Homework #6 (10/18/2017)

Transcription

1 Homework #6 (0/8/207). Let G be the set of compound gambles over a finite set of deterministic payoffs {a, a 2,...a n } R +. A decision maker s preference relation over compound gambles can be represented by utility function v : G R. Let g G, and let probability p i be associated to the corresponding payoff a i. Finally, consider that the decision maker s utility function v( ) is given by v(g) = ( + a ) p ( + a ) p 2...( + a n ) pn = n ( + a i ) p i (a) Show that this is not a von Neumann-Morgenstern (vnm) utility function. Since v(g) is not linear in the probabilities, then v(g) cannot be a vnm expected utility function, with general form v(g) = N p i u(a i ) (b) Show that the decision maker has the same preference relation as an expected utility maximizer with von-neumann Morgenstern utility function u(g) = n p i ln ( + a i ). Since ln( ) is a monotonic transformation of v( ), both functions represent the same preference relation. Applying the monotonic transformation u(g) = ln [v(g)] to the original function v(g), we obtain ( n ) n ln ( + a i ) p i = p i ln ( + a i ) which represents the initial preference relation over lotteries, and it is linear in the probabilities. Hence, it is a vnm utility function. (c) Assume now that the decision maker you considered in part (b) has utility function u(w) = ln ( + w) over wealth w 0. Evaluate his risk attitude (concavity in his utility function). Additionally, find the Arrow-Pratt coeffi cient of absolute risk aversion, r A (w, u). How does r A (w, u) change in wealth?

2 Given that u(w) = ln( + w), where w 0, then the first and second order conditions with respect to w are u (w) = + w > 0 and u (w) = ( + w) 2 < 0, which implies that the utility function is concave, as depicted in figure, and that the decision maker is risk-averse. Figure. Utility function u(w) = ln( + w) Let us now obtain the Arrow-Pratt coeffi cient of absolute risk-aversion, r A (w, u), as follows (+w) 2 +w r A (w, u) = u (w) u (w) = = + w Finally, we want to know how this coeffi cient of absolute risk aversion varies with wealth, r A (w, u) w = ( + w) 2 which is negative for all wealth levels w 0. Hence, the agent becomes less risk-averse as he becomes more wealthy. Figure 2 illustrates this coeffi cient, 2

3 r A (w, u), evaluated at different wealth levels. Figure 2. Coeffi cient of absolute risk aversion. 2. Consider the set of deterministic payoffs {a, a 2,...a n } R +. Studies in regret-based decision making often consider the following utility function: first, define the highest deterministic payoff that could be reached in gamble g by using function h(g) = max {a k : k {, 2,..., n} and p k > 0}. Subtracting h(g) from all deterministic outcomes and computing its expected value yields the utility level v(g) = n p i (a i h(g)) = n p i a i h(g) Intuitively, after event i realizes (which provides a payoff a i to this individual), the regretful decision maker compares such monetary payoff with respect to the highest possible payoff he could have obtained from playing this lottery, h(g). Utility functions of this type hence reflect regret, as individuals experience a disutility from not receiving the highest possible monetary payoff in the lottery. (a) Compute the expected value of the following two gambles: g = ( 0,, 2; 3, 3, ) 3 and g 2 = (, 4, 5; 2, 3, ) 6 First, note that h(g ) = max {0,, 2} since all these events can occur with strictly positive probability in lottery g. Then, h(g ) = 2, and therefore the Recall that h(g) is unaffected by probability p i, as it describes the highest payoff receiving a positive probability. 3

4 individual s expected utility from playing the first gamble, g, is v(g ) = 3 (0 2) + 3 ( 2) + (2 2) = 3 Similarly, we can find the expected utility from playing the second gamble, g 2. In particular, in this case the highest payoff of the lottery is h(g 2 ) = max {, 4, 5} = 5, implying that the expected utility from this gamble is v(g 2 ) = 2 ( 5) + 3 (4 5) + 6 (5 5) = 7 3 Note that the individual experiences a lower expected utility from playing the second than the first lottery. Intuitively, this happens because: () the distribution of payoffs in the second lottery is more spread than in the first lottery, and this makes the lower payoffs on the second gamble to be compared to a higher possible payoff h(g 2 ); and (2) because the lowest payoffs on the second gamble are more likely than in the first and, as a consequence, the individual assigns a higher weight in the expected utility calculation to those monetary payoffs in which he is experiencing the biggest regret. (b) Show that all deterministic outcomes (outcomes with probability 00%) yield the same utility level. That is, v(a ) = v(a 2 ) =... = v(a n ). Let us represent by v(a i ) the individual s utility level from a certain deterministic outcome a i, i.e., p ai =. But if outcome a i occurs with certainty, there is no potential regret. In particular, function h(g) can only find the maximum among all outcomes of the lottery whose probability is strictly greater than zero. Since p ai =, then all other outcomes of the lottery receive probability zero, and hence h(g) = max {a k : k {, 2,..., n} and p k > 0} = max {a i } = a i Therefore, the individual s expected utility becomes v(a i ) = n p i (a i h(g)) = (a i a i ) = 0 Thus, v(a ) = v(a 2 ) =... = v(a n ) = 0, regardless of the monetary payoff associated to outcome a i. If there is just one event to be regretful about, my expected utility is zero! 4

