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Методички
Given a ratio Z = /VTi, the expressions for the time 2 coefficients satisfying Equation (A10) provide a2 = 1 - ft (A16) 02£>= VO2 + 2рЦ2 + v\)z2 - IhoRsis, + {Vp + V*)Z + £0#2(1 - pss, + + hois, + *2)22 (A17) Y2£> = hsisiiVp + F22) + M3(l - P2)sisiV\Vl - hoRpsis, + sVxV2 (A18) 52£> = -MKp + VI) Z + M3(l - P2)*22 + hoRss, + ) К2 (A19) where D = h0R2s2s2 (V\ + 2pVxV2 + F1)Z2 - 2hRsxs2sx + 52)(V1F2P + i)Z + R2(l - p*)sxs2V\ VIIsa + h0(Sl + 52)] + h0(Sl + 52)2F? Equating Equations (A18) and (A19) provides Equation (A15). Equality (A15) and the ratio of (A17) and (A18) provides (A14). Appendix B. A Myopic-Investor Equilibrium This appendix shows that an equilibrium exists and that technical analysis has value in every myopic-investor economy. The results of this appendix rely on the expressions for risky asset demands [Equations (6a) and (6c)]. These expressions are sensible only if var(P2S/i) and o\ are nonzero. The existence result, Theorem 1, demonstrates that yx and b2 are each greater than zero in equilibrium and that this implies var(JP2S/i) > 0 and <j\ > 0. The results of this appendix also rely on the fact that each of the arguments of Appendix A is applicable to the myopic-investor economy except for the derivations of first-period demands and price coefficients. Proof of Theorem 1 Given linear conjectures (5), the derivation of these coefficients of the price P2 proceeds as it does in Proposition A5. These coefficients must satisfy Equations (Al6) to (A19). Given demands (6c), market clearing at time 1 requires V - Pi , 7?var(P2 Ел) Using Equations (A20) and (A4) and the definition of 77, one finds the coefficients of the time 1 price satisfy 1 = 1 - 0i (A21) -A <a22) Т.-Жгау (Л23) The values var(P21 E/i), 02, and 03 can be calculated as functions of the times 1 and 2 price coefficients; see Proposition Al. The ratio of Equations (A22) and (A23) implies that the ratio Z = $Jyx satisfies 0 = hoRsblVl + 282y2pVxV2 + y22V\)Z3 - 2boRs102(62pV1V2 + y2V\)Z2 + [52#(1 - p2)Oi + h0)V2V22 + 520pViV2 + P2h0Rs2Vx + 72W] 02W (A24) A linear rational expectations equilibrium exists if and only if Equation (A24) obtains for some real number Z and var(P2Ert) > 0 and al > 0. From Equations (A17) to (A19), 02, 7г> and b2 are seen to be functions of Z. Upon substitution, Equation (A24) becomes a fifth-degree polynomial in Z which is hereby labeled F{Z). Proposition Al implies that the variances varCP2E,i) and al are positive when 71 and S2 are nonzero. Given the definition of Z, yx is nonzero when Zis finite. Equilibrium exists if and only if a finite Z exists such that F(Z) = 0 and 52(Z) Ф 0 where 52(Z) is given by Equation (A19). We begin to argue the existence of a satisfactory Z by collecting a few results. The values 02(Z), 72(Z), 52(Z), that is, Equations (A17) to (A19), are each finite when Z= 0. Also 02(O) > 0. Hence F(0) = -$2KV\ < 0. Let Z° satisfy 82(Z°) = 0; note that if pV2 + Vx = 0, then Z° does not exist, otherwise sign(Z°) = sign( Vx + pV2). Also, when Z° exists, 02(Z°) = Z°72(Z°) and, therefore, F(Z°) = 0. Differentiating F(Z), one finds h0RKl ~ P)2(l + p)2s\slV\V\ {h0[R2slS2(V2 + 2pVJK2 + VI) + 5l + %] = [VKp - 1) - 5! - 5,] - + pF2)2} Vi + pV2 The numerator is negative so that sign[F(Z°)] = -sign( Vx + pV2). Finally, limz .oo02(Z) = 1 and limz .oo72(Z) = limz .0052(Z) = 0, so that lim F(Z) = 00. Because F(Z) is continuous in Z, F(0) < 0 and lim*, F(Z) = 00, the intermediate value theorem [Rosenlicht (1968, p. 82)] implies the existence of 0 < Z* < 00 such that F(Z*) = 0. If V, + pV2 < 0, then Z° < 0 and, using the linearity of 52(Z), 52(Z*) Ф 0, so an equilibrium exists. If Vx + p V2 = 0, then 52(Z) 0, for all Z, so an equilibrium exists. If Vx + p V2 > 0, then F(Z°) = 0, F(Z°) < 0, and Z° > 0. Hence, using the intermediate value theorem and the linearity of 52(Z), there exists 0 < Z* < Z° such that F(Z*) = 0 and 52(Z*) Ф 0. It follows that a linear, two-period rational expectations equilibrium exists when investors are myopic. Proof of Theorem 2 Expression (A15) for the ratio Z = /3a/7i must obtain whenever ТА has no value in the myopic-investor equilibrium. The ratio Zmust simultaneously solve Equation (A24) in an equilibrium. Equality (A24), upon substitution of Equation (A15) into its right-hand side, becomes b0(l - p2)(s\ + sJV.Vs2 + 5, + R2s1s2(V2 + 2pVxVl + V2)]T biVl + 2pVxV2 + VDH&ss, + 40i + %>] -{V\ + 2pVxV2 + V2) + £0Oi + h)2)2 (A25) where T = hR2 + T2)(R2T5 + T4) + R2T5(R2T6 + Г7) г, s 5i52(f2 + гр + v2) T2 = 5j ~Г S2 П (v? + pv,v2){v\ + 2pviv2 + vi) Г4 - (s, + s2)[SlV\ + PV1V2(.sl - s2) - s2Vl] Г5 - s*$ (Vj + 2pVxV2 + VI) T6 ш Sls2 (v? + pViVjiVj + 2pVtV2 + VI) T7 m SlV* + pV1V2(.s1 - s2) - s2V\ Label as 0 that subset of W meeting the parametric restrictions, that is, *i> s2, V\, V\, h0, and R each greater than zero and \p\ < 1. Note that Equation (A25) obtains if and only if T = 0. Also note that each of the values ОРГц + T2), (R2 + T4), R2T5, and {P?T6 + T7) is never zero on 0. Thus, T is zero on 0 if and only if h0 is equal to -R2T5(R2T6 + Г7) - (R2T1+ T2)(R?T} + T4) J The value of h is never zero on 0 and, because each of the 7} is continuous on 0, h is continuous on 0. Inspection of h shows its value is negative whenever V, = V2 and sx = s2. Therefore, h0 and h cannot be equal and, consequently, ТА has value at all points in 0. 1 2 3 4 5 6 [ 7 ] 8 |