2. Consider a two-period economy with many identical household who supply labor inelastically. The utility of the representative household is function of consumption in the present and future (C, cf ): u(c) + Bu(c) (1) where u(.) is an increasing and concave utility function. The economy's resource constraints are Y = AF(K, N) = c+I+G Kf = (1 -d) K + I Yf = A'F(K', NJ) = of + Gf - (1 - d)kf, where AF(K, N) is constant returns to scale production function, and (A, A') are exogenous total factor productivity in the present and future. (G, G/) are exogenous government pur- chases of goods in the present and future, d is a constant depreciation rate, and K is initial capital stock which is predetermined historically. (a) Show that the resource constraint implies C = AF(K, N) + (1 - d)K - Kf - G (2) = AF(K', NS) + (1 d)kf Gf. (3) (b) Show that the competitive equilibrium with lump-sum taxes achieves the efficient allo- cation, i.e., K/ is chosen to maximize the utility (1) subject to the resource constraints (2, 3)