Differentiating |
 | piqriβ(q) = 1 |
|
with respect to q gives |
 |
piqriβ(q)(ln(pi) +
ln(ri) dβ/dq) = 0 |
|
Solving for dβ/dq, |
dβ/dq = -( |
 |
piqriβ(q)(ln(pi)))/( |
 |
piqriβ(q)(ln(ri))) |
|
Because each piq > 0, riβ(q) > 0,
ln(pi) < 0, and ln(ri) < 0, we see dβ/dq < 0. |
For q = 1, and hence β(1) = 0, we have each riβ(q) =
riβ(1)
= ri0 = 1, and piq = pi1
= pi. Consequently, |
alpha = -dβ/dq = (p1ln(p1) + ... +
pNln(pN))/(p1ln(r1)
+ ... + pNln(rN)) |