Some properties of a production function
Constantin Chilarescu
TL;DR
The paper investigates a production function with variable elasticity of substitution, specified as $f(k)=A k^\theta [\alpha k^\psi + \beta]^\omega$, to determine when elasticity $\sigma(k)$ can be below one and to establish Inada conditions under reasonable parameter restrictions. It derives $\sigma(k)$ and the derivatives $f'(k)$, $f''(k)$, and proves Inada conditions hold for parameter sets $\Phi$, yielding $\lim_{k\to 0} f'(k)=+\infty$ and $\lim_{k\to\infty} f'(k)=0$, with asymptotic behavior linked to the signs of $\psi$ and $\omega$. Under Inada, the function converges to an asymptotic Cobb–Douglas form with endogenously determined exponents, as shown by $\alpha_z$, $\beta_z$, $A_z$, and $B_z$ calculations. Numerical simulations for two cases ($\omega>0$ and $\omega<0$) illustrate how elasticity $\sigma(k)$ can exceed or fall below one and demonstrate convergence of trajectories as $k$ grows, highlighting distinct transitional dynamics.
Abstract
We examine the new production function developed by Chilarescu, and prove that under certain restrictions, the values of the elasticity can also be less than one. We will also prove that under certain restrictions on the parameters, the production function satisfies the Inada conditions.