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The Mistakes of the Marginal Productivity Theory of Income Distribution

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The Mistakes of the Marginal Productivity Theory of Income Distribution

by Dimitrios Nomidis*

Abstract

The debate that took place at the end of 19th and the beginning of 20th century on the

neoclassical income distribution theory based on the marginal productivity of the production

factors is well known. The debate evolved especially around the question whether the product

is exactly exhausted through its distribution to the factors of production according to the value

of their marginal products. This question is now considered resolved and closed by the proofs

presented by Wicksell, Walras and later on by other distinguished economists (Chapman,

Hicks etc).

The purpose of this paper is to demonstrate that the proofs which were presented to

support the product exhaustion theorem are mistaken and consequently the theory of income

distribution on the basis of the marginal productivity of the production factors is wrong.

Furthermore, this paper attempts to detect and explain the profounder reasons that presumably

led to these mistakes, as well as to identify and propound the new relations that replace the

wrong ones of the product exhaustion theorem. Last, it attempts to formulate the equilibrium

of the whole economic system (demand, supply, production, factors of production etc)

through a holistic-way equation system.

* Dimitrios Nomidis, Athens University of Economics and Business

National Technical University of Athens

E-mail: [email protected]

Author's ID: https://ssrn.com/author=2246677

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1. Introduction

As is known, the neoclassical marginal productivity theory of income distribution

states that under perfect competition the factors of production are rewarded with the value of

their marginal product. Also well known is the debate that took place at the end of 19 th and the

beginning of 20th century on this issue and especially on the question whether the product is

exactly exhausted by its distribution to the factors of production according to the value of

their marginal products. More specifically, on the question whether the following equation is

valid:

q ( L , K ,....)= ∂ q ∂ L

L+ ∂ q ∂ K

K +.... =MPL·L+MPK·K+.... where:

q (L , K,....) the quantity of production, a function of the production factors L (labor), K

(capital) and possibly other production factors (land etc).

L the quantity of labor employed in the production.

K the quantity of capital used in the production.

MPL=∂q/∂L the marginal product of labor of quantity L.

MPK=∂q/∂Κ the marginal product of capital of quantity Κ.

We shall remind this debate here very briefly1. Clark is considered the Father of this

idea (1889, 1891, 1899) (although germs of the theory were already contained in the Walras's

book “Elements of Pure Economics”, chapter 36, 1874, reissue 1896). But Clark did not

provide any mathematical proof of the product exhaustion problem. The first attempt of

mathematical proof was done by Wicksteed (1894), but it was quite inadequate since it was

based on specific production functions, namely on homogeneous of first degree production

functions, for which the proof of the product exhaustion problem is based on a simple

application of the Euler's theorem for mathematical functions homogeneous of first degree.

Next, Wicksell (1900, 1901, 1902) gave a proof of the product exhaustion theorem for

the point where the production function presents the minimum average cost of production and

which constitutes the equilibrium point of the market under pefect competition in the long

run. At this point the production function -whichever the form of the production function is-

presents constant returns to scale and thus the properties of a homogeneous function of first

degree (i.e. if the factors of production increase proportionally, then production increases at

1 Accounts of the debates surrounding marginal productivity abound. Those of Joan Robinson (1934), George Stigler (1941: Ch. 12) and John Hicks (1932a,b) are probably the best. Also worthwhile are the accounts by Henry Schulz (1929), Dennis H. Robertson (1931) and Paul Douglas (1934).

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the same proportion), and consequently the Euler's theorem can apply there. Walras (1874,

reissue 1896) had expressed a rationale similar to that of Wicksell, but in a manner a bit

confusing and not so explicit.

The resolution of Wicksell closed, in essence, the question of the product exhaustion

through its distribution to the factors of production on the basis of their marginal products and

constituted the mathematical (and at the same time ethical) foundation of the economic theory

of income distribution based on the marginal productivity of the production factors, which has

been scientifically accepted and established till today.

