The Ohio State University Department of Economics. Homework Set Questions and Answers
|
|
- Emerald McDaniel
- 6 years ago
- Views:
Transcription
1 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. Consumer 's utility function and endowments are given by: u (x, x ) = a log(x ) + log(x ) ω = (,). Consumer 's utility function and endowments are given by: u (x, x ) = log(x ) + a log(x ) ω = (,). The parameters, a and a, are both positive. (a) Compute the aggregate excess demand function for this economy, and verify by direct computation that Walras Law holds. (b) Define a competitive equilibrium for this economy. (c) Calculate the competitive equilibrium price and allocation. (d) For what values of a and a will consumer be a net purchaser of good at the competitive equilibrium? ANSWER: (a) Solving consumer 's utility maximization problem, we simultaneously solve the marginal rate of substitution condition, a x / x = p, and the budget equation holding with equality, p x +x = p +, where prices are normalized so that the price of good is p and the price of good is. We get x = a (p+)/[p(a +)] and x = (p+)/(a +). Solving consumer 's utility maximization problem, we get the first order conditions: x / (a x ) = p and p x + x = p +. Solving for consumer 's demand functions, we get: x = (p+)/[p(+a )] and x = a (p+)/(+a ). Therefore, the excess demand function is: z(p) = ( a (p+)/[p(a +)] + (p+)/[p(+a )] -, (p+)/(a +) + a (p+)/(+a ) - ). To verify that Walras Law holds, we must calculate pz (p) + z (p), which equals a (p+)/(a +) + (p+)/(+a ) - p+ (p+)/(a +) + a (p+)/(+a ) - = (p+) + (p+) - p - = 0. It checks. (b) A competitive equilibrium is a (normalized) price vector (p,) and an allocation (x, x, x, x ) such that:
2 (i) (x, x ) solves max a log(x ) + log(x ) s.t. p x +x! p + x " 0 and (x, x ) solves max log(x ) + a log(x ) s.t. p x + x! p + x " 0. (ii) x + x! x + x!. Note: because both utility functions are strictly monotonic, budget and market clearing inequalities must hold with equality. (c) To calculate the competitive equilibrium price and allocation, we look for a price where z(p) = 0. Looking at market, we must solve: (p+)/(a +) + a (p+)/(+a ) - = 0. This can be rewritten as: (+a )(p+) + (a +)a (p+) = (a +)(+a ). Therefore, p( + a + a + a a ) + (+a + a + a a ) = (a +)(+a ). p( + a + a a ) = + a + a a, which implies the equilibrium price, p* is p* = ( + a + a a ) / ( + a + a a ). Simplifying the equilibrium allocation is tedious, but you can simply express the answer as: x = a (p*+)/[p*(a +)], x = (p*+)/(a +), x = (p*+)/[p*(+a )], and x = a (p*+)/(+a ), where p* = ( + a + a a ) / ( + a + a a ). (d) Consumer will be a net purchaser of good when x >. a (p*+)/[p*(a +)] > can be rewritten as a (p*+) > p*(a +), or a p* + a > a p* + p*. Thus, the condition is: p* < a /. Plugging in the value of p*, we have ( + a + a a ) / ( + a + a a ) < a /.
