CONF COURSE IN APPLIED MATH
CONF COURSE IN APPLIED MATH M 393C
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Lecture 2 CONTINGENT MARKET EQUILIBRIA 1 of 4 Course M393C equilibrium theory Term Spring 2007 Instructor Gordan Zitkovic Lecture 2 FINANCE ECONOMIES AND CONTINGENTMARKET EQUILIBRIA This lecture is based on sections 6 of MQQ li 21 TWOperiod nance economies The timeset of the economy consists of two points It 0 and t l we call them dates At t 1 each of S outcomes can occur The S outcomes at date 1 together with date 0 are called states and are labeled by s 0liiiS the state 8 0 corresponding to date 0 There is a single good and a nite number of consumers labeled 239 l i i i L Each consumed has a preference relation 5139 de ned over the consumption space R1 where n S 1i The elements I 011p i i 15 of X will be called consumption streams We assume that there are continuous utility functions ui R3 A R compatible with jiielui1i Assumption 21 Three sets of assumptions postulated for all i E 1 i i i I are offered 11 strong monotonicity 1 ii is strictly monotonei 11 strong monotonicity and strict quasi concavity 1 ii is strictly monotonei 2 ui is strictly quasi concavei M smooth preferences 1 ui is continuous on R and C00 on Ri I 6 RC 2 I gtgt 0 2 the39sets 1 E R z 2 satisfy Q R for all z E Rii 3 Vulz E Riur for all z E Riur 4 the Hessian is negative de nite when restricted to the tangetnt space h E R Vuiz h 0 for all z E Rii Example 22 T39 39 p t39 ddquot 1 p H A f Let 1 R 000 A R beacontinuous strictly concave strictly increasing function which is C00 on 0 00 with limgcno 1 00 Let m i i i p5 be a vector with 251 p5 1 ps 2 0 for all s and let 0 lt 6 lt 1 be a constant De ne S vzo 6 Zp5vzsi One can check that u satis es all parts of Assumption Mm except maybe see Problem 21 Together with hisher preference relation each consumer receives income both at date 0 and at date 1 The income at ate 1 is uncertain and depends on the outcome Consumer is income is called the initial endowment and is denoted by of 01301 qwls E R1 De nition 23 l A nance economy uw is a pair uw where u 111 i ui is a vector of utility functions and w w i i i wI a vector of initial endowmentsi 2 A vector 1 11 i i II of consumption streams is called an allocation for the economy 69u w Last Updated January 25 2007 Lecture 2 CONTINGENT MARKET EQUILIBRIA 2 of 4 3 An allocation 1 is said to be feasible if I Zn 7 mi 0 for all s 0 S 21 i1 The set of all feasible allocations will be denoted by F Fuw Remark 24 A requirement that a statement should hold for all s 0 S will usually be signaled by omitting the subscript s For example the feasibility of an allocation as in 21 above would be denoted as I 2w 7 at g 0 22 i1 Alternatively and equivalently one can think of 22 as a vector inequality where S is really 7 and 0 000 Note that the notion of feasibility depends on the initial endowments w1 wI only through their sum w Zi wi The vector w is called the aggregate endowment De nition 25 An allocation i is said to be Pareto optimal for the economy uw if 1 i is feasible i E Fuw and 2 whenever I E Fuw has the property that 2 for all i then I E 22 Contingent market equilibria A contingent contract for the state 3 E 0 S is a promise to deliver 1 unit of the good in the state s and nothing otherwise If there is a price for such a contingent contract payable at units of account at date t 0 it will be denoted by as E R If there is a liquid market for all contingent contracts with prices 7r 7n rrs then consumer i can sell hisher initial endowment of for S moi Z midi 50 here moi is just a shortcut for the inner product 7Twi in R The set of all consumption streams that agent i can exchange hisher initial endowment of for given the prices 7r for the complete system of contingent markets is called the budget set and is denoted by B7rwi Clearly B7rwi I E R3 7rz iwi 0 De nition 26 A contingentmarket CM equilibrium for the economy uw is a pair fir consisting an allocation i and a vector of prices 7 such that 1 ii 6 argmax z z E B7Twi for all i and lt2 ELM 7w 0 23 