Markus K. Brunnermeier and Jonathan Parker. October 25, Princeton University. Optimal Expectations. Brunnermeier & Parker. Framework.

Transcription

1 Optimal Markus K. and Jonathan Parker Princeton University October 25, 2006

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 Overview: Three Main Elements 1 Felicity at t: Ê t [U (c 1,..., c T )] Agents care about utility flow today and expected utility flows in the future happier if more optimistic 2 No split personality Distorted beliefs distort actions better outcomes if more rational 3 Optimal beliefs balance these forces [ Beliefs maximize well-being 1 T E T ] t=1 Êt [U (c 1,..., c T )]

9 Overview: Three Main Elements 1 Felicity at t: Ê t [U (c 1,..., c T )] Agents care about utility flow today and expected utility flows in the future happier if more optimistic 2 No split personality Distorted beliefs distort actions better outcomes if more rational 3 Optimal beliefs balance these forces [ Beliefs maximize well-being 1 T E T ] t=1 Êt [U (c 1,..., c T )]

10 Overview: Three Main Elements 1 Felicity at t: Ê t [U (c 1,..., c T )] Agents care about utility flow today and expected utility flows in the future happier if more optimistic 2 No split personality Distorted beliefs distort actions better outcomes if more rational 3 Optimal beliefs balance these forces [ Beliefs maximize well-being 1 T E T ] t=1 Êt [U (c 1,..., c T )]

11 Outline 1 Optimal 2 3 Related 4 5

12 Actions: The At each t agent chooses c t to maximize felicity t given subjective beliefs ˆπ ( s t s t 1), and resource constraints. Felicity at t: Êt[U(c 1,..., c T )] with time-separable exponential discounting equals t 1 β τ u (c τ ) τ=1 }{{} memory utility + β t u (c t ) + Ê t [ T τ=t+1 β τ u (c τ ) ] } {{ } expected utility Note: βs for past consumption could be replaced with δ.

13 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

14 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 Well-being ª

15 Beliefs: At t = 0 optimal beliefs are ˆπ OE ( ) s t s t 1 ˆπ that maximize [ Well-being: W = 1 T E T ] t=1 Êt [U( )] subject to: agent behavior given these beliefs ˆπ OE ( s t s t 1) are probabilities ˆπ OE ( s t s t 1 ) = 0 if π ( st s t 1 ) = 0

16 Two-period Example with Consumption at t = 2 t = 1 t = 2 felicity in period 1 βê[u(c 2)] felicity in period 2 βu(c 2 ) Actions maximize felicity: βê[u(c 2 )] Beliefs maximize well-being: W = 1 2 βê[u(c 2 )] βe[u(c 2)]

17 Two-period Example with Consumption at t = 2 t = 1 t = 2 felicity in period 1 βê[u(c 2)] felicity in period 2 βu(c 2 ) Actions maximize felicity: βê[u(c 2 )] Beliefs maximize well-being: W = 1 2 βê[u(c 2 )] βe[u(c 2)]

18 1 Subjective probabilities are chosen once and forever Bayes Rule (LIE) holds, Can be interpreted as choice of priors 2 If beliefs are objective, wellbeing = felicity Only incentive to distort beliefs is anticipatory utility gain 3 Rational expectations are optimal only if anticipatory utility does enter felicities or anticipatory utility does not enter well-being W. 4 Different memory discounting in felicity Paper s results hold qualitatively for any memory discounting But can introduce additional incentives to bias beliefs

19 1 Subjective probabilities are chosen once and forever Bayes Rule (LIE) holds, Can be interpreted as choice of priors 2 If beliefs are objective, wellbeing = felicity Only incentive to distort beliefs is anticipatory utility gain 3 Rational expectations are optimal only if anticipatory utility does enter felicities or anticipatory utility does not enter well-being W. 4 Different memory discounting in felicity Paper s results hold qualitatively for any memory discounting But can introduce additional incentives to bias beliefs

20 1 Subjective probabilities are chosen once and forever Bayes Rule (LIE) holds, Can be interpreted as choice of priors 2 If beliefs are objective, wellbeing = felicity Only incentive to distort beliefs is anticipatory utility gain 3 Rational expectations are optimal only if anticipatory utility does enter felicities or anticipatory utility does not enter well-being W. 4 Different memory discounting in felicity Paper s results hold qualitatively for any memory discounting But can introduce additional incentives to bias beliefs

