GAO Feng (Tsinghua SEM)
HE Ping (Tsinghua SEM)
HE Xi (MIT Economics)
Preface

 A long time ago, a visitor from out of town came to a tour
in Manhattan. At the end of the tour they took him to the
financial district. When they arrived to Battery Park the
guide showed him some nice yachts anchoring there, and
said, "Here are the yachts of our bankers and
stockbrokers." "And where are the yachts of the
investors?" asked the naive visitor.
Investment

 Investment decisions are intertemporal choices involving
tradeoffs among costs and benefits occurring at different
times, which not only affect one's health, wealth, and
happiness, but may also determine the economic
prosperity of nations
 Fisher (1930): investment is not an end in itself but rather
a process for distributing consumption over time
 Major concerns: return, risk, information acquisition, lifecycle, liquidity constraint, risk preference, background
risk, etc.
Related Studies

 Samuelson (1969, REStat), Merton (1969, REStat): dynamic
programming with uncertainty
 Ehrlich and Hamlen (1995, JEDC): precommitment
strategy with intermittent revision
 Campbell and Viceira (1999, QJE): time varying
investment opportunities
 Viceira (2001, JF): background risk, life-cycle
 Gollier (2002, JME): Liquidity constraint, decreasing
aversion to risk on wealth
 Chacko and Viceira (2005, RFS): incomplete market with
stochastic volatility
Empirical Facts

 Investment behaviors are more complex than what most
standard theories could explain
 Barber and Odean (2002, RFS): online trading make
investors trade more actively but less profitable
 Barnea, Cronqvist and Siegel (2010, JFE): genetic factor is
critical for investor behavior
 He and Hu (2010, RBF): horizon effect
 Mastrobuoni and Weinberg (2009, AEJ-EP): consumptions
are not smoothed
 Meier and Sprenger (2010, AEJ-AE): individuals with
present-biased preference over-borrow on their credit cards
Behavioral Theories and Investment

 Barberis & Huang (2001, JF): mental accounting and loss
aversion
 Angeletos, Laibson, Repetto, Tobacman and Weinberg
(2001, JEP); Harris & Laibson (2001, Econometrica);
Salanie and Treich (2006, EER): hyperbolic discounting
 Grenadier & Wang (2007, JFE): real options investment
model with hyperbolic discounting entrepreneurs
 Munk (2008, JEDC): habit formation
Hyperbolic Discounting

 Frederick, Loewenstein and O’Donoghue (2002, JEL): given two
similar rewards, humans show a preference for one that arrives sooner
rather than later, but valuations fall very rapidly for small delay
periods, but then fall slowly for longer delay periods
Facts Related to Hyperbolic Discounting

 Time inconsistent preferences, implying a motive for
consumers to constrain their own future choices
(Laibson, QJE 1997)
 Under-saving (Laibson, EER 1998; Diamond and
Koszegi, JPubE 2003; Salanie and Treich, EER 2006)
 Over-borrowing (Heidhues and Kőszegi, AER 2010)
 Use of commitment device (Basu, AEJ-Micro 2011)
The Role of Intermediaries for Investors

 Information production: He (2007, RFS); Gorton and He
(2008, RES)
 Monitoring: Diamond (1984, RES)
 Screening: Bernanke and Blinder (1988, AER)
 Liquidity provider: Diamond and Dybvig (1983, JPE)
 Risk transformation: Diamond (1984, RES)
 Maturity transformation: Diamond and Dybvig (1983, JPE)
 Payment methods: He, Huang and Wright (2005, IER)
Goal of This Paper

 Time inconsistent preference generates a liquidity
shortage for the investor who invests on his own
 Financial intermediaries make investments on behalf
of the investors and provide liquidity for
unsophisticated investors
 The financial intermediaries in our model can be
interpreted as banks, pension funds, mutual funds,
etc.
Related Works

