Download Interest and the Marginal Product of Capital: Böhm

Interest and the Marginal Product of Capital:
Böhm-Bawerk versus Samuelson
Robert P. Murphy∗
Hillsdale College
August 2004
Abstract
It is quite common to associate the equilibrium rate of interest with the marginal
product of capital. However, not all economists have endorsed this practice, and indeed
many have challenged the explanation of interest according to marginal principles.
In this paper I offer a resolution to this dispute by first agreeing with the critics of
mainstream models, but then demonstrating that the "justification" of interest is quite
straightforward, once we have remembered the (elementary) distinction between rents
and interest. To underscore my belief that the historical controversy was ultimately
due, not to marginal principles, but to more modern techniques that are valid only
under special circumstances, I have framed the paper as a pseudo-debate between
(nineteenth century theorist) Eugen von Böhm-Bawerk and Paul Samuelson.
Keywords: Interest theory, capital theory, capital controversy, Böhm-Bawerk, growth
theory
JEL Classification: .
∗
Department of Economics, Hillsdale College, 33 E. College St., Hillsdale, MI 49242.
E-mail:
[email protected]. I would like to thank Paul Dower and Ryuichi Tanaka for helpful comments.
1
1
Introduction
The association of the equilibrium real rate of interest with the marginal product of capital is
a staple of modern mainstream economics. Indeed, when graduate students are asked to find
a typical model’s equilibrium values of the real wage and interest rate, there is apparently
nothing more natural than calculating the derivative of the production function with respect
to labor and capital, respectively.
This seems to make perfect economic sense, because
under competitive conditions, the laborer gets paid the marginal product of his labor, while
the capitalist gets paid the marginal product of his capital.
Despite its popularity, not all economists have always endorsed this practice. Indeed,
during the celebrated capital controversy of the 1950s and ’60s, many prominent critics questioned the validity of the marginal productivity explanation of interest payments.1 These
writers suggested that the orthodox theory proceeded in a vicious circle, because (unlike
labor) there was really no such thing as units of abstract "capital"; the only way to amalgamate heterogeneous units of physical machinery, factories, and goods in process into a
single number, was to sum their present market values, a calculation which itself relied on
an antecedent rate of interest. Consequently, the moral legitimacy of interest income was
cast into doubt, since (apparently) such payments were not really a return to any productive
factor, after all.
The present article seeks to resolve the dispute by acknowledging the partial merits of
both sides. First, the critics of orthodox neoclassical theory are correct; in general it really
isn’t coherent to view interest as a return to the marginal product of capital.
Second,
the neoclassical apologists for capitalism are also correct; the distribution of income in a
competitive market economy proceeds according to marginal principles, and the income
earned by the owners of capital goods is just as legitimate (in this respect) as that earned
by laborers or landlords. These two apparently contradictory positions will be reconciled
1
Some of the standard works on this front are Robinson (1953-1954), Sraffa (1960), and Pasinetti (1966).
2
(and hopefully rendered obvious) after a review of some basic economic truisms.
Although certainly appropriate to historians of economic thought, the present article is
intended for the general economist. Consequently, I have avoided extensive surveys of the
literature, and instead have couched my arguments in terms of a "debate" between Eugen von
Böhm-Bawerk (a nineteenth century giant in capital and interest theory) and Paul Samuelson
(an obvious representative of the modern mainstream approach to capital and interest). As
I hope to convince my reader, the mainstream neoclassicals left themselves open to their
Cambridge critics not because of any fundamental flaw with marginalist theory, but rather
because their fascination with mathematical models led them to forget the insights of earlier
thinkers.
To illustrate this subtle aspect of my thesis, the choices of Böhm-Bawerk and
Samuelson are quite fitting.
2
Böhm-Bawerk’s Critique of the "Naive" Productivity Theory
In the late nineteenth century, Böhm-Bawerk (1959 [1881]) offered a critique of (what he
called) the "naive productivity theory" of interest, a theory which explained (and justified)
the capitalists’ income as a return to the "productivity of capital." Böhm-Bawerk famously
argued that this theory was inadequate, since it only explained why capital goods possessed
market value; by itself, the physical productivity of a capital good could not explain why its
present purchase price was lower than the future revenues it was expected to yield. Referring to a tribal fisherman who initially lives hand-to-mouth before using a more capitalistic
technique (an example familiar to economists of the time), Böhm-Bawerk argued:
Now let us turn to the second interpretation of which the naive productivity
theory is capable. Here the productive power ascribed to capital is, in the first
instance, to be understood as physical productivity only, that is to say, a capacity
3
on the part of capital to furnish assistance which results in the production of
more goods or of better goods than could be obtained without its help. But it
is assumed as self-evident that the increased product, besides replacing the costs
of capital expended, must include a surplus of value. Just how convincing is this
interpretation?
