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Abstract. This note proves that the representation of the Allen elasticity of substitution obtained by Uzawa for linear homogeneous functions holds true for nonhomogeneous functions. It is shown that the criticism of the Allen-Uzawa elasticity of substitution in the works of Blackorby, Primont, Russell is based on an incorrect example.
Key words: elasticity of substitution, nonhomogeneous production functions.
2010 Mathematics Subject Classification: 91B02, 91B38.
In the book by Allen the concept of the two-variable elasticity of substitution (ES) proposed by Hicks has been extended to an arbitrary number of factors. Later, (the books by Allen and Hicks were first published in 1938 and 1932 respectively) Uzawa noticed that for homogeneous functions of degree $k=1$, known as linear homogeneous functions, the value of Allen elasticity of substitution (AES) can be expressed in terms of the cost function and its derivatives. The vector of the input factors $x\in \mathbb{R}^{n}_{++}$ from Allen's definition needs to be linked to the price vector $p\in \mathbb{R}^{n}_{++}$ from Uzawa's form of AES in such a way that the minimum cost of input factors for the given level of output $y>0$ is attained at the given point $x$.
In his paper Uzawa solves the problem of describing the class of all linear homogeneous functions with constant AES. Representation of AES for such functions through the cost function is an auxiliary result of this article, achieved by means of certain relationships established by Allen for linear homogeneous functions. Specifically, on page 292 of his paper Uzawa references page 598 of Allen's book containing the equality $F_0 = 0$, which is true for linear homogeneous functions, but does not hold even for homogeneous functions of degree $k=2$.
We are not aware of any publication where Uzawa's form for the AES is provided with proof for nonhomogeneous functions. The main purpose of this article is to present such proof.
Section (ref) of this paper contains key definitions, as well as the concepts from mathematical analysis used in later sections. Some lemmas and proofs from linear algebra have been relegated to appendices.
In section (ref), we formulate and prove the main result of the paper on Uzawa's form of AES.
In section (ref), it is shown that in examples from Blackorby1989882, Blackorby2007203 the production function for all points of $\mathbb{R}^{3}_{++}$ either has a singular bordered Hessian or has no partial derivatives. Thus, for such a function AES is not defined at all. Interestingly, the cost function is nevertheless defined for all $p \in \mathbb{R}^{3}_{++}$ and belongs to the class $C^{\infty} $.
The final section (ref) references some publications that discuss variations of ES which differ from AES.
If $f\colon X \subset \mathbb{R}^n \to \mathbf{R}$ is a function of class $C^2$ in a neighborhood of some point $x$, then we denote by $f_i(x)$ and $f_{ij}(x)$ the first derivatives with respect to $x_i$, $i = 1,\dots,n$, and second order derivatives for the pairs of variables $x_i,x_j$, $i,j =1,\dots,n$, at that point. As usual, the gradient of $f$ at the point $x$ is a vector $\nabla f(x)$ of all first-order partial derivatives $f_i(x)$, the Hessian matrix $H_f(x)$ of $f$ at the point $x$ consists of elements $f_{ij}(x)$. The bordered Hessian $f$ at the point $x$ is the matrix of size $(n + 1)\times (n + 1)$ of the following form
The determinant of ((ref)) is denoted by $F$; the cofactor of any element $f_{ij}$ in this matrix will be denoted by $F_ {ij}$. Allen assumes the positivity of all the first partial derivatives and the alternation of the signs of the leading principal minors $\Delta_2, \dots, \Delta_{n + 1}$ of the bordered Hessian matrix, which is equivalent to requiring the strong concavity of $f$ (see. \Citet[p. 414] {Diewert1981397}). For our purposes it is sufficient to have a single requirement that $\Delta_{n + 1} =F$ is nonzero.
Following Allen, the partial elasticity of substitution of factors $i$ and $j$ (or simply AES of factors $i$ and $j$) is the number
The bordered Hessian matrix is symmetric for functions of class $C^2$. Therefore, $\sigma^ A_{ij} =\sigma^ A_{ji}$ for all $i,j = 1,\dots, n$.
