i'm trying to figure out a generic way to figure out pricing of a
photograph for a variety of different usages. the prices and figures
given in the following example are (obviously) subject to change.

i have a base price, $300 for a 1/4 ad in a magazine with 1000 circulation.
the parameters that would alter the price are the size and the circulation.

x-axis = price
y-axis = circulation
z-axis = size

if i have two other prices in mind that I'd like to be paid in given situations.
   - $2000 for full page with 5000 circulation
   - $10,000 for 1/2 page with 20,000 circulation

question #1. is there a formula that can be described that would allow
me to find the price if i input size and circulation? (it's clear to
me that i'm not looking for a linear algorithm)

question #2. how many "points" are needed to construct the algorithm?
(i assume the algorithm is a description of a curve in 3 dimensional
space)

question #3. what is the algorithm?

---

Request for Question Clarification by answerguru-ga on 04 Apr 2005 00:35 PDT

Hi a3m-ga,

I could come up with a formula that would be able to reflect the value
of photograph based on size and circulation. In order to do this
logically though, some additional information is necessary:

1. Which factor (size or circulation) has a larger impact on cost?

2. What are the 'boundary conditions' in terms of size and
circulation? In other words, what are the minimum and maximum values
that each of these variables could take?

3. Are there any 'points of interest' in your business in terms of
size and circulation? An example would be that distribution becomes
less costly when circulation is above a certain threshold.

I should mention that if we were to only consider the three (price,
size, circulation) data points that you provided, it is fairly
straightforward to calculate a system of equations which would yield
something like:

Cost = A*Size + B*Circulation (where A and B are known constants)

The problem with this, though, is that the formula is being derived
from the example points. There may be additional information that is
being left out.

I look forward to hearing back from you - this could be an interesting one.

Thanks,
answerguru-ga

---

Clarification of Question by a3m-ga on 04 Apr 2005 10:06 PDT

hey answerguru,

as a point of clarification, there is no single formula that i'm
looking for. i want a photographer to be able to give me some target
prices that he wants to be paid for a photo, and then be able to
interpolate the pricing in between those points. in the simple
2-dimensional case (e.g. price and size), i would ask him for how much
he'd like to be paid for a 1/4 page ad and a 2-page ad. he says "$300"
and "$2400". in this simple linear case, if some one asked him how
much he wants for a 1-page ad, the answer would be $1200. but if the
photographer provided a third data point (e.g. 1/2 page ad for 800),
then the solution is now a curve.

if we factor in a third dimension, circulation, then we potentially
have two curves, and i think (i have poor math skills) i'm looking for
an equation that describes a curved surface.

the problem is that i'd like to be able to ask any photographer "what
do you want to be paid for this photo" and come up with an equation
for that individual.

other factors might include: duration (for example, a 1/4 page ad in a
weekly magazine with a circulation of 10,000). in my analysis of
pricing, there is a discount placed when you run an ad for a long
period of time. 4 weeks isn't just 4*1 week rate.

in trying to determine the algorithm, i want to ask the photographer
the minimum subset of information needed to interpolate all points of
pricing. if end points are necessary, then i could ask that.

some photographers might provide data points that yield a linear
solution. some might provide data points that yield a non-linear
solution, and some photographers might provide data points that yield
no solution. i'm looking to try to bring a logical solution to the
issue of pricing, so that there is a mathematical solution.

don't know if i clarified or obfuscated the issue. ask more questions
if you need follow-up.

Estimating functions of multiple variables can be done with the "Latin
Hypercube" method as found in "Numerical Recipies" books.  I suspect
that isn't really the way to deal with your problem.

I think you should also read about differential pricing.
Some beginners' economics links are here:`
http://www.cl.cam.ac.uk/~rja14/#Econ