Optimal Financial Transaction Taxes

Transcription

1 Optimal Financial Transaction Taxes Eduardo Dávila NYU Stern Advances in Price Theory Conference Becker Friedman Institute 12/04/ / 23

2 Motivation Should financial transactions be taxed? 2 / 23

3 Motivation Should financial transactions be taxed? Verbal arguments Tobin 72 Stiglitz 89, Summers and Summers 89, Ross 89 Little formal analysis 2 / 23

4 Motivation Should financial transactions be taxed? Verbal arguments Tobin 72 Stiglitz 89, Summers and Summers 89, Ross 89 Little formal analysis This paper: welfare-theoretic analysis of transaction taxes 2 / 23

5 Preview: 3 roles of financial markets Exchange Economy Fundamental Trading 3 / 23

6 Preview: 3 roles of financial markets Exchange Economy Risk Sharing/Transfer Life Cycle Informed Trading Fundamental Trading 3 / 23

7 Preview: 3 roles of financial markets Exchange Economy Risk Sharing/Transfer Life Cycle Informed Trading Fundamental Trading Non-fundamental Trading 3 / 23

8 Preview: 3 roles of financial markets Exchange Economy Risk Sharing/Transfer Life Cycle Informed Trading Fundamental Trading Non-fundamental Trading Speculation/Betting Belief Disagreement 3 / 23

9 Preview: 3 roles of financial markets Exchange Economy [OBPS] Risk Sharing/Transfer Life Cycle Informed Trading Fundamental Trading Tax τ > 0 Non-fundamental Trading Speculation/Betting Belief Disagreement 3 / 23

10 Preview: 3 roles of financial markets [OBPS] Fundamental Trading Tax τ > 0 Production Non-fundamental Trading 3 / 23

11 Preview: 3 roles of financial markets [OBPS] Fundamental Trading Tax τ > 0 Production q-theory Non-fundamental Trading 3 / 23

12 Preview: 3 roles of financial markets [OBPS] Fundamental Trading Tax/ Subsidy τ 0 Tax τ > 0 Production q-theory Non-fundamental Trading 3 / 23

13 Preview: 3 roles of financial markets [OBPS] Fundamental Trading Tax/ Subsidy τ 0 Tax τ > 0 Production Non-fundamental Trading 3 / 23

14 Preview: 3 roles of financial markets [OBPS] Fundamental Trading Tax/ Subsidy τ 0 Tax τ > 0 Production Non-fundamental Trading Information aggregation Irrelevance results Information acquisition Less information acquired Trading Costs and Informational Efficiency Davila/Parlatore (2015) 3 / 23

15 Roadmap 1. Baseline model: static exchange economy Positive analysis Normative analysis (main results) 4 / 23

16 Roadmap 1. Baseline model: static exchange economy Positive analysis Normative analysis (main results) 2. Extensions Static model (several) Dynamics Production 3. Conclusion Literature 4 / 23

17 Baseline model: CARA-Normal (Lintner 69) Single trading stage, t = {1, 2} Distribution of investors F (i) - CARA utility A i 5 / 23

18 Baseline model: CARA-Normal (Lintner 69) Single trading stage, t = {1, 2} Distribution of investors F (i) - CARA utility A i Two assets - Unrestricted portfolios 1. Riskless asset: Gross rate R = 1 - Elastic supply - Irrelevant endowment 2. Risky asset: Price P 1 > 0 - Fixed supply Q - Pays dividend D Initial position X 0i - Choose X 1i 5 / 23

19 Baseline model: CARA-Normal (Lintner 69) Single trading stage, t = {1, 2} Distribution of investors F (i) - CARA utility A i Two assets - Unrestricted portfolios 1. Riskless asset: Gross rate R = 1 - Elastic supply - Irrelevant endowment 2. Risky asset: Price P 1 > 0 - Fixed supply Q - Pays dividend D Initial position X 0i - Choose X 1i (True) Dividend distribution D N(E[D], Var[D]) Heterogeneous dogmatic beliefs (disagreement about means) D N(E i [D], Var[D]) 5 / 23

20 Baseline model: CARA-Normal (Lintner 69) Single trading stage, t = {1, 2} Distribution of investors F (i) - CARA utility A i Two assets - Unrestricted portfolios 1. Riskless asset: Gross rate R = 1 - Elastic supply - Irrelevant endowment 2. Risky asset: Price P 1 > 0 - Fixed supply Q - Pays dividend D Initial position X 0i - Choose X 1i (True) Dividend distribution D N(E[D], Var[D]) Heterogeneous dogmatic beliefs (disagreement about means) D N(E i [D], Var[D]) Stochastic endowment, E 2i, correlated with D, Cov[E 2i, D] 0 5 / 23

21 Baseline model: CARA-Normal (Lintner 69) Single trading stage, t = {1, 2} Distribution of investors F (i) - CARA utility A i Two assets - Unrestricted portfolios 1. Riskless asset: Gross rate R = 1 - Elastic supply - Irrelevant endowment 2. Risky asset: Price P 1 > 0 - Fixed supply Q - Pays dividend D Initial position X 0i - Choose X 1i (True) Dividend distribution D N(E[D], Var[D]) Heterogeneous dogmatic beliefs (disagreement about means) D N(E i [D], Var[D]) Stochastic endowment, E 2i, correlated with D, Cov[E 2i, D] 0 Four reasons to trade 1. [Fundamental] Different hedging needs Cov[E 2i, D] 2. [Fundamental] Different risk aversion A i 3. [Fundamental] Different initial conditions X 0i 4. [Non-fundamental] Different beliefs E i [D] 5 / 23

22 Policy instrument Linear anonymous tax: single instrument Paid by buyers and sellers on the dollar value of the transaction Revenue: 2τP 1 X 1i 6 / 23

23 Policy instrument Linear anonymous tax: single instrument Paid by buyers and sellers on the dollar value of the transaction Revenue: 2τP 1 X 1i Assumption: No tax avoidance 6 / 23

24 Policy instrument Linear anonymous tax: single instrument Paid by buyers and sellers on the dollar value of the transaction Revenue: 2τP 1 X 1i Assumption: No tax avoidance Lump-sum rebate: T 1i = τp 1 X 1i (for simplicity) 6 / 23

25 Policy instrument Linear anonymous tax: single instrument Paid by buyers and sellers on the dollar value of the transaction Revenue: 2τP 1 X 1i Assumption: No tax avoidance Lump-sum rebate: T 1i = τp 1 X 1i (for simplicity) Ex-ante lump-sum transfers (Kaldor/Hicks: focus on efficiency) 6 / 23

26 Investors problem Linear anonymous tax: single instrument Paid by buyers and sellers on the dollar value of the transaction Revenue: 2τP 1 X 1i Assumption: No tax avoidance Lump-sum rebate: T 1i = τp 1 X 1i (for simplicity) Ex-ante lump-sum transfers (Kaldor/Hicks: focus on efficiency) CARA utility: U i (W 2i ) = e A iw 2i max X 1i E i [U i (W 2i )] max E i [W 2i ] A i X 1i 2 Var [W 2i] 6 / 23

27 Investors problem Linear anonymous tax: single instrument Paid by buyers and sellers on the dollar value of the transaction Revenue: 2τP 1 X 1i Assumption: No tax avoidance Lump-sum rebate: T 1i = τp 1 X 1i (for simplicity) Ex-ante lump-sum transfers (Kaldor/Hicks: focus on efficiency) CARA utility: U i (W 2i ) = e A iw 2i max X 1i Return/Budget constraint: E i [U i (W 2i )] max E i [W 2i ] A i X 1i 2 Var [W 2i] W 2i = E 2i + X 1i D + X 0iP 1 X 1i P 1 τp 1 X 1i + T }{{ 1i } Tax/Rebate 6 / 23

