Interest Rate Unequal To Marginal Product Of Capital (Part 3 Of 4)

3.0 A Simple Two-Good Counterexample

The question investigated here is whether Equation 9 is an implication of neoclassical microeconomics. I claim Equation 9 is not an implication of neoclassical microeconomics.

It is sufficient to demonstrate this negative conclusion by describing an example compatible with neoclassical microeconomics, but in which Equation 9 does not hold. The existence of such a counterexample demonstrates the use in macroeconomics of models in which the marginal product of capital and the interest rate are equal cannot claim full generality. It is up to the users of such models to state their assumptions and justify their use of these special cases.

How is the counter-example constructed? Assume we observe that in our economy only two goods are produced, steel and wheat, each measured in their own physical units, tons and bushels, respectively. We also observe the physical quantity flows in each industry, which I am going to write in a somewhat cryptic manner. Define d(0) as in Equation 12:

(12)

Suppose the physical quantity flows, on a per worker basis, are as Table 1. All quantities in the table are positive and:

(13)

Notice that the sum of person-years across industries is unity, as promised. Also notice that the sum of the inputs of steel is equal to the steel produced by the steel industry. As a further clarification, we observe that these inputs are purchased at the beginning of the year, and the outputs become available at the end of the year. Furthermore, the steel input is totally used up in these production processes. The output of the steel industry just replaces the steel used up in the economy. The net output consists solely of wheat, which we observe to be a consumption good.

Table 1: Quantity Flows Per Worker

INPUTS HIRED AT START OF YEAR	STEEL INDUSTRY	WHEAT INDUSTRY
Labor (Person-Years)
Steel (Tons Steel)
OUTPUTS

Assume constant returns to scale. This means we can express the observed quantity flows as in Table 2. This explains the puzzling notation in Table 1. The parameters reflect unit (gross) outputs in both industries.

Table 2: Quantity Flows Per Unit Output

INPUTS HIRED AT START OF YEAR	STEEL INDUSTRY	WHEAT INDUSTRY
Labor	Person-Years	Person-Years
Steel	Tons Steel	Tons Steel
OUTPUTS	1 Ton Steel	1 Bushel Wheat

Notice nothing has been assumed about the available technology other than constant returns to scale and that these proportions are possible. Based on our observations of the quantity flows actually used in this little model economy, we can draw no conclusions about what outputs will be produced when the inputs of either industry are in different proportions. It might even be the case that wheat can be used as a capital good for some other technology, or that copper or some other capital good might be used at a different set of prices.

We have observed that tons steel per worker are used in the economy, but this is not the value of capital per worker, k, used in Equation 5. The units are different. If aggregate output per head, y, is measured in units of bushels wheat per capita, capital per head, k, must also be measured in bushels wheat per capita in the aggregate production function framework. We have to figure out how many bushels of wheat this quantity of steel represents. But that's what prices are for.

Suppose we observe that prices are unchanged over the year in which we are making our observations, and that the wage w is paid at the end of the year. Suppose that we also observe that competition has brought about the same rate of interest in both industries. Let wheat be numeraire. Then the following price equations obtain:

(14)

(15)

I have not specified enough equations to fully define the price system. Thus, we can solve for two of the price variables in terms of the third, say the rate of interest. Define d(r) as in Equation 16:

(16)

The price of a ton of steel as a function of the interest rate is then given by Equation 17:

(17)

Hence, the quantity of steel, when measured in bushels wheat, is:

(18)

This value quantity of steel varies with the interest rate.

The wage can also be found as a function of the rate of interest:

(19)

Equation 19 expresses the factor price curve [6] associated with the observed technique. A different technique, with its own factor price curve, may be preferred at a different rate of interest. All these curves can be graphed on the same diagram with the wage as the ordinate and the interest rate as the abscissa. The cost-minimizing technique(s) at any interest rate will correspond to the technique(s) with the highest wage at that interest rate [7]. The factor price frontier is thus formed from the outer-envelope curve of the factor price curves corresponding to each individual technique. Points on this frontier that lie on two or more curves for individual techniques are known as switch points. The optimal cost-minimizing technique is unique at interest rates for non-switching points.

