计算机代考 1. The Lagrangean – cscodehelp代写

1. The Lagrangean
Contingent Claims Solutions
L(x1;x2;) = x1x2 (p1x1 +p2x2 M) Lx1(x1;x2;) = x2p1=0
Lx2(x1;x2;) = x1p2=0

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L(x1;x2;) = p1x1+p2x2M=0
Express x1 = p2 and x2 = p1 and substitute into the third equation.
2p1p2 = M=)= M 2p1 p2
x1 = M;x2=M 2p1 2p2
 M2 du(x1;x2)M u(x1;x2) = 4pp =) dM =2pp =:
This illustates the derivation on the slides 26-27 on Optimization. That derivation proves that du(x1;x2) =  is always true.
2. Exercise 2.2 from Lengwiler.
3. Exercise 2.3 from Lengwiler. I give a. here. The solution at the end of the book is a bit too terse.
The budget constraint connects the two periods. Assume p1 = 1 and p2 = p: (!x1) = px2 ()x1+px2 =!
L(x1;x2;) = lnx1 +lnx2 (x1+px2 !) L x 1 ( x 1 ; x 2 ;  ) = x1   = 0
L x 2 ( x 1 ; x 2 ;  ) = x  p = 0 2
L(x1;x2;) = x1+px2!=0 (1) Divide the last two to eliminate 
x2 = =)px2 =x1subinto(1). x1 p
x1+x1 = !=)x1 = ! whynohere? 1+
savingss=!x1 =  !andx2 = ! 1+ (1+)p

4. Consider consumers A and B with uilities
uA xA1 ;xA2  = lnxA1 +(1 )lnxA2 uB xB1 ;xB2  = lnxB1 +(1 )lnxB2 :
Suppose the total endowment of good 1 in this economy is ! and of good 2  1
it is !2: Suppose !A1 ;!A2 is owned by consumer A and !B1 ;!B2 is owned by consumer B: We have !A1 + !B1 = !1 and !A2 + !B2 = !2 by deÖnition.
(a) What is the exchange economy equilibrium here.
(b) What is a Pareto e¢ cient allocation? The social welfare is maximized by allocating the goods to consumers A and B; and the allocation has to respect the resourse constraints. Formulate the search for such allocations as an optimization program.
(c) What can be said about the equilibrium prices at that allocation?
Solutions.
a. Let p = p1=p2: This is equivalent to setting p1 = p and p2 = 1: That is, consumer A solves
uAxA1;xA2 = lnxA1 +(1 )lnxA2 !max subjectto pxA1 +xA2 = p!A1 +!A2:
The Lagrangean here
L= lnxA1 +(1 )lnxA2 pxA1 +xA2 p!A1 !A2 The First order conditions
Eliminate :
@L = p = 0: @xA1 xA1
@L = 1 =0: @xA2 xA2
xA2 = p: Hence pxA = xA: 1xA 1 12
Take that into the BC
xA2 +xA2 = p!A1 +!A2:
xA2 = (1 )p!A1 +!A2:
xA1 =pp!A1+!A2

Similarly for consumer B
xB2 = (1 )p!B1 +!B2 :
xB1 =pp!B1+!B2:
To Önd the equilibrium p we impose market clearing conditions on one of the
goods. Either good will do, choose the one that looks easier to deal with.
xA2 +xB2 = (1 )p!A1 +!A2+(1 )p!B1 +!B2=!A2 +!B2: (1 )p!A1 +!B1  = !A2 +!B2 
Only the total endowments here matter for the relative price p; recall that this is
p = !2 : 1 !1
p1=p2. One may substitute this into xA and xB to obtain the equilibrium allocations. xA2 = (1 )p!A1 +!A2= !2!A1 +(1 )!A2:
xA1 = p p!A1 +!A2 = !A1 +(1 )! !A2
Recall that we have denoted !A1 + !B1 = !1 and !A2 + !B2 = !2: With these
xA2 = !2!A1 +(1 )!1!A2=!1 xA1 = !2!A1 +(1 )!1!A2=!2 xB2 = !2!B1 +(1 )!1!B2=!1 xB1 = !2!B1 +(1 )!1!B2=!2
= !2 !A1 +!B1 +(1 )!1 !A2 +!B2 =!2 = = ( !2!1 +(1 )!1!2)=!2 =!1 =!A1 +!B1
xA2+xB2 =!2=!A2+!B2
b. We maximize the utility of A subject to ìnot hurtingîconsumer B: That is
lnxA1 +(1 )lnxA2 ! max subjectto
u  lnxB1 +(1 )lnxB2:
u  ln!1 xA1 +(1 )ln!2 xA2 : 3