I grant without ado that capital actually possesses the physical productivity ascribed to it, that is to say, that more goods can actually be produced
with its help than without. I will also grant. . . that the greater amount of goods
produced with the help of capital has higher value than the smaller amount of
goods produced without it. But there is not one single feature in the whole set
of circumstances to indicate that this greater amount of goods must be worth
more than the capital consumed in its production. And that is the feature of the
phenomenon of excess value which has to be explained.
To put it in terms of Roscher’s familiar illustration, I readily admit and
understand that with the assistance of a boat and net one catches 30 fish a day,
while without this capital one would have caught only 3. I readily admit and
understand, furthermore, that the 30 fish are of higher value than the 3 were.
But that the 30 fish must be worth more than the pro rata portion of boat and net
which is worn out in catching them is an assumption which the conditions of the
problem do not prepare us for, or even cause to appear tenable, to say nothing
of making it obvious. If we did not know from experience that the value of the
return to capital is regularly greater than the value of the substance of capital
consumed, the naive productivity theory would not furnish a single reason for
regarding such a result as necessary. It might very well be quite otherwise. Why
should not capital goods that yield a great return be highly valued on that very
account and indeed, so highly that their capital value would be equal to the value
of the abundance of goods which they yield? Why, for instance, should not a boat
4
and net which, during the time that they last, help to procure an extra return of
2,700 fish be considered exactly equal in value to those 2,700 fish? But in that
event, in spite of the physical productivity, there would be no excess value. (I,
pp. 93-94, italics original)
Böhm-Bawerk’s fundamental insight was that the rate of interest was not, in the first
instance, a matter of product, but instead a ratio of the value of product relative to the
value of the capital asset yielding it. An engineer can tell us that a certain machine will
yield $1000 in output for each of five years, after which the machine will be useless. But
these technical facts alone do not determine the rate of return to a capitalist who purchases
such a machine; if the initial price is (roughly) $4,329.48, the rate of return will be 5 percent
per year, but if the initial price is $5,000, then the rate of return on the investment will be 0
percent.2 The physical productivity of the machine explains the magnitude of its periodic
rental payments, but it is the relation between the initial capitalized value of the machine
and these future cash flows that determines the internal rate of return.
Of course, no modern economist would deny these truisms.
When a mathematical
economist sets the equilibrium rate of interest equal to the "marginal product of capital" in
a formal model, this is not to commit the fallacy involved in the so-called naive productivity
theory. However, even though it is free from internal contradiction in its formal exposition,
the modern approach certainly invites the very misconceptions that Böhm-Bawerk took pains
to refute. More serious, by drawing general conclusions from these quite limited models,
the mainstream apologist for capitalism leaves him or herself open to critics of a Sraffian
persuasion.
In the following sections, I will take two examples where Samuelson entirely
misses the significance of Böhm-Bawerk’s insight, by assuming away the very problems that
2
For a rate of return of 0 percent, the calculations are straightforward. For a rate of return of 5 percent,
consider that the machine at the beginning of its fifth year has a spot price of $952.38 (since it will yield
$1000 in marginal product at the end of the year). Working backwards, we find that the first and second
year prices are $4329.48 and $3545.95. Consequently, a capitalist who buys a brand new machine must pay
$4329.48. After one year, he will have $1000 in marginal product, plus a machine that can now be sold for
$3545.95, for a net return on his investment of ($1000+$3545.95)−$4329.48
≈ 5%.
$4329.48
5
so worried Böhm-Bawerk (and later the Cambridge critics).
3
Example One: Samuelson Assumes Steady State
In this section, our point of departure will be Samuelson’s discussion of von Thünen, whom
Böhm-Bawerk had criticized for failing to see that interest involved more than simply the
"productivity of capital." As we shall demonstrate, Samuelson’s modern exposition of von
Thünen involves an expression for the rate of interest that is valid only in the steady state,
and thus misses the point of Böhm-Bawerk’s critique.
3.1
Samuelson’s Discussion of von Thünen
In his article celebrating the work of von Thünen, Samuelson writes:
Thünen is indicted and convicted by Böhm as holding a "productivity theory of
interest": in allegedly merely assuming that capital projects exist that do yield
a saving of labor greater than the labor needed to produce the capital goods
themselves, Thünen is accused by Böhm of the logical fallacy of petitio principii—
he is begging the question that needs to be faced and resolved.
However, as
Böhm’s admirer Knut Wicksell had to point out, the scientist has no choice but
to beg every such question in the sense that ultimately a Newton must accept
how it is that apples do fall; and that it is not in the power of an Einstein to
deduce why Mercury must lag in the way that telescopes confirm it does and
that the truth of general relativity theory would entail. (1983, p. 1469, i.o.)