In the case of two variables, the definition of ((ref)) is consistent with the definition of the elasticity of substitution introduced by Hicks (HES). Indeed, in this case $\sigma^A_{12}(x_1,x_2)$ equals
that matches the formula for the HES in Allen up to notation.
In Blackorby1981147, Blackorby1989882, Blackorby2007203 one of the advantages of the Morishima elasticity of substitution (MES) is considered to be the possibility to represent MES as the so-called logarithmic derivative.
Preceding the discussion of this definition in section (ref), let us point to the possibility of such representation of $\sigma^A$ in the case of two factors.
At the level curve $f (x_1, x_2) = y$, where $y$ is a predetermined number, provided $f_1\neq 0$ it is possible to express variable $x_1$ as the function of class $C^2$ of variable $x_2$ per the implicit function theorem. While the derivative $\frac{\partial x_1}{\partial x_2}= - \frac {f_2}{f_1}$ can be treated as the rate of substitution of variable $x_1$ for variable $x_2$ consider the logarithmic derivative, which Hicks calls the elasticity of substitution
The formula ((ref)) can be obtained from the formula ((ref)) using the implicit function theorem for the following system of equations
In vector notation, this is the equation of the form $\Phi(u, v, x_1, x_2)=0$. Columns of the Jacobi matrix of function $\Phi$
corresponding to the partial derivatives with respect to $u, x_1, x_2$, form the matrix of non-zero minor of order $3$. It is immediately verified that the
where $F$ is the determinant of the bordered Hessian.
Here and later in this paper, we need to apply the implicit function theorem \footnote{See. \Citet[p. 490]{Zorich1}} for equations of the form $f(x,y)=0$, where $f\colon M\subset \mathbf{R}^n\times \mathbf{R}^m\to \mathbf{R}^m$, $f\in C^k, k\ge 1$. In such cases it is convenient to represent Jacobi matrix of $f$ as composed of two blocks $D_x f$ and $D_y f$, where $D_x f$ consists of the first $n$ columns of the Jacobi matrix and $D_y f$ consists of the last $m$ columns.
Therefore, in the equation $\Phi(u,v,x_1,x_2)=0$ the vector of variables $(u,x_1,x_2)$ can be expressed through the variable $v$ and
Hence, the expression for $D_v u$ obtained here matches ((ref)).
The function $f\colon X\subset \mathbf{R}^n\to \mathbf{R}$ is called homogeneous of degree $k\in \mathbf{R}$ if for all numbers $t>0$ and any $x\in X$ $tx\in X$ and $f(tx)=t^k f(x)$. If the equality $f(tx)=t^k f(x)$ holds for all numbers $t$ in some neighborhood of $1$ in some neighborhood of $x$, then the function is called locally homogeneous of degree $k\in \mathbf{R}$ function in the neighborhood.
For homogeneous and locally homogeneous degree $k$ functions of class $C^1$ the Euler identity is true $$ x_1f_1(x)+\dots+x_nf_n(x)=kf(x). $$ The partial derivatives of these functions are homogeneous or locally homogeneous functions of degree $(k-1)$.
Homogeneous degree $k = 1$ functions are often called linear homogeneous functions. Euler's identity for the first derivatives of linear homogeneous functions of two variables $x_1f_{11}+x_2f_{12}=0$, $x_1f_{21}+x_2f_{22}=0$ allows to convert the formula ((ref)) to the form
Allen notes that it is in the form ((ref)) that the elasticity of substitution was first defined by Hicks.
The function
is the Lagrange function of the problem of minimizing $\sum\limits_{i=1}^n p_ix_i$ subject to $f (x)-y=0$ where the number $y$ and vector $p\in \mathbf{R}^n_{++}$ are given parameters. Critical points of the Lagrange function ((ref)) are defined as the solutions $(x, \lambda)$ of equations $$ \left\{
\right. \quad or \quad \left\{
\right. $$
For any point $\bar{x}\in \mathbf{R}^n_{++}$ at $p_i=f_i(\bar{x})$ and $y=f(\bar{x})$ the point$(\bar{x},\bar{\lambda})$, where $\bar{\lambda}=-1$, is a critical point of the Lagrange function ((ref)).
