Economics 201b Spring 2010 Solutions to Problem Set 1 John Zhu
|
|
- Blanche Waters
- 5 years ago
- Views:
Transcription
1 Economics 201b Spring 2010 Solutions to Problem Set 1 John Zhu 1a The following is a Edgeworth box characterization of the Pareto optimal, and the individually rational Pareto optimal, along with some relevant curves. Key curves of 1 budget frontier for prices p = q,q Figure 1: 1a-d Pareto optimal individually rational Pareto optimal equilibrium allocation x 1 = 2,2 and x 2 = 2,2 b If either agent has a boundary allocation then his utility is 0, in which case, the pareto optimal allocation is to give everything to the other agent. There are two such pareto optimal exact and they correspond to the lower left and upper right corners of the Edgeworth box. Since the common utility function is strictly increasing in both variables in the interior, then any pareto optimal allocation involving interior must also be exact. The marginal rate of substitution of the function U = x 1i x 2i is x 2i x 1i. So for any exact allocation w 1 =, x 21, w 2 = 4, 4 x 21 to be Pareto optimal it must be the case that their marginal rates of substitution MRS at this allocation equal: x 21 = 4 x 21 4 The only solutions are when = x 21. Thus the Pareto optimal are { {x1 = s, s, x 2 = 4 s, 4 s} s [0, 4] } The individual rationality condition requires that s 2 3 = U 1 ω 1 and 4 s 2 3 = U 2 ω 2. So the set of individually rational Pareto optimal are { {x1 = s, s, x 2 = 4 s, 4 s} s [ 3, 4 3] } 1
2 c Define q = p 1 = so that p = q, q. Agent 1 s wealth is 4q and his demand D 1 p is argmax x 21 s.t. + x 21 q 4q,x 21 The constrained maximization problem has a unique solution 2, 2. By symmetry, D 2 p = 2, 2. So {q, q, x 1 = 2, 2, x 2 = 2, 2} is an equilibrium. d Given prices p = p 1,, agent 1 s wealth is p and his demand D 1 p is argmax x 21 s.t. p 1 + x 21 p 1 + 3,x 21 [ ] p1 argmax Taking the derivative and solving, we get p1 + 3 D 1 p =, p and similarly The excess demand function is Ep = D 2 p = 3p1 +, 3p 1 + 4p 4, 4p 4 Setting this equal to 0, we get p 1 =, in which case we have the equilibrium of part c. Thus there are no other equilibria. e Given prices p = p 1,, first assume p 1, 0. Agent 1 s wealth is p and his demand D 1 p is argmax min{, x 21 } s.t. p 1 + x 21 p 1 + 3,x 21 Since more of one good than the other provides no extra utility but does cost something, a necessary condition for utility maximizing behavior is that = x 21, so the constrained maximization problem can be simplified to so and similarly argmax s.t. p D 1 p = D 2 p = p1 + 3, p p1 +, 3p 1 + 2
3 Key equilibrium curves of 1 budget frontiers Figure 2: 1e and the excess demand function is 4p Ep = 4, 4p 4 = 0 So we ve found some equilibria. Now assume the price is p 1, 0. Then agent 1 s budget frontier is the vertical line through ω 1 and therefore his demand correspondence is D 1 p = { 1, x 21 x 21 1 } and D 2 p = { 3, x 22 x 22 3 } Since the social is 4, 4, one can see the only way for the excess demand to be 0 is if x 21 = 1 and x 22 = 3. We can do a similar analysis for the case when prices are 0,. Collecting all the results together, we can express the set of equilibria as follows { {p1,, x 1 = p 1 + 3, p 1 + 3, x2 = 3, 3 } p = p1, R 2 + } f Most of the calculations translate easily over from d. First assume p 1, 0. Then p1 + 3 D 1 p =, p p1 + D 2 p =, 4p 1 + and the excess demand function is 5p Ep = 5, 5p 4 = 3 p2, p 1 0
4 curves of 1 These two curves don' t intersect, demonstrating that the price vector p = p 1, where p 1 > 0 does not support an equilibrium. Figure 3: 1f A p = p 1, budget frontier where p 1 > 0 equilibrium The p = 0, budget frontier So any equilibrium must have one of the prices as zero. So first suppose that the scarcer good is free i.e. = 0. Then D 1 p = { 1, x 21 x 21 1 } and D 2 p = { 4, x 22 x 22 4 } Notice there is no way x 21 + x 22 4, so markets can t clear for good 2 and this price supports no equilibrium. The last type of price is when the more common good is free i.e. p 1 = 0. In which case D 1 p = {, 3 3 } and D 2 p = { x 12, 1 x 12 1 } It is certainly possible for + x 12 5 and so the set of equilibria is { {0, p2, x 1 = 3 + r, 3, x 2 = 2 r, 1 } r [0, 1]} Thus, up to normalization there is now a unique equilibrium price p = 0, 1. 2a Using an argument similar to that found in exercise 1, we can show all Pareto optimal are exact. Agent 2 achieves her maximal utility, 4, with her ω 2. So a necessary condition for an exact allocation x 1 =, x 21, x 2 = 5, 5 x 21 to be Pareto optimal 4
