Asset Economy Solutions
1. Exercise 3.2 from Lengwiler.
Either proceed directly that is solve for the portfolio (z1; z2) that delivers 4000 in either state
4000 = 20z2
Copyright By PowCoder代写 加微信 powcoder
4000 = 100z1 + 35z2
Then z2 = 200; z1 = 30:
Or realize that you need to hold 4000 of Arrow securities for either state. The question is how to compose the portfolios that correspond to these Arrow securities. You can refer to the example on slides 12-13, but the idea is that the columns of the matrix r 1 are the portfolios that correspond to Arrow securities. Here
020 1 7 1 r = hence r = 400 100
100 35 1 0 20
Youneed4000( 7 + 1 )= 30unitsofasset1and4000 1 =200ofasset 400 100 20
2. Exercise 3.3 from Lengwiler.
3. Exercise 3.4 from Lengwiler.
3.3 and 3.4 are solved in the Lengwilerís book, you may refer to the deÖnitions from the slides 8-10 as well.
4. Consider an economy with one good and two states.
(a) Suppose we have two Önancial assets and the asset markets are complete.
Show that a consumerís budget constraints for s = 0; 1; 2,
p0x0+q1z1+q2z2 = p0!0
p1 x1 = p1 !1 +r1 z1 +r12 z2
p2 x2 = p2 !2 +r21 z1 +r2 z2 can be collapsed into one budget constraint
Q1 x1 +Q2 x2 +x0 =Q1 !1 +Q2 !2 +!0
Determine Q1 and Q2: (you may use some particular values for r11;r12; etc.)
a.Letr=r1 r12 =1 3:Alsoletq=(q;q)=(1;1)tokeepthings
r21 r2 2 1 1 2 simple. First, present
p1 x1 p2 x2
r1 z1 +r12 z2 r21 z1 +r2 z2
= p1 !1 +r1 z1 +r12 z2 = p2 !2 +r21 z1 +r2 z2
= 1z1 +3z2 =p1 (x1 !1) = 2z1 +1z2 =p2 (x2 !2)
We proceed by treating these as a system of two linear equations in two unknowns z = (z1;z2): We need to solve it for z = (z1;z2): What is written is a system of equations in matrix form. (Follow the video on matrices posted before)
rz = d;where
d = d1 =p1(x1 !1):
d2 p2 (x2 !2)
For any matrix r; the product r 1 r = I; where I is the identity matrix. Then the solution to the above system is
z=r 1d: 1 3
With the values for r as above the inverse matrix r 1 = 51 2 1 : Then
z1 = 15 (3p2 (x2 !2) p1 (x1 !1));andz2 = 15 (2p1 (x1 !1) p2 (x2 !2)): These expressions can be substituted into
p0 x0 +z1 +z2 =p0 !0
p0x0+51 (3p2 (x2 !2) p1 (x1 !1))+51 (2p1 (x1 !1) p2 (x2 !2))=p0!0 p0 x0 +52p2 (x2 !2)+51p1 (x1 !1)=p0 !0
p0 x0 +51p1 x1 +52p2 x2 =p0 !0 +15p1 !1 +52p2 !2 b. Arrow security prices can be determined if r is complete, i.e. r 1 exists by
the formulae on page 19 of the slides on asset economy.
=( 1; 2)=(q1;q2)r 1: With r 1 as above and (q1; q2) = (1; 1) ;
(1;2)=51(1;1) 1 3= 15;25: 2 1
Note that the budget constraint in a. is e§ectively
p0x0+ 1p1x1+ 2p2x2 =p0!0+ 1p1!1+ 2p2!2
Which is the budget constraint on page 28 of the asset economy slides. The interpretation is that Arrow security prices 1 and 2 give the exact prices of transferring wealth into (or from) states 1 and 2.
c. Suppose we have one Önancial asset hence the asset markets are not complete. Nowyouhaveoneassetwithpayo§matrixr=r1 Showthataconsumerís
budget constraints for s = 0; 1; 2,
p0 x0 +q1 z1 p1 x1 p2 x2
= p1 !1 +r1 z1 = p2 !2 +r2 z1
can be collapsed into two budget constraint
P1 x1 +x0 = P1 !1 +!0 P2 x2 +x0 = P2 !2 +!0
(a) Determine P1 and P2: (you may use some particular values for r1; etc.) Solution. Express z1 from the time 0 budget constraint
z1 = p0 (!0 x0): q1
Now sub it into the state 1 and 2 budget constraints
Rewrite them as
p1x1 = p1!1+r1p0 (!0 x0) q1
p2x2 = p2!2+r2p0 (!0 x0) q1
r1p0 x0+p1x1 = p1!1+r1p0 !0 q1 q1
r2p0 x0+p2x2 = p2!2+r2p0 !0 q1 q1
These can be further polished and manipulated but the main point here is that such system does not collapse into one budget constraint as opposed to the system in a.
