Re: No Mainstream Economist Will Acknowledge Here Math In This Post Is Correct

From: Andy F (aft627_at_aol.com)
Date: 07/01/04 

Date: 01 Jul 2004 12:25:36 GMT

The numbers may be correct but your interpretation of them isn't. What you
produce is an incomplete model where prices and interest rates are
indeterminate.To determine the interest rate you would need to make further
assumptions.
Since you don't produce a value for the interest rate, your example neither
proves nor disproves the mainstream theory.

Robert Vienneau wrote:

> "The marginal product of capital will in equilibrium be equal to the
> real interest rate adjusted for market perceptions of risk. If one
> investment offers a rate of return higher than others, capital will
> flow to that investment and away from others, lowering the marginal
> productivity of additional investment in the former and raising it
> in the latter."
> -- Don Dale making a mistake, 10 February 1997
>
> "Neither let us forget that the real interest rate is equivalent
> to the marginal product of capital."
> -- Edward Flaherty making a mistake, 4 February 1995
>
> "Suppose the interest rate were less than the marginal product of
> capital. Borrowing and building capital would earn a positive return
> so simple arbitrage would drive them together. Suppose the interest
> rate were above the marginal product of capital. How could loans to
> finance capital purchases then ever be repaid? So investment would
> fall, allowing depreciation to raise the marginal product of capital
> until it reached the interest rate."
> -- Mark Witte making a mistake, 10 February 1995
>
> "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 homogeneoues 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, "On Some Debates in Capital Theory". Economica.
> V. 41, August 1974.
>
>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.
>
> I have explained this before. Several economists have mistakenly
>asserted this argument has a simple technical flaw, although they don't
>all agree where that flaw lies. In fact, the length of my exposition
>here results from my attempting to clarify several points of confusion
>exhibited by economists responding to previous versions.
>
> This argument is well-established in the literature ([1], [2]). 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 [3]. 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.
>
>2.0 SOME RELATIONSHIPS AMONG AGGEGATE VARIABLES
>
> Consider a very simple capitalist economy in which the value of all net
>output is distributed as wages or profits:
>
> Y = W + P (1)
>
>where Y is net national income, W is total wages, and P is total profits
>(or interest charges). The term "profits" is used in some economic
>traditions to mean what neoclassical economists think of as "interest."
>If this causes confusions, read "profit" as "interest" throughout this
>essay.
>
> 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:
>
> W = w L (2)
>
>where L is the number of person-years employed. Similarly, if the capital
>stock used up in a year, K, can be valued in the same units as output,
>total profits relate to the interest rate r as in Equation 3:
>
> P = r K (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 manipulation of accounting identities, we can transform
>Equation 1 to Equation 4:
>
> Y = w L + r K (4)
>
>Or
>
> y = w + r k (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 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:
>
> y = f( k ) (6)
>
>The function f is supposed to satisfy certain assumptions. 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:
>
> r = f'( k ) = dy/dk (7)
>
> w = f( k ) - k f'( k ) (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 [4]. I intend to challenge Equation 7 in any
>truly multicommodity 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 [5].
>
> If only one fixed-coefficients (Leontief) technique is known, the
>real Wicksell effect will be zero. But the price Wicksell effect may
>be non-zero. So the assumption of a Leontief technique is no obstacle
>to finding a nonzero variation in the value of capital per head with
>the interest rate.
>
> 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 [6]:
>
> k = - dw/dr (9)
>
>Proof:
>
> The total differential of Equation 5 is Equation 10:
>
> dy = r dk + k dr + dw (10)
>
>Thus, the interest rate is equal to the marginal product of capital
>(Equation 7) if and only if Equation 11 holds:
>
> k dr + dw = 0 (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) will demonstrate that the interest rate need
>not be equal to the marginal product of capital (Equation 7).
>
>3.0 A SIMPLE TWO-GOOD COUNTEREXAMPLE
>
> The question to be investigated here is whether Equation 9 is
>an implication of neoclassical microeconomics. I claim Equation 9
>is not an implication of neoclassical microeconomic.
>
> 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 ) = a12 a01 + ( 1 - a11 ) a02 (12)
>
>Suppose the physical quantity flows are as in the following table
>on a per worker basis:
>
> INPUTS STEEL INDUSTRY WHEAT INDUSTRY
> Labor [a01 a12/d(0)] person-years [(1 - a11) a02/d(0)] person-years
> Steel [a11 a12/d(0)] tons steel [(1 - a11) a12/d(0)] tons steel
>
> OUTPUTS [a12/d(0)] tons steel [(1 - a11)/d(0)] bushels wheat
>
>where all quantities are positive and
>
> 0 < a11 < 1 (13)
>
>Notice that the sum of person-years in both 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.
>
> Assume constant returns to scale. This means we can express the
>observed quantity flows as follows:
>
> a01 person years & a11 tons steel PRODUCE 1 ton steel