21 1 Subjective probabilities are chosen once and forever Bayes Rule (LIE) holds, Can be interpreted as choice of priors 2 If beliefs are objective, wellbeing = felicity Only incentive to distort beliefs is anticipatory utility gain 3 Rational expectations are optimal only if anticipatory utility does enter felicities or anticipatory utility does not enter well-being W. 4 Different memory discounting in felicity Paper s results hold qualitatively for any memory discounting But can introduce additional incentives to bias beliefs

22 ... 5 Frictionless Extreme 6 Why optimal expectations? It is optimal: as if interpretation Parents/Upbringing affects (prior) beliefs Neuroscientific story : prefrontal cortex exerts effort to reduce overoptimism (subconscious process) 7 Payoff: biases are endogenous biases are small when distort behavior a lot large when provide the most expected future utility

23 ... 5 Frictionless Extreme 6 Why optimal expectations? It is optimal: as if interpretation Parents/Upbringing affects (prior) beliefs Neuroscientific story : prefrontal cortex exerts effort to reduce overoptimism (subconscious process) 7 Payoff: biases are endogenous biases are small when distort behavior a lot large when provide the most expected future utility

24 ... 5 Frictionless Extreme 6 Why optimal expectations? It is optimal: as if interpretation Parents/Upbringing affects (prior) beliefs Neuroscientific story : prefrontal cortex exerts effort to reduce overoptimism (subconscious process) 7 Payoff: biases are endogenous biases are small when distort behavior a lot large when provide the most expected future utility

25 Related 1 Adam Smith (1776) That the chance of gain is naturally overvalued,... That the chance of loss is frequently undervalued,... 2 Anticipatory utility ( Pleasure of Expectation ): Bentham, Hume, Böhm-Barwerk, Marshall, Loewenstein, Geanakopolis-Pearce-Stacchetti, Caplin-Leahy 3 Models of belief distortions: cognitive dissonance (Akerlof-Dickens), agents choose beliefs (Yariv and Landier), intrapersonal (confidence) games (Bénabou-Tirole), cognitive dissonance and overconfidence (Gervais-O Dean), procrastination (O Donoghue-Rabin),... follow up: link to prospect theory (Gollier), (Glaeser)

26 Related 1 Adam Smith (1776) That the chance of gain is naturally overvalued,... That the chance of loss is frequently undervalued,... 2 Anticipatory utility ( Pleasure of Expectation ): Bentham, Hume, Böhm-Barwerk, Marshall, Loewenstein, Geanakopolis-Pearce-Stacchetti, Caplin-Leahy 3 Models of belief distortions: cognitive dissonance (Akerlof-Dickens), agents choose beliefs (Yariv and Landier), intrapersonal (confidence) games (Bénabou-Tirole), cognitive dissonance and overconfidence (Gervais-O Dean), procrastination (O Donoghue-Rabin),... follow up: link to prospect theory (Gollier), (Glaeser)

27 Related 1 Adam Smith (1776) That the chance of gain is naturally overvalued,... That the chance of loss is frequently undervalued,... 2 Anticipatory utility ( Pleasure of Expectation ): Bentham, Hume, Böhm-Barwerk, Marshall, Loewenstein, Geanakopolis-Pearce-Stacchetti, Caplin-Leahy 3 Models of belief distortions: cognitive dissonance (Akerlof-Dickens), agents choose beliefs (Yariv and Landier), intrapersonal (confidence) games (Bénabou-Tirole), cognitive dissonance and overconfidence (Gervais-O Dean), procrastination (O Donoghue-Rabin),... follow up: link to prospect theory (Gollier), (Glaeser)

28 Portfolio choice preference for skewed returns equilibrium endogenous heterogenous prior beliefs equity premium puzzle versus long shot phenomena Consumption-savings problem with stochastic income optimism and overconfidence in future income consumption profiles concave due to news choose incomplete consumption insurance Optimal timing of a single task procrastination, planning fallacy, context effect

