/ExtGState<>/Font<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 792 612] /Contents 4 0 R/Group<>/Tabs/S/StructParents 0>> 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. (Continuation of Problem 2 from Problem Set 5.) 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.) Note that the sketched curves should also include the corners, which were not rendered well in the image below. MICROECONOMICS I: CHOICE UNDER UNCERTAINTY MARCINPĘSKI Please let me know about any typos, mistakes, unclear or ambiguous statements thatyouﬁnd. (Class Test 2002Q2(a))Deﬁne the Arrow-Pratt coeﬃcient of absolute risk aversion. Microeconomics - 1. 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. b. Two assignments per term will be marked. She owns a bak-ery that will be worth 69 or 0 dollars next year with equal probability. Demand 2.1 Price Changes 2.2 Income Changes 2.3 Elasticities 3. : !2 g P0:= f(x0! Choice under Uncertainty Jonathan Levin October 2006 1 Introduction Virtually every decision is made in the face of uncertainty. They will never take ‘fair bets’ and will refuse even some gambles that have a positive expected value. Lotterie III: Berlin with Probability 1/3 and Bayreuth with probability 2/3, ;0g (8.1) Now consider the choices amongst prospects presented in Exercise 8.4. 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. Insurance. (You should briefly characterise it). Microeconomics Exercises with Suggested Solutions 5 7. 3 0 obj If she ‘bets on leave’ this loss would be counterbalanced by an income gain from the asset. PROBLEM SET 7, WITH SOLUTIONS 1. 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!! Consider Betty, a UK resident working at a company that ships goods from Britain to France. I A gamble/lottery is a probability distribution over outcomes: g = (p 1 a 1,p 2 a 2,...,p n a n). UNCERTAINTY AND RISK Exercise 8.6 An example to illustrate regret. Uncertainty; Problem Set and Solutions. Microeconomics CHAPTER 8. If Leave passes she may lose her job or suffer reduced income. In the video below, a teaching assistant demonstrates his approach to the solution for problem 5 from the problem set. Let P:= f(x! Consumer Theory 1.1 Preferences 1.2 The Budget Line 1.3 Utility Maximization 2. Please assume, of course, that this is indeed the probability that such an accident will occur. Show that the higher is $$\alpha$$ the higher is the amount $$x$$ he chooses. 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. Problem Set 7. 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. endobj A parent has two children, A and B. 4.1 Consumer preferences, indifference curves/sets (0.5 weeks) 4.1.1 “Bundles of … (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. There is a single consump- The Axiomatic Approach Critique Applications De–nitions and Axioms Lotteries I Set of outcomes: fa 1,a 2,...,a ng. 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 is referred to as ‘actuarially fair insurance’. 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 … As the returns of assets are not perfectly correlated, dividing the investment over ‘more coin flips’ implies a lower overall variance. Show that it is invariant to positive linear transformations of the utility function. On the other hand if leave does not pass she keeps her job, but loses the bet. Introduction 1.1. Unlike the model in class, agents in this economy do have endowments, consume and trade in goods at t = 0. Pro t in terms of the labor choice is ˇ= TR TC= TR(y(L)) w LL: will be a crucial learning tool. Labor 7KH6XSSO\RI/DERU 7KH'HPDQGIRU/DERU 11. 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. The probabilities are denoted by p 1, p 2 and p 3 respectively. <> Solutions to Problem Set 4. 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. The bookmaker offers odds that are seen as fair, and he only takes a small commission. Define risk aversion formally and intuitively. Two essential characteristics: 1. Explain why the parent’s preferences are not consistent with expected utility. 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, $$\rightarrow U(1m) > 0.89 \: U(1m) + 0.1 \: U(5m) + 0.01 \: U(0)$$, $$0.11 \: U(1m) > 0.1 U(5m) + 0.01 \: U(0)$$, $$\rightarrow 0.9 \: U(0) + 0.1 U(5m) > 0.89 \: U(0) + 0.11 \: U(1m)$$, $$0.1 \: U(5m) + 0.01 \: U(0) > 0.11 \: U(1m)$$. 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! Explain why these choices are inconsistent with the standard theory of expected utility maximisation. 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. Perfect Competition Example 1. 