Assume the observed technique is a non-switching point. In the case of a discrete technology, the factor price curve for the selected technique is tangent to the factor price frontier at this rate of interest. The desired derivative, dw/dr, in Equation 9 is the slope of the factor price frontier at the observed rate of interest. From this tangency relationship one can see that the slope can be found by differentiating Equation 19, the factor price curve for the observed technique (Figure 2).

Figure 2: Factor-Price Curve For Cost-Minimizing Technique At Non-Switching Point

It seems useful to provide an aside on a correct understanding of marginal productivity relationships before proceeding with this differentiation. The analysis of the choice of technique in long run equilibrium through the construction of the factor price frontier is completely general in circulating capital models. It applies to a Leontief technique, a choice among several Leontief techniques, or continuously differentiable micro-economic production functions in which all inputs are specified in physical units (e.g. tons, bushels, person-years). In the last case, all points along the frontier are non-switching points, although the chosen technique varies continuously with the interest rate [8]. Price Wicksell effects, as explained in this essay, result in the difficulty in defining a unit of "capital" in any case. Marginal productivity is another method of analyzing the choice of technique in the continuously differentiable case. When correctly applied, marginal productivity does not determine the distribution of income, and no equation analogous to Equation 7 arises [9]. If my example is supplemented by the needed assumptions, one can show that the price of a ton of steel is equal to the value of the marginal product of (ton) steel in both sectors. Since time discounting is used in this relationship, the interest rate appears in the mathematical statement of these equalities. But these equalities clearly differ from Equation 7, for capital is measured in the same units as output in Equation 7. The two good model has other properties that differ from the simple one good model.

Now we can return to our problem of examining Equation 9 for this simple two-good model. From Equation 19, the slope of the factor price frontier is given by Equation 20:

(20)

We can now compare the value of capital with the additive inverse of the tangent to the factor price frontier. The right hand sides of Equations 18 and 20 do not look like additive inverses of one another. As a matter of fact, assuming a positive interest rate, the interest rate is equal to the marginal product of capital at any given interest rate if Equation 21 holds:

(21)

So if neoclassical theory is compatible with a steady state in which Equation 21 does not hold, macroeconomic models in which the interest rate is equated to the marginal product of capital are not the general case.

Equation 21 is quite interesting. It implies that equilibrium prices are proportional to labor values, as defined in classical economics. As a matter of history, the reliance of the labor theory of value on this sort of extremely special case was thought to be a major weakness. If neoclassicals find this condition too extreme for the labor theory of value, they can hardly find it general enough as a defense of neoclassical macroeconomics [10].

Perhaps the solution lies in adopting another method of evaluating the physical quantity of capital in the same units as net output. Champernowne has proposed a chain index measure of capital that will restore the macroeconomic equality [11]. However, this measure only works under special cases, too. Burmeister has shown that the macroeconomic equality can be established with this chain-index if and only if real Wicksell effects are always negative. However, as was shown by the Cambridge Capital Controversy, this assumption of "negative real Wicksell effects" is not a general case either. In fact, nobody has determined what are necessary assumptions on technology to ensure the desired conclusion will follow [12]. Finally, if this index is used to express an aggregate production function in per capita terms, the wage is no longer equal to the marginal product of labor as that marginal product is typically expressed in such functions [13].

But, some may object, aggregate production functions work empirically. So even though economists cannot state their assumptions, they may say, this empirical success justifies the continual use of aggregate production functions. This is an extremely weak defense. Income distribution has been stable over much of the period in which macroeconomists have been using aggregate production functions. Franklin Fisher has shown through simulation that the supposed empirical success of aggregate production functions can arise under these conditions even in cases where the needed assumptions do not hold. Thus, this supposed empirical success of aggregate production functions fails to test the models with the unstated assumptions of aggregate neoclassical theory or to test among alternative theories. In fact, economists who rely on this defense seem to be confusing their empirical results with another accounting relationship [14].

