Notes V General Equilibrium: Positive Theory. 1 Walrasian Equilibrium and Excess Demand


 Alfred Rich
 1 years ago
 Views:
Transcription
1 Notes V General Equilibrium: Positive Theory In this lecture we go on considering a general equilibrium model of a private ownership economy. In contrast to the Notes IV, we focus on positive issues such as existence, uniqueness, or stability of equilibria. The notes are based on MWG, chapter Walrasian Equilibrium and Excess Demand I > 0, J > 0, L > 0, all of them are finite. ({X i, i } I i=1, {Y j } J j=1, {(ω i, θ i1,..., θ ij )} I i=1). An economy is defined by Definition 1 A Walrasian (or pricetaking) equilibrium is an allocation (x, y ) and a price vector p = (p 1,..., p L ) if (i) For every j, y j Y j maximizes profits in Y j : p y j p y j for all y j Y j. (ii) For every i, x i X i is maximal for i in the respective budget set {x i X i : p x i p ω i + j θ ijp y j }. (iii) i x i = i ω i + j y j. For a while, let us consider exchange economies. Most results obtained from this exercise easily carry over to economies with production. Definition 2 An exchange economy is defined by E ({X i, i } I i=1, Y 1 = R L +, {ω i } I i=1). We assume that preferences are continuous, strictly convex, and locally nonsatiated (before long: strictly monotone). Moreover, we assume: i ω i 0. Notice also that we assume free disposal, which is taken into account by allowing for one firm whose only available technology is that of free disposal: Y 1 = R L + y l1 0 for all l = 1,...L. Query V.1 What is the relationship between the assumption of free disposal and prices? In the setting of an exchange economy, an allocation (x, y ) and a price vector p constitute a Walrasian equilibrium if and only if (i) y 1 0, p y 1 = 0, p 0, (ii) x i = x i (p, p ω i ) for all i, and (iii) i x i = i ω i + y 1. Ronald Wendner V1 v1.1
2 Notice that (i) we will prove in class. However, by (i) and (iii) aggregate demand cannot exceed aggregate supply of a commodity (as y 1 0). Thus, by (i), a price p l is zero if and only if aggregate demand is smaller than aggregate supply (i.e., only free goods can be in excess supply). Proposition 1 Suppose, preferences in E are strictly convex and locally nonsatiated. Then, p is a Walrasian equilibrium price vector if and only if: i (x i(p, p ω i ) ω i ) 0. Query V.2 Prove Proposition 1. Definition 3 Consumer i s excess demand function is z i (p) = x i (p, p ω i ) ω i, where x i (p, p ω i ) is her Walrasian demand function. The aggregate excess demand function is z(p) = i z i(p). From here on, we ll state most results in terms of the excess demand (rather than Walrasian demand). Definition 4 E + defines an exchange economy where i are strictly monotone, continuous, and strictly convex. In E +, p constitutes a Walrasian price vector if and only if z(p) 0. If, moreover, preferences are strictly monotone this we ll assume from here on a Walrasian price vector has the property that p 0. Query V.3 If i are strongly monotone for all i, why must a Walrasian price vector be strictly positive: p 0? Under strong monotonicity of preferences, p is a Walrasian equilibrium price vector if and only if z l (p) = 0 for every l = 1,..., L. I.e., z(p) = 0. Proposition 2 Consider an economy E + where i ω i 0. Then z(p) is defined on p 0 and satisfies: (i) z(p) is continuous. (ii) z(p) is HD 0. (iii) Walras law: p z(p) = 0. (iv) z(p) is bounded below. I.e., there is some number s : z l (p) > s for every l = 1,..., L and all p. (v) If p n p, where p 0 and p l = 0 for some l, then: {max{z 1 (p n ),..., z L (p n )}} n=1. Ronald Wendner V2 v1.1
