us consider a Malthusian economy populated at time t with an aggregate population of size Nt- II. Estimating Land and Productivity Growth. (25 points) population law of motion is described by the following equation: Let us assume that before 1830 the Canadian output per capita at time t was described by the ollowing aggregate production function: Nt+1 = () Nt yt = ZtVXt/Nt where Z- stands for the aggregate productivity level at time t, Xt denotes the stock of land used in agriculture at time tand Nt represents the aggregate population at time t ere u > 0 and a E(0,1) are parameters and yt denotes the income/output per capita at time t a. Derive a relationship between the growth rate of the output per capita from time tto t+1, and ined as: the growth rate of the aggregate population from time t to t+1 under the assumptions that productivity grows at a constant rate a and land grows at a constant rate b. (5 points) Yt y. = Ne Download the historical statistics Angus Maddison's Historical Statistics of the World Economy: 1- 2008 AD available from to answer question b. : ere Yt denotes the aggregate level of income/output at time t which can be described by the b. Compute the average annual growth rates of the Canadian GDP per capita from 1700 to 1830 and the average annual growth rates of the Canadian population from 1700 to 1830. (5 owing aggregate Cobb-Douglas production function: points) c. Using the equation obtained in question a. and the average annual growth rates found in Yt = Z+XBL)-B question b., derive the average annual growth rate of productivity from 1700 to 1830 a under the assumption that the aggregate stock of land used in agriculture remained constant: b = 0. (5 points) ere B E(0,1) is a production parameter, Zt > 0 stands for the aggregate productivity level at time d. Using the equation obtained in question a. and the average annual growth rates found in denotes a fixed aggregate stock of land used in agriculture and Lt > 0 stands for the aggregate question b., derive the average annual growth rate of land b from 1700 to 1830 under the ulation of workers which is assumed to be a fraction 1 - & of the aggregate population: assumption that productivity remained constant: a = 0. (5 points) e. Using the equation obtained in question a. and the average annual growth rates found in Lt = (1 - =) Nt question b., derive the average annual growth rate of productivity and land from 1700 to 1830 under the assumption that they both grew at the same rate: a = b. (5 points) ere & E(0,1) denotes the fraction of the aggregate population that is part of an elite class of size at time t : III. The Output per Capita Gap between Canada and India. (15 points) Using the Penn World table version 10.0: ( Et = ENt download the following data for 2009 and 2019 for Canada and India: "Output-side real GDP at chained PPPs (in mil. 2017US$)" denoted by rgdpo. "Population (in millions)" denoted by pop. xt stand for the land per capita at time t defined as: X a. Compute the real GDP per capita in Canada and in India in 2009 and in 2019 and derive the ratio of the real GDP per capita in Canada in 2019 to the real GDP per capita in India in 2019 Xt - Nt (5 points) b. ute the average annual growth rates of the real GDP per capita in both countries from 2009 to 2019. (5 points) Under the assumption that both Canada and India will continue to growth at the same average rates found in question b., how many years will it takes for the output per capita gap between these two countries found in a. to be cut in half? (5 points)
