> <u>ddgΨR</u><sup>(g0)=0=⇒a{1 −α(1 −p)]A(g0+ αs0)a−1[(1 −τ</sup><sup><u>)</u></sup> <u>w</u> − CR(g<sup>0</sup> )]<sup>1−a</sup> − (1 − a)A(g<sup>0</sup> + αs<sup>0</sup> )<sup>a</sup> [(1 − τ <u>)w</u> − CR(g<sup>0</sup> )]<sup>a dC</sup> dg<sup><u>R</u>(g0) = 0 =⇒</sup>

> <u>dCdgR</u><sup>(g0) = [1 −α(1 −p)]</sup> 1−a a <u>(1−τg)</u><sup>0</sup> <u>w+−αsCR</u><sup>0</sup> <u>(g</u><sup>0</sup> <u>)</u> .

Condition (ii) requires that the government’s objective function be maximized. We can simplify the FOC to this problem by noticing that the second term of this function is proportional to the sum of each group’s utility, and the derivative of this sum is zero, as seen above. Then we have:

> <u>dGdg</u><sup>(g0) = 0 =⇒x</sup> <u>hλR(p)(1 −</u> p)<sup><u>dC</u></sup> dg<sup><u>R</u>(g0) + λP(p)p</sup><sup><u>dC</u></sup> dg<sup><u>P</u>(g0)</sup> i = 0 =⇒ λR(p)(1− p)<sup><u>dC</u></sup> dg<sup><u>R</u>(g0) = −λP (p)p</sup><sup><u>dC</u></sup> dg<sup><u>P</u>(g0).</sup> Using the previous results, we may thus obtain: λR(p)(1−p)[1−α(1−p)] 1−a a <u>(1−τg)</u><sup>0</sup> <u>w+−αsCR</u><sup>0</sup> <u>(g</u><sup>0</sup> <u>)</u> = λP (p)p(1−p) 1−a a <u>(1−τ )ws−</u><sup>0</sup> CP (g<sup>0</sup> <u>)</u> =⇒ <u>(1−τg)</u><sup>0</sup> <u>w+−αsCR</u><sup>0</sup> <u>(g</u><sup>0</sup> <u>)</u> =<sup>λ</sup> λ<sup>P</sup> R<sup><u>(</u></sup> (p<sup><u>p</u></sup> )<sup><u>)</u></sup> 1−α(1p −p) <u>(1−τ )ws−</u><sup>0</sup> CP (g<sup>0</sup> <u>)</u> . The sufficiency of this FOC is assured just as in the proof of Proposition 3. <u>1</u> (ii) It is easy to check that h 1−α(1p −p) <u>BA</u> i 1−a < 1−α(1p −p)<sup>,becauseofour</sup> parametric assumptions of 0 < α < 1 and B < Aα<sup>a</sup> , which imply B < A. Expressions (3) and (4) could therefore be equal only if λR(p) and λP (p) are exactly such as to compensate for that difference. As they are simply parameters, this could only happen by coincidence: the truthful equilibrium allocation will thus be efficient only for a zero-measure set of parameters. Hence we prove that it is almost always inefficient. Proof. Corollary

It is a mere consequence of the fact mentioned in the proof of Proposition <u>1</u> 2, that h 1−α(1p −p) <u>BA</u> i 1−a < 1−α(1p −p)<sup>.</sup> Proof. Lemma 1 It is enough to take the partial derivative of θ with respect to p. The first termlocation,of θ,haswhicha derivativecorrespondsof 1to−1αtheh 1−parameterα(1p −p) i 1−aaassociated<sup>¡</sup> <u>BA</u> ¢ 1−1 a [1−withα1(1−α−pthe)]<sup>2</sup> efficient<sup>>0,which</sup> alis the efficiency-effect. The derivative of the second term, associated with the political equilibrium, may be divided in two components, by the rule of product differentiation: −<sup>λ</sup> λ<sup>P</sup> R<sup><u>(</u></sup> (<sup><u>p</u></sup> p)<sup><u>)</u></sup> [1−α1(1−α−p)]<sup>2<0,whichcomesfromdifferentiating</sup> 1−α(1p −p)<sup>(atermthatappearsinthepoliticalequilibriumbecauseofthenum-</sup> ber of poor and rich contributing to the government and on the social welfare function), and − 1−α(1p −p) h λλPR0((pp))<sup>−</sup> λ<sup>λ</sup> R<sup>P</sup> (<sup><u>(</u></sup> p<sup><u>p</u></sup> )<sup><u>)2</u>λR0(p)</sup> i > 0, which comes from differentiating<sup>λ</sup> λ<sup>P</sup> R<sup><u>(</u></sup> (p<sup><u>p</u></sup> )<sup><u>)</u>(atermthatrepresentsthepoliticalweightsassociatedwith</sup> each group’s ease of coordination). Those are the participation-effect and the coordination-effect, respectively. Proof. Proposition 3 (i) From Lemma 1 we know that 27