29 Portfolio choice preference for skewed returns equilibrium endogenous heterogenous prior beliefs equity premium puzzle versus long shot phenomena Consumption-savings problem with stochastic income optimism and overconfidence in future income consumption profiles concave due to news choose incomplete consumption insurance Optimal timing of a single task procrastination, planning fallacy, context effect

30 Portfolio choice preference for skewed returns equilibrium endogenous heterogenous prior beliefs equity premium puzzle versus long shot phenomena Consumption-savings problem with stochastic income optimism and overconfidence in future income consumption profiles concave due to news choose incomplete consumption insurance Optimal timing of a single task procrastination, planning fallacy, context effect

31 Portfolio choice preference for skewed returns equilibrium endogenous heterogenous prior beliefs equity premium puzzle versus long shot phenomena Consumption-savings problem with stochastic income optimism and overconfidence in future income consumption profiles concave due to news choose incomplete consumption insurance Optimal timing of a single task procrastination, planning fallacy, context effect

32 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 > 2 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

33 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 > 2 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

34 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 > 2 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

35 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 > 2 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

36 Stage 2: Agent max α β S s=1 ˆπ su (R + αz s ) Stage 1: FOC: 0 = S ˆπ s u (R + αz s ) Z s s=1 Choose ˆπ s to maximize well-being α (ˆπ) 1 S 2 β ˆπ s u (R + α Z s ) + 1 S 2 β π s u (R + α Z s ) s=1 s=1 }{{}}{{} felicity at t = 1 average utility at t = 2 FOC: β 2 (u S u s ) = β S π s u (R + α dα Z s ) Z s }{{} 2 d ˆπ s=1 s }{{} benefits of anticipation costs of changed behavior

37 Stage 2: Agent max α β S s=1 ˆπ su (R + αz s ) Stage 1: FOC: 0 = S ˆπ s u (R + αz s ) Z s s=1 Choose ˆπ s to maximize well-being α (ˆπ) 1 S 2 β ˆπ s u (R + α Z s ) + 1 S 2 β π s u (R + α Z s ) s=1 s=1 }{{}}{{} felicity at t = 1 average utility at t = 2 FOC: β 2 (u S u s ) = β S π s u (R + α dα Z s ) Z s }{{} 2 d ˆπ s=1 s }{{} benefits of anticipation costs of changed behavior

38 Stage 2: Agent max α β S s=1 ˆπ su (R + αz s ) Stage 1: FOC: 0 = S ˆπ s u (R + αz s ) Z s s=1 Choose ˆπ s to maximize well-being α (ˆπ) 1 S 2 β ˆπ s u (R + α Z s ) + 1 S 2 β π s u (R + α Z s ) s=1 s=1 }{{}}{{} felicity at t = 1 average utility at t = 2 FOC: β 2 (u S u s ) = β S π s u (R + α dα Z s ) Z s }{{} 2 d ˆπ s=1 s }{{} benefits of anticipation costs of changed behavior

39 (i) (ii) Proposition Excess risk taking due to optimism Agents are optimistic about states with high portfolio payou S if α > 0, (ˆπ s π s ) u (R + α Z s ) Z s > 0; if α < 0, s=1 S (ˆπ s π s ) u (R + α Z s ) Z s < 0. s=1 Agents go even more long (short) than agent with RE or in the opposite direction if E[Z] > 0, then α > α RE > 0 or α < 0; if E[Z] < 0, then α < α RE < 0 or α > 0;

40 Preference for Skewed Returns Empirical Phenomena: Horse race long shots: Golec and Tamarkin (1998) Lottery demand: Garrett and Sobel (1999) Security design? Swedish lottery bonds, PS-Lotteriesparen Setup 2 states with payoffs: Z 1 < 0 < Z 2, hold variance and mean fixed and E[Z] < 0 π 1

41 Preference for Skewed Returns Empirical Phenomena: Horse race long shots: Golec and Tamarkin (1998) Lottery demand: Garrett and Sobel (1999) Security design? Swedish lottery bonds, PS-Lotteriesparen Setup 2 states with payoffs: Z 1 < 0 < Z 2, hold variance and mean fixed and E[Z] < 0 increase skewness

