Modeling concave curves - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T16:32:21Z http://mathoverflow.net/feeds/question/100177 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/100177/modeling-concave-curves Modeling concave curves Sam 2012-06-20T20:33:10Z 2012-06-20T21:15:58Z <p>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)</p> <p>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).</p> <p>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 [1]. This is pretty easy, I would just fit a quadratic equation to the data I'm observing.</p> <p>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 [2].</p> <p>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:</p> <p>V = (P ^ a) (P - MC + NMV) + FC</p> <p>or simplifying and ignoring FC:</p> <p>V = P ^ (a + 1) - Z * P (where Z is a constant NMV - MC)</p> <p>Am I wrong?</p> <p>=====</p> <p>Links:</p> <ol> <li><a href="http://www.wolframalpha.com/input/?i=V+%3D+Q+" rel="nofollow">http://www.wolframalpha.com/input/?i=V+%3D+Q+</a>*+P+where+P+%3D+1+-+Q</li> <li><a href="http://www.wolframalpha.com/input/?i=X+%3D+Y+%5E+-0.75+from+0+to+100" rel="nofollow">http://www.wolframalpha.com/input/?i=X+%3D+Y+%5E+-0.75+from+0+to+100</a></li> </ol>