Existence of CM equilibria Lemma 27 Assume 11 Then the following three statements are equivalent 1 The problem A max over all z E B7Twi has a solution ie the max is nite and attained 2 7 6 R1 3 B7 ro is compact Under the stronger condition 11 uniqueness can be added to I as well Proof No surprises here D Last Updated January 25 2007 Lecture 2 CONTINGENT MARKET EQUILIBRIA 3 of 4 The following theorem can be generalized to the case when preferences are assumed to be quasi concave and not merely strictly quasi concave Theorem 28 Existence of CM equilibria Suppose that ll holds and that w 2114 E RiJr Then there exists at least one CM equilibrium 77 The key concept in the proof of Theorem 28 is the one of the aggregate demand function f 1R3r A R given y I f7r 21W i1 where the demand functions fi 1R3r A R are given by argmax z z E B7rwi A close relative of the aggregate demand function is the excess aggregate demand function Z ELF A R given I I M 1 7 Di 20 7 w Clearly7 the following proposition holds Proposition 29 A pair in is a CM equilibrium if and only if 7r 6 Riy Z7T 0 and ii fiir for all i Proof Everything is just a restatement of the definition in terms of new notation7 except7 maybe7 the fact that 7 E RiJr in equilibrium This7 however7 follows directly from Lemma 27 The quest is7 really7 for a zero of the function Z on R To make some progress with it7 here are some useful properties of the demand functions Proposition 210 Suppose that the assumptions of Theorem 28 hold Than each of the functions f i l7 I has the following properties 1 E R for all 7r 6 Riy 2 is continuous on R 3 fia7r for all 7r 6 Riy and all a gt 0 4 7r 7 mi 0 Walras Law lt5 rmwame Hfiorw 00 Proof Properties l4 are left as an exercise see Problem 22 5 is quite technical and messy Ways around dealing with it will be discussed later D Remark 211 In terms of the excess demand function Z7 the properties in Proposition 210 can be restated as 1 Z7r 2 7w for all 7r 6 RZJH Boundedness from below 2 Z is continuous on Riu Continuity 3 Za7r Z7r for all 7r 6 RZJH and all a gt 07 Homogeneity 4 7rZ7r 07 VV39alras7 Law 5 limwnaki m 00 Boundary behavior Thanks to the homogeneity of the excess demand function7 one is free to normalize prices7 and7 for the present proof7 the most useful choice is the following one restrict the function Z onto the n 7 ldimensional strictly positive unit sphere S YWER1 Zn1 50 Last Updated January 257 2007 Lecture 5 REPRESENTATIVE AGENT 1 of 3 Course M393C equilibrium theory Term Spring 2007 Instructor Gordan Zitkovic Lecture 5 PARETO OPTIMALITY7 THE SUP CONVOLUTION AND THE REPRESENTATIVE AGENT We are going back to MQQG as a source for this lecture 51 Pareto optimality Continuing with the discussion of the CM equilibrium we turn to some qualitative properties of the equi librium prices and allocations Proposition 51 Assume M Let 7rz be a CMequilibrium Then the allocation 1 zlxz is Pareto optimal Proof Suppose to the contrary that it is not That means that we can nd another feasible allocation i i1ixin with the property that 2 for all i and for some i0 gt In particular Triio gt Trzio because zio is optimal for agent i0 under prices 7r and Tr 2 Trzi for all it Using the strict monotonicity assumption we conclude that each agent saturates hisher budget in equilibrium iei Trzi mail Summing over i we get mo7rE wl7rE zllt7rE ii i i which is in contradiction with the feasibility of i D The following simple cliaiacteii atiou of t t39 l quot quot in the quotE 39 case A A quot M will shed some light on the concept of the representative agent introduced belowi Proposition 52 Assume M An allocation 1 is Pareto optimal and only ifz is feasible Ii 6 RiJr for all i and there exists 7 E RiJr and constants A1 gt 0 A1 gt 0 such that AiVuiz 7T Vilxi1i Proof All Pareto optimal allocations have the property that gt 0 for all i and s why We can and will therefore assume that Ii 6 RiJr for all allocations z in the proof Note the following characterization i is Pareto optimal if and only if for each i E 1 I i maximizes over all allocations z 11p 