00Thaler_FM i-xxvi.qxd

Unfortunately, we are not yet finished because we do not have closed

form formulas for the expectations in this expression. To provide

sufficient conditions, we need to bound these quantities. In the re-

mainder of the proof, we construct a number k

• where

This makes p 1 <p 2 k

• a sufficient condition for (A.2). Of course, this
  assumes that p 2 ≥0, and so we impose this as an additional con-
  straint to be satisfied. In practice, we find that for the ranges of πL,
  πH, λ 1 , and λ 2 allowed by the model, p 2 is always positive. However,
  we do not attempt a proof of this.
  The first step in bounding the expression
  is to bound E(q). To do this, note that

Consider the function g(q) defined on [q

• ,q–]. The idea is to bound
  this function above and below over this interval by straight lines,
  parallel to the line passing through the endpoints of g(q), namely (q


• 
  

  ,
  g(q

  
  


• 
  

  )) and (q–,g(q–)). In other words, suppose that we bound g(q)
  above by g–(q)=a+bq. The slope of this line is

  
  

and awill be such that

Given that

Eq(g(q)−q)=0,

we must have

Eq(g–(q)−q)≥ 0

or

E( )

E( ) ,

abqq

q

a

b

+−≥

≤

−

0

1

inf ( ( )).

qqq[,]

abqgq

∈

+− = 0

b

gq gq

qq

qq q q

qq

=

−

−

=

−− +

−

<

() () ( ) (() ())

,

1

2

1

∆∆

E( ) E( ) E (E( ))

E(( ( )) (()))

E(()).

qq qq

qq qq

gq

ttqtt

qt t t t

q

t

t

==

=++−

=

++ 11

1

2

1

2

∆∆

E( )qqq++^12 E ( ( )q∆∆( ))

kq

q qq

<+

+

E( )

E( () ())

.

∆∆

2

456 BARBERIS, SHLEIFER, VISHNY