3 Property (i) follows from the fact that x i (p, p ω i ) is continuous. Query V.4 From which property about i does it follow that x i (p, p ω i ) is continuous? Property (ii) follows from the fact that x i (p, p ω i ) is HD 0, (iii) comes from strong monotonicity of i, (iv) stems from the fact that demand cannot be negative. Finally, (v) we ll show in class. 2 Some Mathematical Prerequisites for Existence Proofs Correspondence. A correspondence is a multivalued function. Suppose our domain is A R N. A (real valued) function f : A R is a rule that assigns to every x A a single value f(x) R (a singleton). In contrast, a (real valued) correspondence ϕ(x) : A R K is a rule that assigns to every x A a set ϕ(x) R K (which is not necessarily a singleton). Obviously, every function is a correspondence. But a correspondence is a function if and only if for every x A we have that ϕ(x) is a singleton. ConvexValuedness of a Correspondence. Suppose, a correspondence ϕ(x) : A R K assigns to every x A a set ϕ(x) R K. This correspondence is convex valued at x if ϕ(x) is a convex set. This correspondence is convex valued if ϕ(x) is a convex set for all x A. Upper Hemicontinuity (uhc) of a Correspondence. Let ϕ : A Y be a correspondence, where A R N, Y R K, both A and Y are closed, and Y is bounded. Consider any two converging sequences {x n } and {y n } such that for all n, y n ϕ(x n ), where x n x and x, x n A, and y n y and y, y n Y for n = 1, 2,... The correspondence ϕ : A Y is said to be uhc at x if y ϕ(x). The correspondence ϕ : A Y is said to be uhc if it is uhc at all x A. Brouwer s FixedPoint Theorem. Suppose that A R N is nonempty, compact, and convex. If f : A A is a continuous function from A to itself, then f(.) has a fixed point; i.e., there is an x A such that: x = f(x). Kakutani s FixedPoint Theorem. Suppose that A R N is nonempty, compact, and convex. If ϕ : A A is an upper hemicontinuous correspondence Ronald Wendner V3 v1.1
4 from A to itself, with ϕ(x) A being nonempty and convex for every x A (i.e., convexvalued ), then ϕ(.) has a fixed point; i.e., there is an x A such that: x ϕ(x). 3 Existence of Equilibrium This is the first (positive) question. We cannot use our GEframework unless there is an equilibrium. The conditions for which an equilibrium exists are clarified in this section. By HD 0 of an excess function, we are allowed to normalize the price vector (e.g., set one price equal to unity, or normalize prices to the unit simplex in R L +). The unit simplex is defined by {p R L + : l p l = 1}. Moreover, denote the interior of by i, and the boundary of the simplex by. Before going to the propositions, please be sure you understand the following concepts: convexvaluedness of a correspondence, (upper) hemicontinuity of a correspondence, Brouwer s FixedPoint Theorem, and Kakutani s FixedPoint Theorem. I start with the general result first, and give a simplified (more special, but probably more instructive) version thereafter. However, for the real existence proof which is also applicable for production economies I ask you to read my Notes VI. Proposition 3 Consider an exchange economy E + with ω 0. There exists a Walrasian equilibrium, i.e., there exists an allocation (x, y ) and a price vector p that constitute a solution to the system of equations z(p) = 0. Proof. First, construct a correspondence f(p) from all p into. Step (i) considers f(p) : i, step (ii) considers f(p) :. (i) Construct a correspondence for all p i : f(p) = {q : z(p) q z(p) q for all q }, which assigns an element (a set) of to every p i. Observe that if z(p) = 0 (i.e., we are having a Walrasian equilibrium), f(p) =. However, if z(p) 0, then f(p). In particular, q l = 0 if z l (p) < max{z 1 (p),..., z L (p)}. (ii) Construct a correspondence for all p : f(p) = {q : p q = 0} = {q : q l = 0 if p l > 0}. As for any p : p p > 0, no fixed point can be represented by a price vector p. (iii) Certainly, a fixed point of f(p) is a Walrasian equilibrium. Notice that Ronald Wendner V4 v1.1
