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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: d.nomidis@yahoo.com
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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