Proof of Corollary to Proposition 1: It is easy to check that $∂λμ∂μ⋗2(h−l−w)(h−l)>0$⁠.

Proof of Proposition 2: (i) The total welfare is given by the sum of welfare in both markets. We have:

Differentiating total welfare with respect to λ, we obtain:

Note that, when $λ=λμ$ there is a discontinuity jump:

(ii) Differentiating total welfare with respect to μ, we obtain:

Proof of Proposition 3: Consider first market for good x. Without the cost to obtain personal data, total welfare in such a market is the same with or without a market for data: in equilibrium both types buy the good and welfare is equal to the sum of willingness to pay for all consumers; i.e., $hλ+(1−λ)l$⁠. Introducing the cost to buy data reduces welfare in the market for good x.