5 a fixed point means p f(p ). In this case, p. Thus, p 0. But if z(p ) 0, then p. Hence, a fixed point represents a Walrasian equilibrium. (iv) The fixed point correspondence is convexvalued and upper hemicontinuous (as will be shown in class). (v) Now we can apply Kakutani s FixedPoint Theorem to establish that there is a fixed point. By (iii), then, there is a Walrasian equilibrium.. W.H.O.W. All right, this was pretty tough. The difficulty in the preceding proof arose from boundary complications, i.e., excess demand is not well defined when p, as the maximum z l (p) is going to infinity. For purely instructive reasons, we proceed as follows. Assume properties (i) to (iii) from Proposition 2, and z(p) is well defined for all nonzero p R L +. 1 Remember that in equilibrium we have z(p) 0. Corollary 1 Consider an exchange economy E with ω 0 and z(p) being well defined for all p R L +. Then there exists a Walrasian equilibrium, i.e., there exists an allocation (x, y ) and a price vector p 0 that constitute a solution to the system of equations z(p) 0. (i) As z(p) are HD0 in prices, we can restrict our attention to the price simplex: = {p R L + l p l = 1}. (ii) Define the function z + l (p) = max {z l (p), 0}. The function z + (p) is continuous, and z + (p) z(p) = 0 implies z(p) 0. (iii) Define α(p) = l (p l + z + l (p)) 1. (iv) f(p) = (p + z + (p))/α(p) is a continuous function from the price simplex to itself. (v) By Brouwer s FixedPoint Theorem there exists a p such that p = f(p ). (vi) By Walras law: 0 = p z(p ) = f(p ) z(p ) = (1/α(p )) (p + z + (p )) z(p ) = (1/α(p )) z + (p ) z(p ). But then, z + (p ) z(p ) = 0, which implies, by (i), that z(p ) 0. W.H.O.W. Query V.5 Show that f(p) :, as claimed in step (iv). 1 Such excess demand functions are not possible with monotone preferences, yet they exist with locally nonsatiated preferences. Ronald Wendner V5 v1.1
6 4 Bonus Stuff Uniqueness A few Results Suppose there exist Walrasian equilibria. The question then is: How many equilibria are there? If there is a (globally) unique equilibrium, we can perform meaningful comparative static analysis. However, if there is more than one equilibrium (i.e., multiplicity) the next best thing is to have a finite number of equilibria. In this case, we have local uniqueness, i.e., at every Walrasian equilibrium (x, y ), there exists an ɛ > 0 and an ɛ ball about (x, y ), B ɛ (x, y ), such that there is no other Walrasian equilibrium within B ɛ (x, y ). More precisely, a Walrasian equilibrium price vector p 0 is locally unique, if there is an ɛ > 0 such that if p p, and p p < ɛ then z(p ) 0. In contrast to local uniqueness, we might encounter indeterminate equilibria, in which case for every ɛ > 0 however small there is an infinite number of Walrasian equilibrium price vectors in p p < ɛ. Indeterminateness is not a desirable property. If the economy is regular, all equilibria are locally unique (determinate). Moreover, an economy is regular, if the Jacobian matrix of price effects Dẑ(p) has rank L 1 (is nonsingular). 2 Query. Suppose, L = 2. Under which condition is E + regular? Under which condition does E + face indeterminate equilibria? We now consider a condition that guarantees global uniqueness of equilibrium. Definition 5 (Gross Substitution) The excess demand function has the gross substitution (GS) property if whenever p and p are such that, for some l, p l > p l and p k = p k for all k l, we have z k (p ) > z k (p) for all k l. Notice that the gross substitution property (as defined above) implies: z l (p ) < z l (p)! In a differential version, GS implies: δ z k (p)/δ p l > 0, i.e., all the offdiagonal entries of Dz(p) are positive. Proposition 4 In E +, there is a globally unique equilibrium, if z(p) satisfies the gross substitution property. 2 Normalize the price vector such that the price of good L = 1: p = (p 1, p 2,..., p L 1, 1). The normalized excess demand function is then: ẑ(p) = (z 1 (p), z 2 (p),..., z L 1 (p)). Then, p 0 is a Walrasian equilibrium price vector if ẑ(p) = 0. Ronald Wendner V6 v1.1