For the proof of Theorem ((ref)) and Corollary ((ref)) see (ref).
In Blackorby1981147,Blackorby1989882,Blackorby2007203 Morishima elasticity of substitution \footnote {The article Morishima1967144 was never translated from Japanese, Blackorbi and Russell argue that they independently came to the MES in 1975} (MES) is defined, similarly to Uzawa form of AES, using the cost function $C(p, y)$ according to formula ((ref)). By the envelope theorem $C_i=x_i$, $C_j=x_j$, and $p_j/p_i=f_j/f_i$ at critical points of the problem of minimizing the cost of a given level of output $y$ at factors' prices $p$. Therefore, it may seem that the logarithmic derivative ((ref)) is calculated anew from the definition of HES. However, this is not true in the case of three or more input factors, as $C(p, y)$ is assumed to be the extreme value of the problem with the full range of factors $x$ and full range of prices $p$. The use of such a function $C(p,y)$ makes keeping relation $f(x) = y$ in calculating the $\sigma^M_{ij}$ unnecessary.
The logarithmic derivative ((ref)) allows for different interpretations, leading to different values of the derivative.
We apply the implicit function theorem to the system of equations
This is a vector equation of the form $\Phi(u,v,p,y)=0$, where Jacobi matrix of the mapping $\Phi$ contains the matrix $ \left(
\right) $ in the first two columns, while the remaining columns have zeros in the second row, except for the columns of the partial derivatives with respect to the variables $p_i$ and $p_j$. To give meaning to differentiation of $u$ with respect to $v$, the matrix of non-zero minor in the implicit function theorem should consist of a pair of columns for the variables $u, p_i$ or the variables $u, p_j$.
In the first case the variables $u, p_i$ can be represented as functions of class $C^1$ on the other variables of this problem, and
Hence
In the second case,
If there are only two inputs, using linear homogeneity of $C$ in $p$ and applying Euler's identity it is possible to transform ((ref)), ((ref)) into the Uzawa form of AES. For three or more factors the values of ((ref)) and ((ref)) are, generally speaking, different. In Blackorby1981147,Blackorby1989882,Blackorby2007203 Blackorbi, Primont and Russell believed $\sigma^M_{ij}$ to be equal to the value ((ref)). Thus, unlike AES, MES is not symmetric. MES also can not be considered as a true logarithmic derivative. It is inaccurate to indicate the representation of MES as a logarithmic derivative as an advantage of MES over AES.
Here is an example of a true logarithmic derivative
Hence, MES can be represented as the difference $\varepsilon_{ji}-\varepsilon_{ii}$ between two logarithmic derivatives with respect to two different variables.
The inference in Blackorby1989882, Blackorby2007203 that AES for three or more factors does not possess any of the essential properties of the original concept of HES is based on the examples of the production function
in Blackorby1989882 and a similar function in Blackorby2007203. AES is calculated in Uzawa's form using the cost function $C(p_1,p_2,p_3,y)=y\left(p_1+2p_2^{1/2}p_3^{1/2}\right)$ corresponding to the function ((ref)).
The result is an example of a function $f$, for which Uzawa's form of AES is defined, but the original AES is not. The proposed function has a singular bordered Hessian in the majority of the points $x\in \mathbf{R}^3_{++}$, and does not have even the first partial derivatives in points with $x_1^2=x_2x_3$. The minimum of $p_1x_1+p_2x_2+p_3x_3$ on the surface $f(x_1,x_2,x_3)=y$ is achieved at the points of the curve, defined by the system of equations
Bordered Hessian for this problem is nonsingular $$ \left|
\right|=-2x_2x_3. $$ Therefore, the minimum value function is subject to the envelope theorem. It is immediately verified that $C_1^2=C_2C_3$ for all $(p,y)$, i.e. the minimum is attained on the curve ((ref)).
