Sustainable Energy Development

Recently, there has been significant political interest in developing sustainable energy in the United States. This paper first outlines the practical constraints and considerations in moving away from fossil fuels. Then, taking these constraints into account, we develop a basic model of energy production with varying levels of pollution in discrete time. We reduce the model to a system of integral equations. Under appropriate assumptions, the solutions to the equations are the optimal level of pollution and energy production in the economy. It is unknown, however, if these equations can be solved. Hence, we pose the problem in continuous time, which yields an analogous system of differential equations. The traditional approaches used to solve these equations have failed. Solving this equation is the next step towards developing rigorous models in mathematical finance.