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Equilibrium in Dynamic Economy with Endogenous Beliefs

In this section, the Hellwig (1980) and Diamond and Verrecchia (1981) noisy rational expectations models are extended to two periods. A riskless asset and one risky asset are exchanged in markets opening at times t = 1 and t=2. Consumption occurs only at t= 3 when each share of the riskless asset pays 1 unit and the risky asset provides a random payoff of u. The riskless rate is assumed to be 0.2 Investors / (i = 1, 2, 3, . . .) are a priori identical and countably infinite in number. Each enters the first period with щ units of the riskless asset,3 and chooses a feasible trading strategy to maximize the expected utility of consumption at time 3:

£[-ехр(-Ли;в)Но] (1)

where R is the common absolute risk aversion parameter and S0 is the common prior information available to traders. Just prior to the opening of the market at time tf each individual receives a private signal, yit, of the time 3 payoff of the risky asset. Competitive trading establishes the risky asset price Pt at each date. The information available to investor / at time 1 and 2 is E* = {S0, ул, Pi) and aa = {Ел, уй, 2}> respectively.

A feasible trading strategy requires that planned asset holdings be measurable with respect to the traders available information and satisfy the individuals budget at each trading date. Let *4 denote individual fs time £ holding of the risky asset. Then the payoff at time 3 is n0 + da(P2 - Рг) + da(u - P2) and the optimal trading strategy is determined by sequentially solving

Ja{dn) s max £{-exp[-/?(wo + da(P2 - Pj

+ da(u - P2))] I 3J (2a)

2 Because consumption occurs only at the final date, investors marginal utilities at t = 1 and t = 2 are indeterminate without an exogenous specification of the riskless rate of interest.

3 As in Hellwig (1980), the (possibly random) endowments of shares of the risky asset are left unspecified. This assumption implies that individuals ignore information the random endowments might provide. In the multiperiod model developed here, the assumption also implies that individuals do not use the risky asset to hedge against future variations in wealth resulting from the random endowment. Brown and Jennings (1988) allow random individual endowments of the risky asset.

Ja = max Е{ГаШ I Ел} (2b)

Define 2, = (Su, a2n . . ) as the information set available to the market at time t A rational expectations equilibrium is a pair of demand functions (4i, dii) for each investor and a pair of equilibrium price functions (Pu P2) that together satisfy the following conditions. First, the P,are functions of 2, through their dependence on investors demands and per capita supplies,4 and, for each realization of 2 traders price conjectures are identical to P,(2,). Second, each traders strategy is feasible and solves Equations (2), when the conjectured price functions are used in Equation (2a). Finally, traders strategies and the equilibrium prices are such that markets clear. Define dt = lim 2<=1 dti/I, and let xu and хг + x denote the random per capita supplies of the risky asset at times 1 and 2, respectively.5 The market-clearing condition is written

x\ + #2 = d-2 (3a)

#i = dx (3b)

where these equalities hold with probability 1.

The exogenous random variables in the economy (the asset supplies and payoffs and the private signals received by investors) are assumed to follow a multivariate normal distribution. Normality of the distribution of the conjectured prices follows from their linear dependence on the exogenous variables in a rational expectations equilibrium.

Individuals homogeneous prior beliefs about и are represented by a normal distribution with mean y0 and variance ho. The private signal observed by individual / at time t is

yit = U + € (4)

where eit ~ N(0, sf) and Е{ши\а0) = 0. The signals errors are independent across investors and time periods. Because the € are distributed normally with finite variances homogeneous across investors, the law of large numbers implies that average signal, yt = limoo 2/=i/7, equals и with probability 1 for each t. The per capita supply increment xt is distributed N(0, Vf) conditional on S0, with independent of wand the private signals. The correlation between the supply increments xx and x is denoted p.

The stochastic behavior of Pt depends on its functional relationship to the exogenous variables xt and u. Individuals conjecture that prices are

4 See Diamond and Verrecchia (1981) for a discussion of this issue.

5 Noise is necessary for prices to be less than fully revealing of private information. Following Grossman and Stiglitz (1980), Diamond and Verrecchia (1981), and Hellwig (1980), we introduce noise via systematic variation in supply.

linear functions of the supply increments and aggregate information:

Pi = a2y0 + p2u - 72*1 b2x2 (5a)

л = ijo + ft и - 7л (5b)

Conjectures are identical across individuals and the coefficients are determined in an equilibrium in which the conjectures are rational. Because the price conjectures (5) are linear functions of normal variates, they are normally distributed.

1.1 A linear noisy rational expectations equilibrium

Appendix A derives the system of equations that equates the coefficients of Equations (5) with the corresponding coefficients in the price functional satisfying Equations (3), assuming a noisy rational expectations equilibrium exists. This derivation is sketched below.

Given the price conjectures (5), the risky asset demands planned by trader i to solve Equations (2) can be written

da = --, , л (6a)

Z?var(w I Ай)

E(P2 I gfl) - P, Ejdg I gfl)(G12 ~ Gu)

Var(wEc), Gu, and G12 are computed in Appendix A using the prescribed covariances of the exogenous variables and the coefficients of the price functions (5). They are constants, identical across investors. Averaging equations (6) over / and imposing the market-clearing conditions (3) provides the potential equilibrium price functions:

P2 = м2 - Raj(x1 + Xz) (7a)

Pi = V +-;--RGuxx (7b)

where м, = lim 2 E(u\ait)/l for t= 1, 2 т/ s limooZi адЕл) a2 = var(wE ), for f = 1, 2

Because and ту are linear functions of y0, u, and xp Equations (7) may be written as

Pi = &гУо + PiU- 72*i - K*i (8a)

л = оцУо + PiU- 7!

(8b) 531

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