Some Math: Difference between revisions
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?<sub>N</sub>=-C<sub>N</sub>>-L<sub>N</sub>-?=0 | ?<sub>N</sub>=-C<sub>N</sub>>-L<sub>N</sub>-?=0 | ||
?<sub> | ?<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>= | |||
Revision as of 02:04, 5 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
?D=-rd-CD-LD+?=0
??=K-S-B+N+D=0
r b-CB=-C B-LS=CN+LN=rd+C D+LD=
TRY TO WRITE THIS DIRECTLY ON THE WEBSITE- PG 43 OF COMPETITION AND CURRENCY