5. Suppose we have two Önancial assets and the asset markets are complete. For-
mulate the portfolio choice problem. That is consider the agent with today
wealth w ; who has access to the Önancial markets with two assets with the 0 r 1 1 r 12
payo§s r = r21 r2 with the prices of the assets (q1; q2) : (you may use some
particular values for r1; etc.) At date 0 the agent can decide on asset hold- ings (z1 ; z2 ) which determines his t = 1 wealth, call those w1 and w2 in the corresponding states.
(a) Derive the agentís budget constraint, that is the combinations of w1 and w2 that are feasible given the markets and the initial wealth.
Solution. Takesomer=r1 r12 =1 2;someq=(q;q)=(2;1) r21r2 31 12
and some w0 = 20: The agent has no wealth endowment in states 1 and 2 and has to transfer the wealth from date 0 using this Önancial market. At date 0 she purchases (maybe sells short) the Önancial assets in the quantities (z1;z2) and obeys the constraint
2z1+z2 = 20; (1) q1z1 + q2z2 = w0 (in general)
Holding the portfolio (z1; z2) allows having wealth w1 = z1+2z2 instate1
w2 = 3z1+z2 instate2.
These two expressions together with (1) is the market span of this economy. Use
the budget constraint (1) to express say z2 = 20 2z1 and substitute into w1 and w2 w1 = z1 +2(20 2z1)=40 3z1
w2 = 3z1 +20 2z1 =20+z1: Now express away z1 as well connecting w1 and w2
w1 + 3w2 = 100: (2) as the budget constraint in the space (w1; w2) :
b. Notice that we could take a more direct approach since the asset market is complete. Calculate the Arrow security prices using
= q r 1: 4
Theinversematrixr 1 =15 1 2 :Withq=(2;1); 1 =51; 2 =35: 3 1
These are the prices for transferring wealth into states 1 and 2. They are both positive hence our economy has no arbitrage. The budget constraint can be written as
1w1+ 2w2=w0: Notice that this produces exactly (2) above.
Note also that this budget constraint has a usual negative slope. Notice also that we restrict w1;w2 0; but we place no such restrictions on z1 and z2: In fact some portions of the budget constraint are only feasible with negative z2 or z1: For example w1 = 100; w2 = 0 is feasible according to (2) and would require a portfolio satisfying
w1 =40 3z1=100 w2 =20+z1=0:
Hencez1 = 20andz2 =20 2z1 =60forw1 =100;w2 =0:Investingallofthe wealthintoasset1allowsw1 =10;w2 =30:Indeed,withw0 =20andq1 =2;one can buy 10 units of this asset. According to the payo§ matrix r; this will produce 10 in state 1 and 30 in state 2. At the same time investing all the wealth into asset 2allowsw1 =40;w2 =20:Similarlogic,q2 =1soonecanbuy20unitsofthis asset. According to the payo§ matrix this produces 40 in state 1 and 20 in state 2. Check that both such combinations are on the budget constraint. All the wealth combinations in between these points require z1 0; z2 0: The points further to the left from w1 = 10; w2 = 30 require z2 < 0; the points further to the right from thepointw1 =40;w2 =20requirez1 <0:
0 10 20 30 40 50 60 70 80 90 100
Finally, imagine that the prices of the assets are (q ; q ) = (1; 3) instead. With 12
thesamer 1 =51 1 2 ;now( 1; 2)=(1;3)15 1 2 = 85; 15:The
3 1 3 1 5
budget constraint is still
1w1+ 2w2=w0
but now it is upward sloping as 2 < 0: The agent can get inÖnite wealth by holding
the appropriate portfolio. Design one such portfolio.
程序代写 CS代考 加微信: powcoder QQ: 1823890830 Email: powcoder@163.com