Professor Erdinç
ECO 402
Homework #3
(Due April 8th in class)
Part 1 (35 Points): Q-Theory of Investment
Consider the Q-Theory of Investment as described in class and assume that
 ( K t )  a  bK t
and
I
C ( I t )  t with a describing general state of the economy or aggregate output, and b is the
2
2
marginal profitability of capital.

a) (10 Points) Find the q  0 locus and the long-run equilibrium value of K t .

b) (10 Points) What is the effect of an increase in a , i.e. the aggregate output on the q  0

and K  0 loci? New Equilibrium?

c) (10 Points) What is the effect of an increase in r , i.e. the interest rate on the q  0 and

K  0 loci? New Equilibrium?
d) (5 Points) Suppose that a new tax code proposed has a 50% chance of passing in the
parliament and increases a to anew . Given this uncertainty, companies take the expected
value of the tax effect, i.e.
anew
when making decisions. Compare the effect of the new
2
tax code on the steady state levels of q and K with (
uncertainty.
anew
) and without ( anew ) this
2
Part 2 (35 Points): Money in the Utility Function
a
Suppose that the instantaneous Utility function is given by
1 a
(ct .mt )1
U ( c, m ) 
1 a
where 0  a  1,   0 with U c  0 , U m  0 , U cc  0 , U mm  0 and U cm  0 .

The production function is Cobb-Douglas with f (kt )  kt and the household can hold its wealth

Mt

in the form of either money or capital. The rate of nominal money growth is given by
Mt
and the government transfers new money to households in the form of payments, xt such that

Mt
 xt
Pt

ct  k t 
each period. The household budget constraint is, then, given by

Mt
 wt  rk t  xt
Pt
. Under perfect competition, wt  f (k t )  k t f ' (k t ) and
rt  f ' (k t ) . Rate of time preference (discount rate) is given as   0 .


a) (5 Points) Verify that product market equilibrium is f (k t )  ct  k t and xt  mt  mt

Pt
where  
.
Pt
b) (15 Points) Set up the current value Hamiltonian and find the equations describing the
steady-state equilibrium. Write down the transversality conditions. Verify that rate of
nominal money growth is equal to the inflation rate and hence, money is neutral at the
steady state. Verify also that money is super-neutral.
c) (10 Points) Show that the real money stock is decreasing in  at the steady state i.e.
ct
mt
 0.
 0 , and the real consumption is increasing in real money stock i.e.
mt

d) (5 Points) Show that optimal growth rate of money,  * is given by the Friedman Rule,
which sets the growth rate of money equal to minus the real interest rate or discounting
rate. Does the return on money equal to the return on capital under Friedman Rule?
Discuss.
Part 3 (30 Points): Tobin Effect in a Production Economy
Suppose that the economy is represented with the following equations:

k  f (k )  c(a) with f '  0 , f ' '  0 , c' 0 and f 'c'  0
m  k. (r   * ) where i  r   * , nominal interest rate,  * is expected inflation and  ' (.)  0 .
a  m  k where a denotes real wealth and comprises both real money and real capital.

m  (   )m where  is the growth rate of money and  is actual inflation.
r  f ' (k )
Assume rational expectations such that expected inflation is equal to actual inflation, i.e.  *  
a) (5 Points) Interpret the economic meaning of each equation.


b) (10 Points) Reduce the system to a dynamic system in m and k . Derive the steady state
equations.


c) (15 Points) Draw the phase diagram carefully identifying the slopes of m and k lines
under the restrictions assumed above. Is the system stable? What is the effect of an
increase in  on the steady state levels of m and k ? Is money super-neutral in this
model? Why? Why not?