AAE635
Fall 2006
Homework #6
Solutions
Chapter 10, problem 4 (p. 310)
Max u(x)
S.t.
y = pTx
where y is income and x is consumption bundle.
The Lagrangean to utility maximization is:

L  u ( x)   y  p T x

The solution to this maximization problem yields the Marshallian demand functions.
Note that Marshallian demand functions are HD0 in p and y. This just says “no money
illusion”. To see this, multiply income and prices by positive constant. Doing so will
alter the constraint equation and hence HD0. If x* is HD0, it follows that the indirect
objective function (indirect utility function), u(x*) is HD0.
Using envelope theorem:
u ( x * ( p, y))
 * ( y, p) is the marginal utility of income.
y
We know that if a function f is HD r in its arguments, then f’ (with respect to the
arguments) is HD r-1. Using this fact: Because u(x*(p,y)) = v(p, y) is HD 0 in p and y,
then * ( y, p) will be HD -1.
Another way to see this using the fact that u(x*) HD 0:
u ( x * ( p, y ))
u ( x * ( p, y ))
 * ( y, p) pi , or * ( y, p) 
pi
xi
xi
We know, from class that Marshallian demands are H.D.0 in (p,y). It follows that:
u ( x * (tp, ty))
u ( x * ( p, y ))
1
*
 (ty, tp) 
tpi 
tpi  * ( y, p) .
xi
xi
t
FONC implies that
1
Chapter 10, problem 5 (p. 310)
The Lagrangean is:


L  x1 1 x 2 2   M  p1 x1  p 2 x 2  .
This yields the FONC:
(1)
w.r.t. x1: 1u x1* M , p   * M , p  p1  0
(2)
w.r.t. x2:  2 u x2* M , p   * M , p  p2  0
(3)
w.r.t. : y  p1 x1* M , p   p2 x2* M , p   0 .
 p
Combing (1) and (2) yields: x1* M , p   1 2 x1* M , p  , which can be substituted into (3)
 2 p1
2M
1 M
to yield : x2*  y, p  
. and x1*  y, p  
1   2  p2
1   2  p1
Substituting the expression of the Marshallian demands for good 1 into either of the first
two FONC yields:
 
  y, p    1 
 p1 
1
*
2

 p2



2
 M

 1   2



1  2 1
.
The indirect objective function will be:
 1M


u ( x*)  v( y, p)  
  1   2  p1 
1
 2M

  1   2  p 2



2
 
  1 
 p1 
1
2

 p2



2
 M

 1   2



1  2
To verify the envelope theorem: take the derivative of the above v (p,M) wrt M i.e
vM , p 
 * M , p  . That yields:
y
 
  y, p    1 
 p1 
*
1
2

 p2



2
 M

 1   2



1  2 1
QED.
Chapter 10, problem 12 (p. 311)
a. For notational clarity, let’s use m1 , m2 instead of x10 , x20 . Then:
Note that the price of a unit of consumption in period 1 is 1, and for the second period the
price will be 1/(1+r). (This is relative price taking the first period as numeraire)
Similarly a unit of income obtained in period 1 has value of 1 and income obtained in
period two will have a value of 1/(1+r)
2
Therefore the budget constraint will be:
1
1
x 2  m1 
m2
1 r
1 r
Rewriting the constraint: 1  r m1  x1   m2  x2  0
x1 
Assume a well behaved utility function over the consumption of the two periods.
The Langragean will be:
L  U x1 , x2   1  r m1  x1   m2  x2  .
.
The primal dual results states that:
 x * 
is symmetric.
* 
  

V - L(x*, *, ) = [Lx(x*, *), L(x*, *)] 

uT [V - L(x*, *, )] u = uT [Lx(x*, *), L(x*, *)] 
 x * 
u  0 for all
* 
  
(k1) vector u satisfying h(x*, ) u = 0.
Now: h(x*, ) u means, in this cases, that:
 u1 
 u1 
 
 
h r   u 2   0 ; or m1  x1  1  r  1 u 2   0 ; or u3  u1 x1  m1   u 2 1  r  . Thus
m 
u 
 m1   u 
 2 3 
 3
we can say that:
u1


*




x

uT [Lx(x*, *), L(x*, *)]  *  u0, for all u of the form: 
u2

  
 u x  m   u 1  r 
1
2
 1 1

 Lr   m1  x1 
 

 

where L   Lm1     1  r   , it follows that Lx : L    0

 

 0

 Lm2  
0
m1  x1 
1  r  
0
1
0

3
u1
0
0

  1

 1


 
 u1  

 
u 
u2
1
1 u~ , where
 0
 u    0
 u x  m   u 1  r  x  m   1  r  2  x  m   1  r 
1
1
1
2

 1

 1 1
  1
~
u is any 21 real vector.
Therefore, the primal dual result tells us that:
*
*
*
0 m1  x1  x1 r x1 m1 x1 m2   1
0 
 
 *

1 0 x1  m1   

*
*
~
u
0
0 1  r   x 2 r x 2 m1 x 2 m2  
0
1 u~  0, u~.


