Optimal Expectations. Markus K. Brunnermeier and Jonathan A. Parker Princeton University

Transcription

1 Optimal Expectations Markus K. Brunnermeier and Jonathan A. Parker Princeton University

2 rational view Bayesian rationality Non-Bayesian

3 rational expectations Lucas rationality rational view Bayesian rationality Non-Bayesian

4 rational expectations Lucas rationality biases: confirmation, optimism, overconfidence... behavioral view rational view Bayesian rationality Non-Bayesian

5 rational expectations Lucas rationality common priors Harsanyi rationality biases: confirmation, optimism, overconfidence... non-common priors behavioral view rational view Bayesian rationality Non-Bayesian

6 rational expectations Lucas rationality common priors Harsanyi rationality biases: confirmation, optimism, overconfidence... non-common priors behavioral view rational view Bayesian rationality Non-Bayesian Critic: no disagreement no-trade theorem

7 rational expectations Lucas rationality common priors Harsanyi rationality biases: confirmation, optimism, overconfidence... non-common priors behavioral view rational view Bayesian rationality Non-Bayesian Critic: no disagreement no-trade theorem everything goes no structure

8 Optimal Expectations Our Goal: Provide structural model of subjective beliefs What is the direction of belief distortion? When are belief distortions large? Provide common framework for different biases 3

9 Three Main Elements 4

10 Three Main Elements 1. Anticipatory utility: 4

11 Three Main Elements 1. Anticipatory utility: Agents care about utility flow today AND expected utility flows in the future (possibly past) happier if more optimistic 4

12 Three Main Elements 1. Anticipatory utility: Agents care about utility flow today AND expected utility flows in the future (possibly past) happier if more optimistic 2. No schizophrenia Distorted beliefs distort actions better outcomes if more rational 4

13 Three Main Elements 1. Anticipatory utility: Agents care about utility flow today AND expected utility flows in the future (possibly past) happier if more optimistic 2. No schizophrenia Distorted beliefs distort actions better outcomes if more rational 3. Optimal beliefs balance these forces Beliefs maximize lifetime well-being 4

14 Outline 1.) The General Framework 2.) Applications and Empirical Implications 3.) Conclusion 5

15 Utility Flow, Felicity and Well-being felicity at t = 1 t=1 t=2 t=3 t=4 t=5 t=6 t=7 t=8

16 Utility Flow, Felicity and Well-being felicity at t = 1 t=1 t=2 t=3 t=4 t=5 t=6 t=7 t=8 felicity at t = 2 sunk t=2

17 Utility Flow, Felicity and Well-being felicity at t = 1 t=1 t=2 t=3 t=4 t=5 t=6 t=7 t=8 felicity at t = 2 sunk t=2 felicity at t = 3 t=3 Lifetime well-being S ª

18 1. The General Framework Felicity at t: }{{} + u(c t ) }{{} }{{} 7

19 1. The General Framework Felicity at t: +V t }{{} T t + u (c t ) + Ê t β τ u ( c t+τ ) } τ=1 {{} expected [ ( utility = )] βê t Vt+1 xt+1 ; s t+1 }{{} =V t (x t ;s t) V t = expected utility from current and future consumption 7

20 1. The General Framework Felicity at t: M t + V t t 1 δ t r u (c t r ) r=1 }{{} memory ( utility ) = M t c t 1 T t + u (c t ) + Ê t β τ u ( c t+τ ) } τ=1 {{} expected [ ( utility = )] βê t Vt+1 xt+1 ; s t+1 }{{} =V t (x t ;s t) V t M t = expected utility from current and future consumption = memory utility from past consumption 7

21 1. The General Framework Felicity at t: M t + V t t 1 δ t r u (c t r ) r=1 }{{} memory ( utility ) = M t c t 1 T t + u (c t ) + Ê t β τ u ( c t+τ ) } τ=1 {{} expected [ ( utility = )] βê t Vt+1 xt+1 ; s t+1 }{{} =V t (x t ;s t) V t M t = expected utility from current and future consumption = memory utility from past consumption Stage 2: At each t choose c t to maximize V t + M t given subjective beliefs ˆπ ( s t s t 1), state, xt, and resource constraints. 7

22 Stage 1: At t = 0 assign optimal beliefs ˆπ OE ( s t s t 1 (conditional probabilities to each branch of event tree) ) ˆπ that maximize Lifetime well-being: W = E [ Tt=1 β t ] (M t + V t ) 8

