*If r1*(H) > r*, then rj(H) = r*. Any higher offer will make "i" change its policy in
the following period by undercutting ("cheating"); any lower offer lowers profits. This is
by definition of r*.
-If r (L) < r(L) and r;(H) > r;(L), then rj (H) will be determined as above.
rj(L) = max{rj (L),r* (L)}, this way the competitor can get its offer accepted by all (or
half if the second is an equality) for whom Si=H and half for whom Sj=L; or rj(L) =
max{ro(L),r* (H) e], when he gets the offer accepted by all those for whom Si=H. It
depends of which bears more profits. In either case the first player will change its
original strategy since it now has negative expected profits; since this case brings about
cheating, it is not to be expected.
If r*(L) < r(L) and r*(H) < r* (L), when r*(H) > rjo(L), then rj(L) = r[(H) -
when r* (H) < rjo (L), the n r (L) = r (L). To see why
*Consider first the case rf (H) = ro(L), since (1 PL) rj (L) PL = 0, then
k[(1 -p ,L) (H) PL] + (1 k)[piH r(H) (1 P =H)] 0 and (1-3) holds.
Then rj (L) = r(L) and at most half of those for whom Si=L could accept its offer (when
r* (L) = r (L)). Any lower interest rate gives negative expected profits. The case of
undercutting is trivial.
If r1*(L) > r(L) then, again, undercutting is profitable given that Sj=L.
(ii) The lender with the informational advantage
Given (ii), the only pricing policy that does not introduce "cheating" in the
following period and that maximizes profits, provided (1-2) does not hold, is r1*(H) =
rj (L), since "i" prefers to get the most out of the few for which Si=H but Sj=L, than to