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...
You assumed there that each client values songs independently. What if this is not the case?
For example, assume that each client has a base price, ranging from $0 to $0.5, and a per song priceranging between $0 and $0.06. The valuation of any given song is now the sum of the two. You can think of the base price as an indicator of how much each client is willing to pay for songs in general, regardless of the specific song. In this setup, bundling won’t allow you to capture almost all the surplus, as it did in your simpler model.
Bundling exploits uncorrelated (or, even better, anti-correlated) preferences. When selling two products A,B, it allows you to extract a little bit of surplus from the client who values A more than average and values B less than average. A client who overvalues both A and B would have bought the two products anyway, and a client who undervalues both A and B won’t buy your bundle.
I agree that if consumer preferences correlate between the things you bundle, the producer won’t be able to capture an approaching-full surplus. (They’ll still be able to capture an increased surplus in many cases, though counterexamples exist.)
Your qualitative explanation about when this works seems spot-on.
You assumed there that each client values songs independently. What if this is not the case?
For example, assume that each client has a base price, ranging from $0 to $0.5, and a per song price ranging between $0 and $0.06. The valuation of any given song is now the sum of the two. You can think of the base price as an indicator of how much each client is willing to pay for songs in general, regardless of the specific song. In this setup, bundling won’t allow you to capture almost all the surplus, as it did in your simpler model.
Bundling exploits uncorrelated (or, even better, anti-correlated) preferences. When selling two products A,B, it allows you to extract a little bit of surplus from the client who values A more than average and values B less than average. A client who overvalues both A and B would have bought the two products anyway, and a client who undervalues both A and B won’t buy your bundle.
I agree that if consumer preferences correlate between the things you bundle, the producer won’t be able to capture an approaching-full surplus. (They’ll still be able to capture an increased surplus in many cases, though counterexamples exist.)
Your qualitative explanation about when this works seems spot-on.