5 and individually rational is that U 2 x 2 = 4. This is equivalent to 5 5 x So such an allocation solves: argmax,x 21 x 21 s.t. 5 5 x 21 4 argmax There is a unique solution = 3, making the exact allocation x = { x 1 = 3, 3, x 2 = 2, 2 } the unique allocation that is Pareto optimal and individually rational for both agents. These two don' t match. budget frontiers curves of 1 boundary of the region of 2 Pareto optimal allocation equilibrium The range of budget frontiers that correspond to equilibrium prices. Figure 4: 2a, b b Given prices p = p 1,, doing the same work as in exercise 1d we know that p D 1 p =, p In equilibrium, after agent 1 has taken what he can afford, the rest of the social must belong to the demand correspondence of agent 2: 9p1 4 5, 5 D 1 p =, 6p 2 p 1 D 2 p = { x 12, x 22 x 12 x 22 4 } 5
6 Let r = p 1 then since 9p 1 4 2p 1 6 p 1 2 = so the problem reduces to finding all r such that 9r 46 r 4r 9r 46 r 16r 9r 46 r 16r 0 3r r 8 0 3r 2 r r [ 2 3, 4] So the set of equilibria is { {p1,, x 1 = p, p, x 2 = 9p 1 4, 6 p 1 } r = p 1 [ 2 } 2p 1 2p 1 3, 4] There are a number of ways to show that none of the of this set is the Pareto optimal allocation x. Perhaps the most geometric way to show this is to note that all the budget frontiers of the equilibrium prices are steeper than the budget frontier that goes through x. Thus from agent 1 s perspective, x 1 is always out of reach too expensive given the equilibrium prices. Indeed the latter budget frontier has slope 1, whereas the budget frontiers corresponding to 3 the equilibrium prices range from 2 down to 4 remember given prices p 3 1, the budget frontier s slope is p 1. See picture. 3a In order to find the offer curves, we need to find each agent s demand. So fix a price p = p 1,. OC 2 budget frontiers Figure 5: 3a, b and also OC 1 The offer curve for agent 1 is just a single - his. This follows from a combination of two facts: one, he only cares about the first good, and two, all he has is the first 6
7 good. Thus no matter the prices, he can t afford to get more of the first good - since it s all he has to begin with! Agent 2 s wealth is 2 and her demand D 2 p is argmax x 22 + log x 12 s.t. x 12,x 22 p 1 x 12 + x 22 = 2 2 p 1 x 12 argmax + log x 12 x 12 x 12 = p 1 x 22 = 1 So D 2 p = p 1, 1 and OC 2 is the horizontal line with height 1 in agent 2 s consumption set R b Recall that the two offer curves must intersect in the Edgeworth box for there to be an equilibrium. Since OC 1 OC 2 = Ø there is no equilibrium. 4a Checking for Pareto optimality is straightforward - since the allocation is in the interior, and utilities are smooth, it suffices to check that the MRS equal. In the first exercise we calculated that the MRS at the allocation x 1i, x 2i is x 2i x 1i. Thus the MRS of each agent under the allocation is 1 and so we have Pareto optimality. b The agents 1 and 2 can get together and split their goods down the middle so that each agent receives the allocation 3.5, 3.5 which is strictly preferred to x 1 and x 2. In other words, these two agents would be mutually strictly better off splitting from the proposed allocation, and trading amongst themselves, leaving out agent 3. 7
Introduction to General Equilibrium
Introduction to General Equilibrium Juan Manuel Puerta November 6, 2009 Introduction So far we discussed markets in isolation. We studied the quantities and welfare that results under different assumptions
More informationThe Fundamental Welfare Theorems
The Fundamental Welfare Theorems The so-called Fundamental Welfare Theorems of Economics tell us about the relation between market equilibrium and Pareto efficiency. The First Welfare Theorem: Every Walrasian
More informationMicroeconomic Theory -1- Introduction
Microeconomic Theory -- Introduction. Introduction. Profit maximizing firm with monopoly power 6 3. General results on maximizing with two variables 8 4. Model of a private ownership economy 5. Consumer
More informationFirst Welfare Theorem
First Welfare Theorem Econ 2100 Fall 2017 Lecture 17, October 31 Outline 1 First Welfare Theorem 2 Preliminaries to Second Welfare Theorem Past Definitions A feasible allocation (ˆx, ŷ) is Pareto optimal
More informationMarket Equilibrium and the Core
Market Equilibrium and the Core Ram Singh Lecture 3-4 September 22/25, 2017 Ram Singh (DSE) Market Equilibrium September 22/25, 2017 1 / 19 Market Exchange: Basics Let us introduce price in our pure exchange
More informationMicroeconomics II. MOSEC, LUISS Guido Carli Problem Set n 3
