HEURISTIC METHODS OF INTEGRATING CYCLE AND GROWTH MODELS

HOWARD SHERMAN.

U N l M R D l l Y OF CAI.IFORNIA. RIVIEWIDL

There arc an enormous number of business cycle models and growth models, and there is even a respectable amount of literature on combined did growth m0dels.I Perhaps the most comprehensive and interesting of the JcaI growth models is Arthur Smithies’ [161. The model by Smithies fits some needs, but it does not provide us with the sim lest and most neral method of combining our knowledge of q d e and growth mo P els in a single r ramework. That is the purpose of the present brief note.

It must be recognized-as Smithies does not always dearly state-that his analysis (like the analysis presented here) develops a very useful heuristic model for under- standing and explanation, but that this is uite diffetent from an econometric pre- diction m ~ d e ~ j nus, smithies’ b i c mo& (like my own) correctly omits any governmental activity, any international relationships, any speafication of the money and credit system, any exogenous factors or random shocks, and any attempt at empirical testing.

In his review of the literature, Smithies points up the difference between those who begin with cycle models and add growth facton, and those who begin with

owth models and add @cal factors. On the one side is the classic cycle model 6 Samuelson,* a second order linear difference quation which (in its homogeneous form) cannot produce growth at the same time as it produces cycles. Smithies notes that Hi& ( [ 81, the furthest development from Samuelson) merely adds an “auton* mous investment” (At) as a dew ex macbina to “arplain” growth. The other fashion in cyclical growth models stems from the simple growth model, such as that of Har- d or Domar [2]. In its s h lest form it results in a first order linear difIerrn~e equation, which can only be u J t o represent the smooth growth of supply. Smithies neatly summarizes his own model: (1) A consumption function which determines consumers’ expenditures in rela-

tion to income and trend variables. (2) An investment function which does the same thing for gross investment

expenditures. ( 3 ) A short-run equilibrating relation which states that consumption and invest-

ment so detemined add up to the GNP in every year without excessive demand or supply.

(4) A relation that determines the increase in full-capacity output in response to net investment [ 16, p. 91.

The strength of Smithies’ model lies in the logical completeness of this framework. Any macrodynamic model of the economy must logically be considered to include all of these relations, explicitly or implicitly.

+The author wishes to thank Professors Bcmard Saffran d.Ro+~,E. -.for some important improvements on an earlier draft, without rn any way ImpllcaUng them m Its prexnt conclusions.

‘The earlier literature is cited and discussed in Smithies [16]. Some later articles include Minsky [12], Kurihara [9], and Pasinetti [131. ’See the classic econometric work by Tinbergen 1171. ‘Samuelson [14]. Another early statement is in H d [71.

276

SHERMAN: CYUE AND GROWTH MODELS 1‘77

SIMPLE GROWTH AND CYCLE MODELS

Within this generalized four-relation framework, it will proye useful to review briefly the familiar Domar and Samuelson models (before using these as compo- nents to build a cyclical growth model). In Smithies’ terms the formal Domar-type growth model may be expressed as follows:

(consumption behavior) c, = (1 - s) P, (1)

(investment behavior) I, = Y, - c, (2)

(income equality) Y, = c, + It (3)

(capacity growth) Y, - Yt-1 = k It-1 (4) - -

where Y is net national product, C is consumption, I is investment, F is full-capa- city product (potential output with fullest efficient use of capacity), t is a time period, J is the long run propensity to save (assumed constant), and k is the long-run output/capital ratio (assumed constant).

