LECTURE 7CONSUMPTION IN THE CYCLE-THEORY MODEL.

SAY’S LAW

JACOB T. SCHWARTZ

EDITED BY

KENNETH R. DRIESSEL

Abstract. We modify the model in Lecture 6 by supposing that

in each production day, a certain fixed amount ej of commodity Cj

is consumed by the manufacturers who collectively constitute our

model and we also add an amount hj of commodity Cj as basic

inventory to the inventory which the manufacturer of commodity

Cj wishes to carry.

1. Mathematical Analysis of Consumption as an Addi-

tional Feature. The Model Cycle.

We saw at the conclusion of the preceding lecture that the inexorable

rise and permanent oversupply of inventories in our model was due to

the fact that no inventory-reducing consumption was introduced in the

model. We now wish to modify the earlier model by supposing that

in each production day, a certain fixed amount ej of commodity Cj is

consumed by the manufacturers who collectively constitute our model.

We shall also make our description of inventory policy somewhat more

realistic by adding an amount hj of commodity Cj as basic inventory

to the inventory which the manufacturer of commodity Cj wishes to

carry; thus, we will now have

desired inventory = basic inventory + cj × day’s sales.

Instead of introducing these assumptions into the disaggregated equa-

tions of Lecture 5, we shall, in order to obtain the simplest aggregative

2010 Mathematics Subject Classification. Primary 91B55; Secondary 37N40 .Key words and phrases. Business Cycles Model, Keynes Theorem, Say’s Law .

1

2 JACOB T. SCHWARTZ

model, assume at once that ej and hj are proportional to the “bal-

anced” eigenvector v of equation (6.1). Making use of the technique

for direct elaboration of aggregative equations explained at the begin-

ning of the previous lecture, we then obtain the following modification

of the recursions (6.11a) and (6.11b):

b(t) = b(t− 1) + (1− γ)a(t− 1)− e(7.1a)

a(t) = min[{((c+ 2)γ − 1)a(t− 1)− b(t− 1)(7.1b)

+ (c+ 2)e+ h}+, (b(t)/γ)].

It should be explained that in writing the final term in (7.1b), we as-

sume that the whole of the inventory available at the beginning of a

“day” is available for production; one may for instance take the sub-

traction from stocks for the purposes of consumption to occur “in the

evening” after the day’s production has taken place. In writing equa-

tion (7.1a) as simply as we do, we are assuming in similar fashion that

(1 − γ)a(t) + b(t) always exceeds e, so that the whole economy is not

eventually devoured by the consumers. It will be necessary for us to

check the correctness of this assumption when we examine the detailed

orbit of our model.

Our analysis may now follow the paths marked out in the preceding

lecture. As long as this transformation keeps us between the “produc-

tion cutoff line” a = 0 and the “scarcity line” a = b/γ of the preceding

lecture (cf. Fig. 2), the equations (7.1a-b) bid us iterate the inhomo-

geneous linear map defined by

b(t) = b(t− 1) + εa(t− 1)− e(7.2a)

a(t) = γa(t− 1)− b(t− 1) + h.(7.2b)

Here we have put ε = 1− γ, γ = (c+ 2)γ − 1, and h = (c+ 2)e+ h. If

this transformation takes us out of the admissible region of Fig. 2, we

are to return to this region in the same way as in Lecture 6 (cf. Fig.

3).

7. CONSUMPTION IN THE CYCLE-THEORY MODEL 3

We may now make use of the simple mathematical principle that

an inhomogeneous linear transformation is simply a homogeneous lin-

ear transformation referred to a displaced center, namely, to the fixed

point of the inhomogeneous transformation. The fixed point of the

transformation defined by (7.2a–b) is the solution [aK , bK ] of the equa-

tions

bK = bK + εaK − e(7.3a)

aK = γaK − bK + h.(7.3b)

Thus

aK = e/ε(7.4a)

bK = (γ − 1)(e/ε) + h.(7.4b)

We note that this Keynes point [aK , bK ] lies above the scarcity line

γa = b, since c > 1 and thus

γ(e/ε) ≤ (cγe/ε)− 2e+ (c+ 2)e(7.5)

≤ (cγe/ε) + (2γ − 2)(e/ε) + (c+ 2)e+ h

= ((c+ 2)γ − 1− 1)(e/ε) + h

= (γ − 1)(e/ε) + h.

