Some Math
From Dickinson College Wiki
S=B=N+D+K
?=rbB- rdD- C- L
where:
?= expected profit
r b =rate on bills
rd=rate on deposits
C= operating costs
L=expected liquidity costs
S=specie
P= % adjustment cost for impending specie deficiency. Assumed to be constant
X= net specie outflow during the given period
P(X? N,D)= the pdf of X given N and D
C= f (S,B,N,D)
L= g (S, N, D)
L= ?s ? p(X-S) P(X? N,D)dX
L(s)<0
L(n)>0
L(d)>0
From these partial derivatives, it follows that expected liquidity costs decrease when S increases. Also, L increases when N and D increase. Finally, let us solve this using a Lagrangian:
TRY TO WRITE THIS DIRECTLY ON THE WEBSITE- PG 43 OF COMPETITION AND CURRENCY
This model was presented by Lawrence H. White.
