These distributions can be approximated to normal distributions for a large n.
Lets use the text book conservative rule that the minimum number of observations, nc,
should be such that min-(hcpHV,ncl PH,)} > 5 for v = r and s.
For n nc, define X, = xL. The Lindberg-Levy Central Limit Theorem implies that
each lender can be (1 a) 100% sure that any particular borrower's Xn-R, if X, e
PHr ZPHr PHr) and Xn-S, when Xn E PHs Za PHs(PHs). Since these two intervals
n- 4- 4 'T
may intercept, accurate estimations of the borrowers type cannot be made until both
intervals are disjoint, which occurs when
PH, -PHr PHr( -PH,) + PHs (1-PHs )].
Let n* = [ P (1PHr) + pH(1- PH.)]2. Hence, n, = maxi[Wh*, nc}.
PHs PHr
Proposition 4: When the accurate lender is separating according to its signal, so
that an observation of L gets a high interest rate offer and an H gets a low one, all
borrowers will approach the accurate lender in the first period. (In case I: 92(02-01) > 1).
S,
Proof Given that the accurate lender is separating according to its signal, safe
borrowers will decide to approach this lender first since the expected interest rate offer
from the accurate lender is less than the one from the other lenders:
p* (k + 01) + (1 p)(k + 02) < pHs (k + 01) + (1 pH)(k + 2)1
Since this is known by the inaccurate lenders, they know all first time applicants
are risky, which means they will charge them k + 02. Since borrowers in this model must
1 The right hand side is the expected offer given the inaccurate lender is separating. This lenders) may also be pooling,
but given the case they would onlybe pooling at k + 02 for all borrowers, which is an even higher offer. The left hand
side is the expected offer from the accurate lender.