A Solow Growth Model (65 Points) Let us consider a production economy endowed with a single perfectly competitive firm renting at every time both labour and physical capital from the current human population at the real rental rates wy, 1y, respectively. In equilibrium at time w, = MPL, andr, = MPK, where MPL; and MPK; denote the marginal product of labour and the marginal product of physical capital, respectively. The aggregate output/income Y; at every time tis produced according to the following Cobb-Douglas production function: Y, = AK}ELE where A>0 stands for the total factor productivity parameter, K, represents the physical capital and L; denotes the total number of workers with (0,1) is for the labour share of output. Let us assume that the total number of workers corresponds to the size of the aggregate human population at every time & L, = N; which grows at a constant rate ne(0,+wx): Ny = (1+n)N, where y, = Y; /N, denotes the output/income per capita at time fand k; = K, /N, stands for the physical capital per capita at time The change in the aggregate physical capital from time tto time + is governed by the following equation of motion: Kioi= 1+ (11 8)K; where e(0,1) represents the physical capital depreciation rate parameter and [; denotes the aggregate investment in physical capital at time which in equilibrium is equal to the aggregate saving: I, = S;. a. Derive at time an expression for the marginal product of labour schedule and an expression for the marginal product of physical capital schedule. (10 points) b. Show that in equilibrium at time the aggregate output/income Y; is the sum of the aggregate labour income: w; L, and the aggregate physical capital income: 1 K;. (10 points) . Write-down the production function in per capita units. (5 points) d. If all the labour income is saved: S; = w;L; at time , then write-down the equilibrium equation of motion for the physical capital per capita from time #to time +1. If all the labour income is saved at every period, then derive the steady-state physical capital per capita and the steady-state output/income per capita. (15 points) e. If all the physical capital income is saved: S; = 1;K; at time , then write-down the equilibrium equation of motion for the physical capital per capita from time to time +1. If all the capital income is saved at every period, then derive the steady-state physical capital per capita and the steady-state output/income per capita. (15 points) f. Derive the steady-state consumption per capita associated with the steady-state solutions found in d. and e. Which of the two steady-state solutions has the highest steady-state consumption per capita? (10 points)