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Barro-Gordon model As Barro and Gordon (1983a, b), assume a social loss function depending on employment l and prices p L = (l l)2 + (p p)2; where l is e cient employment and p is the price level consistent with optimal inﬂation. An economist would advise a risk-averse investor to ‘diversify’ her investments, no matter how risk averse she is … as long as she is at least a little bit risk-averse, she will prefer to minimise the variance of the return (for a given expected return). 3 0 obj
Assume that \(\lambda\) makes this profit zero, so that \(\lambda = 1/p\). However, this depends what our purpose is in making this comparison across individuals – if we want to compare how risk averse they are given their current wealth, this may not be a problem. Solutions to Problem Set 4. Problem Set Questions (PDF) Problem Set Solutions (PDF) Problem Solving Video. endobj
Problem Set #3: Solutions 1. Uncertainty Lotteries Expected Utility Money Lotteries Stochastic Dominance Lotteries A simple lottery can be represented as a point in simplex. 1. %PDF-1.5
and show that if he is strictly risk-averse he rejects the offer. Thus both the gains and losses are reduced by making this bet; i.e., the variance is reduced. Perfect Competition Describe one choice that a risk neutral person might make that a risk-averse person would never make. Explain why the parent’s preferences are not consistent with expected utility. . Define risk aversion formally and intuitively. Econ 100B: Economic Analysis – Macroeconomics Problem Set #6 – Solutions Due Date: August 7, 2020 General Instructions: • Please upload a PDF of your problem set to Gradescope by 11:59 p.m. • Late homework will not be accepted. All lower case letters denote logarithmic terms. Would the advice be the same for any risk-averse investor, or would it vary depending on her level of risk-aversion? A risk-averse person (a person with risk averse preferences) will always prefer a sure thing to a gamble with the same expected monetary value. Problem Set 5 Solution Microeconomic Theory Chapters 11 and 12 ECON5110 | Fall 2019 1. The Axiomatic Approach Critique Applications De–nitions and Axioms Lotteries I Set of outcomes: fa 1,a 2,...,a ng. Social Links Twitter Facebook Flickr Instagram LinkedIn YouTube She can optimise along this margin by ‘optimally diversifying’, buying assets in proportion to their representation (relative value) in the market. Please see lecture notes on Allais paradox, Allais paradox illustrated by a scenario such as. Problem Set #1: Solutions 1. If she ‘bets on leave’ this loss would be counterbalanced by an income gain from the asset. One possibility is that it is too complicated and analytical for most people to handle or to take seriously given low stakes. Uncertainty Advanced Microeconomics I Andras Niedermayer1 1Department of Economics, University of Mannheim Fall 2009 Chapter 3: Individual Choice Under Uncertainty Fall 2009 1 / 76. An exchange economy has two dates t =0,1 and two states of nature s =1,2 which will be revealed at date 1. ;0g (8.1) Now consider the choices amongst prospects presented in Exercise 8.4. This allows her to reduce the variance of her returns for a given expected return, or increase the expected return for a given variance. Problem Set 10 (graded) S O L U T I O N S T O A S S I G N M E N T S. Solutions to Problem Set 1. Explain why or why not, referring to equations and diagrams as needed. Let P:= f(x! More formal definitions, depictions, and intuition is given in this web book above. Advanced Microeconomics ProblemSet2: ChoiceunderRisk andUncertainty Exercise 2.1 Consider the following pairs of lotteries over weekend trip destinations: Lotterie I: Berlin with probability 1, vs. Lotterie II: Berlin, Bayreuth, and Munich with probability 1/3 each. Deﬁnition: The set ∆ = {p ∈ R+N: P pi = 1} is called a N-dimensional simplex. They will never take ‘fair bets’ and will refuse even some gambles that have a positive expected value. PROBLEM SET 7, WITH SOLUTIONS 1. or consider the measurement of risk (certainty equivalent, Arrow-Pratt measures, etc.). Microeconomic Theory I: Choice Under Uncertainty Parikshit Ghosh Delhi School of Economics September 8, 2014 Parikshit Ghosh Delhi School of Economics Choice Under Uncertainty. maxfx0 x! Choice Under Uncertainty: Problem Set 1. Problem Set 5 Prof. Dr. Gerhard Illing, Jin Cao January 29, 2011 1. <>
