The economy is described as follows: ( ) ( ) ( ) 1 1 1 1 , 1 1 , 1 , 0, q a a q d - - + = - = = - + = > = " t t t t t t t t t t t t c u c Y K h L k k i h bk b LLt (a) Describe how human capital, as given by ht , accumulates in this economy. Do firms or individuals directly invest in improving the level of human capital? Or, is it simply a side effect of physical capital accumulation? Given your answer, do you believe that the social planner and decentralized competitive equilibrium will coincide in this model?

(b) Write out and solve the social planner's problem in this economy. (Assume that the discount factor is b (0,1) ) (i) What is the growth rate of consumption in the economy? (ii) What is the optimal savings rate? (Assume a linear savings function)

(c) Solve for the competitive equilibrium in this economy. (For simplicity, assume that the labor supply of individuals is exogenous and equal to 1. Moreover, assume that the borrowing constraint is never binding.) (i) Prove the household budget constraint can be written as: ct + + k b t t + + 1 1 = wtht t + + (1 1 r -d ) kt + + ( R b t t ) where bt are government bonds that have return Rt (ii) What is the arbitrage condition between risk-free bonds and capital? (iii) Use the FOCs of the household to find the Euler condition. (iv) Using the firm's profit maximizing behavior, what is the equilibrium interest rate for physical capital and the equilibrium wage rate? (Remember, the firm takes human capital as exogenous). (v) What is the growth rate of consumption?

(d) Compare the growth rate of the social planner's equilibrium from Part B to that of the decentralized equilibrium in Part C. How are they different, and why?

(e) Now suppose that the government subsidizes the private cost of capital for firms. In particular, assume that the private cost of capital for firms is now given by (1 ) t -t r and the government pays for this subsidy via lump sum taxes, T , on individuals. Resolve for the competitive equilibrium of this economy. (i) What is the subsidy, t , the government should set such that the competitive equilibrium growth rate of consumption coincides with the social planner's outcome we found in Part A? (ii) Why does the government want to subsidize investment in this model?