Some Math
The following explanatory model was presented by Lawrence H. White.
These are the terms used in the formulation
?= expected profit
r b =rate on bills
rd=rate on deposits
C= operating costs
L=expected liquidity costs
N=notes
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
S=B=N+D+K
?=rb- rdD- C- L
C= f (S,B,N,D)
(costs are function of the entries in the balance sheet)
L= g (S, N, D)
(in case of exhaustion of specie)
L= ?s? p(X-S) P(X? N,D)dx
(Holding notes and deposits constant, the expected liquidity costs decrease when the amount of specie increases)
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. We obtain the following equimarginal equations
?(S,B,N,D,K)= rbB-rdD-C-L+ ? (K-S-B+N+D)
?s=-Cs-Ls-?=0
?B=rb-Cb-?=0
?N=-CN>-LN-?=0
?D=-rd-CD-LD+?=0
??=K-S-B+N+D=0
r b-CB=-C B-LS=CN+LN=rd+C D+LD
Finally, we can conclude that the profit optimization condition for free banking is the following:
The marginal net benefit from holding specie should be equal to the marginal cost of maintaining notes in circulation
White, Lawrence. "Competition and Currency: Essays on Free Banking and Money." New York: New York University Press, 1989.