Before proceeding to the technical details, let us pause to note that Samuelson has
apparently misunderstood Böhm-Bawerk’s critique. As a re-reading of the passage quoted
in this paper demonstrates, Böhm-Bawerk did not expect productivity theorists to explain
particular facts of nature (i.e. the net technical productivity of capitalistic processes). What
6
worried Böhm-Bawerk was that too many writers thought that these brute technical facts
alone explained the existence of a positive interest rate. What Böhm-Bawerk sought was a
satisfactory explanation of how such technical facts could give rise to a premium on present
versus future consumption.
Now perhaps von Thünen did give such an explanation, or perhaps he didn’t; this is a
matter best left to history of thought journals. Either way, we must be clear on what BöhmBawerk’s objection was.3 Because Samuelson has apparently misunderstood it, his formal
defense of von Thünen is unsatisfying. In an appendix entitled "Vindicating Thünen’s Logic
on Capital," Samuelson writes:
[NOTE TO EDITOR AND REFEREES: For simplicity during the various revisions
of the present paper, I have simply attached the relevant page 1485 from Samuelson in its
entirety at the end.]
3.2
A General Derivation
We will now derive the general expression4 for the equilibrium interest rate in the discreteperiod model (i.e. the model in Samuelson’s footnote 13). First, consider an individual in
period t who has b−1 units of labor.
He must be indifferent between selling his labor to
the machine industry or to the grain industry, and so the wages in each industry must be
identical. Consequently (because we assume zero profits) the marginal product of his labor
in either industry must be of equal exchange value (whether reckoned in period t or in period
t + 1).
The individual’s b−1 units of labor can be used (in period t) to produce either 1
machine, or b−1 F2 [K(t), LQ (t)] units of grain, in period t + 1. Thus, we have established
that in period t + 1, the (spot) grain-price of 1 machine must be b−1 F2 [K(t), LQ (t)].
By
similar reasoning, we know that in period t + 2, the (spot) grain-price of a machine must be
3
Böhm-Bawerk’s specific remarks on von Thünen (I, pp. 111-116) do not differ importantly (for our
purposes) from his arguments presented at the beginning of this paper.
4
It goes without saying (literally, in the case of Samuelson) that we are concerned only with an interior
solution, i.e. we are characterizing the equilibrium interest rate and real wages if labor is being used (in
every period) in the production of both machines and grain.
7
b−1 F2 [K(t + 1), LQ (t + 1)].
Now consider an investor in period t + 1 who wishes to purchase one machine. From
above, we know that he must spend b−1 F2 [K(t), LQ (t)] units of real consumption on it. He
can then use it (in combination with hired laborers) to yield F1 [K(t + 1), LQ (t + 1)] units
of grain (which will not be available until period t + 2) that are directly attributable to
the productivity of this machine. Because of (physical) depreciation, the investor will also
have 1 − δ units of machinery remaining. (Recall that the grain-price of a new machine in
t + 2 is b−1 F2 [K(t + 1), LQ (t + 1)].) Since we know the market value (in grain) of his initial
investment, and the total market value (in grain) of the resulting assets in the following
period, we can express the investor’s implicit net real rate of interest in period t + 1 by the
following:
i(t + 1) =
=
market value of investment in period t + 2
−1
market value of investment in period t + 1
(1 − δ)b−1 F2 [K(t + 1), LQ (t + 1)] + F1 [K(t + 1), LQ (t + 1)]
− 1.
b−1 F2 [K(t), LQ (t)]
In the special case where the number of machines per worker in the grain industry (i.e.
K(t)
)
LQ (t)
is constant, we can write
i(t + 1) =
=
F1 [K(t + 1), LQ (t + 1)]
−δ
b−1 F2 [K(t), LQ (t)]
F1 [K(t)/LQ (t), 1]
− δ,
F2 [K(t)/LQ (t), 1]b−1
which is of course identical to Samuelson’s expression for (what he labels) z.
Thus,
Samuelson’s expression for the rate of interest is valid only in the steady state. The only
mention Samuelson makes of this assumption is when he remarks that his equation (A10)
8
is "valid both in steady states and in general." It is apparently then left to the reader to
realize that any other equations are valid only in steady states.
Let me emphasize that the issue here is not simply that Samuelson failed to enumerate
all of his assumptions. Remember that Böhm-Bawerk had accused von Thünen of begging
the question by assuming (in his treatment of capital’s productivity and the rate of interest)
what he was trying to prove. Samuelson then feels he has adequately defended von Thünen
by formally modeling steady state growth. But notice that in such a steady state, where
capital per worker in the grain industry is constant, the spot grain-price of machines is also
constant.5 In other words, Samuelson has imposed from the outset that the rate of return
cannot be influenced by fluctuations in the (market) value of the capital goods from period
to period. Thus, Samuelson has merely formalized von Thünen’s argument without really
demonstrating that it should be acquitted of Böhm-Bawerk’s charges.
