1. A Solow Growi Model Iwith Automation (65 points} Let us consider a Solow growth model in which the aggregate stock of physical capital at every time r :epoted by H, can be use either as a labour-complementing machine: M; or as a labour-replacing robot: :- K; = M! + R, Let us consider that labour can be performed by humans and robots as follows: L, = EI + R; where L, 3::- 0 represents the aggregate labour input at time t, E, :: II] denotes the size of the aggregate population of workers at time {which is assumed is) be a constant fraction y 54%;?) of the aggregate human population NE: E: = FM: The aggregate output at every time I: 11 is produced according to the following Cobb-Douglas aggregate pro-Cluciion unction: i; = AMELIH where Er E {0.1) stands forthe output elasticity with respectto the aggregate stock of machines and A M? denotes the aggregate productivity level. The law of motion for the aggregate stock of physical capiial is described by the following equation: mH=s+umm where i, denotes the aggregate investment in physical capital and 6 E {0,1} represents its rate of depreciation. In equilibrium, die aggregate investment is a constant fraction 5 E (0.1] of the aggregate output: I: = 51'; The aggregate human population grows at a constant rate n : [II per period: NI+1 : [I + \"J \"E. Let my E 5 denote the aggregate stock of machines per capita, let rI E :7: stand for the aggregate stock .vt of robots per capita, let 3.1 ENL: represent the aggregate output per capita and let kl E % represent the aggregate physical capital per capita. a. Under which condition(s) production is possible without human labour: (y = 0) given a e (0,1) A, > 0? (5 points) b. Write down the per capita production function at time & in terms of the aggregate stock of machines per capita and the aggregate stock of robots per capita at that period. (5 points) c. If all the physical capital is used in production: K, = M, + Ro, then show that the output per capita expression at time & found in question b can also be written as a function of the aggregate stock of physical capital per capita and the aggregate stock of robots per capita at time t. Is it possible to use too much robot per capita given k,? (5 points) d. Identify an equation showing how the output per capita at time & and the aggregate physical per capita at time t affects the growth rate of the aggregate physical capital from time tto t+1 in equilibrium. (5 points) e. Derive the steady-state aggregate physical capital per capita: Kt+1 = Kt = kss, the steady- state income per capita: It+1 = Vt = Vss if there is no automation: 1 = n = 0 at every time t. (10 points) f. Derive the steady-state aggregate physical capital per capita: Kt+1 = Kt = kss, the steady- state income per capita: )+1 = Vt = )ss if the aggregate stock of robots per capita is a fraction e E (0,1) of the aggregate stock of physical capital per capita: n = 0k, at every time t. (10 points) g. Using the per capita production function found in question c, derive an expression for the aggregate level of robot per capita at time & maximizing the income per capita level at that period. Is it always optimal to use robots? (15 points) h. Using your answer to g. show that the output per capita at time t under optimal automation is no longer governed by the law of diminishing marginal returns to the aggregate physical capital per capita. (10 points)