Questions by bjenkins - Page 34

Set up: The production function is given by: Y = AKg (Nht)1-a where N is a constant population level and he is the per-capita level of human capital. 1 Y = Ct + St Law of motion for per capita physical capital is given by: kt+1 = (18)kt + syt Law of motion for per capita human capital is given by: ht+1 = (1 8)ht + QYt Tt ht is the ratio of human to physical capital at time t kt Questions: (a) Divide the laws of motion for per-capita physical capital and per-capita human capital by kt and ht respectively. Express the growth rate of per capita physical capital (9k) and the growth rate of per capita human capital (gn) solely as a function of parameters A, 6, a.s, q and rt. S (b) Consider the case in which rt = r = for all t (the economy starts out with the long run ratio of human to physical capital). Rewrite the per-capita production function just in terms of per-capita capital. Does the production function exhibit diminishing returns in capital anymore? (c) Using the result from part b, re-derive the Solow-Swan equation. That is, find the expressions that gives you kt+1 as a function of k, and the parameters. (d) What is different about this version of the Solow-Swan equation? Is your law of motion for per capita capital still concave or is it linear? What does this imply for the evolution of per-capita capital? (e) Draw a phase diagram where kt is on the x-axis and kt+1 is on the y-axis. What can you say about the steady state in this model? Are there any? What does this model predict about growth of GDP per capita? (Hint: You will have 3 cases, 1) where d = sArl-a, 2) where 8 < sArl-a and 3) > sAr-a. Comment on the existence and stability of steady states if any?)
Solow-Swan Model with human capital In class, we laid down the basics of the Solow-Swan Model with human capital. You will now solve it completely to determine steady states, growth rates etc that result from this extension. Set up: The production function is given by: Y-AK? (Nh)- where N is a constant population level and he is the per capita level of human capital. 1 Y = C + S Law of motion for per capita physical capital is given by: k+1 = (1 6)k + 8y Law of motion for per capita human capital is given by: h+1 = (1 8)h+q 7,- is the ratio of human to physical capital at time t Questions: (a) Divide the laws of motion for per-capita physical capital and per-capita human capital by k, and h, respectively. Express the growth rate of per capita physical capital (92) and the growth rate of per capita human capital (g) solely as a function of parameters A, 8, a.s, q and r. (b) Consider the case in which r- r- for all t (the economy starts out with the long run ratio of human to physical capital). Rewrite the per capita production function just in terms of per-capita capital. Does the production function exhibit diminishing returns in capital anymore? (c) Using the result from part b, re-derive the Solow-Swan equation. That is, find the expressions that gives you k+1 as a function of k, and the parameters. (d) What is different about this version of the Solow-Swan equation? Is your law of motion for per capita capital still concave or is it linear? What does this imply for the evolution of per-capita capital? (e) Draw a phase diagram where k, is on the x-axis and k+1 is on the y-axis. What can you say about the steady state in this model? Are there any? What does this model predict about growth of GDP per capita? (Hint: You will have 3 cases, 1) where 8 = sAr, 2) where 8 sAr-. Comment on the existence and stability of steady states if any?) (f) Assume that the rate of return for capital is equal to capital's marginal product, MP - K Does this model help explain low rates-of-return to capital in poor countries relative to rich countries? (g) Revisit the law of motion for human capital, h+1 = (1-8)h +qy. Do you think it's reasonable to include depreciation (6) in the equation? Explain. (h) Remember that he stands for per-person human capital. Under certain conditions, does the law of motion imply that he can grow indefinitely? Is that reasonable? (i) Bonus question: Suppose we changed the law of motion of human capital to include an upper limit. If he