On the Fallacy of Geometrical Proofs of the Marginal Theory of Distribution

The marginal revolution of the late 19th century arose from the attempt to apply the ricardian theory of land rent arising from marginal productivity to all factor returns.

This immediately led to the ‘product exhaustion’ or ‘adding up’ problem; would the total returns of each marginal factor add up to the total returns?  The neo-classical solution deriving from Wicksteed. Wicksell and Barone working independently relied (though non realised it at the time) on Eulers  theorem of homogeneous functions, so that once paid their marginal revenue the total returns would exactly add up to total returns.  John Pullens magisterial history of the theory recounts these controversies.

The theory was immediately attacked for it initial assumptions.  The Euler theorum approach relied on partial derivatives of each factor, which assumes full independence of each factor.  This is unrealistic, the employment of an additional shepherd for example may require the use of an addition crook.  In the short run factor proportions are fixed. This lead for example Wicksteed and eventually Wicksell to abandon support for the theory, none the less it had gotten into the text books.

Modern general equilibrium theory has no requirement at all for marginal productivity theory, and following the capital theory controversies of the 50s and 60s there are less mentions of the ‘marginal productivity’ of capital.  This leads to the odd result that mainstream neo-classical theories of economics lacks a satisfactory theory of distribution.

Hence there have been a number of attempts at graphical representations of the marginal productivity theory.  These date back over a century but the most celebrated example is the Samuelson/Hicks model.

A brief recourse to the assumptions of the theory is necessary.  The theory assumes perfect competition and hence no profits.  It also assumes constant returns to generate a linearly homogeneous production function.  Finally the theory is not one of prices and so is not even a theory of factor returns, rather it is a theory of factor costs (supply curves) and so only becomes a theory of factor returns assuming general equilibrium- in this case one of perfect competition and full employment.

In this case returns are rental returns on labour, land and capital goods only, there is surplus, no profit and no interest.

It is worth noting the similarity of this system with Schumpeter’s stationary state earning no interest, and where all costs can be imputed to ‘stored up’ land and labour. In a single good world with zero interest this is a pure corn model where at the extensive margin of land rent all costs are equal to discounted labour values.

Not surprisingly these assumptions are somewhat at variance with reality.  The first to be faced was the assumption of constant returns. Of course firms generally face U shaped cost curves of increasing returns to the point where adding labour is generally less productive per marginal unit of input.  Wicksell modified the theory to incorporate U shaped cost curves, where under perfect competition firms produce at the bottom of the U of their cost curves.  This is a consequence of the assumption of perfect competition and implies an intersection at that point of the demand curve, and so also implies general equilibrium.

This is the situation implied in the Samuelson/Hicks model of factor returns shown below.

Assume two factors of production capital goods K and labour L.  (there are hidden assumptions here regarding a single capital good only and absence of consumption goods – to which we will return)


TR=TC= P x Q

On the input side P x Q = wL + rK where w and r are the marginal ‘own rates’ of productivity of each factor.



Therefore  P x Q = L(P.MPPl)+K(P.MPPk)

Dividing both sides with P

Q=L(MPPl) = K(MPPk)

So taking any given price the factor returns of their MPPs to K and L would exactly equal to TR.

At this point there is product exhaustion and it is not necessary to assume a linear homogeneous production function.

Of course this approach is nowhere nearly as closely defined in the neoclassical textbooks, we still have loose references to ‘profits’ being the capitalists return, and references to Eulers Thoerum, whilst the assumptions of gneral equlibrium and perfect competition are often glossed over or not mentioned at all.  Profits and interests remain unexplained.

Factor returns in  a three factor model has nothing to do with profits.

Under this models assumption of perfect competition every factor of production productive factor receives its marginal product, and this exhausts all output; there is no separate category of income called “interest” or “profit” that is distinct from payments to the owners of labor, land, and capital goods

Fisher (1930)

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 might be objected that partial differentials have come in through the back door.  The Samuelson/Hicks theory assumes cost and revenue curves.  These must be derived and at the point of intersection one can take the partial derivatives.  However this is not valid the geometrical method assumes no function for the derivation of price and quantity, you could have for example a leontiff type system of fixed technical coefficients determining price and quanity with fixed rations of capital and labour inputs, indeed ANY input-output system.

All the geometrical system is doing is expressing an accounting relationship under conditions of perfect competition and general equilibrium.  No causation is implied – somehow a firm has reached the bottom of its average costs curve that is all.   As such it all the outputs of the production function is doing is expressing the accounting relationship of the capital/labour share in inputs.  It therefore fall victim to Shaikh’s criticism of the ‘Humbug’ production function.

So under these restrictions the geometrical approach is valid at determining marginal productivity of labour determines prices of goods.  This is easy to show in an economy with no capital goods as P.Q=L(MPPl)

Once we introduce time, profits and investments however we reach a situation that no rational firm or investor would wish to acheive.  One which is incompatible with capitalism.

Look at a firm with increasing returns – at the beginning of its cost curve.  Providing the market can absorb the output at a rate of profit which is increasing a ‘capital maximising’ firm will increase output to the point where the marginal increase in the rate of profit is zero, and where the marginal return to the rate of profit in other investments is not higher.  If it overshoots the bottom of the cost curve it will reduce production and labour to try to hit the bottom of the curve.  If the period of return of investment accounting for depreciation is greater than the current stock of capital goods this will be decreased and production reduced, the converse will lead to investment and increase in capital goods.