5 (c) Show that the preference relation does not satisfy monotonicity if outcomes are deterministic. From the definition of monotonicity, we have that (a, a n ; α, α) (a, a n ; β, β) for all α, β [0, ] if and only if α > β. So if we make α = and β = 0, then the above condition on monotonicity becomes (a, a n ;, 0) (a, a n ; 0, ) Then, clearly a a n and a a n, which implies that a a n. However, in part (b) we have shown that the individual s utility is the same (and equal to zero) when outcomes are deterministic. In other words, he is indifferent between gambles whose outcomes are deterministic, i.e., a a 2... a n. But this contradicts that a a n. Therefore, this regretful preference relation cannot satisfy monotonicity. 3. Exercise 6.B.4 MWG 6.B.4 2 The purpose of this exercise is to illustrate how expected utility theory allows us to make consistent decisions when dealing with extremely small probabilities by considering relatively large ones. Suppose that a safety agency is thinking of establishing a criterion under which an area prone to flooding should be evacuated. The probability of flooding is %. There are four possible outcomes:. [(A)] 2. No evacuation is necessary, and none is preferred. 3. An evacuation is performed that is unnecessary. 4. An evacuation is performed that is necessary. 5. No evacuation is performed, and a flood causes a disaster. 2 Note the following errata in the first print of MWG, for which the lottery of A (instead of B) with probability q and D with probability -q. Also, for criterion 2, it results in ldots an unnecessary evacuation in 5% of the cases in which no flooding occurs 5

6 Suppose that the agency is indifferent between the sure outcome B and the lottery of A with probability p and D with probability p, and between the sure outcome C and the lottery of A with probability q and D with probability q. Suppose also that it prefers A to D and that p (0, ) and q (0, ). Assumes that the conditions of the expected utility theorem are satisfied.. [(a)] 2. Construct a utility function of the expected utility for the agency. 3. Consider two different policy criteria: (a) This criterion will result in an evacuation in 90% of the cases in which flooding will occur and an unnecessary evacuation in 0% of the cases in which no flooding occurs. (b) This criterion is more conservative. It results in an evacuation in 95% of the cases in which flooding will occur and an unnecessary evacuation in 5% of the cases in which no flooding occurs. First, derive the probability distributions over the four outcomes under these two criteria. Since the agency prefers A to D, we normalize u A = and u D = 0. Therefore, since the agency is indifferent between the sure outcome B and the lottery of A with probability p and D with probability p, u B = p u A + ( p) u D = p Analogously, since the agency is indifferent between the sure outcome C and the lottery of A with probability q and D with probability q, u C = q u A + ( q) u D = q Then, by using the utility function in (a), decide which criterion the agency would prefer. Given that the probability of flooding is %, we denote p (F ) = 0.0 and p(f ) = p (F ) = 0.99, which represent the probability of flooding and no flooding, respectively. Next, we denote. [(A)] 2. p A = p(e, F ) for the probability that no evacuation is necessary, and none is preferred. 6

7 3. p B = p(e, F ) for the probability that an evacuation is performed that is unnecessary. 4. p C = p (E, F ) for the probability that an evacuation is performed that is necessary. 5. p D = p(e, F ) for the probability that no evacuation is performed and a flood causes a disaster. In this context, we evaluate the two criteria.. This criterion will result in an evacuation in 90% of the cases in which flooding will occur, such that p (E F ) = 0.9 and p(e F ) = p (E F ) = 0.. For an unnecessary evacuation in 0% of the cases in which no flooding occurs, p(e F ) = 0. and p(e F ) = p(e F ) = 0.9. Therefore, (a) [(A)] (b) p A = p(e, F ) = p(e F )p(f ) = = 0.89 (c) p B = p(e, F ) = p(e F )p(f ) = = (d) p C = p (E, F ) = p (E F ) p (F ) = = (e) p D = p(e, F ) = p(e F )p (F ) = = 0.00 The expected utility of the agency in this case becomes E (U) = p A u A + p B u B + p C u C + p D u D = p q 2. This criterion is more conservative. It results in an evacuation in 95% of the cases in which flooding will occur, such that p (E F ) = 0.95 and p(e F ) = p (E F ) = For an unnecessary evacuation in 5% of the cases in which no flooding occurs, p(e F ) = 0.5 and p(e F ) = p(e F ) = Therefore, (a) [(A)] (b) p A = p(e, F ) = p(e F )p(f ) = = (c) p B = p(e, F ) = p(e F )p(f ) = = (d) p C = p (E, F ) = p (E F ) p (F ) = = (e) p D = p(e, F ) = p(e F )p (F ) = =