Microeconomics II MOSEC, LUISS Guido Carli Problem Set n 3 Problem 1 Consider an economy 1 1, with one firm (or technology and one consumer (firm owner, as in the textbook (MWG section 15.C. The set of
More informationThe 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
More informationSecond Welfare Theorem
Second Welfare Theorem Econ 2100 Fall 2015 Lecture 18, November 2 Outline 1 Second Welfare Theorem From Last Class We want to state a prove a theorem that says that any Pareto optimal allocation is (part
More informationANSWERS TO ODD NUMBERED EXERCISES IN CHAPTER 3
John Riley 5 Setember 0 NSWERS T DD NUMERED EXERCISES IN CHPTER 3 SECTIN 3: Equilibrium and Efficiency Exercise 3-: Prices with Quasi-linear references (a) Since references are convex, an allocation is
More informationThe Debreu-Scarf Theorem: The Core Converges to the Walrasian Allocations
The Debreu-Scarf Theorem: The Core Converges to the Walrasian Allocations We ve shown that any Walrasian equilibrium allocation (any WEA) is in the core, but it s obvious that the converse is far from
More informationThe Ohio State University Department of Economics. Homework Set Questions and Answers
The Ohio State University Department of Economics Econ. 805 Winter 00 Prof. James Peck Homework Set Questions and Answers. Consider the following pure exchange economy with two consumers and two goods.
More informationIn the Name of God. Sharif University of Technology. Microeconomics 1. Graduate School of Management and Economics. Dr. S.
In the Name of God Sharif University of Technology Graduate School of Management and Economics Microeconomics 1 44715 (1396-97 1 st term) - Group 1 Dr. S. Farshad Fatemi Chapter 10: Competitive Markets
More informationAdvanced Microeconomics
Advanced Microeconomics Partial and General Equilibrium Giorgio Fagiolo giorgio.fagiolo@sssup.it http://www.lem.sssup.it/fagiolo/welcome.html LEM, Sant Anna School of Advanced Studies, Pisa (Italy) Part
More informationIntroduction to General Equilibrium: Framework.
Introduction to General Equilibrium: Framework. Economy: I consumers, i = 1,...I. J firms, j = 1,...J. L goods, l = 1,...L Initial Endowment of good l in the economy: ω l 0, l = 1,...L. Consumer i : preferences
More informationEconomics 501B Final Exam Fall 2017 Solutions
Economics 501B Final Exam Fall 2017 Solutions 1. For each of the following propositions, state whether the proposition is true or false. If true, provide a proof (or at least indicate how a proof could
More informationLecture Notes October 18, Reading assignment for this lecture: Syllabus, section I.
Lecture Notes October 18, 2012 Reading assignment for this lecture: Syllabus, section I. Economic General Equilibrium Partial and General Economic Equilibrium PARTIAL EQUILIBRIUM S k (p o ) = D k k (po
More informationDifferentiable Welfare Theorems Existence of a Competitive Equilibrium: Preliminaries
Differentiable Welfare Theorems Existence of a Competitive Equilibrium: Preliminaries Econ 2100 Fall 2017 Lecture 19, November 7 Outline 1 Welfare Theorems in the differentiable case. 2 Aggregate excess
More information2.9. V = u(x,y) + a(x-f(l x,t x )) + b(y-g(l y,t y )) + c(l o -L x -L y ) + d(t o -T x -T y ) (1) (a) Suggested Answer: V u a 0 (2) u x = -a b 0 (3)
2.9 V = u(,) + a(-f(, )) + b(-g(, )) + c( o - - ) + d( o - - ) (1) (a) Suggested Answer: V u a 0 (2) u = -a V u b 0 (3) u = -b V f a d 0 (4) V V V b g d 0 (5) a f c 0 (6) b g c 0 (7) g c/ b c c/a f (b)
More informationAdvanced Microeconomics Problem Set 1
dvanced Microeconomics Problem Set László Sándor Central European University Pareto optima With Cobb-Douglas utilities u x ; x 2 ; x 3 = 0:4 log x 2 + 0:6 log x 3 and u x ; x 2 ; x 3 = log x 2 + log x
More informationThe Fundamental Welfare Theorems
The Fundamental Welfare Theorems The so-called Fundamental Welfare Theorems of Economics tell us about the relation between market equilibrium and Pareto efficiency. The First Welfare Theorem: Every Walrasian
More informationProblem Set 1 Welfare Economics
Problem Set 1 Welfare Economics Solutions 1. Consider a pure exchange economy with two goods, h = 1,, and two consumers, i =1,, with utility functions u 1 and u respectively, and total endowment, e = (e
More informationRecitation #2 (August 31st, 2018)
Recitation #2 (August 1st, 2018) 1. [Checking properties of the Cobb-Douglas utility function.] Consider the utility function u(x) = n i=1 xα i i, where x denotes a vector of n different goods x R n +,
More informationx 1 1 and p 1 1 Two points if you just talk about monotonicity (u (c) > 0).
