Some Math

S=B=N+D+K

?=rbB- rdD- C- L

where:

If :

?= 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.