Summer 2003 (420 2)

Transcription

1 Microeconomics 3 Andreas Ortmann, Ph.D. Summer 2003 (420 2) andreas.ortmann@cerge-ei.cz Week of May 12, lecture 3: Expected utility theory, continued: Risk aversion Key reading: MWG 6C Supplementary sources: Binmore (1992), Kreps (1990) Assignments: [for May 20] MWG 6D [for May 20] Problem set # 2 ~ downloadable Wednesday noon from home.cergeei.cz/babicky/micro3. There you will also find Thursday afternoon the lecture notes for the week. [for May 20] Brown Kruse and Thompson (2001), A comparison of salient rewards in experiments: money and class points. Economics Letters 74, [for May 20] Holt and Laury (2002), Risk Aversion and Incentive Effects. American Economic Review 92, [for May 20] Harrison, Johnson, McInnes, and Rutstroem (2003), Risk Aversion and Incentive Effects: Comment. Manuscript. [for May 20] Berg, Dickhaut, and Rietz (1999), On the performance of the Lottery Procedure for Controlling Risk Preferences. Manuscript. [for May 22] 6.E., 6.F. [for May 22] Starmer (2000), Developments in Non-Expected Utility Theory: The Hunt for a Descriptive Theory of Choice under Risk. Journal of Economic Literature 38, [leisure reading] Hertwig and Ortmann (2001), Experimental practices in economics: A methodological challenge for psychologists? Behavioral and Brain Sciences 24, [Includes commentaries of 34 economists and psychologists and our response to them.] New articles to write four-page paper on: Bellemare and Kroeger (2003), On Representative Trust. Manuscript. Brandts and Charness (2003), Truth or Consequences: An experiment. Management Science, 49.1.,

2 2 [soon on reserve] [Commandments for 4-page paper, see on my website the course materials section.] Guiding questions for Cox & Grether (1996), The preference reversal phenomenon: Response mode, markets and incentives. 1. What is the preference reversal phenomenon (PRP from here on)? What two kinds of bets does it involve? The lotteries or lotteries or bets involved are the P bet and the $ bet. The P-bet is a gamble that has a high probability of winning a small amount of money. The $ bet is a gamble that has a low probability of winning a large amount of money. Subjects are asked to choose between the two gambles (and they typically end up choosing the P bet). Subjects are also asked to through various modes of elicitation such as the BDM method to place monetary values on the two bets (and they typically end up putting the higher value on the$ bet).. 2. What is a predicted reversal as opposed to an unpredicted reversal? [Hint: See pp. 382, 386] A predicted reversal occurs if subjects select the P bet but place a higher monetary value on the $ bet. See p. 386 for the (two) pairs of gambles used in this study. Note that that s not many. You can think of this as a framing effect. 3. What puzzle motivates Cox & Grether to study this phenomenon? [Hint: p. 382] What puzzles them is that while individual choice behavior seems often as in the case of PRP inconsistent with EUT in ways that are systematic and replicable, in market experiments behavior typically seems in line with predictions derived from models that often are based on EUT (e.g., auctions - see the Gode and Sunder reference ) even if financial incentives are low-powered. 4. What could be the reason for this curious and somewhat disturbing discrepancy (that robust observations of deviations from the benchmark EUT model are not robust in market experiments)? What do Cox & Grether find? [Hint: See pp and summary and conclusions.] Feedback, repetition, psychologically different tasks, institutions, information, economic incentives. Response mode. 5. What does the response mode in the title of the Cox & Grether piece refer to? How does the nonmarket choice task work? How does the BDM procedure differ from the one that I used to elicit your risk preference in decision task 1 on decision questionnaire 1? [Hint: See pp ] How do the auctions work that Cox & Grether employ? And what exactly is their function? To the various nonmarket (choosing most preferred item from each of three pairs... ; BDM

