Some Math: Difference between revisions
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The following explanatory model was presented by Lawrence H. White. | |||
These are the terms used in the formulation | |||
?= expected profit | ?= expected profit | ||
r <sub>b</sub> =rate on bills | |||
r<sub>d</sub>=rate on deposits | |||
C= operating costs | C= operating costs | ||
L=expected liquidity costs | L=expected liquidity costs | ||
N=notes | |||
S=specie | S=specie | ||
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X= net specie outflow during the given period | 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<sub>b</sub>- r<sub>d</sub>D- C- L | |||
C= f (S,B,N,D) | C= f (S,B,N,D) | ||
L= g (S, N, D) | (costs are function of the entries in the balance sheet) | ||
L= g (S, N, D) | |||
(in case of exhaustion of specie) | |||
L= ?<sub>s</sub><sup>?</sup> 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(s)<0 | ||
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L(d)>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 | 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<sub>b</sub>B-r<sub>d</sub>D-C-L+ ? (K-S-B+N+D) | |||
?<sub>s</sub>=-C<sub>s</sub>-L<sub>s</sub>-?=0 | |||
?<sub>B</sub>=r<sub>b</sub>-C<sub>b</sub>-?=0 | |||
?<sub>N</sub>=-C<sub>N</sub>>-L<sub>N</sub>-?=0 | |||
?<sub>D</sub>=-r<sub>d</sub>-C<sub>D</sub>-L<sub>D</sub>+?=0 | |||
?<sub>?</sub>=K-S-B+N+D=0 | |||
r <sub>b</sub>-C<sub>B</sub>=-C <sub>B</sub>-L<sub>S</sub>=C<sub>N</sub>+L<sub>N</sub>=r<sub>d</sub>+C <sub>D</sub>+L<sub>D</sub> | |||
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. | |||
Latest revision as of 06:42, 8 December 2006
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.
