(See discussion under ‘benefits of diversification’ He is made the following offer. Two essential characteristics: 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 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 \(\frac{1}{2}\). Explain why an economist would advise a risk-averse investor to `diversify’ her investments. 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. Use s = 0 to denote the date-event pair corresponding to date 0. as well as the discussion of the CAPM model). Describe the ‘Allais paradox’, giving a specific example of a set of choices that illustrate this paradox. Two assignments per term will be marked. Econ 101A, Microeconomic Theory Fall 2009. What sort of preferences would Betty have to have to make this advice worth following? Gamble D: a 90 percent chance of winning nothing and a 10 percent chance of winning £ 5 million. 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’. Would the advice be the same for any risk-averse investor, or would it vary depending on her level of risk-aversion? 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!! • Please put your name, student ID & your GSI’s name at the upper right corner of the front page. A risk averse person always prefers the expected monetary value of a gamble to the gamble itself. 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. Deﬁnition: The set ∆ = {p ∈ R+N: P pi = 1} is called a N-dimensional simplex. PROBLEM SET 7, WITH SOLUTIONS 1. Please assume, of course, that this is indeed the probability that such an accident will occur. <>/ExtGState<>/Font<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 792 612] /Contents 4 0 R/Group<>/Tabs/S/StructParents 0>> A parent has two children, A and B. and show that if he is strictly risk-averse he rejects the offer. Insurance. Problem Set 8. Microeconomics Exercises 6 Suggested Solutions 1. Define ‘risk aversion’. PROBLEM SET 6, WITH SOLUTIONS 1. 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. Gamble C: an 89 percent chance of winning nothing and an 11% chance of winning 1 million. In May 2016, an economist (Al) advises Betty that if the UK votes ‘leave’ in the June referendum, this may reduce trade with France. Because the individual paid \(x\) and the insurer must compensate him \(\lambda x\) with probability \(p\). Problem sets will be provided and answers to selected problems will be discussed during classes. Breakdown of points: 10 for setting up the objective function correctly, 10 for solving the optimization problem correctly. ;ˇ ) : !2 g be two prospects available to an individual. 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! . 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. Describingtheuncertainty. Solutions to Problem Set 2. 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. A parent. Problem Set Questions (PDF) Problem Set Solutions (PDF) Problem Solving Video. 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). A sure pro–t of $240. 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. What would justify the economist’s advice to buy this asset? 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: Note that the sketched curves should also include the corners, which were not rendered well in the image below. All lower case letters denote logarithmic terms. Amherst College 220 South Pleasant Street Amherst, MA 01002. 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 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. 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. (Continuation of Problem 2 from Problem Set 5.) 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. An individual has wealth \(w\) and is afraid that an accident will occur with probability \(p\) that will cause him a loss of \(D\). A company develops a product of an unknown quality. (Think of these as millions of dollars if you like.) Other measures include specific empirical elicitations/comparisons as those done in experiments, such as Holt and Laury discussed here. 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). The level set for Alex is also depicted. The bookmaker offers odds that are seen as fair, and he only takes a small commission. 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 . On the other hand if leave does not pass she keeps her job, but loses the bet. They will never take ‘fair bets’ and will refuse even some gambles that have a positive expected value. Under uncertainty, the DM is forced, in eﬀect, to gamble. 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. 2. This is referred to as ‘actuarially fair insurance’. One possibility is that it is too complicated and analytical for most people to handle or to take seriously given low stakes. Problem Set 5 Solution Microeconomic Theory Chapters 11 and 12 ECON5110 | Fall 2019 1. 2. Demand 2.1 Price Changes 2.2 Income Changes 2.3 Elasticities 3. Microeconomics Exercises with Suggested Solutions 5 7. 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. Let P:= f(x! UNCERTAINTY AND RISK Exercise 8.6 An example to illustrate regret. 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’. 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. Pro t in terms of the labor choice is ˇ= TR TC= TR(y(L)) w LL: Show that the higher is \(\alpha\) the higher is the amount \(x\) he chooses. These are also arbitrary and may be sensitive to the experimental framing. 