Mathematics for Economists

Liquidity modelling

There is a demandDfor liquid resources, cash or euro in a bank or in a

teller machine, with distribution functionF.There are two costs:his the

holding cost andpis the penalty for not satisfying the demand. What is

the level of optimal supplyS?Our goal function is

J(S)=h
E



(SD)+



+p
E



(DS)+



.

If there is a density function then

E



(DS)+



=

Z∞

S

(xS)f(x)dx

E



(SD)+



=

Z

R

(Sx)+dF(x)=

Z

R

(Sx)+f(x)dx=

=

ZS

0

(Sx)f(x)dx.