28 Investors problem: solution when τ = 0 X 1i = E i [D] A i Cov [E 2i, D] P 1 A i Var [D] 7 / 23

29 Investors problem: Inaction + Dampening E i [D] A i Cov[E 2i,D] P 1 (1+τ) A i Var[D] ; X 1i > 0 Buyer X 1i = X 0i ; X 1i = 0 No Trade E i [D] A i Cov[E 2i,D] P 1 (1 τ) A i Var[D] ; X 1i < 0 Seller 7 / 23

30 Investors problem: Inaction + Dampening E i [D] A i Cov[E 2i,D] P 1 (1+τ) A i Var[D] ; X 1i > 0 Buyer X 1i = X 0i ; X 1i = 0 No Trade E i [D] A i Cov[E 2i,D] P 1 (1 τ) A i Var[D] ; X 1i < 0 Seller X 1i X 1i X 0i (Net Change in Asset Holdings) X 1i > 0 (Buyer) Inaction Region ( E i[d] A icov[e 2i,D] P 1 A ivar[d] Zero Tax Optimal Portfolio ) (Seller) X 1i < 0 X 0i (Initial Asset Holdings) 7 / 23

31 Investors problem: Inaction + Dampening E i [D] A i Cov[E 2i,D] P 1 (1+τ) A i Var[D] ; X 1i > 0 Buyer X 1i = X 0i ; X 1i = 0 No Trade E i [D] A i Cov[E 2i,D] P 1 (1 τ) A i Var[D] ; X 1i < 0 Seller X 1i X 1i X 0i (Net Change in Asset Holdings) Inaction Region τ ( E i[d] A icov[e 2i,D] P 1 A ivar[d] Zero Tax Optimal Portfolio ) X 0i (Initial Asset Holdings) Convex problem 7 / 23

32 Equilibrium Standard equilibrium definition - Market clearing X 1i df (i) = Q 8 / 23

33 Equilibrium Standard equilibrium definition - Market clearing X 1i df (i) = Q Equilibrium price P 1 8 / 23

34 Equilibrium Standard equilibrium definition - Market clearing X 1i df (i) = Q Equilibrium price P 1 Positive Results: Lemma 1: dp 1 Lemma 2: dx 1i can be positive/negative/zero Figure is negative for buyers (positive for sellers) dx 1i = X 1i τ + X 1i dp 1 P 1 8 / 23

35 Equilibrium Standard equilibrium definition - Market clearing X 1i df (i) = Q Equilibrium price P 1 Positive Results: Lemma 1: dp 1 Lemma 2: dx 1i can be positive/negative/zero Figure is negative for buyers (positive for sellers) dx 1i Aligned with empirical evidence = X 1i τ + X 1i dp 1 P 1 8 / 23

36 Normative analysis: Welfare criterion How to assess welfare with heterogeneous beliefs? 9 / 23

37 Normative analysis: Welfare criterion How to assess welfare with heterogeneous beliefs? My approach: 1. Solve planner s problem using a single belief E[D] 9 / 23

38 Normative analysis: Welfare criterion How to assess welfare with heterogeneous beliefs? My approach: 1. Solve planner s problem using a single belief E[D] 2. Characterize conditions under which the solution to the planner s problem is independent (!) of the belief chosen 9 / 23

39 Normative analysis: Welfare criterion How to assess welfare with heterogeneous beliefs? My approach: 1. Solve planner s problem using a single belief E[D] 2. Characterize conditions under which the solution to the planner s problem is independent (!) of the belief chosen Paternalism? 1. Philosophy - Does the planner respect investors beliefs? 9 / 23

40 Normative analysis: Welfare criterion How to assess welfare with heterogeneous beliefs? My approach: 1. Solve planner s problem using a single belief E[D] 2. Characterize conditions under which the solution to the planner s problem is independent (!) of the belief chosen Paternalism? 1. Philosophy - Does the planner respect investors beliefs? No 9 / 23

41 Normative analysis: Welfare criterion How to assess welfare with heterogeneous beliefs? My approach: 1. Solve planner s problem using a single belief E[D] 2. Characterize conditions under which the solution to the planner s problem is independent (!) of the belief chosen Paternalism? 1. Philosophy - Does the planner respect investors beliefs? No 2. Constrained Efficiency - Does the planner need to know more than the investors? 9 / 23

42 Normative analysis: Welfare criterion How to assess welfare with heterogeneous beliefs? My approach: 1. Solve planner s problem using a single belief E[D] 2. Characterize conditions under which the solution to the planner s problem is independent (!) of the belief chosen Paternalism? 1. Philosophy - Does the planner respect investors beliefs? No 2. Constrained Efficiency - Does the planner need to know more than the investors? Not always. No informational advantage 9 / 23

43 Normative analysis: Welfare criterion How to assess welfare with heterogeneous beliefs? My approach: 1. Solve planner s problem using a single belief E[D] 2. Characterize conditions under which the solution to the planner s problem is independent (!) of the belief chosen Paternalism? 1. Philosophy - Does the planner respect investors beliefs? No 2. Constrained Efficiency - Does the planner need to know more than the investors? Not always. No informational advantage Different from welfare criteria papers (complementary) BSX15: convex combination of beliefs BCEST15: worst case scenarios over set of possible beliefs GSS14: axiomatic 9 / 23

44 Normative analysis: Welfare criterion How to assess welfare with heterogeneous beliefs? My approach: 1. Solve planner s problem using a single belief E[D] 2. Characterize conditions under which the solution to the planner s problem is independent (!) of the belief chosen Paternalism? 1. Philosophy - Does the planner respect investors beliefs? No 2. Constrained Efficiency - Does the planner need to know more than the investors? Not always. No informational advantage Different from welfare criteria papers (complementary) BSX15: convex combination of beliefs BCEST15: worst case scenarios over set of possible beliefs GSS14: axiomatic Behavioral Welfare Economics O Donoghue-Rabin, Chetty, Farhi-Gabaix, Campbell 2016, etc 9 / 23

45 Normative analysis: Planner s problem Social welfare V (τ) - Pareto frontier - Welfare weights λ i V (τ) = λ i V i df (i) with V i E [U i (X 1i )] 10 / 23

46 Normative analysis: Planner s problem Social welfare V (τ) - Pareto frontier - Welfare weights λ i V (τ) = λ i V i df (i) with V i E [U i (X 1i )] 1. X 1i chosen by investors 2. Planner uses E, instead of E i 10 / 23

47 Marginal tax change Proposition 1a: General case dv = dv i λ i df (i) 11 / 23

48 Marginal tax change Proposition 1a: General case dv = dv i λ i df (i) 11 / 23

49 Marginal tax change Proposition 1a: General case dv = dv i λ i df (i) dv i = E [ U i (W 2i ) ] }{{} Expected Marginal Utility d ˆV i }{{} Change in Certainty Equivalent 11 / 23

50 Marginal tax change Proposition 1a: General case dv = λ i E [ U i (W 2i ) ] d ˆV i df (i) 11 / 23

51 Marginal tax change Proposition 1a: General case dv = λ i E [ U i (W 2i ) ] d ˆV i df (i) 11 / 23

52 Marginal tax change Proposition 1a: General case dv = λ i E [ U i (W 2i ) ] d ˆV i df (i) d ˆV i = E [D] E i [D] + sgn ( X }{{} 1i ) P 1 τ }{{} Belief distortion Fundamental distortion dx 1i X dp 1 1i }{{ } Terms-of-trade Detailed derivation 11 / 23

53 Marginal tax change Proposition 1a: General case dv = λ i E [ U i (W 2i ) ] d ˆV i df (i) d ˆV i = E [D] E i [D] + sgn ( X }{{} 1i ) P 1 τ }{{} Belief distortion Fundamental distortion dx 1i X dp 1 1i }{{ } Terms-of-trade Detailed derivation 11 / 23

