Professor Erdinç
ECO 402
Final Examination
Part 1 (35 Points): Q-Theory of Investment
Consider the Q-Theory of Investment as described in class and assume that the representative
firm produces output according to the following production function, i.e.
Y (k t )   k t
and cost
I t
of installing new capital, C ( I t ) 
with  describing general state of the economy, and  is
2
2
a constant cost parameter capturing the degree of adjustment costs. The representative firm

maximizes the following objective function by choosing I t ,    e
0
t

I t 2 
 k t  I t 
 dt
2 


subject to k t  I t . Assume q is the multiplier of the current value Hamiltonian.
a) (10 Points) Set up the current value Hamiltonian for this problem. Derive the optimality


conditions. Find q  0 and k  0 loci at the steady state. Find the solution value for the
steady state capital stock and the slopes of the lines on q and k space.

b) (10 Points) What is the effect of an increase in  on the steady state, and q  0 and

K  0 loci? New Equilibrium? (Hint: Show how the lines will shift on q and k space.)
c) (10 Points) What is the effect of an increase in  , i.e. the discount factor on the steady


state and q  0 and K  0 loci? New Equilibrium? (Hint: Show how the lines will shift,
if any.)
d) (5 Points) Suppose that  is indirectly measuring the degree of credit imperfections in
the banking system and hence, a larger value implies greater difficulty in obtaining credit
for the firm and hence, a larger cost of installing capital. Does  affect the steady state
value of kt ? Does it affect anything at all? Is this intuitive? Explain. (Hint: No need for
derivation, just interpret verbally.)
Part 2 (35 Points): Money in a Log Utility Function
Suppose that the utility function is given by:
U (c, m)  a l o cgt )( (1  a) l o m
g t ()
where 0  a  1 , with U c  0 , U m  0 , U cc  0 , U mm  0 . The production function is Cobb
Douglas with f (kt )  kt and the rate of nominal money growth is given by  . Rate of time
preference (discount rate) is given as   0 .
Also, assume that the following optimality conditions were derived based on the maximization of


M
utility with respect to ct and mt subject to the constraint given by ct  k t 
 wt  rt k t  xt .
P
The optimality conditions are:
1.
H c
 Uc    0
ct
2.
H
   c    rt  
k t
3.
H
   c    U m    
mt
4.
5.
c *  k * (At the steady state when k  0 )
rt  f ' (k t )
6.
m t  (   )mt
Uc  





   rt




U

    m





a) (20 Points) Differentiate the utility function with respect to c t and mt to find the
functional form for U c and U m . Find the equations describing the steady-state



equilibrium when k  m    0 . 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 at the steady state. (Hint: Solve for the steady state value of k .
Does it depend on  ?)
b) (5 Points) Show that the real money stock is decreasing in  at the steady state i.e.
m *
 0.

*
c) (10 Points) Determine the optimal growth rate of money,  given by the Friedman Rule,
and show that it implies a zero nominal interest rate. Why do contemporary central banks
don’t follow the Friedman Rule? Cite two reasons.
Part 3 (30 Points): A Model of Credit Market Imperfections
Suppose that you are given the following loan supply derived based on the profit maximization
behavior of a representative bank in class.
LS 
p (rL )rL  brD
where p(rL ) is the “probability of loan payment” as a function of loan
q
interest rate with p' (rL )  0 , rL is the loan interest rate, rD is the deposit interest rate, b is the
deposit/loan ratio (positively related to the reserve requirement ratio) and q is the cost parameter
for servicing loans (higher its value, less efficient is banking).
a) (10 Points) Show that the loan supply is backward bending by deriving the slope of this
loan supply on rL and LS space? (Hint: Differentiate the loan supply with respect to rL
and invert, holding rD , b and q constant.)
b) (10 Points) Assume that there exists credit rationing in the loan market such that loan
supply and loan demand do not intersect. What is the effect of an increase in b on the
location of the loan supply? Does this enhance or reduce the incidence of credit rationing
in banking? What is the effect of an increase in q on the location of the loan supply?
Does this enhance or reduce the incidence of credit rationing in banking? (Hint: How do
these changes shift the loan supply? And why?)
c) (10 Points) How do the economists interpret the meaning of “adverse selection” and
“moral hazard” in the context of credit markets? Be succinct. (Hint: How is the
“probability of default” affected by these two conditions? Provide a verbal analysis.)
BONUS (10 Points): Is Money Super neutral? (When money is not part of real
wealth)
Suppose that the economy is represented with the following equations:

k  f (k )  c(k ) 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 .
where real wealth consists only the physical capital stock.

m  (   )m where  is the growth rate of money and  is actual inflation.
r  f ' (k )
Rational expectations hold:  *  
The reduced form of this model is represented with the following two equations:

1. k  f (k )  c(k )
2. m  k . ( f ' (k )   )

3. m  (   )m


Draw the phase diagram, carefully identifying the slopes of m and k lines under the restrictions
assumed above on m and k space. 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?