# Giffen Goods and Market Making

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16 Appendix The following lemma, which is useful to compute conditional expected values, adapts a standard result from normal theory to the present context (see e.g. DeGroot (1969), Theorem 1, section 9.9). Lemma 1 Suppose that X 1, X 2,..., X n is a random sample from a multivariate normal distribution with unknown mean vector M and specified precision matrices Σ i, i = 1, 2,..., n. Suppose also that the prior distribution of M is multivariate normal with mean vector µ o and precision matrix Σ o such that µ o IR K and Σ o is a symmetric positive definite matrix. Then the posterior distribution of M when X i = x i (i = 1, 2,..., n) is a multivariate normal with mean vector µ 1 and precision matrix Σ 1 = Σ o + n i=1 Σ i, where µ 1 = Σ 1 1 (Σ o µ o + ( n i=1 Σ i) x) and x = ( n i=1 Σ i) 1 ( n i=1 Σ ix i ). Proof. For M = m and X i = x i (i = 1, 2,..., n), the likelihood function f n (x 1, x 2,..., x n m) satisfies the following relation: f n (x 1, x 2,..., x n m) exp However, it can be verified that { (1/2) } n (x i m) Σ i (x i m). (5.4) i=1 n i=1 (x i m) Σ i (x i m) = (m x) ( n i=1 Σ i ) (m x) + n (x i x) Σ i (x i x). i=1 Since the last term in the previous equation does not involve m, we can rewrite (5.4) as follows: { ( n ) } f n (x 1, x 2,..., x n m) exp (1/2)(m x) Σ i (m x). (5.5) i=1 The prior p.d.f. of M satisfies ϕ(m) exp { (1/2)(m µ o ) Σ o (m µ o )}, (5.6) and the posterior p.d.f. g( x 1, x 2,..., x n ) of M will be proportional to the product of the functions specified by (5.5) and (5.6). However, one can verify that ( n ) (m µ o ) Σ o (m µ o ) + (m x) Σ i (m x) = (m µ 1 ) Σ 1 (m µ 1 ) i=1 + terms not involving m. 16

17 Hence, we can write g(m x 1, x 2,..., x n ) exp { (1/2)(m µ 1 ) Σ 1 (m µ 1 )}. (5.7) The p.d.f. specified by (5.7) is that of a multivariate normal distribution for which the mean vector and the precision matrix are as specified in the statement of the lemma. QED Proof of proposition 2 When submitting his demand, a trader i has available the vector of signals s i. In any linear equilibrium, due to the strong law of large numbers, private and public information are conditionally independent. Therefore, the trader s strategy depends both on s i, and on the equilibrium price vector p. Assume the agent submits a vector of demand schedules X Ii (s i, p), indicating the position desired in each asset k at every price vector p, contingent on the available information, and restrict attention to linear equilibria. Market makers thus observe the vector of aggregate order flows L( ) = 1 0 X Iidi + u. Consider a candidate symmetric, linear equilibrium X Ii (s i, p) = As i + φ(p), where A and φ( ) are respectively the matrix of trading intensities and a linear function of current prices. Also, assume for the time being that the matrix A is nonsingular (in equilibrium this assumption turns out to be correct). Owing to linear strategies, the aggregate order flow is then: L( ) = z + φ(p), where z = Av + u, is the vector of order flows informational contents. Because of competition for each order flow and risk neutrality, market makers set the equilibrium price equal to the conditional expectation of the pay-off vector given the informational content of the order flows. Notice that A 1 z v N(v, A 1 Π 1 u (A ) 1 ), hence we can apply lemma 1 with n = 1, m = v and x 1 = A 1 z to obtain: p = E[v z] = Π 1 (Πv v + A Πuz) = Λ RN z + (I Λ RN A) v, where Π = (Var[v z]) 1 = Πv + A ΠuA and Λ RN = Π 1 A Πu. Given the formula for the market price I can now solve for the the demand function of a generic trader i. Since in equilibrium the parameters of the price are known, the trader conditions his estimation of the pay-off vector on p and s i. utility function and multivariate normality gives: The assumption of a CARA X Ii (s i, p) = γ(var[v s i, z]) 1 (E[v s i, z] p). (5.8) Notice again that s i v N(v, Π 1 ɛ ) and that p in equilibrium is observationally equivalent to A 1 z which we know satisfies A 1 z v N(v, A 1 Π 1 u (A ) 1 ). Applying again lemma 1 with n = 2, x 1 = s i, x 2 = A 1 z and m = v gives E[v s i, z] = Π 1 i (Πp + Πɛs i ), where Π i = 17

