金融经济学基础Chapter 5

4
FOC:
X
!2
i ! @ ui @ui
i!
!(ci o
!@(ccii
! o
@ci o
ci ci
!) !)
= =
i! io
8!
()
i
P
!2
!
@ ui !(ci o ci !)
@ci !
i
!
@ui
!(ci @ci
o o
ci
!)
=
! o
8! 2 (1
)
RHS Independent of i ) LHS Independent of i ) LHS ratio same for all agents.
oxr!de!r)=
1
2
:::
;1
) Payo matrix looks like (Prove X Invertible)
0 BBBBBBB@
x0!1 0.. 0
x!2x;0!.. 2x!1 0
xx!!33x;;!.. 3xx!!12 0
::: :::
.: .: .:
:::
x! x! x!
x!
;
;..
Example: Replicate, and Price A/D 1 0 0] Choose f g s.t.
X j xj ! = 1(!=1)
j
14
Matrix Notation 0 X = 1 0 0]
Since Inverse 9 0 = 1 0 0] X;1
To Price:
V = X j Sj
8
In Practice, Don't Have A/D Securities Have Complex Securities
Q? Have Pareto-E cient Allocation?? A! Typically, Only if Market Complete
# Independent Securities = # States of Nature Intuition: With Complete Markets:
1
Given:
1) State Prices (Complete Market)
2) 3)
IAngdgivriedguaatli'osnECnodnowstmraeinnttsffeCi oo
eCi !!gg
0
!
Individuals Objective:
ci
mo facix!
g
X
!2
i ! ui !(ci o ci !)
xx!!12
; x! ;1
1 CCCCCCCA
Clearly, such a market does not exist in practice.
16
Restrict Economy To:
3
However, The Economy Constraints + (Individual Constraints) Linearly Dependent
) Only Relative Pricing Typically Choose o = 1 (Numeraire)
vs. Partial Equilibrium For Individual Problem (Prices f !g Given): ( + 2) Equations ( + 2) Unknowns ci o fci !g i
s.t. Where:
ci
mo faxi j
g
X
!=1
i ! ui !(ci o ci !)
ci o + XN i j Sj = ei o + XN ^i j Sj
j=1
j=1
ci ! = Wi ! = XN i j xj !
j=1
Note:
@ui !(ci o ci !)
@ ij
=
@ui !(ci o ci !) @ci !
mi = X;1 S
RHS Indep. of i ) LHS Indep. of i
Have Pareto E ciency ) Wealth Only Constraint
Have Solved for Ratio of Marginal Utilities ) mi! = m! = !
13
Implications
A securities Market is a complete market i # linearly independent securities = # of states ) Pareto optimal allocation achieved in competitive
equilibrium with individuals trading only in complex securities. Why? Because A/D securities can be replicated if securities market is complete
) No Pareto Optimality
)
Due
to
P
!=1
12
De ne
@ui !
mi!
i
!
P @ci !
!=1 i !
@ui @ci
!
0
Then
X xj ! mi! = Sj
!=1
Or, in Matrix Notation
Xmi = S
If Markets Complete ) X Invertible:
@ci ! @ i j
=
@
ui
!(ci
@ci
o !
ci
!)
xj
!
11
Solve FOC
X
!=1
i
!
@ui @ci
! !
xj
Leabharlann Baidu
!
=
Sj
X
!=1
i
!
@ui @ci
!
0
=
Hence, in contrast to (*):
X
!=1
@ui !
i
!
P @ci !
!=1 i !
@ui @ci
!
0
xj
!
=
Sj
Does Not Indicate that MRS Between Present and Future Consumption Are Identical Across Individuals
ui !5
i=1 !=1
I(1 + ) Choice Parameters (I fci o ci !g)
( + 1) Market Clearing Constraints
L=
XI
i
2 4
X
3
i ! ui !5 +
i=1
!2=1 0 4C0
;
XI
ci
3 o5
+
X
!