. (a) (8 points) What does it mean for observations x and p... x T and p T to be rationalized by a monotone utility function? Notice that this is a one good economy. For all t, p t x t function. p t x
More informationDepartment of Economics The Ohio State University Final Exam Questions and Answers Econ 8712
Prof. Peck Fall 20 Department of Economics The Ohio State University Final Exam Questions and Answers Econ 872. (0 points) The following economy has two consumers, two firms, and three goods. Good is leisure/labor.
More informationAdding Production to the Theory
Adding Production to the Theory We begin by considering the simplest situation that includes production: two goods, both of which have consumption value, but one of which can be transformed into the other.
More informationCompetitive Equilibrium
Competitive Equilibrium Econ 2100 Fall 2017 Lecture 16, October 26 Outline 1 Pareto Effi ciency 2 The Core 3 Planner s Problem(s) 4 Competitive (Walrasian) Equilibrium Decentralized vs. Centralized Economic
More informationNotes on General Equilibrium
Notes on General Equilibrium Alejandro Saporiti Alejandro Saporiti (Copyright) General Equilibrium 1 / 42 General equilibrium Reference: Jehle and Reny, Advanced Microeconomic Theory, 3rd ed., Pearson
More information(ii) An input requirement set for this technology is clearly not convex as it
LONDON SCHOOL OF ECONOMICS Department of Economics Leonardo Felli 32L.4.02; 7525 Solutions to Assignment 5 EC487 Advanced Microeconomics Part I 1. Sketch of the answers: (i) The map of isoquants for this
More informationPh.D. Preliminary Examination MICROECONOMIC THEORY Applied Economics Graduate Program June 2016
Ph.D. Preliminary Examination MICROECONOMIC THEORY Applied Economics Graduate Program June 2016 The time limit for this exam is four hours. The exam has four sections. Each section includes two questions.
More informationEquilibrium and Pareto Efficiency in an exchange economy
Microeconomic Teory -1- Equilibrium and efficiency Equilibrium and Pareto Efficiency in an excange economy 1. Efficient economies 2 2. Gains from excange 6 3. Edgewort-ox analysis 15 4. Properties of a
More informationThe B.E. Journal of Theoretical Economics
The B.E. Journal of Theoretical Economics Topics Volume 9, Issue 1 2009 Article 43 Simple Economies with Multiple Equilibria Theodore C. Bergstrom Ken-Ichi Shimomura Takehiko Yamato University of California,
More informationQuestion 1. (p p) (x(p, w ) x(p, w)) 0. with strict inequality if x(p, w) x(p, w ).
