxpx(1−p)n−(x)=0otherwise4
[0,1].Intuitively,thismeansthateachpointin[0,1]isequallylikelytobechosen;rigorously,{f(x)=1for0≤x≤(ortheirlogarithms)“nature”suchasintelligencetestscores,humanheights,(x)=√1x−µexp{−12()}for−∞<x<+∞2piσ2σwhereµ∈Randσ2>µ=0andσ2==x−µσ.WegiveasamplecalculationusingthischangeofvariableforarandomvariableXwhichhasanormaldensitywithparametersµandσ∫aP[X≤a]=√1x−µexp{−12()}dx−∞2piσ2σ∫a−µσ=√1exp{−z2}dz−∞,σ2ratherthanσ
(orexpectedvalueormean)ofXandisdenotedbyE(X).,x2,...anddensityfunctionf,wedefine∑E(X)=xif(xi),wedefine∫+∞E(X)=xf(x)dx−∞{1,2,...,6}andtheprobabilitydensityfunctionisf(x)=16forx=1,2,...,,wecompute∑E(X)=xif(xi)i=111(1)()+(2)()+...+(6)()666=1(1+...+6)6=[0,1](x)=1for0≤x≤,wecompute∫+∞E(X)=xf(x)dx−∞∫1=xdx0=126
-sityfunctionµf2(x)=√1exp{−1x−()}for−∞<x<+∞2piσ2σwhereµ∈Randσ2>=x−µσ,∫+∞1E(X)=x√exp{−1x−µ2()}dx∫−∞2piσ2σ+∞=(σz+µ)√1exp{−z2}dz−∞2pi2∫+∞∫+∞=1σz√exp{−z2}dz+µ√1z2exp{−}dz−∞2pi2−∞2pi2=1+2Theintegralin1=0sincetheintegrandzexp{−z22}=,wehavethatE(X)=µThisistobeexpected(nopunintended)sincethedensityfunctionissymmetricaboutµ.Sincethedefinitionofexpectationisgivenintermsofeitherasumoranintegral,,thenE(X)=,thenE(X)≥αandβarerealnumbers,thenE(αX+βY)=αE(X)+βE(Y).(asmeasuredbytheexpecta-tion)-ingthedifferencebetweentworandomvariablesishowspreadout(probabilistically)2Forarandomvariablewhichhasconstantvalue,
(X2)=E((X−E(X)))Analgebraicallyequivalentformulaisusuallymoreconvenientforcomputationalpurposes;(X)=E(X2)−(E(X2))√ThesquarerootVar(X)ofthevarianceiscalledthestandarddeviation;ithasthesameunitsasXandE(X):•Theexpectation(alsoknownasthemeanortheexpectedvalue)E(X)oftherandomvariableXisoftencalledX.•ThevarianceVar(X)ofarandomvariableXisoftencalledσ2X;thestandarddeviationofXisoftencalledσ,weomitthesubscriptandjustwriteσ2andσ(thesameexamplesforwhichex-pectationswerecomputedabove.)NotethatcomputingtheexpectationofafunctionφofarandomvariableXmaybedoneb∑yusingtheappropriateformulaE(φ(X))=φ(xi)f(xi)ior∫+∞E(φ(X))=φ(x)f(x)dx−∞foradiscretervorcontinuousrv,(X)=(X2)=E((X−X))=1222252(((−5)+(−32)+(−1)+1()+3()+())6222222=
[0,1].WefoundabovethatE(X)=∫+∞E(X2)=x2f(x)dx−∞∫1=x2dx0=13Thus,Var(X)=E(X2)−(EX2)=1−12()32=−µf(x)=√1exp{−12()}for−∞<x<+∞2piσ2σwhereµ∈Randσ2>(X)=µ.Wecomputeusingthestandardchangeofvariableforthenormaldensity∫+∞Var(X2)=(x−µ)f(x)dx∫−∞+∞=x−µ(x−µ2)√1exp{−12()}dx−∞2piσ2σ∫+∞=σ2√1z2exp{−z2}dz2pi−∞2Wesetu=zandv=zexp−(X)=σ2Notethatthetwoparametersthatoccurinthedefinitionofanormaldensityfunctionare,infact,,inanynormaldensitytheinterval[µ−σ,µ+σ]containsroughly68%ofthetotalprobability;[µ−2σ,µ+2σ],roughly95%ofthetotalprobability;and[µ−3σ,µ+3σ],over99%
,theprobabilisticpropertiesofXandYconsideredtogetheraredescribedbyajointdensityfunctionf(x,y)whichdescribes(justasinthecaseofonerv)howlikelythevalue(x1,x2),,,inaportfolioconsistingof20stocksthereare20returnsofinterestand,hence,20randomvariables–,wewouldliketocombinetheserandomvariablessothatwemayanalyzetheoverallreturnR=w1R1+···+(w1R1+···+wnRn)=w1E(R1)+···+wnE(Rn)Sinceweintendtousethevarianceofreturnsasaproxyforrisk,weneedtoaskwhatcanbesaidaboutthevarianceVar(w1R1+···+wnRn).Formultiplicationbyrealnumbers,theformuladefiningvariancetellsusimme-diatelythatforarealnumberaandanrvXVar(aX)=a2Var(X)Letuscomputefromthedefinitionthevarianceofthesumoftworv’s:()Var(X+Y)=E{(X+Y)−(X+Y)}2()=E{(X+Y)−(X+Y)}2()=E{(X−X)+(Y−Y)}2()=E2(X−X)+(Y−Y2)+2(X−X)(Y−Y)()()()=E2(X−X)+E(Y−Y2)+2E(X−X)(Y−Y()=Var(X)+Var(Y)+2E(X−X)(Y−()Cov(X,Y)=E(X−X)(Y−Y)Anequivalentformwhichisusefulforcomputationisalsoavailable:Cov(X,Y)=E(XY)−E(X)E(Y)10
