Let us assume that all economies were at the steady-state in 1820 and the output per capita at time (was described by the following aggregate production function: X 3": = At (E) where A, stands for the aggregate productivity level at time t; X denotes the xed stock of land used in agriculture and N, represents the aggregate papulation size at time t: 1/3 a. Derive a relationship between the growth rate of the output per capita from time tto HI, and the growth rate of the aggregate population from time tto t+1 under the assumptions that productivity grows at a constant rate a. (5 points) Download the historical statistics Angus Maddison's Historical Statistics of the World Economy: 1- 2003 AD available from htt : www. c.n m ison oriin ex.htm to answer the next question. b. Compute the growth rates of the GDP per capita from 1000 to 1320 and the growth rates of the population from 1000 to 1820 for 30 w. Europe, w. Offshoots, L. America, Asia and total Africa? (10 points) c. Using the equation obtained in question a. and the growth rates found in question b., derive the growth rates of productivity for 30 w. Europe, w. Offshoots, L. America, Asia and total Africa from 1000 to 1820. In which part of the world, productivity grew the fastest? (10 points) d. Display using a scatter plot the relationship between the standard of living in 1820 and the productivity growth rate from 1000 to 1820. Is the graphical relationship found in d. consistent with the prediction of a Maltiiusian growth model with productivity growth? (10 points)