The value function for a borrower of type i is
V(t) = max['d(t)7T(R),SE[V (t + 1)|f(t)]}.
Equilibrium Analysis
The solution to the search problem from the borrowers' perspective consists of a
sequence of reservation values below which the offer is accepted, and above which it is
rejected. So, if Rf < -t then d(t) = 1; otherwise, d(t)=0. According to this behavior,
lenders decide their offers.
Let's analyze first how each type would behave in a separate market where there
is only one type of borrower. A stationary equilibrium yields a time independent
reservation rate R so that V[Q(t), R] = E{V[Q(t+1), R]}. A stationary reservation rate is
such that the borrower is indifferent between accepting today and waiting another
period. Formally, rr(RIO) = 5 E[V(t + 1)1fl(t), R]
Hence,
g(k + 0 R) = g(k + 0 R)
which means, R = k + 0.
Then, if both types were in separated markets, all projects would make zero
profits.
Note that the reservation rate for the safe borrower, R,, is smaller than that of the
risky borrower, 7R,. Consequently, there is no reason to ever make offers less than 9Rs
since both borrowers will accept this amount (provided no other lender will offer a
smaller rate either).
Now consider again both borrowers participating in the same market. For the risky
borrower it is not possible to find a stationary reservation value since a positive