a. Write-down the equation governing the relationship between the growth rate of the aggregate population from time t to t+1 and the income per capita at time t. (5 points)b. Plot the relationship found in a. in a two-dimensional diagram. (5 points)c. Show that the output per capita at time t can be written as a function of the productivity level at time t and the land per capita at time t. (5 points)

Let us consider a Malthusian economy populated at time t with an aggregate population of size Nt. The population law of motion is described by the following equation: Nt+1 = (It ) Nt where u > 0 and a E(0,1) are parameters and yt denotes the income/output per capita at time t defined as: Yt yt = Nt where Y denotes the aggregate level of income/output at time t which can be described by the following aggregate Cobb-Douglas production function: Yt = Z+XBL]-B where BE(0,1) is a production parameter, Zt > 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 population of workers which is assumed to be a fraction 1 - & of the aggregate population: Lt = (1 - E) Nt where & E(0, 1) denotes the fraction of the aggregate population that is part of an elite class of size Et at time t : Et = &Nt Let xt stand for the land per capita at time t defined as