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Advances in Economics and Business 10(1): 1-13, 2022 http://www.hrpub.org
DOI: 10.13189/aeb.2022.100101
A Theory for Building NEO-Classical
Production Functions
1,* 2
Oscar Orellana , Raúl Fuentes
1Department of Mathematics, Federico Santa María Technical University, Chile
2Department of Industries, Federico Santa María Technical University, Chile
Received October 17, 2021; Revised January 14, 2022; Accepted February 8, 2022
Cite This Paper in the following Citation Styles
(a): [1] Oscar Orellana, Raúl Fuentes , "A Theory for Building NEO-Classical Production Functions," Advances in
Economics and Business, Vol. 10, No. 1, pp. 1 - 13, 2022. DOI: 10.13189/aeb.2022.100101.
(b): Oscar Orellana, Raúl Fuentes (2022). A Theory for Building NEO-Classical Production Functions. Advances in
Economics and Business, 10(1), 1 - 13. DOI: 10.13189/aeb.2022.100101.
Copyright©2022 by authors, all rights reserved. Authors agree that this article remains permanently open access under the
terms of the Creative Commons Attribution License 4.0 International License
Abstract In this study, we propose a mathematical production function in relation to the aggregation problem
theory for building neoclassical production functions with usually found in macroeconomics has been a topic of
homogeneous inputs in both aggregate and per capita terms. theoretical and empirical discussion (or debate) in the field
This theory is based on two concepts: Euler’s equation and of economics. Whereas prominent researchers such as
Cauchy’s condition for first-order partial differential Robert Solow, Paul Samuelson, and Franco Modigliani
equations. The analysis is restricted to functions that have defended economic orthodoxy (neoclassical
exhibit constant returns to scale (CRS). For the function to synthesis), other researchers, including Nicholas Kaldor,
meet the law of diminishing marginal returns, we present Joan Robinson, Piero Sraffa, Richard Khan, and Pierangelo
the necessary and sufficient conditions to be satisfied by Garegnani, among others, are advocates of heterodoxy
the curve that defines Cauchy’s condition. In this context, (post Keynesianism). To date, there has been no agreed
we also discuss the Inada conditions. We first present upon result of this debate in relation to our study. Instead,
functions that depend on two inputs and then extend and two almost divergent agendas (schools of economic
discuss the results for functions that depend on several thought) that assign a very different role to the production
inputs. The main result of our research is the provision of a function in their respective methodologies have resulted.
clean and clear theory for constructing neo-classical From the orthodox perspective, the production function is a
production functions. We believe that this result may cornerstone of its methodological construct. For
contribute to closing the huge methodological gaps that heterodoxy, its use is generally avoided. At this point, it is
separate schools of economic thought that defend or reject important to mention that part of the economic heterodoxy
the use of production functions in economics. inherited from the Cambridge controversy categorically
Keywords NEO-Classical Production Functions, rejects both the theoretical and empirical possibility of the
Partial Differential Equations, Euler’s Equation, Cauchy’s existence (use) of the production function, particularly, the
Condition, Homogeneous Inputs aggregated production function. Theoretically, supporters
of this position assert that it is impossible to overcome the
JEL Classification: C02; E13 dimensions, measures, and switching and reswitching
problems of capital identified by Robinson and her
colleagues during the Cambridge debate. Empirically, it is
alluded that production functions in a macroeconometric
1. Introduction analysis hide a simple accounting identity. This type of
Since the famous Cambridge capital controversy1, the criticism originated with Shaik’s (1974) provocative article,
which has been extended more recently by J. Felipe and
1 We refer to the capital theory and production function controversy
that took place among the most important economists from the that was initiated by Joan Robinson’s famous article, “The production
universities of Cambridge (UK) and Cambridge (Massachusetts, USA) function and the theory of capital” (1953-1954).
2 A Theory for Building NEO-Classical Production Functions
J.S.L. McCombie (2014), among others 2 . From an data in the form of a time series, which is used to
economic orthodoxy point of view, it is well known that 7
parametrize the corresponding production function . Once
the production function is one of the key concepts of the protoproduction function is realized, the necessary and
mainstream neoclassical theories. The theory of production sufficient conditions to satisfy the law of diminishing
function is even a substantial part of both pre- and marginal returns are obtained. In terms of noncompliance,
postgraduate micro- and macroeconomics courses. The we inquire into the conditions under which the production
Cobb–Douglas function, which was inspired by the function built up to this stage satisfies the well-known
empirical work of these authors during the 1920s, is by far Inada conditions, which are the usual properties required
the most widely used theoretical and empirical neoclassical for the formulation of per capita production functions.
