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