# First Welfare Theorem

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 First Welfare Theorem Econ 2100 Fall 2015 Lecture 17, November 2 Outline 1 First Welfare Theorem 2 Preliminaries to Second Welfare Theorem

2 Last Class Definitions A feasible allocation (x, y) is Pareto optimal if there is no other feasible allocation (x, y ) such that x i i x i for all i and x i i x i for some i. An allocation (x, y ) and a price vector p R L are a competitive equilibrium if 1 for each j = 1,..., J: p y j p yj for all y j Y j ; 2 for each i = 1,..., I : xi i x i for all x i B i (p ) = {x i X i : p x i p ω i + ( ) j θij p yj } ; 3 and i x i = i ωi + j y j. What is the relationship between competitive equilibrium and Pareto effi ciency? Is any competitive equilibirum Pareto effi cient? First Welfare Theorem. This is about excluding something can Pareto dominate the equilibrium allocation. Is any Pareto effi cient allocation (part of) a competitive equilibrium? Second Welfare Theorem. This is about finding prices that make the effi cient allocation an equilibirum.

3 First Welfare Theorem: A Picture Things seem easy O 2 ω x * O 1

4 How do we rule examples like this out? Need consumers preferences that are locally non satiated: there is always something nearby that makes a consumer better off. First Welfare Theorem: Counterexample An Edgeworth Box Economy Consider a two-person, two-good exchange economy. Person a has utility function U a (x 1a, x 2a ) = 7 and person b has utility function U b (x 1b, x 2b ) = x 1b x 2b. The initial endowments are ω a = (2, 0) and ω b = (0, 2). CLAIM: x a = (1, 1), x b = (1, 1), and prices p = (1, 1) form a competitive equilibirum. a s utility is maximized. b s utility when her income equals 2 is maximized (this is a Cobb-Douglas utility function with equal exponents, so spending half her income on each good is optimal). x a + x b = (2, 2) = ω. Is this allocation Pareto optimal? No: ˆx a = (0, 0) and ˆx b = (2, 2) Pareto dominates xa, xb since consumer a has the same utility while consumer b s utility is higher.

5 Local Non Satiation Definition A preference ordering i on X i is satiated at y if there exists no x in X i such that x i y. Definition The preference relation i on X i is locally non-satiated if for every x in X i and for every ε > 0 there exists an x in X i such that x x < ε and x i x. Remark Remember: y z = two points. L l=1 (y l z l ) 2 is the Euclidean distance between If i is continuous and locally non-satiated there exist a locally non-satiated utility function; then, any closed consumption set must be unbounded (or there would be a global satiation point).

6 Local Non Satiation and Walrasian Demand Lemma Suppose i is locally non-satiated, and let xi be defined as: xi i x i for all x i {x i X i : p x i w i }. Then and x i i x i implies p x i w i x i i x i implies p x i > w i In words, if a consumption vector is preferred to a maximal consumption bundle (i.e. an element of the Walrasian demand correspondence) it must cost more. Something strictly preferred to a maximal bundle must not be affordable (or the consumer would have chosen it).

7 First Welfare Theorem Theorem (First Fundamental Theorem of Welfare Economics) Suppose each consumer s preferences are locally non-satiated. Then, any allocation x, y that with prices p forms a competitive equilibrium is Pareto optimal. There are almost no assumptions... Local non-satiation has bite: there is always a more desirable commodity bundle nearby. Among the economic assumptions implicit in our definition of an economy one is important for this result: no externalities. Externalities are present if one person s consumption influences another person s preferences For example: in a two-person, two-good exchange economy person a has utility function U a(x a1, x a2 ) = x a1 x a2 x b1.

8 Proof of the First Welfare Theorem (by contradiction) Suppose not: there exists a feasible allocation x, y such such that x i i xi for all i, and x i i xi for some i. By local non satiation, x i i xi implies p x i p ω i + j θij (p yj ) x i i xi implies p x i > p ω i + j θij (p yj ) Therefore, summing over consumers p x i > p ω i + θ ij (p yj ) accounting = p ω + p yj Since each y j maximizes profits at prices p, we also have p yj p y j. Substituting this into the previous inequality: p x i > p ω + p yj p ω + p y j But then x, y cannot be feasible since if it were we would have x i = ω + y j p x i = p ω + p y j which contradicts the expression above.