However (advanced point) if she cannot borrow/lend at the risk-free rate she cannot choose along the ‘market line’ and thus may not want to diversify quite as much; buying the ‘market basket’ may then be too-risky/too-safe relative to her preferences. (To fully answer this last part it will help to have read into the ‘CAPM’ model: see, e.g., the hypothes.is annotated Wikipedia entries on referred to above). Contact Us (413) 542-2000 Contact Us Map & Directions. The consumers will reject any proposed exchange that does not lie in their shaded superlevel set s. line 400 800 line 600 200 These measures, and the intuition for them, are discussed above. Overview of module & rules, discussion/background, Intuition for ‘risk aversion iff concave value function. Problem Set 2 Solutions Intermediate Microeconomics Mark Dean February 4, 2016 Question 1 (Indi erence Curves) 1.Assume that the consumer only gains utility from plants in plant pots. Note that \(\lambda\) will determine, in effect, the ‘price’ of the insurance, per unit of compensation in the event of an accident. : !2 g P0:= f(x0! MWGchapter6.A.Kreps“NotesontheTheoryofChoice”, chapters4and7(theﬁrstpartonly). This is referred to as ‘actuarially fair insurance’. In ation dynamics under optimal monetary policy. will be a crucial learning tool. Textbooks The course will draw mainly on the textbook: Riley, Essential Microeconomics, Cambridge University Press, 2012. Show that if the individual is risk-averse he optimally chooses \(x = pD\) , so that he is fully insured: [implying that] his net wealth is the same whether or not he has an accident. Choice under Uncertainty Gamble C: an 89 percent chance of winning nothing and an 11% chance of winning 1 million. Does this depend on whether she can borrow or lend at the ‘risk-free’ rate? In May 2016, an economist (Al) advises Betty that if the UK votes ‘leave’ in the June referendum, this may reduce trade with France. The ‘coefficient of absolute’ risk aversion is one measure but it may not be constant within the range of an individual’s income; thus some normalisation or averaging would be required to make this comparison across individuals. ;��J*��d� �}����sI���'���Y�V��E�b1�U��U}ɔh����5�-�ǹ|S!yy�pOw�t���EͯHyY���E ? Insurance. A sure pro–t of $240. Introduction 1.1. Solutions to selected exercises from Jehle and Reny (2001): Advanced Microeconomic Theory Thomas Herzfeld September 2010 Contents 1 Mathematical Appendix 2 Under uncertainty, the DM is forced, in eﬀect, to gamble. What sort of preferences would Betty have to have to make this advice worth following? 4.1 Consumer preferences, indifference curves/sets (0.5 weeks) 4.1.1 “Bundles of … %����
Calculators: The production function for a firm in the business of calculator assembly is given by q = √ l, where q denotes finished calculator output and l denotes hours of labor input. ECON 302 - Microeconomic Theory II: Strategic Behavior IRYNA DUDNYK Tutorial 7. A parent. Note: I can probably improve the notation in the above video. Use s = 0 to denote the date-event pair corresponding to date 0. At each 45 line the steepness of the Respective sets are both 1 2 S S. Therefore 1 2 MRS MRSB B A A( ) ( ) S ZZ S!! 1.2. PROBLEM SET 6, WITH SOLUTIONS 1. 2. (a) Suppose her rm is the only asset she has. Note to Beem101 2020: this was part of a question on a previous exam. Oligopoly 8.2 The Cournot Model 8.3 The Bertrand Model 9. Note: Here you are being asked to depict the lottery he faces in net including the lottery \(p\), which may have any number of prizes, as well as the additional ‘coin flip’ lottery mentioned above. Al advises Betty to buy an asset (a ‘bet on leave’ with a bookmaker) that will pay off in the event that the UK votes for ‘leave’. 2 0 obj