11 with the following two properties 1 E S EM an 2 W102 MW for j a 2 Fix i and use the Lagrange Multiplier Theorem it will work without any dif culties thanks to smoothness and convexity conditions in M to conclude that the following equalities 7 7r 7 0 0 077r 7 a asuuzj 0 z j 30MSj7 i 51 5 must hold for some constants 7r 2 0 and a E R j it Setting 7 n1 and A1 l M lai j a i establishes the claimi Conversely it is enough to realize that the rstorder conditions 51 are actually necessary and sufficient this is due to concavity of the objective function D Last Updated February 19 2007 Lecture 5 REPRESENTATIVE AGENT 2 of 3 Corollary 53 Equilibrium pricing If n71 is a CM equilibrium then the relative prices usen51 of the contingent contracts for the states 8 so and s sl equals to ratio of marginal utilities for so and 317 ie7 7750 7 asouiltzigt n51 asluizi 7 for all i 52 The supconvolution Let f17 7 1 be real functions de ned on some convex subset C of R De nition 54 The supconvolution f17 7 fm denoted by f1ltgtf2ltgt ltgtfm is the function f Cm A R given by fr supf1rlfmzm zlrm 17 11771m E C7 where Cm zeRn zzlzm7 for some 1177zm E C Remark 55 The typical choices for the domain C are C Riur of C R1 ln both cases Cm C7 for all m Proposition 56 Assume M De ne u RiJr A R by u A1u1ltgtltgtIu1 For I E RiJr the following two statements are equivalent 1 A1u1zl AIuIzI 2 z 11 z and there exists 7r 6 Riy st AiVuizi 7r for all i Proof Use the Lagrange Multiplier Theorem see Problem 52 D 53 The representative agent The idea of a representative agent is one of the central ideas in equilibrium analysis The central idea can be described loosely as follows Each CM equilibrium can be thought of the result of the following procedure assign each agent a numerical weight M gt 07 imagine a super77agent whose utility function is u A1u1ltgt ltgt AIui and who receives all the initial endowments from all the agents 3 have the super77agent redistribute the total initial endowment back to the agents according to the solution to hisher supconvolution problem smile mysteriously AA to H VV A g V The art is in the choice of the weights A17 7 A17 of course ln order to give a more rigorous proof of the loose statement above7 assume that n71 is a CM equilibrium By Proposition 517 z is a Paretooptimal allocation7 so7 by Proposition 52 there exist weights A17 7A such that Aim Finally7 Proposition 567 z is exactly the allocation that the represen tative agent would end up enforcing The converse direction works in the same way Why is this characterization important First of all7 the representativeagent approach provides us with an explicit parametrization of all Paretooptimal allocations Also7 the number of agents I will typically be much smaller than the number of states n7 and the representativeagent approach reduces an ndimensional problem to an I dimensional one This will be very important when we move on to the in nitedimensional case n 00 Finally7 it has a number applications in economics I will leave those to the economists A natural question one can ask is the following what is the meaning of the weights A17 7A The answer is quite informative in equilibrium7 the relative weight AiAj equals the ration of marginal utilities in any state s M 7 E 85u139zj Last Updated February 197 2007 Lecture 5 REPRESENTATIVE AGENT 3 of 3 Loosely speaking7 the more an agent stands to gain from a small additional amount of consumption7 the higher the weight heshe receives Don7t get your hopes up7 though This is only in equilibrium7 and you need the weights to compute the equilibrium in the rst place 54 Problems Problem 5 1 Use Proposition 51 to provide another proof of Proposition 51 under the set of assumptions Problem 52 1 Show that f f1ltgt ltgtfm is a concave function even if fl7 i i i 7 fm are not assumed to be concavei 2 Prove Proposition 56 REFERENCES MQQG Michael Magill and Mannie Qujnzii Them y of incomplete markets volume 1 MIT Press Cambridge and London 996 Last Updated February 197 2007 Lecture 1 PREFERENCES 1 of 4 Course M393C equilibrium theory