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

On Quasi-Homogeneous Production Functions

“…and taking into account that ∑ n i=1 g i = 0, we conclude that f takes the form of Equation (14). Hence, we have Case (a) of the statement.…”

“…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]. This treatment is based on the fact that one can associate a graph hypersurface to any production function and it is remarkable that one can relate basic concepts from production theory with some differential geometric invariants (intrinsic and extrinsic) of the associated hypersurface [26,27].…”

On some geometric properties of quasi-product production models

“…But using ( 21) and ( 22), we obtain that the Hessian matrix of a composite function of the form (11) has the determinant expressed by [4] (38)…”

Isotropic geometry of graph surfaces associated with product production functions in economics

“…then the metric on M 2 induced from I 3 is given by g * = d u 2 1 + d u 2 2 . This implies that M 2 is always a flat space with respect to the induced metric g * .…”

“…which yields g g ′′ = d , d = 0. Considering this in (3.6) implies2 d g ′ g ′′ = 0 and it is a contradiction. Thereby we can rewrite (3.5) as −…”