54 Marginal tax change Proposition 1a: General case dv = λ i E [ U i (W 2i ) ] d ˆV i df (i) d ˆV i = E [D] E i [D] + sgn ( X }{{} 1i ) P 1 τ }{{} Belief distortion Fundamental distortion dx 1i X dp 1 1i }{{ } Terms-of-trade Detailed derivation 11 / 23

55 Marginal tax change Proposition 1a: General case dv = λ i E [ U i (W 2i ) ] d ˆV i df (i) d ˆV i = E [D] E i [D] + sgn ( X }{{} 1i ) P 1 τ }{{} Belief distortion Fundamental distortion dx 1i X dp 1 1i }{{ } Terms-of-trade Detailed derivation 11 / 23

56 Marginal tax change Proposition 1a: General case dv = λ i E [ U i (W 2i) ] E [D] E i [D] + sgn ( X 1i ) P 1 τ dx 1i }{{} }{{}}{{} Welfare weight Belief distortion Fundamental distortion X 1i dp 1 }{{} Terms-of-trade df (i) 11 / 23

57 Marginal tax change Proposition 1a: General case dv = λ i E [ U i (W 2i) ] E [D] E i [D] + sgn ( X 1i ) P 1 τ dx 1i }{{} }{{}}{{} Welfare weight Belief distortion Fundamental distortion X 1i dp 1 }{{} Terms-of-trade df (i) 11 / 23

58 Marginal tax change Proposition 1a: General case dv = λ i E [ U i (W 2i) ] E [D] E i [D] + sgn ( X 1i ) P 1 τ dx 1i }{{} }{{}}{{} Welfare weight Belief distortion Fundamental distortion X 1i dp 1 }{{} Terms-of-trade df (i) Assumption [NR]: No Redistribution λ i E [ U i (W 2i ) ] is constant i 11 / 23

59 Marginal tax change Proposition 1a: General case dv = λ i E [ U i (W 2i) ] E [D] E i [D] + sgn ( X 1i ) P 1 τ dx 1i }{{} }{{}}{{} Welfare weight Belief distortion Fundamental distortion X 1i dp 1 }{{} Terms-of-trade df (i) Assumption [NR]: No Redistribution λ i E [ U i (W 2i ) ] is constant i Kaldor/Hicks efficiency Quasilinearity Maximization of certainty equivalents (ex-ante transfers) 11 / 23

60 Marginal tax change Proposition 1b: [NR] holds When [NR] holds: [ dv = [(E [D] E i [D]) + sgn ( X 1i ) P 1 τ] dx ] 1i dp 1 X 1i df (i) 11 / 23

61 Marginal tax change Proposition 1b: [NR] holds When [NR] holds: [ dv = [(E [D] E i [D]) + sgn ( X 1i ) P 1 τ] dx ] 1i dp 1 X 1i df (i) Market clearing implies X 1i df (i) = 0 11 / 23

62 Marginal tax change Proposition 1b: [NR] holds When [NR] holds: [ dv = [(E [D] E i [D]) + sgn ( X 1i ) P 1 τ] dx ] 1i df (i) Terms-of-trade drop out 11 / 23

63 Marginal tax change Proposition 1b: [NR] holds When [NR] holds: [ dv = [(E [D] E i [D]) + sgn ( X 1i ) P 1 τ] dx ] 1i df (i) Market clearing implies dx1i df (i) = 0 11 / 23

64 Marginal tax change Proposition 1b: [NR] holds When [NR] holds: dv = [ E i [D] + sgn ( X 1i ) P 1 τ] dx 1i df (i) Planner s belief drops out Identical optimal policy for any belief (!) Consistency 11 / 23

65 Marginal tax change Proposition 1b: [NR] holds When [NR] holds: dv = [ E i [D] + sgn ( X 1i ) P 1 τ] dx 1i df (i) Planner s belief drops out Identical optimal policy for any belief (!) Consistency Key assumptions 1. No redistribution 2. Fixed supply 11 / 23

66 Marginal tax change Proposition 1c: Sign around τ = 0 When [NR] holds: dv = [ E i [D] + sgn ( X 1i ) P 1 τ] dx 1i df (i) 11 / 23

67 Marginal tax change Proposition 1c: Sign around τ = 0 When [NR] holds: dv = E i [D] dx 1i df (i) τ=0 11 / 23

68 Marginal tax change Proposition 1c: Sign around τ = 0 When [NR] holds: dv = τ=0 E i [D] dx ( 1i df (i) = Cov F E i [D], dx 1i ) τ=0 11 / 23

69 Marginal tax change Proposition 1c: Sign around τ = 0 When [NR] holds: dv = τ=0 E i [D] dx ( 1i df (i) = Cov F E i [D], dx 1i ) τ=0 Assumption [OBPS]: Optimists Buyers/Pessimists Sellers ( Cov F E i [D], dx ) 1i < 0 τ=0 11 / 23

70 Marginal tax change Proposition 1c: Sign around τ = 0 When [NR] holds: dv = τ=0 E i [D] dx ( 1i df (i) = Cov F E i [D], dx 1i ) τ=0 Assumption [OBPS]: Optimists Buyers/Pessimists Sellers ( Cov F E i [D], dx ) 1i < 0 τ=0 Two justifications: disagreement drives trading 1. Theoretical - X 1i = f 1 (E i [D]) + f 2 (Cov [E 2i, D], A i, X 0i ) 2. Empirical 11 / 23

71 Marginal tax change Proposition 1c: Sign around τ = 0 When [NR] holds: dv = τ=0 E i [D] dx ( 1i df (i) = Cov F E i [D], dx 1i ) τ=0 Assumption [OBPS]: Optimists Buyers/Pessimists Sellers ( Cov F E i [D], dx ) 1i < 0 τ=0 11 / 23

72 Marginal tax change Proposition 1c: Sign around τ = 0 When [NR] and [OBPS] hold: dv > 0 τ=0 11 / 23

73 Marginal tax change Proposition 1c: Sign around τ = 0 When [NR] and [OBPS] hold: dv > 0 τ=0 Intuition: start from τ = 0, τ Less trading Less Fundamental trading (2nd order loss) Less Non-fundamental trading (1st order gain) 11 / 23

74 Marginal tax change Proposition 1c: Sign around τ = 0 When [NR] and [OBPS] hold: dv > 0 τ=0 Intuition: start from τ = 0, τ Less trading Less Fundamental trading (2nd order loss) Less Non-fundamental trading (1st order gain) τ can be negative (a subsidy) if [OBPS] doesn t hold 11 / 23

75 Optimal tax τ Proposition 2: Optimal tax 1. When [NR] and [OBPS] hold, τ > 0 12 / 23

76 Optimal tax τ Proposition 2: Optimal tax 1. When [NR] and [OBPS] hold, τ > 0 2. When [NR] holds, the optimal tax is given by: τ = Ω B Ω S 2 12 / 23

77 Optimal tax τ Proposition 2: Optimal tax 1. When [NR] and [OBPS] hold, τ > 0 2. When [NR] holds, the optimal tax is given by: τ = Ω B Ω S 2 With Ω B i B ωb i E i [D] P 1 df (i), equivalently Ω S 12 / 23

78 Optimal tax τ Proposition 2: Optimal tax 1. When [NR] and [OBPS] hold, τ > 0 2. When [NR] holds, the optimal tax is given by: τ = Ω B Ω S 2 With Ω B i B ωb E i [D] i P 1 df (i), equivalently Ω S Sufficient statistics 1. Beliefs (Pigovian principle) 2. Equilibrium portfolio derivatives (drop out with symmetry) 12 / 23

