Solution to question 1.
a: Let p2 = p:
xA1 + px
A
2 = 12 for A
xB1 + px
B
2 = 4 + 6p for B
Since multiplying both sides of the BC by a constant makes an equivalent BC,
only the ratio p2=p1 can be recovered in equilibrium, hence normalizing p1 = 1 does
not change anything.
b. The slope of the indi§erence curve should be equal to the slope of the BC at the
optimal point (x1; x2). This can be stated either as MRS at (x1; x2) which is the slope
of the indi§erence curve = price ratio which is the slope of the BC. Or the vector
orthogonal to the indi§erence curve, i.e., the indi§erence curve gradient ru (x1; x2)
at (x1; x2) is collinear with the price vector (1; p) ; which itself is orthogonal to the
BC.
MRSA (x1; x2) = "MU1 (x1; x2)
MU2 (x1; x2)
= " x
A
2
3xA1
= "1
p
for A
ru (x1; x2) =
!
1
4xA1
;
3
4xA2
"
collinear to (1; p)
c. The Edgeworth box is a rectangle with sides 16 and 6. The x side repre-
sents allocations and endowments in state 1, the y side represents allocations and
endowments in state 2. Since consumer B has the whole endowment in state 2, the
endowment point is on the bottom edge of the box, allocating 12 to A and 4 to B.
The BC goes through this point ortogonal to the vector (1; p) : See the Ögure in f:
below.
d. For consumer A the Lagrangean
L
#
xA1 ; x
A
2 ; 4
$
=
1
4
ln
#
xA1
$
+
3
4
ln
#
xA2
$" 4 #xA1 + pxA2 " 12$
For consumer B the Lagrangean
L
#
xB1 ; x
B
2 ; 4
$
=
1
2
ln
#
xB1
$
+
1
2
ln
#
xB2
$" 4 #xB1 + pxB2 " 4" 6p$
e. For consumer A the Örst order conditions
1
4
1
xA1
= 4
3
4
1
xA2
= 4p
xA1 + px
A
2 " 12 = 0
1
Eliminate 4 from the Örst two equations
3
xA1
xA2
= p;=) pxA2 = 3xA1 ; use this with A0s BC
xA1 = 3; x
A
2 = 9=p
For consumer B the Örst order conditions
1
2
1
xB1
= 4
1
2
1
xB2
= 4p
xA1 + px
A
2 " 4" 6p = 0
Eliminate 4 from the Örst two equations
xB1
xB2
= p;=) pxB2 = xB1 ; use this with B0s BC
xB1 = 2 + 3p; x
B
2 = 2=p+ 3:
f . Either good can be used to determine p through the market clearing conditions
xA1 + x
B
1 = 3 + 2 + 3p = 16 =) p = 11=3
xA2 + x
B
2 = 2=p+ 3 + 9=p = 6 =) p = 11=3
g.
xA1 = 3; x
A
2 = 27=11
xB1 = 13; x
B
2 = 2=p+ 3 = 6=11 + 3 = 39=11
f . We can maximize the utility for A subject to the utility for B being at least
/u: The corresponding Lagrangean
L
#
xA1 ; x
A
2 ; x
B
1 ; x
B
2 ; 4
$
=
1
4
ln
#
xA1
$
+
3
4
ln
#
xA2
$" 4!/u" 1
2
ln
#
xB1
$" 1
2
ln
#
xB2
$"
You may consult a similar question in homework 2. Using resource constraints xA1 +
xB1 = 16; x
A
2 + x
B
2 = 6 the above can be written as
L
#
xA1 ; x
A
2 ; 4
$
=
1
4
ln
#
xA1
$
+
3
4
ln
#
xA2
$" 4!/u" 1
2
ln
#
16" xA1
$" 1
2
ln
#
6" xA2
$"
:
The Örst order conditions with respect to
#
xA1 ; x
A
2
$
1
4xA1
= 4
1
2 (16" xA1 )
for xA1
3
4xA2
= 4
1
2 (6" xA2 )
for xA2
2
Simplify and cross multiply the Örst two equations
1
2xA1
= 4
1
16" xA1
;
3
2xA2
= 4
1
6" xA2
16" xA1 = 24xA1 ;
3
#
6" xA2
$
= 24xA2
Eiminate 4 from these equations
16" xA1
3 (6" xA2 )
=
xA1
xA2
16xA2 " xA1 xA2 = 18xA1 " 3xA1 xA2
8xA2 + x
A
1 x
A
2 = 9x
A
1
xA2 =
9xA1
8 + xA1
This is the equation for the contract curve, note that both
#
xA1 ; x
A
2
$
= (0; 0) and#
xA1 ; x
A
2
$
= (16; 6) belong to this curve. This curve is the thick line Edgeworth box
is below
0 2 4 6 8 10 12 14 16
0
1
2
3
4
5
6
The straight line is the budget costraint with p = 11=3 as per the equilibrium.