7 Observe that the GS property is sufficient, not necessary! Excess Demand in Economies with Production Definition 6 An economy with production is defined by P ({X i, i } I i=1, {Y j } J j=1, {(ω i, θ i1,..., θ ij )} I i=1). Let P + be an economy with production, where all production sets are closed, strictly convex and bounded. Consider an economy P. The productioninclusive excess demand is given by: z(p) = i x i(p, p ω i + j θ ijπ j (p)) i ω i j y j(p). Proposition 5 Consider an economy P +. (i) to (v), as given by Proposition 2. See Exercise 17.B.4 (MWG, p.642). Local Nonsatiation and Positivity of Prices Then, z(p) satisfies properties Notice that local nonsatiation implies that there is at least one desirable good, otherwise 0 would be a global satiation point. Thus, p x i = p ω i. Proposition 6 Suppose, preferences in E are strictly convex and locally nonsatiated. Then, p is a Walrasian equilibrium price vector if and only if: z(p) i (x i(p, p ω i ) ω i ) 0. Proof (Sketch). It can easily be shown that z(p) 0 [(y 1 0, p y 1 = 0, p 0) & (x i = x i (p, p ω i ) for all i) & ( i x i = i ω i + y 1)]. Proposition 7 Let p be a Walrasian price vector in E. Then, no commodity has a negative price: p l 0 for all l = 1,..., L. Proof (direct). Because of the possibility of free disposal, there are no transactions with a negatively priced commodity (nobody is willing to sell). So there are is no trade with such commodities hence, no good has a negative price. Proposition 8 Let p be a Walrasian price vector in E. Then, p 0. Proof (direct). Suppose p = 0. By local nonsatiation there is a desirable commodity, say l. But then, as the budget set is unbounded, there exists no maximal element, x i, in the budget set. From HD0 of the excess demand functions, and from Proposition 3, we can normalize prices without loss of generality: l p l = 1. Ronald Wendner V7 v1.1
8 Corollary 2 Let p be a Walrasian price vector in E. every desirable commodity l is strictly positive: p l > 0. Then, the price of Corollary 3 Let p be a Walrasian price vector in E. If all commodities are desirable (strong monotonicity), p 0. Proposition 9 Let p be a Walrasian price vector in E. If some commodities are not desirable, i.e., z l (p) < 0, then, p l = 0 and the price vector is not strictly positive. Proof (direct). Suppose first, all goods are desirable. Then, p 0. As p z(p) = 0 we have z(p) = 0, i.e., z l (p) = 0 for all l = 1,..., L. Next, suppose that l is not desirable, i.e.: i z l i(p) < 0. Define z (p) = (z 1, z 2,...z l 1, z l +1,..., z L ), and p = (p 1,...p l 1, p l +1,..., p L ). Then, p l z l (p)+ p z (p) = 0. By Corollary 1, p 0. Moreover, z (p) = 0, as all those goods are desirable (and z (p) 0). Thus, p z (p) = 0. As z l (p) < 0, we must have p l = 0. The argument can easily be extended to the case with several bads. Ronald Wendner V8 v1.1
Economics 200B Part 1 UCSD Winter 2015 Prof. R. Starr, Mr. John Rehbeck Final Exam 1
Economics 200B Part 1 UCSD Winter 2015 Prof. R. Starr, Mr. John Rehbeck Final Exam 1 Your Name: SUGGESTED ANSWERS Please answer all questions. Each of the six questions marked with a big number counts
More informationConvex analysis and profit/cost/support functions
CALIFORNIA INSTITUTE OF TECHNOLOGY Division of the Humanities and Social Sciences Convex analysis and profit/cost/support functions KC Border October 2004 Revised January 2009 Let A be a subset of R m
More informationLecture 8: Market Equilibria
Computational Aspects of Game Theory Bertinoro Spring School 2011 Lecturer: Bruno Codenotti Lecture 8: Market Equilibria The market setting transcends the scenario of games. The decentralizing effect of
More informationWalrasian Demand. u(x) where B(p, w) = {x R n + : p x w}.
Walrasian Demand Econ 2100 Fall 2015 Lecture 5, September 16 Outline 1 Walrasian Demand 2 Properties of Walrasian Demand 3 An Optimization Recipe 4 First and Second Order Conditions Definition Walrasian
More informationFirst Welfare Theorem
First Welfare Theorem Econ 2100 Fall 2015 Lecture 17, November 2 Outline 1 First Welfare Theorem 2 Preliminaries to Second Welfare Theorem Last Class Definitions A feasible allocation (x, y) is Pareto
More informationUCLA. Department of Economics Ph. D. Preliminary Exam MicroEconomic Theory
UCLA Department of Economics Ph. D. Preliminary Exam MicroEconomic Theory (SPRING 2011) Instructions: You have 4 hours for the exam Answer any 5 out of the 6 questions. All questions are weighted equally.
More information1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
More informationName. Final Exam, Economics 210A, December 2011 Here are some remarks to help you with answering the questions.
