## Nonlinear Models 31 The Goodwin Model

Both the Hansen-Samuelson version and the Hicks linear version of the accelerator and multiplier models are unsatisfactory from the point of view of business cycle theory because of the nature of the possible states in which the economy may be. 'By dropping the highly restrictive assumption of linearity we neatly escape the rather embarrassing special conclusions which follow (in linear theory). Thus, whether we are dealing with differences or differential equations, so long as they are linear, they either explode or die away with the consequent disappearance of the cycle or the society.'12 The suggestions that non-linear relations be used as a substitute for the linear form of the simplest theory has appeared in two forms: one a 'formal exposition' by Goodwin and the second a longer informal exposition by Hicks. In what follows we shall follow Goodwin's formal exposition through the essentials of his first three models. The essential difference between the Hicks and the Goodwin non-linear models is that in Goodwin the accelerator coefficient's non-linearity is determined in the first instance by the relation between actual and desired capital whereas in Hicks the non-linearity arises due to a 'full employment' ceiling to output, which results, once full employment is reached, in making Yt of the order of Yt_v This results in but little induced investment even though the accelerator coefficient is unchanged. Hicks' non-linearity depends upon a full employment ceiling to income whereas Goodwin's non-linearity, at least in his third model, depends upon the 'generating relation for the accelerator coefficient'. As such, nothing in Goodwin's third model is inconsistent with this thesis.13

'The central difficulty with the acceleration principle is that it assumes that actual realized capital stock is maintained at the desired relation with output. We know in reality this is seldom so, there being now too much and now too little capital stock.'14 The significant relation to Goodwin is the relation between the actual capital stock k and the capital stock desired for a given output Y. Inasmuch as Y is being produced, and it is possible to produce Y without having the desired capital stock, the production function for output as a whole which Goodwin uses is one that involves alternative combinations of factors capable of producing a given output: Goodwin's production function for output as a whole involves substitution among factors. This is in and of itself an improvement upon most accelerator doctrines. The investment decision depends upon the relation between existing capital and the desired capital for a given realized output: we can assume therefore that the investment decision is based upon the difference between the 'present plant' and the 'plant which can produce today's output at the least cost'. But the 'plant which can produce today's output at the least cost' is dependent, in a production function with substitution, upon the relative prices of the factors of production. If investment costs are 'high', the amount of investment induced by a given output greater than the 'best' output for a given plant will be smaller than if investment costs are 'low'. The amount of investment necessary to bring a realized capital stock into the desired relation with output depends upon the rate of interest, among other determinants of the best way to produce a given output.

The reason given by Goodwin for the divergence between the actual quantity of capital and the desired stock of capital during the period in which there is too little capital stock is that 'the rate of investment is limited by the capacity of the investment goods industries'.15 This is, of course, a judgement about the nature of the world as one could construct a model in which the limited capacity of the capital goods producing industries is just sufficient to satisfy the demand derived from the difference between the desired and the existing capital stock. Also, if the desired increase in capital stock is greater than the limited productive capacity of the capital goods industries, the price of capital goods can be expected to change. However, Goodwin ignores these possibilities, so we have that the maximum amount of investment possible is a constant, k*, per period, and if the desired capital is i, the actual capital is k, maximum investment will take place for - k]/k* periods. An additional gratuitous observation by Goodwin that 'entrepreneurial expectations are such that, even if it were possible to expand plant in the boom, there would be great resistance to it' is made.16 Aside from being a casual empiric assertion about entrepreneurial behavior, it would also, if taken seriously, make any investment theory of the business cycle impossible - for during the boom entrepreneurs resist investment, therefore no investment takes place at high incomes. This flies in the face of any casual observation of business cycle behavior.

The reason for the divergence between the actual quantity of capital and the desired quantity of capital during the period in which there is too much capital is that 'Machines, once made, cannot be unmade, so that negative investment is limited to attrition from wear, from time, and from innov-ation'.17 This ignores, as is typical in such models, disinvestment in working capital which is not limited to wear, time or innovation, but is limited to the level of consumption and investment purchases or by the level of such stocks (for example, if output drops to zero, and all sales are out of stocks). This observation also applies to investment: the maximum amount of investment possible is not equal to the 'capacity of investment goods industries', but is equal to the production of all commodities which can be stored. Continuing to follow Goodwin, we can define a negative investment rate k** per period, and if the desired capital is i, and the actual capital k,wehave that maximum disinvestment will take place for [k -i]/k** periods. (k >i for disinvestment.) We can also define a situation in which k = £, which implies zero net investment. Because of these limitations we have that 'capital stock cannot be increased fast enough in the upswing, nor decreased fast enough on the downswing, so that at one time we have shortages and rationing of orders and at the other excess capacity with idle plants and machines'.18

The problem remains of defining the desired capital stock. Goodwin defines i = KY; i = desired capital stock, Y = income, K a constant. This is a linear relationship between desired capital stock and output. We therefore have a production function in which there is substitution in the short run: for Y can be produced with a capital stock not equal to i, but the long run production function has fixed proportions.19 Once the substitution is admitted, there seems to be no reason why it should not also be allowed in the long run, writing K = K(PL, PK), the accelerator coefficient K becoming a function of the relative price of labor and capital for example. We therefore would have that i = K(PL, PK)Y. If the price ratios of the factors change so that capital becomes relatively more expensive, the quantity of capital desired to produce a given income decreases, whereas if capital becomes relatively cheaper, the desired capital stock increases. Such a relation between the accelerator coefficient and the relative prices of the factors could be utilized to integrate money market phenomena with the accelerator-multiplier type model. For example, if the amount of financing ability in a community is inelastic, so that when induced investment is high the price of financing rises relatively faster than the price of labor, the accelerator coefficient will decrease. Conversely, if the price of financing falls more rapidly than the price of labor, the result may be a high desired relation between capital stock and output. This would tend to shorten the period [k — i]/k** during which income is falling.20

