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Методички
Proposition Al. Given the conjectures (5), the coefficients of
<яяг/ quantities VAR [и, yi2, Р2)\Ац] anda22 = var(u\ai2) are constants conditional on S. Proposition Al follows from the properties of multivariate normal random variables. Computing an individuals time 1 demands in Proposition A3 below requires knowledge of the first row Gx = [Gu, G12] of the matrix G = (2N + W 1) 1, where W = VAR[(P2, мй)Ел] and where N is de- fined by <r2 2 -<r2 2 -<r22 a22 Note that W is invertible whenever s2, V\, and 82 are nonzero. Proposition A2. Given conjectures (5), Gn and G12 are constants conditional on S0. This proposition follows from the definition of N and the properties of multivariate normal random variables. Proposition A3. The optimal strategy (dw di2) for investor i given the conjectured price functional is E(P2 I 8fl) ~ Л , E(da I 8Д)(СЬ - Gu) d ~--+-- (A7) Proof The representation (A6) is the solution to the end-of-the-horizon problem [Equation (2a)] and its derivation can be found in Diamond and Verrecchia [1981, see their equation (10)]. Note that <г is nonzero when Vi> VI, Si> $2, 7ь and 82 are nonzero. Substituting Equation (Аб) into the maximand of Equation (2a), one finds the time 1 derived utility of time 2 wealth is Ji2 = -exp\-R[no + da(P2 - Pi)]---- Define Ц = (-#4i> 0), M\ = (P2) дп), and = £(M,. E ). Then, using knowledge of multinormal variables, ЕЛ) = E[-exp(-Rno + + ZJAf, - MtSM} E,J = -W 2N + W-M-expt- + Mi Л + Ш - QQt + (1/2X4 2Q;.N)(2N + W-1) 1 2NQ,)] The first-order condition with respect to dn is Px - Е{Рг\% + dnRGu - 2(?iNQ,= 0 Algebra provides dn = R-G[E{Pa\nd -Ptl] + IR-GWQ, Algebra and the definitions of Gly N, and Qt provide Equation (A7). Proposition A4. Given conjectures (5), the optimal risky asset demands dtl and di2 can be written as linear functions: da = *пУо + %тУп + *13Л da = *2iJo + %2Уп + *тэУа + *мЛ + *25Л Proposition А4 follows from Propositions Al and A3. Proposition A5. Given the price conjectures (5) and optimal demands (6), the price functions that satisfy the market-clearing conditions (3) are Рг = &2У0 + &2U ~ 72*i - 2*2 (A8) Л = &1У0 + iM - 7i*i (A9) where the coefficients (a2, P2, y2, b2i аь $ь у J are constants conditional on So and are functions of the conjectured parameter values (without carats). Proof Given demands (6), which are derived in Proposition A3, market clearing at time 2 requires that P2 satisfy X2 ~\~ x d2 = [XlMl + \2u + \,(Р2 - ij) - P2)R~la22 (A10) The second equality follows from Equation (A6) and the definitions of d2 and ix2. The third equality follows from Equation (A5) and the definitions of Mi, y2, and г). Relations (Al) and (A4) and the definitions of Mi and rj provide Mi = аУо + bu + cPx (All) V = вгУо + в2и + в5Рх (A12) Rearranging Equation (A10) as a definition of P2, then substituting for Mi, ry, and Px using Equations (All), (A12) and conjectures (5b), respectively, provides the constant coefficients of Equation (A8). Market clearing at time 1 requires x1 = 1=-+(Ml ?y)(Gl2 Gll) (A13) 1 1 RGU RGU RGua22 The second equality follows from Equation (A7) and the definitions of Mi and г]. Substituting in Equation (A13) for Mi and rj using Equations (All) and (A12), respectively, and algebra provides the constant coefficients of Equation (A9). Proposition A6. Technical analysis has value in equilibrium if and only ifb2 and у2 are unequal. Furthermore, when 82 and y2 are equal in equilibrium, the ratios fiy2 andfij/yt must satisfy 02 = Ь + Sx y2 Rs ft (sx + s1)V1(V1 + рУ2) 7i RsVl + pVxV2+ VI) Proof. The expressions for y2 and d2 derived from the market-clearing condition (A10) satisfy 72 2 = 7i(#2c Y2e,)(K2s2 - Г,) 1 where H2 = var(P2Ert), Y2 = cov(w, Р2ЕЛ), K2 = VARO , Р2ЕД), and с and 03 are defined in Equations (Al) and (A4), respectively. Hence in equilibrium (i.e., when y2 = y2, etc.), y2 and 82 are unequal if and only if H2c and Y265 are unequal. ТА has value if and only if di2 varies with Px. Examination of the right side of Equation (A6) shows that da varies with Px if and only if K2ls2H2c and K2ls2Y205 are unequal; to see this result use Equations (Al), (A4), and (A5). Hence, ТА has value if and only if y2 and 82 are unequal. (A14) (A15) 1 2 3 4 5 [ 6 ] 7 8 |