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  • Manh-Hung Nguyen (a1) and Phu Nguyen-Van (a2)


This paper considers an optimal endogenous growth model where the production function is assumed to exhibit increasing returns to scale and two types of resource (renewable and nonrenewable) are imperfect substitutes. Natural resources, labor, and physical capital are used in the final goods sector and in the accumulation of knowledge. Based on results in the calculus of variations, a direct proof of the existence of an optimal solution is provided. Analytical solutions for the planner case, balanced growth paths, and steady states are found for a specific CRRA utility and Cobb–Douglas production function. It is possible to have long-run growth where both energy resources are used simultaneously along the equilibrium path. As the law of motion of the technological change is not concave, reflecting the increasing returns to scale, so that the Arrow–Mangasarian sufficiency conditions do not apply, we provide a sufficient condition directly. 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 resource growth rate, which are tested based on OECD data on production and energy consumption.


Corresponding author

Address correspondence to: Phu Nguyen-Van, BETA, CNRS and Université de Strasbourg, 61 avenue de la Forêt Noire, 67000 Strasbourg, France; e-mail:


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