Appendix on Solution Procedures and Derivation of Variances
Models based on intertemporal optimization can generally not be solved analytically. The basic problem is that some endogenous variables, like for example the shadow price of capital q, are conditioned on the entire future path of all variables in the model. There is also a number of predetermined variables like the capital stock K which depend on the past history of the forward-looking variables. While the initial values of the backward-looking variables are known, all that can be done initially with respect to the forward-looking variables is to impose certain terminal conditions. This poses a problem usually referred to as a two-point boundary value problem.
In the present model, the main problem is to find the exact path for the capital stock and the shadow price of capital consistent with optimizing the firm’s objective function. Once this problem has been solved, the rest of the model can be solved recursively. The solution to the firm’s decision problem is characterized by saddle-point instability. Initially small deviations from the optimal path result in a sequence that eventually diverges away from steady state. It can be shown, however, that a certain path for the shadow price of capital results in an optimal path for the capital stock which gradually approaches steady state. This path can only be approximated through iteration. The extended path method by Fair and Taylor (1983) has proved to be very useful in this regard. 11/ The basic steps involved in solving the model can be described as follows.
1. The first step is to calculate a steady state growth path solution which can be used to define terminal conditions for the forward-looking variables. Such a solution requires that all real variables grow at the same rate as the exogenous growth in productivity and that all relative prices are constant. Imposing these conditions yields a steady state value for the shadow price of capital qS. With endogenous terms of trade, the steady state price of capital must be determined simultaneously with the price of output. It is difficult to do this analytically. However, the model can be iterated to yield a combination of the two prices consistent with a steady state growth path. 12/
2. The second step is to choose a terminal period T. The terminal period must be chosen so as to allow the model to be solved a sufficient number of periods beyond the actual simulation period. Guess an initial path i=qt,qt+1,…,qT and assume that qT=qS. This yields a starting point for further iterations.
3. The third step is to solve the model in each period using the initial path and the starting values of all predetermined variables. With endogenous terms of trade, this includes finding the price which clears the market for the domestic good in each period. As already noted, this is by itself not an entirely straightforward exercise as it involves finding the market-clearing price through iteration.
4. Using the initial path to represent the expected next-period value of the shadow price of capital in each period, the fourth step is to calculate a new path i+1=qt,qt+1,…,qT which replaces the old path and is used to represent expectations in the next iteration. Go back to the previous step and iterate until the difference between any two estimates for the same period of two consecutive iterations is sufficiently small to meet a convergence criterion.
5. The fifth step is to choose a new terminal period T’>T and repeat the third and fourth steps until convergence is reached for the new extended simulation period. The solution period is extended until the difference between any two estimates for the same period of two consecutive solutions for the original solution period 0-T is small enough to meet a convergence criterion.
After a solution has been found, deriving the variance properties of each variable is simple. This involves calculating a set of moving average weights which describes each variable’s dynamic response pattern in log differences relative to the underlying productivity process. For example, the variance of the change in productivity ΔZt=lnZt-lnZt-1 is given by:
where MAZ1, MAZ2, …, MAZT are the moving average coefficients and
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The views expressed in this paper are those of the author and do not necessarily reflect the views of the International Monetary Fund.
See for example Flavin (1984). Recent empirical studies of the role of liquidity constraints include Campbell and Mankiw (1989) and Japelli and Pagano (1989). See Hayashi (1985) for a survey of earlier work.
A number of researchers have argued that consumption may be subject to “taste shocks,” movements in consumption which are unrelated to news about interest rates or incomes, in which case this interpretation may no longer be valid.
There is by now a large and growing literature on saving and investment behavior in equilibrium models. Examples of two-country models include Backus, Kehoe and Kydland (1992), Baxter and Crucini (1992), Finn (1989) and Stockman and Tesar (1990). For an overview of the real business cycle research program, see McCallum (1989) and Plosser (1989).
Finite horizons increase the sensitivity of consumption to changes in current income and tend to tilt the response to productivity shocks into the future. A convenient alternative to this specification is the Campbell and Mankiw (1989) setup with current income consumers who set consumption equal to their disposable income.
In a more elaborate framework, the time series processes of government taxes and expenditures could be estimated econometrically.
Needless to say, all variables are naturally excessively correlated with output. This would of course change if one would allow for simultaneous shocks in other variables, for example terms of trade, the real interest rate or employment.
The rate of time preference is adjusted to make it possible to generate a steady state path according to which consumption grows in line with output.
The income accruing to the household sector (net of investment expenditures, government taxes and interest expenditures) is split into two parts. One part goes to permanent income consumers who determine consumption on the basis of intertemporal optimization. The other part goes to current income consumers. Total consumption is then a weighted sum of each group’s consumption.
Serial correlation may of course also simply reflect time averaging. If the basic permanent income-life cycle model holds in continuous time, then measured consumption is the time average of a random walk. In this case, the change in consumption will be serially correlated even if the underlying model is accepted.
While the expanded path method remains popular, the increased demand for numerical solution methods for non-linear rational expectations models has stimulated rapid growth in alternative solution techniques. For a recent survey, see Taylor and Uhlig (1990).
The model can be solved both with exogenous and endogenous terms of trade. The steps taken to solve the model are basically the same. However, with endogenous terms of trade, the price of the domestic good is determined by market-clearing. Generally, the market-clearing price can only be calculated through iteration. This makes the solution procedure somewhat more complicated and therefore more time-consuming than in the simple case with exogenous terms of trade.