In an economy, let S(t) denote the stock of the fuel and E(t) the rate of extraction of fuel (and energy use) at any time t, with S(0) = So. Then, we have $ = dSE. Energy use, E, makes possible the production of goods and services for consumption, C, which creates utility but generates a flow of pollution, and P, which creates disutility. The consumption and pollution functions are C = C(E) = E + 1 and P = P(E) = E. The social utility function depends on consumption and pollution: U = U(C, P) = ln (C) P = In(E +1) E. An Energy Board is appointed to plan and chart the optimal time path of the energy-use variable E to maximize the social utility over a specified period [0, 1]. 1. Write the Board's dynamic optimization problem. [10 points] 2. Write the Hamiltonian function of the Board's problem and check whether the conditions of the maximum principle are sufficient for the social utility maximization problem. [20 points] dt 3. Solve the Board's problem and discuss how the optimal control, state, and costate vari- ables evolve with time. [20 points]|