9 Competitive Equilibrium and the Core Theorem Any competitive equilibrium is in the core. Proof. Homework. This is very very similar to the proof of the First Welfare Theorem. A converse can be established in cases in which the economy is large, that is, it contains many individuals. That is called the core convergence theorem and I do not think we will have time for it.

10 Second Welfare Theorem: Preliminaries This is a converse to the First Welfare Theorem. The statement goes: under some conditions, any Pareto optimal allocation is part of a competitive equilibrium. Next, we try to understand what these conditions must be. We state and prove thw theorem next class. In order to prove a Pareto optimal allocation is part of an equilibrium one needs to find the price vector that works for that allocation, since an equilibrium must specify and allocation and prices. First, we see a simple sense in which this cannot work: Pareto optimality disregards the budget constraints. This is fixed by adjusting the definition of equilibrium. Then we see two counterexamples that stress the need for convexities. These are fixed by assuming production sets and better-than sets are convex. Finally, we see an example showing that an equilibrium may not exist, and therefore there is no way the theorem holds. This is fixed by, again, adjusting the definition of equilibrium.

11 O 1 Pareto Optimal Allocation x ω Initial Endowment O 2 O 2 The way to make the Pareto optimal allocation an equilibirum is to stick a price vector tangent to both indifference curves. This is not enought, however, since the Pareto optimal allocation violates the budget constraint of consumer 2. From the Edgeworth box, you can see that there is no hope of going from any Pareto optimal allocation to prices compatible with maximization and initial endowment. O 1 Pareto Optimal Allocation x ω Initial Endowment Income Transfers There is hope, however, to keep the slope of the budget constraint fixed and adjust income so that things work out. One needs to adjust income appropriately. We achieve this by modifying the definition of equilibrium.

12 Equilibrium With Transfers Definition ) Given an economy ({X i, i } I, {Y j } J, ω, an allocation x, y and a price vector p constitute a price equilibrium with transfers if there exists a vector of wealth levels w = (w 1, w 2,..., w I ) with w i = p ω + p yj such that: 1 For each j = 1,..., J: p y j p y j for all y j Y j. 2 For each i = 1,..., I : xi i x i for all x i {x i X i : p x i w i } 3 = I ω i + J x i Aggregate wealth is divided among consumers so that the budget constrants are satisfied. How is the wealth of each consumer effected? They get a positive or negative transfer relative to the value of their endowment at the equilibrium prices. A competitive equilibrium satisfies this: set w i = p ω i + j θ ij (p y j ). y j

13 Remark Equilibrium With Transfers The income transfers (across consumers) that achieve the budget levels in the previous definition are: T i = w i p ( ω i + θ ij p yj ) Summing over consumers, we get T i = w i p ω i + i i = w i p ω + p yj = 0 θ ij ( p y j Transfers redistribute income so that the aggregate budget balances. This is important: in a general equilibirum model nothing should be outside the economy )

14 Counterexample I to Second Welfare Theorem Need convex preferences for the Second Welfare Theorem x* p Õ 2 Õ 1 x* is Pareto optimal, but one can see it is not an equilibrium at prices p

15 Counterexample II to Second Welfare Theorem One Producer One Consumer Economy (Robinson Crusoe) x is the preferences maximizing bundle at the given prices x x does not maximizes profits in production set Y at the given prices (not even locally) Y Õ 1 Need convex production sets for Second Welfare Theorem

16 Definition Convexity A preference relation on X is convex if the set (x) = {y X y x} is convex for every x. If x and x are weakly preferred to x so is any convex combination. Convexity implies existence of an hyperplane that supports a consumer s better than set. Definition In an exchange economy, an allocation x is supported by a non-zero price vector p if: for each i = 1,..., I x i x i = p x p x i Convexity also yields an hyperplane that supports all producers better than set at the same time. This hyperplane is the price vector that makes a Pareto optimal allocation an equilibirum.

