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1. Consider the continuous time Solow growth model with population growth but without technological progress, ie. n>0 and g = 0. Looking at economic data we find a negative correlation between income and population growth, that is higher income is associated with lower population growth rates.. This problem will ask you to analyze how this affects our model in a simple way. Suppose there is a threshold of capital per effective worker kr, below which a country has a population growth of ni and above which its population grows at rate n2. Mathematically, the following conditions hold: ni > n2 n=n1 for k < kr n=n2 for k > kT Furthermore the following is satisfied: sf (kr) < (8 + n1)kT sf (kT) > (8 + n2)kT Note that the last two conditions represent inequalities that hold at the threshold level and do NOT represent conditions that hold for every k. (a) Show graphically how the above conditions affect the Solow model. Specifically, draw the savings function, the production function and the depreciation and dihution function (the latter includes population growth) all in per-efficiency-units-of-labor terms. Also identify clearly all possible steady states. (b) For each of the steady states identify if it is (locally) stable or unstable. (c) If the economy starts with a capital stock per worker that is smaller than kr, in which steady state will the economy end up?

1. Consider the continuous time Solow growth model with population growth but without technological progress, ie. n>0 and g = 0. Looking at economic data we find a negative correlation between income and population growth, that is higher income is associated with lower population growth rates.. This problem will ask you to analyze how this affects our model in a simple way. Suppose there is a threshold of capital per effective worker kr, below which a country has a population growth of ni and above which its population grows at rate n2. Mathematically, the following conditions hold: ni > n2 n=n1 for k < kr n=n2 for k > kT Furthermore the following is satisfied: sf (kr) < (8 + n1)kT sf (kT) > (8 + n2)kT Note that the last two conditions represent inequalities that hold at the threshold level and do NOT represent conditions that hold for every k. (a) Show graphically how the above conditions affect the Solow model. Specifically, draw the savings function, the production function and the depreciation and dihution function (the latter includes population growth) all in per-efficiency-units-of-labor terms. Also identify clearly all possible steady states. (b) For each of the steady states identify if it is (locally) stable or unstable. (c) If the economy starts with a capital stock per worker that is smaller than kr, in which steady state will the economy end up?

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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4. Consider the contimuous time Solow growth model with population growth but without technological progress, i.e. n > 0 and g = 0. Looking at economic data we find a negative correlation between income and population growth, that is higher income is associated with lower population growth rates. This problem will ask you to analyze how this affects our model in a simple way. Suppose there is a threshold of capital per effective worker kT, below which a country has a population growth of n1 and above which its population grows at rate n2. Mathematically, the following conditions hold: ni > n2 n=n1 for k < kT n=n2 for k > kT Furthermore the following is satisfied: sf (kr) < (8 +n1)kT sf (kr) > (8 + n2)kT Note that the last two conditions represent inequalities that hold at the threshold level and do NOT represent conditions that hold for every k. (a) Show graphically how the above conditions affect the Solow model. Specifically, draw the savings function, the production function and the depreciation and dihution function (the latter includes population growth) all in per-efficiency-units-of-labor terms. Also identify clearly all possible steady states. (b) For each of the steady states identify if it is (locally) stable or unstable. (c) If the economy starts with a capital stock per worker that is smaller than kr, in which steady state will the economy end up? 2 (d) Suppose the country is in the steady state that you identified in (c). What can the country do to reach the steady state with the highest capital stock per worker among those steady states you found in (a)? (e) What is the growth rate of output per worker in the steady state from part (c)? What is it for the steady state the country tries to reach in part (d)?
Transcribed Image Text:4. Consider the contimuous time Solow growth model with population growth but without technological progress, i.e. n > 0 and g = 0. Looking at economic data we find a negative correlation between income and population growth, that is higher income is associated with lower population growth rates. This problem will ask you to analyze how this affects our model in a simple way. Suppose there is a threshold of capital per effective worker kT, below which a country has a population growth of n1 and above which its population grows at rate n2. Mathematically, the following conditions hold: ni > n2 n=n1 for k < kT n=n2 for k > kT Furthermore the following is satisfied: sf (kr) < (8 +n1)kT sf (kr) > (8 + n2)kT Note that the last two conditions represent inequalities that hold at the threshold level and do NOT represent conditions that hold for every k. (a) Show graphically how the above conditions affect the Solow model. Specifically, draw the savings function, the production function and the depreciation and dihution function (the latter includes population growth) all in per-efficiency-units-of-labor terms. Also identify clearly all possible steady states. (b) For each of the steady states identify if it is (locally) stable or unstable. (c) If the economy starts with a capital stock per worker that is smaller than kr, in which steady state will the economy end up? 2 (d) Suppose the country is in the steady state that you identified in (c). What can the country do to reach the steady state with the highest capital stock per worker among those steady states you found in (a)? (e) What is the growth rate of output per worker in the steady state from part (c)? What is it for the steady state the country tries to reach in part (d)?
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