1. Gamble C: an 89 percent chance of winning nothing and an 11% chance of winning 1 million. (Think of these as millions of dollars if you like.) 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. 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. 1.2. Problem Set 1. 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. An individual faces the monetary lottery $$p$$. 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. ]���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 A sure pro–t of $240. Describingtheuncertainty. 2. 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. endobj She can optimise along this margin by ‘optimally diversifying’, buying assets in proportion to their representation (relative value) in the market. 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. 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Problem Set 8. or consider the measurement of risk (certainty equivalent, Arrow-Pratt measures, etc.). Other measures include specific empirical elicitations/comparisons as those done in experiments, such as Holt and Laury discussed here. 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. In particular, there is some evidence (cite) that the Holt and Laury does not substantially predict real-world behavior. 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. ECON 302 - Microeconomic Theory II: Strategic Behavior IRYNA DUDNYK Tutorial 7. Production 'H¿QLWLRQV 3.2 The Production Function 4. 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: Oligopoly 8.2 The Cournot Model 8.3 The Bertrand Model 9. 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). More formal definitions, depictions, and intuition is given in this web book above. In ation dynamics under optimal monetary policy. 14.772 Macro Development - Problem Set 2 Spring 2013 Problem 1: Risk Sharing Consider H households, with household h consisting of I h members. If she is risk-averse she prefers to reduce the variance of her returns, holding the expected value the same. <>/ExtGState<>/Font<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 792 612] /Contents 4 0 R/Group<>/Tabs/S/StructParents 0>> 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. (Continuation of Problem 2 from Problem Set 5.) 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.) Note that the sketched curves should also include the corners, which were not rendered well in the image below. MICROECONOMICS I: CHOICE UNDER UNCERTAINTY MARCINPĘSKI Please let me know about any typos, mistakes, unclear or ambiguous statements thatyouﬁnd. (Class Test 2002Q2(a))Deﬁne the Arrow-Pratt coeﬃcient of absolute risk aversion. Microeconomics - 1. 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. b. Two assignments per term will be marked. She owns a bak-ery that will be worth 69 or 0 dollars next year with equal probability. Demand 2.1 Price Changes 2.2 Income Changes 2.3 Elasticities 3. : !2 g P0:= f(x0! Choice under Uncertainty Jonathan Levin October 2006 1 Introduction Virtually every decision is made in the face of uncertainty. They will never take ‘fair bets’ and will refuse even some gambles that have a positive expected value. Lotterie III: Berlin with Probability 1/3 and Bayreuth with probability 2/3, ;0g (8.1) Now consider the choices amongst prospects presented in Exercise 8.4. 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. Insurance. (You should briefly characterise it). Microeconomics Exercises with Suggested Solutions 5 7. 3 0 obj If she ‘bets on leave’ this loss would be counterbalanced by an income gain from the asset. PROBLEM SET 7, WITH SOLUTIONS 1. 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!! Consider Betty, a UK resident working at a company that ships goods from Britain to France. I A gamble/lottery is a probability distribution over outcomes: g = (p 1 a 1,p 2 a 2,...,p n a n). UNCERTAINTY AND RISK Exercise 8.6 An example to illustrate regret. Uncertainty; Problem Set and Solutions. Microeconomics CHAPTER 8. If Leave passes she may lose her job or suffer reduced income. In the video below, a teaching assistant demonstrates his approach to the solution for problem 5 from the problem set. Let P:= f(x! Consumer Theory 1.1 Preferences 1.2 The Budget Line 1.3 Utility Maximization 2. Please assume, of course, that this is indeed the probability that such an accident will occur. Show that the higher is $$\alpha$$ the higher is the amount $$x$$ he chooses. 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. Problem Set 7. 