The last object is the utility of B that incorporates the resourse constraints, !1=xA1 +xB1;!2=xA2 +xB2:
The Lagrangean here
L= lnxA1 +(1 )lnxA2 u ln!1 xA1(1 )ln!2 xA2 The Örst order conditions
@L= =0 @ x A1 x A1 ! 1 x A1
@L=11 =0 @ x A2 x A2 ! 2 x A2
Form these two equations
Cross multiply
x A2 = ! 2 x A2 : x A1 ! 1 x A1
xA2 !1xA1 = xA1 !2xA2 x A2 ! 1 = x A1 ! 2
Hence !2 x A2 = x A1 ! :
1 AA A!2 Now let us calculate the gradient of uA at any such allocation x1 ; x2 = x1 !1 .
c. Notice that
ruAxA1;xA2=uxA;uxA= ;1 !1: 12 xA1xA1!2
p 1 = ! 2 = u x A1 ; p2 1!1 uxA2
where p1=p2 comes from a; and uxA1 from b: This states that the equilibrium price uxA2
vector p is collinear (aligned) with the gradient of the utility of consumer A at any Pareto e¢ cient allocation (irrespective of u). This is special for this economy with identical preferences for all consumers.
5. Consider the same consumers A and B that live for two periods and suppose now !1 = 6 and owned by A. This is the entire crop of this economy at date 1. Suppose also that !2 = 12 and owned by B. This is the entire crop of this economy at date 2. The good is perishable and cannot store.

(a) Suppose A and B invented a way to trade over times 1 and 2. Formulate the consumer choice problems for A and B when such trade is possible.
(b) What is the equilibrium?
(c) Verify that trading over time allows to achieve Pareto e¢ ciency.
Solutions.
a. Again let p = p1=p2: Then, consumer A solves
uAxA1;xA2 = lnxA1 +(1 )lnxA2 !max subjectto pxA1 +xA2 = p!1:
The Lagrangean here
L= lnxA1 +(1 )lnxA2 pxA1 +xA2 p!1 The First order conditions as before
Eliminate :
@L = p = 0: @xA1 xA1
@L = 1 =0: @xA2 xA2
xA2 = p: Hence pxA = xA: 1xA 1 12
Take that into the BC
Similarly for consumer B
xA2+xA2 =p!1:
xA2 =6(1)p:
xB2 =12(1): xB1 =12p:
b. To Önd the equilibrium p we impose market clearing conditions on one of the goods. Either good will do, choose the one that looks easier to deal with.
xA2 +xB2 = 6(1 )p+12(1 )=12: 6(1 )p = 12

