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 \(\frac{1}{2}\). She can optimise along this margin by ‘optimally diversifying’, buying assets in proportion to their representation (relative value) in the market. An exchange economy has two dates t =0,1 and two states of nature s =1,2 which will be revealed at date 1. More formal definitions, depictions, and intuition is given in this web book above. Microeconomics - 1. 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. He is made the following offer. 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). Problem Set 2. Gamble D: a 90 percent chance of winning nothing and a 10 percent chance of winning £ 5 million. Choice under Uncertainty Jonathan Levin October 2006 1 Introduction Virtually every decision is made in the face of uncertainty. 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. (See discussion under ‘benefits of diversification’ A risk averse person always prefers the expected monetary value of a gamble to the gamble itself. Definition: The set ∆ = {p ∈ R+N: P pi = 1} is called a N-dimensional simplex. (Class Test 2002Q2(a))Define the Arrow-Pratt coefficient of absolute risk aversion. 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. Lotterie III: Berlin with Probability 1/3 and Bayreuth with probability 2/3, Solutions to Problem Set 5 Problem Set 8. 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. General Equilibrium 'H¿QLWLRQV (I¿FLHQW3URGXFWLRQ 12. All lower case letters denote logarithmic terms. A right decision consists in the choice of the best possible bet, not simply in whether it is won or lost after the fact. ;0g (8.1) Now consider the choices amongst prospects presented in Exercise 8.4. 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. 14.772 Macro Development - Problem Set 2 Spring 2013 Problem 1: Risk Sharing Consider H households, with household h consisting of I h members. Solutions to Problem Set 3. The … Introduction 1.1. 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. Perfect Competition In ation dynamics under optimal monetary policy. Choice Under Uncertainty: Problem Set 1. 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 In May 2016, an economist (Al) advises Betty that if the UK votes ‘leave’ in the June referendum, this may reduce trade with France. Gamble C: an 89 percent chance of winning nothing and an 11% chance of winning 1 million. • Please put your name, student ID & your GSI’s name at the upper right corner of the front page. ‘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). 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. 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 2a-b from the problem set. 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. UNCERTAINTY AND RISK Exercise 8.6 An example to illustrate regret. Social Links Twitter Facebook Flickr Instagram LinkedIn YouTube The level set for Alex is also depicted. ;ˇ ) : !2 g be two prospects available to an individual. 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, \(\rightarrow U(1m) > 0.89 \: U(1m) + 0.1 \: U(5m) + 0.01 \: U(0)\), \(0.11 \: U(1m) > 0.1 U(5m) + 0.01 \: U(0)\), \(\rightarrow 0.9 \: U(0) + 0.1 U(5m) > 0.89 \: U(0) + 0.11 \: U(1m)\), \(0.1 \: U(5m) + 0.01 \: U(0) > 0.11 \: U(1m)\). (Think of these as millions of dollars if you like.) 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. If Leave passes she may lose her job or suffer reduced income. . 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. or consider the measurement of risk (certainty equivalent, Arrow-Pratt measures, etc.). Solutions Problem 1. One possibility is that it is too complicated and analytical for most people to handle or to take seriously given low stakes. Problem Set 7. 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. Solutions to selected exercises from Jehle and Reny (2001): Advanced Microeconomic Theory Thomas Herzfeld September 2010 Contents 1 Mathematical Appendix 2 (You should briefly characterise it). Assume that \(\lambda\) makes this profit zero, so that \(\lambda = 1/p\). This is referred to as ‘actuarially fair insurance’. J Problem Set 2 - Solution. J.1 Two-period Intertemporal Optimization; K Problem Set 3 - Solution. Define risk aversion formally and intuitively. 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. The probabilities are denoted by p 1, p 2 and p 3 respectively. Problem Set 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. As the returns of assets are not perfectly correlated, dividing the investment over ‘more coin flips’ implies a lower overall variance. Explain why or why not, referring to equations and diagrams as needed. Explain why an economist would advise a risk-averse investor to `diversify’ her investments. Thus both the gains and losses are reduced by making this bet; i.e., the variance is reduced. Oligopoly 8.2 The Cournot Model 8.3 The Bertrand Model 9. 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. Under uncertainty, the DM is forced, in effect, to gamble. Show that it is invariant to positive linear transformations of the utility function. Ana’s utility function is U = p w, where wis her wealth. Other measures include specific empirical elicitations/comparisons as those done in experiments, such as Holt and Laury discussed here. ECON 302 - Microeconomic Theory II: Strategic Behavior IRYNA DUDNYK Tutorial 7. 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. A company develops a product of an unknown quality. 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!! On the other hand if leave does not pass she keeps her job, but loses the bet. 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. Pro t in terms of the labor choice is ˇ= TR TC= TR(y(L)) w LL: endobj