Name Final Exam, Economics 210A, December 2011 Here are some remarks to help you with answering the questions. Question 1. A firm has a production function F (x 1, x 2 ) = ( x 1 + x 2 ) 2. It is a price
More informationMSc Economics Economic Theory and Applications I Microeconomics. General Equilibrium. Dr Ken Hori,
MSc Economics Economic Theory and Applications I Microeconomics General Equilibrium Dr Ken Hori, k.hori@bbk.ac.uk Birkbeck College, University of London October 2004 Contents 1 General Equilibrium in a
More informationMTH4100 Calculus I. Lecture notes for Week 8. Thomas Calculus, Sections 4.1 to 4.4. Rainer Klages
MTH4100 Calculus I Lecture notes for Week 8 Thomas Calculus, Sections 4.1 to 4.4 Rainer Klages School of Mathematical Sciences Queen Mary University of London Autumn 2009 Theorem 1 (First Derivative Theorem
More informationLecture 3: Growth with Overlapping Generations (Acemoglu 2009, Chapter 9, adapted from Zilibotti)
Lecture 3: Growth with Overlapping Generations (Acemoglu 2009, Chapter 9, adapted from Zilibotti) Kjetil Storesletten September 10, 2013 Kjetil Storesletten () Lecture 3 September 10, 2013 1 / 44 Growth
More informationGame Theory: Supermodular Games 1
Game Theory: Supermodular Games 1 Christoph Schottmüller 1 License: CC Attribution ShareAlike 4.0 1 / 22 Outline 1 Introduction 2 Model 3 Revision questions and exercises 2 / 22 Motivation I several solution
More informationWorking Paper Series
RGEA Universidade de Vigo http://webs.uvigo.es/rgea Working Paper Series A Market Game Approach to Differential Information Economies Guadalupe Fugarolas, Carlos HervésBeloso, Emma Moreno García and
More informationNo: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics
No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results
More information6.207/14.15: Networks Lecture 15: Repeated Games and Cooperation
6.207/14.15: Networks Lecture 15: Repeated Games and Cooperation Daron Acemoglu and Asu Ozdaglar MIT November 2, 2009 1 Introduction Outline The problem of cooperation Finitelyrepeated prisoner s dilemma
More informationAdvanced Microeconomics
Advanced Microeconomics Ordinal preference theory Harald Wiese University of Leipzig Harald Wiese (University of Leipzig) Advanced Microeconomics 1 / 68 Part A. Basic decision and preference theory 1 Decisions
More information2.3 Convex Constrained Optimization Problems
42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions
More informationWhat is Linear Programming?
Chapter 1 What is Linear Programming? An optimization problem usually has three essential ingredients: a variable vector x consisting of a set of unknowns to be determined, an objective function of x to
More informationExample 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum. asin. k, a, and b. We study stability of the origin x
Lecture 4. LaSalle s Invariance Principle We begin with a motivating eample. Eample 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum Dynamics of a pendulum with friction can be written
More informationProblem Set II: budget set, convexity
Problem Set II: budget set, convexity Paolo Crosetto paolo.crosetto@unimi.it Exercises will be solved in class on January 25th, 2010 Recap: Walrasian Budget set, definition Definition (Walrasian budget
More informationGains from Trade. Christopher P. Chambers and Takashi Hayashi. March 25, 2013. Abstract
Gains from Trade Christopher P. Chambers Takashi Hayashi March 25, 2013 Abstract In a market design context, we ask whether there exists a system of transfers regulations whereby gains from trade can always
More information1 Local Brouwer degree
1 Local Brouwer degree Let D R n be an open set and f : S R n be continuous, D S and c R n. Suppose that the set f 1 (c) D is compact. (1) Then the local Brouwer degree of f at c in the set D is defined.
More informationFixed Point Theorems
Fixed Point Theorems Definition: Let X be a set and let T : X X be a function that maps X into itself. (Such a function is often called an operator, a transformation, or a transform on X, and the notation
More informationLecture 13 Linear quadratic Lyapunov theory
EE363 Winter 289 Lecture 13 Linear quadratic Lyapunov theory the Lyapunov equation Lyapunov stability conditions the Lyapunov operator and integral evaluating quadratic integrals analysis of ARE discretetime
More information4: SINGLEPERIOD MARKET MODELS
4: SINGLEPERIOD MARKET MODELS Ben Goldys and Marek Rutkowski School of Mathematics and Statistics University of Sydney Semester 2, 2015 B. Goldys and M. Rutkowski (USydney) Slides 4: SinglePeriod Market
More informationSeparation Properties for Locally Convex Cones
Journal of Convex Analysis Volume 9 (2002), No. 1, 301 307 Separation Properties for Locally Convex Cones Walter Roth Department of Mathematics, Universiti Brunei Darussalam, Gadong BE1410, Brunei Darussalam
More information1 Norms and Vector Spaces
008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)
More informationBANACH AND HILBERT SPACE REVIEW
BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but
More informationRi and. i=1. S i N. and. R R i
The subset R of R n is a closed rectangle if there are n nonempty closed intervals {[a 1, b 1 ], [a 2, b 2 ],..., [a n, b n ]} so that R = [a 1, b 1 ] [a 2, b 2 ] [a n, b n ]. The subset R of R n is an
More informationNash Equilibrium. Ichiro Obara. January 11, 2012 UCLA. Obara (UCLA) Nash Equilibrium January 11, 2012 1 / 31
Nash Equilibrium Ichiro Obara UCLA January 11, 2012 Obara (UCLA) Nash Equilibrium January 11, 2012 1 / 31 Best Response and Nash Equilibrium In many games, there is no obvious choice (i.e. dominant action).