Where it intersects with the contract curve is the equilibrium allocation in g. For
every /u; the equation /u = 1
2
ln
#
16" xA1
$
+ 1
2
ln
#
6" xA2
$
deÖnes the indi§erence curve
for agent B: It intersects the contract curve at some point, this is the Pareto e¢cient
allocation that corresponds to this /u: The dashed red lines correspond to the indif-
ference curves with /u = 1 and /u = 2: For some lucky choice of /u the Pareto e¢cient
allocation will coincide with the equilibrium allocation in g:; but generically it does
not.
3
Solution to question 2.
a: The payo§ matrix is invertible, its determinant is 1. This means that NO
asset is redundant and combining assets in the right way the investor can transfer
the wealth into either one of the tomorrow states independenty of the other state.
Suppose further the prices of the assets (a1; a2) are (q1 = 3:5; q2 = 2) :
b: Arbitrage is a possibility of holding a portfolio that requires no initial investment
but allows strictly positive payo§ in at least one of the future states and non-negative
payo§ in every future state. Arrow securities allow tranferring wealth selectively into
particular states tomorrow. Whe the prices of all Arrow securities are positive,
arbitrage opportunities do not exist. Intuitively it means that transferring money
into any future state has its positive price.
c: Two important formulaes
q = 8 $ r and
8 = q $ r"1
Here r is the matrix of payo§s, r =
%%%% 2 13 2
%%%% in this case. q is the vector of real assetsí
prices, q = (3:5; 2) ; 8 = (81; 82) is the vector of Arrow securities prices. r"1 is the
inverse matrix to matrix r , r"1 = $
%%%% 2 "1"3 2
%%%%. Check that r $ r"1 = r"1 $ r = I =%%%% 1 00 1
%%%%. Using 8 = q $ r"1; follows 8 = #1; 12$ : Both 81; 82 are positive, there is no
arbitrage.
d. The frontier is given by the equation
81y1 + 82y2 = w0
y1 +
1
2
y2 = w0
Set w0 = 7 just to draw the graph. y2 = 2 (7" y1)
0 2 4 6 8 10 12 14
0
5
10
15
x
y
4
The straight line is the frontier, the dots are the assets, the one to the left is asset
a2: Investing all the wealth into asset a2 one can buy 7/2 of the asset. According
to the payo§ matrix this portfolio will pay 7/2 in state 1 and 7 in state 2. This
combination of wealths is the left blue dot on the graph. Investing all the wealth into
asset a1 one can buy 2 of the asset. According to the payo§ matrix this portfolio will
pay 4 in state 1 and 6 in state 2. This portfolio is the right blue dot on the graph.
Note that the portfolios on the graph above are close to each other, yet allow the
entire line of combinations of the future wealth. This is achieved by selling assets
short i.e.e by holding z1 < 0 or z2 < 0:
e. The inÖnitely risk averse investor will hold a perfectly sure portfolio with the
payo§ y1 = y2 = y, the same in both states. Solve for this portfolio using
y +
1
2
y = w0
y = 2
3
w0:
This is the level of wealth she will receive. On the graph above this point is the
intersection of the 45 degree dashed line with the frontier. Using the payo§ matrix r
y1 = 2z1 + z2 =
2
3
w0
y2 = 3z1 + 2z2 =
2
3
w0:
We need to solve this for (z1; z2) to determine the actual portfolio. Subtract eq 2
from eq 1, "z1" z2 = 0; hence z2 = "z1; use this to obtain z1 = 23w0 and z2 = "23w0:
f: When he investor is risk neutral, his indi§erence curve is a straight line orthog-
onal to the vector (>1; >2) : If this indi§erence line is steeper than the frontier, that
is when
>1
>2
>
81
82
the risk-neutral investor will aim to achieve wealth combination of the frontier with
y2 = 0: Hence 3z1 + 2z2 = 0; z2 = "32z1: Together with the budget constraint
3:5z1 + 2z2 = w0; this gives 3:5z1 " 3z1 = w0; z1 = 2w0; z2 = "3w0: This portfolio is
(y1; y2) = (w0; 0) :
If the indi§erence line is áatter than the frontier she will aim at y1 = 0: Then z2 =
"2z1: Together with the budget constraint 3:5z1+2z2 = w0; this gives 3:5z1"4z1 = w0;
z1 = "2w0; z2 = 4w0: This portfolio is (y1; y2) = (0; 2w0) :