Ja A 1

Our experience with the variable term t2 should enable us to generalize the method to the case of ct". In the new trial solution, there should obviously be a term Bnt", to correspond to the given variable term. Furthermore, since successive differencing of the term yields the distinct expressions t, and 50

(constant), the new trial solution for the case of the variable term ct" should be written as

But the rest of the procedure is entirely the same.

It must be added that such a trial solution may also fail to work. In that event, the trick—already employed on countless other occasions—is again to multiply the original trial solution by a sufficiently high power of t. That is, we can instead try y, = t(B0 + 5,; + B2t2 + ■ ■ • + Bnt"). etc.

Higher-Order Linear Difference Equations

The order of a difference equation indicates the highest-order difference present in the equation; but it also indicates the maximum number of periods of time lag involved. An wth-order linear difference equation (with constant coefficients and constant term) may thus be written in general as

(17.37) yt + n + a]y,+n_l + ■■■ + a„_xyt+x + anyt = c

The method of finding the particular integral of this does not differ in any substantive way. As a starter, we can still try y, = k (the case of stationary intertemporal equilibrium). Should this fail, we then try y, = kt or yt = kt2, etc., in that order.

In the search for the complementary function, however, we shall now be confronted with a characteristic equation which is an n th-degree polynomial equation:

(17.38) b" + axb"~1 + ■ ■ • + an _ ,b + an = 0

There will now be n characteristic roots bi (i = 1,2,..., n). all of which should enter into the complementary function thus:

i= l provided, of course, that the roots are all real and distinct. In case there are repeated real roots (say, b, = b2 = b3), then the first three terms in the sum in (17.39) must be modified to

Moreover, if there is a pair of conjugate complex roots—say, bn_x, bn—then the last two terms in the sum in (17.39) are to be combined into the expression

A similar expression can also be assigned to any other pair of complex roots. In case of two repeated pairs, however, one of the two must be given a multiplicative factor of tR' instead of R'.

After y and yc are both found, the general solution of the complete difference equation (17.37) is again obtained by summing; that is, y, = yP + yc

But since there will be a total of n arbitrary constants in this solution, no less than n initial conditions will be required to definitize it.

Example 3 Find the general solution of the third-order difference equation

By trying the solution yt = k. the particular integral is easily found to be yp = 32. As for the complementary function, since the cubic characteristic equation can be factored into the form the roots are ft, = ft2 = 2 and b3 = — j. This enables us to write

Note that the second term contains a multiplicative t; this is due to the presence of repeated roots. The general solution of the given difference equation is then simply the sum of yc and yp.

In this example, all three characteristic roots happen to be less than 1 in their absolute values. We can therefore conclude that the solution obtained represents a time path which converges to the stationary equilibrium level 32.

Convergence and the Schur Theorem

When we have a high-order difference equation that is not easily solved, we can nonetheless determine the convergence of the relevant time path qualitatively without having to struggle with its actual quantitative solution. You will recall that the time path can converge if and only if every root of the characteristic equation is less than 1 in absolute value. In view of this, the following theorem—known as the Schur theorem*—becomes directly applicable:

The roots of the «th-degree polynomial equation a0b" + axbn~l + • • • + an_ ,ft + an = 0

will all be less than unity in absolute value if and only if the following n

* For a discussion of this theorem and its history, see John S. Chipman, The Theory of Inter-Sectoral Money Flows and Income Formation, The Johns Hopkins Press, Baltimore, 1951, pp. 119-120.

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