(a) Suppose her rm is the only asset she has. Econ 101A, Microeconomic Theory Fall 2009. Define ‘risk aversion’. The consumers will reject any proposed exchange that does not lie in their shaded superlevel set s. line 400 800 line 600 200 Equivalently, a risk averse person will always reject a fair gamble. Two essential characteristics: 1. An individual has wealth \(w\) and is afraid that an accident will occur with probability \(p\) that will cause him a loss of \(D\). Solutions to Problem Set 2. An individual faces the monetary lottery \(p\). Problems with solutions, Intermediate microeconomics, part 1 Niklas Jakobsson, nja@nova.no Katarina.Katz@kau.se Problem 1. PROBLEM SET 7, WITH SOLUTIONS 1. 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. <>
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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. %PDF-1.5
The parent has in hand only one gift. 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. 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’. Describe the ‘Allais paradox’, giving a specific example of a set of choices that illustrate this paradox. 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���! Consider Betty, a UK resident working at a company that ships goods from Britain to France. 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: 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 … In particular, there is some evidence (cite) that the Holt and Laury does not substantially predict real-world behavior. Choice under Uncertainty Please see (and present and give intuition for) formal presentations as given above. Microeconomic Theory I: Choice Under Uncertainty Parikshit Ghosh Delhi School of Economics September 8, 2014 Parikshit Ghosh Delhi School of Economics Choice Under Uncertainty. 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. Microeconomics Exercises with Suggested Solutions 5 7. %����
Explain why these choices are inconsistent with the standard theory of expected utility maximisation. Describe the lottery \(q\) that he faces if he accepts the offer. 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. 1 0 obj
Monopolistic Competition 10. The bookmaker offers odds that are seen as fair, and he only takes a small commission. 1. Problem Set 6. The Axiomatic Approach Critique Applications De–nitions and Axioms Lotteries I Set of outcomes: fa 1,a 2,...,a ng. Problem Set 9. 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.) These are also arbitrary and may be sensitive to the experimental framing. Problem Set 3. <>/ExtGState<>/Font<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 792 612] /Contents 4 0 R/Group<>/Tabs/S/StructParents 0>>
She owns a bak-ery that will be worth 69 or 0 dollars next year with equal probability. 1. 2. Please see lecture notes on Allais paradox, Allais paradox illustrated by a scenario such as. 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. (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). What sort of preferences would Betty have to have to make this advice worth following? Unlike the model in class, agents in this economy do have endowments, consume and trade in goods at t = 0. If she is risk-averse she prefers to reduce the variance of her returns, holding the expected value the same. In the video below, a teaching assistant demonstrates his approach to the solution for problem 5 from the problem set. Problem Set #1: Solutions 1. Does this depend on whether she can borrow or lend at the ‘risk-free’ rate? A lottery between a pro–t of $1000 with probability 25% and 0 with probability 75%. Choice under Uncertainty (cont’d). stream
Would the advice be the same for any risk-averse investor, or would it vary depending on her level of risk-aversion? Game Theory %DVLF&RQFHSWV 7.2 Games on Normal Form 7.3 Games on Extensive Form 8. 2 0 obj
