prior distribution, i.e., is completely uninformative. Naturally, the larger the
value of q the more informative a given signal is. The paper assumes that the
degree of informativeness of a given signal q is a choice variable. Agents are
free to choose how well informed they become. However, a given choice of q is
accompanied with a cost in terms of utility C (q2). The cost function satisfies
standard properties
C (0) = 0, C' (q2) ≥ 0 and C'' (q2) ≥ 0. (27)
Note that agents can choose to be fully informed. They select q to be equal
to one. However, if they do so they must pay the cost C (1). On the other hand,
agents can rely only on their priors, i.e., they can set q to be equal to zero and
save on information acquisition as C (0) = 0.
An agent who decides at time t to purchase a signal of quality qt receives a
signal ν and then forms the corresponding posterior distribution. Specifically,
if she receives a signal νxt+1 for variable xt+ι then the corresponding posterior
distributions is obtained using the Bayes rule and is given by Fxt+1 (xt+ι ∣νxt+1 ) .
In the following step the posterior distributions are used in the process of the
determination of optimal behavior.
Recall that under the assumption of perfect foresight the optimal level of
investment is given by
⅛t+ι = 2 (1 - Xt+1)
and is independent of the future consumption price pt + ι and the future rental
costs rt+ι. Moreover, given the assumption of logarithmic utility the two vari-
ables remain irrelevant even in the case of uncertainty and expected utility
maximization approach. In order to preserve complete analytic tractability the
paper deals with uncertainty by assuming certainty equivalence behavior or
more precisely by assuming that economic agents treat future variables as if
they were equal to their means. Under this assumption the level of investment
is given by
⅛t+1 =2 (1 - E (xt + 1∣νxt + ι )) . (28)
This form of behavior implies that realized utility conditional on signal νxt + 1 is
given by
U = At + log((1 + xt + ι)2 — (xt + 1 — E (xt + ι ∣Vχt + 1 )) ),
where At = — logpt + log (wt) + log (pt ) + log(ɪ(+-)) — 2 log(2). Note that the
level of realized utility depends both on the future price of consumption pt+ι
and on future rental cost rt+ι. Of course it also depends on the realized value
of xt+ι. Clearly, as long as the true realized value differs from the expected one
economic agents pay a cost in the form of
ε2 = (xt+1 — E (xt+ι ∣Vχt + 1 ))2
as the consumption profile is not smoothened out optimally. Note that a priori
economic agents can avoid this cost by purchasing a perfectly informative signal.
15