79 Optimal tax τ Proposition 2: Optimal tax 1. When [NR] and [OBPS] hold, τ > 0 2. When [NR] holds, the optimal tax is given by: τ = Ω B Ω S 2 With Ω B i B ωb E i [D] i P 1 df (i), equivalently Ω S Sufficient statistics 1. Beliefs (Pigovian principle) 2. Equilibrium portfolio derivatives (drop out with symmetry) Alternative implementation: volume as intermediate target Volume 12 / 23

80 Optimal tax τ Proposition 2: Optimal tax 1. When [NR] and [OBPS] hold, τ > 0 2. When [NR] holds, the optimal tax is given by: τ = Ω B Ω S 2 With Ω B i B ωb E i [D] i P 1 df (i), equivalently Ω S Sufficient statistics 1. Beliefs (Pigovian principle) 2. Equilibrium portfolio derivatives (drop out with symmetry) Alternative implementation: volume as intermediate target Volume Measurement Recovering heterogenous beliefs, Borovicka/Davila (in progress) 12 / 23

81 Numerical examples 1. First Example: Only disagreement trading 2. Second Example: All cases 13 / 23

82 Numerical examples 1. First Example: Only disagreement trading 2. Second Example: All cases Simplifications Identical risk aversion A i = 1 Identical initial positions X 0i = Q = 1 Trading motives [Non-fundamental] Different beliefs E i [D] [Fundamental] Different risk-sharing needs Cov [E 2i, D] 13 / 23

83 Example 1: Only disagreement trading E [D] = 100 and Var [D] = / 23

84 Example 1: Only disagreement trading E [D] = 100 and Var [D] = 16 Optimists E H [D] = 106 Cov [E 2i, D] = 0 Optimistic Buyers (50%) Pessimists E L [D] = 96 Cov [E 2i, D] = 0 Pessimistic Sellers (50%) 14 / 23

85 Welfare 1.2 Optimists Equilibrium Allocation X 1i Optimists % Deviation Social Welfare Pessimists Shares Pessimists 0 86 Tax τ Equilibrium Price P Tax τ 0.16 Volume Dollars Shares Tax τ Example 1: Only non-fundamental trading - τ = 5.98% - Gain 0.86% Tax τ

86 Example 3: Optimists/Pessimists/Fundamental investors E [D] = 100 and Var [D] = / 23

87 Example 3: Optimists/Pessimists/Fundamental investors E [D] = 100 and Var [D] = 16 Optimists E H [D] = 106 Correct E[D] = 100 Pessimists E L [D] = / 23

88 Example 3: Optimists/Pessimists/Fundamental investors E [D] = 100 and Var [D] = 16 Optimists E H [D] = 106 Cov [E 2i, D] = 0 Optimistic Buyers (30%) Correct E[D] = 100 Pessimists E L [D] = / 23

89 Example 3: Optimists/Pessimists/Fundamental investors E [D] = 100 and Var [D] = 16 Optimists E H [D] = 106 Cov [E 2i, D] = 0 Optimistic Buyers (30%) Correct E[D] = 100 Pessimists E L [D] = 96 Cov [E 2i, D] = 0 Pessimistic Sellers (30%) 16 / 23

90 Example 3: Optimists/Pessimists/Fundamental investors E [D] = 100 and Var [D] = 16 Optimists E H [D] = 106 Cov [E 2i, D] = 0 Optimistic Buyers (30%) Cov [E 2i, D] < 0 Correct Buyers (20%) Correct E[D] = 100 Pessimists E L [D] = 96 Cov [E 2i, D] = 0 Pessimistic Sellers (30%) 16 / 23

91 Example 3: Optimists/Pessimists/Fundamental investors E [D] = 100 and Var [D] = 16 Optimists E H [D] = 106 Cov [E 2i, D] = 0 Optimistic Buyers (30%) Cov [E 2i, D] < 0 Correct Buyers (20%) Correct E[D] = 100 Cov [E 2i, D] > 0 Optimistic Sellers (20%) Pessimists E L [D] = 96 Cov [E 2i, D] = 0 Pessimistic Sellers (30%) 16 / 23

92 Example 3: Optimists/Pessimists/Fundamental investors E [D] = 100 and Var [D] = 16 Optimists E H [D] = 106 Cov [E 2i, D] = 0 Optimistic Buyers (30%) Cov [E 2i, D] < 0 Correct Buyers (20%) Correct E[D] = 100 Cov [E 2i, D] > 0 Optimistic Sellers (20%) Pessimists E L [D] = 96 Cov [E 2i, D] = 0 Pessimistic Sellers (30%) 16 / 23

93 Welfare 1 Optimists Buyers Equilibrium Allocation X 1i Correct Buyers 0.5 Pessimists Sellers 1.2 % Deviation Social Welfare Correct Buyers Shares Optimists Buyers Optimists Sellers 1.5 Optimists Sellers 0.7 Pessimists Sellers Tax τ Equilibrium Price P Tax τ 0.16 Volume Dollars Shares Tax τ Example 3: 35% Non-fundamental trading - τ = 2.01% - Gain 0.11% Tax τ

94 1.5 Welfare Optimists Buyers 1 % Deviation Pessimists Sellers Social Welfare Correct Buyers 1.5 Optimists Sellers Tax τ

95 Remarks τ = Ei [D] dx 1i df (i) P 1 sgn ( X1i ) dx 1i df (i) = Ω B Ω S 2 Remark 1: Investors who stop trading are inframarginal for τ Meaningful non-convexity Figure 18 / 23

96 Remarks τ = Ei [D] dx 1i df (i) P 1 sgn ( X1i ) dx 1i df (i) = Ω B Ω S 2 Remark 1: Investors who stop trading are inframarginal for τ Meaningful non-convexity Figure Remark 2: Harberger 64 revisited (money metric respecting beliefs) dx 1i Upper bound on welfare loss: L(τ) = 2τP 1 df (i) i B }{{} Change in volume 18 / 23

97 Remarks τ = Ei [D] dx 1i df (i) P 1 sgn ( X1i ) dx 1i df (i) = Ω B Ω S 2 Remark 1: Investors who stop trading are inframarginal for τ Meaningful non-convexity Figure Remark 2: Harberger 64 revisited (money metric respecting beliefs) dx 1i Upper bound on welfare loss: L(τ) = 2τP 1 df (i) i B }{{} Change in volume Remark 3: Allocation changes (volume) determine social welfare Intuition: price changes are only redistributional 18 / 23

98 Remarks τ = Ei [D] dx 1i df (i) P 1 sgn ( X1i ) dx 1i df (i) = Ω B Ω S 2 Remark 1: Investors who stop trading are inframarginal for τ Meaningful non-convexity Figure Remark 2: Harberger 64 revisited (money metric respecting beliefs) dx 1i Upper bound on welfare loss: L(τ) = 2τP 1 df (i) i B }{{} Change in volume Remark 3: Allocation changes (volume) determine social welfare Intuition: price changes are only redistributional Remark 4: Derivatives dx 1i Diamond 73 appear because of second-best problem 18 / 23

99 Extensions 1. Multiple (J) risky assets: weighted average J τ = ω j τj j=1 19 / 23

100 Extensions 1. Multiple (J) risky assets: weighted average J τ = ω j τj 2. Preexisting trading costs: τ formula unchanged, as long as they are a compensation for the use of economic resources j=1 19 / 23

101 Extensions 1. Multiple (J) risky assets: weighted average J τ = ω j τj 2. Preexisting trading costs: τ formula unchanged, as long as they are a compensation for the use of economic resources 3. Portfolio constraints: modeled as g (P 1 ) X 1i g (P 1 ) τ formula unchanged if price independent (short-sale constraints) Corrected formula if price dependent (borrowing constraints) j=1 19 / 23

