The Social Value of Private Information

Transcription

1 The Social Value of Private Information Tarek A. Hassan 1, Thomas M. Mertens 2 1 University of Chicago, NBER and CEPR 2 New York University Weihnachtskonferenz December 19, / 27

2 Motivation Much of the information relevant for economic decisions is dispersed across individuals. A cornerstone of modern economics is the idea that prices aggregate this dispersed information: Allocation of resources in competitive markets maximize welfare because prices aggregate dispersed information (Hayek, 1945). In an efficient market asset prices aggregate both public and private information (Fama, 1970). Nevertheless, standard models of information aggregation in financial markets do not allow for welfare analysis. Lack the formal tools for policy analysis. 2 / 27

3 Contribution Model Propose a workhorse model of information aggregation in financial markets which allows for policy analysis. The market s capacity to aggregate information depends on investors near-rational errors. Findings When the asset traded is in fixed supply, the social value of private information is almost always negative. In a competitive market with endogenous capital, the social value of private information is ambiguous. 3 / 27

4 The Standard Model Take the standard Noisy Rational Expectations model (e.g. Hellwig, 1980) Two time periods (t = 0, 1). Two assets: Z units of a risky asset, which pays X N( X, σ X ) in t = 1. storable risk-free asset (the numéraire) in inelastic supply. Investors i [0, 1] receive a private signal about X s i = X + ι i ν i, ν i N (0, 1) ; observe the equilibrium price of the risky asset, p. Based on this information, investors beliefs are E 1i [X ] = E[X s i, p] 4 / 27

5 The Standard Model Investors own the endowment of the risk-free asset and choose z i = arg max E 1i exp [ ρw 1i ] subject to w 1i = e + z i (X p). Investors choose the demand for the risky asset z i = E 1i[X ] p ρvar 1i [X ] The conditional variance Var 1i [X ] = Var[X s i, p] reflects the amount of learning from prices and signals. 5 / 27

6 The Standard Model Noise Traders j [0, 1] own the endowment of the risky asset and choose z j = µε, ε N (0, 1) subject to w 1j = pz + z j (X p). An equilibrium consists of optimal demands by investors, noise traders demand, and prices that satisfy the market clearing condition 1 0 z i di z j dj = Z. 6 / 27

7 Expected Utility in the Standard Model 1. Standard model contains class of non-maximizing Noise Traders. Can ignore Noise Traders and define expected utility for Investors [ 1 ] E[U] = E U i di - Market s capacity to aggregate information is determined by a parameter (µ). 2. Expected utility driven by wealth transfers from noise traders to investors. - Utility of investors decreases with the information content of p! Welfare analysis doesn t make much sense. 0 7 / 27

8 Expected Utility in the Standard Model E U Μ 8 / 27

9 Modifying the Standard Model 1. Kill noise traders: Investors own all assets initially. w 1i = e + pz + z i (X p) 2. Market s capacity to aggregate information determined by agents: Near-rational behavior ε N (0, 1) shifts the posterior distribution of investors beliefs about X. Investor i makes near-rational errors of size µ i. Ê 1i (X ) = E 1i X + µ i ε. 9 / 27

10 Modifying the Standard Model 1. Kill noise traders: Investors own all assets initially. W 1 = e + pz + Z (X p) 2. Market s capacity to aggregate information determined by agents: Near-rational behavior ε N (0, 1) shifts the posterior distribution of investors beliefs about X. Investor i makes near-rational errors of size µ i. Ê 1i (X ) = E 1i X + µ i ε. 9 / 27

11 Modifying the Standard Model 1. Kill noise traders: Investors own all assets initially. W 1 = e + ZX 2. Market s capacity to aggregate information determined by agents: Near-rational behavior ε N (0, 1) shifts the posterior distribution of investors beliefs about X. Investor i makes near-rational errors of size µ i. Ê 1i (X ) = E 1i X + µ i ε. 9 / 27

12 Modifying the Standard Model 1. Kill noise traders: Investors own all assets initially. w 1i = e + pz + z i (X p) 2. Market s capacity to aggregate information determined by agents: Near-rational behavior ε N (0, 1) shifts the posterior distribution of investors beliefs about X. Investor i makes near-rational errors of size µ i. Ê 1i (X ) = E 1i X + µ i ε. 9 / 27

13 Modifying the Standard Model 1. Kill noise traders: Investors own all assets initially. w 1i = e + pz + z i (X p) 2. Market s capacity to aggregate information determined by agents: Near-rational behavior ε N (0, 1) shifts the posterior distribution of investors beliefs about X. Investor i makes near-rational errors of size µ i. Ê 1i (X ) = E 1i X + µ i ε. Posterior Distribution (Densities) Rational Near Rational ǫ E i(η) E i(η) 9 / 27

14 E it (X ) = A 0 + A 1 s i + A 2 p Investor i p, Stock market Investor j E jt (X ) = A 0 + A 1 s j + A 2 p 10 / 27

15 Rational Expectations Eq. Investor i Investor j E it (X ) = A 0 + A 1 s i + A 2 p trades in stock market trades in stock market E jt (X ) = A 0 + A 1 s j + A 2 p p= π 0 + A 1 1 A 2 X 10 / 27