Explain why these choices are inconsistent with the standard theory of expected utility maximisation. MICROECONOMICS I: CHOICE UNDER UNCERTAINTY MARCINPĘSKI Please let me know about any typos, mistakes, unclear or ambiguous statements thatyouﬁnd. Problem Set Questions (PDF) Problem Set Solutions (PDF) Problem Solving Video. A company develops a product of an unknown quality. Microeconomics Exercises 6 Suggested Solutions 1. Lecture: TuTh 9:30-11AM, 60 Evans Hall Instructor: Professor Stefano DellaVigna Office: 515 Evans Hall E-mail: sdellavi@econ.berkeley.edu Office Hours: Thursday 12-2pm . Many people choose B over A and choose D over C: This contradicts Expected Utility theory: (Note: Suggested answers provided to Beem101 students, not to be posted on the web by request of O-R. Beem101 students can consult the Class Notebook, or the direct link HERE). An individual has wealth \(w\) and has to choose an amount \(x\), after which a lottery is conducted in which with probability \(\alpha\) he gets \(2x\) and with probability \(1 − \alpha\) he loses \(x\). The solution keys for problem set 5 are uploaded.” 2008/07/07, “Solution keys for problem set 4 are uploaded.” 2008/07/01, “There is a problem set due on July 8.” 2008/06/25, “We have a final exam on July 22 from 10:35-12:05” Important! Ana’s utility function is U = p w, where wis her wealth. For the upcoming midterm, I would probably add an additional challenging element to such a question, e.g., asking you to formally specify her preferences in some way (concavity of value function, etc.) If utility is differentiable we can define risk aversion in terms of a diminishing marginal utility of income (or in general, concavity). 4 0 obj
2. Demand 2.1 Price Changes 2.2 Income Changes 2.3 Elasticities 3. Problem Set 1. Note that the sketched curves should also include the corners, which were not rendered well in the image below. Because the individual paid \(x\) and the insurer must compensate him \(\lambda x\) with probability \(p\). General Equilibrium 'H¿QLWLRQV (I¿FLHQW3URGXFWLRQ 12. (See discussion under ‘benefits of diversification’ Production 'H¿QLWLRQV 3.2 The Production Function 4. The probabilities are denoted by p 1, p 2 and p 3 respectively. Lotterie III: Berlin with Probability 1/3 and Bayreuth with probability 2/3, As the returns of assets are not perfectly correlated, dividing the investment over ‘more coin flips’ implies a lower overall variance. She owns a bak-ery that will be worth 69 or 0 dollars next year with equal probability. Solutions Problem 1. A right decision consists in the choice of the best possible bet, not simply in whether it is won or lost after the fact. Problem Set 3. Consumer Theory 1.1 Preferences 1.2 The Budget Line 1.3 Utility Maximization 2. Solutions to Problem Set 5 On the other hand if leave does not pass she keeps her job, but loses the bet. If Leave passes she may lose her job or suffer reduced income. Example 1. While we often rely on models of certain information as you’ve seen in the class so far, many economic problems require that we tackle uncertainty head on. (Continuation of Problem 2 from Problem Set 5.) The level set for Alex is also depicted. The … A parent has two children, A and B. Show that it is invariant to positive linear transformations of the utility function. 14.772 Macro Development - Problem Set 2 Spring 2013 Problem 1: Risk Sharing Consider H households, with household h consisting of I h members. Note that expected utility requires the ‘independence’ property. endobj
Gamble D: a 90 percent chance of winning nothing and a 10 percent chance of winning £ 5 million. Advanced Microeconomics 1 (Part 1), Fall 2017 Problem Set 5: Possible Answers Exercise 1 Tversky and Kahneman (1986) report the following experiment: each partic- ipant receives a questionnaire asking him to make two choices, the –rst from fa;bgand the second from fc;dg: a. Solutions to Problem Set 2. How much depends on the grader’s discretion. If she is risk-averse she prefers to reduce the variance of her returns, holding the expected value the same. Microeconomics CHAPTER 8. An individual faces the monetary lottery \(p\). (You should briefly characterise it). endobj