Samuelson feels he has vindicated von Thünen, because his (Samuelson’s) system of
equations is free of any vicious circularity. But as we have demonstrated above, this is only
true because Samuelson’s system of equations is only valid in the steady state—i.e. when
we have already assumed the conclusion under dispute! Yes, if Samuelson’s equations were
true in general, then Böhm-Bawerk’s criticism would indeed be captious.
But since the
more general expression for the equilibrium interest rate allows for precisely the type of
value fluctuations that worried Böhm-Bawerk, Samuelson’s techniques in no way allow us
to award von Thünen "highest marks for getting all this right."6 Böhm-Bawerk was trying
to explain the phenomenon of interest in general, and not merely in steady states.
The
modern exponents of the marginal productivity explanation of interest should take little
comfort from the fact that their explanation is perfectly sound in the steady state.
That is, constant LK(t)
implies constant b−1 F2 [K(t − 1), LQ (t − 1)], which is the spot grain-price of a
Q (t)
machine in period t.
6
Again, I am not here concerned with whether von Thünen’s model is ultimately sound or not. After
all, someone can "beg the question" and still be right.
5
9
4
Example Two: Samuelson Assumes One Good
In the previous section, we saw that Samuelson dismissed Böhm-Bawerk’s concerns with the
productivity approach by a focus on the steady state, without (apparently) realizing that
this tactic assumed away the problem. In this section, we will see Samuelson follow a similar
strategy in his criticism of Schumpeter’s famous zero-interest doctrine. Rather than merely
imposing a steady state, Samuelson here will postulate an economy consisting of only one
good.
This too will (obviously) eliminate the possibility of changes in the market value
of capital goods, and hence the one-good model is particularly irrelevant in arguments over
capital valuation.7
4.1
Samuelson’s Discussion of Schumpeter
One of Joseph Schumpeter’s most (in)famous theories was that, absent innovation, the rate
of interest in an economy in long-run equilibrium would necessarily be zero. In a train of
thought reminscent of Böhm-Bawerk’s critique8 of the naive productivity theory, Schumpeter
argued that (absent innovation) there would be no net return in any line of production,
because however great the productivity of capitalistic processes, the value of the final product
would be imputed back to the land and labor factors that went into the capital goods.
(Schumpeter 1912) Against this reasoning Samuelson replies:
...Schumpeter does not reconcile his doctrine that all value is decomposable
into land rent and labour wages with a possible technological case: 100 rice ripen
into 110 rice in one year’s time without need for any labour or scarce land...
7
Cohen (1989) reaches a similar conclusion.
The reader who refers to Samuelson’s article may be puzzled, since Samuelson often claims to be taking
Böhm-Bawerk’s side (versus Schumpeter’s). However, in the context of the present paper, Samuelson’s
reliance on the single-good model to refute Schumpeter shows that Samuelson does not fully appreciate the
logic of Böhm-Bawerk’s critique of the naive productivity theory.
8
10
Schumpeter’s answer might start out in terms of these words (which I put into
his mouth): ’With more rice next year, the price of each grain falls. Thus, 110
then sell for the total of marks 100 now will sell for. So 100 marks today still get
you 100 marks tomorrow. The rate of interest is zero sans innovations. Q.E.D.’
Around p. 170, Schumpeter (1912) repeatedly makes the point that (when the
interest rate is truly zero) the greater magnitude of the forest is already imputed
back in value to the saplings: so these foreseen changes in time only conserve the
already calculated value of the process. Without labour and land, zero Kuznet
product is being produced and it is correctly decomposable into zero real rent
and zero real wages.
But this is pure self-deception. Real rice is being produced net. Kuznets
can measure it. You can eat 10 rice every year and still not impair your circularflow income. With land redundant and labour not needed, Kuznets measures
national income of zero in terms of primary-factors’ income.
To this he adds
permanent real interest income of 10 rice per year. No hocus-pocus of backward
imputation—of forest to sapling, or rice grain to rice grain—evades the naive fact
of productive interest.
Empirically, Schumpeter may deny...that there always exists positive
technical productivity of capital at the margin.
But logically he must throw
in the towel: when 100 rice as input yields 110 rice output at the end of the year,
no steady-state (real!, ’own’ !) rate of interest can obtain other than 10% per
year. A zero equilibrium rate becomes a contradiction (and settling down of the
system to a steady state may no longer occur). (Samuelson 1981, pp. 22-23)
As with Samuelson’s discussion of von Thünen, he has here sought to establish the
"naive fact of productive interest" by analyzing the steady state,9 and in particular an
9
Also as with the discussion of von Thünen, I am not taking sides with Samuelson or Schumpeter in
regard to the particular dispute. Schumpeter himself was discussing a long-run steady state, and hence
Samuelson’s analysis may be quite suitable for his purpose.
