JIANG WANG INTERTEMPORAL ASSET PRICES 259 4.4.The uninformed investor's optimization problem Let w"be an uninformed investor's wealth,X"his holding of the stock and c"his consumption.His optimization problem is Maxx".c" s.t.dW"=(rW“-c“)dt+X“dQ.(4.12) The solution to the optimization problem can,in general,be complicated.The uninformed investor's consumption-investment policy is a function of their information set,which contains the whole history of dividends and prices.Given the processes assumed for the primary state variables as well as the price,however,the information structure generated by"(t)has an equivalent representation which is the one generated by w.Here,w is the innovation process of the filters which is a Wiener process with respect to.The filters provide a sufficient statistic for(t).Using this equivalent representation of the information structure,we can restate the uninformed investor's optimization problem as a standard Markovian one with the filters being the effective state variables and the innovation process w generating the dynamics.It is then formally similar to the informed investors'problem.Thus,we have a situation in which the Separation Principle applies.24 LetJ“(W“;⑥；t)be the value function.Since E[△F]=0,from equation(4.7)the expected excess return of an uninformed investor is determined only by 6.Thus,his value function only depends on in addition to his wealth and time.J"satisfies the following Bellman equation: 0=Maxc".x"{-et-e“+E[dJ"(w";⑥；t)l"(t)]/d (4.13) s.t.dW"=(rW"-c“)dt+X“dQ, lim,+oE[J“(W“;⑧；t+r)川“(t)]=0, The solution to equation(4.13)is given in Theorem 4.4. Theorem 4.4.The program given by equation(4.13)has a solution of the form J“(W"；©；t)=-epl-rw"-v"() (4.14) where V"(©)=ΨuTv“业“，业r=(l,©).The optimal demand of the stock is X“=业". (4.15) Here,v"and f"=(fo,fe)are respectively (2x2)and (1x2)constant matrices. Proof.See Appendix B. 4.5.Market clearing Under the assumed form of the price function,the demand of stock by individual informed and uninformed investors is given respectively by equation (4.11)and(4.15).When the market clears,they must sum to 1+.Thus (1-ω)[f6+f6Θ+fa△]+w[f6+f88]=1+0. (4.16) 24.The Separation Principle states that under certain conditions,the optimal control problem with partial observed state variables can be solved in two stages:first solve for the optimal estimation problem for unobserved variables and then solve the control problem using the estimates as the state variables.For a detailed discussion on the Separation Principle,see Fleming and Rishel (1975)
260 REVIEW OF ECONOMIC STUDIES Using equation(4.3),we have the following equations: 1=(1-w)f0+of6, (4.17) 1=(1-w)f6+ωf&, (4.18) 0=(1-w)paf&-ωpmf6 (4.19) The solution to equation(4.17-4.19)determines the coefficients po,Pe,Pa in the price function.25 This completes our proof of the Proposition.26 5.PRICE VARIABILITY,RISK PREMIUM AND RETURN AUTOCORRELATION In what follows,we analyze how the underlying information structure affects stock prices, the risk premium,price volatility and serial correlation in returns. In the current model,the parameter characterizes the information structure of the economy.Most of the comparative static analysis is concerned with the effect of changing @As discussed in Section 2,captures two aspects of the information structure of our model.One aspect is the imperfection in some investor's information.As w increases, more investors have imperfect information and the total amount of information in the market decreases.The other aspect is information asymmetry.As changes,the extent to which information is asymmetric among investors also changes.For example,when is slightly less than 1 the investors are no longer homogeneous and information is asymmetric among them.Hence,as we change we are changing these two aspects of the information structure at the same time.The net effect will be the sum of the two. The effect of imperfect information is best illustrated by comparing the two extreme cases:@=0 and @=1.The former corresponds to the case of perfect information while the latter the case of pure imperfect information.Comparing the result for in the vicinity of the extreme case (@=0 or 1)with the extreme case illustrates the effect of information asymmetry. 5.1.Stock prices As stated in the Proposition in Section 4,the equilibrium price of the stock is P=Φ+(P+PoΘ)+Pa△. (5.1) where is given in Theorem 3.1. Two extreme cases deserve our attention:the benchmark case of @=0 in Section 3 and the case when @=1,i.e.when all investors are uninformed.When @=0,the equilibrium price is P*=(+p)+pD+pnI+p as derived in Section 3.Let p** be the equilibrium price when @=1.In this case,II drops out of the price function since it is not in any investors'information set and in equilibrium the price reveals the true value of P**has the following form: P**=(+p*)+pD+pin+pa*, (5.2) where po*and p are the solutions to equation(4.17-4.19)when =1.It can be shown that po*<p<o and p*<0. 25.Equation (4.17-4.19)is a set of algebraic equations.The proof of existence of a solution is available from the author on request. 26.We do not have an analytical form of the solution to equations(4.17-4.19).Numerical solutions are used in most of the analysis to follow