3 procedure) and market (English clock auction; second price sealed bid auction) choice and pricing tasks, respectively, that Cox & Grether used. The nonmarket choice task consisted of subjects having to choose the most preferred from each of three pairs: {P bet, $ bet}, {P bet, $ X}, {$ bet, $X} where $X was set equal to the midpoint of the certainty equivalents which themselves were elicited through BDM mechanism and the two auctions. BDM procedure - subjects were given the right to play a gamble and asked to state their minimum selling price. Random prices were determined by drawing balls from a bingo cage. (Why is that procedure incentive compatible?) The BDM procedure is finer tuned in that it allows to compute the degree of risk aversion. Of course, one could construct series of gambles as Holt & Laury did and get a similar result. Auctions - questions? Note that while both auctions are market mechanisms, the response mode in the clock auction consists of choices (because each subject must decide whether to choose to play the bet or to stay in the auction), whereas the response mode in the sealed bid auction consists of stating prices. 6. What does the incentives reference in the title of the Cox and Grether piece refer to? What curious problem did arise initially? The three levels of subject payments: full, diluted, and flat fee. See p. 387, see also p. 392 about what went wrong initially with the third payment mode. 7. Looking at Table 1 (p. 388):.1.Compare the PRs for BDM1, CR1, CR.5, and CR0. What do you note?.2. Compare the PRs for BDM5, CR1, CR.5, and CR0. What do you note?.3. Compare your preceding two observations. What do you note?.4. Now compare the PRs for SPA1, CR1, CR.5, and CR0. What do you note?.5. Now compare the PRs for SPA5, CR1, CR.5, and CR0. What do you note?.6. Compare your preceding two observations. What do you note?.7. Finally, compare the PRs for ECA1, CR1, CR.5, and CR0. What do you note?.8. Finally, compare the PRs for ECA5, CR1, CR.5, and CR0. What do you note?.9. Compare your preceding two observations. What do you note?.10. Under what conditions is the effect of high-powered incentives the largest? 8. What are the key insights to take from this paper? Response mode matters. Markets (repetition and feedback matters!) Incentives matter. But they don t always matter in straightforward and simple ways. Specifically, (Asymmetric!) PRP was observed consistently in all response modes on the first repetition, even in a market setting (second price auction) with immediate feedback, both with and without financial incentives (confirming earlier experimental results, especially those for 3

4 4 other response modes such as BDM mechanism). After five repetitions of the auction, the subjects bids were generally consistent with their choices the number of PRs clearly went down and the asymmetry between the rates of predicted and unpredicted reversals had disappeared. Same for BDM mechanism. The repetitive nature of the tasks in conjunction with feedback is an important factor. Monetary incentives played less of a role than is typically assumed. The results presented in this paper support the view that the nature of the market institutions and the information generated by the markets, together with the feedback and the repetitive nature of market tasks, account for the generally positive results of market experiments. (p. 403)

5 Note 1: Assume that lotteries have prizes (i.e., monetary outcomes) that are continuous. Denote prizes by the continuous variable x. Define the cumulative distribution function F = F(x) = I f(t)dt for all x: œ -> [0,1] is the probability that the realized payoff is less than or equal to x. (MWG take the space of all lotteries to be the set of all distribution functions over nonnegative amounts of money. This is obviously just for pedagogical reasons. Introduction of negative amounts of money creates interesting problems though empirically and theoretically: -> prospect theory. Note 2: We work with distribution function because they make life easier (e.g., allow us a general formulation for discrete and continuous outcomes) Note 3: F preserves the linear structure of lotteries: the distribution of final prizes induced by the compound lottery (L 1,..., L K ; " 1,..., " K ) is F(x) = 3 k " k F k (x). Note 4: The extension of the expected utility theorem to the continuous case is straightforward: The VNM utility function becomes U(F) = I u(x) df(x) = I u(x) f(x) dx where u(x) is the assignment of utility values to monetary outcomes. Obviously U(F) is the mathematical expectation, over the realizations of x, of the values of u(x). Abusing notation, u(x) corresponds to u(x i ) and f(x) corresponds to p(x i ) in the discrete case. U is linear in F as U was linear in p. Note 5: MWG call U(@) the VNM expected utility function (which is defined over lotteries); they call u(@) the Bernoulli utility function (which is defined over sure amounts of money). In the current context u(@) is assumed to be increasing and continuous. Note 6: (St. Petersburg paradox) Are there any restrictions on utility functions? Here s a classic example that shows that there has to be some boundedness. Consider the lottery that pays you $2 k if on the kth trial (of tossing a fair coin repeatedly) heads shows up for the first time. How much would you be willing to pay for that lottery? Assume that each toss of the coin is independent. Then the prob of H happening on the first trial is prob(h) = 1/2, the prob of H happening on the second trial is prob(th) = 1/4, and so on. The expectation in dollars of that lottery is hence E(L) = $2 prob(h) + $4 prob(th) +... = $2x1/2 + $4x1/ = $1 + $ But this means that the expected dollar value of the lottery is.... How much would you offer? Should you be willing to liquidate your entire worldly wealth in order to buy a ticket to participate in the lottery? [Discussed in class] 5

