Sixteen

DISCRETE TIME: FIRST-ORDER DIFFERENCE EQUATIONS

In the continuous-time context, the pattern of change of a variable y is embodied in the derivatives y'(t), y"(t), etc. The time change involved in these is infinitesimal in magnitude. When time is, instead, taken to be a discrete variable, so that the variable t is allowed to take only integer values, the concept of the derivative obviously will no longer be appropriate. Then, as we shall see, the pattern of change of the variable 7 must be described by so-called "differences," rather than by derivatives or differentials, of y(t). Accordingly, the techniques of differential equations will give way to those of difference equations.

When we are dealing with discrete time, the value of variable y will change only when the variable t changes from one integer value to the next, such as from t = 1 to t = 2. Meanwhile, nothing is supposed to happen to y. In this light, it becomes more convenient to interpret the values of t as referring to periods—rather than points—of time, with t = 1 denoting period 1 and t = 2 denoting period 2, and so forth. Then we may simply regard y as having one unique value in each time period. In view of this interpretation, the discrete-time version of economic dynamics is often referred to as period analysis. It should be emphasized, however, that "period" is being used here not in the calendar sense but in the analytical sense. Hence, a period may involve one extent of calendar time in a particular economic model, but an altogether different one in another. Even in the same model, moreover, each successive period should not necessarily be construed as meaning equal calendar time. In the analytical sense, a period is merely a length of time that elapses before the variable y undergoes a change.

16.1 DISCRETE TIME, DIFFERENCES, AND DIFFERENCE EQUATIONS

The change from continuous time to discrete time produces no effect on the fundamental nature of dynamic analysis, although the formulation of the problem must be altered. Basically, our dynamic problem is still to find a time path from some given pattern of change of a variable y over time. But the pattern of change should now be represented by the difference quotient Ay/At, which is the discrete-time counterpart of the derivative dy/dt. Recall, however, that t can now take only integer values; thus, when we are comparing the values of y in two consecutive periods, we must have A/ = 1. For this reason, the difference quotient Ay/At can be simplified to the expression Ay\ this is called the first difference of y. The symbol A, meaning difference, can accordingly be interpreted as a directive to take the first difference of (>')■ As such, it constitutes the discrete-time counterpart of the operator symbol d/dt.

The expression A y can take various values, of course, depending on which two consecutive time periods are involved in the difference-taking (or "differencing"). To avoid ambiguity, let us add a time subscript to y and define the first difference more specifically, as follows:

(16.1) A y,=yl+]-y, where y, means the value of y in the t th period, and yt, , is its value in the period immediately following the rth period. With this symbolism, we may describe the pattern of change of y by an equation such as

Equations of this type are called difference equations. Note the striking resemblance between the last two equations, on the one hand, and the differential equations dy/dt = 2 and dy/dt = -0.lv on the other.

Even though difference equations derive their name from difference expressions such as Ayn there are alternate equivalent forms of such equations which are completely free of A expressions and which are more convenient to use. By virtue of (16.1), we can rewrite (16.2) as

For (16.3), the corresponding alternate equivalent forms are (16.3') ^,-0.9^ = 0

The double-prime-numbered versions will prove convenient when we are calculating a y value from a known y value of the preceding period. In later discussions, however, we shall employ mostly the single-prime-numbered versions, i.e., those of (16.2') and (16.3').

It is important to note that the choice of time subscripts in a difference equation is somewhat arbitrary. For instance, without any change in meaning, (16.2') can be rewritten as yt - yt_, = 2, where (t - 1) refers to the period which immediately precedes the rth. Or, we may express it equivalently as

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