 DellaVigna and Malmendier (2004, QJE): contract
design with time inconsistency (monopoly firm)
 Heidhues and Kőszegi (2010, AER): credit contract
with time inconsistency (competitive firm)
 Basu (2011, AEJ-Micro): individuals join rotational
savings and credit associations (roscas) to fund
repeated purchases of nondivisible goods without
defect even when there is no punishment, roscas
serves a commitment device
Agenda

 Basic model
 Competitive equilibrium
 Linear contract and term premium
 Discussions
 Conclusion
Investment Technology

 Three dates (t=0,1,2)
 Each agent is endowed with 1 unit of good at date 0,
and consumes at date 1 and 2
 The good can be stored with 0 return, or invested in
a project at date 0 with a return R > 1 at date 2, if it is
liquidated at date 1, one can get 1
Time Inconsistent Preferences

 Self 0’s utility is u(c1) + u(c2)
 Self 1’s utility is u(c1) + βu(c2)
 Self 0 believes that self 1’s utility
u (c )  βˆu (c )
1
2
β  βˆ  1
perfect sophistica tion : βˆ  β
complete naivety : βˆ  1
First Best

 We measure welfare using long-run self-0 utility
 The first best solution does not depend on degree of
time-inconsistency
max c1 ,c2 u (c1 )  u (c2 )
s.t. c1  c2 / R  1
u ' (c1fb )
FOC :
R
fb
u ' (c2 )
Diagram of Proof

c1
1
(c1fb , c2fb )
u(c1 )  u(c2 )
R
c2
Autarky

 Investors cannot commit, and liquidation has no
cost, so they will liquidate some of the investment
for consumption at date 1 based on their date 1
preference regardless what they believe at date 0
max c1 ,c2 u (c1 )  βu (c2 )
s.t. c1  c2 / R  1
u ' (c1at )
at
fb
at
fb
FOC :

β
R

R

c

c
,
c

c
1
1
2
2
u ' (c2at )
Diagram of Proof

c1
1
(c1at , c2at )
(c1fb , c2fb )
u(c1 )  βu(c2 )
u(c1 )  u(c2 )
R
c2
Ineffective Market

 In the autarky case, if we allow for trading at date 1, that
is, an investor can trade his date 2 consumption from his
investment for date 1 consumption, investors will have
the same consumptions as in autarky case
 Proof: The price of date 2 consumption, p, must be 1/R,
otherwise either (1,0) or (0,R) will dominate all other
points on the budget line and it cannot be equilibrium
max c 0 ,c 0 ,c ,c u (c1 )  βu (c2 )
1
s.t.
2
1
2
c1  pc2  c10  pc20
c10  c20 / R  1
Diagram of Proof

c1
pR
p > 1/R
1
p = 1/R
p < 1/R
R
1/p
c2
Role of Intermediary

 At its own best interest, an intermediary can improve
the welfare of an investor with time inconsistent
preference by offering a contract that punishes early
withdraw
Incentive Compatible Contract

 Assume there are finite β’s among people, with β1 <
β2 < … < βN, and βˆ {βˆ1 , βˆ2 ,..., βˆ N } , and financial
intermediaries offer a finite menu of repayment
options C = {(c1s, c2s)}s  S .
 An incentive compatible map (c1(.), c2(.)): {β1, β2, …
,βN} {βˆ1 , βˆ2 ,..., βˆ N }R+ satisfies the following
condition:
u (c1 ( β ))  βu (c2 ( β ))  u (c1 )  βu (c2 )
β { β1 , β2 ,..., βN }  { βˆ1 , βˆ2 ,..., βˆ N } and (c1 , c2 )  C
Equilibrium Definition

 We define a competitive equilibrium as a contract C
offered by the financial intermediaries with an
incentive compatible map (c1(.), c2(.)) that satisfies the
following properties:
1. Zero-profit
2. No profitable deviation, there exists no contact C’
with incentive-compatible map (c1‘(.), c2‘(.)) such that
for some β, u(c1‘(β)) + βu(c2‘(β)) > u(c1(β)) + βu(c2(β)),
and C’ yields positive profits
3. Non-redundancy
Observable Naïve Investors

