5. (26 points) Consider a one-period closed economy model with government. The aggre- gate production is given by Y = ZN where N represents the employment and z is the total factor productivity. The consumer's preferences are given by U (C, l) = In (C) + 1n (l) subject to the following budget constraint C=w(hl)+7rT 2 where 7r denotes prots. The consumer has h units of time to split between labour supply and leisure. The government spending is denoted by G' and lumpsum taxes are denoted by T. The government's budget constraint is G = T. (a) Solve the rm's maximization problem and derive the labour demand curve. (b) Graph the labour demand curve and explain its shape. (c) Show that prots are equal to zero, 7T 2 0. ((1) Set up the Lagrangian, solve the consumer's optimization problem and derive the labour supply curve. (e) Graph the labour supply curve and explain its shape using the substitution and income effect. (f) Solve for the competitive equilibrium and show that the optimal labour is N * = 3112'? and real wage rate is 10* = 2. (g) Derive the equilibrium solution for output, 3\"\