1. The Lagrangean

Contingent Claims Solutions

L(x1;x2;) = x1x2 (p1x1 +p2x2 M) Lx1(x1;x2;) = x2p1=0

Lx2(x1;x2;) = x1p2=0

Copyright By cscodehelp代写 加微信 cscodehelp

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 )

程序代写 CS代考 加微信: cscodehelp QQ: 2235208643 Email: kyit630461@163.com