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

r_d=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

?=r_b- r_dD- 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)= r_bB-r_dD-C-L+ ? (K-S-B+N+D)

?_s=-C_s-L_s-?=0

?_B=r_b-C_b-?=0

?_N=-C_N>-L_N-?=0

?_D=-r_d-C_D-L_D+?=0

?_?=K-S-B+N+D=0

r _b-C_B=-C _B-L_S=C_N+L_N=r_d+C _D+L_D

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.