2. The Confutation of the Classic Theory of Competition and Distribution

The purpose of this paper is to demonstrate that the above theory of income

distribution is wrong. Before we adduce the proof that follows in the next section, we must

mention that the mistake lies firstly in the fact that the equilibrium point of the market under

perfect competition does not lie at the minimum point of the average cost curve (as the

neoclassical economic theory argues), where the Euler theorem for the product exhaustion

could be applied. As it is indisputably proved and extensively analyzed in the works of the

writer (Nomidis 2015a, 2015b, 2016a, 2016b, 2018a, 2018b), the market equilibrium point

under perfect competition is not determined by the intersection of the total supply and the

total demand (see Figure 1, point A), as the conventional theory argues (which intersection, in

fact, occurs at the minimum average cost), simply because that point does not maximize the

profits of the firms, whereas it should maximize them according to the conventional theory

itself (even if the economic profits -i.e. the profits beyond normal- become zero under perfect

competition).

Conversely, in order for this basic condition of free market, i.e. profits' maximization,

to be accomplished, the market equilibrium point under perfect competition is determined by

the intersection M of the total supply (marginal cost) and the marginal revenue that comes

from the total demand (and not the total demand itself), which gives the equilibrium point E.

But then the equilibrium does not take place at the minimum point a of the average cost

curve, as the conventional theory argues, but at a higher cost and smaller production (point e,

corresponding to the intersection of marginal cost LMC=SMC and marginal revenue mr, for

profit maximization); while the individual demand curve for the firm dd (which is not

horizontal, as the neoclassical theory says) becomes (with the entry of new firms due to

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perfect competition and the elimination of economic profit) tangent to the average cost curve

at this equilibrium point. These things unavoidably entail a monopolistic character of the

market even under perfect competition and zero economic profit, which results in the

equilibrium point not being at the minimum of the average cost curve, where the Euler's

theorem for the product exhaustion could be applied.

3. The Mistake of the Conventional Distribution Theory

But even if we ignore all of the above that confute the conventional economic theory

for the equilibrium of the firm and the market and examine the issue of income distribution in

the framework of this conventional economic theory, even then the neoclassical distribution

theory based on the marginal productivity of the production factors is wrong. Because in order

to prove the product exhaustion theorem it applies a proportional variation of the production

factors, while it is known that the variations in the quantities of the production factors should

always follow the rule that equalizes the marginal ratio of technical substitution (MRTS) of

those inputs with the ratio of their prices (MRTS=MPL/MPK=w/r, where w the price-reward of

labor (wage) and r the price-reward of capital (rate of return)), which is not consistent with a

proportional variation of the inputs. More spesifically:

Let's suppose that the production function contains as inputs the two basic factors of

production, labor and capital:

q=q ( L , K ) where:

q (L , K) the quantity of production as a function of the production factors L (labor) and

K (capital).

L the quantity of labor employed in the production.

K the quantity of capital used in the production.

As the conventional microeconomic theory itself teaches, when the entrepreneur varies

his production (e.g. he increases it), then he varies the quantities of the employed production

factors in a way that the ratio of their marginal products be always equal to the ratio of their

prices (i.e. their rewards, which are exogenously determined at the markets of labor and

capital) in order that the minimum cost be always obtained (and therefore profit

maximization) for the new production level. Specifically:

MPL/MPK = w/r where:

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MPL=∂q/∂L the marginal product of labor of quantity L.

MPK=∂q/∂Κ the marginal product of capital of quantity K.

w the reward of labor.

r the reward of capital.