 The financial intermediary solves
( β  βˆ )
max cˆ1 ,cˆ2 ,c1 ,c2 (1  c1 ) R  c2
u (cˆ1 )  u (cˆ2 )  u
s.t.
(paticipat ion constraint , PC)
u (cˆ1 )  βˆu (cˆ2 )  u (c1 )  βˆu (c2 )
(perceived - choice constraint , PCC)
u (c1 )  βu (c2 )  u (cˆ1 )  βu (cˆ2 )
(incentive - compatible constraint , IC)
 u is the perceived utility from the perspective of date 0 if
she accepts the contract
Equilibrium Outcome

 PC must be binding
 IC must be binding
 PCC is equivalent to cˆ1  c1 and cˆ2  c2
 Perceived date 1 consumption is zero: cˆ1  0
 Competitiveness will drive the financial
intermediary’s profits to zero
 The problem is equivalent to setting the profit to be
zero with PC binding through lifting u
Equilibrium Contract

 For a naïve investor, the competitive-equilibrium
contract has two repayment options, with the
investor expecting to choose cˆ1  0, and cˆ2  0, and
actually choosing c1 and c2 satisfying
(1  c1 ) R  c2  0  c1  c2 / R  1
u ' (c1 )
 βR
u ' ( c2 )
 Equivalent to the autarky case
Diagram of the Result

c1
1
u (c1 )  βu (c2 )  u (cˆ1 )  βu (cˆ2 )
where u (cˆ1 )  u (cˆ2 )  u with cˆ1  0
u(c1 )  βu(c2 )
(c1 , c2 )
(cˆ1  0, cˆ2 )
(1  c1 ) R  c2  0
R
c2
Intuition

 At date 0, the intermediary will offer a perceived-choice
contract with very high date 2 consumption but zero date
1 consumption while expecting the investor with a need
of immediate gratification at date 1 will switch to a
contract with early withdraw of date 2 consumption
despite of a high penalty
 The more date 2 perceived-consumption, the greater drop
in utility when date 1 comes, the more desperate the
investor is, and the less the intermediary needs to offer in
an alternative contract
Observable Sophisticated Investors

 The financial intermediary solves ( β  βˆ )
max c1 ,c2 (1  c1 ) R  c2
s.t. u(c1 )  u(c2 )  u (PC)
Equilibrium Outcome

 PC must be binding
 Competitiveness will drive the financial
intermediary’s profits to zero
 The problem is equivalent to setting the profit to be
zero with PC binding through lifting u
Equilibrium Contract

 For a sophisticated investor, the competitiveequilibrium contract has a single repayment option
satisfying
(1  c1 ) R  c2  0  c1  c2 / R  1
u ' (c1 )
R
u ' ( c2 )
 Equivalent to the first best case
Diagram of the Result

c1
1
Equilibrium consumption for naïve investors
Equilibrium consumption for sophisticated investors
u(c1 )  βu(c2 ) (smaller slope)
u(c1 )  u(c2 )
(1  c1 ) R  c2  0
R
c2
Intuition

 A sophisticated investor rationally expect his own
preference change and his choice at date 1, which is
the only relevant choice for his utility at date 0
Summary for Observable Preference

 For a naïve investor, the financial intermediary offers a
contract with a punishment for early withdraw, the
welfare of a naïve investor is NOT improved
 However, if liquidation is costly, then the financial
intermediary can improve welfare as it avoids costly
liquidation
 For a sophisticated investor, the first best is achieved, and
his welfare is strictly improved
 If everyone else is as naïve as you are, or everyone knows
that you are naïve, making investment through a zeroprofit intermediary does not help nor hurt
Unobservable Preference

 Again we study the most simple case: all investors has the
same β̂ at date 0, and investors are naïve ( βn  βˆ ) with
probability π, and investors are sophisticated (βs  βˆ ) with
probability 1 – π
 All investors choose the same contract (c1s, c2s) at date 0,
but naïve investors will switch to (c1n, c2n) at date 1
max c ,c ,c ,c π (1  c1n ) R  c2 n   (1  π )(1  c1s ) R  c2 s 
1n
2n
1s
2s
u (c1s )  u (c2 s )  u (PC)
s.t. u (c1n )  βnu (c2 n )  u (c1s )  βnu (c2 s ) (IC n )
u (c1s )  βs u (c2 s )  u (c1n )  βs u (c2 n ) (IC s )
Equilibrium Outcome