16 Ê it (X ) = A 0 + A 1 s i + A 2 p+µ ε Investor i p= π 0 + A 1 1 A 2 X Investor j Ê jt (X ) = A 0 + A 1 s j + A 2 p+µ ε 10 / 27

17 Ê it (X ) = A 0 + A 1 s i + A 2 p+µ ε Investor i Investor j trades in stock market p= π 0 + A 1 1 A 2 X +µ ε trades in stock market Ê jt (X ) = A 0 + A 1 s j + A 2 p+µ ε 10 / 27

18 Investor i Investor j Ê it (X ) = A 0 + A 1 s i + A 2 p+µ ε trades in stock market p= π 0 + A 1 1 A 2 X +µ ε trades in stock market Ê jt (X ) = A 0 + A 1 s j + A 2 p+µ ε 10 / 27

19 Investor i Investor j Ê it (X ) = A 0 + A 1 s i + A 2 p+µ ε trades in stock market p= π 0 + A 1 1 A 2 X +µ ε trades in stock market Ê jt (X ) = A 0 + A 1 s j + A 2 p+µ ε 10 / 27

20 Investor i Investor j Ê it (X ) = A 0 + A 1 s i + A 2 p+µ ε trades in stock market trades in stock market Ê jt (X ) = A 0 + A 1 s j + A 2 p+µ ε p= π 0 + A 1 1 A 2 X A 2 µ ε 10 / 27

21 Investor i Investor j Ê it (X ) = A 0 + A 1 s i + A 2 p+µ ε trades in stock market trades in stock market Ê jt (X ) = A 0 + A 1 s j + A 2 p+µ ε p= π 0 + A 1 1 A 2 X A 2 µ ε 1. The price is linear in fundamentals and near-rational errors. 2. Near-rational errors are amplified as investors inform on equilibrium stock price. 10 / 27

22 Investor i Investor j Ê it (X ) = A 0 + A 1 s i + A 2 p+µ ε trades in stock market trades in stock market Ê jt (X ) = A 0 + A 1 s j + A 2 p+µ ε p= π 0 + A 1 1 A 2 X A 2 µ ε 1. The price is linear in fundamentals and near-rational errors. 2. Near-rational errors are amplified as investors inform on equilibrium stock price. - Amplification larger the more households rely on p. 10 / 27

23 Near-Rational Expectations Eq. Investor i Investor j Ê it (X ) = A 0 + A 1 s i + A 2 p+µ ε trades in stock market trades in stock market Ê jt (X ) = A 0 + A 1 s j + A 2 p+µ ε p= π 0 + A 1 1 A 2 X A 2 µ ε 1. The price is linear in fundamentals and near-rational errors. 2. Near-rational errors are amplified as investors inform on equilibrium stock price. - Amplification larger the more households rely on p. 10 / 27

24 Near-Rational Expectations Eq. Investor i Investor j Ê it (X ) = A 0 + A 1 s i + A 2 p+µ ε trades in stock market trades in stock market Ê jt (X ) = A 0 + A 1 s j + A 2 p+µ ε p= π 0 + A 1 1 A 2 X A 2 µ ε 1. The price is linear in fundamentals and near-rational errors. 2. Near-rational errors are amplified as investors inform on equilibrium stock price. - Amplification larger the more households rely on p. 3. Information content of equilibrium price falls as investors adjust A 0, A 1 and A / 27

25 Conditional variance Var 1 i X Var X Μ Figure: Example for σ X = 1, χ = 2, ρ = 5, X = 2, e = 1, Z = 1,ι = 10 Direct mapping between near-rational errors and the market s capacity to aggregate information. 11 / 27

26 Welfare Analysis The model generates information aggregation without the presence of non-maximizing agents. In a symmetric equilibrium, market s capacity to aggregate information depends on two simple variables: near-rational errors by every agent µ and the noise in private information ι. Near-rational errors induces only small losses in individual expeced utility(envelope theorem) but determines the market s capacity to aggregate information. Agents are thus near-rational. Can now perform welfare analysis. 12 / 27

27 Social Welfare 1. Social planner: A social planner dictates near-rational errors and private information. The result is a constrained efficient allocation. 2. Private incentives: An investor changes the amount of errors and quality of information given others choices. Comparison with social planner isolates effects of externalities. The social welfare function is defined as SWF [ι, µ] = = E[U i [ι i, µ i, ι i, µ i ]]di E[U i [ι, µ, ι, µ]]di The social planner chooses equal ι and µ across the population. 13 / 27

28 SWF Welfare with Near-Rationality (1/2) Μ Figure: Example for σ X = 1, χ = 2, ρ = 5, X = 2, e = 1, Z = 1,ι = 10 As the information aggregation worsens, social welfare falls monotonically. 14 / 27

29 Welfare with Near-Rationality (2/2) The Social Value of Information: SWF Example for σ X = 1, χ = 2, ρ = 5, X = 2, e = 1, Z = 1,µ = Ι 15 / 27