42 Proposition Skewness An agent with an unbounded utility function holds some of the asset even though its mean payoff is negative if the payoff is sufficiently skewed. Remark: Agent goes long for large π 1 even though E[Z] < 0, since there is not much room to short and distort beliefs shorting becomes very risky

43 Empirical Phenomena: betting & gambling high trading volume (stock and FX market) home bias endogenous heterogenous prior beliefs? negatively skewed: equity premium puzzle positively skewed: IPO underperformance, long-shots Setup: The portfolio choice problem with A continuum of agents with identical endowments A fixed supply of bonds with normalization R = 1 The risky asset in zero net supply: 1 + Z s = 1+εs P e

44 Empirical Phenomena: betting & gambling high trading volume (stock and FX market) home bias endogenous heterogenous prior beliefs? negatively skewed: equity premium puzzle positively skewed: IPO underperformance, long-shots Setup: The portfolio choice problem with A continuum of agents with identical endowments A fixed supply of bonds with normalization R = 1 The risky asset in zero net supply: 1 + Z s = 1+εs P e

45 Proposition Hetereogeneous Priors For S > 2 agents split into two groups with different beliefs (i) Optimists with Ê i [ Z OE ] > 0 and α OE,i > 0 = α RE (ii) Pessimists with Ê j [ Z OE ] < 0 and α OE,j < 0 both groups trade against each other and {ˆπ i } {π} {ˆπ j }. Example u (c) = 1 1 γ c1 γ with γ = 3, π 1 = 0.25, π 2 = 0.75, ε 1 = 0.6, ε 2 = 0.2 so P RE = 1.

46 Proposition Hetereogeneous Priors For S > 2 agents split into two groups with different beliefs (i) Optimists with Ê i [ Z OE ] > 0 and α OE,i > 0 = α RE (ii) Pessimists with Ê j [ Z OE ] < 0 and α OE,j < 0 both groups trade against each other and {ˆπ i } {π} {ˆπ j }. Example u (c) = 1 1 γ c1 γ with γ = 3, π 1 = 0.25, π 2 = 0.75, ε 1 = 0.6, ε 2 = 0.2 so P RE = 1.

47 Figure: Wellbeing as a function of subjective beliefs, ˆπ 2

48 In this example, as we vary the economic environment, beliefs change... P OE > P RE = 1 if payoff is positively skewed (long-shots, IPO) P OE < P RE = 1 if payoff is negatively skewed (stock market). Conjecture For multi-asset case with positive net supply: Heterogeneity in beliefs is less pronounced. Agents invest in different skewed assets (forgo diversification benefits to hold skewed assets.) Complicates Aggregation: Representative agent has different preference structure from individual (possibly identical) investors.

49 In this example, as we vary the economic environment, beliefs change... P OE > P RE = 1 if payoff is positively skewed (long-shots, IPO) P OE < P RE = 1 if payoff is negatively skewed (stock market). Conjecture For multi-asset case with positive net supply: Heterogeneity in beliefs is less pronounced. Agents invest in different skewed assets (forgo diversification benefits to hold skewed assets.) Complicates Aggregation: Representative agent has different preference structure from individual (possibly identical) investors.

50 Empirical Phenomena: households expect upward sloping consumption profile (Barsky et al. (1997)) actual average consumption growth is non-positive and profiles are concave (Gourinchas (2002)) Setup: Finite-lived agent, quadratic utility u(c t ) = ac t 1 2 bc2 t, one risk-free asset, Rβ = 1, i.i.d. income: Objective prob.: Subjective prob.: ) y t independent over time Π (y t ȳ t 1 = dπ (y t ) > 0 for all y [ y, ȳ ] ). ˆΠ (y t ȳ t 1 0 for all y [ y, ȳ ]

51 Optimal Consumption Euler equation: c t ( A t, ȳ t ) = Ê [c t+1 (A t+1, ȳ t+1 ) ȳ t ] Consumption rule: ( ( ) ct y = 1 R 1 T t A t 1 R (T t) t + y t + R τ Ê Note: c t τ=1 [ y t+τ ȳ t ] ) ] depends only on [y Ê t+τ ȳ t (not higher moments)

