Page:Cournot Theory of Wealth (1838).djvu/45

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m_{1,2}c_{1,2}=m_{2,1}, \mbox{ or } c_{1,2}=\frac{m_{2,1}}{m_{1,2}}

We have in general c_{2,1}=\frac{1}{c_{2,1}},
and in this particular case c_{2,1}=\frac{m_{1,2}}{m_{2,1}}

Whenever, therefore, the ratio \frac{m_{2,1}}{m_{1,2}} only differs from unity by a quantity less than the cost of transportation of a monetary unit from one place to the other, the balance between the two places will be settled without actual transportation of money and by the mere effect of the rate of exchange.

Let us now suppose any number of places in correspondence, and let mi,k express generally the total of the sums annually due from the place (i) to the place (k), and ci,k the coefficient of exchange from (i) to (k). The number of these coefficients for a number of places (r) will be r(r—1); but, as in general, c_{i,k}=\frac{1}{c_{k,i}}, the number of coefficients to be determined is reduced at the outset to \frac{r(r-1)}{2}.

Nor are all these coefficients mutually independent; for if, for instance,

c_{i,k} > c_{i,l} \times c_{l,k},

any one having to convey money from (k) to (i), instead of getting draft from (k) on (i), will find it cheaper to get one from (k) on (l) which he can exchange for one from (l) on (i). For the same reason it is impossible to have

c_{i,k} < c_{i,l} \times c_{l,k},