The Keynes point [aK , bK ] is by definition the level of inventory and

production at which production exactly balances consumption, and

actual inventory is exactly equal to desired inventory. If we now let

a(t) = a(t) − aK and b(t) = b(t) − bK denote the deviations of actual

production and inventory at time t from the Keynes point, we find

from (7.3a–b) that (7.2a–b) are equivalent to

b(t) = b(t− 1) + εa(t− 1)(7.6a)

a(t) = γa(t− 1)− b(t− 1).(7.6b)

These, however, are exactly the recursions which were governing in the

preceding lecture. We find in consequence that the pattern of orbits

in our new model is exactly like the pattern of orbits in the preceding

4 JACOB T. SCHWARTZ

model, except that the center of the oblique node of Fig. 5 is shifted

from the origin to the Keynes point. We are consequently examining

the very same node sections of which are portrayed in Figs. 5, 6,

and 9, but looking at a different section of the picture! This remark

enables us to construct the corresponding figures for our new model

with a minimum of effort. We have again a dichotomy between an

“expansive case” and a “depressive case” as in the preceding lecture.

The case corresponding to the expansive case portrayed in Fig. 7,

defined by the property that the small-eigenvalue eigenvector of the

transformation defined by (7.6a–b) points into the angle between the

production cutoff and the scarcity lines, appears as in the following

Figure 10. (Compare Fig. 7.)

The “recovery point” marked in Fig. 10 is at the level of inventory

where sales for the purpose of consumption bring about a rise in pro-

duction from the zero level; according to equation (7.1b), this is at

the level b = (c + 2)e + h of inventory. Geometrically, this point is

the unique point at which an orbit curve proceeding down from the

Keynes point is tangent to the b-axis. The “danger point” marked on

the diagram is the point at which full employment of existing inventory

in production just barely yields a surplus of e; it is consequently at the

level b = γe/ε. If inventories ever fall below this danger level, they

will inevitably be reduced without limit by persistent consumption,

and eventually the whole economy will be devoured by the consumers

(in our oversimplified model; in a more realistic model this would lead

to “inflation” and the restriction of consumption). Points above the

danger point on the scarcity line proceed upward in successive periods

of time; points below the danger point proceed downwards. In order

that our model, with fixed (nonproductive) consumption in the sense

we have assumed, be reasonable, it is then necessary that (c+ 2)e+ h

should distinctly exceed γe/ε.

We have marked a typical orbit in Fig. 10; it shows a recession be-

ginning, production falling to zero to permit the decline of inventories

to recovery levels, recovery, and, in accordance with the basically “ex-

pansive” nature of the case under examination, a permanent prosperity

7. CONSUMPTION IN THE CYCLE-THEORY MODEL 5

Fig. 10. Consumption in an expansive case.

thereafter, with inventories engaged in a perpetual race to keep up with

each other, and consumption, having once led to recovery, playing an

ever smaller role. This illustrates the mechanism of recovery, but evi-

dently not that of recession. To study this latter mechanism, we must

examine the case corresponding to the depressive case portrayed in Fig.

9, in which the small-eigenvalue eigenvector of the transformation de-

fined by (7.6a–b) points out of the angle between the production cutoff

and the scarcity lines. Here we have Fig. 11 (cf. Fig. 9).

The indicated “recovery point” and “danger point” have here the

same significance as previously. The new feature is the occurrence of a

“recession point,” which may be defined as that point where, even at

prosperity levels of production, existing inventory is just barely desired.