18 (Var[v s i, z]) 1 = Πv + A ΠuA + Πɛ. Plugging these expressions in (5.8) and simplifying, X Ii = γπɛ(s i p). Identifying the demand components: A = γπɛ and φ(p) = Ap. Notice that A = A = γπɛ, hence the assumption that A is nonsingular is correct in equilibrium. To determine a market maker s demand, consider the equilibrium condition 1 0 X Ii di + u X MMj dj = z Ap + X MMj = 0. Solving for z from the equilibrium price, substituting it in the above equation and isolating X MMj (p) gives: X MMj (p) = (Λ 1 RN A)( v p). QED Proof of proposition 3 Along the same lines of the previous proof, owing to CARA utility functions and multivariate normality X Ii (s i, p) = γ I (Var [v s i, z]) 1 (E [v s i, z] p) (5.9) = γ I Πɛ(s i p) + γ I Π (E [v z] p), and X Uj (p) = γ U (Var [v z]) 1 (E [v z] p) (5.10) = γ U Π (E [v z] p). Imposing market clearing, p = Λ U z + (I Λ U A) v, where A = γ I Πɛ, Λ U = (A + γπ) 1 (I + γaπu) and γ = γ I + γ U. Solving for z in the equilibrium price gives z = Λ 1 U p (I + γaπu) 1 γπv v, and substituting it in (5.9) and (5.10) gives the demand functions for informed and uninformed traders displayed in proposition 3. QED References Admati, A. R. (1985). A noisy rational expectations equilibrium for multi-asset securities markets. Econometrica 53, Barlevy, G. and P. Veronesi (2002, April). Rational panics and stock market crashes. Journal of Economic Theory (forthcoming). 18

19 Battacharya, U. and M. Spiegel (1991). Insiders, outsiders, and market breakdowns. Review of Financial Studies 4 (2), Bhattacharya, U., P. J. Reny, and M. Spiegel (1995). Destructive interference in an imperfectly competitive multi-security market. Journal of Economic Theory 65 (1), Caballé, J. and M. Krishnan (1992). Insider trading and asset pricing in an imperfectly competitive market. Econometrica 62, Cespa, G. (2001). A comparison of stock market mechanisms. UPF Working Paper 545. Cespa, G. (2002). Short-term investment and equilibrium multiplicity. European Economic Review 46, DeGroot, M. H. (1969). Optimal Statistical Decisions. McGraw-Hill. Diamond, D. and R. E. Verrecchia (1981). Information aggregation in a noisy rational expectations economy. Journal of Financial Economics 9, Gennotte, G. and H. Leland (1990). Market liquidity, hedging, and crashes. American Economic Review 80 (5), Grossman, S. and J. Stiglitz (1980). On the impossibility of informationally efficient markets. American Economic Review 70, He, H. and J. Wang (1995). Differential information and dynamic behavior of stock trading volume. Review of Financial Studies 8 (4), Hellwig, M. F. (1980). On the aggregation of information in competitive markets. Journal of Economic Theory 22, Kyle, A. (1985). Continuous auctions and insider trading. Econometrica 53, Vives, X. (1995a). Short-term investment and the informational efficiency of the market. Review of Financial Studies 8 (1), Vives, X. (1995b). The speed of information revelation in a financial market. Journal of Economic Theory 67,

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