Footnotes

[6] See:

• Heinz D. Kurz, "Factor Price Frontier," The New Palgrave.

[7] Textbook treatments of the connection between cost minimization and the factor price frontier can be found in (Woods 1990) or

• Heinz D. Kurz and Neri Salvadori (1995). Theory of Production: A Long Period Analysis, Cambridge University Press.

[8] These properties of the factor price frontier in the "continuous substitution" case were brought out by Luigi Pasinetti in correcting a technical mistake by Robert Solow. See:

• Luigi Pasinetti (1969). "Switches of Technique and the 'Rate of Return' in Capital Theory," Economic Journal: 508-513.

It seems worth pointing out, since many may be confused on this point, that reswitching and capital reversing are possible when the optimal technique varies continuously with the interest rate. See:

• P. Garegnani (1970). "Heterogeneous capital, the Production Function and the Theory of Distribution," Review of Economic Studies, v 37, (June): 407-36.

[9] See:

• Frank Hahn (1982). "The neo-Ricardians," Cambridge Journal of Economics, V. 6: 353-374.

[10] Hence, Paul Samuelson's defense of aggregate production functions is inadequate. This defense can be found in:

• Paul A. Samuelson (1962). "Parable and Realism in Capital Theory: The Surrogate Production Function," Review of Economic Studies: 193-206.

[11] D. G. Champernowne (1953-1954), "The Production Function and the Theory of Capital: A Comment," Review of Economic Studies, V. 21: 112-35.

[12] See the reference in footnote 4.

[13] Salvatore Baldone (1984), "From Surrogate to Pseudo Production Functions," Cambridge Journal of Economics, V. 8: 271-288. Baldone also shows Burmeister's claims are problematic when used to compare quasi-stationary economies with a positive rate of growth, instead of just stationary economies.

[14] Anwar Shaikh (1990). "Humbug Production Function," The New Palgrave: Capital Theory, Macmillan.

Interest Rate Unequal To Marginal Product Of Capital (Part 2 Of 4)

2.0 Some Relationships Among Aggregate Variables
Consider a very simple capitalist economy in which the value of all net output is distributed as wages or (accounting) profits:

(1)

where Y is net output, W is total wages, and P is total profits. The term "profits" is used in some economic traditions to mean what some neoclassical economists call "interest". If this causes confusions, read "profit" as "interest" throughout this essay, except where "economic profit" is used.

If there is some homogeneous unit in which to measure the labor force (person-years), the wage w is related to total wages as in Equation 2:

(2)

where L is the number of person-years employed. Similarly, if the capital K used up in a year can be valued in the same units as output, total profits relate to the interest rate r as in Equation 3:

(3)

Equations 1, 2, and 3 are accounting identities, true by definition. No assumptions have been made yet about how any of these variables are determined.

Continuing with the manipulation of accounting identities, one can transform Equation 1 to Equation 4:

(4)

Or:

(5)

where y is net output per head and k is the value of capital per head. Note that the value of output per head, the wage, and the value of capital per head are all measured in the same units, say bushels of wheat. The interest rate is a percentage rate with no units attached (other than, perhaps, an implicit time dimension).

Some neoclassical economists relate net output to inputs of labor and capital by means of an aggregate production function, which, when written in per capita form looks like Equation 6:

(6)

The function f is supposed to satisfy certain assumptions that characterize Constant Returns to Scale and diminishing marginal returns to each factor. Given these assumptions and perfect competition, cost minimization (or the maximization of economic profit) is supposed to ensure the equilibrium conditions given by Equations 7 and 8:

(7)

(8)

Equation 7 shows the interest rate is equal to the marginal product of capital, while Equation 8 shows an equality between the wage and the marginal product of labor [3]. I intend to challenge Equation 7 in a framework that includes Equations 5, 6, 7, and 8.

The argument for the aggregate production function, when written in per capita form, is the value of capital per head. How can the value of capital per head vary? Consider a multi-commodity model in a steady state. Suppose the same technique is adopted at different interest rates. The corresponding price structure will vary with the interest rate. Even though the same capital goods may be used at different interest rates, the value of capital per head will differ with the interest rate. This variation in the value of a given set of capital goods with the interest rate is known as a price Wicksell effect.