8 The expected utility of the agency in this case becomes E (U) = p A u A + p B u B + p C u C + p D u D = p q As a result, the agency would prefer criterion to criterion 2 if he obtains a higher utility from criterion, which happens when p q p q 99 99p + q And on the contrary, the agency would prefer criterion 2 to criterion if he obtains a higher utility from criterion 2, which happens when p q < p q 99 < 99p + q As seen in figure, the agency would prefer criterion to criterion 2 in the lower trapezium capped by the four lines, q = 0, q =, p = 0 and q = 99 ( p). Otherwise, the agency would prefer criterion 2 to criterion in the upper triangle capped by the three lines, q =, p = and q = 99 ( p), which happens when the agency faces extreme values of p and q. In particular, substituting q = into the above isoquant, we find that 98 < p in order for 99 the agency to prefer criterion 2 to criterion. Therefore, for 0 p 98, the agency would 99 prefer criterion to criterion 2, which implies that the agency prefers a less conservative evacuation policy in most of the cases. 8

9 Figure : Agency s preference for policy criteria 6.B.5 The purpose of this exercise is to show that the Allais paradox is compatible with a weaker version of the independence axiom. We consider the following axiom, known as the betweenness axiom [see Dekel (986)]: For all L, L and λ (0, ), if L L, then λl + ( λ) L L. Suppose that there are three possible outcomes.. [(a)] 2. Show that a preference relation on lotteries satisfying the independence axiom also satisfies the betweenness axiom. L L λl λl λl + ( λ) L λl + ( λ) L by the independence axiom λl + ( λ) L L which satisfies the betweenness axiom 3. Using a simple representation for lotteries similar to the one in Figure 6.B.(b), show that if the continuity and betweenness axioms are satisfied, then the indifference curves of a preference relation on lotteries are straight lines. Conversely, show that if the indifference curves are straight lines, then the betweenness axiom is satisfied. Do these straight lines need to be parallel? 9

10 From part (a), if L L, then the betweenness axiom implies that L λl + ( λ) L L Intuitively, if I am indifferent between lotteries L and L, then I am also indifferent between any convex combinations of L and L. Hence, the indifference curve connecting L and L must be a straight line passing through all the convex combinations "in between" L and L, i.e., the composite lottery λl+( λ) L for all λ [0, ], therefore coinciding with the bottom line of the triangle as depicted in figure 2. Conversely, if the indifference curves are straight lines, then the indifference curve coincides with all the convex combinations of the simple lotteries forming the end points of the indifference curve, therefore satisfying the betweenness axiom. Since we do not impose the Indifference axiom, by the Continuity axiom, we have either or 2 L + 2 L 2 L + 2 L 2 L + 2 L 2 L + 2 L Suppose, without loss of generality, we consider the case that 2 L + 2 L 2 L + 2 L, then there exists an α ( 2, ] such that 2 L + 2 L αl + ( α) L. 3 Intuitively, there exists a composite lottery αl + ( α) L, which, by assigning more weights to L and less weights to L, would make me just as indifferent as choosing 2 L + 2 L. Figure 2: Indifference Curves 3 Continuity of the preference relation does not guarantee the existence of an indifferent composite lottery on the line connecting L and L, but we assume that it exists for expository purposes. 0

11 By the betweenness axiom, 2 L + 2 L αl + ( α) L implies that for all λ [0, ], [ λ 2 L + ] 2 L + ( λ) [αl + ( α) L ] 2 L + 2 L λ 2 L + α ( λ) λ + 2 ( α) ( λ) L + L 2 2 L + 2 L λ 2 L + α ( λ) ( α) (2 λ) L + L 2 2 L + 2 L Analogously, λ 2 L + α ( λ) L + ( α)(2 λ) 2 L αl + ( α) L, such that the indifference curve connecting the two end points, 2 L + 2 L and αl + ( α) L, contains all the convex combinations therein, represented by the dotted line in figure 2. As seen, the dotted line, which represents the indifference relationship 2 L+ 2 L αl + ( α) L, is not parallel to the bottom solid line, which represents the indifference relationship L L, though both indifference curves are straight lines on their own. 4. Using (b), show that the betweenness axiom is weaker (less restrictive) than the independence axiom. From Figure 6.B.5. (MWG, p. 75), the independence axiom imposes straight and parallel indifference curves. However, any preference represented by straight, but not parallel indifference curves satisfies the betweenness axiom but not the independence axiom. Hence, the former is weaker than the latter. 5. Using Figure 6.B.7, show that the choices of the Allais paradox are compatible with the betweenness axiom by exhibiting an indifference map satisfying the betweenness axiom that yields the choices of the Allais paradox. Since Allais paradox violates the parallel indifference curve requirement, it does not satisfy the independence axiom. Nevertheless, given straight indifference curves, figure 6.B.7 is an example of a preference relation and its indifference map that satisfies the betweenness axiom and yields the choice of the Allais paradox.