2. 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"(Wh;)'-" where / is a constant population level and he is the per capita level of human capital. . Yi = Ci + S . Law of motion for per capita physical capital is given by: kit - (1 - 6)k, + sy . Law of motion for per capita human capital is given by: hat = (1 - 8)h, + qy . r = is the ratio of human to physical capital at time f 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 (9:) and the growth rate of per capita human capital (ga) solely as a function of parameters A, 6, o.s,q and r. (b) Consider the case in which ry - r = ; for all t (the cconomy 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 kat 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 & is on the x-axis and kit, 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, I) where 6 = sArk -", 2) where 6 sArk-". Comment on the cristence and stability of steady states if any?) (f) Assume that the rate of return for capital is equal to capital's marginal product, MP = ax 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, hit - (1-6, ), +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 by 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 by