Geometrically, it is the unique point at which an orbit curve proceed-

ing up from the Keynes point is tangent to the scarcity line. Noting

that desired production is {γa(t− 1)− b(t− 1) + h}+, and that along

the scarcity line we have a = b/γ, it follows that the recession-point

inventory level is determined by

[{(γ − 1)/γ} − 1]b = −h, i.e., b = [γ/(γ − γ + 1)]h(7.7)

= γh/[2− (c+ 1)γ].

6 JACOB T. SCHWARTZ

Fig. 11. Consumption in a depressive case.

We have marked a significant orbit in Fig. 11: it shows recession,

inventory reduction, recovery, inventory buildup, and new recession.

Examination of the configuration of orbits in Fig. 11 shows that (except

for orbits beginning at dangerously low inventory levels), any initial

motion will, after a first recession, trace out the cyclic motion which

has been marked. In the terminology of orbit-theory, the orbit marked

is a stable limit cycle. This conclusion is true to the extent that the

small inaccuracies in the motions at the production-cutoff line and the

scarcity line occasioned by our use of continuous curves rather than

discrete sequences of points in Fig. 11 may be ignored.

2. A Qualitative Account of the Preceding Results

Our cycle-theory model is, of course, a monstrous oversimplification

of the complex of factors which play a rule in the cycles of the actual

economy. Perhaps our greatest error is to assume instantaneous and

sharp reaction in all sections of the economy to a change in conditions,

which, coupled with the assumption of instantaneous transmission of

7. CONSUMPTION IN THE CYCLE-THEORY MODEL 7

orders and shipments from one point of the economy to another and

with the assumption of a standard production-period in all lines of

industry and of a perfect “lockstep” coordination of industries gives

our model recession an all-encompassing violence foreign to the actual

recession. Nevertheless, our model uncovers a central mechanism whose

significance is generally acknowledged. Let us review the main features

of this mechanism, as they follow from our model and as they would

follow from any similar model.

1. Once a recession has begun, the desired inventory levels fall below

actual inventory levels ; a rapid falling-off of production leads to a cor-

respondingly rapid fall of desired inventory. The lowered production

levels and the continuation of a reasonable level of consumption imply

the slower but progressive decrease of inventories. The recession will

continue until a certain surplus of inventory is “burnt off.” When this

point is reached, recovery will begin.

2. Widespread attempts to sustain inventory then begin, leading to

a pick-up in sales, and to a consequent upward revision of notions as to

what constitutes desirable levels of inventory. In an effort to build up

inventories, production is rapidly advanced, desirable inventories rise

still more, and shortages (symptomatized, say, by difficulty in getting

quick delivery) develop. Production then continues at a high level, in-

ventories gradually being built up. Production is sustained beyond the

requirements of consumption by a general desire to increase inventories.

Prosperity will continue until inventories reach their falling-off point.

3. As inventories increase, the part of inventory size justified by con-

sumption grows relatively smaller, a larger fraction of their size being

justified by the collective efforts to increase inventory. Eventually this

unstable sustaining force collapses, and a new recession begins. (Note

that the “expansive” and the “depressive” cases which we have been

led to distinguish are defined precisely by the stability vs. unstability

of this sustaining force.)

The following quotations from the December, 1960, Survey of Cur-

rent Business will indicate the extent to which the inventory cycle

mechanism of our model is observed empirically:

8 JACOB T. SCHWARTZ

Production is currently being held back to pare inventories. While stocks in

trade channels have undergone little net change in the aggregate since midyear,

manufacturers’ holdings have been reduced moderately. The cutback—amounting

to about $800 million from June through October—has been concentrated so far

in working stocks; this reduction was not significantly different on a relative basis

from the 4 per cent decline in sales over this period. Accumulation of finished goods

at the factory level has continued through 1960, and in view of the current volume

of sales, business seems to regard them as being on the high side at the present

time.

