The Aftalion-Clark Accelerator

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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 )²K_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