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where the coefficients (with carats) can be expressed as functions of the coefficients in Equations (5) (without carats). A rational expectations equilibrium with linear price functions exists for the rational-investor economy if there exists a solution to the set of simultaneous equations relating the coefficients of Equations (8) to their counterparts in Equations (5). Existence can be demonstrated for limiting cases of the rational-investor economy in which a fully revealing equilibrium price exists at time 2. Assuming such a price, one finds that fi2 = 1> аг = 7г = К = 0> ft = (sx№V\hQ + A0)/i2i(A0 + *i) + *0], i = l- ft, and = sfr represent an equilibrium. When the time 2 price is fully revealing, the time 1 price coefficients are identical to those in Hellwigs (1980) single-period model. Sufficient conditions for the time 2 price to be fully revealing are that the private signals at time 1 or 2 are noiseless (sj = 0 or s2 = 0) or that p = - 1 and V\ = V\. In the first case, и is fully revealed by each investors time 2 private information set. In the second, the aggregate supply disturbance is identically zero so P2 is not perturbed by noise. In either case, the time 2 price is fully revealing regardless of whether or not investors recall the historical price Pv Under the following alternative conditions P2 is not fuiiy revealing, but Px and P2 are jointly fully revealing. First, suppose that V\ = 0 and the other variances are finite and nonzero. This implies that x2 = 0 and, as may be seen from the correctly conjectured equilibrium price functional (5), Pi and P2 jointly reveal и and xx. Alternatively, suppose that the second-period supply increment is proportional to the first periods supply increment: x2 = qxx. Then, as before, Px and P2 jointly reveal и and xx. In either of these cases, P2 = и to avoid arbitrage.6 In the one-period models of Grossman (1976) and Grossman and Stiglitz (1980) without a random per capita supply, equilibrium price reveals a sufficient statistic for the asset payoff. As is well known, this implies that each investors signal is redundant so that there is no incentive for investors to condition their beliefs and demands on their private information. In our model, if Px and P2 jointly reveal u, then the absence of arbitrage implies that P2 = u, that is, current price fully reveals the payoff u. Yet current price P2 is fully revealing only because the two prices jointly reveal и and xx\ when historical price is forgotten by every investor, current price is not fully revealing. Thus, there is no incentive for any individual to expend resources to observe past price (which is public information) because each investor is conditioning on past price. Beyond the special cases discussed above, there are no results establishing existence of an equilibrium in the rational-investor economy. For this reason, we turn to a more simple economy in which investors behave myopically. General existence conditions are established for this economy. 6 Equilibrium price coefficients also can be derived when there is no time 2 signal (s1 = 0) and there is no time 2 supply shock (V\ = 0); see Grundy and McNichols (1989) for this result. 1.2 Equilibrium in a myopic-investor economy The myopic-investor economy is identical to that presented in the previous section except that the first-period demands E(P21 5fl) - Pl d tfvarCP2S*) are substituted for the demands (6b) in the determination of the market-clearing price (8b).7 Note that the primary difference in demands (6c) and (6b) is the elimination of a hedging demand that is proportional to the time 1 expectation of time 2 demand. Subject to this difference, the equilibrium price coefficients are determined in the myopic-investor economy as they are in the rational-investor economy. As others, for example, Singleton (1985), have found, the elimination of the investors hedging demands simplifies the analysis. Indeed, it is possible to prove Theorem 1. Given that Vj, V22, slf s2) and R are each greater than zero and that \p\ < 1, a linear, noisy rational expectations equilibrium exists in the myopic-investor economy, and each of the price coefficients other than (possibly) at and y2 is nonzero. Proof. The proof proceeds by deriving expressions for the coefficients a2i Pi, 72> and 52 of Equation (5a) as functions of the ratio Z = $Jy Substituting these expressions into ft and уг of Equation (8b) provides a fifth-degree polynomial in Z [defined by Equation (A24) of Appendix В]. An equilibrium exists if and only if there exists a finite zero to this polynomial satisfying b2 Ф 0. A solution, not necessarily unique, is shown to exist such that a2i /?2, and 52 are nonzero. The details are provided in Appendix B. 1.3 The conditions for technical analysis to have value Risky asset demands in Equations (6) depend on the contemporaneous price and conditional moments of u. The linearity of ixt2 = E(u\At2), = £(wE;i), and Е(Р2\йц) as functions of the elements of the information sets Ей and Efl allows demands to be written as linear functions of those elements: da = *2d>o + *2яУл + *гъУк + *24Л + *252 (9a) dn = *пУо + *12Уп + *1зЛ (9b) These linear forms obtain whether investors are myopic or rational, although the values of the coefficients differ in the two economies. ТА has no value in equilibrium if and only if the investors time 2 beliefs 7 The formal results in Appendix A are applicable to the myopic-investor economy except for the derivation of the time 1 demands in Proposition A4 and the derivation of the time 1 price coefficients in Proposition Аб. and demands do not vary with Px after conditioning on the other variables in Ec; that is, if and only if \F24 = 0. Equation (6a) implies that Яг24 = 0 if and only if the investors time 2 expectation of the risky asset payoff \xt2 does not vary with Px and, hence, with xx. In other words, ТА has no value if and only if cov(w, PX\P2, уш yn, 20) = 0. Equations (7a) and (8a) imply that /xt2 does not vary with xx if and only if the time 2 price coefficient on xlt 72, is equal in value to the coefficient on x2l 52. Thus, the conditions, (1) tf24 = 0, (2) cov(w, Рг\Р2, yt2, yn, So) = 0, (3) 72 = в2, and (4) ТА has no value, are equivalent. Using expressions in Appendix A for /?2, y2, and 82 as functions of Z = it can be shown that y2 = 52 and that ТА has no value if and only if Ё1=Ь±± (10a) 72 Я*1 *2 Pi (* + %)v;(vi + Pv2) yx RsVj + pVxV2 + VI) (10b) Condition (10b) defines a subset of the parameter space in which ТА has no value, while (10a) follows from (10b) and the necessary conditions for an equilibrium. Conditions (10) permit us to make a strong statement regarding the value of ТА in the myopic-investor economy. Theorem 2. Under the parametric restrictions of Theorem 1, ТА has value in every linear, two-period rational expectations equilibrium of the myopic-investor economy. Proof. See Appendix B. This result follows from a demonstration that the ratio (10b) defining Zis not a zero of the fifth-degree polynomial equilibrium condition. In the following section is a numerical analysis of the rational-investor economy equating the coefficients of Equations (8) with those in Equations (5). The results demonstrate the existence of equilibria in which Equations (10) do not obtain and ТА is of value. The results also quantify this value for a range of parameter values. 2. The Value of Technical Analysis in the Rational-Investor Economy The value of ТА is assessed in two ways. First, the relation between the optimal time 2 demand for the risky asset by an individual and the time 1 price is examined. Second, the value to an otherwise uninformed investor of observing the time 1 price is computed. The equilibrium price distributions are held constant as this comparison is made in order to provide a measure of the private value of technical analysis. 1 [ 2 ] 3 4 5 6 7 8 |