6 Definition 6.C.1 (Risk preferences, without presuming expected utility formulation): A decision maker is a risk averter if for any lottery F(@) the degenerate lottery that yields the amount I x df(x) with certainty is at least as good as the lottery F(@) itself. If the decision maker is always [i.e., for any F(@)] indifferent between these two lotteries, we say that he or she is risk neutral. Finally, we say that he is strictly risk averse if indifference holds only when the two lotteries are the same [i.e., when F(@) is degenerate]. Note 0: This sounds at first very confusing... but isn t really. It s just a fancy way of formalizing decision problem 1 on decision questionnaire 1. Note 1: First interpretation of risk aversion: The decision maker is risk averse if and only if I u(x) df(x) # u(i x df(x)) for all F(@) if preferences admit an expected utility representation with Bernoulli utility function. I.e., a decision maker is risk averse if and only if her or his Bernoulli utility function is concave. Strict concavity describes strict risk aversion. Equality reflects risk neutrality. [Discussion Figure 6.C.2(a)(b)] Note 2: Second interpretation of risk aversion: Strict concavity means that MU of money is decreasing. Hence, at any level of wealth of x, the utility gain from an extra dollar is smaller than (the absolute value of) the utility of having a dollar less. [Discussion Figure 6.C.2 (a)(b)] Note 3: Third and fourth interpretation of risk aversion: Definition 6.C.2 (certainty equivalent, probability premium): Given a Bernoulli utility function u(@), (i) the certainty equivalent of F(@), denoted c(f,u), is the amount of money for which the individual is indifferent between the gamble F(@) and the certain amount c(f,u); that is, u(c(f,u)) = Iu( x) df(x). (ii) for any fixed amount of money x and positive number,, the probability premium denoted by B(x,,,u), is the excess of the winning probability over fair odds that makes the individual indifferent between the certain outcome x and a gamble between the two outcomes x+, and x-,. That is u(x) = (1/2 + B(x,,,u))u(x+,) + (1/2 - B(x,,,u))u(x-,) [Discussion Figure 6.C.3 (a)(b)] Note 0: The terminology here can be confusing. E.g., Hey (1979), as does much of the preceding literature and also later literature (e.g., Kreps 1990, pp. 84-5), talks of risk premium to describe the wealth that an individual is willing to forgo so as to change a risky gamble into a certain one. That is, in other words, the difference between the expected value and the certainty equivalent. Note 1: The risk premium quantifies the trade-off between return and certainty (risk) that a 6

