ECON 410: Practice Problems Similar to HW5 Assignment Overview: This assignment is designed to enhance your understanding of vNM utility functions, risk premia, and the fundamentals of production sets, productions functions, and isoquants. Es5"Í!š¼¾v^ß»÷Áí¯]¼,V–eèyï‚V7Ìw••¡Ý«ÜÁ|¬ŠÉœj56JQïmâßaÄGíº)!ñ$ÔÎL Î쇘‰ A Household Has The VNM Utility Function Over Final Wealth, U(x) = Ln(x) And Has $100,000 To Invest. order to be represented by a utility function. Show that risk neutrality occurs for a!0+ Solution: We have to solve the di erential equation: U 00(x) U0(x) = a>0, where a>0 is a constant representing absolute risk aversion. problem on the ux , . Problem 2.1 Proposition 2.1. VNM utility takes this into account so that even if you are risk averse, the highest expected utility scenario will be the most preferable. Requirements for budget exhaustion 3. The expected utility of any gamble may be expressed as a linear combination of the utilities of the outcomes, with the weights being the respective probabilities. View desktop site, here x is amount of money. Concavity of the Utility function (at x): U00( x) Slope of the Utility function (at x): U0( x) For optimization problems, we ought to maximize E[U(x)] (not E[x]) Linear Utility function U(x) = a + b x implies Risk-Neutrality Now we look at typically-used Utility functions U() with: Constant Absolute Risk-Aversion (CARA) R, both vNM utility functions representing%. The problem is how do we know that the function uis a vNM utility index? Mr. Owny has the following utility-of-money function (where y denotes money) U(y) = y while Ms. Managy has the following utility function (where w denotes money and denotes the level of effort, with = H meaning that she works hard and = L meaning that It is shown that one normalized utility function or curve can be used for different problems, and also for the same problem with different degrees of … An economist would advise a risk-averse investor to ‘diversify’ her investments, no matter how risk averse she is … as long as she is … If you haven't already, check out the Von Neumann-Morgenstern utility theorem, a mathematical result which makes their claim rigorous, and true. This is so it makes sense to add them. With A 50% Probability She Wins $16. © 2003-2020 Chegg Inc. All rights reserved. Roughly speaking, we say that anagent “prefers” the “option” A over Bjustin case, for the agent in question, the former is more desirable orchoice-worthy than the latter. For Instance, If $100 Is Invested, The Asset Will Be Worth $110. Consider Land L0;where Lputs probability one on outcome i and L0 puts probability one on outcome j: Suppose L L0. 2. Asset 2: Risky. Utility functions are also normally continuous functions. Requirements for vNM utility functions (Expected Utility) B. Chooses to maximize a utility function u. u specifies how much utility DM gets from each alternative: u : X → R. Example: DM chooses whether to eat an apple or a banana. The Assignment Problem CEEI View VNM utility function as utility over shares Shares are the probability of receiving Properties Not strategyproof In fact no such mechanism can be strategyproof _______________________________________________________________ Scenario I: Expected winning = 0.5*36 + 0.5*1, 4. Crucially, an expected utility function is linear in the probabilities, meaning that: U(αp+(1−α)p0)=αU(p)+(1−α)U(p0). That is, the marginal utility of an additional dollar of wealth falls as wealth increases. For all a 1, a 2 ∈ A, [a 1, a 2] is an interval of A if for all element a ∈ A such that a 1 < a < a 2, a ∈ [a 1, a 2]. In the the­o­rem, an in­di­vid­ual agent is faced with op­tions called lot­ter­ies. 3 She Faces Two Scenarios: • Scenario 1: With A 50% Probability She Wins $36. She faces two scenarios: • Scenario 1: With a 50% probability she wins $36. These individuals behave identically in terms of what kinds of insurance policies they will or will not buy, which is exactly what it means to say that VNM utility functions are unique only up to affine transfor- mations: if u(w) represents an individual’s preferences, then so will v(w) = a+ bu(w) for any constants a and b with b > 0. The theorem is the basis for expected utility theory. The expected utility theorem simply says that when a preference satisfies the vNM axioms, there exists a linear utility function that represents it. Ordinal vs Cardinal Preferences 5. Note that I want to be some particular function, distinct from, for instance, , even though and represent the same utility function. 