3 Cross-multiplying: a + a a < a + a a + (a ) a, or (a ) a + a a - 4 a - 4 > 0. Factoring the above expression, we have: (a +)[a a - 4] > 0. Since a is positive, the condition for consumer to be a net purchaser of good is: a a > 4.. Construct an example of an economy (a specification of the consumers, the utility functions, and the endowments) that does not have a competitive equilibrium. Carefully and clearly explain your answer. Which assumptions from our existence theorem are violated in your example? ANSWER: The easiest answers involve dropping the assumption of strict quasiconcavity, and drawing a carefully labeled Edgeworth box. Alternatively, you could specify functional forms. For example, suppose there is just one consumer and goods. u(x,x ) = (x ) + (x ) and ω = (,). At any price vector, this consumer will spend all of his/her income on one commodity or the other, whichever is relatively cheaper. Normalizing p = p / p, we have excess demand for good when p <, excess demand for good when p >, and excess demand for one of the goods (whichever the consumer chooses) when p =.. Assume that we have a pure exchange economy with n consumers and k goods, in which endowments are strictly positive for each consumer and each commodity. Assume also that all utility functions satisfy strict quasiconcavity, strict monotonicity, and continuity. (A) (B) Show that if the economy has more than one competitive equilibrium allocation, then the initial endowments cannot be Pareto optimal. Dropping the assumption of strict quasiconcavity while maintaining the other assumptions, give an example of an economy that has more than one competitive equilibrium allocation, but where the initial endowment is Pareto optimal. A carefully drawn and labeled Edgeworth Box is good enough. ANSWER: (A) Suppose not. Then the initial endowments are PO and there is at least one other distinct PO allocation, x* (the other CE allocation FFTWE). In the utility maximization problems determined by (p*,x*), each consumer can afford his/her * endowment. Therefore, ui( xi ) ui( ω i) for all i. But since endowments are PO, the inequality cannot be strict for any consumer, or else x* would Pareto ** * dominate ω. But now the allocation, xi = ( xi +ωi)/ must provide at least as much utility to each consumer as x* or ω, and strictly higher utility for any * consumer where xandω are distinct. Thus, x** Pareto dominates x* and ω. # i i
4 (B) Suppose there are two consumers and two goods. Let each consumer have the utility function, ui( xi) = xi + xi and let each consumer have the endowment (/,/). The initial endowment is PO, and any allocation on the budget line defined by a price ratio of is a CE allocation. The indifference curves are all straight lines with a slope of This question concerns a pure exchange economy with K commodities and n consumers, where all utility functions are strictly monotonic, strictly quasi-concave, and continuous. For each of the following statements, if the statement is true, then prove it. If the statement is false, then provide a counterexample. A carefully drawn, labeled, and explained Edgeworth Box diagram is enough for a counterexample. (A) If x* and x** are strongly Pareto optimal, then u ( x ) u ( x ) for all i. i i i i (B) If x* and x** are strongly Pareto optimal, then u ( x ) u ( x ) for some i. i i i i (C) If u ( x ) = u ( x ) for all i, and if x* $ x**, then x* cannot be strongly Pareto optimal. i i i i ANSWER: (A) This statement is false. Starting with just about any two Pareto optimal points will be a counterexample. For example, two consumers, two goods, Cobb-Douglas utility functions, and an aggregate endowment of unit of each good. The contract curve is the diagonal of the Edgeworth box, so let x* be given by x * = (/4, /4) and x * = (/4, /4), and let x** be given by x ** = (/4, /4) and x ** = (/4, /4). The statement is false, since consumer strictly prefers x ** to x *. (B) This statement is true. If not, then every consumer strictly prefers x**, but then x** would Pareto dominate x*, contradicting the Pareto optimality of x*. (C) This statement is true. If ui( xi ) = ui( xi ), then the two bundles are on the same indifference curve. Consider instead the allocation, x%, in which everyone receives the midpoint of the line segment connecting x* and x**, x% = (x* + x**)/. Since x* and x** are feasible, then x% is feasible. If x i * = x i **, then obviously this consumer receives the same utility under x i * and x i %, since it is the same bundle. For all consumers i such that x i * $ x i **, and we know there is at least one such consumer, then consumer i strictly prefers x i %, because the line segment cuts through the upper contour set, by strict quasi-concavity. Therefore, x% Pareto dominates x*, so x* cannot be Pareto optimal.