Two assignments per term will be marked. Problems with solutions, Intermediate microeconomics, part 1 Niklas Jakobsson, nja@nova.no Katarina.Katz@kau.se Problem 1. Solow model in continuous time. On the other hand, if we want to make a comparison (e.g., between men and women) to say something about genetic or culturl predisposition to risk-seeking, then the issue of ‘differing baseline incomes’ may be important. She can then move to her desired point on the risk/return frontier, aka the ‘market line’, by either leveraging (borrowing) or putting some of her investment in a risk-free asset. Problem Set 7. Problem Set 6: Solutions ECON 301: Intermediate Microeconomics Prof. Marek Weretka Problem 1 (Insurance) (a) Ben’s a ordable bundle if there is no insurance market is his endowment: Explain why an economist would advise a risk-averse investor to `diversify’ her investments. An individual has wealth \(w\) and is afraid that an accident will occur with probability \(p\) that will cause him a loss of \(D\). Problem Set 2 Welfare and Allocation Nov 11 Reading: JR Chapter5 Reference: Varian Chapter10; General Equilibrium Nov 11 Problem Set 2: Solution Reading: Koopmans Chapter 1 Exercise 3 Production Economy Nov 18 Reading: Laffont Introduction; Time and Uncertainty Nov 18 Problem Set 3 Problem Set 3: Solution K Problem Set 5 - Solution K.1 Gregory N. Mankiw - NYT - Nov 30, 2008 According to Gregory N. Mankiw, the factors contributing to hold back consumption are low consumer confidence and “wait and see” behavior caused by falling house price values, shrinking 401(k) balances (due to the fall of the stock market, my addition) and increased unemployment. write a lottery as a set {xi: pi}N i=1 and denote by L the set of all simple lotteries over the set of outcomes C.) 2 / 31. stream
Describe a particular measur} of risk-aversion that would allow us to rank individuals according to their level of risk aversion, considering the strengths and weaknesses of this measure. Pro t in terms of the labor choice is ˇ= TR TC= TR(y(L)) w LL: Econ 101A, Microeconomic Theory Fall 2009. The greater the curvature (relative to the slope) of the VnM utility function, the more risk averse, at least by the popular ‘Arrow-Pratt’ measures. J Problem Set 2 - Solution. ]���1/��. Problem sets will be provided and answers to selected problems will be discussed during classes. De–ne the expected regret if the person chooses P rather than P0as X!2 ˇ! Please assume, of course, that this is indeed the probability that such an accident will occur. In the video below, a teaching assistant demonstrates his approach to the solution for problem 5 from the problem set. Two essential characteristics: 1. ECO 317 – Economics of Uncertainty – Fall Term 2009 Problem Set 2 – Due October 15 Question 1: (30 points) Consider a situation of uncertainty with three possible outcomes, namely money rewards of amounts 1, 2 and 3. Other measures include specific empirical elicitations/comparisons as those done in experiments, such as Holt and Laury discussed here. Costs 4.1 Costs in the Short Run 4.2 Costs in the Long Run 5. x���]o�6���?�RZ�$J��^t�Z�*"+��X�ly@����%��|�7�: sӇ���sHy�j߷�Uݳ\����~h��v��}�c����Y~�6mW���[~=���W?7պ���{��
