1. A Solow Growth Model Augmented with Automation (65 points) Let us consider a Solow growth model in which the aggregate stock of physical capital at every time t denoted by Kt can be either a labour-complementing machine: M, or a labour-replacing robots: Rt. Kt: Mt+Rt The aggregate output at every time t'. Yt is produced according to the following CobbDouglas aggregate production function: Yt = mug\"[eNt + zRJ" where 1 a E (0,1) stands for the output elasticity with respect to the aggregate stodt of machines, N, > 0 represents the human population, e>0 denote the constant time allocated to work and 11,2 >0 denote productivity parameters. The change in the aggregate stodt of physical capital from time tto time t+1 is described by the following law of motion: Kt+1 _ Kt = It _ 6K1: where 1; denotes the aggregate investment and 6 e (0.1) represents the depreciation rate. In equilibrium, the aggregate investment is a constant fraction y E (0,1) of the aggregate output: It = IVY; The aggregate human population grows at a constant rate n > 0 per period: Nr+1 : (1 +71) Nt Let mt E 23: denote the machine per capita, Iet rt 2 2: stand for the robot per capita, let yt E ;: represent the output per capita and let kt 5 : represent the aggregate physical capital per capita. a. Would production possible at time t: (Yt > 0) without machines: (Mt = 0)? Under which condition production would be possible at time t: (Y, > 0) without human labour: (e = U)? (5 points) b. Write-down the aggregate production function in per capita units. (5 points) c. Using the physical capital resource constraint: Kt = M. + R; and your answer to b., show that output per capita can be written as a function of kt and r,. (5 points) d. Write-down in per capita units the equilibrium law of motion for the aggregate physical capital. (5 points) e. In the situation in which automation does not exist: (2 = 0), derive the steady-state physical capital per capita and the steady-state output per capita. (10 point) f. Derive an expression for the level of robot per capita maximizing the output per capita. Display in a two-dimensional diagram the output per capita maximizing level of robot per capita as a function of the physical capital per capita. For whid1 levels of physical capital per capita the use of robot becomes optimal? (15 points) 9. Using your answer to f. show that the resuiting expression for the maximized output per capita is linear with raped to the physical capital per capita. (10 point) h. If it is optimal to use robots, does the economy grow forever or converge to a steady-state solution? Please justify your answer(s). (10 points) 2. Quantitative Problem using the Penn World Data Set (35 Points) Let us consider a Solow growth model in which the aggregate production function time tfor country i is given by