# How to create parametric price estimating equations

The basic principle behind parametric price estimating is that you can use basic properties of the item you are going to buy (size, power, temperature, etc) and find some regression curve that relates to price. A simple example might be the price of a tank P=(V/V0)^.65 where P is price, V is the volume of the tank you want to buy, and V0 is a basis volume given for this equation. The .65 is the constant found to best match this regression, and is often just given to you in text books.

My question is, how did they get to .65? I am trying to develop my own parametric equations using data I have collected myself. However I can't figure out which constants to use or how to come up with best regression curves. I know there must be a method for doing this.

Prices tend to vary in rough proportion to the price itself. The is, the deviations in price are proportional to price. To convert a multiplication problem into an addition problem, we take its logarithm. In this case, $\ln (P)=0.65[\ln(V)-\ln(V0)]$. Rewriting 0.65 as some unknown constant $m$, we then write $y=\ln(P),\ x= \ln(V),\ b=-m \ln(V0).$ Then we regress $y=m x+b$ to find $m,\ b$ from which $e^{-b/m}=V0$, where $m\rightarrow0.65$ or some other constant is found by linear regression.