Econ 4660 1st EditionExam # 1 Study Guide LecturesAn Alternative Model: Two Phases of Choice Process: Editing Evaluation () Prospect Theory Spring 2015 2 / 7Prospect Theory: Editing Stage Editing Stage: organize & reformulate the problem What is going on? A choice problem is described to you, and then you transform it into the lotteries that you will evaluate. For instance: Coding:code outcomes as gains & losses relative to some reference point. Cancellation: discard shared components. Simplification: rounding of probabilities. Eliminating dominated alternatives. () Prospect Theory Spring 2015 3 / 7Prospect Theory: Evaluation Stage A person evaluates a prospect (x, p; y, q) according to: V (x, p; y, q) = π (p) v (x) + π (q) v (y ). Reminder: EU theory says evaluate according to: U (x, p; y, q) = pu (w + x) + qu (w + y ) + (1 p q) u (w) What is new? π () is the probability-weighting function. v () is the value function. () Prospect Theory Spring 2015 4 / 7Prospect Theory: Value Function Three key features of the value function v (): The carriers of value are changes in wealth (v (0) = 0). Diminishing sensitivity to the magnitude of changes (v 00 (x) < 0 for x > 0, v 00 (x) > 0 for x < 0). Loss aversion: losses loom larger than gains. A functional form thatís often used: v (x) = x α if x 0 λ(x) β if x 0 () Prospect Theory Spring 2015 5 / 7Prospect Theory: Probability-Weighting Function Some key features of the probability-weighting function π (): [Recall: EU theory says π (p) = p.] Natural assumptions: π (0) = 0, π (1)= 1, and π is increasing. Subcertainty: π (p) + π (1 p) < 1. Subproportionality: π (pq) /π (p) π (pqr) /π (pr) for p, q,r 2 (0, 1). For small p, π (p) > p. () Prospect Theory Spring 2015 Four Themes that Emerged from Prospect Theory 1. Non-linear decision weights. 2. Reference dependence & loss aversion. 3. Framing e§ects & mental accounting.Prospect Theory For many years, expected utility has been used by economists to capture risk preferences ó indeed, it is still used in almost all applications. BUT economists are starting to recognize that some behaviors are hard to interpret in terms of expected utility; and for many such behaviors, prospect theory provides a natural interpretation. To illustrate, weíll consider 8 examples. () Applications of Prospect Theory Spring 2015 2 / 20Application #1: The Samuelson Bet Example due to Samuelson (1963) Consider the following bet: win $200 with prob 1/2 lose $100 with prob 1/2 Samuelsonís colleague turned down this bet, but announced that he would accept 100 plays of the same bet. Samuelson proved that his colleague was irrationally ó by proving that it is inconsistent with expected-utility theory to turn down the single bet but accept 100 such bets. But WAS his colleague ìirrationalî? () Applications of Prospect Theory Spring 2015 3 / 200 03 0.04 0.05 0.06 0.07 0.08 0.09 Probability Histogram for the Samuelson Bet 0 0.01 0.02 0.03 -10000 -5000 0 5000 10000 15000 20000 Amount Application #1: The Samuelson Bet Consider an alternative ìmodelî: Suppose that a person evaluates bets according to the value function v (x) = x if x 0 2.5x if x 0 Consider the single bet y = [200, .5; 100, .5]. Consider taking two such bets, which means you face aggregate gamble z = [400, .25; 100, .5; 200, .25]. Point: Unlike EU, loss aversion can lead a person to reject one play of the bet but to accept multiple plays of the bet. () Applications of Prospect Theory Spring 2015 4 / 20An Important Issue: Mental Accounting Mental accounting: the process a person uses to interpret a choice situation. Any application of prospect theory requires a mental-accounting assumption. Sometimes, what is required is an assumption about how people decide what are the objects for evaluation. E.g., Kahneman & Tversky interpret the isolation effect as people ignoring seemingly extraneous parts of the problem. E.g., to explain the behavior of Samuelsonís colleague, we assumed that the person collapses the aggregate bet into a single lottery and decides whether to accept that lottery. Sometimes, what is required is an assumption about when and how people code outcomes as gains and losses. . () Applications of ProspectTheory Spring 2015 5 / 20Application #2: Risk Aversion From Rabin & Thaler (JEP 2001): Risk Aversion:People tend to dislike risky prospects even when they involve an expected gain. E.g.: A 50-50 gamble of losing $100 vs. gaining $110. Economists explanation: EU with a concave utility function. Rabin & Thaleríspoint: Calibrationwise, this explanation doesnít work, because according to EU, anything but virtual risk neutrality over modest stakes implies manifestly unrealistic risk aversion over large stakes. Furthermore, loss aversion is a useful alternative. () Applications of Prospect Theory Spring 2015 6 / 20Application #2: Risk Aversion Suppose you have wealth $20,000, and you turn down a 50-50 bet to win $110 vs. lose $100. Suppose you have a CRRA utility function u(x) = (x) 1ρ 1 ρ . What values of ρ are consistent with you rejecting this bet? Reject if 1 2 (20, 110) 1ρ 1 ρ + 1 2 (19, 900) 1ρ 1 ρ < (20, 000) 1ρ 1 ρ With a little work, one can show that rejecting the bet implies that ρ > 18.17026. () Applications of Prospect Theory Spring 2015 7 / 20Application #2: Risk Aversion Suppose you have ρ = 19. Again suppose you have wealth$20,000, and consider how you feel about a 50-50 bet to lose Y vs. win X? For Y = $100, accept if and only if X > For Y = $200, accept if and only if X > For Y = $500, accept if and only if X > For Y = $750, acceptif and only if X > For Y = $1000, accept if and only if X > Point: The degree of risk aversion required to explain your rejection of the moderate-stakes gamble implies ridiculous behavior for larger-stakes gambles. () Applications of Prospect Theory Spring 2015 8 / 20Application #2: Risk Aversion In fact, need not assume anything about the functional form for u. Here is another example: Suppose Johnny is a risk-averse EU maximizer (u 00 0). Suppose that, for any initial wealth, Johnny will reject a 50-50 gamble of losing $100 vs. gaining $110. Consider a 50-50 gamble of losing $1000 vs. gaining $X. What is the minimum X such that Johnny might accept? Answer: X = ∞ ó that is, Johnny will reject for any X. () Applications of Prospect Theory Spring 2015 9 / 20Application #2: Risk Aversion Two plausible features ofpreferences consistent with loss aversion: 1. How you