Typically, though, the cost-minimizing technique will also vary with the interest rate. Consider the prices ruling at a given interest rate, where that interest rate is a switch point. That is, at least two techniques are cost minimizing at the given interest rate. We can then consider variations in capital goods resulting from a variation in the usage of two cost minimizing techniques. The resulting variation in the value of capital per head at the given prices is known as a real Wicksell effect. The chain-rule for differentiation shows how the price and real Wicksell effects combine to determine the total variation in the value of capital per head with the interest rate [4].

For completeness, I note there is a third manner in which the value of capital per head can vary, namely if the composition of final output varies, for example, due to a difference in the rate of growth. This possibility is not important to my argument.

Now I want to prove a theorem by some simple formal manipulations. Given Equation 5, the marginal product of capital is equal to the interest rate (Equation 7) if and only if Equation 9 holds [5]:

(9)

Proof: Equation 10 gives the total differential of both sides of Equation 5:

(10)

Thus, the interest rate is equal to the marginal product of capital (Equation 7) if and only if Equation 11 holds:

(11)

Equation 9 follows. Q.E.D.

A demonstration that the value of capital per head need not be equal to the additive inverse of the slope of the factor price frontier (Equation 9) demonstrates that the interest rate need not be equal to the marginal product of capital (Equation 7).

Footnotes
[3] A more general statement of these relations, abstracting from price Wicksell effects, is given by Equations 7' and 8' in terms of left-hand and right-hand derivatives:

(7')

(8')

In the discrete case without price Wicksell effects, the neoclassical aggregate production function is supposed to resemble the function in Figure 1.

Figure 1: Interest Rate Bounded By Marginal Products

[4] A good explanation of price and real Wicksell effects can be found in:

• Edwin Burmeister, "Wicksell Effects," The New Palgrave.

Burmeister writes:

"The value of capital, however, is not an appropriate measure of the 'aggregate capital stock' as a factor of production except under extremely restrictive assumptions. Wicksell (1893, 1934) originally recognized this fact, which subsequently was emphasized by Robinson (1956)."

[5] Equation 9 follows from Equation 7' at a non-switching point in the discrete case. Consider a second set of values of y, w, r, and k, related as in Equation 5:

(I)

The difference between these two sets of values is given by Equation II:

(II)

Or, in obvious notation:

(III)

Assume Equation 7' holds. Two cases arise.

Case 1: . By diminishing marginal productivity, one also has . Ignoring higher order terms, the production function is:

(IV)

From Equation 7', one has:

(V)

Or:

(VI)

Substitute from Equation III:

(VII)

A little algebra yields:

(VIII)

Take the limit as the value of capital per head approaches k from above (and therefore the interest rate approaches r from below):

(IX)

Case 2: and thus Ignoring higher order terms, one has:

(X)

The inequality in Display V follows, once again, from Display 7'. An argument parallel to the first case yields the inequality in Display XI:

(XI)

The left-hand and right-hand derivatives of the factor-price frontier are equal at a non-switching point in the discrete case. The two cases establish that that derivative is equal to the value of capital per head:

(XII)

which was to be shown.

Interest Rate Unequal To Marginal Product Of Capital (Part 1 Of 4)

"Then Buddha said: ... But tell me, Subhuti, do you really believe that having only one homogeneous capital good will permit you to derive a rate of profit purely from the technical relationship between homogeneous capital and output?

Subhuti replied: Thus it is said in some venerable books.