Factory Orders and Output Drop

Incoming orders to manufacturers declined in October following a two-month

spurt due largely to accelerated defense order placement. For both durable and

nondurable goods producers, new orders were once more close to earlier lows. With

incoming new business flowing at a seasonably adjusted pace about equal to sales

in the last several months, backlogs were unchanged at a volume $5 billion, or 10

per cent below a year ago. . . .

The reduction of inventories of finished steel in the hands of consumers and

producers has been under way for the past 6 months or so. For producers, the

inventory liquidation has been moderate with the book value of current stocks of

finished goods inventories—intermediate and finished steel products as well as other

finished materials—just moderately under the high point. Though actual figures

are not available, it appears that the reduction has been on a much larger scale for

the metal fabricating industries.

3. The Keynes Theorem

The cycle, regarded as an entity, is then a mechanism preventing

the unrestricted rise of inventories; thus, on the average, adjusting

production to consumption. We can obtain a more satisfactory view

of the over-all significance of this principle by using the tautologous

relation between inventory and production levels, incorporated in our

model, but relatively independent of its special features:

(7.8) bj(t)− bj(t− 1) = aj(t− 1)−n∑

i=1

ai(t− 1)πij − ej.

7. CONSUMPTION IN THE CYCLE-THEORY MODEL 9

If we sum this equation from t = 1 to t = K and divide by K, we find

that

(7.9) a(K)j −

n∑i=1

a(K)i πij = ej +K−1(bj(K)− bj(0));

here, a(K) denotes the average of the quantity a(t) over the period t = 0

to t = K − 1. In situations like the one we have been examining, in

which there exists an obstacle to the unbounded increase of inventories,

so that inventories will remain bounded (or increase relatively slowly)

over long periods of time, we get the essential force of this last rela-

tionship by letting K → ∞, and writing aj for the long-time average

of aj(t); doing this, we obtain the Keynes Theorem:

(7.10) aj −n∑

i=1

πij ai = ej;

matricially expressed

(7.11) (I − Π′)a = e.

This shows the way in which, abstracting from the dynamic details of

the business cycle, consumption determines production. In a model

in which consumption e was not fixed but variable, and in which we

also had investment f , this relationship, under the same assumption of

bounded inventories, would be modified to

(7.12) (I − Π′)a = e + f .

If we solve this equation for (a) we get

(7.13) a = (I − Π′)−1(e + f),

the first of a number of multiplier relations which we shall discuss.

Let us remark that the prototype of these relationships is the equation

aK = e/ε which in our preceding discussion located one coordinate of

the Keynes point.

10 JACOB T. SCHWARTZ

4. General Reflections on Theories of the Business Cycle.

Say’s Law

The success of a model like the one that has been presented has, if

we are willing to take it seriously, implications for the theory of the

business cycle. In the first place, in our model cycle all industries are

at all times in perfect proportionate balance: thus the cycle is not a

cycle of disproportions, but a cycle of general overproduction. This

may justly make us skeptical of the host of theories which insist that a

recession is only a correction of disproportions that have developed in a

previous boom period. Of course, in empirical fact, the various lines of

production will behave differently and disproportions will continually

develop and be corrected; nevertheless, the basic movement is that of

a general aggregate of production. Our model indicates no reason why

the correction of disproportions should characterize recessions more

than periods of boom.

It will be noted that our discussion of cycle theory has been carried

out entirely in “real” terms, no question of prices entering. That, with

this exclusion of all monetary considerations, any theory is possible,

leads us to infer that the business cycle is not primarily a monetary

phenomenon (though it surely has important monetary aspects). Since

this conclusion is disputed by various “monetary theorists” of the busi-

ness cycle, we may regard it as nontrivial. In order to strengthen our

inference, it is well to have at least a glance at the neglected monetary

side of the phenomenon visible in our model. We may ask: do the

exchanges which we have assumed in our model lead to a progressive

deficit at one or another point, until one or another manufacturer runs

out of means of payment?

