[Paper Review] Geometry of quasi-sum production functions with constant elasticity of substitution property

Bang‐Yen Chen|arXiv (Cornell University)|Jul 15, 2013

Process Optimization and Integration10 references22 citations

TL;DR

This paper classifies quasi-sum production functions with constant elasticity of substitution (CES) and proves that their graphs have vanishing Gauss-Kronecker curvature (i.e., are flat) if and only if they are either linearly homogeneous generalized ACMS or linearly homogeneous generalized Cobb-Douglas functions. The study connects geometric curvature properties in differential geometry to economic production function structures, offering a differential-geometric characterization of key microeconomic production models.

ABSTRACT

A production function $f$ is called quasi-sum if there are strict monotone functions $F, h_1,...,h_n$ with $F'>0$ such that $$f(x)= F(h_1 (x_1)+...+h_n (x_n)).$$ The justification for studying quasi-sum production functions is that these functions appear as solutions of the general bisymmetry equation and they are related to the problem of consistent aggregation. In this article, first we present the classification of quasi-sum production functions satisfying the constant elasticity of substitution property. Then we prove that if a quasi-sum production function satisfies the constant elasticity of substitution property, then its graph has vanishing Gauss-Kronecker curvature (or its graph is a flat space) if and only if the production function is either a linearly homogeneous generalized ACMS function or a linearly homogeneous generalized Cobb-Douglas function.

Motivation & Objective

• To classify quasi-sum production functions that satisfy the constant elasticity of substitution (CES) property.
• To investigate the geometric condition under which the graph of such a production function has vanishing Gauss-Kronecker curvature.
• To establish a necessary and sufficient condition linking the geometric flatness of the graph to the functional form of the production function.
• To demonstrate that only two specific classes of production functions—linearly homogeneous generalized ACMS and linearly homogeneous generalized Cobb-Douglas—satisfy both the CES property and flat graph curvature.
• To provide a differential-geometric characterization of fundamental economic production models using curvature invariants.

Proposed method

• The analysis begins with the general form of a quasi-sum production function: $ f(\mathbf{x}) = F(h_1(x_1) + \cdots + h_n(x_n)) $, where $ F' > 0 $ and each $ h_i $ is strictly monotone.
• The constant elasticity of substitution (CES) property is imposed via the Hessian-based elasticity formula $ H_{ij}(\mathbf{x}) = \sigma $, a constant for all $ i \neq j $.
• The Gauss-Kronecker curvature of the graph of $ f $ is computed using the Hessian determinant and first derivatives, leading to a curvature condition involving $ F'' $ and $ F' $.
• A differential equation is derived for $ F $ by equating the Gauss-Kronecker curvature to zero, which is solved under the CES constraint.
• The solution is analyzed under two cases: homothetic generalized ACMS and homothetic generalized Cobb-Douglas functions.
• The proof uses substitution and transformation techniques to reduce the curvature condition to solvable ODEs, yielding explicit functional forms for $ F $.

Experimental results

Research questions

• RQ1 Which quasi-sum production functions with constant elasticity of substitution (CES) have graphs with vanishing Gauss-Kronecker curvature?
• RQ2 What geometric condition on the graph of a quasi-sum CES production function implies flatness (zero Gauss-Kronecker curvature)?
• RQ3 Are there only specific functional forms that satisfy both the CES property and zero curvature in the graph?
• RQ4 Can the class of linearly homogeneous generalized ACMS and Cobb-Douglas functions be uniquely characterized by their curvature properties?
• RQ5 Is the flatness of the graph equivalent to the production function being either a linearly homogeneous generalized ACMS or a linearly homogeneous generalized Cobb-Douglas function?

Key findings

• The graph of a quasi-sum production function has vanishing Gauss-Kronecker curvature if and only if it is a linearly homogeneous generalized ACMS function of the form $ f(\mathbf{x}) = \gamma \left( \sum_{i=1}^n a_i^\rho x_i^\rho \right)^{1/\rho} $ with $ \gamma, a_i, \rho \neq 0 $.
• The graph has vanishing Gauss-Kronecker curvature if and only if it is a linearly homogeneous generalized Cobb-Douglas function of the form $ f(\mathbf{x}) = \gamma x_1^{\alpha_1} \cdots x_n^{\alpha_n} $ with $ \sum_{i=1}^n \alpha_i = 1 $ and $ \gamma, \alpha_i \neq 0 $.
• For a two-input quasi-sum function of the form $ f = F(x_2/x_1) $, the Gauss-Kronecker curvature cannot vanish unless $ F' = 0 $, which contradicts the strict monotonicity of $ F $, so such functions are excluded from the flatness condition.
• The differential equation $ F'(u) = (\sigma - 1)u F''(u) $ arises from the curvature condition and leads to a power-law solution $ F(u) = \alpha u^{\sigma/(\sigma-1)} $, characterizing the generalized ACMS form.
• For the generalized Cobb-Douglas case, the curvature condition yields $ F'(u) = \gamma u^{1/\alpha - 1} $, leading to $ f(\mathbf{x}) = \gamma (x_1^{\alpha_1} \cdots x_n^{\alpha_n})^{1/\alpha} $, which reduces to a linearly homogeneous form when $ \sum \alpha_i = 1 $.
• The converse holds: both the generalized ACMS and generalized Cobb-Douglas functions with degree of homogeneity one have flat graphs, confirming the characterization.