I. A Malthusian Model with Productivity Growth (50 points) Let us consider a Malthusian economy in which the aggregate level of output at time t can be described by the following aggregate Cobb-Douglas production function: Y = AXxN-x where ae(0,1), A denotes the productivity level at time t, X > 0 stands for a fixed aggregate stock of land used in agriculture and Nt represents the aggregate population. Let us assume that productivity grows at a constant rate a = (0,1): At+1 - At At = a Let x+ stand for the land per capita at time t defined as: X xt = Nt At time t, let us consider that the birth rate: bt and the death rate: d can be written as: b = B(y+) d = 1 -8(y+) where , 8 > 0, = (0,1) are parameters and yt denotes the output per capita at time t defined as: Yt= Nt The rate of population growth is given by: Nt+1 Nt Nt - = bt - dt 1