Find determinant of matrix of second partials (regularities of product life cycle)

[ShubhamRathi]

Consider the Equations.:
g'(rd*)G = 1 ... (3)
[ Q _it-1 (Q_t/ Q _t-1) + Δ q_it* ] l (rc_it*)=1 ... (4)

Its then said, Differentiating (3)-(4) with respect to Q_{it - 1} and rearranging yields:

d rc*/d Q _it-1 = g’’ (rd_it) G m’’ (Δ q_it) l’ (rc_it*) x (Q_t/ Q _t-1) /D ... (5)
d Δ q_it* / d Q _t-1 = g’’ (rd_it) G l’ (rc_it )² x (Q_t/ Q _t-1) /D ... (6)
​

where D is the determinant of the matrix of second partials.
​

** What is D. How does one calculate it? Can someone please explain the calculus that is happening here?**
Just if someone wants to get in more details, the paper is here. This result is Lemma 2 on page 10/23.

 

[stapel]

Consider the Equations.:
g'(rd*)G = 1 ... (3)
[ Q _it-1 (Q_t/ Q _t-1) + Δ q_it* ] l (rc_it*)=1 ... (4)

Its then said, Differentiating (3)-(4) with respect to Q_{it - 1} and rearranging yields:

d rc*/d Q _it-1 = g’’ (rd_it) G m’’ (Δ q_it) l’ (rc_it*) x (Q_t/ Q _t-1) /D ... (5)
d Δ q_it* / d Q _t-1 = g’’ (rd_it) G l’ (rc_it )² x (Q_t/ Q _t-1) /D ... (6)
​

where D is the determinant of the matrix of second partials.
​

** What is D. How does one calculate it? Can someone please explain the calculus that is happening here?**
Just if someone wants to get in more details, the paper is here. This result is Lemma 2 on page 10/23.

There is no "D" in Lemma 2 (on page 9 of 23). Do you maybe mean Lemma 3? If so, then D is defined as:

D is the determinant of the matrix of second partials of \(\displaystyle \,E\left(\Pi_{it}\right)\,\) with respect to rd_it, rc_it, and \(\displaystyle \,\Delta q_{it},\,\) evaluated at \(\displaystyle \,rd_{it}^{*},\, rc_{it}^{*},\, \) and \(\displaystyle \,\Delta q_{it}^{*}.\)