To characterize the IMRS and the behavior of asset returns, a solution for the log price-consumption ratio, [z.sub.t], and the log

price-dividend ratio, [z.sub.m,t], is needed.

Because of the apparent lack of low-frequency movements in either excess returns or real dividend growth, it is not possible to identify which of these is more important in producing long swings in the

price-dividend ratio. More formally, we show that the data cannot distinguish a model in which there are small permanent changes in dividend growth from one in which there are small permanent changes in excess returns.

In the context of (2.1), the consumption-wealth ratio sends dividend growth and returns in the same direction, so its effects on the

price-dividend ratio offset.

However, since there is a lag between physical investment and increased output, the variation in

price-dividend ratios due to demographics can be as high in the Diamond model as in the exchange model with fixed land.

These expectations imply that rapid dividend growth increases stock prices more than proportionally, so that the

price-dividend ratio rises when dividends are growing strongly.

The long-run equilibrium adjustment is given by the demeaned log

price-dividend ratio, [y.sub.t] = [d.sub.t] - [p.sub.t].

Other commonly used valuation ratios include price-sales ratios, price-to-book ratios, and

price-dividend ratios.

Here, the prime after the P is used as it was in Section 3 above, to denote the expectation of the present value given only the information [[chi].sub.t], i.e., the

price-dividend ratio implied by the dynamic Gordon model (6) and the time series model (10).

Agents investing in pyramid [[DELTA].sub.1] face a more risky--if more favorable--market than agents investing in pyramid [[DELTA].sub.2], because the return [D.sub.t+1] + [Q.sub.t+1] depends more on the capital value term [Q.sub.t+1] when the

price-dividend ratio is expected to be high, and more on the dividend [D.sub.t+1] when the

price-dividend ratio is expected to be low.

Alternatively, investors might use past levels of the

price-dividend ratio to forecast future dividend growth because this ratio has been found to successfully predict stock prices in the empirical literature.

Consider the bivariate model consisting of the first difference in log of stock prices, [Delta][p.sub.t], and the log stock

price-dividend ratio or spread, st, used in Campbell and Shiller [1987] and Lee [1995].(5)

The model also implies a negative sensitivity of

price-dividend ratios to expected excess returns, and the magnitude of the sensitivity is substantially larger for more persistent exposure.