production function in practice3. The CES function, which Once this objective is achieved, the objective of
was introduced to economics by Arrow, Chenery, Minhas formulating a two-factor neoclassical production function
and Solow (1961), is well-reputed but less used in is also met. Finally, we extend this theory to more than two
neoclassical economic practice4. In contrast, some variable production factors. In each step, we complement our theory
elasticity of substitution (VES) production functions have with examples.
emerged, but their use has remained quite sparse in To clarify, in our article, we do not attempt to support
theoretical and empirical literature 5 . The use of one economic vision or the other. However, it is clear that,
neoclassical production functions in economics is very in part, our work begins to challenge the position of Shaik
diverse, from uses in microeconomics to macroeconomics. and his successors, which denies the theoretical possibility
They have been widely used in microeconomic to study, of the existence of production functions. From a strictly
among other things, a firm’s short term and long term mathematical point of view (which should not be confused
behaviors, and are a cornerstone of the whole theory of with the practical and empirical points of view), it is worth
economic growth, as well as of the theory of international mentioning that our theory is well posed in the sense of
trade, in macroeconomics6. Hadamard. That is, (a) there is a solution, (b) the solution is
However, in our opinion, all of these theories suffer from unique, and (c) the solution continuously depends on the
an important declivity: they presume the theoretical data (in our case from Cauchy’s condition).
existence of the production function. Once this function is The remainder of this paper is given as follows. In
given, different properties and characteristics are studied. Sections II and III, we present the theory for obtaining
This presumption opens a very large space to explore the protoproduction functions for two inputs. In Section IV, we
conditions that are required to generate production obtain the necessary and sufficient conditions required for
functions, which is precisely the objective of our research these functions to satisfy the law of diminishing marginal
agenda. Given the ambitious nature of this agenda, in this returns. In Section V, we present and discuss the
article, we search for the minimum conditions required for relationship between the self-similar solution and the per
a mathematical formulation in the form = capita production function and Inada conditions. In Section
( , , ⋯, , ) to be considered a neoclassical
1 2 VI, we generalize the theory to several independent and
production function. In other words, a formulation that homogeneous production factors. Section VII concludes
satisfies the properties of constant returns to scale (CRS) this work.
and the law of diminishing marginal returns. To achieve
the stated objective, we propose a three-step methodology
restricted to the analysis of homogenous inputs. First, we 2. Basic Definitions
define what we call a protoproduction function, that is, a
two-input function that satisfies both Euler’s equation (EE) In this section, we present, develop, and discuss a
and Cauchy’s condition. This function must first comply mathematical theory of the protoproduction functions
with the CRS property, which is very usual and easy to capable of being represented in the form =
interpret in economic terms. To the best of our knowledge, ( , , ⋯, , ). This function satisfies both EE and
1 2
the second requirement is not used in the current theory of Cauchy’s condition. As applied to economics, EE is
production functions. In this context, we postulate that equivalent to the property of CRS. Lesser known in
Cauchy’s condition represents the necessary economic economics, Cauchy’s condition for first-order partial
differential equations is given by:
2 The link of this identity has a crystalline demonstration in the case of ( , τ , …,τ )
γ τ
the Cobb–Douglas and constant elasticity of substitution (CES) functions. 1 2
( ) ( ) ( )
=F(α ,⋯, , ⋯,α ,⋯, , β , ⋯ , )
3 See Cobb and Douglas (1928). We also suggest reading the recent work 1 1 1 1
by Biddle (2010) for a historical and modern perspective of this function.
4 The acronym CES represents the constant elasticity of substitution The economic intuition underlying Cauchy’s condition
between factors. While the Cobb–Douglas function assumes an elasticity is as follows. Let us consider production functions that
of substitution between productive factors equal to 1, the CES function
allows values that differ from 1. See Mishra (2007) for more technical and depend only on two factors: capital (K) and labor (L). Then,
historical details on the CES production function.
5 See for example Revankar (1971), Karagiannis et al. (2005) and Tach
(2020).
6 For readers unfamiliar with this subject, it is highly recommended to 7 Additionally, we will show that from this condition we can build
visit the works by Barro and Sala-i-Martin (2003), Mas-Colell, Whinston theoretical support for the problem of aggregation found in
and Green (1995) and Krugman and Obstfeld (2002). macroeconomics, which is a very attractive result of our theory.
