A Welfare Criterion for Models with Distorted Beliefs
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1 A Welfare Criterion for Models with Distorted Beliefs Markus Brunnermeier, Alp Simsek, Wei Xiong August 2012 Markus Brunnermeier, Alp Simsek, Wei Xiong () Welfare Criterion for Distorted Beliefs August / 22
2 Belief distortions have no e ect on recent nancial regulations This research: Welfare criterion for models with heterogeneously distorted beliefs. Markus Brunnermeier, Alp Simsek, Wei Xiong () Welfare Criterion for Distorted Beliefs August / 22
3 A true story of famous economists Markus Brunnermeier, Alp Simsek, Wei Xiong () Welfare Criterion for Distorted Beliefs August / 22
4 They decide to bet, at some cost Markus Brunnermeier, Alp Simsek, Wei Xiong () Welfare Criterion for Distorted Beliefs August / 22
5 They realize that this is Pareto optimal Markus Brunnermeier, Alp Simsek, Wei Xiong () Welfare Criterion for Distorted Beliefs August / 22
6 The end of the story is unknown Markus Brunnermeier, Alp Simsek, Wei Xiong () Welfare Criterion for Distorted Beliefs August / 22
7 Heterogeneous beliefs might come from two distinct sources Subjective beliefs (Di erences in priors): Savage: Beliefs as representation of preferences. Spurious unanimity problem (Mongin, Gilboa, Schmeidler) Distorted beliefs (Mistakes in Bayesian updating): Biases that distort people s beliefs: Overcon dence, representativeness, limited attention... Pareto not appropriate. Need to use objective belief. Challenge: Planner might not know the objective belief. Markus Brunnermeier, Alp Simsek, Wei Xiong () Welfare Criterion for Distorted Beliefs August / 22
8 Our solution: Listen to all beliefs Chinese proverb: Listen to all, pluck a feather from every passing goose, but, follow no one absolutely. Our belief-neutral criterion: An allocation is (in)e cient, if it is (in)e cient under any reasonable belief. Reasonable: Any convex combination of agents beliefs. Same belief must be used to evaluate everyone s utility. Criterion is necessarily incomplete but works surprisingly well! Outline of the talk: Formalization and applications. Markus Brunnermeier, Alp Simsek, Wei Xiong () Welfare Criterion for Distorted Beliefs August / 22
9 Consider a generic dynamic economy Dynamic model with periods t 2 f0; 1; ::; T g. Uncertainty captured by state, s t. Agents i 2 f1; ::; Ng with di erent beliefs i = i t;s. A social allocation: x = xt i (s T ). Agent i s expected utility under own belief: E0 i ui st ; xt i (s T ). Set of reasonable beliefs: Any convex combination of agents beliefs: h = P i hi i, where h i 0 and P i hi = 1. Agent i s expected utility under belief h: E0 h ui st ; xt i (s T ). We o er two versions of our criterion... Markus Brunnermeier, Alp Simsek, Wei Xiong () Welfare Criterion for Distorted Beliefs August / 22
10 Belief-neutral criterion: Welfare function version Suppose the planner uses a xed social welfare function: W E0 h [u 1 ] ; E0 h [u 2 ] ; :::; E0 h [u N ] : E.g.: Bergsonian: P i i E h 0 [u i ] with i 0, Utilitarian: P i E h 0 [u i ]. De nition (V1: Expected social welfare) An allocation x is belief-neutral inferior to another allocation ^x if it leads to a lower social welfare under any reasonable belief h, i.e.: W E0 h u1 s T ; xt 1 (s T ) i ; :::; E0 hu h N s T ; xt N (s T ) W E0 h u1 s T ; ^x T 1 (s T ) i ; :::; E0 hu h N s T ; ^x T N (s T ). Markus Brunnermeier, Alp Simsek, Wei Xiong () Welfare Criterion for Distorted Beliefs August / 22
11 Belief-neutral criterion: Welfare function version Back to the story: Suppose planner uses utilitarian function: W = E h [u Joe ] + E h [u Bob ]. Planner assumes probability of natural is between 0:1 and 0:9. For any belief: Expected welfare from status quo = $100 + $100. Expected welfare from the bet = $100 + T h + $100 - T h - $50. The bet is belief-neutral inferior to the status quo. Key insight: Transfer has no e ect on the sum =) Negative sum. Markus Brunnermeier, Alp Simsek, Wei Xiong () Welfare Criterion for Distorted Beliefs August / 22
12 Belief-neutral criterion: Pareto version Suppose the planner uses a generalized Pareto criterion. De nition (V2: Belief-neutral Pareto) An allocation x is belief-neutral Pareto ine cient (resp. e cient) if, under any reasonable belief h, there exists (resp. does not exist) an alternative feasible allocation ^x s.t. E h 0 ui s T ; x i T (s T ) E h 0 ui s T ; ^x i T (s T ) for each agent i, with strict inequality for at least one i. Markus Brunnermeier, Alp Simsek, Wei Xiong () Welfare Criterion for Distorted Beliefs August / 22
