In using the Mill ratio1, we assert that
With the previous inequality, we can deduce that, for inequality [A2.7]:
By transforming inequality [A2.8], we see that:
when:
Therefore, expression [A2.8] is always superior to 0 for or
By transforming inequality [A3.7], we find that:
This relationship will be positive if:
Because of this, [A3.7] will always be positive for companies when (the value of company assets is at least equal to the discounted nominal value of its debt) or likewise, when d1 ≥ 0.
By defining we know that h(d) is always positive. We can demonstrate that h’(d) > 0 regardless of the value of d. This means that h(d) is a monotonic function of d and strictly a growth function. If d1 > d2, then h(d1) − h(d2) > 0.
Given that we can deduce that for inequality [A3.8]: