lecture 7 consumption in the cycle-theory model. … 07.pdffrom (7.3a{b) that (7.2a{b) are...
TRANSCRIPT
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?