Contact Us (413) 542-2000 Contact Us Map & Directions. b. Problem Set Questions (PDF) Problem Set Solutions (PDF) Problem Solving Video. 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’. 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. 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). How much depends on the grader’s discretion. and show that if he is strictly risk-averse he rejects the offer. Microeconomics Exercises 6 Suggested Solutions 1. 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. Insurance. Problem Set 5 Prof. Dr. Gerhard Illing, Jin Cao January 29, 2011 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 . I A gamble/lottery is a probability distribution over outcomes: g = (p 1 a 1,p 2 a 2,...,p n a n). Solutions to Problem Set 4. These measures, and the intuition for them, are discussed above. Production 'H¿QLWLRQV 3.2 The Production Function 4. Consumer Theory 1.1 Preferences 1.2 The Budget Line 1.3 Utility Maximization 2. Overview of module & rules, discussion/background, Intuition for ‘risk aversion iff concave value function. Show that the higher is \(\alpha\) the higher is the amount \(x\) he chooses. Problem Set 5 Solution Microeconomic Theory Chapters 11 and 12 ECON5110 | Fall 2019 1. MICROECONOMICS I: CHOICE UNDER UNCERTAINTY MARCINPĘSKI Please let me know about any typos, mistakes, unclear or ambiguous statements thatyoufind. PROBLEM SET 6, WITH SOLUTIONS 1. This allows her to reduce the variance of her returns for a given expected return, or increase the expected return for a given variance. De–ne the expected regret if the person chooses P rather than P0as X!2 ˇ! Labor 7KH6XSSO\RI/DERU 7KH'HPDQGIRU/DERU 11. 4.1 Consumer preferences, indifference curves/sets (0.5 weeks) 4.1.1 “Bundles of … Textbooks The course will draw mainly on the textbook: Riley, Essential Microeconomics, Cambridge University Press, 2012. Suggestedreadings. MWGchapter6.A.Kreps“NotesontheTheoryofChoice”, chapters4and7(thefirstpartonly). Problem Set 4 (graded) Problem Set 5. Uncertainty Advanced Microeconomics I Andras Niedermayer1 1Department of Economics, University of Mannheim Fall 2009 Chapter 3: Individual Choice Under Uncertainty Fall 2009 1 / 76. 2. Amherst College 220 South Pleasant Street Amherst, MA 01002. 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). Note that \(\lambda\) will determine, in effect, the ‘price’ of the insurance, per unit of compensation in the event of an accident. Two assignments per term will be marked. as well as the discussion of the CAPM model). maxfx0 x! There is a single consump- Risk aversion: The extent to which uncertainty of an outcome (holding the expected material or monetary value constant) implies an individual values it less. Solow model in continuous time. ]���1/��. Note: I can probably improve the notation in the above video. Therefore there are gains to be made from trading state claims. Note that expected utility requires the ‘independence’ property. will be a crucial learning tool. 1.2. 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. What would justify the economist’s advice to buy this asset? A parent has two children, A and B. endobj
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Describe one choice that a risk neutral person might make that a risk-averse person would never make. Use s = 0 to denote the date-event pair corresponding to date 0. If she ‘bets on leave’ this loss would be counterbalanced by an income gain from the asset. Let P:= f(x! 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. A parent. Uncertainty; Problem Set and Solutions. : !2 g P0:= f(x0! <>
If utility is differentiable we can define risk aversion in terms of a diminishing marginal utility of income (or in general, concavity). Note to Beem101 2020: this was part of a question on a previous exam. 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. This should hold in a perfectly competitive insurance market if there are no moral hazard or asymmetric information issues, no transactions costs, etc. 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 \(x\). Describingtheuncertainty. A choice must be made among various possible courses of actions. Problem Set Questions (PDF) Problem Set Solutions (PDF) Problem Solving Video. Breakdown of points: 10 for setting up the objective function correctly, 10 for solving the optimization problem correctly. Note that the sketched curves should also include the corners, which were not rendered well in the image below. Microeconomics CHAPTER 8. 4 0 obj
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. ;��J*��d� �}����sI���'���Y�V��E�b1�U��U}ɔh����5�-�ǹ|S!yy�pOw�t���EͯHyY���E ? Demand 2.1 Price Changes 2.2 Income Changes 2.3 Elasticities 3. <>>>
They will never take ‘fair bets’ and will refuse even some gambles that have a positive expected value. 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. Explain why the parent’s preferences are not consistent with expected utility. Problem Set #3: Solutions 1. Problem sets will be provided and answers to selected problems will be discussed during classes. Example 1. ;ˇ !) Because the individual paid \(x\) and the insurer must compensate him \(\lambda x\) with probability \(p\). Uncertainty Lotteries Expected Utility Money Lotteries Stochastic Dominance Lotteries A simple lottery can be represented as a point in simplex. Costs 4.1 Costs in the Short Run 4.2 Costs in the Long Run 5. A sure pro–t of $240. 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J Problem Set 5 Prof. Dr. Gerhard Illing, Jin Cao January 29, 2011 1 the DM is,. P ∈ R+N: p pi = 1 } is called a N-dimensional simplex: Riley, Essential,! Various uncertainty microeconomics problem set solution courses of actions what would justify the economist ’ s advice to buy this asset 2 -..