Cost per item. Diminishing marginal discount, if you will. (Bigger discount for first few items) Optimal number of units to buy?

Question

The graph above shows price per unit. Say they are cupcakes. When you buy a higher quantity, you get a lower price per unit. Say it levels off like this graph. Obviously, buying 2 nets a nice discount. You can buy one for a friend or save one for tomorrow. But, the marginal discount decreases with each additional unit. At some quantity (about 9), you're paying $4 each, regardless of how many more you buy.

What concept will help to determine the "optimal" number of units to buy? Where you're minimizing the diminishing utility of incremental discount. You want to maximize the discount, while minimizing the additional number of units you need to buy. (B/c you don't really want to have so many cupcakes lying around, but you can't bring yourself to pay $9 for just 1)

Informally, by eyeballing the graph, it seems like you might as well buy about 3 or 4. You've gotten the best deal while minimizing the number of additional units needing to be purchased. It's not exactly inflection point, but something similar, right?

What's the more formal approach? If you had an equation for this curve, you could take the derivative. The first derivative would be very negative and slowly start approaching zero. How can I identify where it starts to flatten out, and it's not worth buying additional units, since the savings are relatively low?

I want to know where the rate of change of the rate of change is dropping, right? Or goes under some threshold? Does the 2nd derivative come into play? Am I on the right track? What is the methodology to arrive at a conclusion like "You should buy 4, b/c that's where most of the savings, per unit, have been realized already"

Answer 1

"Optimal number of units to buy" is a vague term in this context. All of my post is from a Calculus perspective. To get an economic perspective (possibly where this post belongs) you need someone who knows more about stackexchange than me, but possibly repost in economics. We could assume

Where you're minimizing the diminishing utility of incremental discount.

is what you're looking for. So let's use that as a working assumption. So yes, you want to know where the rate of change levels off. This is determined by the second derivative.
The goal is to minimize f', not minimize f. However, since we don't have an equation, perhaps drawing a graph of the derivative and maybe even a graph of the 2nd derivative. It appears to be a piecewise linear function, meaning it's portions of lines. So maybe draw a first derivative graph by estimating the slope between each pair of consecutive points? If that doesn't work out well, how else might you create a graph of the derivative?

I suspect since the original graph has no obvious points of inflection it, you'll find that there is no point one can definitively point to where the first derivative levels off. Unfortunately without a more mathematical definition of utility, "where it starts to flatten out", "not worth buying additional units", "savings are relatively low" or relationship between utility to cost a definitive mathematical answer may not be possible.

Answer 2

Normally, I would have suggested an inflection point as the solution. But it does not work here. I have created a method (which is not formal as reported in the literature) but my own methodology.

From the graph, I have paired (units and price) and have used them in our discussion as the curve does not have an equation. The idea behind this methodology is for an arbitrary number of units (that you would like to consume), you compute the price. This way the I can caculate the discount that I realize in percentage compared to the maximum total price. Similarly for a fixed but aribitrary income, you compute the number of units would get for that amount. Normalize the prices that you get for a fixed units(that takes into account the discount). Similarly normalize the total units that you get for a fixed income(again that takes into account the discount). At this point it will be clear that normalized prices would be rank ordered (in comparison to the highest) from the highest of 1 to the lowest. Simlarly the normalized units would be rank ordered from the lowest to the highest being 1. From the table it is also evident that (unit, pair) (12,4) will get you the highest number of units for the lowest price. But that is exactly what you do not want,because you consider that units that are purchased above the optimal level (if any) would lead to a burden in terms of storage, perishability or umpteen reasons you can think of.

Ideally you would want to maximize the discount and minimize the additional quantity purchased. This can be translated to Maximize(%Discount + %Burden) (%Burden being negative). The pair of (price, unit) that achieves this level is the most optimum. In my methodology, I have computed this % Discount as $\frac{(\text{Max total price - Total price})}{\text{Max total price}}$. Similary I have computed % Burden as $\frac{(\text{Min Units -TotalUnits})}{\text{Max Units}}$. Now the optimal unit, price combination would be the one which maximizes (%Discount + %Burden) In your graph, the (3, 5.2) is the optimal (unit, price) that will maximize this metric.

Let me know if this makes sense.

Thanks

Satish