I'm looking to find a best-fit line through a set of data that follows a 'concave up' curve. The curve is not necessarily quadratic. The curve is defined as: V = Q * (P - MC + NMV) - FC (economically-speaking, this is a value curve)
Over time I can get several observations of Q and P (for the purposes of this question imagine I have 3 or 4 observations). I already know MC (marginal cost), NMV (non-monetary-value) and FC (fixed cost).
If we assume that the relationship between P an Q (this is the demand curve) is P = 1 - Q than the value curve is quadratic . This is pretty easy, I would just fit a quadratic equation to the data I'm observing.
However, we can't always assume that relationship. A coworker feels like the relationship between P and Q is something closer to: log(Q) = a * log(P), or Q = P ^ a. This makes sense. With a negative value for 'a' the curve looks something like what one would expect for a demand curve .
Because I can observe Q and P and I already have a guess that the relationship is Q = P ^ a my coworker feels like I should just find a best-fit line that solves for a, and then solve for V. However, it seems like it shouldn't be any harder to find a best-fit line for V itself (since MC and FC are fixed and known), essentially solving for a in the equation:
V = (P ^ a) (P - MC + NMV) + FC
or simplifying and ignoring FC:
V = P ^ (a + 1) - Z * P (where Z is a constant NMV - MC)
Am I wrong?