Let us consider a Malthusian economy populated at time t with N, individuals. At time t, the aggregate number of births: B- and the aggregate number of deaths: D+ can be written as: Bt = b(yt) Nt Dt = [1 - d(yt)"]Nt where o E(0,1), b, d >0 are parameters and yt = Yt/Nt represents the income/output per capita at time t. The law of motion from time t to time t+1 for the aggregate population is: Nt+1 = Bt - D+ + Nt The aggregate level of income/output at time t denoted by Y, can be described by the following aggregate Cobb-Douglas production function: Yt = AtX1-BLE where BE(0,1) is a production parameter, At > 0 stands for the aggregate productivity level at time t, X denotes a fixed aggregate stock of land used in agriculture and Lt > 0 stands for the aggregate number of workers at time t, which is assumed to be a fraction y E(0,1) of the aggregate population: Lt = yNt Let us assume that productivity grows at a constant rate a 2 0: At+1 - At _ = a At Let xt stand for the land per capita at time t defined as: Xt = X/Nt. a. Show that the growth rate of the aggregate population from time t to t+1 can be written as a function of the income/output per capita at time t. (5 points) b. In a two-dimensional diagram, plot the relationship found in a. and identify the level of income/output per capita compatible with no growth in the aggregate population. (10 points) c. Write down the production function in per capita units. (5 points) d. Using the production function per capita found in c., derive a relationship between the growth rate of the income/output per capita from time t to t+1 and the growth rate of the aggregate population from time t to t+1 given the productivity growth rate a and B. (10 points)