23 Two-period example with consumption at t = 2 t = 1 t = 2 t=1-self s felicity βê[u(c 2 )] t=2-self s felicity E[u(c 2 )] Well-being: W = βê[u(c 2 )] + βe[u(c 2 )] 9

24 2. Why Optimal Expectations? 10

25 2. Why Optimal Expectations? A. It is optimal as if interpretation Scientific method 10

26 2. Why Optimal Expectations? A. It is optimal as if interpretation Scientific method B. Evolution Happiness may lead to better health or marriage prospects (Taylor and Brown (1988)) 10

27 2. Why Optimal Expectations? A. It is optimal as if interpretation Scientific method B. Evolution Happiness may lead to better health or marriage prospects (Taylor and Brown (1988)) C. Parents choose Parents have the objective of optimal expectations 10

28 4. Applications 4a.) Portfolio choice preference for skewed returns 11

29 4. Applications 4a.) Portfolio choice preference for skewed returns 4b.) General equilibrium endogenous heterogenous prior beliefs 11

30 4. Applications 4a.) Portfolio choice preference for skewed returns 4b.) General equilibrium endogenous heterogenous prior beliefs 4c.) Consumption-savings problem with stochastic income unexpected decline in consumption profile 11

31 4. Applications 4a.) Portfolio choice preference for skewed returns 4b.) General equilibrium endogenous heterogenous prior beliefs 4c.) Consumption-savings problem with stochastic income unexpected decline in consumption profile 4d.) Optimal timing of a single task Planning Fallacy, procrastination, context effect 11

32 4a. Portfolio choice Setup: 1. Two period problem: invest in period 1, consume in period 2 12

33 4a. Portfolio choice Setup: 1. Two period problem: invest in period 1, consume in period 2 2. Two assets: a risk-free asset, return R; a risky asset, return R + Z 12

34 4a. Portfolio choice Setup: 1. Two period problem: invest in period 1, consume in period 2 2. Two assets: a risk-free asset, return R; a risky asset, return R + Z 3. Uncertainty: S states, π s > 0 for s = 1 to S, Z s < Z s+1, Z 1 < 0 < Z S 12

35 4a. Portfolio choice Setup: 1. Two period problem: invest in period 1, consume in period 2 2. Two assets: a risk-free asset, return R; a risky asset, return R + Z 3. Uncertainty: S states, π s > 0 for s = 1 to S, Z s < Z s+1, Z 1 < 0 < Z S 4. c 0 in all states 12

36 Stage 2: Agent max w β S s=1 ˆπ s u (R + wz s ) FOC: 0 = S s=1 ˆπ s u (R + wz s ) Z s w (ˆπ) 13

37 Stage 2: Agent max w β S s=1 ˆπ s u (R + wz s ) FOC: 0 = S s=1 ˆπ s u (R + wz s ) Z s w (ˆπ) Stage 1: Choose ˆπ s to maximize lifetime well-being β S ˆπ s u ( R + w ) Z s + β } s=1 {{} expected utility at t = 1 S π s u ( R + w Z s ) } s=1 {{} utility flow at t = 2 13

38 Stage 2: Agent max w β S s=1 ˆπ s u (R + wz s ) FOC: 0 = S s=1 ˆπ s u (R + wz s ) Z s w (ˆπ) Stage 1: Choose ˆπ s to maximize lifetime well-being β S ˆπ s u ( R + w ) Z s + β } s=1 {{} expected utility at t = 1 S π s u ( R + w Z s ) } s=1 {{} utility flow at t = 2 FOC: β (u S u s ) }{{} marginal expected utility = β S π s u ( R + w Z s ) Zs dw dˆπ s } s=1 {{} marginal cost of distortion 13

39 Proposition Excess risk taking due to optimism 14

40 Proposition Excess risk taking due to optimism (i) (ii) Agents are optimistic about states with high portfolio Agents go even more long (short) than agent with RE or even in the opposite direction if E[Z] > 0, then w RE > 0, and w > w RE or w < 0; if E[Z] < 0, then w RE < 0, then w < w RE or w > 0; 14

41 When Do agents buy asset with E[Z] < 0? Empirical Phenomena: Preference for Skewness Horse race long shots: Golec and Tamarkin (1998) Lottery demand: Garrett and Sobel (1999) Security design: LYONs, EPNs, ELNs, Swedish lottery bonds Setup: 2 states with payoffs: Z 1 < 0 < Z 2, hold mean E[Z] < 0 and variance V ar[z] fixed the higher π 1, the more skewed (like lottery ticket) π 1 Z 1 0 Z 2 15

