This paper studies an optimal endogenous growth model using physical capital, labor and two kinds of natural resources in the final goods sector and employing labor to accumulate knowledge. Based on results in calculus of variations, a direct proof of existence of optimal solution is provided. Analytical solutions for the planner case and the balanced growth paths are found for a specific CRRA utility and Cobb-Douglas production function. Transitional dynamics to the steady state from the theoretical model are used to derive three convergence equations of output intensity growth rate, exhaustible resource growth rate and renewable growth rate, which are tested based on data on production and energy consumption in 27 OECD countries.