1. Answer this question using calculus. Consider a version of the two period general equilibrium model with pro duction, as seen in the notes. Suppose that there is a negative pollution externality that takes the same form as the externality in section 7.2 of the chapter 4 notes. That is, let P denote the aggregate level of pollution. The household gets utility from three goods: consumption, leisure, and clean air. This formula tion allows us to maintain the assumption that the household's preferences are monotonic and convex. The household's per period utility function be- comes u(Ct,lt, Pt), where u1(-),u2(-), and u;;(-) are all positive due to the fact that consumption, leisure, and clean air are all goods, not bads. Pollu tion enters the utility function with a negative sign, because an increase in pollution decreases the cleanliness of the air that the household breathes. Answer the following questions about the household's problem. (Note: you are not asked to characterize the entire equilibrium for this question.) (a) Write down the household's problem, and generate rst order conditions. [3 points] (b) Interpret the household's rst order conditions. You only need to in- terpret the intertemporal and intratemporal conditions once each. [3 points] (c) Explain why the rational expectations assumption is needed in order to solve household's problem. [3 points]