Equation ( 3 ) , the short-run equilibrium of supply and demand, is present without change in all models. In this model, however, equations (I), ( 2 ) and (4) form a separate set which reduce to the equation :

T, = (l+kS)Tt-, (5) Equation ( 5 ) says that the rate of growth of full-capacity output depends exclusively on the output-capital ratio and the propensity to save. Furthermore, what is saved is the same as what is invested because equation (2) says that all planned saving (at the full-capacity level of output and income!) becomes planned investment. Mathe- matically, we note that in equations ( 3 ) and (1) actual output and full-capacity output are both equal to the same quantity. Thus, by substitution:

Yt = (l+ks)Yt-, ( 5a) This equation says that actual output will follow the same path as full-capacity output (since planned investment equals planned saving at the full-capacity level of output); and that there will be a steady growth of output if k and I are positive constants.*

The Samuelson-type cycle model has no long-run growth, and ussumes that full- capacity output is constant. It uses the accelerator coeflicient (v), which indicates the investment reaction to a change in output demanded. Its usual consumption function is C = u + b Y, where b is the marginal propensity to consume and u is a given “minimum” consumption level. In the growth context we are investigating, the “minimum” level is better represented by a fixed proportion a of the full-capacity ou ut (so that r2 becomes aU), since full-capacity output is constant in the cycle

The simplest business cycle model of the Samuelson-type may then be represented as foIlows:

mo ’5; el.

(consumption behavior) C, = a y t 4- byt-, ( la)

(investment behavior) It = v(Yt.1 - Yt-2) (2a) (income equality) Yt = c, + I, ( 3 )

(full-capacity output) Y, - Y,-, = 0 (4a) - -

‘The mathematics of this type of model is detailed in Allen [ I , pp. 64-68, 74-78, and 183-861.

2‘6 :XVJSTERN ECONOMIC JOURNAL

where a, 6, and u are constants. The time path of output in this model is given by the equation:

Yt = (b+v)Yt.l - vY,-, + a y t (5b)

If r, u, b, and v are considered to be constants, then this is a linear second order difference equation.’ In its present form it can give rise to either growth M cycles, but not both. Within the range usually assumed for the p m - e rpnge where u is between (l-v‘m2 and ( 1 + V n ) , - t h e r e are cycles in output. “be constant, a y, enters only into the determination of the (long-run) level around which oscillation occurs.

THE SIMPLEST CYCLICAL GROWTH MODEL

The simplest way to construct a cyclical growth model, which will be useful for heiiristic purposes, is to combine the Domar and Samuelson models. The resulting consumption and investment functions allow for both Iong-run growth of capacity and short-run fluctuations of actual output below the capacity level.

We shall examine each of the component relationships in detail, but it is perhaps most helpful to begin by stating the entire model of cyclical g r o w t h e as follows:

where p is a constant.

Equation (lb) shows the determination of consumption. Here, we use both the short-run and long-nm propensities to consume, which many empirical studies have shown to be quite different.’ In this equation, the potential of fullcapacity may be taken to represent the long-run trend or the recipients’ views of “permanent” income.8

Of course, if we simply add together the short-run consumption estimate (aY 4- by) and the long-run consumption estimate (y - SQ, the total consump- tion estimate would be on the order of twice the correct estimate. Hence, both estimates must be scaled down before they are added together. For this purpose we need a weighting device to give the proper weight or proportion ( p ) of short-run estimate to long-run estimate. Let p be a constant between zero and one. Then if we multiply the shm-run consumption estimate by p and the long-run consumption estimate by I-p, the totd estimate must always be of the ti& magnitude. Obvionsly, if p is equal to one, the result is a short-run consumption function; while if p is equal to zero, the result is a purely long-run consumption function. For any p be- tween zero and one, we will have short-run cyclical consumer reactions, but also a built-in andogmous “rachb’ e f f d leading to further growth.

YThe mathematics of this model is detailed in Allen [ l , pp. 79-82 aad 187-2201. *An earlier vmion of this model (with some difTercnces and some inaccuracies) is briefly

‘See Kuzncts [lo, p. 1191. 9ee Fricdmrn 141. ’See the classic discussion by Duesenberry 131.

discussed in a very different context in She- [15, Ap~~ndix ] .

SHERMAN: CYCLE AND GROWTH MODELS 2 79

Fquation (2b) shows the determination of investment. Here, the short-run accel- erator (or capital-output) coefficient is applied to the short-run change in actual income in order to get the short-run investment estimate. To obtain the quite different long-m investment estimate, we apply the long-run saving ratio to the long-run full-capacity income. Once again, mere addition of the short-run and long-run esti- mates would result in a total estimate of investment that would be on the order of twice the correct estimate. Hence, once more, both estimates must be scaled down before they are added together. We need another weighting device to give the proper weight or proportion of the short-run investment estimate to the long-run investment estimate.