11
economy with only one good (which serves as both capital and consumption).
Although
internally consistent, the analysis of such a model can be very misleading when it comes to
the possible relation between interest and the productivity of capital goods, because all of
the complexities are assumed away. For example, the mere fact that one consumption good
possesses an own-rate of interest of 10 percent does not, in general, imply a positive rate
of return to capitalists who invest in such a good.10
I will demonstrate this claim in the
following subsection, and then I will offer a more general model to illustrate the irrelevance
of one-good models in this debate.
4.2
Two-Good Counterexample
The simplest counterexample to Samuelson’s contention is a case where there are not one
but two goods, each of which can be either consumed in the present period, or used for
(re)production in the next period.
(There are no other inputs and no other production
technologies.) Following Samuelson, assume that the first good is rice, and that it contains
a net technical productivity of ten percent: 100 units of rice this period will physically
transform into 110 units of (equally useful) units of rice next year. Now, following Irving
Fisher (1977 [1930] pp. 191-192), imagine that the second consumption good consists of a
stock of figs, and that 100 units of uneaten figs this period will physically transform (due to
rotting) into 50 units of figs next year. Thus the own-rate of interest on rice is 10 percent,
while the own-rate of interest on figs is negative 50 percent. In such an economy, what will
be the real rate of return to capitalists?
Before answering this question, we must first specify the relevant "basket" of commodities, the total price of which will serve as our consumer price index. Let us assume that
the basket consists of 1 unit of rice and 1 unit of figs. Further suppose that the representaXT
tive consumer has utility function Uτ =
ln Rt + ln Ft , where Rt and Ft represent the
t=τ
amounts of rice and figs consumed in period t. Finally, suppose that the initial stocks of
10
For a verbal critique of Samuelson’s argument against Schumpeter, see Kirzner (1996) p. 142.
12
rice and figs are such that the representative consumer optimally chooses R0 = F0 in the
initial period 0.
In this (extremely particular) example, 100 units of rice in period 0 trade for 100 units
of figs in period 0. Because of their net technical productivities, 100 units of rice in period
0 trade for 110 units of rice in period 1, while 100 units of figs in period 0 trade for 50 units
of figs in period 1. In period 1, the relative price of rice versus figs has changed, due to the
growth in rice consumption and decline in fig consumption: 100 units of rice in period 1 now
trade for approximately 45.45 units of figs. Therefore, a capitalist who had the purchasing
power of 100 baskets (i.e. 100 rice and 100 figs) in period 0 and invested in either good (or
any combination) would have only 68.75 baskets of purchasing power in period 1, for a real
rate of interest of negative 31.25 percent.11
As this extreme example shows, the mere presence of a consumption good with a positive
technical net productivity does not guarantee a positive interest rate, because relative prices
of consumption goods can change over time.12
In the next subsection, I generalize the
argument and show more formally how the assumption of a single-good economy completely
masks the conceptual problems with the marginal productivity explanation of interest.
11
It may clarify to use some hypothetical dollar prices. Suppose that the spot price of both rice and figs
is $1 per unit in period 0, while the period 1 spot price of rice is $0.4545... and the spot price of figs remains
at $1. A capitalist in period 0 can then invest $200 in purchasing 100 of the commodity baskets (i.e. 100
units of rice and 100 units of figs). If the capitalist refrains from consuming any of his purchases, in the next
year his investment will become 110 units of rice and 50 figs. At the stipulated spot prices, the capitalist
can sell these products for a total of $100. The nominal money rate of return is hence negative 50 percent.
However, this nominal figure understates his return, because prices in general have fallen; it is now cheaper
to buy a basket of commodities. Indeed, because the spot price of rice has fallen to $0.4545..., the $100 in
period 1 can purchase 68.75 units of rice and 68.75 units of figs, i.e. 68.75 baskets.
12
In fairness to Samuelson, he explicitly acknowledges (in a different paper) the importance of constant spot
prices for his critique of Schumpeter: "[I]t is definitely in contradiction to the usual notion of equilibrium to
state that the price of corn is constant over time, and yet one hundred units of corn are today worth as much
as one hundred and ten bushels are worth tomorrow. But this is what is implied in the Schumpeterian
assertion that there will be reflected in today’s corn the full value of tomorrow’s output stemming from
it....Equilibrium coexistent with a zero money rate of interest would be possible only if prices violated
the constancy postulated of the stationary state. If capital in general had a continuing, real, net, own
productivity, the money rate of interest could be zero only if prices were falling at a percentage rate equal
to that of the productivity." (Samuelson 1943, p. 65)
13
4.3
The More General Case: Distinct Capital and Consumption
Good
In this subsection, I hope to render quite explicit the drawbacks to the single-good model
for resolving any disputes over capital productivity and interest. In both Böhm-Bawerk’s
critique of the "naive" productivity theory, and in the more general attacks on the marginalist
approach coming from the Cambridge critics, the central issue was the valuation of capital
goods. Yet this is precisely the feature that is obliterated by working with a one-good model.
