This question arises from economic research attempting to model and estimate demand curves relating the quantity consumed Q of goods to their price P. Assume P > 0, $Q \ge 0$, and as P increases, Q smoothly decreases when Q > 0 and, if it reaches 0, remains at 0 for all higher P. It is desired to express Q as a function of P, the function being differentiable with respect to its parameters over the full range of P, and ideally linear in its parameters (to facilitate estimation by regression).
When P is sufficiently high it is expected that Q will either be 0 or very close to 0. There seem to be 3 cases:
The curve is asymptotic to the P axis. This is straightforward, eg Y = b / P or ln Y = a – bP are differentiable with respect to the parameters a, b.
There is some value k of P such that Q = 0 when $P \ge k$ and Q > 0 when P < k, and the curve is smooth at k.
As for 2 above, but kinked at k.
On the assumption that k is unknown and must be considered one of the parameters, is there a way to model cases 2 and/or 3 with functions which are differentiable with respect to their parameters?
For case 2 I considered the function:
$Y = b(k - P)^2$ if P < k and = 0 if $P \ge k$
However this does not work – it’s differentiable with respect to b but not with respect to k.