Notation:oftenCov(X,Y)iswrittenasσXY;andifthevariablesaresubscripted,thenCov(X1,X2)isoftenwrittenasσX1X2orσ1,[−1,+1]bydividingthecovarianceoftworandomvariablesbythestandarddevia-tionsofthetworv’s(thusmakingthismeasureofjointspreadindependentoftheunitsoftheindividualrv’s).WedefinethecorrelationcoefficientoftherandomvariablesXandYtobeρXY=σXYσXσYAfundamentalpropertyofcovarianceandthecorrelationcoefficientoftherandomvariablesXandYcanbeexpressedintwoequivalentways:•thecovariancesatisfies|σXY|≤σXσY•thecorrelationcoefficientsatisfies−1≤ρXY≤+1ThecorrelationcoefficientofXandYisameasureofhowcloselythetworandomvariablessatisfyalinearrelationship.•WhenρXY=1,therandomvariablesareperfectlycorrelated;thevariablessatisfyastraightlineY=aX+bwithslopea>0.•When0<ρXY<1,therandomvariablesarepositivelycorrelated;thevariableshaveanapproximatelinearrelationship(withpositiveslope)whosestrengthismeasuredbyhowcloseρisto1.•WhenρXY=0,therandomvariablesareuncorrelated;thevariableshavenoapproximatelinearrelationshipatall.•When−1<ρXY<0,therandomvariablesarenegativelycorrelated;thevariableshaveanapproximatelinearrelationship(withnegativeslope)whosestrengthismeasuredbyhowcloseρisto−1.•WhenρXY=−1,therandomvariablesareperfectlynegativelycorrelated;thevariablessatisfyastraightlineY=aX+bwithslopea<,-dent,whichmeansthattheprobabilitiesofonerandomvariabletakingaspecific11
,whenacoinistossedtwice,,iftwodifferentcoinsaretossedatthesametime,’sXandYareindependent,thenE(XY)=E(X)E(Y).Consequently,ifXandYareindependent,thenCov(X,Y)=,wehavetheimportantspecialcase:ifXandYareindependent,thenVar(X+Y)=Var(X)+Var(Y).12
Ec2030:MathematicsforEconomistsHandout1HarvardUniversity26September2002ProblemSet1Due:Thursday,3October(inclass).,−1f(x)=xiskxforanyrationalk((a)′derivativeofthefunctionf(x)=xisf(x)=(b)(c)(d)*:sin(x)2(a)limx→0x2(b)limxln(x)Hint:UseL’Hospital’sRulex→0()52(c)limexp1+n→∞:x=103x=x+1n+1n4
Ec2030Handout1:ProblemSet123(a)(b)(c)(d)Isthesequenceconverging?Whatisthelimit?=ax+ax+..+ax(1)n1n−12n−2mn−mwherex,x,..,=k(2)(a)Showthatifthereissomesolutionofthetypex=kthenkhastonsatisfymm−1m−2k=ak+ak+..+a(3)12m4(b)(c)Makeaninformalargumentthatforsomegiveninitialvaluesx,..x1m(fromwhichtheentiresequencecanbegenerated):Howmanysolutionsdoesapolynomialequationofordermhavetypically?Howmanysolutionsdoesaquadratichave?4(d)Findaclosed-formsolutionfortheFibonaccisequence:0,1,1,2,3,5,8,13,..:√5(a)50
Ec2030Handout1:ProblemSet13√145(b)(9997)55(c)()(x)convergetolnandalloddelementsconvergetol,(x)suchthatfisincreasing′andf(0)=0,f(0)>1andf(10)<>0.
Ec2030:MathematicsforEconomistsHandout3HarvardUniversity10October2002ProblemSet2Due:Thursday,17October(inclass).,[a,b]whichhasrightandleft-handderivativesatx∈(a,b,):Supposefisa′′(x)≥a≥f(x),welrhavef(y)−f(x)≤a(y−x).-Neumann-Morgensternutilityfunctionu(w),(c)+u(c).Heearnsa212fixedincomeyinperiod1andastochasticincomey˜˜=y+xwhereyishisexpectedincomeinthesecondperiodand222xisasmallmean-zeroerrortermwhichtakesvalue and− σ= .23(a)(b)Showthattheoptimalconsumptiondecisionoftheconsumerinperiod1hastosatisfytheEulerequation:∫′′u(c)=u(y+y−c+x)f(x)dx(1)(c):Showthattheobjectivefunctionofyourmaximizationproblemisconcave!
