A Malthusian Model with Endogenous Productivity Growth (60 points) Let us consider that the law of motion of the aggregate population can be...

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Represent on a graph the relationship between the income per capita at time and the aggregate population at time implied by the production function and the relationship between the productivity and the aggregate population

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I.A Malthusian Model with Endogenous Productivity Growth (60 points)Let us consider that the law of motion of the aggregate population can be described by thefollllowing equation:Nt+1 =Ntwhere o > 0 and & =(0,1) are parameters, N, represents the aggregate population at time t, andy, denotes the income per capita at time t defined as:yt =YtZThe aggregate level of output at time t can be described by the following aggregate Cobb-Douglas production function:Ye = AtX&Ltwhere aE(0,1), At stands for the productivity level at time t, X denotes a fixed stock of land usedin agriculture and Lt represents the aggregate population of worker which is a fraction BE(0,1) ofthe aggregate population:Lt = BNLet x, stand for the land per capita at time t defined as: