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econ6021 vector
创建日期：2024-05-24 16:48:35
所属分类：数学mathematics
标签： econ6021
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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 …gure in f:
below.
d. For consumer A the Lagrangean
L
xA1 ; x
A
2 ;
=
1
4
ln
xA1
+
3
4
ln
xA2
xA1 + pxA2 12
For consumer B the Lagrangean
L
xB1 ; x
B
2 ;
=
1
2
ln
xB1
+
1
2
ln
xB2
xB1 + pxB2 4 6p
e. For consumer A the …rst order conditions
1
4
1
xA1
=
3
4
1
xA2
= p
xA1 + px
A
2 12 = 0
1
Eliminate from the …rst two equations
3
xA1
xA2
= p;=) pxA2 = 3xA1 ; use this with A0s BC
xA1 = 3; x
A
2 = 9=p
For consumer B the …rst order conditions
1
2
1
xB1
=
1
2
1
xB2
= p
xA1 + px
A
2 4 6p = 0
Eliminate from the …rst 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 ;

=
1
4
ln

xA1

+
3
4
ln

xA2
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 ;

=
1
4
ln

xA1

+
3
4
ln

xA2
u 1
2
ln

16 xA1
1
2
ln

6 xA2

:
The …rst order conditions with respect to

xA1 ; x
A
2

1
4xA1
=
1
2 (16 xA1 )
for xA1
3
4xA2
=
1
2 (6 xA2 )
for xA2
2
Simplify and cross multiply the …rst two equations
1
2xA1
=
1
16 xA1
;
3
2xA2
=
1
6 xA2
16 xA1 = 2xA1 ;
3

6 xA2

= 2xA2
Eiminate 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

de…nes 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 5. This means that the assets
are not redundants and combining them 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 = 5; q2 = 3) :
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 = r and
= q r1
Here r is the matrix of payo¤s, r =
4 13 2
in this case. q is the vector of real assets’
prices, q = (5; 3) ; = (1; 2) is the vector of Arrow securities prices. r1 is the
inverse matrix to matrix r , r1 = 1

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FN1024 MATH0094 MANG 6554 ECOS3003 ELEC6252 C2511C PHY328 EC1060 ECON32202 COMP4002-E1 MATH3063 EFIMM0124 QBUS2310 ECON4411 FUNDAMENTALS ACCT20001 18YX RPM COMP 2711 QBUS6600 BUSS5221 ST3189 FIT1047 MAT344 EGB375 MATH60013 MATH96011 COMP3557 MAST20029 ECM159 EAE1011 ENSCEN 105 SECTION A STA 141C SOSE 2022 PAM006 COMP0048 M40007 ISOM2500 CMPE 142 A06472 MCC150 ACST 8083 GEOG6087 ECON920 FINM7008 ENVENG 4110 COMPSCI 361 ENGRCEE 21 PHYS 1112 ACF2100 ECMM108 COMPS267F MAT00050M MAST30025 CSIT970 MSCI 224 INT202 PUBL0054 MSDM5004 BMAN 21020 BMAN21020 MGMT1002 MATH11150 LAWS7023 ACTL30002 CIVL3340 MATH1002 ECON 8069 COMPSCI 351 IBUS7302 STAT4528 CHEM1101 STAT3925 BUSN 7008 Comp 3120 ECOS2040 SWEN90010 ENGN6528 ECM158 ECON20022 COMP6452 Calculation CITS1401 S1889112 BUS 351 ACTL30008 FIT2091 ECON6002 IBUS6020 PHI 001 ECON30290 MATH7501 ECON10004 MFIN7017 ISYS90043 LAWS7012 ECON30025 COMP30023 COMM1190 ECOM20001 GSBS6009 STAT 337 EMET8002 ECOM90001 ECOM30001 FINC 2012 BUSINESS ENGCV512 ECO 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