Ec2030Handout3:ProblemSet223(d)Assumethedecisionmakerconsumesc(0)σthenhiscon-()()22sumptioniscσ.Wecallthedifferencec(0)−cσ-form(approximate)solutionfortheamountofprecautionarysavingoftheconsumerinperiod1.()()223(e)Showthattheprecautionarysavingspσareapproximatelypσ=′′′u(c(0))121σ.Findasufficientconditionontheutilityfunctionwhichmakesa′′2u(c(0))1decisionmakerprudent,(f)*Generalizeyourresultstogeneralsmallerrortermsxwithmean0and2finitevarianceσ.(w).Thesocialplanneroffersto’noiseup’hisincomealittlebit:foreachrealizationofincomewthesocialplanneraddssomemean-zeronoisexsuchthattheincomeofthedecisionmakerisw+’(page57),thatifUandVaresubspacesofthevectorspaceWthenU+VandU∩∪,..xand1nnequations:ax+ax+..+ax=b1111221mm1ax+ax+..+ax=b2112222mm2...ax+ax+..+ax=bn11n22nmmn(2)
Ec2030Handout3:ProblemSet238(a)Showthatthisequationsystemcanbewritteninthefollowingformm∑ax=b(3)iii=(b)Showthatifm>(c)-productsatisfythefollowingconditions:theinnerproduct<x,y>is(i)bilinearinxandy,(ii)commutativeand(iii)satisfies<x,x>≥0and<x,x,>=0iffx=(a)Forsomearbitraryvectorsxandyandanyrealθshowthat:2<x,x>+2θ<x,y>+θ<y,y>≥0(4)Hint:Lookat<x+θy,x+θy>.29(b)*Rememberthataquadraticequationax+bx+chasoneornosolutions2iffb−4ac≤(c)∑nwithmarketsharessuchthats==1∑n21asH=(lowesti=1inconcentration)and1(highestconcentration).-centration?
Ec2030:MathematicsforEconomistsHandout5HarvardUniversity29October2002ProblemSet3Due:Tuesday,5November(inclass).,=1,2,...Robinsoncanusehisendowmentofgraintogrowwheataccordingtosomeproductionfunctiony=f(k)wherefsatisfiestheInadatt−1′conditions:itisstrictlyincreasingandconcaveandf(0)=∞.Furthermore,fisboundedabovesuchthatf(k)≤yforallk(becausethetotallandareaoftheislandisbounded).Hecanconsumepartorallofhisannualproductionandsavetherestwhichbecomesthenhisendowmentofgrainforthenextperiod,∑∞+k=δu(c)fromconsumingwheattt+1ttt=0(weassumeforsimplicitythatRobinsonlivesforeverandstaysforeveronhisboringisland).Theperperiodutilityfunctionuisincreasingandconcaveand0<δ<,butthenremembersthathetookEc2030inhisbetterdays,’(a)Let’sdenotethevalueofhavinganendowmentkwithV(k).Showthattheoptimalconsumptiondecisionc(k)hastosatisfytheBellmanequation:V(k)=max[u(c)+δV(f(k)−c)](1)c∈[0,f(k)]1(b)ShowthatRobinson’∈[0,y].[0,y].1(c)DefinetheBellmanoperatorTasthemapT(U)=WwhereU,W∈C[0,y]andW(k)=max[u(c)+δU(f(k)−c)](2)c∈[0,f(k)]ShowthatTindeedmapsC[0,y]intoitself().
Ec2030Handout5:ProblemSet321(d)*ShowthatTisacontractionmapping,.′′||T(U)−T(U)||≤δ||U−U||(3)Recall,that0<δ<11(e)(f)StartwithsomearbitraryV∈C[0,y]anddefineasequenceV=01T(V),..,V=T(V),..ShowthatthesequenceisCauchyandconverges0n+1n∗tosomeV∈C[0,y].∗1(g)(h)DescribehowyoucanusetheaboveargumenttosolveRobinson’(i)UsingtheBellmanequationandtheenvelopetheoremfindthefollowingfirst-orderconditions:′′0=u(c)−δV(f(k)−c)′′′V(k)=δ[V(f(k)−c)f(k)](4)∗1(j)Solveforthesteadystatewherek=k=+1tShow,that∗∗∗f(k)=c+k1′∗f(k)=(5)δShow,thattheInadaconditionsensurethatsuchanequilibriumexists(agraph-icalproofisfine).1(k)*Usinglinearizationsolvefortheequilibriumoptimalconsumptionde-cisionc(k)>:Maximizen∏x=xx..xi12ni=1subjecttox+x+..+x=Sandx≥0,x≥0,..,x≥
Ec2030Handout5:ProblemSet332(a)Showthatnoneoftheconstraintsx≥(b)Set-uptheLagrangianforthisprogramandfindtheuniquesolution∗∗∗x=(x,..,x)(c)Deducefromb)thattheuniquesolutionistheglobalmaximumoftheconstrainedproblem.(Iexpecthereasimplebutlogicalmathematicalreason-ing).2(d)Deducefromc)thatforanynumbersx≥0,..,x≥0,wehave:1n()1/n∑nn∏xii=1x≤ini=(x,x)=x−2x+x+1121122subjecttox+x≤0andx−4≤-sumer’sutilityforconsumingqunitsofthegoodattotalcostT(q)tothemisθu(q)−T(q)whereθ:highvalueconsumer(θ)andlow-valueconsumers1(θ<θ).Themonopolistcanchargeconsumersatwo-parttariff,:T(q)=F+(a)Show,thatthemonopolistwouldpriceatmarginalcostifhecouldper-fectlydistinguishbetweenconsumers,(b)-handmarkets?4(c),(q)andT(q)
