Goodwin's Non-Linear Accelerator

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One of the first, formal non-linear models of the cycle was presented by Richard Goodwin (1951) in a model similar in spirit to the Harrod-Hicks multiplier-accelerator model, but entirely different in structure. The main difference was that Goodwin's accelerator was non-linear (or, rather, piecewise linear) - which enabled him to obtain cycles for any parameter constellation and without exogenous ceilings, floors, or shocks. Goodwin's 1951 article was more an illustrative exercise of the importance of non-linearity rather than an attempt at a comprehensive theory of the trade cycle - thus its rather simple structure should be in this light.

The explanation for the non-linear accelerator related to a familiar problem in the theory of investment: namely, how to reconcile investment (flow) decisions and capital (stock) decisions. If firms choose and achieve their optimal capital, then net investment flows are zero. Thus, following Friedrich Hayek (1941), Abba Lerner (1944) and others, to have positive investment, one must delay the movement from a given amount of capital stock to optimal capital stock via internal or external marginal adjustment costs to investment. Goodwin's argument rested partly on this - arguing for the limited capacity and rising supply price of the capital goods industry. We take an oversimplified version of Goodwin's model to illustrate how he obtains cycles.

Let optimal or desired capital stock be determined as a proportion of output, i.e. K* = vY where v is some optimal capital-output ratio. Let d denote depreciation (or, more appropriately, scrapping/disinvestment rate), which we shall assume constant, thus net investment dK/dt = I - d where I is gross investment. Let the capacity of the capital goods industry be denoted I_M and let us presume that it either produces I_M or it produces nothing (think of a shipbuilding firm/industry which only has the capacity to produce one ship at a time, does not deliver partial ships). Thus, net investment decision is:

i.e. firms purchase I_M from the shipbuilder if desired capital stock exceeds actual capital stock (so capital increases by I_M - d); purchase nothing and scrap nothing if at the optimal capital stock (so net capital does not change); and disinvest/scrap at the constant depreciation rate if capital exceeds desired capital. Thus, firms behave as bang-bang extremists, trying to buy as much capital as possible or scrap as much as possible to reach their desired rate quickly.

Now, desired capital stock is K* = vY and goods market equilibrium requires that Y = C + dK/dt where C = c₀ + cY. Thus, combining these with the piecewise linear investment function:

For the sake of simplicity, let q = vc₀/(1-c) so that if K < K*, then K* = q +(I_M-d)q ; if K = K*, then K* = q and, finally, if K > K*, then K* = q -dq. The dynamics of the model can begin to be seen in Figure 1.

Suppose we begin at equilibrium, K = K*. Then K* = q and dK/dt = 0 - thus we are at point E in Figure 1. Suppose now it is displaced slightly upwards so that K > K*. Then, immediately dK/dt = -d and desired capital, K*, switches discountinuously from K* = q to the lower amount K* = q -dq . Thus, we jump from point E to point A. Obviously, as K is still K, then it will be now far above K* as the desired capital stock is much lower. Consequently capital will decline (a little more slowly) until it reaches the desired amount, i.e. until K = q -dq . Thus, we now move slowly from point A to point B.

However, once at B, K = K* again - thus desired capital stock switches discontinuously again to K* = q . However, as now we are at K = q -dq , then obviously K < K*. But if K < K*, then we know that optimal capital stock is not really q any longer but rather K* = q + (I_M-d )q - thus there is a two-step switch and firms are seriously behind. This implies that firms now try to invest as much as possible, i.e. dK/dt = I_M - d , thus they jump from point B to point C in Figure 1. From C, they glide towards D at that pace.

However, once D is achieved, then K = K* and, again, the desired capital stock switches discontinuously to K* = q - so now actual capital stock K = q + (I_M-d )q is too much and K > K*. However, if K > K*, then we switch further to K* = q - q d, thus firms have to back quite a long way. Thus, they scrap as quickly as possible, i.e. dK/dt = -d , thereby jumping from point D to point F and (more slowly) make their way back to point B. The process then continues again as at point B, K = K*, thus they make a two-step switch in K* and jump to C, etc.  The resulting dynamics of the economy depicted in Figure 1 are captured over time in Figure 2.

The letters (B, C, D, F) in Figure 2 correspond to the points in the earlier box in Figure 1. Obviously, investment dK/dt has the bang-bang discontinuous pattern outlined earlier -- and income Y(t), of which investment is a component, also exhibits the same bang-bang discontinuity. The capital trajectory, K(t), however, moves more slowly as noted earlier, between the values of q-dq and q+(I_M-d)q. Obviously, this simplistic piecewise linear accelerator is but a special, extreme case of more general non-linear accelerators as Goodwin (1951) suggests. But it is a simple exhibition of the value non-linearities have for generating endogenous cycles without relying on structurally unstable parameters, exogenous shocks, etc. as illustrated, no less forcefully, by Kaldor (1940).

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