Aditya Goenka, and Manh-Hung Nguyen, “Existence of Competitive Equilibrium in an Optimal Growth Model with Elastic Labor Supply and Smoothness of the Policy Function”, TSE Working Paper, n. 09-064, July 20, 2009.
We prove the existence of competitive equilibrium and the moothness of policy function in an optimal growth model with elastic labor supply by using a simple method. Our approach is based on the result of existence of Lagrange multipliers and their representation as a summable sequence due to Le Van and Saglam  to define the sequence of prices and wages. The proof of existence of equilibrium we give is more simple than in Le Van and Vailakis  and requires less stringent assumptions (neither Inada conditions for the utility function and the production function nor constant return to scale for the production function nor strict concavity). We also prove the differentiability of the policy function at a stationary optimal stock where the derivative of the policy function equals the smaller characteristic root in absolute value associated with Euler equation. Conditions for differentiability of the policy function have so far been assumed in the literature.