Ec2030Handout5:ProblemSet344(d)(e)(convex)functionsisconcave(convex).Decideinthefollowingexampleswhetherthefunctionsareconcave,quasi-concave,(x,x)=ln(x)+2∗ln(x)(6)12122f(x,x)=exp(ln(x)+2∗ln(x))=x∗x(7)12121222f(x,x)=xx(8)1212√22f(x,x)=(xx)(9)1212
Ec2030:MathematicsforEconomistsHandout7HarvardUniversity19October2002ProblemSet4Due:Thursday,28November(inclass).,,(k)=,hisper-periodutilityfunctionisu(c)=−exp(−c)andhisdiscountfactorisδ=(a)(b)(c)(d)
Ec2030Handout7:,
Ec2030Handout7:ProblemSet433(c)NowthatyouknowV(0)andW(0)calculatethevaluefunctionsforn<
Ec2030:MathematicsforEconomistsHandout8HarvardUniversity12December2002ProblemSet5Due:Wednesday,(253),=(flow)utilityfunctionisln(c)λexp(−λT).Hecaresaboutthewelfareofhisoffspring:ifbeleavesthemabequestbattimeThisutilityfromdoingsoαisexp(−rT)bwhere0<α<(a)ShowthathisprobabilityofbeingaliveattimeTisexp(−λT).Hint:NotethatifarandomvariableXhasdensityfunctionfthentheprobability∫xP(X≤x)=f(x)dx.−∞1(b):∫∫∞∞−(λ+r)tα−(λ+r)tU=ln(c)edt+λkedt(1)001(c)Showthattheoptimalpolicyhastosatisfythefollowingsystemofdif-ferentialequations:c˙α−1+r+λ=αλckc˙k=−c(2)1(d)Showthattheoptimalconsumptionpolicysatisfiesα−(λ+r)t−λkc=Ee(3),discusstheroleofbequestsontheoptimalconsumptionpath.
Ec2030Handout8:ProblemSet521(e)
Ec2030Handout8:ProblemSet533(d)Howdoestheoverweightingofproducersurplusaffectthegovernment’spolicy?:x˙=−2x+x113x˙=x+2x223x˙=−2x−x(6)3231withinitialconditionsx=(a)(b)
Ec2030:(x+h)−f(x)1(a)Foraconstantfunctionf(x)=xwehaveφ(h)==0-hencehthelimitash→0iszero,(x)=xwehaveφ(h)=(b)Theclaimholdsfork=,thatitholdsk+1′kk−(x)=x+kx∗x=(k+1)+1aswell.−k11(c)Wesimplydifferentiatef(x)=x=.kxmrnm1(d)*Ifr=andf(x)=xthenwehaveg(x)=(f(x))=:n−1′m−1n(f(x))f(x)=mx′r−1f(x)=rx(1)(x)cos(x)2(a)lim=lim=1x→0x→0x11ln(x)x2(b)limxln(x)=lim=lim=0x→0x→01x→01−x2x()(())552(c)limexp1+=explim1+=en→∞n→∞(a)Looksincreasing1,,..3(b)Wewanttoshowthatx>=+−x=(x−x)>+2nn+1n4establishesthatthestatementisalsotrueforn+1.
Ec2030Handout2:SolutionsforProblemSet123(c)Let’,+(d):3limx=limx+1(2)n+1nn→∞n→∞4Sincethesubsequence(x)convergestothesamelimitas(x)wehaven+1nlimx=→∞=ax+ax+..+ax(3)n1n−12n−2mn−mwherex,x,..,=k(4)(a)Justplugtheexpressionx=(b)Sincethedefinitionofthesequenceislinear,linearcombinationsofso-lutionsarealsosolutions:if(x)and(y)aresolutionssoisαx+β(c)Asolutionoftheabovetypehastosolveapolynomialequationoforderm-typicallytheyhavemsolutions(justasthequadratichastwosolutions,thelinearequationhas1solutionetc.).Assumethatthesesolutionsarek,..,∑mnThenpossiblesolutionsareoftheformx=α=1iinitialconditionswegetalinearequationsystemconsistingofmequationsinmunknownsα,..,α.Thishastypicallyauniquesolutionandwearedone!1m4(d)Thesequenceisdefinedbyx=x+xwithx=0andx=−1n−2012Thecorrespondingpolynomialequationsatisfiesk−k−1−1suchthatk=1√√1+51−5andk=.Hencewearelookingforsolutionsoftheform222()()√n√n1+51−5x=α+β(5)n22
Ec2030Handout2:SolutionsforProblemSet13Usingourinitialconditionswehave:0=α+β1=αk+βk(6)121√Fromthisweobtain:α=−β=.:√√√11√5(a)x+h≈x+h-hence50≈7+.2x14√−(b)(x+h)≈x+-hence(9997)≈10− .ThereissomeNsuchthatforalln>Nwehave11|x−l|< .ThereissimilarlysomeNsuchthatforalln>Nwehave2n+122|x−l|< .TakeN=max(2N+1,2N).Nowforalln>Nwehave2n12|x−l|< .(x)=f(x)−,g(0)=0andg(10)<0and′g(0)>-sothereexistssome >0∗suchthatg( )>∈( ,10)such∗∗∗1thatg(x)=0whichimpliesthatf(x)=[0,10].Buttheonlyzerocouldbeatx=,youneedthe argumentsomewhereinyourproof.