This rule establishes an interrelation between L and K, which in the cartesian level of L

and K generates the so called “expansion path”, which gives the combinations of L and K that

obtain the least average cost for each production level. It is this expansion path from where

the curve of the long run average cost (LAC) comes (see Figure 1), which gives the average

cost of production for each point of the expansion path, that is the least average cost for each

production level (the term “long run” is introduced to show that for each variation of the

production, some time is necessary for the production factors -and especially capital- to be

adjusted to the new production level). This curve of the long run average cost (LAC), which

gives the least average cost for each production level, has a minimum point, where the

average cost of production becomes the minimum possible (minimum of minimum) among all

the combinations L, K of the production factors and the respective production levels. This

minimum point of LAC is, according to the classic (conventional) theory, the equilibrium

point of the firm and the market and at that point the production function indeed presents

constant returns to scale, that is it presents the properties of a homogeneous function of first

degree and consequently the Euler's theorem for the product exhaustion could be applied.

But the previously mentioned cost minimization condition MPL/MPK=w/r does not

allow for the Euler's theorem for the product exhaustion to be applied, because the latter

requires a proportional change of the production factors while according to the former the

change of the production factors generally is not proportional. More analytically:

The product exhaustion theorem states that:

q=MPL·L+MPK·K or

(1) 1=MP L L q +MPK

K q

Proof according to the neoclassical distribution theory

Suppose that the production function contains as variable inputs the two basic

production factors, labor (L) and capital (K):

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q=q ( L , K )

dq= ∂ q ∂ L

dL+ ∂ q ∂ K

dK or dq=MP L dL+MP K dK or

dq q

=MP L L q

dL L

+MP K K q

dK K

or

(2) 1=MP L L q

dL / L dq / q

+MP K K q

dK / K dq / q

For the proof of the product exhaustion theorem, then, in its previous form (1), it

should hold: dL/ L dq / q

=1= dK / K dq / q

If we symbolize by C=C(q) the total cost of labor and capital, then at the minimum

point of the long run average cost curve (LAC=C/q) we have:

d (C / q) dq

=0 or

C ΄ q−Cq ΄

q 2

=0 or C ΄ q−Cq ΄ =0 and since q ΄ = dq dq

=1 :

C ΄ q−C=0 or C ΄ = C q

that is:

dC dq

= C q

or dC C

= dq q

Consequently, at the minimum point of LAC the previous relation (2) becomes:

(3) 1=MP L L q

dL / L dC / C

+MP K K q

dK / K dC /C

At this point the classic theory, in order to prove the product exhaustion theorem, says

that if the variations of labor and capital were proportional to their initial quantities (i.e.

dL/L=dK/K), then:

dL L

= dK K

= wdL wL

= rdK rK

= wdL+rdK

wL+rK =

dC C

and therefore: dL/ L dC / C

=1= dK / K dC /C

Hence 1=MP L L q

dL / L dC / C

+MP K K q

dK / K dC /C

= MPL L q +MP K

K q

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Ergo q=MP L L+MP K K

and in this way the classic theory prooves the product exhaustion theorem at the minimum

point of LAC.

But here a very significant mistake has been inserted. The variations of labor and

capital cannot be proportional to their initial quantities (as they were considered), since they

must meet the rule of least cost throughout the substitution process between inputs

(MPL/MPK=w/r), so that the new point of production lie on the expansion path2. In other

words, by simply applying the Euler's theorem to prove the product exhaustion theorem at the

minimum point of the average cost curve LAC (as Wicksell and Walras did), we have not

taken into consideration the condition of cost minimization, while we should (function

extremum under condition).

Therefore, the theorem of product exhaustion through its distribution to the production

factors on the basis of their marginal products does not hold even at the minimum point of the

average cost curve LAC of the long run equilibrium under perfect competition as the classic

(conventional) economic theory argues.

4. The Mistake of the Chapman's Proof

Chapman (1906) attempted to present a diagrammatic proof of the product exhaustion

theorem (see Figure 2).

