Icosian Reflections

…a tendency to systematize and a keen sense

that we live in a broken world.

(Naïve) microeconomics of bundling goods

Originally posted to LessWrong.

Junk fees are in the news from the 2023 State of the Union address, get picked up by Matt Yglesias, and Zvi responds in Junk Fees, Bundling and Unbundling. Matt and my former colleague propose an economic framing of "bundling versus unbundling", and Zvi identifies four win-win advantages of bundling, and four 'advantages' of unbundling (two win-wins, one (company win)-(customer lose), and one mixed win-lose).

I think Zvi is broadly right on the points he makes, but he and Matt both skip over the basic, conventional econ-101 analysis of bundling goods on prices and customer welfare. I think that, for a broader audience, it's worth covering the "naïve microeconomics" perspective as background for the customer-behavioral story (which, admittedly, is more juicy and fun). Rather than responding to the whole conversation, this post will restrict its focus to the econ-101 microeconomics story of bundling, and ignore the behavioral / political / moral dimensions that Zvi, Matt, and others are discussing.

I'm going to assume no formal economic background, and build up an explanation why a world of perfect economic agents should have sellers preferring to sell bundles of goods instead of individually-priced items.

(1) Selling sandwiches

Let's say that you're a sandwich-seller. Every day, 110 people come by your sandwich stall, look at the price you put on the menu, and buy a sandwich if they value it more than the price. (No bargaining or haggling; you don't have time for such things.)

Sandwiches cost you $1 to make, and people value sandwiches somewhere between $0 and $11, fairly uniformly:

[ed: The first-best version of this post includes interactive charts with sliders that let you explore the tradeoffs described in the text. I think there's a large chance that if I wait until I finish them this post will be delayed until "never". So I'm pushing out a second-best version where I just describe the charts and leave them to your imagination. If I do get around to producing them, I'll drop them in and remove this notice.]

[TK: interactive chart - radio buttons let you switch between: "customers at price" is a bar chart with one customer at each $0.1 increment from $0.05 to $10.95 / "customers below price" is a bar chart with X customers at each price X*$0.10.]

How should you price the sandwiches?

Well, if you price them at $1, then you'll sell to 100 out of every 110 potential customers. (The remaining 10 value it less than $1, and so really they don't matter.) And every $0.10 that you raise the price, you'll lose one customer. Your total profits come to ($X-$1)*(100-($X/$0.10)):

[TK: interactive chart: dragging "price" from $0 to $11 shows that you sell 110-(price/$0.1) sandwiches and profit (price-$1)*(110-(price/$0.1)).]

By charging $6 per sandwich, you sell to 50 customers and net $250 of profits per day.

In your spare time, in between making sandwiches, you dream of moving to the next city over, where nearly everyone values sandwiches the same, at $5.50 each. If only you lived there, you could sell to every potential customer at $5.45 and make $490 of profits per day.

(2) Selling songs

Let's say you're an online song store. Every quarter, 1.1 million new people come to your store, look at your ten thousand songs, and buy each song that they value more than the price you're selling it at. (No charging people different prices based on search history; that's illegal.)

Songs cost you $0.01/sale in royalties to the artist, and people value songs between $0 and $0.11 uniformly (and independently), so how should you price the songs?

[TK: interactive chart: dragging "price" from $0 to $0.11 shows that you sell 1.1mln*10k*(1-(price/$0.11)) songs and profit (price-$0.01)*1.1mln*10k*(1-(price/$0.11)).]

Yep, same story, same outcome. You price them at $0.06 each and sell ~4,545 songs to each customer for a total of $273 (that's $227 of profits), so $250 million of profits for the quarter. You buy a $6 sandwich from the shop downstairs and when you get back, you learn you've been promoted to Chief Commercial Officer. (The previous Chief Commercial Officer has retired to live on a beach.)

(3) Selling bundles of songs

Let's say you're a research assistant to the Chief Commercial Officer of an online song store. Your mission (which it is mandatory to accept) is to analyze the CEO's pet idea of the week -- what if we sold our entire library as a bundle? The CCO is off looking at beachfront properties to buy this week, so you're on your own for this one.

While munching on a sandwich from the shop downstairs, you ponder how selling libraries is different from selling individual songs. While people's valuations of different songs vary from $0 to $0.11, their valuations of a whole library are a lot more consistent, averaging $550 with a standard deviation of just $0.92...

[TK: interactive chart: dragging "price" (which also displays "price per song") shows that you sell 1.1mln*(1-cdf(Normal(550,0.92))) [ed: this means $548->97.8%; $549->84%; $550->50%; $551->16%; $552->2.2%;...] packages for profits of (price-$100)*1.1mln*(1-cdf(Normal($550,$0.92))).]

So, if you price your "all-access" library at $545, you can sell to nearly everyone, make $445 of profits per customer, and the company will net nearly $490 million of profits per quarter.

You send the details to your boss and cc the CEO. The next day, at all-hands, you learn that the CCO will be leaving the company to more spend time with their family, and also the company will be bundling its song sales into a $45/month all-access pass "to make it easier than ever to explore new music while keeping access to all your own favorites".

(4) Discussion

Without any cutesy stories, what's going on here? I admit that the seller's surplus from bundling seemed like dark magic to me when it was introduced to me using bundles of two or three goods at a time (which is how I originally learned it). It's only when thinking about the ten-thousand-song case that things come into better focus for me.

Both the seller of sandwiches and the song store wish they could apply price discrimination and sell each good at 1% below what that buyer would be willing to pay for it. Then they could sell to every profitable opportunity at the maximum price, which comes out to an overall average sale price of $6/sandwich or $0.06/song.

Unfortunately for them, they can't. As a result, they end up picking a compromise price that gets about half the potential profits by selling at an average-ish price to about half the people. (It's a coincidence that the seller-maximizing price is the average of the profitable customers' valuations, by the way -- if you try other distributions this doesn't happen.)

What the song store can do (that the sandwich-seller can't) is exchange the per-customer market for songs -- which has high variance in customer valuations -- for the per-customer market for song bundles, which has much lower variance because of the law of large numbers. Then, when the customers' valuations all cluster around the average of $550 per library = $0.055 per song, the seller can price just below that and sell to nearly everyone at around the average price. This is better for the seller than pricing each individual good at an average-ish price and selling to half the people.

(5) Welfare effects; Conclusion

Perhaps sadly, this kind of bundling is worse for the buyer: rather than paying $300 for five thousand songs you value at $425, you're paying $545 for ten thousand songs you value at $550. It is, however, better for the artists (instead of totaling $55 million of royalties per quarter, the artists total $110 million of royalties), and better for the store ($490 million of profits instead of $250 million).

I don't want to over-stress this econ-101 perspective, which argues that we should see maximum bundling of all things (and is therefore in contrast to the actual world we observe). Zvi's post on bundling raises very important customer-behavioral considerations, many of which are first-order here and explain why we see far from maximal bundling.

I also don't want to go into the aesthetics, morality, legality, or politics of bundling and unbundling in this post, so this seems like a fine place to stop.