102 Extensions 1. Multiple (J) risky assets: weighted average J τ = ω j τj 2. Preexisting trading costs: τ formula unchanged, as long as they are a compensation for the use of economic resources 3. Portfolio constraints: modeled as g (P 1 ) X 1i g (P 1 ) τ formula unchanged if price independent (short-sale constraints) Corrected formula if price dependent (borrowing constraints) 4. Asymmetric taxes/multiple instruments: First-best requires investor specific taxes: j=1 τ i = sgn ( X 1i ) F E i [D] P 1, F R 19 / 23

103 Extension: Dynamics General dynamic model: arbitrary utility/general disagreement 20 / 23

104 Extension: Dynamics General dynamic model: arbitrary utility/general disagreement 1. Approximation With constant marginal utility, the optimal CARA+Normal tax is recovered (Arrow-Pratt) 20 / 23

105 Extension: Dynamics General dynamic model: arbitrary utility/general disagreement 1. Approximation With constant marginal utility, the optimal CARA+Normal tax is recovered (Arrow-Pratt) 2. High frequency investors are more affected Intuition: forward-looking behavior Dynamics only modifies weights General dynamic model 20 / 23

106 Extension: Production 2. Production: new first-order effect Introduce producer who can vary S 1k New decision: how many trees to plant 21 / 23

107 Extension: Production 2. Production: new first-order effect Introduce producer who can vary S 1k New decision: how many trees to plant (E [D] E i [D]) dx1i df (i) = CovF [ E i [D], dx1i ] } {{ } Belief dispersion + (E [D] E F [E i [D]]) ds 1k }{{} Aggregate distortion Investment response Production 21 / 23

108 Extension: Production 2. Production: new first-order effect Introduce producer who can vary S 1k New decision: how many trees to plant (E [D] E i [D]) dx1i df (i) = CovF [ E i [D], dx1i ] } {{ } Belief dispersion + (E [D] E F [E i [D]]) ds 1k }{{} Aggregate distortion Investment response Production 21 / 23

109 Extension: Production 2. Production: new first-order effect Introduce producer who can vary S 1k New decision: how many trees to plant (E [D] E i [D]) dx1i df (i) = CovF [ E i [D], dx1i ] } {{ } Belief dispersion + (E [D] E F [E i [D]]) ds 1k }{{} Aggregate distortion Investment response Production 21 / 23

110 Broader price-theoretic agenda in Macro-Finance Three examples 1. Optimal Bankruptcy Exemptions How much should bankrupt borrowers keep? 2. Optimal Deposit Insurance What is the optimal level of deposit insurance coverage? 3. Fire Sales Externalities Which variables determine welfare losses associated with price changes in economies with financial frictions? 22 / 23

111 Conclusion This paper: microfounded welfare analysis of FTT Different welfare effects on fundamental vs non-fundamental trading Irrelevance of planner s belief 23 / 23

112 Conclusion This paper: microfounded welfare analysis of FTT Different welfare effects on fundamental vs non-fundamental trading Irrelevance of planner s belief Practical implications: 1. Belief dispersion: force towards positive tax 2. With production: wrong average beliefs needed 23 / 23

113 Literature 1. Tobin Tax: Proposals: Tobin 72/78, Summers-Summers 89, Stiglitz 89 Empirical: Campbell-Froot 94, Habermeier-Kirilenko 03 Theory: Subrahmanyam 98, Dow-Rahi 00, Buss et al 14, Adam et al Transaction Costs: Amihud-Mendelson 86, Constantinides 86, Vayanos 98, Mamaysky-Lo-Wang 04, Vayanos-Wang Belief Disagreement: Lintner 69, Miller 77, Harrison-Kreps 78, Scheinkman-Xiong 03, Hong-Stein 03, Geanakoplos 09, Simsek 12,13 4. Behavioral Welfare Economics: Belief Disagreement: Morris 95, Brunnermeier-Simsek-Xiong 12, Blume-Cogley-Easley-Sargent-Tsyrennikov 13 O Donoghue-Rabin 06, Bernheim-Rangel Information Diffusion/Acquisition: Grossman-Stiglitz 80, Diamond-Verrechia 81, many others Back to text 24 / 23

114 Identifying beliefs - Volume as intermediate target Volume decomposition: X 1i df (i) i B } {{ } Volume = Θ }{{} F + Θ }{{ NF } Fundamental Non-fundamental Θ }{{} τ, Tax induced reduction 25 / 23

115 Identifying beliefs - Volume as intermediate target Volume decomposition: X 1i df (i) i B } {{ } Volume = Θ }{{} F + Θ }{{ NF } Fundamental Non-fundamental Alternative implementation (conditions needed): Θ }{{} τ, Tax induced reduction Φ NF = Φ τ τ 25 / 23

116 Identifying beliefs - Volume as intermediate target Volume decomposition: X 1i df (i) i B } {{ } Volume = Θ }{{} F + Θ }{{ NF } Fundamental Non-fundamental Alternative implementation (conditions needed): Θ }{{} τ, Tax induced reduction Φ NF = Φ τ τ Back to text 25 / 23

117 Identifying beliefs - Volume as intermediate target Fundamental ( Cov [E2i, D] 2Ω F = + X 0i + S Var [D] ( Cov [E2i, D] + X 0i + Var [D] B ) P 1 A i Var [D] P 1 A i Var [D] df (i) ) df (i) 26 / 23

118 Identifying beliefs - Volume as intermediate target Fundamental ( Cov [E2i, D] 2Ω F = + X 0i + S Var [D] ( Cov [E2i, D] + X 0i + Var [D] B Non-Fundamental 2Ω NF = B E i [D] df (i) A i Var [D] S ) P 1 A i Var [D] P 1 A i Var [D] df (i) ) df (i) E i [D] df (i) A i Var [D] 26 / 23

119 Identifying beliefs - Volume as intermediate target Fundamental ( Cov [E2i, D] 2Ω F = + X 0i + S Var [D] ( Cov [E2i, D] + X 0i + Var [D] B Non-Fundamental 2Ω NF = Tax ( 2Ω τ = τ B B E i [D] df (i) A i Var [D] S ) P 1 A i Var [D] P 1 A i Var [D] df (i) ) df (i) E i [D] df (i) A i Var [D] ) P 1 P 1 df (i) + df (i) A i Var [D] S A i Var [D] 26 / 23

120 General dynamic model Environment Investors solve max C ti,x ti,y ti E i [ T ] β t 1 U i (C ti ) t=1 27 / 23

121 General dynamic model Environment Investors solve Subject to max C ti,x ti,y ti E i [ T ] β t 1 U i (C ti ) t=1 C ti = E ti + X t 1i (P t + D t ) X ti P t τp t X ti + T }{{ ti +RY } t 1i Y ti Tax/Rebate 27 / 23

122 General dynamic model Environment Investors solve Subject to max C ti,x ti,y ti E i [ T ] β t 1 U i (C ti ) t=1 C ti = E ti + X t 1i (P t + D t ) X ti P t τp t X ti + T }{{ ti +RY } t 1i Y ti Tax/Rebate Beliefs: Z ti Radon-Nikodym at each node/state Can be stochastic but cannot depend on endogenous variables 27 / 23

123 General dynamic model Environment Investors solve Subject to max C ti,x ti,y ti E i [ T ] β t 1 U i (C ti ) t=1 C ti = E ti + X t 1i (P t + D t ) X ti P t τp t X ti + T }{{ ti +RY } t 1i Y ti Tax/Rebate Beliefs: Z ti Radon-Nikodym at each node/state Can be stochastic but cannot depend on endogenous variables E ti and D t arbitrary distributions 27 / 23

124 General dynamic model Environment Investors solve Subject to max C ti,x ti,y ti E i [ T ] β t 1 U i (C ti ) t=1 C ti = E ti + X t 1i (P t + D t ) X ti P t τp t X ti + T }{{ ti +RY } t 1i Y ti Tax/Rebate Beliefs: Z ti Radon-Nikodym at each node/state Can be stochastic but cannot depend on endogenous variables E ti and D t arbitrary distributions Planner Single linear tax - Commitment No need to solve (hard) problem 27 / 23