 PC must be binding
 ICn must be binding
 ICs implies c1s < c1n and c2s > c2n
 Competitiveness will drive the financial
intermediary’s profits to zero
 The problem is equivalent to setting the profit to be
zero with PC binding through lifting u
Equilibrium Contract

 Suppose all investors has the same β̂ at date 0, and
investors are naïve (βn  βˆ) with probability π, and
investors are sophisticated (βs  βˆ) with probability 1 – π,
the competitive-equilibrium contract has two repayment
options. All investors choosing the same contract (c1s, c2s)
at date 0, but naïve investors will switch to (c1n, c2n) at date
1. We have
π (1  c1n ) R  c2 n   (1  π )(1  c1s ) R  c2 s   0

u ' (c1n )
u ' (c1s )
π u ' (c1s ) 

 βn R,
 R1  (1  βn )
u ' (c2 n )
u ' (c2 s )
1  π u ' (c1n ) 

Interpretation of the Results

 “Efficiency-at-the-top”: the repayment schedule of
naïve investors is similar to the case with known
preference, but this is not the case for the
sophisticated investors, who get a more back-loaded
repayment schedule
 There is a discontinuity at full sophistication
Cross-Subsidy Effect

 Suppose all investors has the same β̂ at date 0, and
investors are naïve βn  βˆ with probability π, and
investors are sophisticated βs  βˆ with probability 1 – π.
In a competitive equilibrium, the intermediary makes
money on the naïve investors but loses money on the
sophisticated investors. Moreover, the sophisticated
investors’ welfare in the competitive equilibrium is
strictly increasing in π
Diagram of the Result


u' (c1n )
u' (c1s )
π u' (c1s ) 

 βn R,
 R1  (1  βn )
u' (c2n )
u' (c2 s )
1  π u' (c1n ) 

c1
u(c1n )  βnu(c2n )  u(c1s )  βnu(c2 s ) (IC n )
(c1n,c2n)
(1  c1s ) R  c2 s  profit s
(c1s,c2s) u (c1 )  βnu (c2 )
(1  c1n ) R  c2 n  profit n
c2
Intuition

 The intermediary offers a contract with very high
long-term return and large penalty upon early
withdraw, and it makes a profit from the naïve
investors, who suffer from the need for immediate
gratification, while losing money to the sophisticated
investors, who enjoy the high long-term return
Summary for Unobservable Preference

 The welfare of a naïve investor is LOWER than the case of
autarky
 For a sophisticated investor, the first best is NOT
achieved, and his welfare is strictly improved upon the
autarky case
 If you are naïve, do not pretend to be sophisticated,
because that will hurt you
 The sophisticated investors are happier if there are more
naïve investors, but they always think their repayment
structure is distorted with the existence of naïve investors
Restricted Linear Contracting

 Our earlier analyses focus on the case in which
investors can only liquidate a predetermined fixed
portion of his investment contract
 In practice, restricted linear contract corresponds to
the case in which investors can liquidate any portion
of his investment contract
 But do more options bring welfare improvement to
the investors? in particular, the naïve investors?
Observable Sophisticated Investors

 The financial intermediary solves ( β  βˆ )
max R~ ,T (1  c1* ) R  c2*
s.t. u (c1* )  u (c2* )  u (PC)
(c1* , c2* )  arg max c1 ,c2 u (c1 )  βu (c2 )
~
s.t. c1  c2 / R  T
Diagram of the Result

c1
1
~
c1  c2 / R  T
u(c1 )  βu(c2 ) (smaller slope)
u(c1 )  u(c2 )
(1  c1 ) R  c2  0
R
c2
Intuition

