ECO3055 Political EconomyFormative Assessment1. Consider a country withn≥3 citizens, wherenis odd.
Suppose that thepreferences of citizeniover a public goodg∈[0,1] and a private goodci∈[0,1]are characterized by the utility functionui=ci+αi√g,whereαi∈[0,1/n] denotes how much citizenivalues the public good. LetαAV Gbe the average andαMEDbe the median of the distribution of valuations.Assume, in addition, that each citizenihas initial resources in private goodri= 1, and that one unit of private good is necessary to produce one unit ofpublic good.
Last, suppose that to finance the production of the public good, thegovernment raises a taxt∈[0,1] on each citizen so that the budget constraintof citizeniisci≤ri−tand the budget constraint of the government isg≤nt.(a) What is the preferred taxt∗(αi) of each citizeni?
D
escribe howt∗(αi)changes withαi(e.g., increases, decreases, non-monotonic).Solution:For each citizeni, the budget constraintci≤ri−tis binding as herutility increases inci, and byri= 1 we obtainci= 1−t.
The budgetconstraint of the governmentg≤ntis binding as utilities increase ing, sog=nt. Then, the utility of citizenican be written asui= 1−t+αi√nt.
The preferred tax rate of each citizeniist∗(αi) = arg maxtui(t)(the tax rate that maximisesui.)Asdui/dt=−1 +αin(nt)−1/2/2, settingdui/dt= 0 we obtaint∗(αi) =nα2i/4Then,t∗(αi) increases inαibecausedt∗/dαi=nαi/2 which is alwaysnonnegative forαi∈[0,1/n].

(b) What is the socially optimal taxtOPTthat maximises the average utilityuAV G?Solution:The average utility can be written asuAV G=1nn∑i=1ui=1nn∑i=1(1−t+αi√nt)=n(1−t)n+∑ni=1αin√nt= 1−t+αAV G√nt.AsduAV G/dt=−1+αAV Gn(nt)−1/2/2, settingduAV G/dt= 0 we obtaintOPT=nα2AV G/4.(c) Consider a two-party competition where each party proposes a differentvalue oftto win the election. Under majority rule, what is the equilibriumtax (preferred tax)tEQ?

Compare this to the social optimum. When doestOPTcoincide withtEQ?Solution:If the median voter theorem applies, under majority rule the equilibriumtaxtEQis the preferred tax of the median voter. We have to first showwhether preferences are single peaked here.

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What is the socially optimal taxtOPTthat maximises the average utilityuAV G?Solution:The average utility can be written asuAV
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•Check ifuiis strictly concave intfor a given value ofαi.

Hence,takingαias a constant, check the sign ofd2udt2_d2udt2=−14αin1/2t−3/2which is always negative sinceα,n,tare all positive. Hence,d2udt2<0anduiis (strictly) concave int∈[0,1] guaranteeing that preferencesare single peaked on a single dimensional domain. Moreover,nis odd,so MVT can be applied.

Hence, social preferences under majority ruleare identical to the preferences of the median voter.But who is the median voter?Recall that the bliss point of each citizen is determined byt∗(αi) =nα2i/4

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