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 hL k k i h bk b L L t

(a) Describe how human capital, as given by t h , 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:

( ) ( ) 1 1 1 1 t t t t t t t t t c k b wh r d k R b + + + + = + + - + +

where t b are government bonds that have return t R

(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?

This question will add taxes to the human capital model discussed in the notes. Specifically, assume the following conditions for the economy: (iv) Using the firms profit maximizing behavior, what are the equilibrium interest rate and wage rate (for human capital)? u (c. ) = 1- ye (V) What is the optimal k/h ratio? What is the growth rate of consumption? Y, = BK, (h,I, ) . HA 7= 7 '00. The government returns the tax as a lump sum subsidy to individuals. (iii) Use the arbitrage condition to show that the household's budget Resolve for the decentralized equilibrium. constraint can be written as: c, +a, =(1+ R)a , where a, =b, + k, +h, , and use the FOCs to find the Euler Condition. (i) Is the optimal ratio of k/ h affected now? Why or why not? (1i) How does the tax affect the equilibrium prices for capital and the growth of consumption