Buddha said: Revere them, Subhuti, but trust them not. Suppose you do get the value of the marginal product of capital in terms of output of consumer goods. In what units will it be expressed? Physical units of additional consumer goods per unit of additional homogeneous capital. But the rate of profit is a pure number. Surely you will need something more in going from the first to the second to reflect the relative price of the capital good vis-a-vis the consumer good. But the equilibrium price of capital in units of consumer goods depends on the rate of profit used for discounting, and a variation of the rate of profit can involve a variation of the value of the same physical capital in units of consumer goods. This difficulty is not eliminated by having one homogeneous good." -- Amartya Sen (1974). "On Some Debates in Capital Theory", Economica V. 41 (August)

1.0 Introduction
This essay demonstrates that the existence of price Wicksell effects can lead to the inequality of the marginal product of capital and the interest rate. The equality being challenged here should be understood as it is used in macroeconomic models with aggregate production functions. That is, macroeconomic modeling with aggregate production functions is inadequately grounded in microeconomic theory. I conclude with some rather far-reaching possibilities.

The length of my exposition here results from my attempting to clarify several points of confusion exhibited by economists responding on Usenet to previous versions. This argument is well-established in the literature [1]. I suggest that those who think this argument mistaken should take a look at some of my references. If my argument were mistaken, demonstrating the mistake would be worthy of a paper.

I claim this argument is not about index number problems or the aggregation of capital [2]. I also do not see how it relates to the aggregation of production functions. Those who believe otherwise are encouraged to be explicit about the connections. Perhaps, the question, from a neoclassical perspective, is how the services of capital goods are related to the quantity of "waiting" they supposedly represent.

Footnotes
[1] Elements of this argument can be found in Joan Robinson's 1953-1954 article "The Production Function and the Theory of Capital," Review of Economic Studies, 1953-4 and Geoff Harcourt (1972), Some Cambridge Controversies in the Theory of Capital. The closest formulation to my argument is in the following papers:

• Amit Bhaduri (1966). "The Concept of the Marginal Productivity of Capital and the Wicksell Effect," Oxford Economic Papers, XVIII: 284-288
• Amit Bhaduri (1969). "On the Significance of Recent Controversies on Capital Theory: A Marxian View," Economic Journal, LXXIX: 532-539

But the argument was also in Sraffa. See, for example:

• Piero Sraffa (1962). "Production of Commodities: A Comment," Economic Journal, V. LXXII (June): 477-9.

Here is one expositor of Sraffian economics:

"Now a major problem existed because capital, unlike either labor or land, is a produced means of production and cannot be measured unambiguously in purely physical terms: the amount of capital can be measured only in value terms. The problem was to establish the idea of a market for capital, the quantity of which could be expressed independently of the price of its service (i.e. the rate of profit)... The basic deficiency with this approach is in its treatment of capital, which cannot be measured independently of the rate of profit. As observed above, the value of capital, like that of all produced commodities, depends on the rate of profit, or interest." -- J. E. Woods (1990). The Production of Commodities: An Introduction to Sraffa, Humanities Press International: 306-307

[2] The divergence between the marginal product of capital and the rate of interest

"is attributable to the fact that it is impossible to find an invariant unit in which to measure the social quantity of capital.

To put the matter another way, we may say that a change in the supply of capital - arising, for example, from new voluntary savings - alters the units in which all the previously existing capital is measured; and it is therefore incorrect to say that the supply of capital as a whole has increased by the amount of the voluntary saving. It is important to emphasize that this problem of measuring the quantity of capital is not an index-number problem. There are, to be sure, numerous index-number problems of the greatest complexity in the theory of capital. But the problem to which I now refer would exist even in the simplest economy in which all output consisted of a single type of consumer's good and firms were exactly alike." -- L. A. Metzler (1950). "The Rate of Interest and the Marginal Product of Capital," Journal of Political Economy, Vol. 53: 284-306

Metzler provides a brief literature review of awareness of this problem going back to Wicksell. My analysis is closest to his comments on Knight's capital theory, though I think my presentation is clearer.

Do Patents And Copyrights Spur Innovation?

An expert on United States law on intellectual property tells me a National Academy of Sciences study found empirical evidence suggesting the answer is "Yes" to the question in the post title. Presumably, he was referring more to the former than the latter of:

• Patents in the Knowledge-Based Economy (2003)
• A Patent System For The 21st Century(2004)

I've downloaded the executive summaries, but I doubt I will purchase these reports.

Apparently if I go longer between posts, commentators will have more to say on each one.