• 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)
• Using the above productivity equation and the relationship found in c., explain how the aggregate population size at time t affects the output per capita at time t ? (5 points)

1. Derive the steady-state output per capita and the steady-state aggregate population size compatible with no income per capita growth? (10 points)
2. According to your answers to d. and e. is it possible for an economy to be "asphyxiated" by its elite class? If so, then what would be the "optimal" fraction of the aggregate population to be part of the elite class? (10 points)
3. Using the production function per capita found in c., derive a relationship between the growth rate of the output per capita from time t to t+1 and the growth rate of the aggregate population from time t to t+1. (5 points)
4. Derive the steady-state population growth rate and the steady-state output per capita level compatible with no income per capita growth? (10 points)
5. Does the economy achieve a higher standard of living in h. than in e.? Explain (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 = (2 ) 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 NE 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 - =) 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 = ENt Let xt stand for the land per capita at time t defined as:X Xt = 7 Nt a. Write-down the equation governing the relationship between the growth rate of the aggregate population from time tto 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) Show that the output per capita at time t can be written as a function of the productivity level at time tand the land per capita at time t (5 points) Let us assume in the next two questions that the productivity level depends on the size of the elite class: Zt = AEP where p E(0,1) is a parameter with p