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���U��6Z=,n�����h��R/rԅ4��]�f���! In the video below, a teaching assistant demonstrates his approach to the solution for problem 2a-b from the problem set. If she is substantially risk-averse, she is willing to sacrifice at least some amount of expected monetary value (i.e., the commission) to reduce the variance. In particular, there is some evidence (cite) that the Holt and Laury does not substantially predict real-world behavior. Describe the lottery \(q\) that he faces if he accepts the offer. Problem Set 9. Define ‘risk aversion’. Solutions to Problem Set 3. Describingtheuncertainty. Gamble A: an 89 percent chance of winning 1 million a 10 percent chance of winning £ 5 million, and a 1 pct chance of winning nothing. What would justify the economist’s advice to buy this asset? b. Another possibility is that the succession of choices presented by HL leads people to consider it in a way they would not naturally have done, to aim for an ‘arbitrary coherence’. Suggestedreadings. Problem Set 4 (graded) Problem Set 5. A choice must be made among various possible courses of actions. as well as the discussion of the CAPM model). Choice under Uncertainty Jonathan Levin October 2006 1 Introduction Virtually every decision is made in the face of uncertainty. Monopolistic Competition 10. J.1 Two-period Intertemporal Optimization; K Problem Set 3 - Solution. Microeconomics Exercises with Suggested Solutions 5 7. UNCERTAINTY AND RISK Exercise 8.6 An example to illustrate regret. 1 0 obj
Amherst College 220 South Pleasant Street Amherst, MA 01002. Exeter students: I cover this question at length in this recorded session, For a ‘state-space’ diagram presenting the insurance problem, please see Joon Song’s video here, Economic models (& maths tools), ‘empirical’ evidence, Preferences under uncertainty (and over time), Consumer preferences, indifference curves/sets, Consumer behavior/Individual (and market) demand functions and their properties, ‘Monopolies and pricing of profit-maximizing price-setting firms’ (especially monopolies), Behavioural economics: Selected further concepts, Supplement (optional): Asymmetric information (Moral hazard, adverse selection, signaling) and applications, \(\rightarrow U(1m) > 0.89 \: U(1m) + 0.1 \: U(5m) + 0.01 \: U(0)\), \(0.11 \: U(1m) > 0.1 U(5m) + 0.01 \: U(0)\), \(\rightarrow 0.9 \: U(0) + 0.1 U(5m) > 0.89 \: U(0) + 0.11 \: U(1m)\), \(0.1 \: U(5m) + 0.01 \: U(0) > 0.11 \: U(1m)\). <>
These are also arbitrary and may be sensitive to the experimental framing. Breakdown of points: 10 for setting up the objective function correctly, 10 for solving the optimization problem correctly. This should hold in a perfectly competitive insurance market if there are no moral hazard or asymmetric information issues, no transactions costs, etc. Show that the higher is \(\alpha\) the higher is the amount \(x\) he chooses. The parent has in hand only one gift. Uncertainty; Problem Set and Solutions. Choice under Uncertainty (cont’d). He is indifferent between giving the gift to either child but prefers to toss a fair coin to determine which child obtains the gift over giving it to either of the children. (Think of these as millions of dollars if you like.) The bookmaker offers odds that are seen as fair, and he only takes a small commission. If you are wrong in your rst setting up, you will get partial but not full credit for a \conditionally correct" solution of the constrained maximization problem. I A gamble/lottery is a probability distribution over outcomes: g = (p 1 a 1,p 2 a 2,...,p n a n). Problem Set 2. A risk averse person always prefers the expected monetary value of a gamble to the gamble itself. Risk aversion: The extent to which uncertainty of an outcome (holding the expected material or monetary value constant) implies an individual values it less. Exercises - uncertainty, finance, time preferences (‘problem set’) Some questions from previous exams (somewhat easier questions) 3.13 From O-R; 4 Consumer preferences, constraints and choice, demand functions. • Please put your name, student ID & your GSI’s name at the upper right corner of the front page. Unlike the model in class, agents in this economy do have endowments, consume and trade in goods at t = 0. Consider the exchange economy described in that problem: two physical goods l =1,2, two consumers that share the same von-Neumann Morgenstern utility function logc1 +logc2, where c i denotes con- sumption of good i; two states of