A Solow Growth Model with Physical and Human Capital Accumulation

1. A Solow Growth Model with Physical and Human Capital Accumulation (42 points)
Let us consider a Solow growth model augmented with human capital. The aggregate output/income ? ” at every time t is produced according to the following production function: ? “=??”&’()ℎ” +?”-( where ?>0, ?,?∈(0,1) are production parameters, ?” stands for the aggregate physical capital, ℎ ” represents the human capital per worker: ℎ”≡?”/?”, ?” denotes the aggregate level of human capital and ?” is the aggregate number of workers. At every time t , the aggregate number of workers is assumed to be a fraction ?∈(0,1) of the aggregate population size denoted by ?” which grows at a constant rate ?>0: ?”>&=(1+?) ?” The change in the aggregate physical capital from time t to time t+1 can be written as: ?”>&− ?”= ?”B− δ?”

where ?∈(0,1) represents the depreciation rate and ?”B denotes the aggregate investment in physical capital which in equilibrium is assumed to be a constant fraction ε Î(0,1) of the aggregate output/income: ?”B=ε? ” The change in the aggregate human capital from time t to time t+1 can be written as: ?”>&− ?”= ?”F− δ?” where ?”F denotes the aggregate investment human capital which in equilibrium is assumed to be a constant fraction µ Î(0,1) of the aggregate output/income: ?”F=µ? ” At every time t , the goods’ market is in equilibrium if: ? “=?”+?”B+?”F where ?” denotes aggregate consumption at time t . Let ?”≡? “/?” denotes the income/output per capita at time t , ?”≡?”/?” stands for the physical capital per capita and ?”≡?”/?” is for the physical capital per capita. a. At what rate does the aggregate population of workers grow? (4 points) b. Show that the output per capita at time t can be written as a function of the physical capital per capita at time t and the human capital per worker at time t . (4 points) c. Show that in equilibrium, the physical capital per capita at time t+1 : ?”>& can be written as a function of ?” and ℎ”: (4 points) d. Show that in equilibrium, the human capital per worker at time t+1 : ℎ”>& can be written as a function of ?” and ℎ”: (4 points) e. Does a steady-state physical capital per capita solution: ?”>&=?”=?LL also require the human capital per worker to be at the steady-state: ℎ”>&=ℎ”=ℎLL? Explain (6 points) f. Derive the steady-state formulas for the physical capital per capita, the human capital per worker, the output/income per capita and the consumption per capita. (12 points) g. How does ε and µ affet the steady-state income per capita? (4 points) h. Under the assumption that ε=µ, derive the investment rate that maximizes the steadystate consumption per capita. (4 points)