The acceleration principle finds its roots in the work of Thomas Nixon Carver (1903), Albert Aftalion (1909), C.F. Bickerdike (1914) and John Maurice Clark (1917). Unlike other theories of investment, the accelerator theory tends to be sparse in its "microfoundations", relying upon its empirical stregth both for its derivation and justification. The accelerator -- or "the Relation", as Roy Harrod (1936) called it -- lay at the heart of the Keynesian business cycle theory of Harrod, Hicks, Goodwin and others.
This, succinctly stated, is the naive theory of the accelerator, perhaps the simplest of investment theories (and, perhaps surprisingly, the most empirically successful!). In the extreme, the idea that investment responds immediately and entirely to changing demand conditions implies a relation in the following form:
It = Kt - Kt-1 = Yt - Yt-1
where Yt is aggregate demand, Kt capital stock and It investment, all at time t. However, demand shocks are many and not all permanent. If, for instance, a firm responds to an aggregate positive demand shock at time t by increasing capacity immediately, it might be faced with a dilemma if, at time t+1, there is a negative demand shock: it increased its capacity permanently, yet, at time t+1, much of that was not utilized.
As such, then, we can propose that a firm, instead of increasing capacity immediately and fully in response to a single demand shock, it will respond only gradually - perhaps increase capacity now by a little bit, see if the demand change is sustained in period t+1, change a little more then and continue in this little process until it converges to the desired level of capacity. In this case, then, the size of a change in capital is a fraction of the size of the change in demand. i.e.
It = Kt - Kt-1 = v(Yt - Yt-1)
where v is a constant known as the "accelerator coefficient" and it is assumed that 0 < v < 1. Of course, v can be thought of as the desired capital-output ratio, v = K/Y. Thus, given a change in aggregate demand, the accelerator gives us the change in capital needed to achieve that desired capital-output ratio. Since v is a fraction, a change in demand will require a smaller change in capital.
The accelerator argument, initially based on the uncertainty of demand movements, establishes then the desired change in capital stock each period. This is not, as Haavelmo (1960: p.8) reminds us, a theory of investment yet. In principle, there is no a priori reason to assume that this desired change is feasible. Simply, we could invoke the arguments of supply-constrained capital-goods industries, delivery lags, etc. in order to propose that only a portion of the desired investment will actually be undertaken (cf. Goodwin, 1951; Chenery, 1952). Letting It* be desired investment determined by the accelerator and It actual investment, we can then impose a linear partial adjustment rule:
It = m It*
where the parameter m lies between 0 and 1. Thus actual realized investment - the actual change in the capital stock - will be a fraction of the desired change. As It = Kt - Kt-1 and It = Kt* - Kt-1, where Kt* is desired capital stock, then we can manipulate this expression to yield:
Kt = m Kt* - (1-m )Kt-1
Now, from the accelerator expression, It* = v(Yt - Yt--1), so:
Kt* = vYt - vYt-1 + Kt-1
or, as v = Kt-1/Yt-1, then we obtain simply that Kt* = vYt. Plugging this into our expression for Kt:
Kt = m vYt + (1 - m )Kt-1
Now, we know that we can express Kt-1 in the same form we expressed Kt, but only lagged one period back. In other words, assuming a constant accelerator and a constant m , then:
Kt-1 = m vYt-1 + (1-m )Kt-2
Iterating back into our original expression, we obtain:
Kt = m vYt + (1-m )[m vYt-1 + (1-m )Kt-2]
= m vYt + (1-m )m vYt-1 + (1-m )2Kt-2
Doing the same for Kt-2, Kt-3, etc. we can continue iterating so that we obtain:
Kt = m vYt + (1-m )m vYt-1 + (1-m )2 m vYt-2 + (1-m )3 m vYt-3 + ....
Kt = m v・/font> i=1･ (1-m )i-1 Yt-i
So, reconstructing this to represent investment (It = Kt - Kt-1) we obtain the simple investment function:
It = m v・/font> i=1･ (1-m )i-1 (Yt-i - Yt-i-1)
which basically states that actual investment at time t (It) will be a fraction (m ) of the desired investment which, in turn, is a fraction (v) of past changes in output/aggregate demand. Note that desired investment is not determined solely by the current change in output but also by earlier changes in output. The geometrically-declining distributed lag form implies that the earlier the output change, the less of an effect it will have on current desired investment. This ensures that the actual capital stock will only gradually approximate the desired capital stock.
There are several problems with such a model. For one, if we were to attempt an empirical estimate of this, the utilization of geometrically-declining distributed lags opens it up to the criticism that its estimators will be biased and inconsistent. Theoretically, and more importantly, no consideration is given to interest rates influencing investment. Such an absence would only be possible if, in addition to the constant returns to scale assumption, we also assumed constant relative prices for factors.
In spite of these reservations, we can intuitively tie the accelerator in with other theories of investment, such as the marginal adjustment cost theory. In this light, the investment engendered by a new desired capital stock will, through the Keynesian multiplier, lead to greater income levels which will, in turn, "accelerate" capital accumulation by shifting the marginal productivity of capital curve to the right. As a result, there will be a new and higher desired capital level (K*) and thus more investment per period is required. In this sort of situation, it is conceivable that the desired capital stock would never really be reached but rather that it would always remain a step ahead of the actual capital stock. In this respect, then, investment could be more or less "continuous", and so may seem more amenable to macroeconomic description. However, such a "marriage" of the accelerator and the marginal adjustment cost theory, reiforces the Hayekian interpretation, as investment remains a movement towards a final destination rather than simply the inherent behavior of capitalists.