Term Spring 2007 Instructor Gordan Zitkovic Lecture 1 PREFERENCES AND UTILITY FUNCTIONS This lecture is based on several sections of MCWGQS 11 Preference relations De nition 11 A relation 5 on a nonempty set X is said to be 1 transitive if z j y and y j 2 together imply z j z for all zy2 E X 2 re exive if z j z for all z E X 3 symmetric if z j y implies y j I for all zy E X 4 antisymmetric if z j y and y j I together imply that z y for all I y E X 5 complete if either I j y or y j I for all zy E X 6 a partial order if it is transitive re exive and antisymmetric 7 a complete order linear order7 chain if it is a partial order and total 8 an equivalence relation if it is re exive symmetric and transitive De nition 12 For a relation 5 on X we de ne 1 the strict relation lt corresponding to j by zltyifandonlyifzjyandz7 y 2 the indifference relation N corresponding to j by zwyifandonlyifzjyandyjzi De nition 13 A pair Xj Where X is a nonempty set and j is a partial order on X is called a partiallyordered seti An element 1 E X is called 1 maximal if there exists no y E X such that z lt y 2 greatest if y j z for all y E Xi Minimal and least elements are de ned analogouslyi De nition 14 A relation 5 on X a 0 is called rational or a preference relation if it is transitive and complete 12 Utility functions De nition 15 Any function u z X A R is called a utility function De nition 16 A utility function u and a relation 5 on X are said to be compatible if z j y if and only if S uy for all Ly E X Proposition 17 Suppose that j is a relation compatible with some utility function Then 5 is necessarily rational Proof Just check the axioms D Last Updated January 17 2007 Lecture 1 PREFERENCES 2 of 4 Example 18 Lexicographical order Set X R X R7 and de ne the relation 5 on X by 11712 5 91792 if and only if 11 lt 917 0r 11 91 and 12 S 92 Suppose that there exists a utility function u R2 A R compatible with j Since u07 0 lt u07 l7 there exists a rational number go 6 u07 07 u07 Using the same idea7 we can construct a function 4 R A Rationals with the property that ux0 lt qx lt uxli For x1 lt x2 we have qx1 lt ux17 l lt ux20 lt Lx27 so 4 is strictly increasing7 and therefore injective A contradiction there is no injection from reals into rationalsi Corollary 19 There exist preference relations compatible with no utility function 13 Preferences on R De nition 110 The standard partial order on X Q R 7 denoted by gm is de ned for x 11 i i In y ylymyyn by I Sny ifand only ile S y1712 S yawn S on Remark ill The strict version see De nition 12 of n will be denoted by ltni An even stricter version can be de ned x lt y if and only if x1 lt yhxg lt y2iuxn lt yn When no confusion can arise7 we drop the index d and simply write x S y7 for vectors x7 y E R De nition 112 A preference relation 5 on X E R is said to be 1 monotone if x g y implies x j y 2 strictly monotone if 1 lt y implies x j y 3 convex if X is convex and x j y and x j 2 imply x j ay l 7002 for all a E 017 and all xy2 E Xi 4 strictly convex if X is convex and x j y and x j 2 imply x lt ay l 7002 for all a E 017 and all xy2 E Xi 5 locally nonsatiated if X is open and for each x E X and each 5 gt 0 there exists y E X such that Hwa seandyjz De nition 113 For a relation 5 on X7 and x 6 X7 we de ne the upper contour set U5 by U5Iy X I 15y and the indifference set 15 by 15x yEX xwy The lower contour set L5 x7 as well as the strict versions U j and L3 are de ned analogously As always7 the subscript 5 will be dropped when there is no possibility of confusion Proposition 114 Let 5 be a relation on X Q R Then 1 j is monotone US N X E U5 x for all x E X 2 j is strictly monotone Ultlt N X E U5 x for all x E X 3 j is convex if X is convex and U5 is a convex subset of R for all x E X 4 j is locally nonsatiated ifx is an accumulation point of U5 5 j is monotone then it is locally nonsatiate De nition 115 l A preference relation 5 on X R x E R z x 2 0 is called homothetic if x Ny ifand only ifax Nay for all a gt 07 Lye Xi Last Updated January 177 2007 Lecture 1 PREFERENCES 3 of 4 2 A preference