We find it easiest to answer this question for the simplest orbit of our

model: the orbit along which production and inventory remain fixed.

Note first that, in models like the one which we have been studying,

the Keynes relation (7.10) implies that only one level of production

may be assigned to such an orbit: the commodity-by-commodity level

of production which exactly supports the assigned level of autonomous

7. CONSUMPTION IN THE CYCLE-THEORY MODEL 11

consumption. We must now see whether the even turnover of material

commodities which characterizes the Keynes point also meets the fiscal

conditions of our model. Let the vector of total consumption have the

components ej. Let the vector describing the consumption of the ith

manufacturer in a single period have the components e(i)j . We have, of

course,∑

i e(i)j = ej. Let aj be the levels of production, so that (7.10)

is satisfied. The difference between sales and expenditures for the ith

manufacturer is

(7.14)n∑

j=1

ajπjipi + piei −n∑

j=1

aiπijpj.

By the Keynes relation (7.10), this may be written as

(7.15) aipi −n∑

j=1

aiπijpj,

which, by (1.9), is equal to

(7.16) ρn∑

j=1

aiφijpj.

Plainly, then, if the ith manufacturer’s autonomous consumption ex-

penditures are restricted by the ordinary budgetary condition

(7.17)n∑

j=1

e(i)j pj = ρ

n∑j=1

aiφijpj,

all the conditions of exchange are met: income balances outgo and

the rate of profit is uniform. If we interpret the quantities ai not

as the invariable Keynes levels of production, but as average levels

of production over the course of the business cycle, we see by this

computation that even for the dynamic motion each manufacturer’s

income and outgo will be in balance over the business cycle.1 Thus no

financial obstacle need arise in our model.

1A closer examination will show that if there are significant disparities in the cap-italisation of different lines of industry, as measured by differences in the ratio(profit/unit sales), there will be a tendency for the highly capitalised industries toaccumulate bank-balances from the less capitalised industries during boom periods,and vice-versa during recessions.

12 JACOB T. SCHWARTZ

The identity of all the expressions (7.14)–(7.16), that is, the equation

cash value of industrial sales = cash value of industrial(7.18)

purchases

may be called Say’s law. The principle properly extrapolated from

this somewhat trivial identity is the principle tentatively put forward

above; that the business cycle is basically a real and not a monetary

phenomenon. This correct inference has often been overextended in

the economic literature to the inference that there can be no cycle of

general overproduction at all. But the most fundamental fact of which

our model informs us is the fact that recessions can exist, indeed, that

in economies approximately describable by our model, which are in

addition depressive rather than expansive, recessions must take place

from time to time. This conclusion, while evident empirically, has in

the past been the subject of theoretical dispute. Let us therefore dwell

on this matter at greater length.

The definition which we have just given to “Say’s law,” and the phe-

nomenon which we observe in the cycle-theory model which we have

just studied, reveals the persuasive fallacy involved in Say’s law as tra-

ditionally interpreted. This traditional if mistaken statement of the

false “law,” which we have repeated just above, amounts to a denial of

the possibility of general overproduction; hence it denies the existence

of the phenomena which we have just examined. We can, in conse-

quence, use these phenomena to discover the defects in the “law.” We

shall quote statements of the false Say’s law from a number of classical

sources. It is worth noting in this connection that the most trenchant

statements of the erroneous law come in the early 1800’s, that is, early

in the history of political economy. The “law,” once stated, seemed

so self-evident as to pass entirely out of the sphere of discussion, and

to become a general, generally unspoken underlying preconception of

economics. We may remark that fundamental scientific errors often

perpetuate themselves in this form—the analogy with Newton’s notion

of absolute space and time and its overthrow by Einstein is striking.