5 5. Consider the following pure exchange economy with consumers and two commodities. Consumer has the endowment vector (,) and the utility function u ( x, x ) = log( x ) + log( x ). Consumer has the endowment vector (,0) and the utility function u ( x, x ) = log( x ) + log( x ). Consumer has the endowment vector (0,) and the utility function u ( x, x ) = log( x ) + log( x ). (A) Define a competitive equilibrium for this economy. (B) Calculate the competitive equilibrium price and allocation. ANSWER: (A) A competitive equilibrium is a price vector, (p, p ), and an allocation, ( x*, x*, x*, x*, x*, x* ), such that: ) x * solves max log( x ) + log( x ) Subject to px + px = p + p (Equality due to monotonicity), x " 0, ) x * solves max log( x) + log( x ) Subject to px + px = p (Equality due to monotonicity), x " 0, ) x * solves max log( x) + log( x ) Subject to px + px = p (Equality due to monotonicity), x " 0, 4) market clearing: x + x + x =, x + x + x =. (B) To make life easier, we will normalize the price of good to be, so the price vector is (p,). We first derive the demand functions for the three consumers. The two relevant equations are the budget equation and the marginal rate of substitution equation (MRS = p). For consumer, we have:
6 x x p = and px + x = p +. Solving for the demands, we have: x = ( p+ )/ p and x = ( p+ )/. For consumer, we have: x x = p and px x p + = Solving for the demands, we have: x = / and x = p/. For consumer, we have: x x = p and px x + = Solving for the demands, we have: x = / p and x = /. Now we pick one of the market clearing equations to solve for p. Good is easier, so we have: (p+)/ + p/ + / =. Solving for p, we get p =. (We could have guessed that, because of the symmetry between goods and, but it is better to solve it.) Now we plug p = to get the final allocation: ( x*, x*, x*, x*, x*, x* ) = (,, /, /, /, /). 6. Consider the following pure exchange economy with 00 consumers and two commodities. For i =,..., 00, consumer i has the utility function i i i i i u ( x, x ) = log( x ) + log( x ). For i =,..., 00, consumer i has the endowment vector (,). For i = 0,..., 00, consumer i has the endowment vector (,). (A) Define a competitive equilibrium for this economy. (B) Calculate the competitive equilibrium price and allocation. ANSWER: (A) A competitive equilibrium is a price vector, (p,p ), and an allocation {(x i,x i ) for i =,... 00} satisfying: () for i =,..., 00, the nonnegative vector (x i,x i ) solves: max log(x i ) + log(x i ) Subject to p x i + p x i! p + p.
7 () for i = 0,..., 00, the nonnegative vector (x i,x i ) solves: max log(x i ) + log(x i ) Subject to p x i + p x i! p + p. () markets clear x i= i i= i 400 and x 500. (B) Solving for the C.E., compute demand functions by solving the budget equation and the equation found by equating marginal rates of substitution to the price ratio, p. (Normalize p to be p, and p to equal.) For i =,..., 00, we have: x i / x i = p and p x i + x i = p +. Solving, we have: x i = (p+)/p and x i = (p+)/. For i = 0,..., 00, we have: x i / x i = p and p x i + x i = p +. Solving, we have: x i = (p+)/p and x i = (p+)/. To find the equilibrium price, use market clearing for one of the markets. Using market, and noticing that there are 00 consumers of the first type and 00 consumers of the second type, we have: 00[ (p+)/ ] + 00[ (p+)/ ] = 500. Solving, we find that p = 5/4. Plugging a price of 5/4 into the demand functions, we find the allocation: (x i,x i ) = (7/5, 7/4) for i =,..., 00 and (x i,x i ) = (/0, /8) for i = 0,..., 00.
