Source
Szeidl and Szucs (2021), Econometrica 89(1), LIT-1315, read in NOTE-1039. Propositions 1 and 2, equation (8), and their proofs in the supplement’s Appendix A.4. The derivation below is the record’s.
What was actually shown
Setup. Outlet i ∈ {1, 2} chooses true slant s̃ᵢ to maximize Vᵢ = β·E[Σⱼ pᵢqᵢⱼ] − (NT/2)(s̃ᵢ − ŝᵢ)². Measured slant is sᵢ = s̃ᵢ + ζ + ξᵢ, and a reference outlet r has s_r = ŝ_r + ζ + ξ_r. Reader shares are logit: Nᵢ/N = exp[αᵢ + γ(sᵢ − s_r)] / Σₗ exp[αₗ + γ(sₗ − s_r)]. Advertiser j, with budget Wⱼ and state-ownership indicator Gⱼ, maximizes Σᵢ log(qᵢⱼ)·Nᵢ·exp[νᵢⱼ]·exp[Gⱼ(θᵢ + δᵢ(sᵢ − s_r))] subject to Σᵢ pᵢqᵢⱼ = Wⱼ. Only outlet 1 is connected: δ₁ = δ > 0 and δ₂ = 0.
Step 1, demand (Proposition 1). With Cobb–Douglas weights wᵢⱼ the budget share is aᵢⱼ = pᵢqᵢⱼ/Wⱼ = wᵢⱼ/(w₁ⱼ + w₂ⱼ). So log(a₁ⱼ/a₂ⱼ) = log(N₁/N₂) + νⱼ + Gⱼ[θ + δ(s₁ − s_r)], with νⱼ = ν₁ⱼ − ν₂ⱼ and θ = θ₁ − θ₂. The logit gives log(N₁/N₂) = α + γ(s₁ − s₂). Together: log(a₁ⱼ/a₂ⱼ) = α + γ(s₁ − s₂) + θGⱼ + δ(s₁ − s_r)Gⱼ + νⱼ.
Step 2, outlet 1’s first-order condition. Write zⱼ for the right-hand side and Λ(z) = eᶻ/(1 + eᶻ), so a₁ⱼ = Λ(zⱼ) and outlet 1’s revenue is Σⱼ Wⱼ a₁ⱼ. Since s₁ = s̃₁ + ζ + ξ₁, ∂zⱼ/∂s̃₁ = γ + δGⱼ, and Λ′ = Λ(1 − Λ). The objective is smooth, defined on all of ℝ and bounded above (the quadratic dominates bounded revenue), so at an optimum β Σⱼ Wⱼ E[a₁ⱼ(1 − a₁ⱼ)](γ + δGⱼ) = NT(s̃₁ − ŝ₁). With masⱼ = E[a₁ⱼ(1 − a₁ⱼ)], s̃₁ = ŝ₁ + (βγ/NT)Σⱼ masⱼWⱼ + (βδ/NT)Σ_{j: Gⱼ=1} masⱼWⱼ.
Step 3, outlet 2. a₂ⱼ = 1 − Λ(zⱼ) and ∂zⱼ/∂s̃₂ = −γ, so ∂a₂ⱼ/∂s̃₂ = γΛ(1 − Λ). The government term does not involve s₂, since δ₂ = 0. Hence s̃₂ = ŝ₂ + (βγ/NT)Σⱼ masⱼWⱼ, with the same masⱼ because a₂(1 − a₂) = a₁(1 − a₁).
Step 4, the gap. Subtracting, and adding back measurement noise (sᵢ − s_r = s̃ᵢ − ŝ_r + ξᵢ − ξ_r): s₁ − s₂ = (ŝ₁ − ŝ₂) + (βδ/NT)Σ_{j: Gⱼ=1} masⱼWⱼ + (ξ₁ − ξ₂). The reader term (βγ/NT)Σⱼ masⱼWⱼ appears in both outlets’ conditions with the same masⱼ and cancels.
What could have failed: the reader term could have differed between the outlets, leaving the gap confounded with audience tastes. It does not, because with two outlets the marginal shares coincide. The paper uses that to argue that audiences cannot explain the change in relative slant.
What this does not say
- Not a closed form or a uniqueness result. masⱼ depends on the equilibrium shares, so (8) is a fixed-point condition. Existence and uniqueness are checked numerically at the estimates, not proved.
- Not that the split is identified from one episode. One gap fixes the sum of the two terms, not each. Separating them takes variation in the exchange term with the ideal points held fixed, which is the paper’s identifying assumption, and the empirical split is THEORY-852.
- Not robust to more than two outlets as stated. With more outlets the marginal shares need not coincide and the reader term need not cancel.
- Not dynamic. Each episode is a separate one-shot equilibrium.
Connections
- THEORY-837. δ, the government’s ability to reward slant, is the continuous analogue of Besley and Prat’s transaction cost τ, here allowed to differ by outlet.
- CLAIM-407. The last clause of the title is an instance of a description closed under ordinary dynamics that does not fix the response to an intervention on the mechanism. The intervention δ → 0 removes one term and not the other.