At the bottom of the curve existing capital goods will depreciate and need to be replaced, if replaced however there are zero returns to the owner of capital goods.   At zero interest it is not even possible to calculate the rate of value depreciation, indeed the concept is meaningless.  Rather we are simply retaining the physical nature of capital goods only. They are retaining their capital stock but making no money. But no rational investor would invest in a firm making zero profits, including those renting capital goods.  Therefore firms will try not to be at the bottom of the cost curve but a point to the left which generates a return at the expected rate of profit. If such a firm is generating profit it is not in perfect competition or general equilibrium. It is producing a good which is in demand and scarce and so generates profits, hence the short side rule applies, the price is set by the shortage in quantity – whichever is less: quantity demanded; or quantity supplied. Q=min{Qd,Qs}. The short side of the market determines the quantity traded.  This profit will be eventually be reduced as the profit is imputed back to the potential for increased factor prices and with new entrants attracted to the market – so this situation is dynamically unstable.

In such a market there is a surplus, a profit can be made, returns exceed marginal productivity of factors.

Is there a way to rescue the geometrical theory with its fixed  assumptions.  No, but we can relax the assumptions and see whether or not it is more robust.

Assume not three factors but four, we add money and those that own money and rent the factor, banks, at an interest rate.  Under competition the own rate of return of capital goods must equal the own rate of return of rental of money (which is interest plus cost of production of money), and both must be positive for investment, accumulation and the capitalist process to proceed.

Samuelson in his ‘rice model’ objected to our Schumpeterian assumption that the steady state has zero interest rates.  He gave the case of a rice field with yield 100 at year zero and 110 at year one – implying a 10% own rate.  However this pure physical productivity does not explain financial interest.  If the field produces 10 surplus and is the subsistence or workers is 4 – the surplus is 6 – which is rent not interest.  If an investor wished to invest in a new rice technique producing 120, or expanding land under cultivation  at the extensive margin, that is profit capable pf capitalization and earning interest.   Samuelson makes the pre-fisherian mistake of confusing rent and interest.  A dead stand of trees still has a value, but the same stand with the same number of trees has a higher value because of agio.  The valuation of the latter at T1 will be imputed back to the value of the land at T0 as indeed Schumpeter pointed out.

Let us assume that owners of capital goods and banks seek to maintain there rate of return, the system reproduces at a steady rate of profit.  A more classical version of the ‘steady state’.  In this case all factor returns are rent and we can produce a ricardian rent diagram .

The diagram makes the simplifying assumption that the capital intensity of the industry is such that the revenue on rent of capital goods is the same as profits.

You will note that here again we have a graphical (Ricardian) illustration of distribution.  Assuming equality of own rates of money and capital goods and reproduction at r no greater or less then the marginal productivity of the capital goods at that interest rate alone determines the capital goods rental factor share – which determines the interest share  (rate) and wages – or rather maximum wages at that interest rate, is a residual.  Maximum wages however is only achieved at full employment, otherwise again the short side rule applies.  There is circularity here – interest determines capital goods investment which determines interest; a Wicksell effect.

However we have assumed competitive equilibrium between the goods market and the loans market so that

the NPV of  (loans – cost of lending) = NPV of (capital returns – depreciation).

In this one good corn-money model it is the NET marginal productivity of capital goods – accounting for depreciation and cost of lending – which determines interest rates, we have closed the system.  This is making the slightly heroic assumption that banks are able to keep up with  this demand.  In most cases lending will be on the short side and market interest rates will be higher leading to production being less that it potential.

Note maximum wages sets a total wages pool of funding not the size of an individual wage.  With technical coefficients fixed in the short term an individual maximum wage can be found which a firm can use as a marker point in advertising positions according to labour market supply conditions.

Consider a case where an entrepreneur innovates, creates a technique which produces a return on capital goods in excess of the return on lending.  This will generate super profits.   This may eventually produce greater investment or impute back to factor prices and induce entrants.  In the short run shower there are super profits.  By the Kalecki profits formula Profits =Investment + capitalist consumption, capitalists here being the producers, renters of capital goods and renters of money en toto,   If we assume that reinvestment occurs up to the point on the cost curve that maintains R then the residual is capitalist consumption.  All factors impute back to a labour value from which rent is extractive. At positive profit they impute back to past present value of labour of production plus future present value of labour of capitalist consumption.  This is equivalent to the modified classical theory of value put forward by Ian Wright.  We can interpret exploitation here in a new light, it is not the totality of surplus value but rather value over and above that necessary to reproduce the system and withdrawn from future investment.

In the long run too NPV Profits= NPV of  (loans – cost of lending) = NPV of (capital returns – depreciation) as investment is attracted.  There is no NPV equivalence re labour.

By abandoning the neo-classical ceritus paribus assumptions and looking at conditions of reproduction we have a more robust theory of distribution.  However note the conditions though relaxed from perfect competition are still restrictive- we have a one good corn-money world and where short of full employment the relative bargaining powers of capital and labour determine whether the maximum labour share is reached.

Introduce two or more goods and multiple techniques with varying composites of capital and labour things become more complicated. We no longer have a straightforward relationship between interest and capital returns. So what.  Even if there is a change in technique and a reverse in the inter-temporal returns of one more ‘dose’ of any factor the identity of production that NPV Money=NPV Capital Goods applies.  You cannot consider past returns on one technique on future returns on another in an investment decision.  Past returns are sunk costs and are irrelevant.  The investment decision must be considered at that moment in time and for all futures moments in time and must be based only on the marginal return of capital advanced and from this distribution flows from the revised and corrected graphical technique we have presented here.

One thought on “On the Fallacy of Geometrical Proofs of the Marginal Theory of Distribution

  1. Pingback: Valueing an Apple Tree is not the Same as Valuing an Orchard – A reply to @MacRoweNick | Decisions, Decisions, Decisions

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