Department 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 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 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 informationIntroduction 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 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 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 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 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 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 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 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 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 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 informationAdvanced Microeconomic Theory. Chapter 6: Partial and General Equilibrium
Advanced Microeconomic Theory Chapter 6: Partial and General Equilibrium Outline Partial Equilibrium Analysis General Equilibrium Analysis Comparative Statics Welfare Analysis Advanced Microeconomic Theory
More informationStructural Properties of Utility Functions Walrasian Demand
Structural Properties of Utility Functions Walrasian Demand Econ 2100 Fall 2017 Lecture 4, September 7 Outline 1 Structural Properties of Utility Functions 1 Local Non Satiation 2 Convexity 3 Quasi-linearity
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 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 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 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 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 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 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 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 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 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 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 informationMarket Equilibrium Price: Existence, Properties and Consequences
Market Equilibrium Price: Existence, Properties and Consequences Ram Singh Lecture 5 Ram Singh: (DSE) General Equilibrium Analysis 1 / 14 Questions Today, we will discuss the following issues: How does
More information; p. p y p y p y. Production Set: We have 2 constraints on production - demand for each factor of production must be less than its endowment
Exercise 1. Consider an economy with produced goods - x and y;and primary factors (these goods are not consumed) of production A and. There are xedcoe±cient technologies for producing x and y:to produce
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 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 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 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 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 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 informationThe Definition of Market Equilibrium The concept of market equilibrium, like the notion of equilibrium in just about every other context, is supposed to capture the idea of a state of the system in which
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 informationPositive Theory of Equilibrium: Existence, Uniqueness, and Stability
Chapter 7 Nathan Smooha Positive Theory of Equilibrium: Existence, Uniqueness, and Stability 7.1 Introduction Brouwer s Fixed Point Theorem. Let X be a non-empty, compact, and convex subset of R m. If
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 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 information4 Lecture Applications
4 Lecture 4 4.1 Applications We now will look at some of the applications of the convex analysis we have learned. First, we shall us a separation theorem to prove the second fundamental theorem of welfare
More informationUnlinked Allocations in an Exchange Economy with One Good and One Bad
Unlinked llocations in an Exchange Economy with One Good and One ad Chiaki Hara Faculty of Economics and Politics, University of Cambridge Institute of Economic Research, Hitotsubashi University pril 16,
More informationWeek 6: Consumer Theory Part 1 (Jehle and Reny, Chapter 1)
Week 6: Consumer Theory Part 1 (Jehle and Reny, Chapter 1) Tsun-Feng Chiang* *School of Economics, Henan University, Kaifeng, China November 2, 2014 1 / 28 Primitive Notions 1.1 Primitive Notions Consumer
More informationWeek 7: The Consumer (Malinvaud, Chapter 2 and 4) / Consumer November Theory 1, 2015 (Jehle and 1 / Reny, 32
Week 7: The Consumer (Malinvaud, Chapter 2 and 4) / Consumer Theory (Jehle and Reny, Chapter 1) Tsun-Feng Chiang* *School of Economics, Henan University, Kaifeng, China November 1, 2015 Week 7: The Consumer
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 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 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 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 informationGeneral Equilibrium with Production
General Equilibrium with Production Ram Singh Microeconomic Theory Lecture 11 Ram Singh: (DSE) General Equilibrium: Production Lecture 11 1 / 24 Producer Firms I There are N individuals; i = 1,..., N There
More informationNote on social choice allocation in exchange economies with Cobb-Douglas preferences
Note on social choice allocation in exchange economies with Cobb-Douglas preferences Takeshi Momi Department of Economics, Doshisha University April, 2011 Abstract In this note we show that in a pure exchange
More informationShort correct answers are sufficient and get full credit. Including irrelevant (though correct) information in an answer will not increase the score.
Economics 200A Part 2 UCSD Fall 2012 Prof. R. Starr, Mr. Troy Kravitz Final Exam 1 Your Name: Please answer all questions. Each of the six questions marked with a big number counts equally. Designate your
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 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 informationLecture 7: General Equilibrium - Existence, Uniqueness, Stability
Lecture 7: General Equilibrium - Existence, Uniqueness, Stability In this lecture: Preferences are assumed to be rational, continuous, strictly convex, and strongly monotone. 1. Excess demand function
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 informationMidterm #1 EconS 527 Wednesday, February 21st, 2018
NAME: Midterm #1 EconS 527 Wednesday, February 21st, 2018 Instructions. Show all your work clearly and make sure you justify all your answers. 1. Question 1 [10 Points]. Discuss and provide examples of
More informationCore. Ichiro Obara. December 3, 2008 UCLA. Obara (UCLA) Core December 3, / 22
Ichiro Obara UCLA December 3, 2008 Obara (UCLA) Core December 3, 2008 1 / 22 in Edgeworth Box Core in Edgeworth Box Obara (UCLA) Core December 3, 2008 2 / 22 in Edgeworth Box Motivation How should we interpret
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 informationEconomics 201b Spring 2010 Solutions to Problem Set 1 John Zhu
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
More informationEconomics th April 2011
Economics 401 8th April 2011 Instructions: Answer 7 of the following 9 questions. All questions are of equal weight. Indicate clearly on the first page which questions you want marked. 1. Answer both parts.