1. Explain why the parent’s preferences are not consistent with expected utility. 1.2. ]���1/��. Solutions to Problem Set 4. 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. 3 0 obj ‘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). Does this depend on whether she can borrow or lend at the ‘risk-free’ rate? 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. Define risk aversion formally and intuitively. Perfect Competition 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. stream Consumer Theory 1.1 Preferences 1.2 The Budget Line 1.3 Utility Maximization 2. Problem Set 7. There is a single consump- 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. 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. 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. 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. Costs 4.1 Costs in the Short Run 4.2 Costs in the Long Run 5. MWGchapter6.A.Kreps“NotesontheTheoryofChoice”, chapters4and7(theﬁrstpartonly). Describe one choice that a risk neutral person might make that a risk-averse person would never make. Solutions to Problem Set 5 ECON 302 - Microeconomic Theory II: Strategic Behavior IRYNA DUDNYK Tutorial 7. Note that expected utility requires the ‘independence’ property. or consider the measurement of risk (certainty equivalent, Arrow-Pratt measures, etc.). 4 0 obj De–ne the expected regret if the person chooses P rather than P0as X!2 ˇ! : !2 g P0:= f(x0! x���]o�6���?�RZ�$J��^t�Z�*"+��X�ly@����%��|�7�: sӇ���sHy�j߷�Uݳ\����~h��v��}�c����Y~�6mW���[~=���W?7պ���{�� [~��������".x�b���W�)��?/Ҳ`���j�q[m���ݱ߮�^��o�&2^*������`*�ˊ��������~�*b;�`��n�O��&���"�v����v��,ڶ5[��V�\_�[ ���U��6Z=,n�����h��R/rԅ4��]�f���! Uncertainty; Problem Set and Solutions. Microeconomics - 1. (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). Suggestedreadings. The probabilities are denoted by p 1, p 2 and p 3 respectively. Choice Under Uncertainty: Problem Set 1. Problem Set Questions (PDF) Problem Set Solutions (PDF) Problem Solving Video. Labor 7KH6XSSO\RI/DERU 7KH'HPDQGIRU/DERU 11. Ana’s utility function is U = p w, where wis her wealth. The consumers will reject any proposed exchange that does not lie in their shaded superlevel set s. line 400 800 line 600 200 Production 'H¿QLWLRQV 3.2 The Production Function 4. 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. endobj Problems with solutions, Intermediate microeconomics, part 1 Niklas Jakobsson, nja@nova.no Katarina.Katz@kau.se Problem 1. Problem Set 3. Problem Set 5 Prof. Dr. Gerhard Illing, Jin Cao January 29, 2011 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. If utility is differentiable we can define risk aversion in terms of a diminishing marginal utility of income (or in general, concavity). Microeconomic Theory I: Choice Under Uncertainty Parikshit Ghosh Delhi School of Economics September 8, 2014 Parikshit Ghosh Delhi School of Economics Choice Under Uncertainty. <> Explain why or why not, referring to equations and diagrams as needed. endobj 14.772 Macro Development - Problem Set 2 Spring 2013 Problem 1: Risk Sharing Consider H households, with household h consisting of I h members. Please see (and present and give intuition for) formal presentations as given above. The parent has in hand only one gift. MICROECONOMICS I: CHOICE UNDER UNCERTAINTY MARCINPĘSKI Please let me know about any typos, mistakes, unclear or ambiguous statements thatyouﬁnd. Microeconomics CHAPTER 8. 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. Please see lecture notes on Allais paradox, Allais paradox illustrated by a scenario such as. Problem Set 1. Problem Set #1: Solutions 1. In the video below, a teaching assistant demonstrates his approach to the solution for problem 5 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. This should hold in a perfectly competitive insurance market if there are no moral hazard or asymmetric information issues, no transactions costs, etc. Problem Set #3: Solutions 1. Choice under Uncertainty Jonathan Levin October 2006 1 Introduction Virtually every decision is made in the face of uncertainty. 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, \(\rightarrow U(1m) > 0.89 \: U(1m) + 0.1 \: U(5m) + 0.01 \: U(0)\), \(0.11 \: U(1m) > 0.1 U(5m) + 0.01 \: U(0)\), \(\rightarrow 0.9 \: U(0) + 0.1 U(5m) > 0.89 \: U(0) + 0.11 \: U(1m)\), \(0.1 \: U(5m) + 0.01 \: U(0) > 0.11 \: U(1m)\). 2 0 obj 1 0 obj An exchange economy has two dates t =0,1 and two states of nature s =1,2 which will be revealed at date 1. 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 … %PDF-1.5 More formal definitions, depictions, and intuition is given in this web book above. 1. Choice under Uncertainty If she ‘bets on leave’ this loss would be counterbalanced by an income gain from the asset. She can optimise along this margin by ‘optimally diversifying’, buying assets in proportion to their representation (relative value) in the market. 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. Equivalently, a risk averse person will always reject a fair gamble. Note: I can probably improve the notation in the above video. (a) Suppose her rm is the only asset she has. Textbooks The course will draw mainly on the textbook: Riley, Essential Microeconomics, Cambridge University Press, 2012.

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