13 Belief-neutral criterion: Pareto version Back to the story: Under Joe s belief: Bet: Expected transfer of $80 from Bob to Joe and a destroyed pillow. Alternative: Transfer $80 from Bob to Joe and save the pillow. Under any reasonable belief: There is a (belief-dependent) transfer that could make everyone better o. The bet is belief-neutral Pareto ine cient. Key insight: There is no belief that can rationalize the destroyed pillow. Markus Brunnermeier, Alp Simsek, Wei Xiong () Welfare Criterion for Distorted Beliefs August / 22
14 The two versions are closely related Common element across V1-V2: Consider all reasonable beliefs. V1 xes a social welfare function. Enables to compare allocations directly. V2 considers all social welfare functions. Proposition: x is belief-neutral e cient (ine cient) i under any reasonable belief h, there exists (does not exist) Pareto weights s.t.: x 2 arg max ^x feasible i=1 NX i E0 h ui s T ; ^x i T (s T ) : With V2, no direct comparisons. But categorizes allocations: 1 Belief-neutral ine cient... 2 Belief-neutral e cient... 3 Neither... Markus Brunnermeier, Alp Simsek, Wei Xiong () Welfare Criterion for Distorted Beliefs August / 22
15 Comments on the criterion We separate distorted beliefs from preferences. Move subjective beliefs into state-dependent utility. We evaluate welfare with heterogeneously distorted beliefs. Not applicable to aggregate mistakes. But negative-sum speculation prevalent in many applications. We detect also positive-sum speculation in relevant applications. Next: Brief discussion of various (negative and positive) applications. Markus Brunnermeier, Alp Simsek, Wei Xiong () Welfare Criterion for Distorted Beliefs August / 22
16 Application 1: Trading costs in bubble models Speculative bubbles: Harrison-Kreps (1978), Scheinkman-Xiong (2003)... Overvaluation and trading costs (brokerage fees, bid-ask spreads). =) Equilibrium is belief-neutral ine cient. Active trading severely undercuts portfolio returns (Barber-Odean...) Markus Brunnermeier, Alp Simsek, Wei Xiong () Welfare Criterion for Distorted Beliefs August / 22
17 Application 2: Overinvestment in bubble models S.bubbles also drive overinvestment: Bolton-Scheinkman-Xiong (2006)... Overvaluation with endogenous supply. Excessive investment according to all beliefs. =) Equilibrium is belief-neutral ine cient. Related app: Cons./savings distortions in macro models (Sims, 2008). Markus Brunnermeier, Alp Simsek, Wei Xiong () Welfare Criterion for Distorted Beliefs August / 22
18 Application 3: Bene ts of speculation in lemons models Speculation can also be positive-sum in lemons models. Overvaluation induces sellers with quality goods to sell. Mitigates the lemons problem. Additional gains from trade. =) Equilibrium is belief-neutral superior to no-trade status quo. Markus Brunnermeier, Alp Simsek, Wei Xiong () Welfare Criterion for Distorted Beliefs August / 22
19 Application 4: Bankruptcy costs in leverage cycle models Leverage cycles: Geanakoplos (2003, 2009), Simsek (2010)... Optimists use collateralized debt to nance their purchases. Might choose risky debt despite bankruptcy costs in bad state. Costs incurred with positive probability according to all beliefs. =) Equilibrium is belief-neutral ine cient. Markus Brunnermeier, Alp Simsek, Wei Xiong () Welfare Criterion for Distorted Beliefs August / 22
20 Application 5: Excessive risk taking with speculation Speculative trading models: Many examples in asset pricing. Previous models assume risk-neutrality for simplicity. But trading increases agents consumption risks. With risk-averse agents, negative-sum in expected utilities. =) Equilibrium is belief-neutral ine cient. Markus Brunnermeier, Alp Simsek, Wei Xiong () Welfare Criterion for Distorted Beliefs August / 22
21 Application 5.5: Speculation and risk sharing trade-o There can also be a risk sharing motive for trade (Simsek, 2012): Asset with payo v, where v N (0; 1). Two traders with MV prefs and endowments: e + v and e v. Common beliefs =) Trading reduces risks. Trader 1 s position and net worth: x 1 = 1 and n 1 = n 2 = e. Heterogeneous beliefs: Trader 1: v N ("; 1), trader 2: v N ( "; 1). Trader 1 s position and net worth: x 1 = 1 + " and n 1 = e + " v. Large disagreements, " > =) Trading increases risks. Equilibrium is belief-neutral ine cient whenever trading increases risks. Markus Brunnermeier, Alp Simsek, Wei Xiong () Welfare Criterion for Distorted Beliefs August / 22
22 Conclusion: A new welfare criterion Welfare criterion for models with heterogeneously distorted beliefs: Belief-neutral (in)e ciency, Listen to all beliefs. Incomplete ranking, but surprisingly powerful. Opens normative analysis for nancial regulation. Avoid negative-sum speculation (trading costs, over-investment, excessive risks...). Also detects positive-sum speculation (mitigate the lemons problem...). Markus Brunnermeier, Alp Simsek, Wei Xiong () Welfare Criterion for Distorted Beliefs August / 22
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