More informationMarket Outcomes: Efficient or Fair?
Market Outcomes: Efficient or Fair? Ram Singh Microeconomic Theory Lecture 14 Ram Singh: (DSE) Market Equilibrium Lecture 14 1 / 16 Fair Versus Efficient Question 1 What is a fair allocation? 2 Is a fair
More informationMathematical models in economy. Short descriptions
Chapter 1 Mathematical models in economy. Short descriptions 1.1 Arrow-Debreu model of an economy via Walras equilibrium problem. Let us consider first the so-called Arrow-Debreu model. The presentation
More informationMicroeconomics, Block I Part 2
Microeconomics, Block I Part 2 Piero Gottardi EUI Sept. 20, 2015 Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, 2015 1 / 48 Pure Exchange Economy H consumers with: preferences described
More informationProblem Set Suggested Answers
Problem Set 3 --- Suggested Answers 1. In chapters 15 18 the treatment is generalized to unbounded production technologies, resulting in the observation that when general equilibrium prices are announced,
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 informationIntroductory Microeconomics
Prof. Wolfram Elsner Faculty of Business Studies and Economics iino Institute of Institutional and Innovation Economics Introductory Microeconomics The Ideal Neoclassical Market and General Equilibrium
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 informationMicroeconomic Theory-I Washington State University Midterm Exam #1 - Answer key. Fall 2016
Microeconomic Theory-I Washington State University Midterm Exam # - Answer key Fall 06. [Checking properties of preference relations]. Consider the following preference relation de ned in the positive
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 informationECON501 - Vector Di erentiation Simon Grant
ECON01 - Vector Di erentiation Simon Grant October 00 Abstract Notes on vector di erentiation and some simple economic applications and examples 1 Functions of One Variable g : R! R derivative (slope)
More informationi) This is simply an application of Berge s Maximum Theorem, but it is actually not too difficult to prove the result directly.
Bocconi University PhD in Economics - Microeconomics I Prof. M. Messner Problem Set 3 - Solution Problem 1: i) This is simply an application of Berge s Maximum Theorem, but it is actually not too difficult
More informationMicroeconomics. Joana Pais. Fall Joana Pais
Microeconomics Fall 2016 Primitive notions There are four building blocks in any model of consumer choice. They are the consumption set, the feasible set, the preference relation, and the behavioural assumption.
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 informationEcon 401A: Economic Theory Mid-term. Answers
. Labor suly Econ 40: Economic Theory Mid-term nswers (a) Let be labor suly. Then x 4 The key ste is setting u the budget constraint. x w w(4 x ) Thus the budget constraint can be rewritten as follows:
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 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 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 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 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 informationEcon 201: Problem Set 3 Answers
Econ 20: Problem Set 3 Ansers Instructor: Alexandre Sollaci T.A.: Ryan Hughes Winter 208 Question a) The firm s fixed cost is F C = a and variable costs are T V Cq) = 2 bq2. b) As seen in class, the optimal
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 informationU b (x b ) = xb 1x b 2 x a 1. means the consumption of good i by an h-type person.
