There are two further ways of obtaining cycles with a Hicksian multiplier-accelerator model - one is by non-linear functions (see below), another is by making our simple system "shock-dependent". It is something of the latter nature that Ragnar Frisch (1933), Eugen Slutsky (1927) and Michal Kalecki (1935, 1954) famously proposed. Namely, given the simplest Hicksian system, we could argue that:
It = I0 + b (Yt-1 - Yt-2) + ut
where ut is an exogenous shock parameter - reflecting, say, innovation in the production sector. Combining this with the standard MA structure, we obtain:
Yt = c0 + I0 + cYt-1 + b (Yt-1 - Yt-2) + ut
Yt - (c + b )Yt-1 + b Yt-2 = (c0 + I0 + ut)
What Frisch-Slutsky demonstrate is that if we have parameter values c and b so as to obtain case VI dynamics (i.e. damped, not explosive, oscillations), then the impulse provided by the stochastic element will be sufficient to maintain regular cycles provided random shocks are continuously or at least intermittently allowed. If we had allowed explosive oscillations, however, these shocks would not essentially affect the system. Thus, we need damped oscillations for the random shocks to yield an essentially regular cycle.
Frisch-Slutsky type of analysis has been resurrected in New Classical "Real Business Cycle" theory (a.k.a. RBC or stochastic growth theory, e.g. Kydland and Prescott, 1982; Long and Plosser, 1983; Prescott, 1986). Effectively, these models employ growth models of the standard Neoclassical variety (e.g. Solow-Swan or Cass-Koopmans) and add Frisch-Slutsky type of stochastic elements to a variety of elements (e.g. production, tastes, spending, etc.) - yielding then continual growth and cycles. However, as they do not employ an multiplier-accelerator structure, we shall pass over them here in silence. However, as we saw, it is obvious that there is no a priori reason why the Frisch-Slutsky methodology would not be compatible with these Keynesian structures.
Finally, recall that we have so far confined ourselves to real-side "income-expenditure" relationships in the Hicksian model. To expand upon the Keynesian roots of this model, we can add monetary factors to the multiplier-accelerator model so as to allow the LM side to enter the essentially IS multiplier-accelerator model. This was considered by John Hicks (1950: Ch.11) and later taken up by Smyth (1963). Essentially, we have, as before:
Ct = c0 + cYt-1
It = I0 + a rt-1+ b (Yt-1 - Yt-2)
but now, note, we have allowed past interest (rt-1) to enter as a determinant of investment along with autonomous investment and the accelerator. It is assumed, of course, that a < 0. For simplicity, we are ignoring the growth rate of autonomous investment. Having this, then, we obtain:
Yt = c0 + I0 + cYt-1 + a rt-1 + b (Yt-1 - Yt-2)
Yt = c0 + I0 + (c + b )Yt-1 - b Yt-2 + a rt-1
for our second-order linear difference equation for the goods market.
However, this is not sufficient as we do not have anything determining the rate of interest. We shall propose the traditional Keynesian liquidity preference theory for this. Thus, money demand is defined as:
Lt = L0 + h Yt + s rt
with the first component being autonomous money demand, the second as the transactions demand for money and the last as the speculative demand. It is assumed that h > 0 and s < 0 so money demand is positively related to income and negative related to interest.
We shall also propose a money supply rule. Namely:
Mt - Mt-1 = m (Yt-1 - Yt-2)
so the change in the money supply is some negative function of past changes in income (m < 0). The logic is simple policy: in recessions, governments expand the money supply whereas in booms, they restrict the money supply. Smyth (1963) proposes to reduce this to:
Mt = M0 + m Yt-1
so money supply at time t is some function of past income and some constant M0. Thus, in equilibrium:
Lt = Mt
L0 + h Yt + s rt = M0 + m Yt-1
so, solving for rt:
rt = [m Yt-1 + M0 - L0 - h Yt]/s
Lagging this once:
rt-1 = [m Yt-2 + M0 - L0 - h Yt-1]/s
which we can place in our original goods market difference equation:
Yt = c0 + I0 + (c + b )Yt-1 - b Yt-2 + (a /s )[m Yt-2 + M0 - L0 - h Yt-1]
Yt - (c + b - a h /s )Yt-1 + (b - a m /s )Yt-2 = [c0 + I0 + (a /s )(M0-L0)]
which is a second order difference equation. Thus, our particular solution (equilibrium) would be:
Yp = [c0 + I0 + (a /s )(M0-L0)]/[s (1 - c) + a (h - m )]
whereas our characteristic equation would be:
r2 - (c + b - a h /s )r + (b - a m /s ) = 0
The eigenvalues to be determined are then:
r1, r2 = [(c + b - a h /s ) ｱ ﾖ [(c + b - a h /s )2 - 4(b - a m /s )]]/2
where roots are real if the discriminant is non-negative i.e. D ｳ 0 or (c + b - a h /s )2 ｳ 4(b - a m /s )] and complex otherwise. So, for real roots:
(c + b )2 + (a h /s )2 - 2(c + b )a h /s ｳ 4(b - a m /s )
As b > 0 and a h /s < 0, then the term on the right is positive. Of course, (c+b )2 > 0 and (a h /s )2 > 0, thus, the term 2(c + b )a h /s is of crucial significance as to whether the inequality holds or not. Namely, the greater the absolute value of c, b , a , h and the smaller the value of s , the greater the possibility of oscillations. Now, by the Schur Criterion, one always obtains damped monotonic dynamics if:
(i) 1 + (c + b - a h /s ) - (a m /s - b ) > 0
(ii) 1 - (c + b - a h /s ) - (a m /s - b ) > 0
(iii) 1 + (a m /s - b ) > 0
Now, the first Schur Criterion (i) can be rewritten as 1 + c - a h /s - a m /s + 2b > 0 thus given that a , m , s < 0 and c, h , b > 0, then obviously a m /s > 0 and a h /s < 0. Thus, the necessary condition for stability following (i) is that:
c > a (m + h )/s - 2b - 1.
Now, the second part of the Schur criterion (ii) can be rewritten 1 - c + a h /s - a m /s > 0 where, as (1-c) > 0 and a h /s > 0 and a m /s < 0, then this is obviously fulfilled. Finally, the third part of the Schur criterion (iii) can be rewritten as b < 1 - a m /s which may or may not be fulfilled. Now, recall that in the original Hicks model, the condition for stability was that b < 1. Thus, this new condition, that b < 1 - a m /s (where, recall, a m /s < 0), makes the region of instability bigger whereas the region of stability is reduced. We should note, heuristically, the the condition that 1 > b + c implies the IS curve is downward sloping. Whereas if b + c - 1 > a h /s implies the IS curve is steeper than the LM curve.
The analysis of the consequent MA dynamics follows conventionally - which we shall skip over here and merely refer to Smyth (1963) for details. Other types of models which include money, growth and cycles along simple Keynesian relationships include Laidler (1976) and "Keynes-Wicksell" monetary growth models (e.g. Stein, 1966, 1971; Rose, 1967).