The name is absent

prices obtained from equation (20) fully satisfy the model’s equilibrium relationships. It
follows that a valid solution for p_t can be obtained from any subset of the variables in y_t and
u⅛ provided that the resulting G_p matrix has rank(G_p) ≥ n2. Although the particular state
variables that enter the timeless perspective equilibrium will depend on which equilibrium
relationships are used when solving for p_t, by construction, they all imply the same welfare
and equilibrium behavior. That timeless perspective equilibria have multiple representations
is also reflected in the fact that although the procedure described above yields an equilibrium
in which u_t is a function of x_t, x_t-i, y_t-i, and ut-i, the approach described in Section 2
would yield an equilibrium in which u_t is a function of x_t, y_t-ι, u_t-i, and u_t-2.⁷

One situation where the rank condition, rank(G_p) ≥ n2, will often fail is when equation
(18) is used in isolation to solve for the shadow prices, as per Tetlow and von zur Muehlen
(2001). They solve for p_t according to

pt = Fp¹ (ut - F_xxt),                                 (22)

where F^-1 denotes the generalized (left) inverse of F_p, and use this expression to eliminate the
shadow prices from the system. The problem with using equation (22) is that, unless p ≥ n2,
the solution for p_t will be generated from a system that is underdetermined. Consequently,
the shadow prices will no longer necessarily obey the model’s equilibrium relationships. In
the Tetlow and von zur Muehlen (2001) application there was one nonpredetermined variable
and one control variable, so p = n2 and no problem arose. But if p <n2, as is invariably the
case, then employing equation (22) will lead to erroneous results.

Two further points are worth emphasizing before leaving this section. First, provided
d_t = u_t, and therefore that G_p = F_p, a representation for the timeless perspective targeting
rule can be recovered by substituting equation (20) back into equation (18). This substitution
leads to the expression

ut = (Fχ - FpGp¹Gx) xt + FpGp¹dt,                      (23)

which describes a relationship that u_t, x_t, and y_t must satisfy along the equilibrium path.
Second, the timeless perspective equilibrium described by equation (21) essentially has two
components. The first component is a transition equation for the predetermined variables,

xt+i = Nχχi xt + Nχχ2xt-1 + Nχ_ddt-ι + εt+ι∙                  (24)

⁷Importantly, this well-known multiplicity of representations makes no material difference for the analysis
or conclusions that follow, since I assume—for consistency—that the conditioning variables satisfy the timeless
perspective equilibrium relationships.

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