I. A Solow Growth Model with Factor Income Invested (65 Points) Let us consider a production economy endowed with a single perfectly competitive firm renting at every time t both labour and physical capital from households at the real rental rates Wt, It, respectively. In equilibrium at time t. Wt = MPL, and It = 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 t is produced according to the following production function: Yt = ZK1-LE where 2>0 stands for the total factor productivity parameter, K, represents the physical capital and L: denotes the number of workers with & c(0,1) stands for the labour share of output parameter. Let us assume that a constant fraction ye(0,1) of the total household population of size N, works at every time t. Lt = YNt The aggregate population of households grows at a constant rate ne(0, + 20): Nt+1 = (1 +n)N The law of motion for the physical capital from time t to time t+1 can be written as: where de(0,1) represents the physical capital depreciation rate parameter and , denotes the aggregate investment in physical capital at time t which is equal to the aggregate saving in equilibrium: It = St Let y, = denote the output/income per capita at time t and let k, = = stand for the Nt physical capital per capita at time t.b. Show that in equilibrium, the aggregate output/income at time & is the sum of the aggregate labour income: w L, and the aggregate physical capital income: ,K. (10 points)