Criticism of a deeply rooted belief \footnote{For example, Varian wrote on the page 13 of the textbook Varian “The elasticity of substitution measures the curvature of an isoquant”, and further “This is a relatively natural measure of curvature”. It is stated in the book Intriligator on page 182 that “The elasticities of substitution characterize the curvature of the isoquant”.} that AES is a measure of the curvature of the isoquant $f(x)=y$ can be found in DeLaGrandville199723. It is demonstrated using simple examples of the two-variable Cobb-Douglas functions which, is well known to have the AES constant and equal to one along the isoquant, while the curvature can vary widely. On the other hand, any curve with a constant curvature (i.e., an arc of a circle) is not a level curve of a CES-function. This indicates that a parallel shift of the isoquant along one of the axes does not change the curvature, but changes the AES. Furthermore, AES is not dependent on the units for each factor, whereas the curvature is.
Since AES and MES match HES in a case of two variables, the aforementioned statement about the curvature of the isoquant is valid for MES. In Blackorby1989882 $\sigma^M_{12}=0$ and $\sigma^M_{21}=1/2$ are calculated for function ((ref)), but it is not stated how this relates to any curvature. Here the surface $f(x_1,x_2,x_3)=y$ is a union of the sets $\{(x_1,x_2,x_3)\in \mathbf{R}^3_{++}\colon x_1\le y,x_2x_3=y^2\}$ and $\{(x_1,x_2,x_3)\in \mathbf{R}^3_{++}\colon x_1= y,x_2x_3\ge y^2\}$ with common points on the curve ((ref)).
The concept of elasticity of substitution was introduced into economic theory in the early 1930s (see Molina). Constant elasticity of substitution (CES) functions were developed and studied by renowned economists Hicks, Arrow, McFadden63.
The research of possibilities to extend HES to the case of many factors at the expense of “freezing” the value of the output, all but two of the prices and production factors, resulted in the introduction of concepts of DES (Direct partial elasticity of substitution) and SES (Shadow partial elasticity of substitution) in McFadden63. The description of the classes of functions with a constant DES and a constant SES derived by McFadden revealed the need to find other measures of substitution, accomodating a wider class of production functions with constant ES.
One of the reasons for AES modification is associated with the transition from the problem of minimum cost to the problem of maximum profit. If $p$, as before, is the vector of input prices and $p_y$ is the output price then per Bertoletti2005183, the Hotelling-Lau elasticity (HLES) is defined as
This measure does not require the constancy of production or any other parameter that depends on the number of factors or prices. Such ES is called gross ES (the opposite being net ES). Examples of a net ES are AES and MES; gross ES version of MES (MGES) is introduced in Davis1996173
In Blackorby2007203 there is a proof that MES at the point $(p,y)$ is identical for all the pairs of factors to MGES at the point $(p,p_y)$, where $y=f(x(p,p_y))$ is the profit-maximizing output, if and only if the production function is homothetic, i.e. it is a composition of linear homogeneous function with an increasing external function. The authors again do not impose requirements on the smoothness of $f$ and nondegeneracy of $H_f$. In the example ((ref)) the values of $\pi(p,p_y)$ are defined only at $p_y=p_1+2\sqrt{p_2p_3}$ and are equal to zero.
In their review papers Stern2004,Stern2011,Mundra mention more than ten variations of ES or the elasticity of complementarity (EC), and describe the results of estimates of some of them using the four factors data set. It turns out, for example, that for some pairs of factors, AES and MES have opposite signs: complementary factors according to AES are substitutes according to MES. It is appropriate to quote from Stern2004 `There are many different legitimate definitions of the ES and the elasticity of complementarity (EC). None of these is the one true ES --- which one is useful depends on what we wish to measure. As their value and even sign can vary dramatically, the choice of the appropriate indicator is important'.