> a02 person years & a12 tons steel PRODUCE 1 bushel wheat
>
>This explains the puzzling notation above. The parameters reflect
>unit (gross) outputs in both industries.
>
> 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 certainly
>haven't assumed a Leontief fixed-coefficients technology.
>
> We have observed that [a12/d(0)] 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:
>
> a11 p (1 + r) + a01 w = p (14)
>
> a12 p (1 + r) + a02 w = 1 (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 ) = a12 a01 (1 + r) + [ 1 - a11 ( 1 + r ) ] a02 (16)
>
>The price of a ton of steel as a function of the interest rate is
>then given by Equation 17:
>
> p = a01 / d( r ) (17)
>
>Hence, the quantity of steel, when measured in bushels wheat, is:
>
> k = [ a12 a01 ] / [ d( 0 ) d( r ) ] (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:
>
> w = [ 1 - a11 (1 + r) ] / d( r ) (19)
>
>Equation 19 expresses the factor price curve [7] associated with the
>observed technique. A different technique 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 [8]. 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 an
>interval in which that technique is selected. Then the factor price
>curve for the selected technique will be 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.
>
> 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 will
>be non-switching points, although the chosen technique will vary
>continuously with the interest rate [9]. 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 will not determine
>the distribution of income, and no equation analogous to Equation 7 will
>arise [10]. 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 will appear 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 can
>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:
>
> dw/dr = - a12 a01 / [ d( r ) d( r ) ] (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 and only if
>Equation 21 holds:
>
> a01 / a11 = a02 / a12 (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 [11].
>
> 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 [12]. 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 [13]. 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 [14].
>
> But, some may object, aggregate production functions work
>empirically. So if economists cannot even 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 [15].
>
>4.0 CONCLUSION
>
> This article has presented a simple explanation of the
>nonequality of the interest rate and the marginal product of
>capital, as that equality is understood in macroeconomic models.
>Thus, neoclassical microeconomics does not imply that equality.
>Various attempts to defend the macroeconomic models considered
>here have been examined and have been found wanting. An interesting
>aspect of this criticism is that it does not seem to be about index
>number problems [16]. Nor has this argument depended on the
>phenomena of reswitching at all, and it depended on capital reversing
>only in criticizing Champernowne's chain index [17]. If the
>demonstration of the theoretical possibility of these phenomena
>are taken as central to the Cambridge Capital Controversy, then
>that controversy should have been about something other than
>aggregate production functions [18]. As far as I can see, this argument
>is fairly well understood on the Cambridge, England side.
>
> I conclude that the CCC was indeed about something else. I happen to
>think the topic under debate was a confrontation of two rival paradigms
>of value and distribution, namely the Classical theory and the so-called
>Neoclassical theory [19]. In particular, it was shown, I think, that
>Neoclassical economists cannot consistently maintain in their equilibrium
>framework that owners of capital goods make any contribution to
>production. Hence interest cannot be a payment for such a
>contribution [20].
>
> Finally, it is curious that economists continue to use aggregate
>production functions despite the clear warnings of this traditional
>argument. Many of those economists who follow Solow or Lucas do not
>seem concerned about their inadequate concept of capital. Although these
>researchers may be interested in technical improvements in their models,
>capital-theoretic problems do not seem high on their agenda. Furthermore,
>much of graduate education in economics seems to leave newly emerging
>economists unaware of capital-theoretic problems with their models. These
>young economists do not seem to possess the analytical tools that were
>forged in the CCC, such as the correct analysis of the choice of
>technique, a correct analysis of the relationship between the theory of
>rent and income distribution, or even how to analyze depreciation and
>the economic life of machines in a framework of joint production.
>
> What explains this apparent continuation of the miseducation of
>economists that Joan Robinson decried over forty years ago [21]? My
>hypothesis is partly ideological. Any advanced treatment of capital
>theory and the appropriate analytical tools [22] would expose the
>student to the Cambridge Capital Controversy. The student would then
>learn about some serious questioning of the internal consistency of
>many claims of neoclassical economists. There are obviously normative
>overtones to this controversy, for example, over the exploitative
>nature of profits and the capitalist system as a whole. Neoclassical
>economics might be claimed to currently fill the social role of "hired
>prize fighters" for capital, what Marx characterized as "vulgar
>economics" [23]. This social role is threatened by the CCC.
>

  

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