17 Counterexample III to Second Welfare Theorem ω is the initial endowment Consumer 1 owns all of good 2 Consumer 2 owns all of good 1 For any p 0, 2 demands ω 2 when her wealth is w 2 ; but ω 1 is never optimal for 1 if p 0 she would demand an infinite quantity of good 1 p=(1,0) is the candidate Equilibrium price vector Consumer 2 only cares about good 1 There is no Walrasian equilibrium in this case Õ 2 Consumer 1 preferences have infinite slope at ω 1 Õ 1 ω is the unique Pareto optimal allocation, hence the only candidate for an equilibrium. The unique price vector that can support it as equilibrium (normalizing the price of the first good to 1) is p = (1, 0). The corresponding wealth is w 1 = (1, 0) (0, ω 2 ) = 0. However, ω 1 is not maximial for consumer 1 at p since she would like more of good 2 (it has zero price, hence she can afford it. There is no competitive equilibrium in this example.

18 Quasi-Equilibrium To fix the existence at the boundary problem, with make a small change to the definition of equilibrium. Definition Given an economy {X i, i } I, {Y j } J, ω, an allocation x, y and a price vector p constitute a quasi-equilibrium with transfers if there exists a vector of wealth levels such that: w = (w 1, w 2,..., w I ) with w i = p ω + p y j 1 For each j = 1,..., J: p y j p yj for all y j Y j. 2 For every i = 1,..., I : 3 x i if = I ω i + J y j x i x i then p x w i Make sure you see why this deals with the problem in the previous slide. Any equilibrium with transfers is a quasi-equilibrium (make sure you check

19 Problem Set 10 (Incomplete) Due Monday 9 November 1 Consider a competitive model with one input, two outputs, and two firms with production functions y 1 = f 1 (l 1 ) = 2l 1 and y 2 = f 2 (l 2 ) = 2l 2 (where l j denotes the amount of the input l used by firm j, and firm j produces only good j). Consumer a is endowed with 25 units of the input l and owns no shares in the firms, and has utility function U a (x a1, x a2 ) = x a1 x a2. Consumer b owns both firms, but has zero endowment, and has utility function U b (x b1, x b2 ) = x b1 + x b2. Find a competitive equilibrium in this model. Is it unique? 2 Prove that any competitive equilibrium is in the core (for an exchange economy). 3 Consider a two-person, two-good exchange economy in which person a has utility function U a (x a1, x a2 ) = x a1 x a2 x b1 and person b has utility function U b (x b1, x b2 ) = x b1 x b2. The initial endowments are ω a = (2, 0) and ω b = (0, 2). 1 Show that p = (1, 1) and x a = (1, 1), x b = (1, 1) is a competitive equilibrium. 2 Prove or provide a counterexample to the following statement: in this economy any competitive equilibrium is Pareto optimal. 4 Consider a two-person, two-good exchange economy in which person a has utility function U a (x a1, x a2 ) = 1 if x a1 + x a2 < 1 and U a (x a1, x a2 ) = x a1 + x a2 if x a1 + x a2 1. Person b has utility function U b (x b1, x b2 ) = x b1 x b2. The initial endowments are ω a = (1, 0) and ω b = (0, 1). 1 Show that p = (1, 1) and x a = ( 1, 1 ), 2 2 xb = ( 1, 1 ) is a competitive equilibrium Is this allocation in the core? Explain your answer. 3 Does the first welfare theorem hold for this economy? Explain your answer. 5 Consider a two-person, two-good exchange economy where the agents utility functions are U a (x a1, x a2 ) = x a1 x a2 and U b (x b1, x b2 ) = x b1 x b2, and the initial endowments are ω a = (1, 5) and ω b = (5, 1). 1 Find the Pareto optimal allocations and the core. Draw the Edgeworth box for this economy. 2 Find the individual and market excess demand functions. Find the equilibrium prices and allocations. 3 Show directly that every interior Pareto optimal allocation in this economy is a price equilibrium with transfers by finding the associated prices and transfers.

### Walrasian 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

### General 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

### ECON 301: General Equilibrium I (Exchange) 1. Intermediate Microeconomics II, ECON 301. General Equilibrium I: Exchange

ECON 301: General Equilibrium I (Exchange) 1 Intermediate Microeconomics II, ECON 301 General Equilibrium I: Exchange The equilibrium concepts you have used till now in your Introduction to Economics,

### Lecture 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

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

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

### 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

### 1. Briefly explain what an indifference curve is and how it can be graphically derived.

Chapter 2: Consumer Choice Short Answer Questions 1. Briefly explain what an indifference curve is and how it can be graphically derived. Answer: An indifference curve shows the set of consumption bundles

