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.

投資のaccelerator 理論によれば、投資は需要状況の変化に反応する。需要が増えれば、財に対する需要が過大になる。そういう状況になったら、企業は二つの選択に直面する。過剰な需要を抑えるために値段を上げるか、供給を増やすことでその需要に対応するかだ。ある状況では、前者が実行されるのも無理はない場合がある。でも、もっとケインジアン的な世界観では、量的な調整のほうが優先される。高い生産に対応するため、企業はプラントや設備に投資することで産出キャパシティを増やす。

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:

I

_{t}= K_{t}- K_{t-1}= Y_{t}- Y_{t-1}

where Y_{t} is aggregate demand, K_{t} capital stock and I_{t}
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.

I

_{t}= K_{t}- K_{t-1}= v(Y_{t}- Y_{t-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 I_{t}* be desired investment determined by the accelerator and I_{t}
actual investment, we can then impose a linear partial adjustment rule:

I

_{t}= m I_{t}*

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 I_{t} = K_{t} - K_{t-1} and I_{t}
= K_{t}* - K_{t-1}, where K_{t}* is desired capital stock, then we
can manipulate this expression to yield:

K

_{t}= m K_{t}* - (1-m )K_{t-1}

Now, from the accelerator expression, I_{t}* = v(Y_{t} - Y_{t--1}),
so:

K

_{t}* = vY_{t}- vY_{t-1}+ K_{t-1}

or, as v = K_{t-1}/Y_{t-1}, then we obtain simply that K_{t}*
= vY_{t}. Plugging this into our expression for K_{t}:

K

_{t}= m vY_{t}+ (1 - m )K_{t-1}

Now, we know that we can express K_{t-1} in the same form we
expressed K_{t}, but only lagged one period back. In other words, assuming a
constant accelerator and a constant m , then:

K

_{t-1}= m vY_{t-1}+ (1-m )K_{t-2}

Iterating back into our original expression, we obtain:

K

_{t}= m vY_{t}+ (1-m )[m vY_{t-1}+ (1-m )K_{t-2}]

= m vY

_{t}+ (1-m )m vY_{t-1}+ (1-m )^{2}K_{t-2}

Doing the same for K_{t-2}, K_{t-3}, etc. we can continue
iterating so that we obtain:

K

_{t}= m vY_{t}+ (1-m )m vY_{t-1}+ (1-m )^{2 m }vY_{t-2}+ (1-m )^{3 m }vY_{t-3}+ ....

or simply:

K

_{t}= m v・/font>_{i=1･ }(1-m )^{i-1 }Y_{t-i}

So, reconstructing this to represent investment (I_{t} = K_{t}
- K_{t-1}) we obtain the simple investment function:

I

_{t}= m v・/font>_{i=1･ }(1-m )^{i-1 }(Y_{t-i}- Y_{t-i-1})

which basically states that actual investment at time t (I_{t})
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.