 1  r 
0 1
 0 0
1  * r * m1 * m2  x1  m1   1  r 


But this is equivalent to stating that:
1
0

 
0 x1  m1   
0
1
 1  r  
 0
 

 0
0
0
m1  x1  x1*

1  r   x 2*
0
r x1* m1 x1* m 2   1
0 

*
*
r x 2 m1 x 2 m 2  
0
1 
 * r * m1 * m 2  x1  m1   1  r 


1
x1* r x1* m1 x1* m 2   1
0 

0  *
*
*
0
1 
x 2 r x 2 m1 x 2 m 2  

0  *
 x  m   1  r 
*
*
1

 r  m1  m 2   1
0
0

 




 x * r  x1* m2 x1  m1 
x1* m1  1  r  x1*
   1
0
0

positive semi-definite. Because 0, this in turn implies that
 x1* r  x1* m2 x1  m1 
x1* m1  1  r  x1* m2

0
0

semi-definite. This tells us that:
(1)
x1* r   x1* m2 x1  m1  0 .

 





m2 
 is symmetric

 is symmetric negative


As can be seen none of the derivatives have definite sign because the parameters are all in
the constraint.
From symmetry of this matrix we know that:
(2)
x1* m1  1  r  x1* m2 = 0. Substituting from (2) into (1), we have:
(3)

x
*
1


 x1 rm  x
r 
*
1
1
1

 m1   0
Note that this does not allow us to sign x1* m1 directly. However, it does rule out
certain scenarios. For example, if x1* r > 0, and x1  m1  < 0 (the consumer is a net
saver), then x1* m1 cannot be negative.
4
More symmetry results from the Primal-Dual matrix.
 x * 
* 
  
[Lx(x*, *), L(x*, *)] 
*
*
*
  0 m1  x1  x1 r x1 m1 x1 m2 


  0 0 1  r   x2* r x2* m1 x2* m2 
 0 0
1  * r * m1 * m2 





  x1* r  m1  x1  * r

1  r  * r


* r







  x1* m1  m1  x1  * m1
1  r * m1 





  x1* m2  m1  x1  * m2 

1  r  * m2


*
 m2

* m1


is symmetric.
This yields the following three symmetry restrictions.
  x1* m1  m1  x1  * m1 = 1  r  * r
  x1* m2  m1  x1  * m2 = * r
1  r  * m2 = * m1 .












b. The change in a consumer’s welfare as r changes is given by U * r , m1 , m2  r .
Now, by the envelope theorem, we have
U * r , m1 , m2  r  * r , m1 , m2  m1  x1* r , m1 , m2  . As 0, this says that an increase
in r makes a consumer better off (worse off) if they are a net saver (borrower).


Chapter 11, problem 5 (p. 364)
We know that the demand functions satisfy homogeneity (1), and the budget constraint
(2), identically:
n
xi*
xi*
p

M 0
i  1,....., n.
(1)

j
M
j 1 p j
n
(2)
px
i 1
i
*
i
M
The proof proceeds in two parts. First note that by substituting from (2) into the
n
definition of
p s
j 1
j ij
and evaluating the result using (1) we get:
n
n
 xi* xi* *  n
xi* xi* n
xi* xi*
*


p
s

p

x

p

p
x

p

M 0




j ij
j
j  
j
j j
j

p

M

p

M

p

M
j 1
j 1
j

1
j

1
j

1
j
j
j


For the second half of the proof, we being by noting that because (2) holds identically, we
can differentiate it with respect to M and pj to yield:
n
n
xi*
xi*
pi
 xj  0
(3)
(4)
pi
1


p j
M
i 1
i 1
n
n
Substituting (3) and (4) into the definition of
ps
i 1
i ij
5
n
n
 xi* xi* *  n
xi*
xi*
*


p
s

p

x

p

x
p
  x *j  x *j (1)  0 .


i ij
i
j   i
j i
M
i 1
i 1
i 1
 p j M  i 1 p j
n
Chapter 12, problem 1 (p. 391)
Note that the budget constraint for an inter-temporal utility maximization problem is
i
 1 
given by:  
 xi  m , where m is the wealth at the beginning of the period; i is
i 1 r 
time period.
Hence the Lagrangian is given by:
i
i

 1 
 1  
 u ( xi )   m   
L   
 xi  .
i 1  
i 1 r 


Taking FONC with respect consumption in consecutive time periods xt and xt+1 yields:
t
xt:
 1  u ( xt* )
 1 


 
 0
*
1 r 
 1    xt
t
t 1
 1  u ( xt*1 )
 1 
xt+1: 

 

*
xt 1
1 r 
1  
These can be combined to yield:
u ( xt*1 ) u ( xt* )  1   
(1)

.
xt*1
xt*
 1 r 
t 1
0
i
 1 
 log xi , u( xi )  log xi ,
In the case of the utility function U ( x)   
i 1  
u ( xi ) xi  1 xi , and therefore (1) becomes:
xt*  1   

  1 iff (discount factor ) -the impatience to consume earlier- exceeds
xt*1  1  r 
the cost of consuming earlier (r).
(2)
i
 1  
 xi
In the case of the utility function U ( x)   
i 1  
 1
u( xi ) xi  xi , and therefore (1) becomes:

0    1 , u( xi )  xi ,
1
x *  1    1
 1 iff (discount factor ) -the impatience to consume earlier(3) *t  

xt 1  1  r 
exceeds the cost of consuming earlier (r).
6