Ec2030:MathematicsforEconomistsHandout4HarvardUniversity29October2002SolutionstoProblemSet2′+f(x+h)=f(x)andlim−f(x+h)=h→0+h→0′f(x).Thisimpliesthat:−′f(x+h)=f(x)+hf(x)+o(h)forh>0+′f(x+h)=f(x)+hf(x)+o(h)forh<0(1)−′′Butthisshowsthat|f(x+h)−f(x)|≤max|f(x),f(x)|h+o(h)<Mhfor−+(x+h)−f(x)>φ(h)=tendstoh′f(x)forh><x+h<(y)−f(x)φ(h)≥(2)y−xNowtakethelimitash→:f(y)−f(x)′f(x)≥(3)ry−x′Note,thaty−x>0andthatf(x)≤(y)−f(x)≤a(y−x).rTheresultcanalsobeobtainedfory<(a)Theconsumerchoosesfirst-periodconsumptiontomaximizelife-timeutilityU:U=u(c)+Eu(y1+y2+x−c)(4)Eu(y1+y2−x−c)istheexpectationoperatorandmeans∫Eu(y1+y2−x−c)=u(y1+y2−x−c)f(x)dx(5)Thefirst-orderconditionoftheproblembecomes′′u(c)=Eu(y1+y2+x−c)(6)Note,thatwedifferentiate’undertheexpectation’whichreallymeansundertheintegral!TheaboveFOCistheEulerequation.
Ec2030Handout4:SolutionstoProblemSet223(b)(c)Bothu(c)andu(y1+y2−x+c)(y1+y2−x+c)(d)Theideaisthatforsmallerrortermsc(0)−c(σ)-11fore,(andsuppresstheo(h)term):′2′2u(c(σ))=Eu(y1+y2+x−c(σ))[][]12′′′2′′′2u[c(0)]+u[c(0)]c(σ)−c(0)+u[c(0)]c(σ)−c(0)=2[[]′′′2Eu[y1+y2−c(0)]+u[y1+y2−c(0)]x+c(0)−c(σ)+][]12′′′2+u[y1+y2−c(0)]x+c(0)−c(σ)2()()22Denoteprecautionarysavingswithsσ=c(0)−cσandrecallthatfrom′′theEulerequationu(c)=u[y1+y2−c(0)]wecandeducethatc(0)=y1+y2−c(0).Wethenobtain:()12′′2′′′2sσu(c(0))=−σu(c(0))2′′′()1u(c(0))22sσ=−σ(7)′′4u(c(0))Inexerciseslikethisitisusefultoredothelinearexpansionafteryousolvedthesimplifiedsystem(whenyouignoredallo(h),theremaining2errorintheEulerequationhastobeordero(h)((σ)inourcase).′′′3(e)Precautionarysavingswillbepositiveifu>,thatitisquitereasonable-itbasicallysays,thatthesecondderivativeisincreasing,(theutilityfunctioniscon-cave),(f)∫xisf(x,w)withmarginalwealthdistributionf(w)=f(x,w).Theexpectedx
Ec2030Handout4:SolutionstoProblemSet23utilityovernoisedupincomeis:[]∫∫∫Eu(w+x)=u(x+w)f(x,w)dxdw=u(x+w)dxdw(8)x,wx,wwxThelastequalityisknownasFubini’stheoreminmathematicsandthelawofiteratedexpectationsinprobabilitytheory:takingtheexpectationwithre-specttotherandomvariablesxandwcanbeaccomplishedbytakingfirsttheconditionalexpectationoverxassumingsomefixedwandthentakingtheexpectationoverwsubsequently:Eu(x+w)=E[E[u(x+w)|w]](9)x,,thatforfixedwthefunctionu(x+w),byJensen’sinequality,wehaveE[u(x+w)|w]≤u(E(x+w|w))=u(w)(10)xxWenowtaketheexpectationoverwonbothsides:E[E[u(x+w)|w]]≤E[u(w)]wxwEu(x+w)≤Eu(w)(11)x,’)isaconcavefunctionandb)and2x1c)areconvex(usesecondordercondition).Ford)notethat=x−1+.x+1x+1Thefirstpartisweaklyconcave,thesecondpartstrictlyconcave,)requiresyoutosolvetheFOC,+VandU∩,thatifx,y∈U+Vthenthereexistsu,u∈Uxyandv,v∈Vsuchthatx=u+vandy=u++u∈Uandv+v∈VbecauseUandVareclosedunderadditionandxyxyhencex+y=(u+v)+(u+v)∈U+,thezeroelementisinxxyybothUandVandhenceinU+VandU∩∪-butthenunionisnotclosedunderaddition(infactthesmallestsubspace2containingbothUandVisRitself!).
Ec2030Handout4:⊆VisnotfinitedimensionalandassumethatthevectorspacehasfinitedimensiondimV=+,..xandnequations:1nax+ax+..+ax=b1111221mm1ax+ax+..+ax=b2112222mm2...ax+ax+..+ax=bn11n22nmmn(12)8(a)Definecolumnvectorsa=(a)andb=(b).Thenwecanwritethejijiequationsysteminvectorform:m∑ax=b(13)iii=(b)Ifm>,theyhavetobelinearlydependentandthereareµsuchthat:i∑µa=0(14)ijiTherearetwopossibilities:(A)λsuchthati∑λa=b(15)iiiαButanysetofcoefficientη=λ+αµforsomeαsolvesthevectoriiiequationaswell:∑αηa=b(16)iiiTherefore,thereareinfinitelymanysolutions.(B),thatalthoughthevectorsaarelinearlydependenttheydonothavetospanthefullspaceinR:thinkoffourvectorsinthree-dimensionalspacewhichliveonasingleplane.