relation 5 on X E R is said to be continuous if Ux and Lx are closed sets for all x E X Proposition 116 A preference relation 5 on X E R is continuous and only if x j y whenever x limxn y limyn and xn j yn for all n E N n n Proof Problem 13 D Proposition 117 Let X R and let 5 be a monotone and continuous preference relation on X Then there exists a continuous utility function u z X A R compatible with 5 Proof See Proposition 3Cl p 47 in MCWGQS D De nition 118 A function u z X A R de ned on some convex subset X of R is called quasiconcave if uax l 7 ay 2 minux for all a 6 01 xy E X 11 A function u is called strictly quasiconcave when the inequality 2 in 11 can be replaced by the strict inequality gt Proposition 119 Let 5 be a monotone continuous preference relation on X R and let u be a utility function compatible with it Then 1 j is convex ifu is quasiconcave 2 j is strictly convex ifu is strictly quasiconcave Proposition 120 1 Let X E R be an open and convex set and let u z X A R be a continuouslydi erentiable function Then u is quasiconcave if and on y i Vuxx 7 x 2 0 whenever ux 2 ux for all xx E X 12 Here denotes the standard inner product on R 2 Let X E R be an open and convex set and let u z X A R be a tWice continuouslydi erentiable function and let denote its Hessian matrix at the point x E A en u is quasiconcave if and only for all x E A is negative semide nite when restricted to the tangent subspace 2 E R Vuxz 0 to the indi erence set x E A ux 14 Problems Problem 11 Find an example of l a relation 5 such that j is symmetric and transitive but not re exive 2 a partially ordered set X j and an element x E X Which is maximal but not greatest Problem 12 Let X be a nite set and let 5 be a preference relation on X Show 5 is compatible With some utility Problem 13 Prove Proposition 116 Problem 14 Show that a continuous preference relation 5 on X is homothetic if and only if it admits a homogeneous utility function A function f z X A R is called homogeneous is fax afx for all x6Xandalla20 Problem 15 1 Find an example of a quasiconcave function Which is not concave Last Updated January 17 2007 Lecture 4 CM BY BROWER AND KKM 1 of 3 Course M393C equilibrium theory Term Spring 2007 Instructor Gordan Zitkovic Lecture 4 CM EQUILIBRIA VIA BROUWER AND KKM This lecture is based on various sources but mainly on lectures of John Geanakoplos Columbia University 2002 41 A trick of Kenneth Arrow One of the problems related to the excessdemand function Z is that it is de ned only in the interior RiJr of the positive orthant R1 In our rst proof involving the Inwardpointing Vector Field Theorem we circumvented this problem by a clever rede nition of Z close to the boundary of the positive unit sphere This idea comes from Kenneth Arrow Nobel Prize for Economics 1972 and one of its variants will be presented belowi The set of assumptions 1 strict monotonicity strict quasiconcavity continuity will be in force through out this lecture Also remember that we are assuming that of E RXJF or a 39 Pick 12 E RiJr with 125 gt ms for all s 0iHS where w 2114 E RiJr is the total aggregate endowmenti Note that no pricesystem 7r 6 RZJF with the property that 2 b for some 8 can be an equilibrium pricei lndeed markets cannot clear when one consumer in at least one state demands more of the good than all the other agents put together can supply Therefore it makes sense to de ne the capped versions of the budget set and the demand and excessdemand functions of Lecture 2 Bb7rwi zeRiJr wzgwwi andzs 125 80HiSi The capped versions fbi of are de ned as in Lecture 2 by simply replacing the uncapped budget set E with Bbi Finally we set I We ZUWWL Z50 MW e we Proposition 41 A pair 7r 1 is a CAIequilibrium if and only Zb7r 0 and Ii 7r for all 239 Proof Suppose rst that 7r 1 is a CMequilibriumi Then w so in particular by positivity of the functions fi S ws lt 125 for all i and all 8 In other words the maximizer of the utility ui in the budget set B7rwi happens to be capped on its own accord and so fbi7r 7r as restricting the domain of optimization