He considered an industry consisting of n homogeneous agricultural units, each of

which is cultivated by the same number of laborers L that are remunerated with their marginal

physical product as presented in Figure 2. The total labor remuneration is OAEL for each

agricultural unit. The total physical product of each unit is OMEL and consequently the land

revenue in each agricultural unit is AME. Chapman attempted to prove that this revenue, that

simply is the remainder of the total product OMEL after the subtraction of the total labor

remuneration OAEL (remunerated with its marginal product), constitutes also the marginal

product of the land. To calculate, though, the marginal product of land, he did not increase

marginally the area of the agricultural unit keeping the number of its laborers constant, but he

considered that a new agricultural unit is added in the industry and kept the number of

2 Marshall, already long before Hicks, had conceived of this concept by what he called “net marginal product”, which he defined as the increase in output that arises from the employment of an extra unit of the varying factor after all the other factors have been adjusted to their new optimal (i.e. profit maximizing ) levels (cf. Marshall, 1890: p.426-30).

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laborers of the whole industry constant (L·n). This total number of laborers is now equally

distributed to n+1 firms of the industry, which thus decreases the previous number of laborers

and the product (AMCL΄) per agricultural unit. Chapman calculated the marginal product of

land as the difference between the total productions of the industry in the previous two states

with n+1 and n firms. The calculation in this way results in fact in:

marginal product of land = BMC+n·CDE

which tends to AME when n increases infinitely, which proves the product exhaustion

theorem.

But by considering the problem in this way, that is by considering the production to

consist of the production of n business units, Chapman has fallen in the error to have taken as

production function a function in which the total production, total labor and total land area are

all proportional to the total number of firms n (agricultural units), that is a function that

clearly presents the distinctive feature (definition) of a homogeneous function of first degree

where the proportionate increase of the production factors increases the production by the

same proportion (with step of one business unit). Consequently, since the production function

is homogeneous of first degree, it is naturally expected the product exhaustion theorem to be

valid, due to the Euler's theorem which is valid for every homogeneous function of first

degree.

5. The Mistake of the Hicks's Proof

Hicks (1932, 1963) proves the product exhaustion theorem at the point of minimum

cost of long run equilibrium in his famous book “The Theory of Wages” (in the mathematical

appendix) without recourse to a constant returns to scale assumption (that is without recourse

to the Euler's theorem), but with a direct differentiation of a production function of general

form and making use of the equality between price and average cost due to perfect

competition. But he also falls in the same error, that is he does not take into consideration the

condition of cost minimization throughout the inputs' substitution (MPL/MPK = w/r), while he

should (function extremum under condition).

The amazing thing in the case of Hicks is that this condition of cost minimization by

means of the substitutability between the production factors is one of the major contributions

of Hicks himself to the neoclassical Distribution Theory and is considered his hallmark in that

theory. Thus, it seems a very strange thing for Hicks not to take into consideration this

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condition, which is considered his discovery, in his proof of the product exhaustion theorem

in the mathematical appendix of his book “The Theory of Wages”. To the contrary, he actually

talks there (p.238) about proportionate variations of the production factors as output varies:

“If, as before, we assume that the prices of the factors are constant, and if we assume

further that the proportions in which the factors are employed remain unchanged as output

varies, we can construct a (very specialised) cost curve for the firm, giving the cost per unit of

producing various outputs.”

6. The complete Confutation of the Neoclassical Distribution Theory

In the case of a market of monopolistic character, as the perfect competition under the

new consideration is (Nomidis 2015a, 2015b, 2016a, 2016b, 2018a, 2018b), the conventional

(neoclassical) distribution theory argues that the factors of production are rewarded not with

the value of their marginal product (p·MP) but with the marginal revenue of their marginal

product (mr·MP) (where mr the marginal revenue at the firm level, see Figure 1), thus

undergoing a monopolistic exploitation by the entrepreneurs. It would be interesting here to

examine whether the product exhaustion theorem holds by applying this form of the marginal

productivity law in the new theory of equilibrium under perfect competition (which does not

take place at the minimum point a of the average cost curve LAC but at the point e, having

therefore a monopolistic character, see Figure 1).