Ec2030Handout4:SolutionstoProblemSet25∑8(c)Iftheaarelinearlyindependentandtherearetwosolutionsλa=biiii∑∑′′andλa=bandhence(λ−λ)a=(a)Thefollowingholds:<x+θy,x+θy>≥02<x,x>+2θ<x,y>+θ<y,y>≥0(17)29(b)Theinequalityaboveisoftheformaθ+bθ+c≥θ+bθ+θ+bθ+c=,ifaθ+bθ+c≥0thereisatmostonesolution(convinceyourselfofthisgraphically!).Sincethe√2−b±b−4acsolutionofthequadraticequationisthereisatmostonesolution2a2iffb−4ac≤:24<x,y>−4<x,x><y,y>≤0|<x,y>|≤||x||||y||(18)QED9(c)TheHerfindahlindexsatisfiesn∑∑∑∑221=s=s×1≤ssiiii=11≤Hn(19)11Therefore,H≥.Theminimumisreachedwhenalls=.It’salsoeasytoin∑n2seethatH≤1becauses=1ands≤.
Ec2030:(a)-(h):∗∗∗f(k)−c=k(1)PluggingthisintotheFOCgivesus:′∗′∗0=u(c)−δV(k)′∗′∗′∗V(k)=δ[V(k)f(k)](2)′∗1∗Fromthelastequationwegetf(k)=.δ∗conditionsensurethatthereexistssomek>,theslopeoffisinfiniteat0andthenconvergesto0fork→∞.1∗Therefore,ithastobeforsomekbetween0and∞.δ∗∗∗ofequationsbeforeyoupluginthesteadystateconditionk=f(k)−:′∗′′∗∗′∗∗′′∗∗′∗∗∗0=u(c)+u(c)(c−c)−δV(f(k)−c)−δV(f(k)−c)[f(k)(k−k)−(c−c(3))]∗∗Nowusethesteadystateconditionsandset∆c=c−cand∆k=k−k:[]1′′∗′′∗u(c)∆c=δV(k)∆k−∆c(4)δThesecondequationgivesus:′∗′′∗′∗∗′′∗∗′∗′∗′′∗V(k)+V(k)∆k=δ[V(f(k)−c)+V(f(k)−c)(f(k)∆k−∆c)][f(k)+f(k)∆(k5])]Ignoringsecondordertermsandusingthesteadystateconditionsweget:11′′∗′′∗′∗′′∗V(k)∆k=δV(k)(∆k−∆c)+δV(k)f(k)∆kδδ1′′∗′′∗′∗′′∗V(k)∆k=V(k)(∆k−∆c)+u(c)f(k)∆kδ[]1−δ′′∗′∗′′∗V(k)∆c−∆k=u(c)f(k)∆k(6)δ′′∗Wehavetolinearequationsin∆kand∆cwhichbothcontainV(k):[]′′∗1−δ′∗′′∗V(k)∆c−∆ku(c)f(k)∆kδ[]=(7)1′′∗′′∗u(c)∆cδV(k)∆k−∆cδ
Ec2030Handout6:SolutionstoProblemSet32′∗′′∗u(c)f(k)Let’ssetβ=whichisapositiveparameter:′′∗u(c)1−δ∆c−∆k∆kδ=β(8)∆k−δ∆c∆cTheeasiestwaytosolvethisequationistorealizethat∆c=γ∆kforsomeγwhichwehavetofind(thisisthelinearizedsolution):[]∆c1−δ∆k−∆k∆kδ[]=β∆k∆c∆c−δ∆c1−δγ−δ=β(9)1−δγThisgivesusaquadraticequationinγ:1−δ2γ−γ=β−βδγδ[]1−δ2γ−γ−βδ=β(10)δ(thereisexactlyonepositiveandonenegativeroot):,wehave:√()21−δ1−δ−βδ+−βδ+4βδδγ=(11)>:Maximizen∏x=xx..xi12ni=1subjecttox+x+..+x=Sandx≥0,x≥0,..,x≥(a)Thisisobvious-otherwisethemaximizedvaluewouldbe0andyoucanScertainlyalwaysdobetterbysettingx=(b)Wewanttomaximize[]nn∑∑L=lnx−λx−S(12)iii=1i=1
Ec2030Handout6:SolutionstoProblemSet33Note,:1=λ(13)xiSButthisimpliesthatallxareidenticalandhencex=.iin2(c).∑nx∑i2(d)Takethosenumbersanddefiney=nsuchthaty==1ii=1thenhave:nn∏∏11y≤=(14)innni=1i=1Butthisgivesus:∑nnn∏(x)ii=1x≤=(15)inni=-youcanonlyminimizetheprobleminsteadofmaximizingit:theobjectivefunctionisasumofparabolaswhichareopenedtothetop,’:[]222L=2x−x−1−x−λ[x+x]−µx−4(16)-fore,:2−2x=λ+2µx11−2x=λ(17)=2andx=−2orx=−2andx=λ=4>µ<0whichisimpossible.