More informationA Simple Model of Price Dispersion *
Federal Reserve Bank of Dallas Globalization and Monetary Policy Institute Working Paper No. 112 http://www.dallasfed.org/assets/documents/institute/wpapers/2012/0112.pdf A Simple Model of Price Dispersion
More informationGeneral Equilibrium Theory: Examples
General Equilibrium Theory: Examples 3 examples of GE: pure exchange (Edgeworth box) 1 producer  1 consumer several producers and an example illustrating the limits of the partial equilibrium approach
More information4.5 Linear Dependence and Linear Independence
4.5 Linear Dependence and Linear Independence 267 32. {v 1, v 2 }, where v 1, v 2 are collinear vectors in R 3. 33. Prove that if S and S are subsets of a vector space V such that S is a subset of S, then
More informationIMPLEMENTING ARROWDEBREU EQUILIBRIA BY TRADING INFINITELYLIVED SECURITIES
IMPLEMENTING ARROWDEBREU EQUILIBRIA BY TRADING INFINITELYLIVED SECURITIES Kevin X.D. Huang and Jan Werner DECEMBER 2002 RWP 0208 Research Division Federal Reserve Bank of Kansas City Kevin X.D. Huang
More information9.2 Summation Notation
9. Summation Notation 66 9. Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with a
More informationMigration with Local Public Goods and the Gains from Changing Places
Migration with Local Public Goods and the Gains from Changing Places Peter J. Hammond Department of Economics, University of Warwick, Coventry CV4 7AL, U.K. p.j.hammond@warwick.ac.edu Jaume Sempere C.E.E.,
More informationTHE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING
THE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING 1. Introduction The BlackScholes theory, which is the main subject of this course and its sequel, is based on the Efficient Market Hypothesis, that arbitrages
More informationInfinitely Repeated Games with Discounting Ù
Infinitely Repeated Games with Discounting Page 1 Infinitely Repeated Games with Discounting Ù Introduction 1 Discounting the future 2 Interpreting the discount factor 3 The average discounted payoff 4
More informationECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE
ECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE YUAN TIAN This synopsis is designed merely for keep a record of the materials covered in lectures. Please refer to your own lecture notes for all proofs.
More informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An ndimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0534405967. Systems of Linear Equations Definition. An ndimensional vector is a row or a column
More informationQuasistatic evolution and congested transport
Quasistatic evolution and congested transport Inwon Kim Joint with Damon Alexander, Katy Craig and Yao Yao UCLA, UW Madison Hard congestion in crowd motion The following crowd motion model is proposed
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More informationCHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.
CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,
More informationx if x 0, x if x < 0.
Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the
More information24. The Branch and Bound Method
24. The Branch and Bound Method It has serious practical consequences if it is known that a combinatorial problem is NPcomplete. Then one can conclude according to the present state of science that no
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More information11 Ideals. 11.1 Revisiting Z
11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(
More informationLECTURE 15: AMERICAN OPTIONS
LECTURE 15: AMERICAN OPTIONS 1. Introduction All of the options that we have considered thus far have been of the European variety: exercise is permitted only at the termination of the contract. These
More informationThe MarketClearing Model
Chapter 5 The MarketClearing Model Most of the models that we use in this book build on two common assumptions. First, we assume that there exist markets for all goods present in the economy, and that
More informationMA651 Topology. Lecture 6. Separation Axioms.
MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples
More informationOverview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model
Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model 1 September 004 A. Introduction and assumptions The classical normal linear regression model can be written
More information1 Introduction. Linear Programming. Questions. A general optimization problem is of the form: choose x to. max f(x) subject to x S. where.
Introduction Linear Programming Neil Laws TT 00 A general optimization problem is of the form: choose x to maximise f(x) subject to x S where x = (x,..., x n ) T, f : R n R is the objective function, S
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationI. GROUPS: BASIC DEFINITIONS AND EXAMPLES
I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called
More informationFurther Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1
Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1 J. Zhang Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing
More informationLinear Programming I
Linear Programming I November 30, 2003 1 Introduction In the VCR/guns/nuclear bombs/napkins/star wars/professors/butter/mice problem, the benevolent dictator, Bigus Piguinus, of south Antarctica penguins
More informationResearch Article Stability Analysis for HigherOrder Adjacent Derivative in Parametrized Vector Optimization
Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 510838, 15 pages doi:10.1155/2010/510838 Research Article Stability Analysis for HigherOrder Adjacent Derivative
More informationProperties of BMO functions whose reciprocals are also BMO
Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a nonnegative BMOfunction w, whose reciprocal is also in BMO, belongs to p> A p,and
More informationFIRST YEAR CALCULUS. Chapter 7 CONTINUITY. It is a parabola, and we can draw this parabola without lifting our pencil from the paper.
