In using the Mill ratio¹, we assert that

[A4.1]

With the previous inequality, we can deduce that, for inequality [A2.7]:

[A4.2]

By transforming inequality [A2.8], we see that:

[A4.3]

when:

Therefore, expression [A2.8] is always superior to 0 for or

By transforming inequality [A3.7], we find that:

[A4.4]

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]:}

[A4.5]