Ec2030Handout6:SolutionstoProblemSet34Ifnoconditionbindsthenλ=µ==1andx=+x≤µ=0andx+x==12121andx=−suchthatλ>λ=0andx=2orx=−µ<=andx=−.(a)Consumerswhichfaceatwo-parttariffchoosetheoptimalquantityqsuchthatθu(q)−qp(18)′ismaximized,=u(q).Note,,thatthefixedfeedoesnotaffectconsumerbehavioraslongastheypreferbuyingsomeoutputtonooutputatall(themonopolistwouldneversetthefeethathigh).ThemonopolistsetsthefeeatF=θu(q)−:pi=θu(q)−qp+q(p−c)=θu(q)−qc(19)′Hisprofitismaximizedifu(q)=c,=(b)-handmarket,,,(c):(1)Eachconsumerdoesnotwanttoimitatetheotherconsumerandchoosethebundleintendedforhiminsteadoftheotherconsumertype.(2)(q,T)tobothconsumersii
Ec2030Handout6:SolutionstoProblemSet35whereT=F+:θu(q)−T≥θu(q)−T111122θu(q)−T≥θu(q)−T222211θu(q)−T≥0111θu(q)−T≥0(20)2224(d)Themonopolistmaximizeshistotalprofitspi=T+T−(q+q)’.(a)Thelowtypeconsumerisjustindifferentbetweenbuyingandnotbuying.(b):θu(q)−T=0222θu(q)−T=θu(q)−T(21)122111ThisgivesusT=θu(q)andT=θu(q)−(θ−θ)u(q).Pluggingthisinto121111122theobjectivefunctiongivesus(wecouldalsodoconstrainedoptimizationwithaLagrangianbutthisiseasier):pi=θu(q)−(θ−θ)u(q)+θu(q)−c(q+q)(22)111222212TheFOCbecome:′θu(q)=c11′′θu(q)−(θ−θ)u(q)=c(23)22122Fromthefirstconditionweseethatqisthesameasin(a).,(e)Intuitively,:,-typeconsumersbecausehewantstodistortthedecisionofthehightypesaslittleaspossible.
Ec2030Handout6:-concavebe-causeit’-concaveforpositivevaluesbecausetheyaremonotonictransformationsaswell.
Ec2030:MathematicsforEconomistsHandout8HarvardUniversity2January2003SolutionstoProblemSet4Problem1.′∗11110√1(a)Insteadystatewehavef(k)=,whichgivesus=and∗δ29k∗∗thereforek≈=−≈(b)Forlinearizationweusetheformulafromthelastproblemset(∆c=γ∆k):[]1−δ2γ−γ−βδ=β(1)δ′∗′′∗1uexp(−)(c)f(k)∗β===.′′∗u(c)exp(−)2γ+γ=(2)−±:γ=.Theinterestingrootisthepositiveone:γ=∆c=∆(c)Anyconsumptionpathwhichcanberealizedatsomekcanbereplicated′fork>(d),=f(x)=(1−sin(x)).4′1Youcanshowthat|f(x)|=|cos(x)|(1−sin(x))<=0forexampleandtheniteratebyx=f(x).0n+1n11Wegetx≈=−exp(−x).Nowwe23′1havef(x)=exp(−x)<1forx>≈
Ec2030Handout8:(a)-piedspacewillbetakenunlessit’(n)=max[−n,(1−α)V(n+1)+αW(n+1)]W(n)=(1−α)V(n+1)+αW(n+1)(3)3(b)Itcanneverbeoptimaltosearchfurtherwhenn≥(n)=−:W(n)=−(1−α)(n+1)+αW(n+1)(4)Applyingthisequationrepeatedlyweget:2W(0)=−(1−α)−α(1−α)2−α(1−α)3−...[]2=−(1−α)1+2α+3α+...[]d2=−(1−α)α+α+..dαdα1=−(1−α)=−(5)dα1−α1−α3(c)Let’scalculateW(−1):αW(−1)=−(6)1−α∗TocalculateV(−1)weconsiderthecutofflevelαwherethedriverisindifferentbetweenacceptingafreespotatn=−1ordrivingon:∗α−1=−(7)∗1−α∗1Wegetα=.Nowconsidern=−,thedrivershouldbemore2∗∗α>α.Atthiscutoffvaluewe22αhaveW(−2)=−(1−α)−(becauseatn=−1youwouldchoosethefree1−αspotforsure).,wehave:2∗(α)∗2−2=−(1−α)−(8)21−α√()1∗1∗1nWegetα=.Goingonthiswayyoucanshowthatα=.2n22
