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# THE PRICE LEVEL: CONCLUDING COMMENTS 2

Appendix
Recall from Equation (4) in the text, that the government’s present value budget constraint holds if and only if

is satisfied. In the FD regime, w, adjusts to satisfy (Al). In the MD regime, (Al) holds for all values of w,. We show below that, under certain conditions, the fiscal rule of Equation (5) in the text is a sufficient condition for a MD regime. We first present the (more transparent) proof under perfect foresight, and then outline the extension allowing for uncertainty.

Using (5) and the flow budget constraint (3) in the text, the dynamics of Wj are governed by

where c* and a. are constants-ensuring that (l-Cj)/aj and аД 1-Cj) remain bounded for all j-and c, is an arbitrarily small positive constant. The condition lim sup
in the following proof. Working directly with (Cl) instead of (СГ) would only entail minor changes in (A6) and (A9) below, without changing the substance of the proof. The proof has three cases.

The infinite series on the right-hand side converges by the “ratio test” [see, for example, Rudin (1976), Theorem 3.34] because

In this case, the right-hand side of (A3) does not converge as T tends to infinity, but (A6) still holds because the exponential function on its right hand side goes to zero faster than |wtVr| can grow.

The infinite series on the right hand-side of (A8) converges and, therefore, the term in the second bracket is finite. The limit in the first bracket is zero by (C3), implying

Stochastic Setup

Let c„ a„ and et be the first three components of a vector (t generated by a first-order Markov process. The other components of £, may include lags of the first three (capturing higher order dynamics) and other relevant random variables observed at or before date t. We assume that Ct E Z for all t, where Z is a compact subset of a Euclidean space, that (Cl1) and (C2) hold with probability one, and that Prob { c,+J > c. | Ct = (} > 0 for all j > 0, some c. > 0, and all С e Z (so that (C3) holds with probability one). Note that for any fixed T > 0, given the current state C, = the random variable wt+T is a bounded function of {C, Ct+i> Ст+м) and Etwt+T=E{ wl+x | Ct = C} is well defined.
Given the state = £ at date t, we can define the probability space (Q, P) so that each point coeQ corresponds to a sample path {C, C,+i, Ct+2 > •••} • The o-algebra & and the probability measure P can be defined so that for any random variable f(C, (t+i…., Ct+т).
For any given qgQ, ignoring sets of measure zero where (СГ), (C2) and (C3) may fail to hold, the real sequence

for all T > 0. Since P is a probability (finite) measure, the right-hand side of (A9) is P-integrable and, therefore, can serve as the dominating function for the Lebesgue Dominated Convergence Theorem [see, for example, Stokey and Lucas (1989), Theorem 7.10 ]. Thus, we have