Economics 701
Professor S. McCafferty
Autumn, 2010
Final Examination
1. Consider the following system of differential equations:
 34
 x1 
 x1 
 x   A x  where : A   1
 2
 2
2
 12 

 12 
a.) Find the eigenvalues of the matrix A. (4 points)
b.) Find the eigenvectors of the matrix A. (4 points)
y 
There exists a transformed set of variables,  1  , such that y 1  1 y1 and y 2  2 y 2 .
 y2 
y 
c.) Express  1  as a specific linear combination of the variables x1 and x 2 .
 y2 
(6 points)
d.) Suppose that x1 (0)  1 and x2 (0)  1 34 . Find the corresponding initial values of
y 1 and y 2 . (3 points)
e.) Find the time functions, x1 (t ) and x 2 (t ) that solve the system of differential
1
equations when x(0)   3  . (2 points)
1 4 
f.) Draw in ( x1 , x 2 ) space the locus of points for which x1  0 . (3 points)
g.) For points to the right of the x1  0 locus, is x1 increasing or decreasing?
Explain briefly. (2 points)
h.) Draw in ( x1 , x 2 ) space the locus of points for which x 2  0 . (3 points)
i.) For points above the x 2  0 locus, is x 2 increasing or decreasing? Explain
briefly. (2 points)
j.) In ( x1 , x 2 ) space draw the path of x1 (t ) and x 2 (t ) for the initial condition,
x1 (0)  1 and x2 (0)  1 34 . (3 points)
k.) Suppose that you are constrained to set x1 (0)  1 , but may freely chose a value
for x 2 (0) . Find the value of x 2 (0) that makes the differential equation system
stable. (3 points)
l.) In ( x1 , x 2 ) space draw the path of x1 (t ) and x 2 (t ) for the appropriately chosen
initial condition. (3 points)
2
2. Consider the problem faced by a crude oil extractor who faces a demand curve given
by:
p  u  , 0    1. ,
where p represents the price of oil and u represents sales of oil per year. The stock of oil
dy
is denoted by y (t ) . Therefore u (t )   . Now assume that the extractor seeks to
dt
solve:
T
Max  p(t )u(t )e t dt .
u (t )
0
a.) Write this problem as an optimal control problem, and identify the appropriate
Hamiltonian. (4 points)
b.) Write down the first-order conditions for the optimal control problem, assuming
an interior solution for the control variable. (4 points)
c.) Find the general solutions for the optimal paths of u (t ) and y (t ) . (8 points)
d.) Assume that T, y(0) and y(T) are all specified. Find the appropriate constant
term(s) in the solutions for u (t ) and y (t ) . (4 points)
e.) Suppose that T and y(0) are given and that y (T ) may be freely chosen subject to
y (T )  0 . Write down the appropriate complementary slackness conditions. Is it
possible that y (T )  0 might hold as a strict equality for finite T? Explain
briefly. (4 points)
f.) Suppose that we also require that u (t )  u max . How should we impose such a
constraint? Is likely to be binding? If this constraint ever binds strictly, when is it
likely to bind? (4 points)
3
3. Consider the following consumption-savings problem. A consumption path is
defined as c (t ) , t  [0, T ] . The problem is:

Max  u (c(t ))e t dt ,
c (t )
0
subject to : a (t )  ra (t )  c(t )
The variable, c (t ) , represents consumption, the variable, a(t ) , represents asset holding,
r is the time-constant real rate of interest, and  represents the constant discount rate.
For this problem, assume that r   . The consumer’s initial level of assets, a (0) , is
given. The consumer is required to set lim a(t )  0 .
t 
Now suppose the instantaneous felicity function, u (c ) , has the following properties:
c  cmin
  ,

u (c)   c, c min  c  cmax
c , c  c
max
 max
Assume that cmin  ra(0)  cmax . Because ra (0)  cmax , the individual cannot set c  cmax
for all t. The individual would never be willing to set c  cmax , because this would limit c
during those times in which he must set c  cmax . Because ra(0)  cmin , the individual
would never need to set c  cmin . Therefore, the solution to the original problem is the
same as the solution to:

Max  c(t )e t dt ,
c (t )
0
subject to : a (t )  ra (t )  c(t ),
c min  c(t )  c max , t  [0,),
a(0) given, and lim a(t )  0
t 
a.) Solve the differential equation, a (t )  ra(t )  cmin . Find the particular solution as
part of your answer, but leave any initial or final value conditions unspecified.
(3 points)
b.) Solve the differential equation, a (t )  ra(t )  cmax . Find the particular solution as
part of your answer, but leave any initial or final value conditions unspecified.
(3 points)
4
c.) Write an expression for the current-value Hamiltonian for this problem. Use the
symbol, Ĥ , to denote the current-value Hamiltonian, and the symbol, q, to denote
the current-value costate variable. Do not explicitly include Lagrange multiplier
terms for the constraints, cmin  c(t )  cmax . (4 points)
d.) Write an expression for
Hˆ
. Consider the first-order condition for an interior
c
Hˆ
 0 . Is this condition likely to provide a solution path for
c
c (t ) ? Is the solution path for c (t ) likely to be governed by (a) corner solution(s)?
Explain briefly. (4 points)
solution for c (t ) ,
e.) Solve for the time path of the current-value costate variable, q(t ) , given the yetto-be-determined value, q (0) . Provisionally assume that q(0)  0 . Does q
increase or decrease over time? Remember to assume that r   . (4 points)
f.) If c  cmin holds at some time during the solution path, is it likely to be in the
neighborhood of t  0 or t   ? Remember that r   . Explain. (2 points)
g.) If c  cmax holds at some time during the solution path, is it likely to be in the
neighborhood of t  0 or t   ? Remember that r   . Explain. (2 points)
If there exists a time tˆ , such that for all t  tˆ , c  cmax , then it must also be the case that
ra(t )  cmax for all t  tˆ . If ra(t )  cmax for some t  tˆ , then c could be set equal to cmax
for some more of the time before tˆ , which would increase utility.
h.) Assume that the optimal path is characterized by c  cmin , 0  t  tˆ and
c  c max , tˆ  t   . Using your answers to parts (a) and (b), and using what you
know must be true of the path of a(t ) , find an expression for tˆ . (6 points)
i.) If c  cmin , 0  t  tˆ and c  c max , tˆ  t   , what must be true of q (tˆ) ?
(2 points)
This problem may also be formulated by adding Lagrange multiplier terms to impose the
constraints cmin  c(t )  cmax .
j.) Set up the current-value Hamiltonian to include such terms. Denote the Lagrange
multipliers as  i (t ) . (4 points)