Ec2030Handout8:SolutionstoProblemSet43∗∗3(d)Foragivenαcalculatensuchthatα≤α≤α.Thisimpliesthatnn+1thedriverwillstartsearchingat−,that11−ln(2)≤lnα≤−ln(2)(9)nn+1Fromthisweget:ln(2)ln(2)−−1≤n≤−(10)lnαlnαln(2)∗Sosimplychoosen=[−]wherethebracketindicatestheintegerpartoflnαthefraction().∗−n<−:∗1∗∗n−(1−α)n−α(1−α)(n−1)−..−α(11)1−αThelasttermissimplythevalueW(0).Youcanusethesametrickasabovetocalculateanexplicitexpressionforthisvariantofageometricseries:()∗∗∗1∗n∗n−1nTC=−(1−α)n1+α+..+α−α(1−α)(1+2α+..nα)−α1−α()∗∗α∗∗1∗n+1∗n+1n+1n=−n(1−α)+(n+1)α−1−α−α(12)1−α1−α3(e)Forn≥’slookatn<,whenyouseeanyopenspotaheadatsomepositionm≤,thefirstrealdecisionyoufaceiswhenyouareatn=−6andhavetodecidewhethertotakeanopenspotatn∗α.=−∗α,−sin(y)=andhencetheequation3()1−sin(y)f(y)=exp−+ln(y)=0(13)(0)=−∞andf(1)>−sin(y)atleastsomey∈[0,1]suchthatf(y)==3alsosatisfiesx∈[0,1].QED
Ec2030:(a)∫Tλexp(−λt)dt=1−exp(−λT)(1)0Hencethesurvivalprobabilityisexp(−λT).1(b)Amica’sobjectiveistomaximizehislife-timeutilityplusthebequestutilityhederiveswhenhedies:∫∫∞∞−(λ+r)tα−(λ+r)tU=ln(c)edt+λkedt(2)001(c)Thecorrespondingstaticproblemwithshadowpricesµis:−(λ+r)tα−(λ+r)tH=ln(c)e+λke+µ(−c)(3)TheFOCare:1−(λ+r)tH=e−µ=0ccα−1−(λ+r)tµ˙=−H=−λαke(4):c˙α−1+r+λ=αλckc1(d)WecanplugintheseconddifferentialequationintotheEulerequationandget:c˙α−1˙+r+λ=−αλkkcNowyoucanintegratebothsides:αln(c)+(r+λ)t+const.=−λk(5)
Ec2030Handout10:SolutionstoProblemSet52Youcansimplifythisequationandget:α−(λ+r)t−λkc=Ee(6)−(λ+r)tabequestmotive:inthiscaseitsimplifiestoc=(e)Weget:α−(λ+r)t−λk˙k=−Ee(7)Wecan’tsolvethisexplicitly(onlynumerically):(1)theinitialcapitallevelattimet=0;(2)thetransversalityconditionwhichmakessurethatcapitalcannotgotoinfinityast→∞.(a)Thestaticproblemattimet<Tisnow:−rtH=ln(c)e+µ(−c)(8)AttimeTwegetthetransversalitycondition(priceofcapitalequalsmarginalreturnofbequest):α−1µ(T)=αkexp(−rT)(9)1−rtc˙Weknowthatµisconstantandµ=+r=−rt˙Thisgivesusimmediatelyc==−cwegetthesecondequation:−rtc=EeE−rtk=D+e(10)r2(b)WeknowthatµµusingtheTVC:[]α−11ErT−rTe=αD+e(11)ErNowusethefactthatinitiallythecapitallevelhastobek:0[]α−1()1ErT−rTe=αk−1−e(12)0Er
Ec2030Handout10:SolutionstoProblemSet532(c)Thelefthandsideisadecreasingfunction:itstartsat∞anddecreasesto0asE→∞.Theright-handsideisincreasinginEandstartsfromapositivevalueandgoesto∞,therehastobeauniquesolution(drawapicturetoconvinceyourself).2(d),withabequestmotive,(c)=ac−αu(c)+(1−α)cp(c)forsome0<α<(t)wherex(t)(a),wehavep(c)=a−(b)(oftengeo-graphicallysuchasUSsugarindustryinLousianaorthetextileindustry).,consumersarewidelydistributedand’small’:222αu(c)+(1−α)cp(c)=α(ac−bc)+(1−α)(ac−2bc)=ac−b(α+2(1−α))c︸︷︷︸ˆb>b(13)Effectively,thegovernmentmaximizestheutilityofaquadraticconsumerwithˆ(c)Therelatedstaticproblemwithapriceµoffishis:()2−rtˆH=ac−bce+µ(gx−c)(14)TheFOCare:()−rtˆH=a−2bce−µ=0cµ˙=−H=−µg(15)x
Ec2030Handout10:SolutionstoProblemSet54−gtThisimpliesthatµ=:(r−g)tˆa−2bc=Ee(r−g)ta−Eec=(16)ˆ(a)Wecanwritethelinearsysteminmatrixform:−201x˙=012x=Ax(17)0−2−1′wherex=(xxx).:(−2−λ)[(1−λ)(−1−λ)+4]=0(18)√√Thethreezerosare:λ=−2,λ=3iandλ=−3i122Thisisgoodnews::′v=(100)1√√′v=(21−33i4+23i)2√√′v=(21+33i4−23i)(19)3Thegeneralsolutionis:221√√√√−2t3it−3itx=C0e+C1−33ie+C1+33ie(20)123√√04+23i4−23iWewanttofocusonrealsolutions:012()()√√√−2tx=C0e+C1cos3t+33sin3t+12√04−2302()()√√√+C1sin3t−33cos3t(21)3√4−23Attheinitialconditionwehave:C+2C=112√C−33C=123√4C+23C=1(22)23
Ec2030Handout10:SolutionstoProblemSet55Wethenget:7C=1175C=217√43C=−(23)3514(b)Bythefundamentaltheoremthesolutionisuniquesincewecanwritex˙=F(x)(c)ConvergencetozerowillonlyoccurifC=C=.
