To compare the two schemes, DPM and DIM, we first note that the two expected welfare functions evaluated at θ = θ are equal (equations (7.7) and (7.12) all give the same values) and that their first derivatives with respect to θ evaluated at θ = θ are negative and equal. Hence, we compare the DPM and DIM schemes by computing the second derivatives of the expected social welfare with respect to θ at θ = θ. Consider first the case of the DIM scheme, where separating incentive mechanisms are chosen by the political majorities. Social welfare when majority 1 decides, W(d ₁, ₁ ), can be written from (7.1) and (7.4) as where W¹(·, ·) is the objective function of majority 1. Using the envelope theorem for W¹(d ₁, ₁ ), we have where, from (7.7), and therefore Similarly, when majority 2 decides, we have yielding

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