 A perfectly sophisticated depositor is fully aware of
her time inconsistency, so it would be profit
maximizing to offer her a contract with an interest
~
rate of R  R / β which aligns self 1's interest with the
self 0’s welfare
 The first best is still achieved
Observable Naïve Investors

 The financial intermediary solves ( β  βˆ )
max R~ ,T (1  c1 ) R  c2
u (cˆ1 )  u (cˆ2 )  u (PC)
(c1 , c2 )  arg max u (c1 )  βu (c2 ) (IC)
~
s.t. c1  c2 / R  T
(cˆ1 , cˆ2 )  arg max u (cˆ1 )  βˆu (cˆ2 ) (PCC)
~
cˆ1  cˆ2 /R  T
Diagram of the Result

c1
~
c1  c2 / R  T
(c1 , c2 )
(cˆ1 , cˆ2 )
(1  c1 ) R  c2  0
u(c1 )  βu(c2 )
u (c1 )  βˆu (c2 )
c2
Diagram of the Result

c1
~
c1  c2 / R  T
u(c1 )  u(c2 )
(c1 , c2 )
(cˆ1 , cˆ2 )
(1  c1 ) R  c2  0
u(c1 )  βu(c2 )
u (c1 )  βˆu (c2 )
c2
Equilibrium Outcome and Intuition

 Naïve investors will benefit from the linear contract
~
R
 The intermediary would set a very high interest rate to
attract the naïve investors, but that will also prevent the
investors from liquidating too much, which lead to a low
profit
 As β  βˆ , the payoff of the investors gets to first best
 For naïve investors, setting a upper limit for interest rate
by regulation will help to improve their welfare
Unobservable Preference

 Again we study the most simple case: all investors has the
same β̂ at date 0, and investors are naïve ( βn  βˆ ) with
probability π, and investors are sophisticated (βs  βˆ ) with
probability 1 – π
 All investors choose the same contract (c1s, c2s) at date 0,
but naïve investors will switch to (c1n, c2n) at date 1
max c1n ,c2 n ,c1s ,c2 s π (1  c1n ) R  c2 n   (1  π )(1  c1s ) R  c2 s 
u (c1s )  u (c2 s )  u (PC)
~
s.t. (c1s , c2 s )  arg max u (c1 )  βs u (c2 ), s.t. c1s  c2 s / R  T (IC s )
~
(c1n , c2 n )  arg max u (c1 )  βnu (c2 ), s.t. c1n  c2 n / R  T (IC n )
Diagram of the Result

c1
(1  c1s ) R  c2 s  profit s
~
c1  c2 / R  T
(c1n , c2 n )
(c1s , c2 s )
(1  c1n ) R  c2 n  profit n
u (c1 )  βnu (c2 )
u(c1 )  βs u (c2 )
c2
Equilibrium Outcomes

 Both sophisticated investors and naïve investors strictly
prefer the unrestricted market, with a higher perceived u,
to the restricted market with linear contract
 When the naïve investors are sufficiently sophisticated,
their welfare and the population-weighted sum of two
types of investors’ welfare are greater in the restricted
market with linear contract
Intuition

 In terms of the perceived date 0 utility, u, if u is higher in
restricted market with linear contract, we know that all
the equilibrium outcomes in restricted market are also
feasible in unrestricted market, but not vice versa
 In terms of welfare, sophisticated investors correctly
predict their future behavior, they are made worse off by
the linear intervention; however, the benefit of this
intervention to not-so-naïve investors outweighs the harm
to sophisticated investors
Summary for Linear Contract

 When preferences are observable, the welfare of a naïve
investor can be strictly improved with linear contract,
while the sophisticated investors can always achieve first
best
 When preferences are unobservable, sophisticated
investor welfare reduces while the naïve investors’
welfare improves, though nobody likes linear contract at
the beginning
 A welfare improving financial innovation might not be
welcomed by anyone, even for those who benefit from it
Term Structure

 The restricted linear contract implies a term structure
of interest rates
c
c2
~
c1  c2 / R  T  1 
1
2
1  i1 (1  i2 )
1  i1  T

