Classification of $h$-homogeneous production functions with constant elasticity of substitution

Abstract

Abstract. Almost all economic theories presuppose a production function, either on the firm level or the aggregate level. In this sense the production function is one of the key concepts of mainstream neoclassical theories. There is a very important class of production functions that are often analyzed in both microeconomics and macroeconomics; namely, h-homogeneous production functions. This class of production functions includes two important production functions; namely, the generalized Cobb-Douglas production functions and ACMS production functions. It was proved in 2010 by L. Losonczi [12] that twice differentiable two-inputs h-homogeneous production functions with constant elasticity of substitution (CES) property are Cobb-Douglas' and ACMS production functions. Lozonczi also pointed out in [12] that his proof does not work for production functions of n-inputs with n > 2.In this paper we settle this classification problem completely by classifying all h-homogeneous production functions which satisfy the CES property. More precisely, we prove that, for arbitrary number of inputs, the only twice differentiable h-homogeneous production functions satisfying the CES property are the generalized Cobb-Douglas production functions and the generalized ACMS production functions. IntroductionIn microeconomics and macroeconomics, a production function is a positive nonconstant function that specifies the output of a firm, an industry, or an entire economy for all combinations of inputs. Almost all economic theories presuppose a production function, either on the firm level or the aggregate level. In this sense, the production function is one of the key concepts of mainstream neoclassical theories. By assuming that the maximum output technologically possible from a given set of inputs is achieved, economists using a production function in analysis are abstracting from the engineering and managerial problems inherently associated with a particular production process.Let R denote the set of real numbers. Let us put R + = {r ∈ R : r > 0} and R n + = {(x 1 , . . . , x n ) ∈ R n : x 1 , . . . , x n > 0}.2010 Mathematics Subject Classification. Primary 90A11; Secondary 91B64.

On Quasi-Homogeneous Production Functions

“…where F is a twice differentiable real valued function of n − 2 variables and " " over u k indicates that u k is omitted. Replacing now Equation (21) into Equation (17), and taking account of Equation (18), we derive that f takes the form of Equation (10). Hence, we have Case (b) of the statement.…”

“…, n}, i = j. Next, from the condition E x i = k i , we deduce from i. that f reduces to Equation (10). Imposing now the conditions E x r = k r , ∀r ∈ {1, .…”

“…Then, we suppose that g i = 0 and g j = 0, for all j = i. Now, from the condition E x i = k i , we deduce from i. that k i = q g i and f reduces to Equation (9). Imposing now the condition E x j = k j , ∀j = i, and taking account of Equations (1) and (9), we get the following system of PDEs…”

“…From this point of view, the main indicators that characterize this behavior are the elasticity of production with respect to an input, the marginal rate of technical substitution and the elasticity of substitution between two inputs [1,2]. The characterization of the production models with constant elasticity of production, with proportional marginal rate of substitution (PMRS) property and with constant elasticity of substitution (CES) property is a challenging problem [3][4][5][6][7] and several classification results were obtained in the last years for different production functions, such as homogeneous, homothetic, quasi-sum and quasi-product production functions [8][9][10][11]. Other notable results concerning the above production models were recently derived using a differential geometric approach [12][13][14][15][16][17][18][19][20][21][22][23][24][25].…”

Hessian determinants of composite functions with applications for production functions in economics

“…Chen's papers [7,8]. In addition, the Hessian matrices have an important geometric interpretation as following.…”