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 marginalproduct of capital. However, not all economists have endorsed this practice, and indeedmany 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 ofmainstream models, but then demonstrating that the "justification" of interest is quitestraightforward, once we have remembered the (elementary) distinction between rentsand interest. To underscore my belief that the historical controversy was ultimatelydue, not to marginal principles, but to more modern techniques that are valid onlyunder 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, growththeoryJEL Classification: .

∗Department of Economics, Hillsdale College, 33 E. College St., Hillsdale, MI 49242. E-mail:Robert.Murphy@hillsdale.edu. I would like to thank Paul Dower and Ryuichi Tanaka for helpful com-ments.

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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 ques-

tioned 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 amal-

gamate 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

1Some of the standard works on this front are Robinson (1953-1954), Sraffa (1960), and Pasinetti (1966).

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(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" Productiv-ity 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. Refer-

ring 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

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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 produc-

tivity 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

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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

2For 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 secondyear 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$4329.48 ≈ 5%.

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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

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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 Böhm-

Bawerk’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 discrete-

period 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−1F2[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−1F2[K(t), LQ(t)]. By

similar reasoning, we know that in period t+2, the (spot) grain-price of a machine must be

3Böhm-Bawerk’s specific remarks on von Thünen (I, pp. 111-116) do not differ importantly (for ourpurposes) from his arguments presented at the beginning of this paper.

4It goes without saying (literally, in the case of Samuelson) that we are concerned only with an interiorsolution, i.e. we are characterizing the equilibrium interest rate and real wages if labor is being used (inevery period) in the production of both machines and grain.

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b−1F2[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−1F2[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−1F2[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

market value of investment in period t+ 1− 1

=(1− δ)b−1F2[K(t+ 1), LQ(t+ 1)] + F1[K(t+ 1), LQ(t+ 1)]

b−1F2[K(t), LQ(t)]− 1.

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)]

b−1F2[K(t), LQ(t)]− δ

=F1[K(t)/LQ(t), 1]

F2[K(t)/LQ(t), 1]b−1− δ,

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)

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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.

5That is, constant K(t)LQ(t)

implies constant b−1F2[K(t − 1), LQ(t − 1)], which is the spot grain-price of amachine in period t.

6Again, I am not here concerned with whether von Thünen’s model is ultimately sound or not. Afterall, someone can "beg the question" and still be right.

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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...

7Cohen (1989) reaches a similar conclusion.8The 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’sreliance on the single-good model to refute Schumpeter shows that Samuelson does not fully appreciate thelogic of Böhm-Bawerk’s critique of the naive productivity theory.

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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 circular-

flow 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

9Also as with the discussion of von Thünen, I am not taking sides with Samuelson or Schumpeter inregard to the particular dispute. Schumpeter himself was discussing a long-run steady state, and henceSamuelson’s analysis may be quite suitable for his purpose.

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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 commodi-

ties, 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 representa-

tive consumer has utility function Uτ =XT

t=τlnRt + lnFt, where Rt and Ft represent the

amounts of rice and figs consumed in period t. Finally, suppose that the initial stocks of

10For 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.

11It may clarify to use some hypothetical dollar prices. Suppose that the spot price of both rice and figsis $1 per unit in period 0, while the period 1 spot price of rice is $0.4545... and the spot price of figs remainsat $1. A capitalist in period 0 can then invest $200 in purchasing 100 of the commodity baskets (i.e. 100units of rice and 100 units of figs). If the capitalist refrains from consuming any of his purchases, in the nextyear his investment will become 110 units of rice and 50 figs. At the stipulated spot prices, the capitalistcan 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 cheaperto buy a basket of commodities. Indeed, because the spot price of rice has fallen to $0.4545..., the $100 inperiod 1 can purchase 68.75 units of rice and 68.75 units of figs, i.e. 68.75 baskets.12In 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 tostate that the price of corn is constant over time, and yet one hundred units of corn are today worth as muchas one hundred and ten bushels are worth tomorrow. But this is what is implied in the Schumpeterianassertion that there will be reflected in today’s corn the full value of tomorrow’s output stemming fromit....Equilibrium coexistent with a zero money rate of interest would be possible only if prices violatedthe constancy postulated of the stationary state. If capital in general had a continuing, real, net, ownproductivity, the money rate of interest could be zero only if prices were falling at a percentage rate equalto that of the productivity." (Samuelson 1943, p. 65)

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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 tthere exists a fixed labor supply L and a variable stock of machineryKt. 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:

13Note that the issue is not one versus a multiplicity of various consumption goods, but rather the dis-tinctness of the capital and consumption good.14Some 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 deploreinterest as "exploitation.") In this section, I will refer to "bankers" and "machine owners," since the morecommon term "capitalists" is often used to mean both groups.

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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 andKc,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 aninterior solution, we know that in equilibrium the following relations must hold:

wt =f2(Kc,t, Lc,t)

1 + it=(πt+1t+1)g2(Km,t, Lm,t)

1 + itand (1)

rt =f1(Kc,t, Lc,t)

1 + it=(πt+1t+1)g1(Km,t, Lm,t)

1 + it, (2)

where f2(·, ·) denotes the partial derivative of f with respect to its second argument andso on. The spot price of consumption has been normalized to one, while πab indicates the

price of a machine in period a, when the machine will be delivered in period b. (Thus πt+1t+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 +πt+1t+1

1 + it. (3)

The equations in (3) require that the period t spot price of a machine equals its imme-

diate yield, rt, plus the current price for a machine available in period t+ 1, i.e. πtt+1. (In

equilibrium πtt+1 =πt+1t+1

1+itbecause of arbitrage.)

Using the first equation in (2), we can substitute for rt in (3) to obtain

πtt =f1(Kc,t, Lc,t)

1 + it+

πt+1t+1

1 + it.

Rearranging yields

it =f1(Kc,t, Lc,t)− (πtt − πt+1t+1)

πtt. (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

15That is, if we define pxy as the period x price of a unit of consumption good delivered in period y, then

it ≡ pttptt+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 one-

for-one against units of consumption, because (obviously) they are units of the same thing.

Thus πaa = 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 πaa = 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 0.1−(2−2)2

= 5 percent.

5 The Distribution of Income

The discussion thus far in this paper has pointed out shortcomings in the standard main-

stream 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"16Thus the effect is not one of mere physical depreciation, as Böhm-Bawerk’s critique of the naive produc-

tivity theory may appear to suggest. Even without physical depreciation, the perfectly intact capital stockcan 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

17Fisher 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

18Of course, to understand why the market places a premium on present consumption, the fact of technicalproductivity is quite significant—but this is of course exactly in agreement with Böhm-Bawerk’s own positivetheory 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

19In the real world, what normally happens is that the "bankers" would sell their consumption goodsin period 0 in exchange for the goods in process (i.e. circulating capital) that were immediately producedby the combination of labor and machines. In the following period, this circulating capital would thenripen into more total consumption goods than were paid to the workers and machine owners in the previousperiod. However, we can equivalently view the situation as a pure consumption loan: As before, the bankersadvance some of their stockpiles of consumption goods in period 0, but now the workers and machine ownersretain full ownership of their circulating capital. They sell the full product in period 1, but before they canconsume they must pay an interest installment on their loan. Interest payments have nothing inherently todo with capitalistic production, save for the effect on the premium placed on present consumption goods.20This 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.

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and McMeel.

[4] Fisher, Irving. (1977 [1930]) The Theory of Interest, Philadelphia: Porcupine Press.

21

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