Advances in Economics and Business 10(1): 1-13, 2022 3
assume the production activity of a particular industry, fall on the manifold data given parametrically by ⃗ =
sector or country can be represented by a production
function in the form Y=F(K,L). Cauchy’s condition is ⃗(1, 2, ⋯ , ) above. From a purely theoretical point of
( ) ( ) ( ) view, this theory works well, and we present some
reduced to () = ( , ), where (), and examples. However, from a practical point of view, we
[ ]
r ∈ 0,
() are functions of the unique paramete now visualize that there are serious empirical difficulties in
(instead of n-parameters 1, 2, ⋯ , 8 ). Hence, if we applying this theory in practice, except for production
interpret τ as time, then α(τ), β(τ) and γ(τ) can be activities that depend on only two independent and
interpreted as continuous time series representations of homogeneous production factors. One epistemologically
capital, labor and output data, respectively, of the given interesting consequence of the theory proposed in this
industry, sector or country. The production function study is that it allows us to define the protoproduction
which is a function of the reduced form of function of a given production activity with CRS as one of
= (, ), ( ) ( ) the two following equivalent forms:
Cauchy’s condition () = ( , ) of two Definition II.1. Given that ≠ 0, a function of the
independent production factors, must go through the ( )
( ) ( ) ( ) form = , , ⋯ , , is a proto-production
3-dimensional data curve , , ∶ ∈ 1 2
[ ] function if it has a parametric representation of the form.
0, obtained from the given production activity. In other
words, the corresponding production function that Then,
represents the given production activity must fully absorb = ( , , ⋯ , , )) = ( , , ⋯ , )
the data curve associated with such production activity. 1 1 1 2 1 1 2
= ( , , ⋯ , , )) = ( , , ⋯ , )
The problem of obtaining the continuous time series 2 2 1 2 ⋮ 2 1 2
representations of capital, labor and output is a matter of = ( , , ⋯ , , )) = ( , , ⋯ , ) (II.1)
econometric and curve adjustments. We make some 1 2 1 2
comments on this in the next sections of this paper. For =(1, 2, ⋯ , , )) = (1, 2, ⋯ , )
now, it is important to remark that if the following = (1, 2, ⋯ , , )) = (1, 2, ⋯ , )
determinant of a Jacobian matrix: { }
⋯ where 1, 2, ⋯ , , are the production input factors,
� 1 2 � Y is the production output, and
⋯
1 1 1 1
( , , ⋯, , , )( , , ⋯ , ) is the parametric
1 2 1 2
1 2 representation of an n-dimensional manifold that
= � 2 2 2 2 �
⋮ ⋮ ⋯ ⋮ ⋮ represents the empirical data of a given production activity.
� 1 2 � Definition II.2. Given that ≠ 0, a function of the form
( , , ⋯, , ) will be called a protoproduction
=
( , ⋯, ) ( , ⋯, ) ( , ⋯, ) ( , ⋯, ) 1 2
1 1 2 1 … 1 1 function if it satisfies the following conditions:
differs from zero, then the Cauchy problem for EE has a + +⋯+ + = (II.2)
unique solution. Moreover, the solution depends 1 2
continuously on the data manifold given parametrically ( 1 ) 2
by: , , ⋯,
1 2( ) ( ) ( (II.3)
=( , , ⋯, , ⋯, , , ⋯, , ,
( ) 1 1 2 1 2 1
⃗ =⃗ 1, ⋯ ,
=( ( , ⋯ , ), ⋯ , ( , ⋯ , ), ( , ⋯ , ), ( , ⋯ , ))
1 1 1 1 1 From (II.1), we can determine the corresponding
( ) ( ) ( ) ( ) restrictions that , , ⋯ , , , and their respective first
which reduces to ⃗ = ⃗ =( , , ) in the 1 2
case of two production factors. Therefore, under the and second derivatives must meet so that the
hypothesis that ≠ 0, Cauchy’s problem for EE is a protoproduction function satisfies the law of diminishing
well-posed problem in the sense of Hadamard9. Hence, marginal returns. Additionally, because ≠ 0, it follows
these are the logical mathematical conditions (abstract and that the first (n + 1) relations of (II.1) are locally invertible.
ideal) for a theory of the ‘homogeneity of degree one’ Therefore, (II.1) is the parametric representation of a
function in the form = ( , , ⋯ , , ) . Another
(HDO) protoproduction functions. 1 2
Moreover, given the solution of the Cauchy problem for interesting consequence of this theory is that because of
EE under the assumption that ≠ 0, we deduce the Cauchy’s condition (II.3), the problem of aggregation
additional conditions that the protoproduction function found in macroeconomics is given some theoretical
must satisfy to comply with the law of diminishing support.
marginal returns. As may be expected, these conditions
3. Euler’s Theorem and Some
8 In this context, T represents the period of time in which the historical Economic Consequences
time series data is registered.
9 The mathematical term “well-posed problem” stems from a definition ( )
given by Jacques Hadamard. He believed that mathematical models of Consider a function in the form = , , where Y is
physical phenomena should have the following properties: a solution the dependent variable that represents the total production
exists, the solution is unique, and the solution’s behavior changes (output), and K and L are the independent variables that
continuously with the initial conditions.
