§8. Counting the Equations and Unknowns

Collecting our various counts of the numbers of equations, we have:

For	Impatience	Principle	A,n(m - 1)	equations
"	"	"	B,n(m - 1)	"
"	Market	"	A,m	"
"	"	"	B,n	"

For	Investment	Opportunity	Principle	A,n equations
"	"	"	"	B,n(m - 1)	"

The sum total of these is 3n(m - 1) + 2n + m, or 3mn + m - n.

To compare this number with the number of unknowns, we note that all the unknowns in the first approximation are repeated;

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the	number	of	f's	being	n(m - 1)
"	"	"	x's	"	mn
"	"	"	i's	"	m - 1

making a total of 2mn + m - n - 1 carried forward from the first approximation.

In addition, the new unknowns, the y's and the r's, are introduced. There is one y for each individual for each year, the total array of y's being

y1', y1'',...,y1(m),
y2', y2'',...,y2(m),
.............
.............
yn', yn'',..., yn(m).

The number of these y's is evidently mn.

There is one r for each individual for each pair of successive years, i.e., first-and-second, second-and-third, etc., and next-to-last-and-last years, the total array of r's being

r1', r1'',..., r1(m-1),
r2', r2'',..., r2(m-1),
.............
.............
rn', rn'',..., rn(m-1).

The number of these r's is evidently n(m - 1).

In all, then, the number of new unknowns, additional to the number of old unknowns carried forward from the first approximation, is mn + n(m - 1), or 2mn - n.

Hence we have:

number of old unknowns, 2mn +m -n - 1,

+ number of new unknowns, 2mn -n,

= total number of unknowns, 4mn +m - 2n - 1,

as compared with 3mn + m - n equations.

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