The Privacy Subsidy in Market Microstructure

Privacy-preserving exchange designs price on a coarsened view of order flow. We show that a market maker committed to informationally efficient (posterior-mean) pricing on a signal strictly coarser than the flow it settles necessarily cedes a closed-form welfare transfer to traders -- the privacy subsidy -- and that no rule restricted to the coarse signal is simultaneously efficient and zero-profit against the settled flow. We establish this impossibility for a general coarsening, then characterise the subsidy in closed form across three canonical microstructure models: single-period Kyle with Gaussian flow noise, Glosten-Milgrom with a binary direction channel, and continuous-time Kyle-Back with a Brownian channel. The subsidy obeys a structural correspondence with Loss-Versus-Rebalancing, both welfare rates factorising as a squared noise driver times a committed-object factor. Gross of fees the subsidy is a pure transfer recovered by a break-even fee; once levied, that fee distorts volume, and the resulting deadweight -- under an explicit allocative value of trade -- is fourth-order in the noise scale while the gross subsidy is second-order, so privacy is welfare-neutral to leading order with a strictly smaller irrecoverable loss. Endogenising the privacy level, a protocol trading a differential-privacy benefit against this deadweight chooses an interior noise scale in closed form; doing so leaves the half-revealing product of price impact and informed intensity intact while unpinning the volatility-elasticity of price impact from its textbook value of one.