University of California, Davis Date: August 24, 2017 Department of Economics Time: 5 hours Microeconomics Reading Time: 20 minutes PRELIMINARY EXAMINATION FOR THE Ph.D. DEGREE Please answer any three
More informationDepartment of Economics The Ohio State University Midterm Answers Econ 805
Department of Economics The Ohio State University Midterm Answers Econ 805 Prof. James Peck Winter 0. (0 points) Consider the following pure-exchange economy with two consumers and two goods. Consumer
More informationDepartment of Agricultural Economics. PhD Qualifier Examination. May 2009
Department of Agricultural Economics PhD Qualifier Examination May 009 Instructions: The exam consists of six questions. You must answer all questions. If you need an assumption to complete a question,
More informationECO 317 Economics of Uncertainty Fall Term 2009 Slides to accompany 13. Markets and Efficient Risk-Bearing: Examples and Extensions
ECO 317 Economics of Uncertainty Fall Term 2009 Slides to accompany 13. Markets and Efficient Risk-Bearing: Examples and Extensions 1. Allocation of Risk in Mean-Variance Framework S states of the world,
More informationLecture #3. General equilibrium
Lecture #3 General equilibrium Partial equilibrium equality of demand and supply in a single market (assumption: actions in one market do not influence, or have negligible influence on other markets) General
More informationPublic Goods and Private Goods
Chapter 2 Public Goods and Private Goods One Public Good, One Private Good Claude and Dorothy are roommates, also. 1 They are not interested in card games or the temperature of their room. Each of them
More information1 Second Welfare Theorem
Econ 701B Fall 018 University of Pennsylvania Recitation : Second Welfare Theorem Xincheng Qiu (qiux@sas.upenn.edu) 1 Second Welfare Theorem Theorem 1. (Second Welfare Theorem) An economy E satisfies (A1)-(A4).
More information1 General Equilibrium
1 General Equilibrium 1.1 Pure Exchange Economy goods, consumers agent : preferences < or utility : R + R initial endowments, R + consumption bundle, =( 1 ) R + Definition 1 An allocation, =( 1 ) is feasible
More informationFirms and returns to scale -1- John Riley
Firms and returns to scale -1- John Riley Firms and returns to scale. Increasing returns to scale and monopoly pricing 2. Natural monopoly 1 C. Constant returns to scale 21 D. The CRS economy 26 E. pplication
More informationEcon Slides from Lecture 10
Econ 205 Sobel Econ 205 - Slides from Lecture 10 Joel Sobel September 2, 2010 Example Find the tangent plane to {x x 1 x 2 x 2 3 = 6} R3 at x = (2, 5, 2). If you let f (x) = x 1 x 2 x3 2, then this is
More informationEconomics 101 Lecture 5 - Firms and Production
Economics 101 Lecture 5 - Firms and Production 1 The Second Welfare Theorem Last week we proved the First Basic Welfare Theorem, which states that under fairly weak assumptions, a Walrasian equilibrium
More informationLecture Notes for January 8, 2009; part 2
Economics 200B Prof. R. Starr UCSD Winter 2009 Lecture Notes for January 8, 2009; part 2 Integrating Production and Multiple Consumption Decisions: A2 2 2 Model Competitive Equilibrium: Production and
More informationSome Notes on Adverse Selection
Some Notes on Adverse Selection John Morgan Haas School of Business and Department of Economics University of California, Berkeley Overview This set of lecture notes covers a general model of adverse selection
More informationEC487 Advanced Microeconomics, Part I: Lecture 5
EC487 Advanced Microeconomics, Part I: Lecture 5 Leonardo Felli 32L.LG.04 27 October, 207 Pareto Efficient Allocation Recall the following result: Result An allocation x is Pareto-efficient if and only
More informationRecitation 2-09/01/2017 (Solution)
Recitation 2-09/01/2017 (Solution) 1. Checking properties of the Cobb-Douglas utility function. Consider the utility function u(x) Y n i1 x i i ; where x denotes a vector of n di erent goods x 2 R n +,
More informationMathematical Foundations -1- Constrained Optimization. Constrained Optimization. An intuitive approach 2. First Order Conditions (FOC) 7
Mathematical Foundations -- Constrained Optimization Constrained Optimization An intuitive approach First Order Conditions (FOC) 7 Constraint qualifications 9 Formal statement of the FOC for a maximum
More informationSometimes the domains X and Z will be the same, so this might be written:
II. MULTIVARIATE CALCULUS The first lecture covered functions where a single input goes in, and a single output comes out. Most economic applications aren t so simple. In most cases, a number of variables
More informationFirms and returns to scale -1- Firms and returns to scale
Firms and returns to scale -1- Firms and returns to scale. Increasing returns to scale and monopoly pricing 2. Constant returns to scale 19 C. The CRS economy 25 D. pplication to trade 47 E. Decreasing
More informationNotes IV General Equilibrium and Welfare Properties