52 Optimal Consumption Euler equation: c t ( A t, ȳ t ) = Ê [c t+1 (A t+1, ȳ t+1 ) ȳ t ] Consumption rule: ( ( ) ct y = 1 R 1 T t A t 1 R (T t) t + y t + R τ Ê Note: c t τ=1 [ y t+τ ȳ t ] ) ] depends only on [y Ê t+τ ȳ t (not higher moments)

53 Optimal Beliefs So Variance only lowers anticipatory utility, but does not affect c OE exhibit no uncertainty for quadratic utility. Therefore [ Ê u ( ct+τ ) ] ]) ȳt = u (Ê [ct+τ ȳ t Note: agents who expect risk have the same behavior and lower felicity

54 Optimal Beliefs So Variance only lowers anticipatory utility, but does not affect c OE exhibit no uncertainty for quadratic utility. Therefore [ Ê u ( ct+τ ) ] ]) ȳt = u (Ê [ct+τ ȳ t Note: agents who expect risk have the same behavior and lower felicity

55 Certainty + Euler equation wellbeing simplifies to 1 T T ψ t E t=1 [ ( ( ))] u ct y t and FOC implies an actual consumption path of ( ) ct y = a t b ψ ( t+τ R τ a [ ( ) ]) ψ t b E ct+τ t+τ y ȳ t ( where ψ t = β t ) T t τ=1 (βτ + (βδ) τ )

56 average consumption path average consumption path for agent with rational expectations T-1 T t Figure: Consumption Path

57 average consumption path overconsumption (overoptimism) consumption at t = 1 for agent with optimal expectations T-1 T t Figure: Consumption Path

58 average consumption path overconsumption (overoptimism) consumption at t = 1 for agent with optimal expectations + expected consumption path for agen with optimal expectations at t = T-1 T t Figure: Consumption Path

59 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 Figure: Consumption Path

60 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 Figure: Consumption Path

61 average consumption path Initial over- consumption (overoptimism) T-1 T t Figure: Consumption Path

62 average consumption path Initial over- consumption (overoptimism) c (t) + + c OE OE (t) T-1 T t Figure: Consumption Path

63 Proposition Undersaving For all t < [ T T ] [ (i) Ê t 1 T τ=0 R τ y t+1+τ ȳ t > E [Ê t 1 τ=0 R τ y t+1+τ ȳ ( ) [ ( ) ] (ii) ct y > E ct+1 y t t+1 ȳ [ ( ) ] [ t ( ) ] (iii) Ê ct+1 t+1 y ȳ t > E ct+1 t+1 y ȳ ( ) ( ) t (iv) as T, ct y ct RE y t t Model predictions optimism and overconfidence consumption profile hump-shaped agent surprised by declining consumption on average overconsumption declines with costs (length of life)

64 Proposition Undersaving For all t < [ T T ] [ (i) Ê t 1 T τ=0 R τ y t+1+τ ȳ t > E [Ê t 1 τ=0 R τ y t+1+τ ȳ ( ) [ ( ) ] (ii) ct y > E ct+1 y t t+1 ȳ [ ( ) ] [ t ( ) ] (iii) Ê ct+1 t+1 y ȳ t > E ct+1 t+1 y ȳ ( ) ( ) t (iv) as T, ct y ct RE y t t Model predictions optimism and overconfidence consumption profile hump-shaped agent surprised by declining consumption on average overconsumption declines with costs (length of life)

65 Rational expectations are sub-optimal: Agents with rational beliefs makes the ex post best decisions but agents that care about the future can be happier with some optimism Utility gain determines biases Optimal expectations is a structural model of non-rational beliefs beliefs are most distorted when decision errors are small beliefs are most distorted when dream benefits are largest excess risk taking due to optimism, preference for skewness endogenous heterogenous beliefs; agreeing to disagree overconfidence, optimism, and lack of consumption insurance subjective procrastination, planning fallacy, context effect

66 Rational expectations are sub-optimal: Agents with rational beliefs makes the ex post best decisions but agents that care about the future can be happier with some optimism Utility gain determines biases Optimal expectations is a structural model of non-rational beliefs beliefs are most distorted when decision errors are small beliefs are most distorted when dream benefits are largest excess risk taking due to optimism, preference for skewness endogenous heterogenous beliefs; agreeing to disagree overconfidence, optimism, and lack of consumption insurance subjective procrastination, planning fallacy, context effect