4 A Theory for Building NEO-Classical Production Functions
represent capital and labor, respectively. From a smooth assumptions) that the integral surface = (,)
mathematical point of view, K, L, and Y are continuous is composed of the so-called characteristic curves that
homogeneous variables. More precisely, we begin with a satisfy the following system of ordinary differential
continuous differentiable function represented by: equations and initial conditions:
+ + +
∶ ⊆ ℝ × ℝ → ℝ
0 ⎧ ( ) ( )
(, ) → = (, ) (III.1) ⎪ = , ℎ ℎ , 0 =()
⎪ ( ) ( ) (III.5)
( ) = , ℎ ℎ , 0 =()
where N is the appropriate domain of = , . As is ⎨
well known in micro- and macroeconomics, one of the very ⎪
usual properties attributable to (III.1) is that it satisfies the ⎪ ( ) ( )
so-called CRS condition. This means that the function ⎩ = , ℎ ℎ , 0 =()
(III.1) has to be HDO, namely: By integrating (III.5), we obtain:
( ) ( )
( ) ( ) (III.2) = , =
, = , = ⎧
which can be proven to be equivalent to the first-order ⎪ ( ) ( ) ∈ [0, ]
linear partial differential equation known as EE. Hence, ⎨ = , = (III.6)
( ) ( ) ⎪ ( ) ( )
under our assumptions we have: ⟺ ⟺ ⎩ = , =
(). The proof of Euler’s theorem is well known and From the implicit function theorem (for two equations),
requires no further details here. More importantly, for our it follows that if the determinant of the Jacobian matrix:
purposes, from Euler’s theorem, we have two a priori (,)
alternatives to solve the problem of determining the �(,) � differs from zero, then the first two equations
( ,0)
function = (, ) . The first strategy (the standard 0
method) is to solve the ‘Cauchy problem of EE’, which, in in (III.6) are locally invertible and = (, ) and
formal terms, is set as follows: = (, ) are defined in the neighborhood of the
( ) ( ) ( ) ( )
⎧ ( ) operation point: 0, 0, 0 =� 0 , 0 , 0 �
+ =, ℎ = , , and such that [ ]
⎪ for some 0 ∈ 0, ], where 0 = (0, 0) ,
⎪ 0 =(0, 0), and 0 = (0, 0).
( ) ( ) ( ) [ For problem (III.3) and considering (III.5), the condition
⎨ =� , � ℎ ∈ 0,), ℎ (III.3) (,)
⎪ ( ) ( ) ( ) ( ) � � ≠0 reduces to:
⎪ ⃗ =⃗ =� , , � (,) ( ,0)
⎩ ℎ . 0
The second strategy is to solve EE requiring additional (0, 0) (0, 0) ′(0) (0)
properties for its solution. For example, Y = F (K, L) = ⎛ ⎞=� �
satisfies the law of diminishing marginal returns for each ⎜ (0, 0) (0, 0)⎟ ′(0) (0) (III.7)
factor of production separately. In this case, the problem ′ ⎝ ′ ⎠
( ) ( )
can be stated as follows: = 0 0 − (0)(0)
( ) ( )
⎧ + = , ℎ = , , From (III.7), it follows that EE at the origin , =
⎪ ( )
and such that (III.4) 0,0 is singular (i.e., = 0). This means that in the
neighborhood of the origin ( ) ( )
⎨ 2 2 , = 0,0 , we cannot
⎪ >0, >0, <0, <0 ( ) ( )
⎩ 2 2 invert the relations = , and = , ;
therefore, there is no integral surface of the form =
However, problems (III.3) and (III.4) are compatible. In ( ) (, )=(0,0).
fact, in section III it is shown that the solution of problem , in the neighborhood of the origin ⃗ ⃗
( )
(III.3) can satisfy (for example) the law of diminishing However, if the projection curve = =
( ) ( ) ( )
marginal returns if the component of the parametric curve � , � does not pass through the origin , =(0,0),
( ) ( ) ( ) ( ) condition (III.7) guarantees the existence and uniqueness
⃗ = ⃗ =� , , � and its first and second
of the solution of problem (III.3) in the neighborhood of
derivatives satisfy certain conditions. Next, we solve the point of operation = ( , , ) =
problem (III.3) by following the standard mathematical 0 0 0 0
method (see [1]) and make some comments. Let us first ( ) ( ) ( )
� , , . Geometrically, problem (III.3) has
assume that the integral surface (production function) 0 0 0 �
= (, ) can be represented parametrically by a unique solution in the neighborhood of the operation
point ( ) ( ) ( ) ( )
= , , =� , , � if the
( ) ( ) 0 0 0 0 0 0 0
= , ; = , ; = (, ), where and t are ( ) ( ( ) ( ) )
[ curve P, defined by ⃗ = ⃗ = , , () , is
only parameters for the time being. However, ∈ 0,). transversal to the characteristic curves that compose the
Using a geometrical interpretation of the first-order partial integral surface (production function) = (,) or the
differential equation, it can be proven (under suitable
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