nature A and B. ;ˇ !) Describe the ‘Allais paradox’, giving a specific example of a set of choices that illustrate this paradox. Consider Betty, a UK resident working at a company that ships goods from Britain to France. ‘Coefficient of relative risk aversion’ is another measure; it also may not be constant throughout the range of income (but that is at least more plausible). Equivalently, a risk averse person will always reject a fair gamble. Problem Set 11: Solutions ECON 301: Intermediate Microeconomics Prof. Marek Weretka Problem 1 (Monopoly and the Labor Market) (a) We nd the optimal demand for labor for a monopoly rm (in the goods market as poposed to the labor market) through the pro t maximization condition. Please see (and present and give intuition for) formal presentations as given above. Microeconomics - 1. The individual has to choose an amount, \(x\), he will pay for insurance that will pay him \(\lambda x\) (for some given \(\lambda\)) if the accident occurs. <>/ExtGState<>/Font<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 792 612] /Contents 4 0 R/Group<>/Tabs/S/StructParents 0>>
Neoclassical microeconomics concieves of and models this using an ‘outcome based’ (Von-Neuman Morgenstern) value function that increases at a diminishing rate, and an individual who tries to maximize the expected value of the outcome as measured by this utility function. For each realization of the lottery another lottery will be executed according to which he will win an additional dollar with probability \(\frac{1}{2}\) and lose a dollar with probability \(\frac{1}{2}\). Labor 7KH6XSSO\RI/DERU 7KH'HPDQGIRU/DERU 11. GSI's: Justin Gallagher, justing@econ.berkeley.edu Office Hours: Friday 2-4pm & Monday 9-10am Location: 608-5 Evans Hall Mariana Carrera, mcarrera@econ.berkeley.edu Office … (Class Test 2002Q2(a))Deﬁne the Arrow-Pratt coeﬃcient of absolute risk aversion. He is made the following offer. Therefore there are gains to be made from trading state claims. A lottery between a pro–t of $1000 with probability 25% and 0 with probability 75%. Note: In answering this question, you can assume that he is an ‘expected utility’ maximiser, and thus the continuity and independence axioms must hold (and by extension, monotonicity). Problem Set 6. Game Theory %DVLF&RQFHSWV 7.2 Games on Normal Form 7.3 Games on Extensive Form 8. Problem Set 8. Her rm is the only asset she has her level of risk-aversion two prospects available an! A lottery between a pro–t of $ 1000 with probability 75 % Exercise 8.6 an example illustrate. Courses of actions Think of these as millions of dollars if you like. ) standard... Various possible courses of actions chooses p rather than P0as X! 2 be..., such as Holt and Laury discussed here of module & rules,,. If leave does not substantially predict real-world behavior: fa 1, p 2 and 3... Be worth 69 or 0 dollars next year with equal probability is too complicated and analytical for people! 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( 413 ) 542-2000 contact Us Map & Directions Holt and Laury not. ’ this loss would be counterbalanced by an income gain from the asset,! Evidence ( cite ) that he faces if he is strictly risk-averse he rejects the offer at. Lottery can be represented as a point in simplex measurement of risk ( certainty equivalent, Arrow-Pratt measures, the! A previous exam = 1 } is called a N-dimensional simplex how much depends on the other hand if passes... Two dates t =0,1 and two states of nature s =1,2 which will be discussed classes... De–Nitions and Axioms Lotteries I Set of choices that illustrate this paradox with... Problem Solving video paradox ’, giving a specific example of a question on a previous.... A ) Suppose her rm is the only asset she has Deﬁne the Arrow-Pratt coeﬃcient of absolute aversion... In goods at t = 0 gains to be made from trading state claims 1/p\ ) justify! - Solution x\ ) he chooses are reduced by making this bet ; i.e., the variance of her,. 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