As we shall see, there is a qualitative leap in moving from an economy with two goods (one
capital and one consumption) to an economy with only one good.13 The modern practice
of focusing either on the steady state or explicitly on a one-good model (as Samuelson has
done in his own work in this area) blinds the modern neoclassical to the subtleties of the
problem, and severely weakens his or her ideological defense of the capitalists’ income.
4.3.1
The Model
Suppose there are an infinite number of time periods, from t = 0 to t = ∞. At any time t
there exists a fixed labor supply L and a variable stock of machinery Kt . In any period, labor
and machinery can be combined to yield, in the next period, units of (extremely perishable)
consumption or units of new machinery. There is no depreciation of machinery in either line
of production. Finally, I explicitly model a bank,14 which provides a loan of consumption
goods to the machine owners and laborers in period t = 0 (since the machines and labor will
not yield consumption goods until t = 1).
The bankers earn a perpetual flow of interest
payments on this principal. We thus have the following relations:
13
Note that the issue is not one versus a multiplicity of various consumption goods, but rather the distinctness of the capital and consumption good.
14
Some third party is necessary in order to complete the model, since (at positive interest rates) the
machine owners and laborers earn their discounted product. (This is what leads many writers to deplore
interest as "exploitation.") In this section, I will refer to "bankers" and "machine owners," since the more
common term "capitalists" is often used to mean both groups.
14
Ct+1 = f (Kc,t , Lc,t ) = Cl,t+1 + Cm,t+1 + Cb,t+1 ,
Kt+1 = Kt + g(Km,t , Lm,t ),
Kt = Kc,t + Km,t , and
Lt ≡ L = Lc,t + Lm,t ,
where Cl,t+1 denotes the consumption of laborers in period t+1, Cm,t+1 the consumption
of machine owners, and Cb,t+1 the consumption of the bankers. Lc,t and Kc,t are the amounts
of labor and machinery devoted in period t to production of the consumption good (which
of course will not be available until the next period), while Lm,t and Km,t are the amounts
of labor and machinery devoted in period t to the production of machinery.
Assuming competitive markets and that f (·, ·) and g(·, ·) are differentiable and yield an
interior solution, we know that in equilibrium the following relations must hold:
)g2 (Km,t , Lm,t )
(πt+1
f2 (Kc,t , Lc,t )
wt =
= t+1
and
1 + it
1 + it
rt =
(π t+1 )g1 (Km,t , Lm,t )
f1 (Kc,t , Lc,t )
= t+1
,
1 + it
1 + it
(1)
(2)
where f2 (·, ·) denotes the partial derivative of f with respect to its second argument and
so on. The spot price of consumption has been normalized to one, while π ab indicates the
price of a machine in period a, when the machine will be delivered in period b. (Thus π t+1
t+1
is simply the spot price of a machine in period t + 1.) Take careful note that rt is not the
rate of interest but rather the rental (or hire) price of a machine; a machine owner rents out
one unit of machinery in period t to immediately receive rt , and then he receives his machine
15
back (in perfect condition, since we assume no physical depreciation) at the start of the next
period. The net real rate of interest is indicated by it , and is defined as the relative premium
on present consumption versus future consumption.15
Relations (1) and (2) require that
labor and machines earn their (discounted) marginal products, and that their returns must
be equal whether they are used to produce consumption or machinery.
Finally, we have the following relations between the prices of capital goods (machinery)
in different periods:
πtt = rt + πtt+1 = rt +
π t+1
t+1
.
1 + it
(3)
The equations in (3) require that the period t spot price of a machine equals its immediate yield, rt , plus the current price for a machine available in period t + 1, i.e. π tt+1 . (In
equilibrium π tt+1 =
πt+1
t+1
1+it
because of arbitrage.)
Using the first equation in (2), we can substitute for rt in (3) to obtain
π tt =
π t+1
f1 (Kc,t , Lc,t )
+ t+1 .
1 + it
1 + it
Rearranging yields
it =
f1 (Kc,t , Lc,t ) − (π tt − π t+1
t+1 )
.
t
πt
(4)
Equation (4) is completely intuitive. It expresses the fact that the net (real) rate of
interest is equal to the net (real) yield of capital divided by the market price of the capital
good.
