the timeless perspective equilibrium is that the shadow prices, pʌ, are not initialized to 0, but
rather behave in the initial period as they do in all subsequent periods. It follows that there
is actually no need to eliminate the shadow prices from the system (the step that leads to mul-
tiple representations) when evaluating equation (39). Instead, one can simply integrate with
respect to the shadow prices conditional on xt. Because the optimal commitment policy has
a unique representation in terms of xt and p/. (under standard and quite general conditions),
so too does the timeless perspective equilibrium, and this unique representation yields unique
values for the mean and variance of p/. conditional on xt .11
4.3 The simple example again
Returning to the simple model, I now evaluate performance using (1 — β)Lt while varying
к, the slope of the Phillips curve, and μ, the weight on gap stabilization.12 Figure 2A
displays the performances for the optimal commitment policy, the timeless perspective policy,
and the time-consistent policy as к is varied between (0, 0.1], holding pu and μ constant at
their benchmark values. Complementing Figure 2A, Figure 2C displays the performances
associated with varying μ between (0,10] while holding pu and к constant at their benchmark
values. In contrast, Figures 2B and 2D are generated allowing both к and μ to vary between
(0, 0.1] and (0,10], respectively, displaying as a percent the fraction of occasions for which the
discretionary policy performs better than the timeless perspective policy against particular
values of к and μ, respectively.
11Without wishing to labor the point, this invariance property is also a feature of unconditional loss, and for
the same reason. Unconditional loss is invariant to the multiplicity of representations because it integrates out
the entire state vector, which includes the auxillary states.
2Consistent with Figure 1, I set к = 0.025, μ = 0.50, ρu = 0.20, and σeu = 1, and the initial state is given
by uo = 0.
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