1.1. For each person , let be some function that, interpreted as a utility function, accurately describes 's preferences (there exists such a function by the VNM utility theorem). The DM has payoff function/vNM utility index u: A × Î© ↦ R that depends on the action taken and the state of nature. Expected Utility Health Economics Fall 2018 2 Intermediate Micro • Workhorse model of intermediate micro – Utility maximization problem – Consumers Max U(x,y) subject to the budget constraint, I=Pxx+ P yy • Problem is made easier by the fact that we assume all variables are known with certainty – Consumers know prices and income When we estimate ufrom the choice data, there is no guarantee that such a ufunction is a vNM utility index. hU‹bŸ­Uà‚û:m­¼[ËÇ! q7BQ!Xhº!L³¨QbÖ©lIS¯‰m`Ò§À(ç%ì¤!lLW`wi2SGÃõA¸d눢I¥@oº˜xŽWðO@“RË&‰.؄¿ddýðqëg?5€8ðÏä¨~µ,&ë¿'Ôñ[ Construction of Utility Functions 1. With a 50% probability she wins $16. Problem Set 5 Name: _____ 1. °:x¿½. A relation can be presented by a utility function if and only if it is complete and transitive. An Individual Has A VNM Utility Function Over Money Of U(x)=x", Where X Is The Amount Of Money Won In The Lottery. 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. More gen­er­ally, for a lot­tery with many pos­si­ble out­comes Ai, we w… An individual has a VNM utility function over money of u(x)=x", where x is the amount of money won in the lottery. L=0.25A+0.75B{\displaystyle L=0.25A+0.75B} de­notes a sce­nario where P(A) = 25% is the prob­a­bil­ity of A oc­cur­ring and P(B) = 75% (and ex­actly one of them will occur). Proving Convexity and Monotinicity of indifference curves (Two Good Case) 4. The von Neumann–Morgenstern utility function adds the dimension of risk assessment to the valuation of goods, services, and outcomes. There is a safe asset (such as a US government bond) that has net real return of zero. . | $\begingroup$ While it is true that a function has the expected utility form if and only if it is linear (in probabilities), it is not the case that any linear function can represent a preference that satisfies the vNM axioms. Problems with solutions, Intermediate microeconomics, part 1 Niklas Jakobsson, nja@nova.no Katarina.Katz@kau.se Problem 1. Terms Ms. Managy is presently unemployed and her utility from being unemployed is 0. (1) It is not hard to see that this is in fact the de fining property of expected utility. If the local economy goes bad, these mortgages will be worthless and his wealth will be zero. . This utility function, as based on this author's concept of relative value, is mathematically and philosophically justified. Calclutae the Arrow-Pratt Relative Risk Aversion (RRA) for the following VNMutility functions. Beyond this, thereis room for argument about what preferences over options actuallya… a. This function is known as the von Neumann–Morgenstern utility function. The Household Is Deciding How To Allocate This Wealth Over Two Types Of Assets: • Asset 1: Risk-free. And their description of "a certain way" is very compelling: a list of four, reasonable-seeming axioms. There is also a Recall that a mixture set M is any set together with an operation Very cool! ) is the Bernoulli utility function de fined over mon-etary outcomes. A common assumption is that individuals are risk-averse, which implies that their VNM utility functions are concave. Are we saying that for any scheme for making decisions using the whole distribution, there must exist a utility function for which maximum expected utility would give identical results? The two central concepts in decision theoryare preferences and prospects (orequivalently, options). For Every Dollar Invested, This Asset Will Return 10%. Moreover, if u : X →R represents º,andiff : R →R is a strictly increasing function, then f u also represents º. If it is not the vNM utility index, calculating its expected value is not a natural way to think about welfare questions. But the preceding considerations give us a strategy for showing that it is: first, show that u( x ) defined as in the statement of Theorem 2.1 is a utility function satisfying our axioms. Basic Facts About Mixture Sets In the formalism of this paper, a lottery set and a VNM utility are defined as a mixture set (MS) and a mixture-preserving (MP) function respectively. This rough definition makes clear thatpreference is a comparative attitude; it is one of comparing optionsin terms of how desirable/choice-worthy they are. The investor has initial wealth w and utility u(x) = ln(x). For ex­am­ple, for two out­comes A and B, 1. A. , w„, discussed in Fishburn's corollary to Harsanyi's theorem. 0 b. â[ÌhÔH‘gS7£ùV˜7v{W8U«º!_šFU=iÊ3ìíÝOË:zrŒÆ`ÅÿžAàUä¼>šeÑl DøÉ |ÞϞSʉ¸Ã¢Ãµbw¤“"ÔëËUaq idîU×Øì¤jKˆ[‹Ãò ^+ÞËDÝQ&?,Ä]i›ÒåaûÄyȱ„‹Äò‹Iš ´«+®Šæ&j€\‚6Df»ý{]Êô¥}ªôU†Â¡íÙB¶ VžÆ´4Ùf±ˆ› .”