30 The Value of Information Central result: Social welfare decreases with better private information over almost the entire range. This result holds in any generic model of information aggregation. To understand the source of this effect, we decompose the social welfare function. 16 / 27

31 Social Welfare Social welfare is given by the unconditional expectation of utility [ 1 ] [ 1 ] E U i di = E e ρ(e+pz+z i (X p)) di 0 0 The exponent consists of products of normally distributed variables. We apply the formula for the expectation of random variables with an inverse Wishart distribution. 17 / 27

32 Social Welfare Social welfare is given by the unconditional expectation of utility [ 1 ] E U i di = E [ e ] ρ(e+pz+z i (X p)) 0 The exponent consists of products of normally distributed variables. We apply the formula for the expectation of random variables with an inverse Wishart distribution. 17 / 27

33 Social Welfare Social welfare is given by the unconditional expectation of utility [ 1 ] E U i di = E [ e ] ρ(e+pz+z i (X p)) 0 The exponent consists of products of normally distributed variables. We apply the formula for the expectation of random variables with an inverse Wishart distribution. 17 / 27

34 Social Welfare Social welfare is given by the unconditional expectation of utility [ 1 ] E U i di = E [ e ] ρ(e+zx +(z i Z)(X p)) 0 The exponent consists of products of normally distributed variables. We apply the formula for the expectation of random variables with an inverse Wishart distribution. 17 / 27

35 Social Welfare Social welfare is given by the unconditional expectation of utility [ 1 ] E U i di = E [ e ] ρ(e+zx +(z i Z)(X p)) 0 The exponent consists of products of normally distributed variables. We apply the formula for the expectation of random variables with an inverse Wishart distribution. 17 / 27

36 Two Determinants of Welfare Decompose Social Welfare into two components: 2 ρ2 σz Cov i 0 (X p,w 1 ) ρσz E[X p] i 2(1 ρ 2 σz 2 σ 2 i X p ) 2(1 ρ 2 σz 2 σ 2 i X p ) SWF = e ρe 0[w 1 ] e 1 2 ρ2 var 0 (w 1 ) e }{{}, = SWF agg, aggregate 1 ρ 2 σz 2 i σx 2 p }{{} = SWF dist, dispersion where w 1 = 1 0 w 1idi and σ zi = var 0 (z i ). Social Welfare depends on the expected level and variance aggregate wealth, as well as on the dispersion of wealth across investors at t = / 27

37 Two Determinants of Welfare SWF ag Μ SWF dist Μ SWF ag Ι SWF dist Ι Information aggregation affects welfare through the dispersion of portfolio holdings. 19 / 27

38 The Value of Information The discussion on informational efficiency is dominated by the free rider problem (Grossman and Stiglitz, 1980). But the previous discussion shows that there are two optima for social welfare when the supply of assets is fixed: The case of full information. The case of no information. In the intermediate range, better private information results in lower welfare. How do private incentives deviate from social incentives? 20 / 27

39 Social vs. Private Incentives Expected utility of an agent is given by E [ U i [ι i, µ i, ι i, µ i ] ] The social marginal product is driven by d dx E [ U i [ι, µ, ι, µ] ] = d SWF [ι, µ] for x = ι, µ dx whereas private incentives are driven by d dx i E [ U i [ι i, µ i, ι i, µ i ] ] for x = ι, µ The difference between private and social marginal product arises due to an informational externality or externalities associated with avoiding near-rational errors. 21 / 27

40 Private vs Social Marginal Product Μ Ι dswf dι dswf dι deui deui dιi dιi Private and social product take opposite signs for any near-rationality and a wide range of informational quality. In summary, the social benefits of better information might well be negative. 22 / 27

41 A Model with Endogenous Capital 1. Investors are not only endowed with the risk-free asset w 1i = e + z i (X p) + Π 2. but also own a technology that transforms the risk-free asset into units of the risky asset in t = 0: K = arg max Π = pk K 1 2 χk 2 = p 1 χ The market for capital clears when K = z i di Perform same experiment: Plot social welfare over µ. 23 / 27

42 Welfare Analysis SWF ag SWF dist Μ Μ SWF ag SWF dist Ι Ι When capital accumulation is endogenous the market s capacity to aggregate information determines the level of consumption and has a positive effect on welfare. 24 / 27

43 Private vs. Social Marginal Product Μ Ι dswf dswf dι deui dι deui dιi dιi When the number of assets are endogenous, the social value of information depends on the market s capacity to transmit information. Learning from prices and a reduction in the risk premium can dominate the negative effects from an increase in dispersion. 25 / 27

44 Interpretation In a competitive market with fixed asset supply, the social value of private information is almost always negative. Information acquisition on the part of investors is privately attractive but leads to a loss in social welfare. Before an IPO or in the medium-run, the social value of information is ambiguous. It depends on the market s capacity to aggregate information. 26 / 27

45 Conclusion Presented a workhorse model of information aggregation in financial markets that allows for policy analysis. The social value of information might be negative while there is still private incentive for more precise signals. Endogenous levels of capital raise the benefits from information. 27 / 27