Combining the above relations, Goodwin obtains his simplest model:

i = KY (desired capital = constant X income)

C = aY+ a0 (a linear consumption function)

the time rate of change of capital, hence investment. We therefore have y = a0/[1 — a] + k/[1 — a]. Goodwin assumes 'that the economy seeks the perfect adjustment of capital to output and that it does so in either of two extreme ways, capacity output of investment goods or zero gross invest-ment'.21 Using the relation between desired investment and income we have i = (K/[1 — a])k + Ka0/[1 — a], and as k = k*, 0, or k** as i^k we would have that i = i*, i0, or i**. We therefore have three possible levels of income, investment and desired capital. The set of values of y = a0/[1 — a], k = 0, and i = i0 = k are equilibrium values. 'It is, however, an unstable equilibrium, since a small displacement in the phase plane leads to a large displacement from which it never returns. For example, if to i0 we add Ai, then k changes from zero to k* and i becomes i*.'22 The model operates in the following manner: for [i* — k]/k* periods the level of income is Y =a0/[1 — a] + k*/[1 —a].Atthe end of this time, i* = k, therefore k = 0. However we have that the desired capital for Y = a0 / [1 — a] is equal to io which is less than the desired capital i* for Y =a0/[1 — a] + k*/[1 — a]. Therefore, investment falls to k** which leads to an income Y = a0/[1 — a] + k**/[1 — a] with a desired capital stock i**. This income lasts for [i* — i**]/k** periods (k = i* when the change in income occurs) at which time k = i**. This leads to Y = a0/[1 — a] > Y = a0/[1 — a] + k**/[1 — a]

which implies k* investment, Y = a0/ [1 - a] + k*/ [1 - a] which implies I* as desired capital. This income continues for [I* - I**]/k* periods (k = I** when the change in income occurs). Inasmuch as k** is assumed to be less than k*, the time spent in the low income state is greater than the time spent in the high income state.

This crude model does illustrate the general characteristics of non-linear models:

A. The final result is independent of the initial conditions.

B. The oscillation maintains itself without any need of outside 'factors' to help in the explanation. In this sense, it is a complete self-contained theory.

C. The equilibrium is unstable and therefore the mechanism starts itself given even the smallest disturbance. Yet in spite of this instability it is a usable theory because the mechanism does not explode or break down but is kept within bounds by the non-linearity.

D. No questionable lags are introduced. The mechanism operates by its own structure.23

The deficiencies of this model are obvious, 'such a crude model cannot claim to be a representation of actual cycles'.24 In particular income has only two levels, and the investment which takes place during the upswing does not increase productive capacity. As an expository device this Goodwin model may suffice, but as a framework for business cycle analysis it is even cruder than the original linear Hansen-Samuelson model.

Goodwin's second model makes allowance for technological progress. 'To make a crude allowance for technological progress, we may assume a steady growth in the desired amount of capital.'25 We therefore write | = at + kY, | = a (the time rate of change of I). In this model, 'no equilibrium exists since k = 0 means that k is constant and hence that I would become greater than k and hence k would cease to be zero'.26

The rate of growth of desired capital I = a, and Goodwin again assumes that the rate of change of capital k has two values. 'If a is greater than k*, the economy can never catch up with its capital needs. Excluding this unrealistic case . . .'.27 The unrealistic case unfortunately may not be so casually excluded during at least a portion of the business cycle. The possibility that during a strong boom the rate of desired growth of capital may outstrip the possible rate of growth of capital cannot be ignored. An inflationary period may be viewed as one in which the supply of resources for capital expansion is less than the demand of resources for capital expansion. An inflationary period may be interpreted as a time during which the financing of investment by sources outside of voluntary real savings are ineffective in yielding as high a rate of investment as entrepreneurs desire. The history of economies which have had long periods of open or suppressed inflation can be interpreted as a 'donkey-carrot' arrangement between actual and desired capital equipment. This may be particularly true in an economy that is beginning its industrial revolution and in an economy which has had a portion of its capital equipment destroyed during a war.

However, by assuming that k* > a, Goodwin achieves a cycle with growth. As k* > a in time t, k = i, so that desired capital falls from i* = a' + K(p/[1 - a] + k*/[1 - a]) to i** = a' + K((/[1 - a] + k**/ [1 - a]). At this stage k is being decreased. However i** increases due to 'technological change' so that in time i* = k, which raises income to (/[1 — a]. This makes . esired capital greater than actual capital and leads to investment at a rate k*. This model therefore operates in essentially the same manner as the simplest model, but it does succeed in reducing the relative length of the low level of income. Even though 'technological change' is taking place, the high level of income Y =(/[1 — a] + k*/ [1 — a] does not rise. Net investment is taking place without any change in attainable income.

Goodwin expands the above models in three directions. He introduces a dynamical multiplier, an investment lag, and he generalizes his non-linear accelerator. Only the third need concerns us - his generalization of the non-linear accelerator. 'The investment, k, consists of an autonomous part, €('), and an induced part 9. About induced investment we may make the less crude (than the previous one) assumption that the acceleration principle i = KY holds over some middle range but passes to complete inflexibility at either end as is shown in (Goodwin's) Figure 4. The upper limit is the k* of the previous models and the lower limit the k**[dq(y)/d(y)] is

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