The equilibrium allocations are
x A1 = 6 ; x B1 = 6 ( 1 )
xA2 = 12 ;xB2 =12(1 ):
c. To verify that this is Pareto e¢ cient calculate the gradients of the utility at
the eq. allocation. They should be collinear.
ruAxA1;xA2 = uxA;uxA= ;1 =1;1 : 12 xA1xA2 612
ruBxB1;xB2 = uxB;uxB= ;1 = ; 1: 12 xB1xB2 6(1)12
Vector ruA xA1 ; xA2  is collinear to ruB xB1 ; xB2  if there is a number  such that r u A x A1 ; x A2  =   r u B x B1 ; x B2  :
That is we seek  such that 1 =  and 1 =  1 hold simultaneously. 6 6(1 ) 12 12
Clearly=(1 )= willdothetrick.
6. Consider the same consumers A and B that live for two periods. At time 0 they do not consume and only plan their consumption at date 1. At date 1 there canbeeitheragooddayforMr. Awith!A1 =6forAand!B1 =2;oragood dayforMrs. Bwith!A2 =0and!B2 =12.
(a) Suppose A and B can write contingent contracts. What are these? At what date are they written? Formulate the consumer choice problems for A and B with contingent claims.
(b) What is the equilibrium?
(c) Verify that contingent claims allow to achieve Pareto e¢ ciency.
Solutions.
a. Let p1 be the price of the contingent claim of one unit of the good available is state 1 and p2 the same for state 2: Let bA1 be the amount of contingent claims for state 1 that A buys (or sells). DeÖne bA2 and bB1 and bB2 similarly. At date 0 only contingent claims can be traded, no physical goods are available yet. Hence for consumer A the budget constraint is
p1bA1 +p2bA2 =0: 6

To get b she has to sell b or vice versa. Consumer A cares about her consumption 12
xA1 ; xA2 : Contingent claims allow to have xA1 = bA1 + !A1 and xB1 = bB1 + !B1 : With the assumption that the prices of the contingent claims match the prices of the physical goods the BC above is equivalent to
p1xA1 + p2xA2 = p1!A1 :
uAxA1;xA2 = lnxA1 +(1 )lnxA2 !max subjectto
p1xA1 +p2xA2 = p1!A1: Normalizing the prices as before
uAxA1;xA2 = lnxA1 +(1 )lnxA2 !max subjectto pxA1 +xA2 = p!A1:
The Lagrangean here
L= lnxA1 +(1 )lnxA2 pxA1 +xA2 p!A1 The First order conditions as before
consumer A solves
Eliminate :
@L = p = 0: @xA1 xA1
@L = 1 =0: @xA2 xA2
xA2 = p: Hence pxA = xA: 1xA 1 12
Take that into the BC
Similarly consumer B solves
uBxB1;xB2 = lnxB1 +(1 )lnxB2 !max subjectto
pxB1 + xB2 = p!B1 + !B2 : The Lagrangean here
L= lnxB1 +(1 )lnxB2 pxB1 +xB2 p!B1 !B2 : 7
xA2 +xA2 = p!A1:
xA2 =6(1)p:

The First order conditions as before
@L = p = 0: @xB1 xB1
@L = 1 =0: @xB2 xB2
xB2 = (1 )(2p+12): xB1 = p(2p+12):
b. To Önd the equilibrium p we impose market clearing conditions on one of the goods. Either good will do, choose the one that looks easier to deal with.
xA2 +xB2 = 6(1 )p+(1 )(2p+12)=12: (1 )8p = 12
The equilibrium allocations are
xA1 = 6 ;xB1 = 2+12=2 +8(1 )
xA2 = 9 ;xB2 =3 +12(1 ):
c. To verify that this is Pareto e¢ cient calculate the gradients of the utility at the eq. allocation. They should be collinear.
ruAxA1;xA2 = uxA;uxA= ;1 =1;1 : 12 xA1xA2 69
ruBxB1;xB2 = uxB;uxB= ;1 = ; 1 : 1 2 xB1 xB2 2 +8(1 ) 3 +12(1 )
Vector ruA xA1 ; xA2  is collinear to ruB xB1 ; xB2  if there is a number  such that
r u A x A1 ; x A2  =   r u B x B1 ; x B2  : Thatisweseeksuchthat 1 = and 1 = 1
neously. Clearly  = +4(1 ) = 43 will do the trick. 33
holdsimulta-
6 2 +8(1 ) 9 3 +12(1 )

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