~
1  i2  R T
 
1/ 2
Term Premium and Impatience

 For sophisticated investors, when they get more
impatient, financial intermediary would offer a
lower one-period interest rate and a higher twoperiod interest rate such that people are committed
to consume the first-best allocation
~
fb
fb ~
R  R / β , T  c1  c2 / R
1  i1  c1fb  βc2fb / R

1/ 2
1  i2  Rc1fb / β  c2fb


Term Premium and Naivete

 Suppose all investors has the same β̂ at date 0, and
investors are naïve βn  βˆ with probability π, and
investors are sophisticated βs  βˆ with probability 1 – π.
In a competitive equilibrium, the term premium is
increasing with the portion of naïve investors in the
economy, π
 The profit an intermediary can make from the naïve
investors shrinks as π increases; the loss from
sophisticated investors increases but is bounded
 The more naïve investors in the economy, the happier the
sophisticated ones and the naïve ones
Diagram of the Result

c1
(1  c1s ) R  c2 s  profit s
~
c1  c2 / R  T
(c1n , c2 n )
(c1s , c2 s )
(1  c1n ) R  c2 n  profit n
u (c1 )  βnu (c2 )
u(c1 )  βs u (c2 )
c2
Tradable Contract

 Suppose all investors has the same β̂ at date 0, and
investors are naïve βn  βˆ with probability π, and
investors are sophisticated βs  βˆ with probability 1 – π.
If contracts are divisible and can be traded on a secondary
market at date 1, in competitive equilibrium all the
depositors get the same allocations as when deposit
contracts are restricted to be linear
Loan Securitization

 If loans can be sold (through securitization) at a high price
instead of being liquidated to get 1, the bank will invest
all proceeds and sell the loan when there is early
withdrawal; the results remain qualitatively the same
 For open-end mutual funds, it requires all the liquidation
value goes to the investor, this corresponds to the case of
restricted linear contract with a slope R/P, with P being
the liquidation value
Diagram of Expanded Budget Constraint

c1
P
1
(1  c1 / P) R  c2  0
R
c2
Transparency of Financial Intermediaries

 If the financial intermediary has to disclose its amount of
investment at date 0 and needs to be able to satisfy all
customers' needs according to the contract, the financial
intermediary cannot remain solvent ex-ante when there is
possibility that all depositors choose that expected
repayment option
 The financial intermediaries make positive profits, and
the naïve investors are worse off
 Transparency does not always benefit the investors
Diagram of with Solvency at Date 0

c1
u(c1 )  u(c2 )
1
(c1 , c2 )
(cˆ1 , cˆ2 )  (c1fb , c2fb )
u(c1 )  βu(c2 )
(1  c1 ) R  c2  0
R
c2
Empirical Evidence in China

 Small banks are more aggressive in attract deposits
 For example, two-year term deposits, 10% higher
interest rate with individual deposits greater than
RMB10,000, for firm deposits greater than
RMB1,000,000
 There is a much higher probability of early withdrawal
for small banks
 For example, some regional branch of ABC, 1.08%;
Everbright (光大), 4%; CMBC (招商), 7.32%(in value);
Minsheng, 9.1%
Empirical Evidence in US

 Gilkeson, List and Ruff (1999, JFSR) found that depositors
withdraw a significant amount of their time deposits
before maturity, 2.4% and 6.4% of the deposit base each
year for shortest and longest maturity type, respectively
 Withdrawals from pension funds for nonretirement
purposes by account holders under 60 amount to $60
billion a year, or 40 percent of the $176 billion employees
put into such accounts each year and nearly a quarter of
the combined $294 billion that workers and employers
contribute
Conclusion

 Naïve investors do suffer from time-inconsistent
preference
 Financial intermediaries subsidize the sophisticated
investors at the cost of naïve investors
 Competition among financial intermediaries might not
help the naïve investors, and regulation on interest rate
might be needed
 Financial innovation like negotiable CD and securitization
help the naïve investors, but this might not necessarily be
the case for regulation on transparency