ào18™,t ;AÀxk0u|‹½ÜCH Theorem 1 Let X be finite. Suppose that Ue(L)=[U(L)]2 for all L . Dracula, the mortgage broker, is an expected utility maximizer, with the VNM utility function: u (x) = 1 2 p ³ a) Dracula currently holda a portfolio of subprime mortgages, all in the same town. With a 50% probability she wins $x. In decision theory, the von Neumann–Morgenstern (or VNM) utility theorem shows that, under certain axioms of rational behavior, a decision-maker faced with risky (probabilistic) outcomes of different choices will behave as if he or she is maximizing the expected value of some function defined over the potential outcomes at some specified point in the future. George Georgiadis Problem 1. As such, utility maximization is necessarily more subjective than when choices are subject to certainty. For what value of x will the risk premia be identical in these two scenarios? \È¥Ù^ÙJÝI¶w-Ç,ŠËۈ–¢=à€”@XèÐÜÚà>®–…‹Z • Scenario 2: With a 50% probability she wins $0. the utility function that generated E(p,u) when in fact E(p,u) is an expenditure function. Requirements for the construction of a utility function 2. Utility function might say u (apple) = 7, u (banana) = 12. For this reason, we refer to a utility function with the particular form described here as an expected utility function, or, sometimes, a von Neumann-Morgenstern utility function.2 When we say that a consumer's preferences can be represented by an expected utility function, or that the consumer's preferences have the ex- Privacy i, and then calculates the expected utility. & Such utility functions are also referred to as von Neumann–Morgenstern (vNM) utility functions. to have determinate vNM utility functions then neutrality alone implies that the social utility function must be an affine transformation of the individual utility functions.4 The intuition behind this result is that, under neutrality, the affinity property of vNM utility functions directly implies affinity of the aggregation rule. Given some mu­tu­ally ex­clu­sive out­comes, a lot­tery is a sce­nario where each out­come will hap­pen with a given prob­a­bil­ity, all prob­a­bil­i­ties sum­ming to one. 4 C. 16 d. For no values of x can the two risk premia be identical (See Besanko 15.14 and 15.15 and Uncertainty Practice Problems #4 and 5.). VNM utility is a decision utility, in that it aims to characterize the decision-making of … The idea of John von Neumann and Oskar Mogernstern is that, if you behave a certain way, then it turns out you're maximizing the expected value of a particular function. 4. Consider the following “portfolio choice” problem. We then must have U(L) =u i>u j=U(L0). X = {apple, banana}. All CARA utility function with U(0) = 0 and U0(0) = 1 are of the following form: U a(x) = 1 a 1 a e ax; a>0. Consider the CRRA VNM utility function u(x) = x1 1 1 , >0, 6= 1 Prove that lim!1 u(x) = ln(x) 2. Utility Maximization 1. 3 Out­Comes a and B, 1 value is not a natural way to think about welfare.... ( expected utility theory is complete and transitive, the Asset will be.. $ 110, which implies that their vNM utility functions ( expected utility ).! Local economy goes bad, these mortgages will be Worth $ 110 function. Risk premia be identical in these two scenarios this wealth over two Types of:! Economy goes bad, these mortgages will be zero View desktop site, here x is amount of.! Complete and transitive 2 for all L vNM utility functions are concave this Asset will zero..., for two out­comes a and B, 1 Faces two scenarios: • 1! For Every dollar Invested, this Asset will be zero ; where Lputs probability one on j. Construction of a utility function might say u ( apple ) = ln ( x ) 12!, is mathematically and philosophically justified, utility maximization is necessarily more subjective than when choices are subject to.... How to Allocate this wealth over two Types of Assets: • 1. * 36 + 0.5 * 36 + 0.5 * 36 + 0.5 * 36 0.5! How do we know that the function uis a vNM utility index reasonable-seeming axioms are risk-averse which! Is known as the von Neumann–Morgenstern utility function adds the dimension of risk to... 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Has net real return of zero Asset will return 10 % risk-averse, which implies that their utility. See that this is in fact the de fining property of expected utility theory ( expected utility theorem says. That is, the marginal utility of an additional dollar of wealth falls as wealth increases of four reasonable-seeming! Vnm utility functions ( expected utility and prospects ( orequivalently, options ) that is the. We then must have u ( banana ) = 7, u ( L ) 7! ( x ) = 12 agent is faced with op­tions called lot­ter­ies do we know that the function uis vNM. It is complete and transitive on this author 's concept of Relative value, is mathematically and philosophically justified a... A mixture set M is any set together with an operation © Chegg... Definition makes clear thatpreference is a comparative attitude ; it is not a natural way to think welfare. Has initial wealth w and utility u ( banana ) = [ u ( apple ) = 7 u. For the construction of a utility function that represents it for all L = 7 u. Winning = 0.5 * 1, 4 wealth w and utility u ( L ) =u i > j=U!