It must, then, be proved that:

p·q=mr·MPL·L+mr·MPK·K

Due to the profit maximization by the entrepreneurs we have mr=MC (=SMC=LMC,

see Figure 1) and due to the perfect competition and zero profit we have p=c, where c=C/q

the average cost of production (LAC). Consequently the above relation can be written:

(1) 1=MP L L q

MC c

+MP K K q

MC c

Proof according to the neoclassical distribution theory

Let the production function be: q=q ( L , K )

dq= ∂ q ∂ L

dL+ ∂ q ∂ K

dK or dq=MP L dL+MP K dK or

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dq q

=MP L L q

dL L

+MP K K q

dK K

or

(2) 1=MP L L q

dL / L dq / q

+MP K K q

dK / K dq / q

For (2) to be equivalent to (1), it should be proved that: dL/ L dq / q

= MC

c =

dK / K dq/ q

or

dL L

c= dq q

MC or

dL L

C q

= dq q

dC dq

or

dL L

= dC C

and likewise dK K

= dC C

If the variations of labor and capital were proportionate (i.e. dL/L=dK/K), then indeed:

dL L

= dK K

= wdL wL

= rdK rK

= wdL+rdK

wL+rK =

dC C

and consequently the initial relation (1) would be fulfilled and hence the product exhaustion

theorem would hold (with the form of the marginal revenue product now).

But the variations of labor and capital cannot be proportionate, since they must meet

the rule of least cost throughout the substitution process between inputs (MPL/MPK = w/r), so

that the new point of production lie on the expansion path (unless on the expansion path holds

dL/L=dK/K, but then the production function is linearly homogeneous) or, in other words,

during the variations of inputs we must take into consideration the condition of cost

minimization.

Therefore, the theorem of product exhaustion by its distribution to the factors of

production on the basis of their marginal revenue product (mr·MP) does not hold in this case

of monopolistic-character perfect competition either, unlike what the classic (conventional)

economic theory says.

7. The Profounder Mistake of the Distribution Theory

Nevertheless, normally and logically the product exhaustion theorem should hold at

the point of long run equilibrium of a competitive economy, where the profit is zero and the

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value of product is distributed to the factors of production by means of their rewards. Also, for

the maximization of profit, the reward of a production factor does indeed meet the rule of its

marginal productivity and equals the product MP·mr of its marginal product times the

marginal revenue (which equals the price of product in perfect competition due to horizontal

individual demand curve for the firm, according to the erroneous neoclassical view, but we

don't discuss this problem now). Where did go wrong, then, the neoclassical theory of

Distribution when it says for the equilibrium point in perfect competition:

p·q=w·L+r·K = mr·MPL·L+mr·MPK·K (general expression) or

p·q=w·L+r·K = p·MPL·L + p·MPK·K (for horizontal demand curve of the firm),

since this relation should be valid, almost by definition, at the equilibrium point of perfect

competition with zero profit?

The profounder mistake of the neoclassical theory of Distribution on the basis of

marginal productivity of the production factors is that the principle of the reward of a

production factor based on its marginal productivity is valid only when this production factor

is the sole variable input in the production process, while all the other factors that participate

in the production remain fixed and with fixed cost, which could probably occur mainly in the

short run. However even in that case, only the variable input would be rewarded with its

marginal product, while the fixed costs for the rewards of the other factors of production

would not meet, in general, the rule of their marginal productivities. That is, the fixed cost of

each fixed factor of production would not equal, in general, the product of the fixed quantity

of the factor times its marginal product, which, besides, varies with the final equilibrium

quantity of the variable factor.