Chapter 9 Welfare Exercise 9. In a two-commodity exchange economy there are two large equalsized groups of traders. Each trader in group a has an endowment of 300 units of commodity ; each person in group
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 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 informationWelfare Economics: Lecture 12
Welfare Economics: Lecture 12 Ram Singh Course 001 October 20, 2014 Ram Singh: (DSE) Welfare Economics October 20, 2014 1 / 16 Fair Vs Efficient Question 1 What is a fair allocation? 2 Is a fair allocation
More information1. Pareto E cient Allocation and Social Welfare Maximization
Spring 2013 1. Pareto E cient Allocation and Social Welfare Maximization! "R $ Consider a private ownership economy E priv = L + ; " i ;e i #i=1;:::;i ; fy jg j=1;:::;j ; f$ i;j g i;j where " i can be
More informationFall Final Examination Solutions Thursday 10 January 2012
EC 20.2 & 20. Fall 202 Deniz Selman Bo¼gaziçi University Final Examination Solutions Thursday 0 January 202. (9 pts) It is the heart of winter the isl of Ludos has been devastated by a violent snowstorm
More informationEconomic Core, Fair Allocations, and Social Choice Theory
Chapter 9 Nathan Smooha Economic Core, Fair Allocations, and Social Choice Theory 9.1 Introduction In this chapter, we briefly discuss some topics in the framework of general equilibrium theory, namely
More informationECON 304 MIDTERM EXAM ANSWERS
ECON 30 MIDTERM EXAM ANSWERS () The short questions: (a) Transitivity says that if y and y z, then z. Note the three bundles in diagram 0.. y because they are on the same indifference curve. y z because
More informationMarket Equilibrium using Auctions for a Class of Gross-Substitute Utilities
Market Equilibrium using Auctions for a Class of Gross-Substitute Utilities Rahul Garg 1 and Sanjiv Kapoor 2 1 IBM T.J. Watson Research Center,USA. 2 Illinois Institute of Technology, Chicago, USA Abstract.
More informationNotes on Consumer Theory
Notes on Consumer Theory Alejandro Saporiti Alejandro Saporiti (Copyright) Consumer Theory 1 / 65 Consumer theory Reference: Jehle and Reny, Advanced Microeconomic Theory, 3rd ed., Pearson 2011: Ch. 1.
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 information1 + x 1/2. b) For what values of k is g a quasi-concave function? For what values of k is g a concave function? Explain your answers.
Questions and Answers from Econ 0A Final: Fall 008 I have gone to some trouble to explain the answers to all of these questions, because I think that there is much to be learned b working through them
More informationUniqueness, Stability, and Gross Substitutes
Uniqueness, Stability, and Gross Substitutes Econ 2100 Fall 2017 Lecture 21, November 14 Outline 1 Uniquenness (in pictures) 2 Stability 3 Gross Substitute Property Uniqueness and Stability We have dealt
More informationTopics in Trade: Slides
Topics in Trade: Slides Alexander Tarasov University of Munich Summer 20 Alexander Tarasov (University of Munich) Topics in Trade Summer 20 / 2 : Rybczynski Theorem (955) How factor endowments affect product
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 informationMicroeconomics II Lecture 4. Marshallian and Hicksian demands for goods with an endowment (Labour supply)
Leonardo Felli 30 October, 2002 Microeconomics II Lecture 4 Marshallian and Hicksian demands for goods with an endowment (Labour supply) Define M = m + p ω to be the endowment of the consumer. The Marshallian
More informationA : a b c d a : B C A E B : d b c a b : C A B D E C : d c a c : E D B C D : a d b d : A D E B C E : a b d. A : a b c d a : B C A D E
Microeconomics II( ECO 50) Questions on the comprehensive exam will be chosen from the list below( with possible minor variations) CALCULATORS ARE ALLOWED Matching. Consider the Gale-Shapley marriage problem
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 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 informationu(c t, x t+1 ) = c α t + x α t+1
Review Questions: Overlapping Generations Econ720. Fall 2017. Prof. Lutz Hendricks 1 A Savings Function Consider the standard two-period household problem. The household receives a wage w t when young
More information