Mathematics for Economists

Liquidity modelling

∆J(S) = hP(DS)pP(D>S)=

= h( 1 P(D>S))pP(D>S)=

= h(h+p)P(D>S).

Obviously ifS=0 then∆J(S)=h(h+p)P(D> 0 ).Also

limS%∞∆(J(S))=h> 0 ,and the expression is not decreasing whenSis

increasing. So the optimal, minimum, solution is

S=min(Sj∆J(S) 0 )=min(SjJ(S+ 1 )J(S)),