Malthusian Economy

Economics

University of Toronto

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I'm working on a macro economics question and need an explanation to help me study.

Solve the 3 questions with proof on how answer was obtained (steps to solving)

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UNIVERSITY OF GUELPH Gordon S. LANG School of Business and Economics Department of Economics and Finance ECON*3810.01: ADVANCED MACROECONOMICS Winter Semester 2021 Homework Assignment 1 (Due: Feb 2nd 2021 11:59PM) I. A Malthusian Economy with Heterogeneous Agents. (60 points) Let us consider a Malthusian economy populated at time t with an aggregate population of size 𝑁" . The population law of motion is described by the following equation: 𝑁"#$ 𝑦" 𝜢 = & ) 𝑁" πœ‡ where πœ‡ > 0 and π›ΌΓŽ(0,1) are parameters and 𝑦" denotes the income/output per capita at time t defined as: 𝑦" ≑ π‘Œ" 𝑁" where π‘Œ" denotes the aggregate level of income/output at time t which can be described by the following aggregate Cobb-Douglas production function: $42 π‘Œ" = 𝑍" 𝑋2 𝐿" where 𝛽 Î(0,1) is a production parameter, 𝑍" > 0 stands for the aggregate productivity level at time t, 𝑋 denotes a fixed aggregate stock of land used in agriculture and 𝐿" > 0 stands for the aggregate population of workers which is assumed to be a fraction 1 βˆ’ πœ€ of the aggregate population: 𝐿" = (1 βˆ’ πœ€)𝑁" where πœ€ΓŽ(0,1) denotes the fraction of the aggregate population that is part of an elite class of size 𝐸" at time t : 𝐸" = πœ€π‘" Let π‘₯" stand for the land per capita at time t defined as: 1 π‘₯" ≑ 𝑋 𝑁" a. Write-down the equation governing the relationship between the growth rate of the aggregate population from time t to t+1 and the income per capita at time t. (5 points) b. Plot the relationship found in a. in a two-dimensional diagram. (5 points) c. Show that the output per capita at time t can be written as a function of the productivity level at time t and the land per capita at time t. (5 points) Let us assume in the next two questions that the productivity level depends on the size of the elite class: > 𝑍" = 𝐴𝐸" where 𝜌Î(0,1) is a parameter with 𝜌 < 𝛽. d. Using the above productivity equation and the relationship found in c., explain how the aggregate population size at time t affects the output per capita at time t ? (5 points) e. Derive the steady-state output per capita and the steady-state aggregate population size compatible with no income per capita growth? (10 points) f. According to your answers to d. and e. is it possible for an economy to be β€œasphyxiated” by its elite class? If so, then what would be the β€œoptimal” fraction of the aggregate population to be part of the elite class? (10 points) Let us assume in the next two questions that productivity grows at every period at a constant exogenous rate: 𝑍"#$ βˆ’ 𝑍" = πœŒπœ€ 𝑍" where πœ€, 𝜌Î(0,1) are parameters. g. 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) h. Derive the steady-state population growth rate and the steady-state output per capita level compatible with no income per capita growth? (10 points) i. Does the economy achieve a higher standard of living in h. than in e.? Explain (5 points) 2 II. Estimating Land and Productivity Growth. (25 points) Let us assume that before 1830 the Canadian output per capita at time t was described by the following aggregate production function: 𝑦" = 𝑍" B𝑋" /𝑁" where 𝑍" stands for the aggregate productivity level at time t, 𝑋" denotes the stock of land used in agriculture at time t and 𝑁" represents the aggregate population at time t. a. 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 under the assumptions that productivity grows at a constant rate π‘Ž and land grows at a constant rate 𝑏. (5 points) Download the historical statistics Angus Maddison’s Historical Statistics of the World Economy: 12008 AD available from http://www.ggdc.net/maddison/oriindex.htm to answer question b.: 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 points) c. Using the equation obtained in question a. and the average annual growth rates found in question b., derive the average annual growth rate of productivity from 1700 to 1830 π‘Ž under the assumption that the aggregate stock of land used in agriculture remained constant: 𝑏 = 0. (5 points) d. Using the equation obtained in question a. and the average annual growth rates found in question b., derive the average annual growth rate of land 𝑏 from 1700 to 1830 under the assumption that productivity remained constant: π‘Ž = 0. (5 points) e. Using the equation obtained in question a. and the average annual growth rates found in 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: π‘Ž = 𝑏. (5 points) III. The Output per Capita Gap between Canada and India. (15 points) Using the Penn World table version 10.0: (https://www.rug.nl/ggdc/productivity/pwt/), download the following data for 2009 and 2019 for Canada and India: β€œOutput-side real GDP at chained PPPs (in mil. 2017US$)” denoted by rgdpo. β€œPopulation (in millions)” denoted by pop. 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. (5 points) b. Compute the average annual growth rates of the real GDP per capita in both countries from 2009 to 2019. (5 points) c. 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) 3 ...
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