### 2. Efficiency and Perfect Competition

General Equilibrium Theory 1 Overview 1. General Equilibrium Analysis I Partial Equilibrium Bias 2. Efficiency and Perfect Competition 3. General Equilibrium Analysis II The Efficiency if Competition The

### Indifference Curves and the Marginal Rate of Substitution

Introduction Introduction to Microeconomics Indifference Curves and the Marginal Rate of Substitution In microeconomics we study the decisions and allocative outcomes of firms, consumers, households and

### MSc 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

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

### MICROECONOMICS AND POLICY ANALYSIS - U8213 Professor Rajeev H. Dehejia Class Notes - Spring 2001

MICROECONOMICS AND POLICY ANALYSIS - U8213 Professor Rajeev H. Dehejia Class Notes - Spring 2001 General Equilibrium and welfare with production Wednesday, January 24 th and Monday, January 29 th Reading:

### Theoretical Tools of Public Economics. Part-2

Theoretical Tools of Public Economics Part-2 Previous Lecture Definitions and Properties Utility functions Marginal utility: positive (negative) if x is a good ( bad ) Diminishing marginal utility Indifferences

### Chapter 6: Pure Exchange

Chapter 6: Pure Exchange Pure Exchange Pareto-Efficient Allocation Competitive Price System Equitable Endowments Fair Social Welfare Allocation Outline and Conceptual Inquiries There are Gains from Trade

### More is Better. an investigation of monotonicity assumption in economics. Joohyun Shon. August 2008

More is Better an investigation of monotonicity assumption in economics Joohyun Shon August 2008 Abstract Monotonicity of preference is assumed in the conventional economic theory to arrive at important

### Answer: TRUE. The firm can increase profits by increasing production.

Solutions to 1.01 Make-up for Midterm Short uestions: 1. (TOTAL: 5 points) Explain whether each of the following statements is True or False. (Note: You will not get points for a correct answer without

### Managerial Economics Prof. Trupti Mishra S.J.M. School of Management Indian Institute of Technology, Bombay. Lecture - 13 Consumer Behaviour (Contd )

(Refer Slide Time: 00:28) Managerial Economics Prof. Trupti Mishra S.J.M. School of Management Indian Institute of Technology, Bombay Lecture - 13 Consumer Behaviour (Contd ) We will continue our discussion

### Preferences. M. Utku Ünver Micro Theory. Boston College. M. Utku Ünver Micro Theory (BC) Preferences 1 / 20

Preferences M. Utku Ünver Micro Theory Boston College M. Utku Ünver Micro Theory (BC) Preferences 1 / 20 Preference Relations Given any two consumption bundles x = (x 1, x 2 ) and y = (y 1, y 2 ), the

### Consumer Theory. The consumer s problem

Consumer Theory The consumer s problem 1 The Marginal Rate of Substitution (MRS) We define the MRS(x,y) as the absolute value of the slope of the line tangent to the indifference curve at point point (x,y).

### Module 2 Lecture 5 Topics

Module 2 Lecture 5 Topics 2.13 Recap of Relevant Concepts 2.13.1 Social Welfare 2.13.2 Demand Curves 2.14 Elasticity of Demand 2.14.1 Perfectly Inelastic 2.14.2 Perfectly Elastic 2.15 Production & Cost

### Lecture Notes 5: General Equilibrium. mrkts

Lecture Notes 5: General Equilibrium We are now ready to analyze the equilibrium in all markets, that is why this type of analysis is called general equilibrium analysis. Recall our graphical overview

### Lecture 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

### MTH4100 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

### Security Pools and Efficiency

Security Pools and Efficiency John Geanakoplos Yale University William R. Zame UCLA Abstract Collateralized mortgage obligations, collateralized debt obligations and other bundles of securities that are

### Economics 401. Sample questions 1

Economics 40 Sample questions. Do each of the following choice structures satisfy WARP? (a) X = {a, b, c},b = {B, B 2, B 3 }, B = {a, b}, B 2 = {b, c}, B 3 = {c, a}, C (B ) = {a}, C (B 2 ) = {b}, C (B

### Topic 7 General equilibrium and welfare economics

Topic 7 General equilibrium and welfare economics. The production possibilities frontier is generated using a production Edgeworth box diagram with the input goods on the axes. The following diagram illustrates

### Name. 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

### U = x 1 2. 1 x 1 4. 2 x 1 4. What are the equilibrium relative prices of the three goods? traders has members who are best off?

