Some Applications to Rational Expectations Models

Let us return to the example of Cagan’s portfolio balance schedule, only now we assume that \(m_t\) is a covariance stationary stochastic process and the log of the price level now expected for next period is the linear least squares projection of \(p_{t+1}\) on information available at time \(t\). We then have the difference equation

(285)\[m_t - p_t = \alpha P_t p_{t+1} - \alpha p_t,\quad \alpha < 0\]

where \(P_t p_{t+1}\) is the linear least squares forecast of \(p_{t+1}\) given information available at time \(t\). This difference equation can be rewritten as

\[ p_t = \left(\frac{-\alpha}{1-\alpha}\right)P_t p_{t+1} - \left(\frac{1}{1-\alpha}\right)m_t \]

or

\[ p_t = \lambda P_t p_{t+1} + (1-\lambda)m_t \]

where \(\lambda = -\alpha/(1-\alpha)\), which implies that \(0 < \lambda < 1\) since \(\alpha < 0\). The stationary solution of the above difference equation obeys[1]

(286)\[p_t = (1 - \lambda)\sum_{j=0}^\infty \lambda^j P_t m_{t+j}\]

Let us assume that \(m_t\) has the autoregressive representation

\[ a(L)m_t = \epsilon_t \]

where \(\epsilon_t\) is fundamental for \(m\), and \(a(L) = 1 - a_1 L - \ldots - a_r L^r\). Then from formula (282) of the preceding section we have that (286) implies

(287)\[p_t = (1-\lambda)a(\lambda)^{-1}\left[1 + \sum_{j=1}^{r-1}\left(\sum_{k=j+1}^r \lambda^{k-j} a_k \right) L^j \right]m_t\]

(288)\[a(L)m_t = \epsilon_t.\]

These two equations express how the stochastic process for \(p_t\) depends on \(m_t, m_{t-1}, \ldots, m_{t-r+1}\) via coefficients that partly reflect the stochastic process (288) that governs \(m_t\). As an example, we set \(a(L) = 1 - a_1 L - a_2 L^2 - a_3 L^3\). Then (287) and (288) become

\[ p_t = (1-\lambda)(1- a_1 \lambda - a_2 \lambda^2 - a_3 \lambda^3)^{-1}[1 + (a_2 \lambda + a_3 \lambda^2)L + (a_3\lambda)L^2]m_t \]

\[ m_t = a_1 m_{t-1} + a_2 m_{t-2} + a_3 m_{t-3} + \epsilon_t \]

Let us reconsider the supply-demand example of Chapter IX where \(x_t\) is now a covariance stationary, indeterministic random process with mean zero and autoregressive representation \(a(L)x_t = \epsilon_t\), where \(\epsilon_t\) is a fundamental white noise for \(x_t\). Our system is naturally modified to become

\[\begin{split} \begin{aligned} C_t &= -\beta p_t, & \beta > 0 \\ Y_t &= \gamma P_{t-1} p_t + x_t & \gamma > 0 \\ I_t &= \alpha(P_t p_{t+1} - p_t), & \alpha > 0 \\ Y_t &= C_t + I_t - I_{t-1}, \end{aligned} \end{split}\]

where \(Y_t\) is production, \(C_t\) demand for consumption, and \(I_t\) holdings of inventories. (A related single-market rational expectations example, in which the suppliers’ and demanders’ Euler equations are solved separately to yield explicit dynamic supply and demand curves, appears in A Difficulty in Interpreting Vector Autoregressions.) Substituting the first three equations into the fourth gives

(289)\[-\alpha P_t p_{t+1} + (\gamma + \alpha)P_{t-1}p_t + (\alpha + \beta)p_t = \alpha p_{t-1} - x_t.\]

Taking projections of both sides against information available at time \(t - 1\) gives

\[ \alpha P_{t-1}p_{t+1} - (\gamma + \beta + 2\alpha)P_{t-1}p_t + \alpha P_{t-1}p_{t-1} = P_{t-1}x_t \]

or

\[ (B^{-1} - \phi + B)P_{t-1}p_t = \alpha^{-1} P_{t-1} x_t \]

where

\[ B^{-1}P_{t-1}z_t \equiv P_{t-1} z_{t+1}, \quad B P_{t-1}z_t \equiv P_{t-1} z_{t-1}, \]

and where

\[ \phi = ((\beta + \gamma)/\alpha) + 2 > 0. \]

Multiplying by \(B\) gives

(290)\[\begin{split}\begin{aligned} (1 - \phi B + B^2)P_{t-1}p_t &= \alpha^{-1}P_{t-1}x_{t-1} \\ (1 - \lambda^{-1}B)(1 - \lambda B)P_{t-1}p_t &= \alpha^{-1}P_{t-1}x_{t-1} \end{aligned}\end{split}\]

where \(|\lambda| < 1\) satisfies \(\lambda + \lambda^{-1} = \phi\). To ensure covariance stationarity of the solution, we shall insist that all lag distributions be square summable. Operating on both sides of (290) with the forward inverse of \((1-\lambda^{-1} B)\) gives

\[ (1 - \lambda B)P_{t-1}p_t = \frac{- \lambda \alpha^{-1}}{1-\lambda B^{-1}}P_{t-1}x_t \]

or

\[ P_{t-1}p_t - \lambda p_{t-1} = \frac{-\lambda}{\alpha}\sum_{i=0}^{\infty}\lambda^i P_{t-1} x_{t+i}. \]

Substituting this solution for \(P_{t-1}p_t\) into (289) gives

(291)\[p_t = \lambda p_{t-1} + \frac{1}{\alpha + \beta - \alpha \lambda}\left[\alpha^{-1} \lambda(\gamma + \alpha)\sum_{i=0}^{\infty} \lambda^i P_{t-1}x_{t+i} - \sum_{i=0}^{\infty} \lambda^i P_t x_{t+i}\right]\]

We have assumed that \(x_t\) has the autoregressive representation \(a(L)x_t = \epsilon_t\). Now by using methods similar to those used to derive (282), it can be established that

\[ P_{t-1}\sum_{j=0}^\infty \lambda^j x_{t+j} = \left(\frac{L^{-1}I - L^{-1} a(\lambda)^{-1}a(L)}{1-\lambda L^{-1}}\right)x_{t-1} \]

Substituting this and (282) into (291) we have the following formula for the equilibrium stochastic process for price \(p_t\) as a function of the \(x_t\) process

\[ p_t = \lambda p_{t-1} + \frac{1}{\alpha + \beta - \alpha \lambda}\left\{\alpha^{-1} \lambda(\gamma + \alpha)\left[\frac{L^{-1}I - L^{-1}a(\lambda)^{-1}a(L)}{1-\lambda L^{-1}}\right]x_{t-1} - \left(\frac{1 - \lambda a(\lambda)^{-1}a(L) L^{-1}}{1-\lambda L^{-1}}\right)x_t\right\} \]

\[ a(L)x_t = \epsilon_t \]

This is the solution to the stochastic difference equation (289) which expresses \(p_t\), as a function of current and lagged \(x\)’s and \(p\)’s, and which gives a covariance stationary process for \(p_t\).

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[1]

In this and the following example we set “transient terms” of the form \(c\lambda^t\) to zero because we are interested in obtaining solutions that are covariance stationary processes.