Some Math: Difference between revisions
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?<sub>s</sub>=-C<sub>s</sub>-L<sub>s</sub>-?=0 | ?<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>B</sub>=r<sub>b</sub>-C<sub>b</sub>-?=0 | ||
?<sub>N</sub>=-C<sub>N</sub>>-L<sub>N</sub>-?=0 | |||
?<sub>B</sub>=r<sub>b</sub>-C<sub>b</sub>-?=0 | |||
Revision as of 22:34, 4 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
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)
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:
?(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
?B=rb-Cb-?=0
TRY TO WRITE THIS DIRECTLY ON THE WEBSITE- PG 43 OF COMPETITION AND CURRENCY
