Question 1: (30 points) Consider the static closed-economy competitive equilibrium model developed in class. For simplicity, assume that government spending G = 0, and therefore lump- sum taxes 7 = 0. A representative consumer chooses consumption C and leisure / to maximize utility subject to the budget constraint C = w/V, + #, and the time constraint I + N. = h. A representative firm chooses labour demand N, to maximize profits * = =KoNI- - wNd, with a E (0, 1). Different from the baseline model, there are negative externalities due to air pollution caused by production. The preferences of the representative consumer are represented by the utility function: U(C, 1, Y) = In(C) + nin(1) - yIn(Y) with n > 0 and y E (0, 1). The last term signifies the utility loss of the consumer as a result of air pollution, and the parameter y captures how much the consumer dislikes air pollution. Note that output Y is not a choice variable for the consumer; i.e. the consumer takes Y as given. 1.a. (6 points) Write down the representative consumer's optimization problem. Write down the associated Lagrangian. Derive the first order conditions with respect to C and I. Combine them to calculate the value of the competitive equilibrium marginal rate of substitution between leisure and consumption MRS, , where CE stands for competitive equilibrium. (Hint: Do not forget that the consumer takes Y as given.) 1.b. (4 points) Write down the firm's optimization problem. Write down the associated La- grangian. Derive the first order condition with respect to Ny. Using the expression, calculate what the marginal rate of transformation MRTc is equal to in the competitive equilibrium