4.3.2
Discussion
As equation (4) demonstrates, the mere fact that capital goods are physically productive by
itself does not guarantee a positive rate of interest, because the periodic rental payments
15
it ≡
That is, if we define pxy as the period x price of a unit of consumption good delivered in period y, then
ptt
ptt+1
− 1.
16
(for the "services of capital") may be partially or (in principle) even completely offset by a
fall in the market value of the machinery.16
But this possibility is completely masked by
assuming an economy with only one good: In this case, a unit of capital always trades onefor-one against units of consumption, because (obviously) they are units of the same thing.
Thus π aa = 1 trivially in any one-good model, and equation (4) reduces to the standard
it = f1 (Kc,t , Lc,t ). It is for this reason that neoclassicals wishing to consider the objections
of the Cambridge critics (as well as the writings of Böhm-Bawerk and Schumpeter) should
not adopt a one-good model as the baseline case, as it involves more than a mere harmless
simplification.
The above model sheds some light even in the case of a steady state, where the price of
the capital good is constant over time. Suppose that π aa = 2; that is, suppose that the spot
price of a machine is always two units of consumption. Further suppose that in some period
t, f1 (Kc,t , Lc,t ) = 0.1; that is, suppose that the marginal unit of machinery will yield an extra
0.1 units of consumption good in the following period.
Since we are in the steady state,
one might carelessly conclude that the real rate of interest in this economy is 10 percent
at period t, because a given unit of machinery will yield itself plus 0.1 units of additional
output in the following period. But as equation (4) tells us, this is incorrect: The correct
rate of interest in period t is only
5
0.1−(2−2)
2
= 5 percent.
The Distribution of Income
The discussion thus far in this paper has pointed out shortcomings in the standard mainstream view of interest, using the work of Paul Samuelson for concrete examples. However,
my analysis does not lead to the typical conclusions (e.g. Garegnani 1970) of those who
side with the critics of the Cambridge debate.
Nothing I have written above in any way
bolsters the claim that the existence of interest or the capitalist class somehow "exploits"
16
Thus the effect is not one of mere physical depreciation, as Böhm-Bawerk’s critique of the naive productivity theory may appear to suggest. Even without physical depreciation, the perfectly intact capital stock
can still become less valuable in terms of consumption goods over time.
17
laborers. To demonstrate this, I do not need to resort to any more models. Ironically, it is
precisely the analysis of mathematical models that led to trouble in the first place! In order
to understand (and hence properly justify) the distribution of income in a market economy,
we would do well to reflect on the insights of two pioneers of neoclassical interest theory,
namely Eugen von Böhm-Bawerk and Irving Fisher.
In a competitive market, every productive factor receives its marginal product, and this
exhausts all output; there is no separate type of income called "interest" that is economically
distinct from payments to the owners of labor, natural resources, and capital goods. Indeed,
to view interest as a type of payment reserved exclusively for "the capitalists" is to revert
to classical fallacies. As Fisher explains:
The hire of human beings is wages; the hire of land is rent. What, then, is the
hire of (other) capital—houses, pianos, typewriters, and so forth? Is it interest?
Certainly not. Their hire is obviously house rent, piano rent, typewriter rent,
and so forth...Rent is the ratio of the payment to the physical object—land, houses,
pianos, typewriters, and so forth—so many dollars per piano, per acre, per room.
Interest, on the other hand, is the ratio of payment to the money value of these
things—so many dollars per hundred dollars (or per cent).
It is, in each case,
the ratio of the net rent to the capitalized value of that rent. It applies to all
the categories—to land quite as truly as to houses, pianos, typewriters.
The
income from land is thus both rent and interest just as truly as the income from
a typewriter or a bond. We can and do capitalize land rent just as truly as we
do house rent. For example, land worth "20 years purchase" yields 5 per cent
interest. (Fisher 1977 [1930], pp. 32-33)
As these elementary observations from Fisher remind us,17 there is really nothing unique
about capital goods and interest; the owner of an automobile factory receives the same rate
17
Fisher himself acnowledges his debt to Frank Fetter, the author of the "capitalization theory" of interest.
See (Fetter 1977).
18
of return (adjusted for risk) on his investment as the owner of farmland receives on hers. The
dividends of the shareholder in a corporation are interest, but they also reflect the marginal
productivity of the various assets of the corporation. And if we consider that workers can
choose to invest in their own education, in a suitably defined model the wages a worker
receives would (a) equal his marginal product and (b) yield the market rate of return on his
"human capital." Viewed from this perspective, there is obviously no question of interest
harming workers, because wages too are ultimately interest dividends.
Surely the reader has abandoned me at this point; it is inconceivable that the resolution
of the controversy could be so simple. After all, isn’t there an obvious sense in which workers
are hurt by interest? Indeed, equation (1) from above tells us that wt =
f2 (Kc,t ,Lc,t )
.