7 decision maker is willing to undergo. Similarly, the probability premium quantifies the odds that a decision maker requires to accept the trade-off between return and certainty (risk). The probability premium is the upward probability shift that induces a risk averse decision maker to trade in a sure payoff for a gamble. Note 2: A positive risk premium can only exist if the decision maker is risk averse; similarly, a positive probability premium can only exist if the decision maker is risk averse. Note 3: It s straightforward to extent the above definitions and concepts to risk proness or risk love. E.g., the decision maker is prone to risk or risk loving if and only if I u(x) df(x) $ u(i x df(x)) for all F(@) if preferences admit an expected utility representation with Bernoulli utility function. I.e., a decision maker is prone to risk or risk loving if and only if her or his Bernoulli utility function is convex. Strict convexity describes strict risk proneness or strict risk love. Or, strict convexity means that MU of money is increasing. Hence, at any level of wealth of x, the utility gain from an extra dollar is greater than (the absolute value of) the utility of having a dollar less. See Fig 6.C.2 (c). And so on. It is useful (and straightforward) to think through the consequences of these definitions for risk premia and probability premia. So, please do. Note 4: The tradeoff between return and risk (and the importance of the impact of risk preferences) is at the heart of portfolio allocation and insurance problems. Proposition 6.C.1. (Equivalency of statements): Suppose a decision maker is an expected utility maximizer with a Bernoulli u(@) on amounts of money. Then the following properties are equivalent. (i) The decision maker is risk averse. (ii) u(@) is concave. (iii) c(f,u)) # I x df(x) for all F(@) (iv) B(x,,,u) $ 0 for all x, [Make sure you understand the little proof of why (iii) is equivalent to (i). Discussed in class] 7

8 8 Note 5: Two well-known measures of risk aversion are Definition 6.C.3 (r A (x)): Given a (twice differentiable) Bernoulli function u(@) for money, the coefficient of absolute risk aversion at x (Arrow-Pratt measure) is defined as r A (x) = - u (x)/u (x). Definition 6.C.3 (r R (x)): Given a (twice differentiable) Bernoulli function u(@) for money, the coefficient of relative risk aversion at x is defined as r R (x) = - xu (x)/u (x). Note 5.1: The difference between the two measures of risk is obvious: they are identical except for the weighting with x. So, r R (x,u) = x r A (x,u). Note 5.2: Both measures exploit the curvature of the Bernoulli utility function and the fact that certainty equivalent (risk premium) and probability premium are intimately tied to the curvature. Since the Bernoulli utility function is only quasi-cardinal, and hence can be changed by way of affine transformation, the second derivative in the numerator is normalized through the first derivative in the denominator. Note 5.3: Differentiating u(x) = (1/2 + B(x,,,u))u(x+,) + (1/2 - B(x,,,u))u(x-,) with respect to, and evaluating at, = 0, we get 4 B (0)u (x) + u (x) = 0. Hence, r A (x) = 4 B (0). Thus, r A (x) measures the rate at which the probability premium increases at certainty with the small risk measured by,.

9 9 [Decision questionnaire 2] Your name:... class point option I prefer the [Circle your choice!] money option This is an experiment in the economics of decision making. If you follow the instructions carefully you can win a significant chance to make either money or earn class points. You are endowed with 2,400 lab dollars and you face a 1 in 6 chance of losing 1,200. You must decide whether you would choose a protective measure or not. If you purchase a protective measure the probability that you will lose the 1,200 drops to 1 in 36. There are six situations: A through F. They are listed below. Each of these situations gives a price for a protective measure and asks you to check YES or NO to buying the protective measure. Once everyone is finished, I will have someone draw a letter (A through F) and roll the dice to determine if the loss case will come up. (The loss case will come up if in case you bought protection rolling a dice twice comes up with a 1" twice, or if in case you did not rolling a dice comes up with a 1".) I will convert the lab dollar balances of three randomly drawn participants to Czech korunas at the rate of 400 lab dollars = 200 korunas or 400 lab dollars = 1 percentage point added to your total grade at the end of the course. => Please decide now if you prefer the class point or the money option. Circle you choice at the top of the page. => Now decide which of the protective measures you want to buy: Situation Price of Protective Measure Buy? Circle YES or NO A 200 lab dollars YES NO B 400 lab dollars YES NO C 600 lab dollars YES NO D 800 lab dollars YES NO E 1,000 lab dollars YES NO F 1,200 lab dollars YES NO

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