42 When Do agents buy asset with E[Z] < 0? Empirical Phenomena: Preference for Skewness Horse race long shots: Golec and Tamarkin (1998) Lottery demand: Garrett and Sobel (1999) Security design: LYONs, EPNs, ELNs, Swedish lottery bonds Setup: 2 states with payoffs: Z 1 < 0 < Z 2, hold mean E[Z] < 0 and variance V ar[z] fixed the higher π 1, the more skewed (like lottery ticket) increase π 1 Z 1 0 Z 2 16

43 Proposition There exists a π such that for all π 1 > π (i.e. if returns are sufficiently skewed), OE agent with an unbounded utility function goes long an asset even though its mean payoff is negative. 17

44 Proposition There exists a π such that for all π 1 > π (i.e. if returns are sufficiently skewed), OE agent with an unbounded utility function goes long an asset even though its mean payoff is negative. Remarks: there is not much room to distort beliefs. shorting becomes very risky. 17

45 4b. General Equilibrium Empirical Phenomena: betting & gambling high trading volume (stock and FX market) endogenous heterogenous prior beliefs home bias puzzle over-investment in employer s stock 18

46 4b. General Equilibrium Empirical Phenomena: betting & gambling high trading volume (stock and FX market) endogenous heterogenous prior beliefs home bias puzzle over-investment in employer s stock Proposition (iii) Heterogeneous prior beliefs In any equilibrium, each agent bets on a different state i believes in heads : ˆπ 1 i > π 1, ˆπ 2 i < π 2, w i < 0, c i 1 > ci 2, and i believes in tails : ˆπ 2 i > π 2, ˆπ 1 i < π 1, w i > 0, c i 2 > c i 1 18

47 4c. Consumption and Saving Empirical Phenomena: households expect upward sloping consumption profile (Barsky et al. 1997) actual average consumption growth is non-positive and profiles are concave (Gourinchas & Parker (2002)) 20

48 average consumption path average consumption path for agent with rational expectations T-1 T t

49 average consumption path overconsumption (overoptimism) consumption at t = 1 for agent with optimal expectations T-1 T t

50 average consumption path overconsumption (overoptimism) consumption at t = 1 for agent with optimal expectations + expected consumption path for agent with optimal expectations at t = T-1 T t

51 average consumption path Reduce consumption since income in t=2 was lower than expected consumption at t = 2 for agent with optimal expectations + + expected consumption path at t = T-1 T t

52 average consumption path Initial over- consumption (overoptimism) consumption at t = 3 for agent with optimal expectations expected consumption path at t = T-1 T t

53 average consumption path Initial over- consumption (overoptimism) T-1 T t

54 average consumption path Initial over- consumption (overoptimism) c (t) + + c OE OE (t) T-1 T t

55 4d. Optimal Timing of a Single Action Empirical Phenomena: planing fallacy: underestimation of time to complete task referee report heavy briefcases for weekend additional options (even when not chosen) alters choice Intuition: Optimal beliefs underestimate how difficult it is to do a task tomorrow (relative to today) 22

56 4d. Optimal Timing of a Single Action Empirical Phenomena: planing fallacy: underestimation of time to complete task referee report heavy briefcases for weekend additional options (even when not chosen) alters choice Intuition: Optimal beliefs underestimate how difficult it is to do a task tomorrow (relative to today) Agents plan to undertake task tomorrow, but when tomorrow comes they postpone it again. 22

57 4d. Optimal Timing of a Single Action Empirical Phenomena: planing fallacy: underestimation of time to complete task referee report heavy briefcases for weekend additional options (even when not chosen) alters choice Intuition: Optimal beliefs underestimate how difficult it is to do a task tomorrow (relative to today) Agents plan to undertake task tomorrow, but when tomorrow comes they postpone it again. Procrastination due to belief distortion and not preference distortion. 22

58 Conclusion 23

59 Conclusion 1. Structural model of priors beliefs are most distorted, when decision errors are small endogenous heterogenous beliefs trade and speculation excess risk taking due to optimism preference for skewness realistic consumption profile 23

60 Conclusion 1. Structural model of priors beliefs are most distorted, when decision errors are small endogenous heterogenous beliefs trade and speculation excess risk taking due to optimism preference for skewness realistic consumption profile 2. Features of procrastination (due to belief distortions) intertemporal preference reversal, context effect 23