cannot increase the value of the maximumi Consequently Zb 7r Z7r 0 lt Conversely suppose that 7r5 0 for all s and Tr solves Zb 7r 0 Let us show that Ii zbi where Ii and If fbi7ri Otherwise we would have lt By strict quasiconcavity uizi l 7 Azbi gt uizbi for all A 6 01 The consumption stream z Ari l 7 Azbi is an element of Bb7rwi for small enough A remember that 125 gt ms for all 3 That is a contradiction however with the fact that zbi and not IA is the maximizer of ui in Bb7rwii Now that we know that Ii 7r fbi7r it remains to show that Z7r 0 This is immediate from the fact that 0 Z500 ZUI VW w 21 w ZW It remains to deal with the case in which 7r5 0 for at least one s 0 i i i St I claim that in that case fbi7r5 125 for all it Indeed the strict monotonicity of uZ will push the agent to consume as much of it as possible Therefore 1125 gt 125 gt ms and Tr cannot be a zero of Zbi Last Updated February 15 2007 Lecture 4 CM BY BROWER AND KKM 2 of 3 What is such a transformed version Z5 of Z good for It retains all the good properties of the original excessdemand function Z but with a larger and more useful domaini lndeed Z5 is an example of an abstract demand function as de ned belowi Unlike in Lecture 2 where we used the quadratic normalization 250 7r l a different and economically more meaningful normalization 250 7r 1 will be used here Moreover from the de nition of the budget sets Bb and the fact that b gt w gt of it is clear that the VV39alras7 law remains valid 7TZ 7r 0 42 CMequilibrium existence Via Brower De ne the function Zl Jr A A R1 Zl Jr 7r Z1Jr 7r i i Z27r where Zf7r maxZ 7r0i De ne the function g z A A A g7r 917r i gn7r by 7r ZETW 1 221 Z50 By Brouwerls theorem 9 is clearly wellde ned continuous and maps A into A there exists 7 E A such that 97 7 ie 9570 Zf 7T Z Z507 7 1 Two things can happen now 0 Zf7 r 0 for all 8 In that case 7 solves Zb7 r Sn 0 0 There exists some 8 such that Zf7 r 0 In that case 7r gt 0 if and only if Zf 7 gt 0 and so 7TZ 7 gt 0 a contradiction with VV39alras7 lawi Therefore Zb7T S 0 In particular fbi7T S of lt bi for each if Strict monotonicity implies now that frfbi7 r frwi why Therefore 7 er7 0 which implies now that Zb7 r5 0 for all s with 7r5 gt 0 How about those 8 with 7r 0 There can7t be any the last part of the proof of Proposition 41 states that Zb7r5 gt 0 when 7r5 0 43 CMequilibrium existence Via KKM We have seen above that all one needs to do is nd a pricevector 7 with Z170 S 0 In order to use the KKM theorem we de ne the following sets F5 NE A Zb7r5 S 0 We would like to show that the sets F5 8 0 i i i S satisfy the assumptions of the theorem of Knaster Kuratowski and Mazurkiewiczi Let I be a nonempty subset of 01HiS and let 7r be a point in conv 7T5 s E I where 7r5 s 1Hin are the vertices of the unit simplexi Then 7r5 0 for all s 6 IC 01H S If Assume that 7r g UseIFi ie that Zf7r gt 0 for all s E If VValras7 law now states that 0 7er7r Zwszb7r5 ZanQr gt 0 5 56 a contradiction The conclusion of the KKM theorem is that there exists 7 E A with Z50 S 0 Following the last part of the proof based on the Brouwerls theorem above we can conclude that Z5 7 0 44 Problems Problem 41 Maximal elements in abstract relations The following theorem can be seen as another incarnation of the theorems of Brouwer or K M Last Updated February 15 2007 Lecture 4 CM BY BROWER AND KKM 3 of 3 Theorem 42 Let K be a compact set in R and let lt be a relation on K having the following properties 1 x g conv Ult here conv Ult denotes the smallest convex set that contains all y E K with x lt y an 2 for each y E K such that x lt y for some x E K there exists 5 gt 0 and x E K such that y lt x for all y E K such that dy y lt 5 Then there exists a ltmaximal element x E K ie Ult 0 and the set of all maximal elements is compact Use the in nite version of the KKM theorem Theorem 38 to prove the theorem above Hint Start With the correspondence Fx K lnt Ult Problem 42 Optional Use the result of the previous problem to give yet another proof of the existence of the CM equilibrium Last Updated February 157 2007