Higherorder Difference Equations

The economic models in the preceding chapter involve difference equations that relate Pt and P,_t to each other. As the P value in one period can uniquely determine the P value in the next, the time path of P becomes fully determinate once an initial value P0 is specified. It may happen, however, that the value of an economic variable in period t (say, y\) depends not only on y, _, but also on j, Such a situation will give rise to a difference equation of the second order.

Strictly speaking, a second-order difference equation is one that involves an expression A2yt, called the second difference of yt, but contains no differences of order higher than 2. The symbol A2, the discrete-time counterpart of the symbol d2/dt2, is an instruction to "take the second difference" as follows:

Thus a second difference of yt is transformable into a sum of terms involving a

* That is, we first move the subscripts in the ( r, ^ i — y,) expression forward by one period, to get a new expression (yl + 2 ~ y,+ i), and then we subtract from the latter the original expression. Note that, since the resulting difference may be written as Ay,+ ] - Ayn we may infer the following rule of operation:

A(>',+ i ->'i) = A>Wl ~ A>'r This is reminiscent of the rule applicable to the derivative of a sum or difference.

two-period time lag. Since expressions like A2j>, and A yt are quite cumbersome to work with, we shall simply redefine a second-order difference equation as one involving a two-period time lag in the variable. Similarly, a third-order difference equation is one that involves a three-period time lag, etc.

Let us first concentrate on the method of solving a second-order difference equation, leaving the generalization to higher-order equations for a later section. To keep the scope of discussion manageable, we shall only deal with linear difference equations with constant coefficients in the present chapter. However, both the constant-term and variable-term varieties will be examined below.

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