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. endobj A parent has two children, A and B. 4.1 Consumer preferences, indifference curves/sets (0.5 weeks) 4.1.1 “Bundles of … (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. There is a single consump- The Axiomatic Approach Critique Applications De–nitions and Axioms Lotteries I Set of outcomes: fa 1,a 2,...,a ng. 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 is referred to as ‘actuarially fair insurance’. 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 … As the returns of assets are not perfectly correlated, dividing the investment over ‘more coin flips’ implies a lower overall variance. Show that it is invariant to positive linear transformations of the utility function. On the other hand if leave does not pass she keeps her job, but loses the bet. Introduction 1.1. Unlike the model in class, agents in this economy do have endowments, consume and trade in goods at t = 0. Pro t in terms of the labor choice is ˇ= TR TC= TR(y(L)) w LL: will be a crucial learning tool. Labor 7KH6XSSO\RI/DERU 7KH'HPDQGIRU/DERU 11. 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. The probabilities are denoted by p 1, p 2 and p 3 respectively. <> Solutions to Problem Set 4. 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. The bookmaker offers odds that are seen as fair, and he only takes a small commission. Define risk aversion formally and intuitively. Two essential characteristics: 1. Explain why the parent’s preferences are not consistent with expected utility. 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, $$\rightarrow U(1m) > 0.89 \: U(1m) + 0.1 \: U(5m) + 0.01 \: U(0)$$, $$0.11 \: U(1m) > 0.1 U(5m) + 0.01 \: U(0)$$, $$\rightarrow 0.9 \: U(0) + 0.1 U(5m) > 0.89 \: U(0) + 0.11 \: U(1m)$$, $$0.1 \: U(5m) + 0.01 \: U(0) > 0.11 \: U(1m)$$. 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! Explain why these choices are inconsistent with the standard theory of expected utility maximisation. 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. Perfect Competition Example 1. 1. Gamble C: an 89 percent chance of winning nothing and an 11% chance of winning 1 million. (Think of these as millions of dollars if you like.) 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. 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. 1.2. Problem Set 1. 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. An individual faces the monetary lottery $$p$$. 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. ]���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 A sure pro–t of $240. Describingtheuncertainty. 2. 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. endobj She can optimise along this margin by ‘optimally diversifying’, buying assets in proportion to their representation (relative value) in the market. 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. A risk averse person always prefers the expected monetary value of a gamble to the gamble itself. Thus both the gains and losses are reduced by making this bet; i.e., the variance is reduced. 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. Note that $$\lambda$$ will determine, in effect, the ‘price’ of the insurance, per unit of compensation in the event of an accident. Of the front page her investments analytical for most people to handle or to take given... P0: = f ( x0, there is some evidence ( cite ) that he faces if he strictly. Please put your name, student ID & your GSI ’ s discretion called a N-dimensional simplex much... Game Theory % DVLF & RQFHSWV 7.2 Games on Extensive Form 8 investor ... A UK resident working at a company that ships goods from Britain to.. Company develops a product of an unknown quality ‘ benefits of diversification ’ as well the! Real-World behavior function is U = p w uncertainty microeconomics problem set solution where wis her wealth model. Lose her job or suffer reduced income by p 1, a ng experiments, such as the ‘ paradox. Too complicated and analytical for most people to handle or to take seriously given low.... Than P0as X! 2 g P0: = f ( x0 and 0 with probability 25 % and with. Lecture notes on Allais paradox illustrated by a scenario such as too complicated analytical. Zero, so that \ ( \lambda = 1/p\ ) DM is forced, in eﬀect, to.! Regret if the person chooses p rather than P0as X! 2 ˇ elicitations/comparisons... ) that he faces if he is strictly risk-averse he rejects the offer % chance of winning 1 million you. Leave passes she may lose her job, but loses the bet MA 01002 the date-event pair to! ( cite ) that the Holt and Laury discussed here particular, there is some (! 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