Obviously, the weighting of the long-run and short-run factors in the investment estimate may be quite different from the weight of the long-run and short-run factors in the consumption estimate. Therefore, in an econometric model using this device, it would be necessary to calculate a different weighting proportion in the investment function than in the consumption function. It would be another constant like p , between one and zero, but usually different from p . Neverthdess, there is nothing 4 priori that says that the difference between the two constants would be very great. Furthermore, their function is identical, since they must stand at one when the model is purely short-run and at zero when the model is purely long-ma. 'Ihercfore, for simplicity of apositio-d strictly as a first approximation, we assume here that p and I - p am the weighting factors for the short- and long-run investment estimates

well as the short- and long-run consumption estimates.

Finslly, equation (4b) stiates the change in full-capacity output as a function of the long-run marginal autput-capital CoefKaent times the amount of net investment. It is again necessary to weight this long-m a d i a e n t by the weighting factor I - p , SO that it is on the same scale as the other equations with which it interacts. We UK (1-p) here rather than p becaw it is a long-m estimate, balanccd against the short-m estimate of the marginal capital-output ratio weighted by p in equation (2b).

It is important to note in this connection that the marginal long-tun outputtapital coefiicient is not the same as the reciprocal of the marginal short-run capital-output cwflident. Empirical investigations*o have shown that the short-run cyclical estimates of the investment and change-of-output relation dif€er quite far from the relatively stable long-run relation of output to capital. The strong empirical support for the no- tion of a significant difference between long-run and short-tun estimates in both the consumption function and the investment function gives a realistic basis to a model based on functions which are a (weighted) sum of the two estimates. Thus, in every quation involving parameters, each of the short-term parameters are weighted by the factor p , while each of the long-term parameters are weighted by the factor I-J. When p quals zero, all of the short-run iduenccs are eliminated, and the m el reduces exactly to the Domar growth equations ( I , 2, 3, and 4) . When p equals one, the long-run influences are eliminated, and the model reduces e x a l p to the Samuelson cyde equations (Iu, 24 3, and 4a).

In the general case for any p between zero and one, the combination of equations (lb), (2b), (3), and (4b) gives the result that:

"See, e.g., Meyer and Kuh [ 111.

280 WESTERN ECONOMIC JOURNAL

where:

m = 1 + p(b+v) + sk(l-p)* n = -p[b + 2v + k(l-p)z (sb + sv - v) - (1-p) (hap ) ]

This is a homogenous third order linear difference equation, in which it is possible to have both p w f h and cycIes. Of course, if p falls to zero, the quation of cyclical growth (5c) reduces to the Domar growth equation (5a); while If p rises to one, quation (5c) reduces to the Samuelson cycle equation (5b) .I1 In the general case, for any p between zero and one, there will be growth if the

quantity (m + n + q) is greater than one; and if (18mnq + n*m') is less than (4m*q + 4na + 27 ), then there will also be qclical fluctuations.U Under

v, and k) will produce both cyda and growth. Assume, for ample, that 1--s (the long-run propensity to consume) equals .%, while b (the short-run marginal pro- pensity to consume) equals 30, and ri (which sets the "minimum" level) equals .lo; so that total short-run and long-run estimates are identical when actual equals fullcapacity output. Assume also that k (the long-run outputcapital ratio) equals .30, while l/u (the short-run autput-capitaI ratio) equals 30 (or u, the urelerator, equals 2.00). Furthu, assume arbitrarily that p is exactly in the middle of the intermediate area, that is, p = .5. Finally, also arbitrarily, let Yo = 95.0, rising to y, = 100.0. Then the time path of actual output, Y, beginning at Yo, is: 95.0, 100.0, 100.0, 95.8, 90.4, 87.1, 87.8, 92.3, 98.3, 103.1, 104.7, 102.9, 99.5, 96.7, 96.5, 99.1, 103.5, 107.6, 109.9, 109.8, 108.0, and onward. Although the form of growth is cyclical, the peaks do rise (from 100.0 to 104.7 to 109.9) in this series.

