Creating non-linear cost functions

Question

From what I understand, the basic cost function looks like $$C(x_i)=\sum_{i=1}^{n}w_ix_i$$

What I'm wondering is whether or not its possible to create a non-linear cost function which adjusts based on the bundle consumed.

Example:

Lets say we have a discrete case where our bundle price changes based on the combination of goods in a non-intuitive way. as seen in the table below. $$\begin{array}{|c|c|c|c|} \hline Bundle\ Price&x_1&x_2&x_3 \\\hline 5&1&1&0 \\\hline 10&2&1&0 \\\hline 15&2&2&0 \\\hline 25&2&3&1 \\\hline \end{array}$$

We see such pricing methods employed in a number of commercial businesses.

How does one model such a case¹?

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_{1. I would appreciate an answer that is more theoretical and does not rely upon regression , however if that is the only way to get an estimate i'll take it.}

Answer 1

I understand that the "bundle price" is cost to us. Then your table depicts a linear system of equations

$$C_j = a_1x_1+a_2x_2+...+a_mx_m,\;\;\; j=1,...,n$$

with $C_j$ and the $x$'s being the known quantities, and we want to determine the unknown alphas.

If it so happens that the bundles on offer are same in number as the $x$'s, ($n=m$), then the system, bar the existence of exact linear dependence, will ave a unique solution for the alphas, and the cost function will be linear.

But in your specific example, the system is "overidentified" ($n>m$) since you have more bundles on offer than inputs.

In such a case, "regression", or better, least-squares approximation enters the picture as a way to obtain a linear approximation, and also, quantify the deviation from actual through the obtained residual series. In your example, one gets, $\hat a_1 = 3, \hat a_2=4, \hat a_3=7$, with estimated cost series $\hat C = 7,10,14,25$ compared to $C = 5,10,15,25$. Essentially the estimated alphas here are the implied approximate separate unit prices per input, maintaining the linear relationship.

The non-linear approach would require to specify a non-linear function form first...and in principle, it would allow us to match the data exactly. The specified form could be found by trial and error, and it could be very complicated, which then would create the well-known issue: "well we matched the data sample exactly, but is this any good for inference/design outside the sample?"

In other words, assume we do obtain a non-linear cost function in such a way. Assume now that we want to produce at a level where we need to purchase $x_1 = 4, x_2=5, x_3=3$. Prior to go to the supplier and ask for a price for this new bundle, how well will the obtained non-linear cost function predict the offer we will get?

Answer 2

When you say non-linear cost function, I assume you aren't referring to the firm's production having non-linear costs, but judging by your example, you rather mean the firm's output prices having weird optimal bundle pricing.

From the abstract of the linked article (Hanson and Martin, 1990):

" Bundle pricing is a widespread phenomenon. However, despite its importance as a pricing tool, surprisingly little is known about how to find optimal bundle prices. "

So this may be a good place for you to start. For an article that is a little more recent/available online, you could look at extensions of the original work in this area, such as bundle pricing for homogeneous items (Grigoriev et.al., no date), or this article correlated valuations for the items (Chen and Ni, 2017).