Notes IV General Equilibrium and Welfare Properties In this lecture we consider a general model of a private ownership economy, i.e., a market economy in which a consumer s wealth is derived from endowments
More informationGeneral Equilibrium. General Equilibrium, Berardino. Cesi, MSc Tor Vergata
General Equilibrium Equilibrium in Consumption GE begins (1/3) 2-Individual/ 2-good Exchange economy (No production, no transaction costs, full information..) Endowment (Nature): e Private property/ NO
More informationExistence, Computation, and Applications of Equilibrium
Existence, Computation, and Applications of Equilibrium 2.1 Art and Bart each sell ice cream cones from carts on the boardwalk in Atlantic City. Each day they independently decide where to position their
More informationBoundary Behavior of Excess Demand Functions without the Strong Monotonicity Assumption
Boundary Behavior of Excess Demand Functions without the Strong Monotonicity Assumption Chiaki Hara April 5, 2004 Abstract We give a theorem on the existence of an equilibrium price vector for an excess
More informationPareto Efficiency (also called Pareto Optimality)
Pareto Efficiency (also called Pareto Optimality) 1 Definitions and notation Recall some of our definitions and notation for preference orderings. Let X be a set (the set of alternatives); we have the
More informationGeneral Equilibrium and Welfare
and Welfare Lectures 2 and 3, ECON 4240 Spring 2017 University of Oslo 24.01.2017 and 31.01.2017 1/37 Outline General equilibrium: look at many markets at the same time. Here all prices determined in the
More informationFundamental Theorems of Welfare Economics
Fundamental Theorems of Welfare Economics Ram Singh Lecture 6 September 29, 2015 Ram Singh: (DSE) General Equilibrium Analysis September 29, 2015 1 / 14 First Fundamental Theorem The First Fundamental
More informationLecture 3 - Axioms of Consumer Preference and the Theory of Choice
Lecture 3 - Axioms of Consumer Preference and the Theory of Choice David Autor 14.03 Fall 2004 Agenda: 1. Consumer preference theory (a) Notion of utility function (b) Axioms of consumer preference (c)
More informationRegularity of competitive equilibria in a production economy with externalities
Regularity of competitive equilibria in a production economy with externalities Elena del Mercato Vincenzo Platino Paris School of Economics - Université Paris 1 Panthéon Sorbonne QED-Jamboree Copenhagen,
More informationThe Existence of Equilibrium in. Infinite-Dimensional Spaces: Some Examples. John H. Boyd III
The Existence of Equilibrium in Infinite-Dimensional Spaces: Some Examples John H. Boyd III April 1989; Updated June 1995 Abstract. This paper presents some examples that clarify certain topological and
More informationLecture 1. History of general equilibrium theory
Lecture 1 History of general equilibrium theory Adam Smith: The Wealth of Nations, 1776 many heterogeneous individuals with diverging interests many voluntary but uncoordinated actions (trades) results
More informationEconS Microeconomic Theory II Homework #9 - Answer key
EconS 503 - Microeconomic Theory II Homework #9 - Answer key 1. WEAs with market power. Consider an exchange economy with two consumers, A and B, whose utility functions are u A (x A 1 ; x A 2 ) = x A
More informationDiscussion Papers in Economics
Discussion Papers in Economics No. 10/11 A General Equilibrium Corporate Finance Theorem for Incomplete Markets: A Special Case By Pascal Stiefenhofer, University of York Department of Economics and Related
More informationSimon Fraser University, Department of Economics, Econ 201, Prof. Karaivanov FINAL EXAM Answer key
Simon Fraser University, Department of Economics, Econ 01, Prof. Karaivanov 017 FINAL EXAM Answer key I. TRUE or FALSE (5 pts each). [The answers below are just examples of correct answers, other possible
More informationAdvanced Microeconomic Analysis, Lecture 6
Advanced Microeconomic Analysis, Lecture 6 Prof. Ronaldo CARPIO April 10, 017 Administrative Stuff Homework # is due at the end of class. I will post the solutions on the website later today. The midterm
More information3. THE EXCHANGE ECONOMY
Essential Microeconomics -1-3. THE EXCHNGE ECONOMY Pareto efficient allocations 2 Edgewort box analysis 5 Market clearing prices 13 Walrasian Equilibrium 16 Equilibrium and Efficiency 22 First welfare
More informationAdvanced Microeconomic Analysis Solutions to Midterm Exam
Advanced Microeconomic Analsis Solutions to Midterm Exam Q1. (0 pts) An individual consumes two goods x 1 x and his utilit function is: u(x 1 x ) = [min(x 1 + x x 1 + x )] (a) Draw some indifference curves
More informationFinal Examination with Answers: Economics 210A
Final Examination with Answers: Economics 210A December, 2016, Ted Bergstrom, UCSB I asked students to try to answer any 7 of the 8 questions. I intended the exam to have some relatively easy parts and
More informationECONOMICS 001 Microeconomic Theory Summer Mid-semester Exam 2. There are two questions. Answer both. Marks are given in parentheses.