2 4C!
;
XI
ci
3 !5
i=1
Endowment of Individual i = fei o, ^i jg Economy has 1) Spot Commodities Market
2) Securities Market Sj Time-0 Price of Security j
10
Individual's Objective:
How can markets be made complete?
1) Assume can create a State Index Portfolio, where payo s x are s.t. x! 6= x!0 8 ! !0 2
2)
Create calls (Place in
with strikes descending
2) Competitive equilibrium is allocationally e cient when markets are complete.
) PO allocation obtains in a competitive economy if markets are complete.
Max E U] s.t. Wealth Constraint With Incomplete Markets:
Max E U] s.t. Wealth, Market Constraints
9
Securities Markets Economy
1 Unit Each of N Securities j = f1 2 Ng Security j Pays xj ! in State ! ) x~j = RV, xj! = Number ! 2 (1 )
5
For Time-Additive, State-Independent Utility: U = ui o(ci o) + X i ! ui(ci !)
!2
Simpli es to (MRS):
i
!
@
ui(ci !) @ci !
@ui o(ci o)
=
! o
8! 2 (1
)
@ci o
If Assume Homogeneous Beliefs i ! = ! 8 i 8 ! :
allocation, 9 (positive) f g s.t. same allocation can
be achieved by social planner maximizing linear
combination
ofUin=divXiIduail24'sXutilitiy!
func3tions.
!=1
i=1
Here, f o !g are Lagrange Multipliers.
Solve, Set
i
=
1
i
) get Eq.( ).
7
Implications
()
i
P
!2
!
@ ui !(ci o ci !)
@ci !
i
!
@ui
!(ci @ci
o o
ci
!)
=
! o
8! 2 (1
)
1) An allocation is e cient, or Pareto Optimal (PO), i for all states ! the MRS between present and future consumption is equal across individuals.
ei o ; ci o + X
!
3 ei ! ; ci ! 5
!2
2
First-Order Conditions: 8 i 2 (1 I)
@L
@ci o
@L
@c@i
! L
@i
= = =
0 0 0
= = =
X
!2
@
i!
o ei
@ui
i!
ui !(ci
@ci o ; ci o
!(ci
@ci
o o
ci
!)
@ ui !(ci !) @ci !
@ui o(ci o) @ci o
=
1
o
! !
8! 2 (1
)
RHS Independent of i ) LHS Independent of i
RHS = State Price Per Unit of Probability
6
Recall 2nd Welfare Thm: For each Pareto-optimal
j
0S
= 1 0 0] X;1 S 1 0 0]
Again we see that in state !
! is price per unit consumption
15
In an incomplete market and arbitrary preferences Pareto E ciency not obtained in general.
Achieve an E cient Allocation.
Typical Answer: Need (Dynamically) Complete Markets for Pareto E ciency. However, in special cases, Pareto E ciency may hold in incomplete markets.
s.t.
o
ci
o
+
X
!2
! ci ! =
o
ei
o
+
X
!2
! ei !
Note: State-Dependent Utility: Compare With ui o(ci o) + X i ! ui(ci !)
!2
Lagrangian:
L = X i ! ui !(ci o ci !)
!2 2
+i4 o
AtlChloeocnaVttaiinvlugeaeEtnitonSceioecfnucSryittaiaetnsed
Ref: H&L Chap 5
2 Questions: 1) What is the Role of Financial Assets in
Allocating Resources Among Individuals? 2) What Types of Securities are Needed to
;
i
o ci !) ;
!
+
X
i !
!
ei !
o ;
8! ci !
!2
Aggregation Constraints:
X ci 0 = C0 X ci ! = C! 8!
i
i
How Many Equations/Constraints?? I( + 2) + ( + 1)
How Many Unknowns (Individuals, Prices)?? I(ci o ci ! i) + ( + 1) = I( + 2) + ( + 1)