Solution of the system of nonlinear PDEs characterizing CES property under quasi-homogeneity conditions

Abstract

The constant elasticity of substitution (CES for short) is a basic property widely used in some areas of economics that involves a system of second-order nonlinear partial differential equations. One of the most remarkable results in mathematical economics states that under homogeneity condition i.e. the production function is a homogeneous function of a certain degree, there are no other production models with the CES property apart from the famous Cobb–Douglas and Arrow–Chenery–Minhas–Solow production functions. In this paper we generalize this classification result to a much wider framework of production functions under quasi-homogeneity conditions, showing in particular the existence of three new classes of production models with the CES property.

Classification of quasi‐product production models with minimal isoquants

“…Then, it follows immediately from (1), (14), and (S1) that the production function 𝑓 is the one given at point (a) in the statement of the theorem.…”

“…Since production functions with $$$ n $$$ inputs can be naturally identified with hypersurfaces of the $$$ \left(n+1\right) $$$‐dimensional Euclidean space (see [2, 3]), the study of production models by means of differential geometric tools has become a fervent topic in recent years (see, e.g., [4–7]). Therefore, we can find many geometric classification results for the basic production models utilized in the economic analysis, namely, homogeneous [8, 9], quasi‐sum [10, 11], homothetic [12, 13], quasi‐homogeneous (or weighted homogeneous) [5, 14], and quasi‐product production models [15, 16]. The proofs of these classification results are highly technical, generally involving a combination of techniques from mathematical analysis, differential geometry, ODE, and PDE.…”

On the minimality of quasi‐sum production models in microeconomics

“…, where d 0 , d 1 are positive constants and a 2 , a 3 , c 2 are nonzero constants; (8) The functions G and X i are given by G = a 0 cosh (kw) + b 0 sinh (kw), X i = a i cosh (ku i ) + b i sinh (ku i ), 1 ≤ i ≤ 3, where a i , b i (0 ≤ i ≤ 3) satisfy the relations (4.67) and k > 0; (9) The functions G and X i are given by G = a 0 cos (kw) + b 0 sin (kw), X i = a i cos (ku i ) + b i sin (ku i ), 1 ≤ i ≤ 3,…”

“…Therefore, both the generalized CD production function and the generalized ACMS production function have the CES property (c.f. Alodan et al 8 and Chen 9 ).…”

Pattern Formation of a Bubbly Fluid Mixture under the Effect of Thermodynamics via Kudryashov‐Sinelshchikov Model

“…Many natural phenomena are determined from NLPDEs of integer order. ese models are used in numerous disciplines of research such as bio-sciences, engineering, and economics [11][12][13]. However, these integer-order models are insufficient without the nonlocal property.…”