Chapter 7 General Equilibrium Exercise 7. Suppose there are 00 traders in a market all of whom behave as price takers. Suppose there are three goods and the traders own initially the following quantities:

### The Market-Clearing Model

Chapter 5 The Market-Clearing 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

### Quiggin, J. (1991), Too many proposals pass the benefit cost test: comment, TOO MANY PROPOSALS PASS THE BENEFIT COST TEST - COMMENT.

Quiggin, J. (1991), Too many proposals pass the benefit cost test: comment, American Economic Review 81(5), 1446 9. TOO MANY PROPOSALS PASS THE BENEFIT COST TEST - COMMENT by John Quiggin University of

### The fundamental question in economics is 2. Consumer Preferences

A Theory of Consumer Behavior Preliminaries 1. Introduction The fundamental question in economics is 2. Consumer Preferences Given limited resources, how are goods and service allocated? 1 3. Indifference

### UNIVERSITY OF VIENNA

WORKING PAPERS Egbert Dierker Hildegard Dierker Birgit Grodal Nonexistence of Constrained Efficient Equilibria when Markets are Incomplete August 2001 Working Paper No: 0111 DEPARTMENT OF ECONOMICS UNIVERSITY

### Schooling, Political Participation, and the Economy. (Online Supplementary Appendix: Not for Publication)

Schooling, Political Participation, and the Economy Online Supplementary Appendix: Not for Publication) Filipe R. Campante Davin Chor July 200 Abstract In this online appendix, we present the proofs for

### Readings. D Chapter 1. Lecture 2: Constrained Optimization. Cecilia Fieler. Example: Input Demand Functions. Consumer Problem

Economics 245 January 17, 2012 : Example Readings D Chapter 1 : Example The FOCs are max p ( x 1 + x 2 ) w 1 x 1 w 2 x 2. x 1,x 2 0 p 2 x i w i = 0 for i = 1, 2. These are two equations in two unknowns,

### General Equilibrium. The Producers

General Equilibrium In this section we will combine production possibilities frontiers and community indifference curves in order to create a model of the economy as a whole. This brings together producers,

### 6.254 : Game Theory with Engineering Applications Lecture 5: Existence of a Nash Equilibrium

6.254 : Game Theory with Engineering Applications Lecture 5: Existence of a Nash Equilibrium Asu Ozdaglar MIT February 18, 2010 1 Introduction Outline Pricing-Congestion Game Example Existence of a Mixed

### UCLA. Department of Economics Ph. D. Preliminary Exam Micro-Economic Theory

UCLA Department of Economics Ph. D. Preliminary Exam Micro-Economic Theory (SPRING 2011) Instructions: You have 4 hours for the exam Answer any 5 out of the 6 questions. All questions are weighted equally.

### servings of fries, income is exhausted and MU per dollar spent is the same for both goods, so this is the equilibrium.

Problem Set # 5 Unless told otherwise, assume that individuals think that more of any good is better (that is, marginal utility is positive). Also assume that indifference curves have their normal shape,

### When the price of a good rises (falls) and all other variables affecting the

ONLINE APPENDIX 1 Substitution and Income Effects of a Price Change When the price of a good rises (falls) and all other variables affecting the consumer remain the same, the law of demand tells us that

### At t = 0, a generic intermediary j solves the optimization problem:

Internet Appendix for A Model of hadow Banking * At t = 0, a generic intermediary j solves the optimization problem: max ( D, I H, I L, H, L, TH, TL ) [R (I H + T H H ) + p H ( H T H )] + [E ω (π ω ) A

### Increasing for all. Convex for all. ( ) Increasing for all (remember that the log function is only defined for ). ( ) Concave for all.

1. Differentiation The first derivative of a function measures by how much changes in reaction to an infinitesimal shift in its argument. The largest the derivative (in absolute value), the faster is evolving.