Microeconomic Theory Summer 206-7 Mid-semester Exam 2 There are two questions. Answer both. Marks are given in parentheses.. Consider the following 2 2 economy. The utility functions are: u (.) = x x 2
More informationPreferences and Utility
Preferences and Utility How can we formally describe an individual s preference for different amounts of a good? How can we represent his preference for a particular list of goods (a bundle) over another?
More information0 Aims of this course
THE UNIVERSITY OF SOUTHAMPTON Paul Klein Office: Murray Building, 35 Email: p.klein@soton.ac.uk URL: http://paulklein.se Economics 31 Topics in Macroeconomics 3 Fall 21 Aims of this course Clear, critical
More informationEconomics 201B Second Half. Lecture 12-4/22/10. Core is the most commonly used. The core is the set of all allocations such that no coalition (set of
Economics 201B Second Half Lecture 12-4/22/10 Justifying (or Undermining) the Price-Taking Assumption Many formulations: Core, Ostroy s No Surplus Condition, Bargaining Set, Shapley-Shubik Market Games
More informationConsider this problem. A person s utility function depends on consumption and leisure. Of his market time (the complement to leisure), h t
VI. INEQUALITY CONSTRAINED OPTIMIZATION Application of the Kuhn-Tucker conditions to inequality constrained optimization problems is another very, very important skill to your career as an economist. If
More informationNotes on Recursive Utility. Consider the setting of consumption in infinite time under uncertainty as in
Notes on Recursive Utility Consider the setting of consumption in infinite time under uncertainty as in Section 1 (or Chapter 29, LeRoy & Werner, 2nd Ed.) Let u st be the continuation utility at s t. That
More informationThe Consumer, the Firm, and an Economy
Andrew McLennan October 28, 2014 Economics 7250 Advanced Mathematical Techniques for Economics Second Semester 2014 Lecture 15 The Consumer, the Firm, and an Economy I. Introduction A. The material discussed
More informationEconS 501 Final Exam - December 10th, 2018
EconS 501 Final Exam - December 10th, 018 Show all your work clearly and make sure you justify all your answers. NAME 1. Consider the market for smart pencil in which only one firm (Superapiz) enjoys a
More informationAxiomatic bargaining. theory
Axiomatic bargaining theory Objective: To formulate and analyse reasonable criteria for dividing the gains or losses from a cooperative endeavour among several agents. We begin with a non-empty set of
More informationEconomics 101. Lecture 2 - The Walrasian Model and Consumer Choice
Economics 101 Lecture 2 - The Walrasian Model and Consumer Choice 1 Uncle Léon The canonical model of exchange in economics is sometimes referred to as the Walrasian Model, after the early economist Léon
More informationDuality. for The New Palgrave Dictionary of Economics, 2nd ed. Lawrence E. Blume
Duality for The New Palgrave Dictionary of Economics, 2nd ed. Lawrence E. Blume Headwords: CONVEXITY, DUALITY, LAGRANGE MULTIPLIERS, PARETO EFFICIENCY, QUASI-CONCAVITY 1 Introduction The word duality is
More informationPublic Economics Ben Heijdra Chapter 9: Introduction to Normative Public Economics
Public Economics: Chapter 9 1 Public Economics Ben Heijdra Chapter 9: Introduction to Normative Public Economics Objectives of this chapter Public Economics: Chapter 9 2 Read Atkinson & Stiglitz (1980,
More informationwhere u is the decision-maker s payoff function over her actions and S is the set of her feasible actions.
Seminars on Mathematics for Economics and Finance Topic 3: Optimization - interior optima 1 Session: 11-12 Aug 2015 (Thu/Fri) 10:00am 1:00pm I. Optimization: introduction Decision-makers (e.g. consumers,
More informationMechanism Design: Basic Concepts
Advanced Microeconomic Theory: Economics 521b Spring 2011 Juuso Välimäki Mechanism Design: Basic Concepts The setup is similar to that of a Bayesian game. The ingredients are: 1. Set of players, i {1,
More informationGraduate Macroeconomics 2 Problem set Solutions
Graduate Macroeconomics 2 Problem set 10. - Solutions Question 1 1. AUTARKY Autarky implies that the agents do not have access to credit or insurance markets. This implies that you cannot trade across
More informationEconomics 200A part 2 UCSD Fall quarter 2011 Prof. R. Starr Mr. Troy Kravitz1 FINAL EXAMINATION SUGGESTED ANSWERS
Economics 200A part 2 UCSD Fall quarter 2011 Prof. R. Starr Mr. Troy Kravitz1 FINAL EXAMINATION SUGGESTED ANSWERS This exam is take-home, open-book, open-notes. You may consult any published source (cite
More information1 Two elementary results on aggregation of technologies and preferences
1 Two elementary results on aggregation of technologies and preferences In what follows we ll discuss aggregation. What do we mean with this term? We say that an economy admits aggregation if the behavior
More informationPh.D. Preliminary Examination MICROECONOMIC THEORY Applied Economics Graduate Program May 2012
Ph.D. Preliminary Examination MICROECONOMIC THEORY Applied Economics Graduate Program May 2012 The time limit for this exam is 4 hours. It has four sections. Each section includes two questions. You are
More informationMechanism Design: Bayesian Incentive Compatibility
May 30, 2013 Setup X : finite set of public alternatives X = {x 1,..., x K } Θ i : the set of possible types for player i, F i is the marginal distribution of θ i. We assume types are independently distributed.