1+it
Here
we see that the worker’s real wage is inversely related to the (real) gross rate of interest.
Nonetheless, I still maintain that this in no sense reflects exploitation. Remember that
the worker is not yielding any product in period t; his or her labor will only yield f2 (Kc,t , Lc,t )
units of additional consumption good in period t + 1. But the worker desires to be paid
immediately, without waiting for his or her contribution to "ripen." It is thus the banker
(in the terminology of my model above) who pays the worker upfront in period t.
This
payment must of course be adjusted to allow for the fact that payment is taking place in a
different period from when the work will actually yield tangible results, and (by definition)
the market exchange rate of present for future goods is given by the real rate of interest. In
this sense, the banker is no more exploiting the worker by paying the discounted marginal
product, than he would be exploiting someone who desires to be paid in US dollars rather
than yen, and has the appropriate exchange rate applied.
The bankers earn interest, therefore, not because of capital’s productivity, but because
they sell present goods in exchange for future goods.18 Recall that the machine owners too
are only paid the discounted marginal product of their machines. What gives the bankers
18
Of course, to understand why the market places a premium on present consumption, the fact of technical
productivity is quite significant—but this is of course exactly in agreement with Böhm-Bawerk’s own positive
theory of capital and interest. (II, pp. 259-289)
19
a perpetual flow of real consumption is thus not capital goods per se, but the fact that the
bankers had a stockpile of consumption goods in the initial period 0 that they were willing to
advance to the workers and machine owners.19 If there is still any question of exploitation,
note that the worker can become a "banker" himself by lending his wages at the prevailing
rate of interest.20
Indeed most workers in modern economies take advantage of positive
interest rates when saving for retirement.
6
Conclusion
The neoclassical habit of equating the real rate of interest with the marginal product of
capital deserves skepticism.
In the first place, the practice is only justifiable in special
circumstances, in which the market value of machinery remains constant.
(In contrast,
the wage rate equals the marginal product of labor in any competitive equilibrium.)
By
focusing on such special cases, the modern economist (not to mention graduate student) is
likely to draw false generalizations regarding what interest is. By analogy, consider this: In
a suitably designed model—perhaps involving reproducible robots, or even human slaves—one
could find that the equilibrium interest rate always equals the "marginal product of labor."
More generally, the modern focus on capital productivity tends to resurrect classical
fallacies—such as the confusion between the rental price of a capital good versus the rate of
return to its owner—that were exploded in the 19th century. (The typical one-good model
of modern macro and growth theory is particularly conducive to such ambiguities.) Such
19
In the real world, what normally happens is that the "bankers" would sell their consumption goods
in period 0 in exchange for the goods in process (i.e. circulating capital) that were immediately produced
by the combination of labor and machines. In the following period, this circulating capital would then
ripen into more total consumption goods than were paid to the workers and machine owners in the previous
period. However, we can equivalently view the situation as a pure consumption loan: As before, the bankers
advance some of their stockpiles of consumption goods in period 0, but now the workers and machine owners
retain full ownership of their circulating capital. They sell the full product in period 1, but before they can
consume they must pay an interest installment on their loan. Interest payments have nothing inherently to
do with capitalistic production, save for the effect on the premium placed on present consumption goods.
20
This argument was one of many Böhm-Bawerk advanced against the "exploitation theory" of interest
(I, pp. 250-280).
20
confusion is deplorable not only for its own sake, but also because it lends ammunition to
the critics of the market economy.
In this paper I have attempted to illustrate the weakness of the standard approach by
citing two examples where Paul Samuelson failed to acknowledge the severe limitations on his
analysis. Were my examples gleaned from a discussion of, say, international trade theory,
they could be dismissed as harmless.
Yet they were occasions on which Samuelson was
arguing over capital and interest theory itself, and consequently his evasions of the central
issue are alarming indeed.
In closing, I urge mainstream economists to carefully reconsider their description of
interest as "the price of capital services" and phrases such as "factor-price frontier." It is
far more accurate to say that interest is, first and foremost, a reflection of the premium
on present versus future consumption.
(After all, there can be interest even in a pure
exchange economy.) Simple one-good models can still be profitably used for certain purposes,
but when it comes to capital and interest theory, only a model with distinct capital and
consumption goods is acceptable.
References
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[3] Fetter, Frank A. (1977) Capital, Interest, and Rent: Essays in the Theory of Distribution, ed. with Introduction by Murray N. Rothbard, Kansas City: Sheed, Andrews,
and McMeel.
[4] Fisher, Irving. (1977 [1930]) The Theory of Interest, Philadelphia: Porcupine Press.
21
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22