### Revealed Preference. Ichiro Obara. October 8, 2012 UCLA. Obara (UCLA) Revealed Preference October 8, / 17

Ichiro Obara UCLA October 8, 2012 Obara (UCLA) October 8, 2012 1 / 17 Obara (UCLA) October 8, 2012 2 / 17 Suppose that we obtained data of price and consumption pairs D = { (p t, x t ) R L ++ R L +, t

### Mathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions

Natalia Lazzati Mathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions Note 5 is based on Madden (1986, Ch. 1, 2, 4 and 7) and Simon and Blume (1994, Ch. 13 and 21). Concave functions

### Week 1 Efficiency in Production and in Consumption and the Fundamental Theorems of Welfare Economics

1. Introduction Week 1 Efficiency in Production and in Consumption and the Fundamental Theorems of Welfare Economics Much of economic theory of the textbook variety is a celebration of the free market

### Definition and Properties of the Production Function: Lecture

Definition and Properties of the Production Function: Lecture II August 25, 2011 Definition and : Lecture A Brief Brush with Duality Cobb-Douglas Cost Minimization Lagrangian for the Cobb-Douglas Solution

### By W.E. Diewert. July, Linear programming problems are important for a number of reasons:

APPLIED ECONOMICS By W.E. Diewert. July, 3. Chapter : Linear Programming. Introduction The theory of linear programming provides a good introduction to the study of constrained maximization (and minimization)

### Introduction. Agents have preferences over the two goods which are determined by a utility function. Speci cally, type 1 agents utility is given by

Introduction General equilibrium analysis looks at how multiple markets come into equilibrium simultaneously. With many markets, equilibrium analysis must take explicit account of the fact that changes

### Prep. Course Macroeconomics

Prep. Course Macroeconomics Intertemporal consumption and saving decision; Ramsey model Tom-Reiel Heggedal tom-reiel.heggedal@bi.no BI 2014 Heggedal (BI) Savings & Ramsey 2014 1 / 30 Overview this lecture

### Moral Hazard. Itay Goldstein. Wharton School, University of Pennsylvania

Moral Hazard Itay Goldstein Wharton School, University of Pennsylvania 1 Principal-Agent Problem Basic problem in corporate finance: separation of ownership and control: o The owners of the firm are typically

### 2.3 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

### Working 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 Hervés-Beloso, Emma Moreno- García and

### Geometry of Linear Programming

Chapter 2 Geometry of Linear Programming The intent of this chapter is to provide a geometric interpretation of linear programming problems. To conceive fundamental concepts and validity of different algorithms

### 7. EXTERNALITIES AND PUBLIC GOODS

7. EXTERNALITIES AND PUBLIC GOODS In many economic situations, markets are not complete. Certain actions of a rm or a consumer may a ect other economic agents welfare but are not market transactions. These

### OLG Models. Jesús Fernández-Villaverde. University of Pennsylvania. February 12, 2016

OLG Models Jesús Fernández-Villaverde University of Pennsylvania February 12, 2016 Jesús Fernández-Villaverde (PENN) OLG Models February 12, 2016 1 / 53 Overlapping Generations The Overlapping Generations

### Lecture 2 Dynamic Equilibrium Models : Finite Periods

Lecture 2 Dynamic Equilibrium Models : Finite Periods 1. Introduction In macroeconomics, we study the behavior of economy-wide aggregates e.g. GDP, savings, investment, employment and so on - and their

### An example of externalities - the additive case

An example of externalities - the additive case Yossi Spiegel Consider an exchange economy with two agents, A and B, who consume two goods, x and y. The preferences of the two agents are represented by

### Choice under Uncertainty

Choice under Uncertainty Part 1: Expected Utility Function, Attitudes towards Risk, Demand for Insurance Slide 1 Choice under Uncertainty We ll analyze the underlying assumptions of expected utility theory

### TOPIC 4: DERIVATIVES

TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the

### Critical 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) =

### THE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING

THE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING 1. Introduction The Black-Scholes theory, which is the main subject of this course and its sequel, is based on the Efficient Market Hypothesis, that arbitrages

### Offer Curves. Price-consumption curve. Y Axis. X Axis. Figure 1: Price-consumption Curve

Offer Curves The offer curve is an alternative way to describe an individual s demand behavior, i.e., his demand function. And by summing up individuals demand behavior, we can also use the offer curve

### Practice with Proofs

Practice with Proofs October 6, 2014 Recall the following Definition 0.1. A function f is increasing if for every x, y in the domain of f, x < y = f(x) < f(y) 1. Prove that h(x) = x 3 is increasing, using