FIRST YEAR CALCULUS WWLCHENW L c WWWL W L Chen, 1982, 2008. 2006. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It It is is
More informationSection 3 Sequences and Limits, Continued.
Section 3 Sequences and Limits, Continued. Lemma 3.6 Let {a n } n N be a convergent sequence for which a n 0 for all n N and it α 0. Then there exists N N such that for all n N. α a n 3 α In particular
More informationSome Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.
Some Polynomial Theorems by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.com This paper contains a collection of 31 theorems, lemmas,
More informationCritical points of once continuously differentiable functions are important because they are the only points that can be local maxima or minima.
Lecture 0: Convexity and Optimization We say that if f is a once continuously differentiable function on an interval I, and x is a point in the interior of I that x is a critical point of f if f (x) =
More informationNotes: Chapter 2 Section 2.2: Proof by Induction
Notes: Chapter 2 Section 2.2: Proof by Induction Basic Induction. To prove: n, a W, n a, S n. (1) Prove the base case  S a. (2) Let k a and prove that S k S k+1 Example 1. n N, n i = n(n+1) 2. Example
More informationK 1 < K 2 = P (K 1 ) P (K 2 ) (6) This holds for both American and European Options.
Slope and Convexity Restrictions and How to implement Arbitrage Opportunities 1 These notes will show how to implement arbitrage opportunities when either the slope or the convexity restriction is violated.
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationLectures 56: Taylor Series
Math 1d Instructor: Padraic Bartlett Lectures 5: Taylor Series Weeks 5 Caltech 213 1 Taylor Polynomials and Series As we saw in week 4, power series are remarkably nice objects to work with. In particular,
More informationIRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction
IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible
More information3. INNER PRODUCT SPACES
. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.
More informationIdeal Class Group and Units
Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals
More informationRecall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula:
Chapter 7 Div, grad, and curl 7.1 The operator and the gradient: Recall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula: ( ϕ ϕ =, ϕ,..., ϕ. (7.1 x 1
More informationMathematical finance and linear programming (optimization)
Mathematical finance and linear programming (optimization) Geir Dahl September 15, 2009 1 Introduction The purpose of this short note is to explain how linear programming (LP) (=linear optimization) may
More informationChapter 7. Sealedbid Auctions
Chapter 7 Sealedbid Auctions An auction is a procedure used for selling and buying items by offering them up for bid. Auctions are often used to sell objects that have a variable price (for example oil)
More informationOPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN. E.V. Grigorieva. E.N. Khailov
DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS Supplement Volume 2005 pp. 345 354 OPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN E.V. Grigorieva Department of Mathematics
More informationLECTURE 5: DUALITY AND SENSITIVITY ANALYSIS. 1. Dual linear program 2. Duality theory 3. Sensitivity analysis 4. Dual simplex method
LECTURE 5: DUALITY AND SENSITIVITY ANALYSIS 1. Dual linear program 2. Duality theory 3. Sensitivity analysis 4. Dual simplex method Introduction to dual linear program Given a constraint matrix A, right
More information10.2 Series and Convergence
10.2 Series and Convergence Write sums using sigma notation Find the partial sums of series and determine convergence or divergence of infinite series Find the N th partial sums of geometric series and
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number
More information{f 1 (U), U F} is an open cover of A. Since A is compact there is a finite subcover of A, {f 1 (U 1 ),...,f 1 (U n )}, {U 1,...
44 CHAPTER 4. CONTINUOUS FUNCTIONS In Calculus we often use arithmetic operations to generate new continuous functions from old ones. In a general metric space we don t have arithmetic, but much of it
More information15 Limit sets. Lyapunov functions
15 Limit sets. Lyapunov functions At this point, considering the solutions to ẋ = f(x), x U R 2, (1) we were most interested in the behavior of solutions when t (sometimes, this is called asymptotic behavior
More informationSolving Linear Systems, Continued and The Inverse of a Matrix
, Continued and The of a Matrix Calculus III Summer 2013, Session II Monday, July 15, 2013 Agenda 1. The rank of a matrix 2. The inverse of a square matrix Gaussian Gaussian solves a linear system by reducing
More informationA Direct Numerical Method for Observability Analysis
IEEE TRANSACTIONS ON POWER SYSTEMS, VOL 15, NO 2, MAY 2000 625 A Direct Numerical Method for Observability Analysis Bei Gou and Ali Abur, Senior Member, IEEE Abstract This paper presents an algebraic method
More informationLecture Notes on Polynomials
Lecture Notes on Polynomials Arne Jensen Department of Mathematical Sciences Aalborg University c 008 Introduction These lecture notes give a very short introduction to polynomials with real and complex
More information1. Prove that the empty set is a subset of every set.
