Figures from the paper
Transitional ratchet of the platform footprint
Yt+1 = (1 − λ) Yt + λ Ye. Move the λ slider to see how the speed-of-adjustment parameter rescales the half-life of convergence while leaving the destination Yss unchanged.
Iso-NS versus welfare in (τ, F) space
Thin curves are iso-welfare contours (labeled by W); the thick curve is the iso-NS locus, the set of (τ, F) pairs yielding the same active-seller count NS at fixed σ. The two families have very different slopes, so the thick curve cuts across many thin ones: the same market-structure observable maps to a range of welfare levels. A regulator watching NS or HHI cannot rank welfare.
Further constructions
Results the paper states analytically rather than plotting, plus two surfaces that are clearer in three dimensions.
Direct vs indirect welfare channels of self-preferencing
Dashed (direct, with τ held fixed at τ*): exactly flat, because the closed form W(τ) does not contain σ. Solid (indirect, along the platform's fee response): downward sloping because self-preferencing pulls τ up, and W is decreasing in τ.
Steady-state footprint comparative statics
The platform's long-run footprint is governed by patience δ and entry cost K. The λ-neutrality of the steady state is documented separately in the transitional-ratchet figure above.
Cap cascade
A fee cap τ̄ below τ* shifts equilibrium to (σFC, τ̄) with σFC < σ*. A non-discrimination mandate σ=0 shifts equilibrium to (0, τND) along bτ.
Profit surface Π(σ, τ)
The platform's profit Π(σ, τ) as a surface. The red marker is the interior maximum (σ*, τ*). The local concavity around the peak verifies the second-order condition. Rendered in 3D because the geometry is clearer than a 2D contour map.
Combined regulation: welfare over (τ̄, σcap)
Consumer welfare W as a function of the fee cap τ̄ and the self-preferencing cap σcap. Three reference policies are marked. The combined cap (low τ̄, σcap=0) sits at the highest level of the surface; either single-instrument regime is dominated. The 3D surface makes the dominance more direct than a 2D heatmap.