More informationHorizontal and Vertical Asymptotes from section 2.6
Horizontal and Vertical Asymptotes from section 2.6 Definition: In either of the cases f(x) = L or f(x) = L we say that the x x horizontal line y = L is a horizontal asymptote of the function f. Note:
More informationPhD Qualifier Examination
PhD Qualifier Examination Department of Agricultural Economics July 26, 2013 Instructions The exam consists of six questions. You must answer all questions. If you need an assumption to complete a question,
More informationLecture 1: Labour Economics and Wage-Setting Theory
ecture 1: abour Economics and Wage-Setting Theory Spring 2015 ars Calmfors iterature: Chapter 1 Cahuc-Zylberberg (pp 4-19, 28-29, 35-55) 1 The choice between consumption and leisure U = U(C,) C = consumption
More informationEconomics 121b: Intermediate Microeconomics Midterm Suggested Solutions 2/8/ (a) The equation of the indifference curve is given by,
Dirk Bergemann Department of Economics Yale University Economics 121b: Intermediate Microeconomics Midterm Suggested Solutions 2/8/12 1. (a) The equation of the indifference curve is given by, (x 1 + 2)
More informationG5212: Game Theory. Mark Dean. Spring 2017
G5212: Game Theory Mark Dean Spring 2017 Adverse Selection We have now completed our basic analysis of the adverse selection model This model has been applied and extended in literally thousands of ways
More informationWalrasian Equilibrium in an exchange economy
Microeconomic Teory -1- Walrasian equilibrium Walrasian Equilibrium in an ecange economy 1. Homotetic preferences 2 2. Walrasian equilibrium in an ecange economy 11 3. Te market value of attributes 18
More informationGARP and Afriat s Theorem Production
GARP and Afriat s Theorem Production Econ 2100 Fall 2017 Lecture 8, September 21 Outline 1 Generalized Axiom of Revealed Preferences 2 Afriat s Theorem 3 Production Sets and Production Functions 4 Profits
More informationCan everyone benefit from innovation?
Can everyone benefit from innovation? Christopher P. Chambers and Takashi Hayashi June 16, 2017 Abstract We study a resource allocation problem with variable technologies, and ask if there is an allocation
More information1.2 Graphs and Lines. Cartesian Coordinate System
1.2 Graphs and Lines Cartesian Coordinate System Note that there is a one-to-one correspondence between the points in a plane and the elements in the set of all ordered pairs (a, b) of real numbers. Graphs
More informationCONSUMER DEMAND. Consumer Demand
CONSUMER DEMAND KENNETH R. DRIESSEL Consumer Demand The most basic unit in microeconomics is the consumer. In this section we discuss the consumer optimization problem: The consumer has limited wealth
More informationMarket Failure: Externalities
Market Failure: Externalities Ram Singh Lecture 21 November 10, 2015 Ram Singh: (DSE) Externality November 10, 2015 1 / 18 Questions What is externality? What is implication of externality for efficiency
More information1 Numbers, Sets, Algebraic Expressions
AAU - Business Mathematics I Lecture #1, February 27, 2010 1 Numbers, Sets, Algebraic Expressions 1.1 Constants, Variables, and Sets A constant is something that does not change, over time or otherwise:
More informationCS 598RM: Algorithmic Game Theory, Spring Practice Exam Solutions
CS 598RM: Algorithmic Game Theory, Spring 2017 1. Answer the following. Practice Exam Solutions Agents 1 and 2 are bargaining over how to split a dollar. Each agent simultaneously demands share he would
More informationHicksian Demand and Expenditure Function Duality, Slutsky Equation
Hicksian Demand and Expenditure Function Duality, Slutsky Equation Econ 2100 Fall 2017 Lecture 6, September 14 Outline 1 Applications of Envelope Theorem 2 Hicksian Demand 3 Duality 4 Connections between
More information