It will also be observed that as p rises from zero toward one, short-run cychcal influences become more important while long-run growth influences become less important. Indeed, as p rises from zero toward one, the amplitude of fluctuations increases while the rate of economic growth declines.

The adjustment process in the general m e of cyclical growth may be indicated here in a few words (since we have already discussed each of the relationships sepa- rately). As national income rises, the rate of growth of consumption declines, restricted primarily by the limited short-run marginal propensity to consume. The importance of consumption in total output leads to a decline in the rate of growth of national income (though the rate of growth is also Iimited by the full-capacity ceiling). Such a decline sets off a decrease of investment, because of the drastic effect of the short- run accelerator coefficient. This then leads directly to the absolute decline of national income.

q pv[I - k(1-S) (I-p)' - (1-P) ( k p ) ]

these criteria, we find %' at almost any reasonable values of the parameters (a, b, 2,

''Amully, wben p equals one, we first Obtaio

Yt= (1 + b f v ) Yt. i - (b + 2v) Yt 2 4- vYt 3 (Sd) But the Samuelson equation without its constnot part (ay, which may be set equal to zero in the short-run) is:

Yt = (b 4- V) Y 1-1 -vY t - a (Se) If we subtract the Samuelson quation (Se) from (Sd), the remainder is simply another equation identical to (Se), but with all components dated one period earlier. It may, then, be proven that equations (x) and (sd) will produce identical time paths from the appropriate initial conditions. See [ 5 , pp. 163-641.

"See for mathematical details Griffiths 16, pp. 29-31].

SHERMAN: CYCLE AND GROWTH MODELS 28 1

The ensuing income decline is stopped far above the trough indicated by the volatile short-run consumption and investment functions. A stable base above that level (a "ratchet" effect) is provided by the long-run expectations of consumption and investment, represented as functions of the potential full-capacity income. It is also realistic to picture the full-ca acity level exerting an immediate upwad pressure (by its unused capacities) during ti e recovery period.

REFERENCES

1. R. G. D. Allen, Ullbematical Economics. London 1957. 2. E. D. Domar, "Expansion and Employmeat," Am. Econ. Rev., Mar. 1947.37, 34-35. 3. J. S. Dueseaberry, Income, Saving, and tbe Theory of Consumer Bebavior. Cambridge, Mass.

4. M. Friedman, A Theory of tbe Consxmption Pxnrtion. Princeton 1957. S. S. Goldbag, Zntroduction to Diffsrence Equations. New York 19S8. 6. L. W. Gr&ths, Zntfduction to the Theory of Eqrationr. New York 1 9 ~ . 7. R. P. H d . Tbe Trade Cycle. London 1936. 8. J. R. Hicks, A Contribution to tbe Tbeory of the Trade Cycle. Oxford l9SO. 9. K. K. Kurihara, '*An Endogenous Model of Cyclical Growth," Oxford &on. Pupers, Oct.

1949.

1960, 12, 243-48. 10. S. S. Kumets. National Product Since 1869. New York 1946. 11. J. R. Mcyer and E. Kuh, Tbe Investment Decision. Cambridge, Moss. 1957. 12. H. P. Minsky, "A Linear Model of Cyclical Growth,'' Rev. Econ. Stat., May 19S9.41, 133-45. 13. L. L. Psinetti, "Cyclical Fluctuations and Economic Growth," Oxford E r a . Paprrs, June

14. P. Somuelson, "Interactions Between the Multiplier Analysis and the Principle of Amlar-

15. H. Sherman, Macrodynamic Economics. Ncw York 1964. 16. A. Smithies. "Emaomic Fluctuations and Growth,'. Economuhir, Jan. 1957,2J, LS2. 17. J. Tinberg-, Sta&icaI Testing of BxJinesr Cycle Tbeorirr, Geneva 1938-39.

1960, 12, 21S-41.

tioa," Rev. &on. Stm., Mny 1939.21, 7S-78.