7. CONSUMPTION IN THE CYCLE-THEORY MODEL 13

Discussion of Say’s law began again, though, in somewhat unsatisfac-

tory form, early in the twentieth century; the decisive overthrow comes

with Keynes’s General Theory (1936). Even today, a residual opinion

clings to the error, as witness Mr. Henry Hazlitt:

Such a hulabaloo has been raised about Keynes’s alleged “refutation” of Say’s law

that it seems desirable to pursue the subject further. One writer has distinguished

“the four essential meanings of Say’s law, as developed by Say and, more fully,

by Mill and Ricardo.” It may be profitable to take her formulation as a basis of

discussion. The four meanings as she phrases them are:

(1) Supply creates its own demand; hence, aggregate overproduction or a “gen-

eral” glut is impossible. . . . I shall contend that . . . 1 is correct, properly understood

and interpreted. . . . There is still need and place to assert Say’s law when anybody

is foolish enough to deny it. It is itself, to repeat, essentially a negative rather than

a positive proposition. It states that a general overproduction of all commodities

is not possible. And that is all, basically, that it is intended to assert.

We quote a number of classical statements of the false Say’s law.

(A) J. B. Say, Treatise on Political Economy, (1801).

It is production that creates a demand for products. To say that sales are dull,

owing to the scarcity of money, is to mistake the means for the cause. Sales cannot

be said to be dull because money is scarce, but because other products are so.

. . . A product is no sooner created than it, from that instant, affords a market for

all other products to the full extent of its own value. Thus the mere circumstance

of the creation of one product immediately opens a vent for other products.

(B) James Mill, Commerce Defended, (1808).

The production of commodities creates, and is the universal cause which creates a

market for the commodities produced. . . . A nation’s power of purchasing is exactly

measured by its annual produce. The more you increase the annual produce, the

more by that very act you extend the national market. The demand of a nation is

always equal to the produce of a nation.

(C) John Stuart Mill, Principles of Political Economy, (1848).

What constitutes the means of payment for commodities is simply commodities.

Each person’s means of paying for the production of other people consist of those

14 JACOB T. SCHWARTZ

which he himself possesses. All sellers are inevitably, and by the meaning of the

word, buyers. Could we suddenly double the productive powers of the country, we

should double the supply of commodities in every market, but we should by the

same stroke, double the purchasing power. . . .

The central confusion here is between ability to purchase and desire

to purchase: each author proves, quite correctly, that an increase in

the scale or production must be matched by a general increase in the

ability of manufacturers to purchase other commodities. This is then

fallaciously identified with a corresponding desire to purchase; so that

the whole mechanism of a (possibly deficient) desire to purchase (on the

part of manufacturers) which is basic to our cycle-model is carelessly

ruled out. We find this error in Say’s first sentence (Does “demand”

mean “ability to purchase” or “desire to purchase”? In the first sense

Say is substantially correct, in the second sense substantially incorrect);

in Say’s third sentence (sales in our model are dull because others

lack the desire to purchase, not the ability); in Say’s fourth and fifth

sentences (ability to purchase vs. desire to purchase again). We note

the same error in James Mill’s too facile transition from “power of

purchasing” to “extend the national market”; and in J. S. Mill’s evident

inference from “purchasing power” to desire to purchase, an inference

whose fallacy is evident in our model.

This confusion is of remarkable persistence. Perhaps its final form

is to be found in the work of Keynes himself, in the sections of the

General Theory where he tries to attach a financial mechanism (based

upon the rate of interest and “liquidity preference”) to the basic real-

term phenomenon, so as to complete his analysis. What is basically

involved, however, is not the inability of manufacturers to produce

without borrowing, but the lack of desire on the part of a manufacturer

in possession of all the elements of production, to go ahead with this

production. Later Keynesians have largely discounted these specifically

fiscal ideas of Keynes, either explicitly or in practice, and either on

empirical or on theoretical grounds.

7. CONSUMPTION IN THE CYCLE-THEORY MODEL 15

Is not all this clear evidence of the theoretical utility of proper math-

ematical method?