125 General dynamic model: Three takeaways 1. Approximation: When marginal utilities are constant, the optimal CARA+Normal tax is recovered 28 / 23

126 General dynamic model: Three takeaways 1. Approximation: When marginal utilities are constant, the optimal CARA+Normal tax is recovered Intuition: Arrow/Pratt approximation - Small risks 28 / 23

127 General dynamic model: Three takeaways 1. Approximation: When marginal utilities are constant, the optimal CARA+Normal tax is recovered Intuition: Arrow/Pratt approximation - Small risks [ T E ] τ t=1 βt E ti [D t+1 + P t+1 ] dx ti df (i) = [ T E ] t=1 βt P t sgn ( X ti ) (1 κ ti ) dx ti df (i) 28 / 23

128 General dynamic model: Three takeaways 1. Approximation: When marginal utilities are constant, the optimal CARA+Normal tax is recovered Intuition: Arrow/Pratt approximation - Small risks [ T E ] τ t=1 βt E ti [D t+1 + P t+1 ] dx ti df (i) = [ T E ] t=1 βt P t sgn ( X ti ) (1 κ ti ) dx ti df (i) 28 / 23

129 General dynamic model: Three takeaways 1. Approximation: When marginal utilities are constant, the optimal CARA+Normal tax is recovered Intuition: Arrow/Pratt approximation - Small risks [ T E ] τ t=1 βt E ti [D t+1 + P t+1 ] dx ti df (i) = [ T E ] t=1 βt P t sgn ( X ti ) (1 κ ti ) dx ti df (i) 28 / 23

130 General dynamic model: Three takeaways 1. Approximation: When marginal utilities are constant, the optimal CARA+Normal tax is recovered Intuition: Arrow/Pratt approximation - Small risks [ T E ] τ t=1 βt E ti [D t+1 + P t+1 ] dx ti df (i) = [ T E ] t=1 βt P t sgn ( X ti ) (1 κ ti ) dx ti df (i) Truth is needed to weight nodes 28 / 23

131 General dynamic model: Three takeaways 1. Approximation: When marginal utilities are constant, the optimal CARA+Normal tax is recovered Intuition: Arrow/Pratt approximation - Small risks [ T E ] τ t=1 βt E ti [D t+1 + P t+1 ] dx ti df (i) = [ T E ] t=1 βt P t sgn ( X ti ) (1 κ ti ) dx ti df (i) [ ] κ ti E Pt+1 ti P t sgn ( X ti ) sgn ( X t+1i ) (Forward looking) 28 / 23

132 General dynamic model: Three takeaways 1. Approximation: When marginal utilities are constant, the optimal CARA+Normal tax is recovered Intuition: Arrow/Pratt approximation - Small risks 28 / 23

133 General dynamic model: Three takeaways 1. Approximation: When marginal utilities are constant, the optimal CARA+Normal tax is recovered Intuition: Arrow/Pratt approximation - Small risks 2. Lower tax with flip-floppers: the optimal tax can be written as τ = T E [w t f t τt ] t=1 28 / 23

134 General dynamic model: Three takeaways 1. Approximation: When marginal utilities are constant, the optimal CARA+Normal tax is recovered Intuition: Arrow/Pratt approximation - Small risks 2. Lower tax with flip-floppers: the optimal tax can be written as τ = T E [w t f t τt ] t=1 τ t Static tax - Weights T t=1 E [w t] = 1 - f t = 1 2 if Buy-and-Sell 28 / 23

135 General dynamic model: Three takeaways 1. Approximation: When marginal utilities are constant, the optimal CARA+Normal tax is recovered Intuition: Arrow/Pratt approximation - Small risks 2. Lower tax with flip-floppers: the optimal tax can be written as τ = T E [w t f t τt ] t=1 τt Static tax - Weights T t=1 E [w t] = 1 - f t = 1 2 if Buy-and-Sell Intuition: tax more powerful with forward-looking investors Larger tax needed to correct for persistent disagreement (ineffective for bubbles) 28 / 23

136 General dynamic model: Three takeaways 1. Approximation: When marginal utilities are constant, the optimal CARA+Normal tax is recovered Intuition: Arrow/Pratt approximation - Small risks 2. Lower tax with flip-floppers: the optimal tax can be written as τ dynamic 1 2 τ static 28 / 23

137 General dynamic model: Three takeaways 1. Approximation: When marginal utilities are constant, the optimal CARA+Normal tax is recovered Intuition: Arrow/Pratt approximation - Small risks 2. Lower tax with flip-floppers: the optimal tax can be written as τ dynamic 1 2 τ static 3. Price covariance matters/not variance 28 / 23

138 General dynamic model: Three takeaways 1. Approximation: When marginal utilities are constant, the optimal CARA+Normal tax is recovered Intuition: Arrow/Pratt approximation - Small risks 2. Lower tax with flip-floppers: the optimal tax can be written as τ dynamic 1 2 τ static 3. Price covariance matters/not variance Tobin/Keynes - Price volatility - Wrong argument Dynamic Harberger / Incomplete markets / Production 28 / 23

139 General dynamic model: Three takeaways 1. Approximation: When marginal utilities are constant, the optimal CARA+Normal tax is recovered Intuition: Arrow/Pratt approximation - Small risks 2. Lower tax with flip-floppers: the optimal tax can be written as τ dynamic 1 2 τ static 3. Price covariance matters/not variance Detailed Argument Back to text 28 / 23

140 Three takeaways: covariance 3. Price covariance matters/not variance 29 / 23

141 Three takeaways: covariance 3. Price covariance matters/not variance Tobin/Keynes - Price volatility - Wrong argument 29 / 23

142 Three takeaways: covariance 3. Price covariance matters/not variance Tobin/Keynes - Price volatility - Wrong argument Incomplete markets Dynamic Harberger - Assume λ i = 1 and β = 1 τ = T t=1 E [ E F [Z i U i (C ] ti) X ti ] dpt [ [ T ]] t=1 E dx E F ξti ti 29 / 23

143 Three takeaways: covariance 3. Price covariance matters/not variance Tobin/Keynes - Price volatility - Wrong argument Incomplete markets Dynamic Harberger - Assume λ i = 1 and β = 1 τ = T t=1 E [ E F [Z i U i (C ] ti) X ti ] dpt [ [ T ]] t=1 E dx E F ξti ti τ is positive when [ ] [ Cov Cov F Zi U i ] dp t (C ti ), X ti, < 0 29 / 23

144 Three takeaways: covariance 3. Price covariance matters/not variance Tobin/Keynes - Price volatility - Wrong argument Incomplete markets Dynamic Harberger - Assume λ i = 1 and β = 1 τ = T t=1 E [ E F [Z i U i (C ] ti) X ti ] dpt [ [ T ]] t=1 E dx E F ξti ti τ is positive when [ ] [ Cov Cov F Zi U i ] dp t (C ti ), X ti, < 0 Hard to disentangle insurance from redistribution Back to text 29 / 23

145 Production (q-theory) Producers - indexed by k 30 / 23

146 Production (q-theory) Producers - indexed by k Produce shares (trees) at a convex cost: Φ (S 1k ) 30 / 23

147 Production (q-theory) Producers - indexed by k Produce shares (trees) at a convex cost: Φ (S 1k ) Output: D (Q + S 1k ) 30 / 23

148 Production (q-theory) Producers - indexed by k Produce shares (trees) at a convex cost: Φ (S 1k ) Output: D (Q + S 1k ) Solve max C 1k,C 2k,S 1k U k (C 1k ) + E [U k (C 2k )] 30 / 23

