an offer of k + 01 is made. If (2-2) does not hold, the offer is k + 02. Let this
strategy be called OS for Optimal Strategy.
Note that since within a period each bank is not competing, banks will try to charge
as loan payment the highest return of the expected borrower's type project.
Assumption 1: PL, > PHr and PHs > PLs
Corollary 1: Given a risky borrower, limit E = 0 and hence, limit mt = 0.
(Pht+1)
Corollary 1 states that as more information becomes available, the probability that
a risky borrower is mis-categorized as safe goes to zero. As a consequence, the
interest rate offers will tend to stay high as time increases.
Proposition 3: For all a < 100, there exists an n, such that after n, observations,
lenders can be a% sure of the type of borrower.
This is an application of the law of large numbers. For a detailed proof and the
value of na see the Appendix.
For a safe borrower, Yt tends to infinity si nce ps 1 and pr 0 as t--*o. This
implies that a safe borrower should keep applying for loans until a favorable offer is
made since the law of large numbers guarantees there is a n such that the offer at n will
be k+01; and a risky borrower that has received all k + 02 offers will accept this interest
rate after certain number of periods searching since for a risky borrower yttends to zero.
In strategy OS, ht is only relevant in what it says about the probability of being
safe. Note that Yt may have the same value for different histories. For example,
(L,H,L,H) and (L,H). Hence, the offer is the same since the information the histories
convey is the same, although t is different.