chapter 1 fisher separation theorem. a.consumption and investment without capital markets...
TRANSCRIPT
Chapter 1
Fisher Separation Theorem
A.Consumption and investment without capital markets
1. Assumptions1) All outcomes from investment are known wit
h certainty, i.e Ri=a1u1+a2u2+…+anun
2) No transaction costs, no exchange
3) No taxes
4) Two-period model
A.Consumption and investment without capital markets
2. Optimal consumption without capital markets1) Consumption
A.Consumption and investment without capital markets
)1(
)(
)(
0
101
i
Bui
ii
r
Cu
CuMRS
BA
BA
rr
MRSMRS
0101
MRS: MRS decreasing:
A.Consumption and investment without capital markets
2) Production
A.Consumption and investment without capital markets
BA
BA
rr
MRTMRT
0101
)1(
)]([
)]([
0
101
i
Bi
ii
r
IWu
IWuMRT
MRT: MRT increasing:
A.Consumption and investment without capital markets
3) Optimal consumption
),(
),(),(),(
:),!(
),,(),(:
),,(),(:?
:
:?
10
101010
2
1010
10101
0
CC
CC
BBDDCC
DD
EEDD
BB
AABB
EE
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CCatnconsumptioOptimal
MRTMRS
CCuCCuCCu
nconsumptioOptimalCtPoinYesu
lessinvestMRTMRS
butCCuCCuDtPoin
moreinvestMRTMRS
butCCuCCuBtPoinu
lessinvestMRSMRTEtPoin
moreinvestMRSMRTAtPoinu
A.Consumption and investment without capital markets
1) Optimal consumption for different investors
201
201
101
101 MRTMRSMRTMRS
Individual 1 prefers consuming more at C1
Individual 2 prefers consuming more at C0
B.Consumption and investment with capital markets
1. Assumptions1) All outcomes from investment are known
with certainty.
2) Inter-temporal exchange rate of consumption bundles, r>0 is known. No transaction costs.
3) No taxes.
4) Two-period model
B.Consumption and investment with capital markets
2. Optimal consumption with capital markets1) consumption
B.Consumption and investment with capital markets
A. Initial endowment, A:
B. Capital market line (CML)
C. Slope of u0 at A=-(1+ri) Slope of CML=-(1+r)
, Invest more consume less at C0
r
yyw
11
00
01 uu
BArr i
B.Consumption and investment with capital markets
intercept slope
w (unchanged), u:u0→u1(↑)
)1(
)1()1(
11
*01
*00
*1
*1*
01
00
rcw
rcrwc
r
cc
r
yyw
B.Consumption and investment with capital markets
2) Production
)1(0101 iii rMRTMRS
A: Personal B: Market -(1+r)
B.Consumption and investment with capital markets
3) Optimal consumption
B.Consumption and investment with capital markets
A. Initial endowment, A(c0,c1)
Invest more at t=0
(or consume less at t=0)
A→B,
ii MRTMRS 0101
?)（unchanged ,,
),(),(
),(),(
10
*1
*010
10*1
*0
why
wccwhere
cccc
ccuccu
B.Consumption and investment with capital markets
B. Optimal consumption without capital markets,
ri>r
markets offer cheaper funds
invest less at t=0
consume more at t=0
borrow more to consume
B→C,
C. Optimal consumption with capital markets,
),( *1
*0 ccB
?),,
),(),(
),(),(
10
**1
**0
*1
*0
*1
*0
**1
**0
whywccwhere
cccc
ccuccu
（
),( **1
**0 ccC
C.Implications1.
2.
3. Fisher separation theorem “Given perfect and complete markets, the production is governed
by an objective market criterion without regard to individual’s subjective preferences that enter into their consumption decision.”
)(
)(
21
32
123
adjustmentselfadjustmentPortfoliouu
LeveragefunctionmarketCapitaluu
uuu
)(
111
0
**1**
0
LeverageincreaseswealthCB
changeswealthNoBAr
cc
r
cc
C.Implications
1) Complete market Basis Span, linear combination Linear independent2) Perfect market
A. No transaction costs, No taxes(Market frictionless)
B. No short sell restrictionC. Perfect competition, price takersD. Perfect information, no information cost,
asymmetric.
C.Implications
3) Optimal production, MRR=ri
Optimal consumption, ri= r
C.Implications
4. Unanimity principle
Same production opportunity set, same investment portfolio,C
MRTrMRSMRSiumInequilibrc
II
tatmoreinvest
rr
rMRTMRSInvestorb
II
consumetotatmoreborrow
rr
rMRTMRSInvestora
)1(,.
0
)1(,2.
0
)1(,1.
201
101
2'2
2
102
102
1'1
1
101
101
C.Implicationsd.Individual delegates investment decision to managers(No individuals’
utility functions involved) Investors’ required rate of return =Market required rate of return Maximization of investors’ wealth Investors’ borrowing or lending according to their utility function
Managers made production decision without regard to the utility of the individual investor
Rf.DeAngelo[1981], Makowski[1983]