# Composite functions with Allen determinants and their applications to production models in economics

## DOI:

https://doi.org/10.5556/j.tkjm.45.2014.1569## Keywords:

Production function, Generalized Cobb-Douglas production function, Composite function, Allen elasticity of substitution.## Abstract

In this paper, we derive an explicit formula for the Allen determinants of composite functions of the form:% \[ f\left( \mathbf{x}\right) =F\left( h_{1}\left( x_{1}\right) \times\cdots\times h_{n}\left( x_{n}\right) \right) . \] We completely classify the composite functions by using their Allen determinants. Some applications of Allen determinants to production models are also given.## References

R. G. Allen, J. R. Hicks, A reconsideration of the theory of value Pt. II, Economica,1(1934), 196--219

K. J. Arrow, H. B. Chenery, B. S. Minhas, R. M. Solow, Capital-labor substitution and economic efficiency, Rev. Econom. Stat., 43(1961), 225--250.

B.-Y. Chen, Geometry of submanifolds, M. Dekker, New York 1973.

B.-Y. Chen, On some geometric properties of h-homogeneous production function in microeconomics, Kragujevac J. Math., 35(2011), 343--357.

B.-Y. Chen, G. E. Vilcu, Geometric classifications of homogeneous production functions, Appl. Math. Comput., 225(2013), 345--351.

B.-Y. Chen, Classification of h-homogeneous production functions with constant elasticity of substitution, Tamkang J. Math., 43(2012), 321--328.

B.-Y. Chen, On some geometric properties of quasi-sum production models, J. Math. Anal. Appl., 392 (2012),192--199.

B.-Y. Chen, Geometry of quasi-sum production functions with constant elasticity of substitution property, J. Adv. Math. Stud., 5(2012), 90--97.

B.-Y. Chen, Classification of homothetic functions with constant elasticity of substitution and its geometric applications, Int. Electron. J. Geom., (2012), 67--78.

B.-Y. Chen, An explicit formula of Hessian determinants of composite functions and its applications, Kragujevac J. Math. 36(2012), 27--39.

B.-Y. Chen,A note on homogeneous production models, Kragujevac J. Math., 36(2012), 41--43.

B.-Y. Chen, Solutions to homogeneous Monge-Ampere equations of homothetic functions and their applications to production models in ecenomics, J. Math. Anal. Appl. 411(2014), 223--229.

C. W. Cobb and P. H. Douglas, A theory of production, Amer. Econom. Rev., 18(1928), 139--165.

P. M. do Carmo, Riemannian Geometry, Birkhauser, Boston, 1992.

C. A. Ioan,Applications of the space differential geometry at the study of production functions, EuroEconomica, 18 (2007), 30--38.

L. Losonczi, Production functions having the CES Property, Acta Math. Acad. Paedagog. Nyhai. (N.S.), 26(2010), 113--125.

A. Mihai, M. Sandu, The use of the h-homogeneous production function in microeconomics. Modelling challenges, Revista Economica, 1 (2012), 465-472.

A. Mihai, A. Olteanu, Applied geometry in microeconomics. Recent developments, Land Reclamation, Earth Observation & Surveying, Enverionmental Enginnering, II(2013), 159-166,

S. K. Mishra, A brief history of production functions, IUP J. Manage. Econom., 8 (2010), 6--34.

H. Uzawa, Production functions with constant elasticities of substitution, The Review of Economic Studies, 29(1962), 291-299.

G. E. Vilcu, A geometric perspective on the generalized Cobb--Douglas production functions, Appl. Math. Lett., 24(2011), 777--783.

A. D. Vilcu and G. E. Vilcu, On some geometric properties of the generalized CES production functions, Appl. Math. Comput., 218(2011), 124--129.

A. D. Vilcu and G. E. Vilcu, On homogeneous production functions with proportional marginal rate of substitution, Mathematical Problems in Engineering (2013), doi.10.1155.

## Downloads

## Published

## How to Cite

*Tamkang Journal of Mathematics*,

*45*(4), 427-435. https://doi.org/10.5556/j.tkjm.45.2014.1569