149 Production (q-theory) Producers - indexed by k Produce shares (trees) at a convex cost: Φ (S 1k ) Output: D (Q + S 1k ) Solve max C 1k,C 2k,S 1k U k (C 1k ) + E [U k (C 2k )] s.t. C 1k + C 2k = E 1k + E 2k + P s 1 S 1k Φ (S 1k ) }{{} Production 30 / 23

150 Production (q-theory) Producers - indexed by k Produce shares (trees) at a convex cost: Φ (S 1k ) Output: D (Q + S 1k ) Solve max C 1k,C 2k,S 1k U k (C 1k ) + E [U k (C 2k )] s.t. C 1k + C 2k = E 1k + E 2k + P1 s S 1k Φ (S 1k ) }{{} Production Optimality conditions U k (C 1k) = E [ U k (C 2k) ] Euler P s 1 = Φ (S 1k ) Supply 30 / 23

151 Production (q-theory) Producers - indexed by k Produce shares (trees) at a convex cost: Φ (S 1k ) Output: D (Q + S 1k ) Solve max C 1k,C 2k,S 1k U k (C 1k ) + E [U k (C 2k )] s.t. C 1k + C 2k = E 1k + E 2k + P1 s S 1k Φ (S 1k ) }{{} Production Optimality conditions U k (C 1k) = E [ U k (C 2k) ] Euler P s 1 = Φ (S 1k ) Supply Social Welfare V (τ) = λ i V i df (i) + λ k V k 30 / 23

152 Production (q-theory) Producers - indexed by k Produce shares (trees) at a convex cost: Φ (S 1k ) Output: D (Q + S 1k ) Solve max C 1k,C 2k,S 1k U k (C 1k ) + E [U k (C 2k )] s.t. C 1k + C 2k = E 1k + E 2k + P1 s S 1k Φ (S 1k ) }{{} Production Optimality conditions U k (C 1k) = E [ U k (C 2k) ] Euler P s 1 = Φ (S 1k ) Supply Social Welfare V (τ) = λ i V i df (i) + λ k V k dv k = E [ U k (C 2k) ] dp 1 S 1k 30 / 23

153 Production (q-theory) Producers - indexed by k Produce shares (trees) at a convex cost: Φ (S 1k ) Output: D (Q + S 1k ) Solve max C 1k,C 2k,S 1k U k (C 1k ) + E [U k (C 2k )] s.t. C 1k + C 2k = E 1k + E 2k + P1 s S 1k Φ (S 1k ) }{{} Production Optimality conditions U k (C 1k) = E [ U k (C 2k) ] Euler P s 1 = Φ (S 1k ) Supply Social Welfare V (τ) = λ i V i df (i) + λ k V k Only terms-of-trade dv k = E [ U k (C 2k) ] dp 1 S 1k 30 / 23

154 Production Optimal tax under [NR] τ = (E [D] Ei [D]) dx 1i df (i) P 1 sgn ( X1i ) dx 1i df (i) 31 / 23

155 Production Optimal tax under [NR] τ = dx 1i df (i) = ds 1k (E [D] Ei [D]) dx 1i df (i) P 1 sgn ( X1i ) dx 1i df (i) 0, exchange economy ds 1k = 0 31 / 23

156 Production Optimal tax under [NR] (E [D] τ Ei [D]) dx 1i df (i) = P 1 sgn ( X1i ) dx 1i df (i) Numerator (first-order effects) [ (E [D] E i [D]) dx1i df (i) = CovF E i [D], dx1i ] } {{ } Belief dispersion + (E [D] E F [E i [D]]) ds 1k }{{} Aggregate distortion Investment response 31 / 23

157 Production Optimal tax under [NR] (E [D] τ Ei [D]) dx 1i df (i) = P 1 sgn ( X1i ) dx 1i df (i) Numerator (first-order effects) [ (E [D] E i [D]) dx1i df (i) = CovF E i [D], dx1i ] } {{ } Belief dispersion + (E [D] E F [E i [D]]) ds 1k }{{} Aggregate distortion Investment response 31 / 23

158 Production Optimal tax under [NR] (E [D] τ Ei [D]) dx 1i df (i) = P 1 sgn ( X1i ) dx 1i df (i) Numerator (first-order effects) [ (E [D] E i [D]) dx1i df (i) = CovF E i [D], dx1i ] } {{ } Belief dispersion + (E [D] E F [E i [D]]) ds 1k }{{} Aggregate distortion Investment response Back to text 31 / 23

159 Production Optimal tax under [NR] (E [D] τ Ei [D]) dx 1i df (i) = P 1 sgn ( X1i ) dx 1i df (i) Numerator (first-order effects) [ (E [D] E i [D]) dx1i df (i) = CovF E i [D], dx1i ] } {{ } Belief dispersion + (E [D] E F [E i [D]]) ds 1k }{{} Aggregate distortion Investment response A decomposition τ = ωτ exchange + (1 ω) τ production, Back to text 31 / 23

160 Production Optimal tax under [NR] (E [D] τ Ei [D]) dx 1i df (i) = P 1 sgn ( X1i ) dx 1i df (i) Numerator (first-order effects) [ (E [D] E i [D]) dx1i df (i) = CovF E i [D], dx1i ] } {{ } Belief dispersion + (E [D] E F [E i [D]]) ds 1k }{{} Aggregate distortion Investment response A decomposition τ = ωτ exchange + (1 ω) τ production, Sign of τ production? Back to text 31 / 23

161 Production Optimal tax under [NR] (E [D] τ Ei [D]) dx 1i df (i) = P 1 sgn ( X1i ) dx 1i df (i) Numerator (first-order effects) [ (E [D] E i [D]) dx1i df (i) = CovF E i [D], dx1i ] } {{ } Belief dispersion + (E [D] E F [E i [D]]) ds 1k }{{} Aggregate distortion Investment response A decomposition τ = ωτ exchange + (1 ω) τ production, Sign of τ production? Perhaps positive τ > 0 Back to text 31 / 23

162 Production Optimal tax under [NR] (E [D] τ Ei [D]) dx 1i df (i) = P 1 sgn ( X1i ) dx 1i df (i) Numerator (first-order effects) [ (E [D] E i [D]) dx1i df (i) = CovF E i [D], dx1i ] } {{ } Belief dispersion + (E [D] E F [E i [D]]) ds 1k }{{} Aggregate distortion Investment response A decomposition τ = ωτ exchange + (1 ω) τ production, Sign of τproduction? Perhaps positive τ > 0 sgn ( ( τproduction ) ( ) ) dp1 = sgn ˆP1 P 1 Back to text 31 / 23

163 Production Optimal tax under [NR] (E [D] τ Ei [D]) dx 1i df (i) = P 1 sgn ( X1i ) dx 1i df (i) Numerator (first-order effects) [ (E [D] E i [D]) dx1i df (i) = CovF E i [D], dx1i ] } {{ } Belief dispersion + (E [D] E F [E i [D]]) ds 1k }{{} Aggregate distortion Investment response A decomposition τ = ωτ exchange + (1 ω) τ production, Sign of τproduction? Perhaps positive τ > 0 sgn ( ( τproduction ) ( ) ) dp1 = sgn ˆP1 P 1 Back to text 31 / 23

164 Production Optimal tax under [NR] (E [D] τ Ei [D]) dx 1i df (i) = P 1 sgn ( X1i ) dx 1i df (i) Numerator (first-order effects) [ (E [D] E i [D]) dx1i df (i) = CovF E i [D], dx1i ] } {{ } Belief dispersion + (E [D] E F [E i [D]]) ds 1k }{{} Aggregate distortion Investment response A decomposition τ = ωτ exchange + (1 ω) τ production, Sign of τproduction? Perhaps positive τ > 0 sgn ( ( τproduction ) ( ) ) dp1 = sgn ˆP1 P 1 Back to text 31 / 23