Namely, the profounder mistake of the neoclassical theory of Distribution on the basis

of marginal productivity of the production factors lies in the fact that it erroneously extented

the law for the reward of a production factor based on its marginal productivity, which is valid

only when this production factor is the sole variable input in the production while all the other

factors remain fixed and with fixed cost, to the case where all the factors of production vary at

the same time, which certainly occurs at least in the long run. More exhaustively, in the above

expressions the marginal products of the production factors (MPL, MPK etc) are the partial

derivatives of the production function with regard to each production factor when all the other

production factors remain constant, which though could perhaps occur in the short run but

certainly does not occur in the long run. For, the partial derivative implies the marginal

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variation of a factor of production, which in the production process automatically entails the

variation of all the other factors of production at least in the long run. For this reason and

since the factors of production vary at the same time (especially in the long run), the previous

expressions of the product exhaustion theorem are not valid. They could be valid if only one

factor of production existed (e.g. labor, whereupon it would indeed hold that

p·q=w·L=mr·MPL·L) or if the other factors of production remained constant and without

reward. The above expressions could also be valid if MPL, MPK etc expressed the marginal

products under the concurrent variation of all the production factors (and exactly under the

variation condition MPL/w=MPK/r=.... for the minimization of the production cost), which

though is not mathematically feasible3.

8. Rebuilding the Distribution Theory

All the above confute thoroughly the neoclassical Distribution theory and certainly the

famous theorem of product exhaustion by its distribution to the factors of production on the

basis of their marginal productivity. It arises then the question:

Which are the new relations that replace the invalid ones and express the equilibrium

of the whole economic system?

Firstly, with regard to the product exhaustion theorem, which constitutes the core of the

Income Distribution theory, this takes now the form (in perfect competition without profit):

p·q = w·L+r·K +.....

accompanied by the condition of inputs' substitution for the minimization of the production

cost:

MPL/w = MPK/r = …..

which leads to a relationship between the factors of production (e.g. L=f(K) or L=f(K,q)).

In the above relations the rewards of the production factors (w, r etc) are determined of

course (as in the conventional theory) exogenously in the markets of the production factors

(based on their supply and demand), but the product exhaustion theorem based on the

marginal productivity of the production factors is not valid in the case of multiple production

3 Marshall had recognized that the marginal product concept can be a bit misleading and for this reason to solve the problem (cf. Hicks 1932: p12-15, Machlup 1937) he proposed the concept of “net marginal product”, which he defined as the increase in output that arises from the employment of an extra unit of the varying factor after all the other factors have been adjusted to their new optimal (i.e. profit maximizing ) levels (see also footnote 2).

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factors (which almost always occurs in praxis).

With regard to the equilibrium of the whole economic system (demand, supply,

production, factors of production etc) under perfect competition in the long run, this is

expressed in the new theory by the following equations' system (for simplification we only

consider the basic two production factors, labor and capital) (see also Figure 1):

(1) Demand Function: p=p(Q)=p(nq)

where Q the demand (=production) for the whole market, q the demand (=production) for

each firm and n the number of firms in perfect competition with zero profit.

Let for simplicity be: p=a-bQ=a-bnq

(2) Production Function: q=q ( L , K )

(3) Condition of Inputs' Substitution for the minimization of cost:

MPL/w= MPK/r or ∂ q /∂ L

w =

∂ q /∂ K r

(4) Zero Profit: p·q = w·L+r·K or (a-bnq)·q = w·L+r·K

(5) Profit Maximization: mr=LMC or a−2bnq= d ( wL+rK )

dq

The above five relations constitute a system of five equations that determine the point

of long run equilibrium (L, K, p, q, n).

9. Conclusions

This paper demonstrates that the proofs which were presented to support the product

exhaustion theorem are mistaken and consequently the theory of income distribution on the

basis of the marginal productivity of the production factors is wrong.

Regardless of this however, first of all the product exhaustion theorem is not valid

because the equilibrium point of the market under perfect competition does not lie at the

minimum point of the average cost curve (as the conventional economic theory argues),

where the Euler's theorem for the product exhaustion could be applied. As it is extensively

analyzed in the works of the writer (Nomidis 2015a, 2015b, 2016a, 2016b, 2018a, 2018b), the

market equilibrium point under perfect competition is not determined by the intersection of

the total supply and the total demand, as the conventional theory argues (which intersection,

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in fact, occurs at the minimum average cost), simply because that point does not maximize the

profits of the firms.

