since b<1. This gives us an upper bound on E(q), which we will call
q–e. A similar argument produces a lower bound q- e
.
The final step before completing the argument is to note that since ∆–(q)
and ∆–(q) are both concave, ∆–(q)+∆–(q) is also concave, so that
where
Therefore,
where
This completes the proof of the proposition.
qqc
qce* eif ,
if.=<
≥
2
20
0E( ) E( ( ) ( )) E( ) (
*
qqqqccqq ccq),
ee
++≥++≥++^12 ∆∆^12121212cqq qq
qqcqq
qq12=−
−=−
−∆∆∆∆() ()
,() ()
.()()∆∆+> ∆() ∆(),−
− +−
−=+qqq
qqqqq
qqqccq 12A MODEL OF INVESTOR SENTIMENT 457