Problem 1: Basic Facts (10 points) Provide a figure that highlights a cross-country pattern related to economic growth. Make sure you clearly include the axis, variable legend, data source, and provide some brief explanation on the pattern. Problem 2: Solow Model (10 points) Consider a Solow model in discrete time with A(t) = L(t) = 1 and the following production function F(K) = if K = (0, K) K _ K if K ( [ K, too) Other parts of the model including the capital depreciation rated and saving rate s are still constant parameters. (a) Prove that for v(6, s) 0 s.t. the model has two steady-state equilibria. (2 points) (b) When there are two steady-state equilibria, find the range ofK (0) s.t. K(t) converges monotonically to at least one of the steady-states. (4 points) (c) Given parameters ,6, Ky, find the range of parameter s, s.t. the economy always fails to grow. (4 points) Problem 3: Ramsay Model (15 points) Consider the following functions forms as an extension of the Ramsay model in our slides F(K (t), N(t), A(t) ) = K(t) ( A(t) N(t) ) 1-a, a e (0, 1 ) ; A (t ) A(t) =g>0, u(c(t)) = Inc(t). (a) Write down the three first order conditions derived from the current-value Hamiltonian of the detrended social planner's problem. (6 points) (b) Solve the steady state saving rate and prove that it is increasing ing. (4 points) (c) Compare the saving rates of U.S. and China via the potentially different values of parameters. Use common sense to compare the parameters. (5 points)