But even if we ignore the above consideration that confutes the conventional economic

theory and examine the issue of income distribution in the framework of this conventional

theory, even then the neoclassical theory of distribution on the basis of the marginal

productivity of the production factors is wrong. Because in order to prove the product

exhaustion theorem it applies a proportional variation of the production factors, while it is

known that the variations in the quantities of the production factors should always follow the

rule that equalizes the marginal ratio of technical substitution (MRTS) of those inputs with the

ratio of their prices (MRTS=MPL/MPK=w/r, where w the price-reward of labor (wage) and r

the price-reward of capital (rate of return)), which is not consistent with a proportional

variation of the inputs.

The profounder mistake, however, of the neoclassical theory of Distribution on the

basis of the marginal productivity of the production factors lies in the fact that it erroneously

extented the law for the reward of a production factor based on its marginal productivity,

which is valid only when this production factor is the sole variable input in the production

while all the other factors remain fixed and with fixed cost, to the case where all the factors of

production vary at the same time, which certainly occurs at least in the long run.

After all of the above, the product exhaustion theorem, which constitutes the core of

the Income Distribution theory, takes now the form (in perfect competition without profit):

p·q = w·L+r·K+.....

accompanied by the condition of inputs' substitution for the minimization of the production

cost: MPL/w = MPK/r = …..

which leads to a relationship between the factors of production (e.g. L=f(K) or L=f(K,q)).

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10. Figures

Figure 1

Long run Equilibrium under Perfect Competition

The long run equilibrium under perfect competition does not take place at the point A (intersection of

total supply and total demand), as the neoclassical theory argues, but at the point E, which maximizes

the profits of firms (as it corresponds to the intersection M of the total supply (marginal cost) and the

marginal revenue MR that comes from the total demand).

Correspondingly at firm level (left graph), the equilibrium does not take place at the minimum point a

of the average cost curve LAC, as the neoclassical theory argues, but at the point e, which maximizes

the profit of firm (as it corresponds to the intersection m of the marginal cost (in the short run SMC

and in the long run LMC) and the marginal revenue mr that comes from the individual demand dd of

the firm). While the individual demand dd of the firm (which is not horizontal, as the neoclassical

theory argues) becomes (by the entry of new firms due to perfect competition and the zeroing of the

economic profit) tangent to the average cost curve LAC at this equilibrium point.

All of them imply unavoidably a monopolistic character of the market even under perfect competition

and zero economic profit, with the consequence for the equilibrium point not to lie at the minimum

point of the average cost curve LAC where the Euler theorem for the product exhaustion could be

applied.

LAC

d

S M

C

e

d

q

p

q

LMC

SAC

m r

SAC

S M

C

D

D

Q Q

M R

p

E

S

S

M

A

pp

m

a

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16

Figure 2

The diagrammatic proof of Chapman for the product exhaustion theorem

Chapman proves by means of a diagram that the marginal product of land in an agricultural

unit which employs L laborers that are remunerated with their marginal product MPL is the remainder

of the total product OMEL after the subtraction of the total labor remuneration OAEL, that is the area

AME, and in this way he proves the product exhaustion theorem. To calculate, though, the marginal

product of land, he did not increase marginally the area of the agricultural unit keeping the number of

its laborers constant, but he considered that a new agricultural unit is added in the industry (consisting

of n agricultural units) and kept the number of laborers of the whole industry constant (L·n).

But by considering the problem in this way, that is by considering the production to consist of

the production of n business units, Chapman has fallen in the error to have taken as production

function a function in which the total production, total labor and total land area are all proportional to

the total number of firms n, that is a function that clearly presents the distinctive feature (definition) of

a homogeneous function of first degree where the proportionate increase of the production factors

increases the production by the same proportion.

O

DC E

LL΄

A B

M

P

MP L

MPL

L

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17

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