Could I have an explanation by letter plz?

Exercise 2 Consider an economy populated by a representative consumer, that has preferences over leisure (1) and consumption (c). The consumer's utility function in this one-period model is: u ( c , 1 ) = =In( c) + , In(1 ) The consumer has 1 unit of time (h = 1) at his disposal to spend between leisure and work. With T being a fixed lump-sum tax, 7 being profits from dividends, and w being wage, the consumer's budget constraint can be written as: ct wl = w+n-T The representative firm's production function is Y = z K1/2 N1/2 Assume that K = 1, 2 = 10, and that the goverment buys two units of consumption good, meaning G = 2. (a) Define and Calculate the marginal rate of substitution of leisure for consumption (MRSuc). Show why it is not optimal for the consumer to choose the allocation c = 5 and / = 0.64 if the wage is w = 10. (b) Show why this firm does not maximize profits 7 if it chooses N = 0.64 when w = 10. What should this firm do? Explain. (c) The Production Possibility Frontier (PPF) is useful to determine the social planner's solution of an economy. Define PPF. Find the PPF for this model. (d) The social planner's solution must also satisfy the condition MRSte = MPN = MRTic. Show that the social planner's solution is c = 3 and / = 0.75 (e) Find the equilibrium wage and the firm's profits in a competitive equilibrium. Using your results, calculate the GDP using the income approach. Justify your answer fully. (f) List all the conditions that an allocation c' and I* must satisfy to be a competitive equilibrium. (g) Suppose that higher production lowers the consumer's welfare because producing more goods leads to more pollution by firms. Explain in which way this new infor- mation is likely to affect the planner's solution found in question (d). Would the planner's solution coincide with the competitive equilibrium in this particular case