165 Production Optimal tax under [NR] (E [D] τ Ei [D]) dx 1i df (i) = P 1 sgn ( X1i ) dx 1i df (i) Numerator (first-order effects) [ (E [D] E i [D]) dx1i df (i) = CovF E i [D], dx1i ] } {{ } Belief dispersion + (E [D] E F [E i [D]]) ds 1k }{{} Aggregate distortion Investment response A decomposition τ = ωτ exchange + (1 ω) τ production, Sign of τproduction? Perhaps positive τ > 0 sgn ( ( τproduction ) ( ) ) dp1 = sgn ˆP1 P 1 Back to text > 0 31 / 23

166 Hayekian production Environment 32 / 23

167 Hayekian production Environment π I Informed investors - Observe θ 32 / 23

168 Hayekian production Environment π I Informed investors - Observe θ π U = 1 π I Uninformed investors - Do not update from prices 32 / 23

169 Hayekian production Environment π I Informed investors - Observe θ π U = 1 π I Uninformed investors - Do not update from prices Firm manager - Chooses production given prices 32 / 23

170 Hayekian production Environment π I Informed investors - Observe θ π U = 1 π I Uninformed investors - Do not update from prices Firm manager - Chooses production given prices ( Dividend Π = D + θ + β θ 2 (θ k ) 2), β 0 32 / 23

171 Hayekian production Environment π I Informed investors - Observe θ π U = 1 π I Uninformed investors - Do not update from prices Firm manager - Chooses production given prices ( Dividend Π = D + θ + β θ 2 (θ k ) 2), β 0 Optimal investment: k = E [θ P 1 ] 32 / 23

172 Hayekian production Environment π I Informed investors - Observe θ π U = 1 π I Uninformed investors - Do not update from prices Firm manager - Chooses production given prices ( Dividend Π = D + θ + β θ 2 (θ k ) 2), β 0 Optimal investment: k = E [θ P 1 ] Assumptions 1. Informed investors do not internalize effect in production 32 / 23

173 Hayekian production Environment π I Informed investors - Observe θ π U = 1 π I Uninformed investors - Do not update from prices Firm manager - Chooses production given prices ( Dividend Π = D + θ + β θ 2 (θ k ) 2), β 0 Optimal investment: k = E [θ P 1 ] Assumptions 1. Informed investors do not internalize effect in production 2. Uninformed investors do not learn 32 / 23

174 Hayekian production Environment π I Informed investors - Observe θ π U = 1 π I Uninformed investors - Do not update from prices Firm manager - Chooses production given prices ( Dividend Π = D + θ + β θ 2 (θ k ) 2), β 0 Optimal investment: k = E [θ P 1 ] Assumptions 1. Informed investors do not internalize effect in production 2. Uninformed investors do not learn There is no trade in equilibrium when θ (τ) θ θ (τ) Sell No Trade Buy θ(τ) 0 θ(τ) 32 / 23

175 Hayekian production Marginal tax change [ ] dv dv = E θ + Ψ(τ) θ }{{} <0 33 / 23

176 Hayekian production Marginal tax change [ ] dv dv = E θ + Ψ(τ) θ }{{} <0 Where Ψ(τ) captures distortions in production efficiency ( ) Ψ(τ) V T (θ) V NT (θ) φ Var[θ] (θ) } {{ } >0 dθ (τ) }{{} <0 ( + ( ) )) V NT θ V T θ } {{ } <0 ) dθ (τ) φ Var[θ] (θ }{{} >0 33 / 23

177 Hayekian production Marginal tax change [ ] dv dv = E θ + Ψ(τ) θ }{{} <0 Where Ψ(τ) captures distortions in production efficiency ( ) Ψ(τ) V T (θ) V NT (θ) φ Var[θ] (θ) } {{ } >0 dθ (τ) }{{} <0 ( + ( ) )) V NT θ V T θ } {{ } <0 ) dθ (τ) φ Var[θ] (θ }{{} >0 33 / 23

178 Hayekian production Marginal tax change [ ] dv dv = E θ + Ψ(τ) θ }{{} <0 Where Ψ(τ) captures distortions in production efficiency ( ) Ψ(τ) V T (θ) V NT (θ) φ Var[θ] (θ) } {{ } >0 dθ (τ) }{{} <0 ( + ( ) )) V NT θ V T θ } {{ } <0 ) dθ (τ) φ Var[θ] (θ }{{} >0 33 / 23

179 Hayekian production Marginal tax change [ ] dv dv = E θ + Ψ(τ) θ }{{} <0 Where Ψ(τ) captures distortions in production efficiency ( ) Ψ(τ) V T (θ) V NT (θ) φ Var[θ] (θ) } {{ } >0 dθ (τ) }{{} <0 ( + Even locally first order Ψ (τ) τ=0 < 0 ( ) )) V NT θ V T θ } {{ } <0 ) dθ (τ) φ Var[θ] (θ }{{} >0 33 / 23

180 Hayekian production Marginal tax change [ ] dv dv = E θ + Ψ(τ) θ }{{} <0 Where Ψ(τ) captures distortions in production efficiency ( ) Ψ(τ) V T (θ) V NT (θ) φ Var[θ] (θ) } {{ } >0 dθ (τ) }{{} <0 ( + ( ) )) V NT θ V T θ } {{ } <0 Even locally first order Ψ (τ) τ=0 < 0 - Learning externality ) dθ (τ) φ Var[θ] (θ }{{} >0 33 / 23

181 Hayekian production Marginal tax change [ ] dv dv = E θ + Ψ(τ) θ }{{} <0 Where Ψ(τ) captures distortions in production efficiency ( ) Ψ(τ) V T (θ) V NT (θ) φ Var[θ] (θ) } {{ } >0 dθ (τ) }{{} <0 ( + ( ) )) V NT θ V T θ } {{ } <0 Even locally first order Ψ (τ) τ=0 < 0 - Learning externality ) dθ (τ) φ Var[θ] (θ }{{} >0 Optimal tax τ = τno-info + τinformation 0 }{{}}{{} >0 <0 33 / 23

182 Hayekian production Marginal tax change [ ] dv dv = E θ + Ψ(τ) θ }{{} <0 Where Ψ(τ) captures distortions in production efficiency ( ) Ψ(τ) V T (θ) V NT (θ) φ Var[θ] (θ) } {{ } >0 dθ (τ) }{{} <0 ( + ( ) )) V NT θ V T θ } {{ } <0 Even locally first order Ψ (τ) τ=0 < 0 - Learning externality ) dθ (τ) φ Var[θ] (θ }{{} >0 Optimal tax τ = τno-info + τinformation 0 }{{}}{{} >0 <0 If τno-info = 0, optimal policy is subsidy, not laissez-faire Back to text 33 / 23

183 89.5 Welfare (in dollars) Dollars Back to text Tax τ

184 Extra Derivations ˆV i = (E [D] A i Cov [E 2i, D] P 1 ) X 1i + P 1 X 0i A i 2 Var [D] (X 1i) 2 d ˆV i = (E [D] A icov [E 2i, D] P 1 A i X 1i Var [D]) dx 1i dp 1 X 1i dv i = [(E [D] E i [D]) + sgn ( X 1i ) P 1 τ] dx 1i dp 1 X 1i Back to text 35 / 23

185 Welfare Welfare Optimists 10 Pessimists 40 % Deviation Social Welfare % Deviation 5 0 Social Welfare Pessimists Optimists 30 Tax τ 3 Welfare Optimists Tax τ 2.5 Welfare Pessimists % Deviation Social Welfare % Deviation Social Welfare Pessimists 0.5 Optimists Tax τ Example 3: 35% Non-fundamental trading - τ = 2.01% - Gain 0.11% Tax τ

186 Intuition on dp 1 P 1 Demand H Supply P 1 Demand H Supply Total Demand Total Demand Demand L Demand L Sold Bought Sold Bought Q Quantity Q Quantity Back to text 37 / 23