### Mathematical Economics: Lecture 15

Mathematical Economics: Lecture 15 Yu Ren WISE, Xiamen University November 19, 2012 Outline 1 Chapter 20: Homogeneous and Homothetic Functions New Section Chapter 20: Homogeneous and Homothetic Functions

### Economics 2020a / HBS 4010 / HKS API-111 FALL 2010 Solutions to Practice Problems for Lectures 1 to 4

Economics 00a / HBS 4010 / HKS API-111 FALL 010 Solutions to Practice Problems for Lectures 1 to 4 1.1. Quantity Discounts and the Budget Constraint (a) The only distinction between the budget line with

### 1 Homogenous and Homothetic Functions

1 Homogenous and Homothetic Functions Reading: [Simon], Chapter 20, p. 483-504. 1.1 Homogenous Functions Definition 1 A real valued function f(x 1,..., x n ) is homogenous of degree k if for all t > 0

### Figure 4.1 Average Hours Worked per Person in the United States

The Supply of Labor Figure 4.1 Average Hours Worked per Person in the United States 1 Table 4.1 Change in Hours Worked by Age: 1950 2000 4.1: Preferences 4.2: The Constraints 4.3: Optimal Choice I: Determination

Advanced International Economics Prof. Yamin Ahmad ECON 758 Sample Midterm Exam Name Id # Instructions: There are two parts to this midterm. Part A consists of multiple choice questions. Please mark the

### Modeling Insurance Markets

Modeling Insurance Markets Nathaniel Hendren Harvard April, 2015 Nathaniel Hendren (Harvard) Insurance April, 2015 1 / 29 Modeling Competition Insurance Markets is Tough There is no well-agreed upon model

### ECON 40050 Game Theory Exam 1 - Answer Key. 4) All exams must be turned in by 1:45 pm. No extensions will be granted.

1 ECON 40050 Game Theory Exam 1 - Answer Key Instructions: 1) You may use a pen or pencil, a hand-held nonprogrammable calculator, and a ruler. No other materials may be at or near your desk. Books, coats,

### Cost Minimization and the Cost Function

Cost Minimization and the Cost Function Juan Manuel Puerta October 5, 2009 So far we focused on profit maximization, we could look at a different problem, that is the cost minimization problem. This is

### IMPLEMENTING ARROW-DEBREU EQUILIBRIA BY TRADING INFINITELY-LIVED SECURITIES

IMPLEMENTING ARROW-DEBREU EQUILIBRIA BY TRADING INFINITELY-LIVED SECURITIES Kevin X.D. Huang and Jan Werner DECEMBER 2002 RWP 02-08 Research Division Federal Reserve Bank of Kansas City Kevin X.D. Huang

### Chapter 15 Introduction to Linear Programming

Chapter 15 Introduction to Linear Programming An Introduction to Optimization Spring, 2014 Wei-Ta Chu 1 Brief History of Linear Programming The goal of linear programming is to determine the values of

### Chapter 4. Duality. 4.1 A Graphical Example

Chapter 4 Duality Given any linear program, there is another related linear program called the dual. In this chapter, we will develop an understanding of the dual linear program. This understanding translates

### Efficiency, Optimality, and Competitive Market Allocations

Efficiency, Optimality, and Competitive Market Allocations areto Efficient Allocations An allocation of resources is areto efficient if it is not possible to reallocate resources in economy to make one

### 88 CHAPTER 2. VECTOR FUNCTIONS. . First, we need to compute T (s). a By definition, r (s) T (s) = 1 a sin s a. sin s a, cos s a

88 CHAPTER. VECTOR FUNCTIONS.4 Curvature.4.1 Definitions and Examples The notion of curvature measures how sharply a curve bends. We would expect the curvature to be 0 for a straight line, to be very small

### Existence of Sunspot Equilibria and Uniqueness of Spot Market Equilibria: The Case of Intrinsically Complete Markets

Existence of Sunspot Equilibria and Uniqueness of Spot Market Equilibria: The Case of Intrinsically Complete Markets Thorsten Hens 1, Janós Mayer 2 and Beate Pilgrim 3 1 IEW, Department of Economics, University

### MICROECONOMICS II. "B"

MICROECONOMICS II. "B" Sponsored by a Grant TÁMOP-4.1.2-08/2/A/KMR-2009-0041 Course Material Developed by Department of Economics, Faculty of Social Sciences, Eötvös Loránd University Budapest (ELTE) Department