1. Prove that the empty set is a subset of every set. Basic Topology Written by MenGen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since
More informationLoad Balancing and Switch Scheduling
EE384Y Project Final Report Load Balancing and Switch Scheduling Xiangheng Liu Department of Electrical Engineering Stanford University, Stanford CA 94305 Email: liuxh@systems.stanford.edu Abstract Load
More informationALMOST COMMON PRIORS 1. INTRODUCTION
ALMOST COMMON PRIORS ZIV HELLMAN ABSTRACT. What happens when priors are not common? We introduce a measure for how far a type space is from having a common prior, which we term prior distance. If a type
More informationTHE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear
More informationAtomic Cournotian Traders May Be Walrasian
Atomic Cournotian Traders May Be Walrasian Giulio Codognato, Sayantan Ghosal, Simone Tonin September 2014 Abstract In a bilateral oligopoly, with large traders, represented as atoms, and small traders,
More informationManipulability of the Price Mechanism for Data Centers
Manipulability of the Price Mechanism for Data Centers Greg Bodwin 1, Eric Friedman 2,3,4, and Scott Shenker 3,4 1 Department of Computer Science, Tufts University, Medford, Massachusetts 02155 2 School
More informationCHAPTER 5. Product Measures
54 CHAPTER 5 Product Measures Given two measure spaces, we may construct a natural measure on their Cartesian product; the prototype is the construction of Lebesgue measure on R 2 as the product of Lebesgue
More informationRotation Rate of a Trajectory of an Algebraic Vector Field Around an Algebraic Curve
QUALITATIVE THEORY OF DYAMICAL SYSTEMS 2, 61 66 (2001) ARTICLE O. 11 Rotation Rate of a Trajectory of an Algebraic Vector Field Around an Algebraic Curve Alexei Grigoriev Department of Mathematics, The
More informationIntroduction to Algebraic Geometry. Bézout s Theorem and Inflection Points
Introduction to Algebraic Geometry Bézout s Theorem and Inflection Points 1. The resultant. Let K be a field. Then the polynomial ring K[x] is a unique factorisation domain (UFD). Another example of a
More informationDuality of linear conic problems
Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least
More informationUnraveling versus Unraveling: A Memo on Competitive Equilibriums and Trade in Insurance Markets
Unraveling versus Unraveling: A Memo on Competitive Equilibriums and Trade in Insurance Markets Nathaniel Hendren January, 2014 Abstract Both Akerlof (1970) and Rothschild and Stiglitz (1976) show that
More informationMoral Hazard. Itay Goldstein. Wharton School, University of Pennsylvania
Moral Hazard Itay Goldstein Wharton School, University of Pennsylvania 1 PrincipalAgent Problem Basic problem in corporate finance: separation of ownership and control: o The owners of the firm are typically
More informationSome stability results of parameter identification in a jump diffusion model
Some stability results of parameter identification in a jump diffusion model D. Düvelmeyer Technische Universität Chemnitz, Fakultät für Mathematik, 09107 Chemnitz, Germany Abstract In this paper we discuss
More informationOn the Efficiency of Competitive Stock Markets Where Traders Have Diverse Information
Finance 400 A. Penati  G. Pennacchi Notes on On the Efficiency of Competitive Stock Markets Where Traders Have Diverse Information by Sanford Grossman This model shows how the heterogeneous information
More informationINDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS
INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problem
More informationt := maxγ ν subject to ν {0,1,2,...} and f(x c +γ ν d) f(x c )+cγ ν f (x c ;d).
1. Line Search Methods Let f : R n R be given and suppose that x c is our current best estimate of a solution to P min x R nf(x). A standard method for improving the estimate x c is to choose a direction
More information6. Cholesky factorization
6. Cholesky factorization EE103 (Fall 201112) triangular matrices forward and backward substitution the Cholesky factorization solving Ax = b with A positive definite inverse of a positive definite matrix
More informationNetwork Traffic Modelling
University of York Dissertation submitted for the MSc in Mathematics with Modern Applications, Department of Mathematics, University of York, UK. August 009 Network Traffic Modelling Author: David Slade
More informationReference: Introduction to Partial Differential Equations by G. Folland, 1995, Chap. 3.
5 Potential Theory Reference: Introduction to Partial Differential Equations by G. Folland, 995, Chap. 3. 5. Problems of Interest. In what follows, we consider Ω an open, bounded subset of R n with C 2
More information