### Unraveling 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

### Gains 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

### International Doctoral School Algorithmic Decision Theory: MCDA and MOO

International Doctoral School Algorithmic Decision Theory: MCDA and MOO Lecture 2: Multiobjective Linear Programming Department of Engineering Science, The University of Auckland, New Zealand Laboratoire

### Lecture Notes. Microeconomic Theory. Guoqiang TIAN Department of Economics Texas A&M University College Station, Texas 77843

Lecture Notes Microeconomic Theory Guoqiang TIAN Department of Economics Texas A&M University College Station, Texas 77843 (gtian@tamu.edu) January, 2004 Contents 1 Overview of Economics 1 1.1 Nature and

### Linear 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

### Preference Relations and Choice Rules

Preference Relations and Choice Rules Econ 2100, Fall 2015 Lecture 1, 31 August Outline 1 Logistics 2 Binary Relations 1 Definition 2 Properties 3 Upper and Lower Contour Sets 3 Preferences 4 Choice Correspondences

### Optimal Security Design

Optimal Security Design Franklin Allen University of Pennsylvania Douglas Gale University of Pittsburgh How should new securities be designed? Traditional theories have little to say on this: the literature

### 14.452 Economic Growth: Lecture 11, Technology Diffusion, Trade and World Growth

14.452 Economic Growth: Lecture 11, Technology Diffusion, Trade and World Growth Daron Acemoglu MIT December 2, 2014. Daron Acemoglu (MIT) Economic Growth Lecture 11 December 2, 2014. 1 / 43 Introduction

### HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!

Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following

### Public Goods : (b) Efficient Provision of Public Goods. Efficiency and Private Goods

Public Goods : (b) Efficient Provision of Public Goods Efficiency and Private Goods Suppose that there are only two goods consumed in an economy, and that they are both pure private goods. Suppose as well

### Fundamentals of Operations Research Prof.G. Srinivasan Department of Management Studies Lecture No. # 03 Indian Institute of Technology, Madras

Fundamentals of Operations Research Prof.G. Srinivasan Department of Management Studies Lecture No. # 03 Indian Institute of Technology, Madras Linear Programming solutions - Graphical and Algebraic Methods

### Equilibrium with Complete Markets

Equilibrium with Complete Markets Jesús Fernández-Villaverde University of Pennsylvania February 12, 2016 Jesús Fernández-Villaverde (PENN) Equilibrium with Complete Markets February 12, 2016 1 / 24 Arrow-Debreu

### Deriving Demand Functions - Examples 1

Deriving Demand Functions - Examples 1 What follows are some examples of different preference relations and their respective demand functions. In all the following examples, assume we have two goods x

### ECON 459 Game Theory. Lecture Notes Auctions. Luca Anderlini Spring 2015

ECON 459 Game Theory Lecture Notes Auctions Luca Anderlini Spring 2015 These notes have been used before. If you can still spot any errors or have any suggestions for improvement, please let me know. 1

### No: 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

### CONSUMER PREFERENCES THE THEORY OF THE CONSUMER

CONSUMER PREFERENCES The underlying foundation of demand, therefore, is a model of how consumers behave. The individual consumer has a set of preferences and values whose determination are outside the

### Definition of a Linear Program

Definition of a Linear Program Definition: A function f(x 1, x,..., x n ) of x 1, x,..., x n is a linear function if and only if for some set of constants c 1, c,..., c n, f(x 1, x,..., x n ) = c 1 x 1

### Optimal Nonlinear Income Taxation with a Finite Population

Optimal Nonlinear Income Taxation with a Finite Population Jonathan Hamilton and Steven Slutsky Department of Economics Warrington College of Business Administration University of Florida Gainesville FL

### Problem Set #3 Answer Key

Problem Set #3 Answer Key Economics 305: Macroeconomic Theory Spring 2007 1 Chapter 4, Problem #2 a) To specify an indifference curve, we hold utility constant at ū. Next, rearrange in the form: C = ū

### The Walrasian Model and Walrasian Equilibrium

The Walrasian Model and Walrasian Equilibrium 1.1 There are only two goods in the economy and there is no way to produce either good. There are n individuals, indexed by i = 1,..., n. Individual i owns