Question

1. The government of Cloneland faces a long-standing constitutional con- straint against levying lump-sum taxes. This is unfortunate, because everyone in Cloneland is identical. The government must raise a given revenue of R per person to pay tribute to a foreign colonial power that conquered Cloneland many centuries ago. The citizens of Cloneland consume two goods (z and r2) and they supply labour (L) to earn enough to cover their consumption purchases. The Clones all have a wage of one; producer prices of the two consumption goods are also fixed and equal to one. The government relies on commodity taxes on goods 1 and 2 at ad valorem (percentage) rates T and T2 to raise the revenue to pay the tribute to the colonial power. The Clones have identical utility functions given by: 1L U T2 1 where 0 and n > 0 are constant preference parameter (a) Determine the ordinary demand functions for the two consump tion goods. (10) (b) Noting that the price of good i is (1+7) because producer prices are fixed at one, show that the own price elasticity of demand for good 1 is and for good 2 it is -n. Show that this can be rearranged to give for good 2. (10) , and similarly a(1+2) a1+n) 11 (c) We want to determine the characteristics of the relative optimal commodity tax rates on these goods from an efficiency perspective (i.e., which maximize V(1+Ti,1+T2) s.t. R = T1xı+T2x2, where V() is the indirect utility function of a representative Clonian, and and 2 are determined by the demand functions from part (a)). To do this it is easier to determine the ratio ()/( rather than. (20) T2

Answer 1

solution: U Given it ito - txist In2! "- L. Uit Tito - 0 @ Demand functions for two consumption goods are: → The utility function is t (ie) EqQ Budget constraint 18 : Pixy +P9.72 =2 1-E + - HiU = In t 2 - Pixi - P272 - 1 Fox for utility maximisation is: (- t)u, 18-P250 : [*i= () 1 89 - ((-) n - 8, so

(b) If price of good i is (HT;) because producer prices are one. for good) ist and showing own price elasticity of demand for good 2 itis -ri Now Pl=14T - P = 1+T2 cas e -+ (H) te no (192--- (+)** Eiga 2-0 se 7 XI (:: P1= ITU) from @waget x a(14 = ) HT mi X from Baveget C: P-147] Hence Price elasticity of demand for goodl=-€ for good 2=-2

D Determination of characteristics of relative optimal Commodity tax rate: is vH"HT) St R =TiNit E7일 I . - Ile l ' +. A U= I.. ' may X 샅 subject to constraint (IAT)P, oxit (1+T2) P2 X 2 = CH-L) | = -L) 1 = (1) ( ) et I | + >[An>R기 +(HTJR -CH-5) - le +3 R (14) = -① + R (IT) = - = -1 + o - T. 부 = ④ title= p (HTY) " = CCT i기

3x3 = [Pill+7)], and wat* = [Pa (1472)5-2 Pemand foeting * = - [Ctr) in (147)*6+(1+4.304 .!") + H 1:48* = H (itri) leplet - (1+ta) ob plan 84,99 = tem -- litije · PL = -t pit (IATJE = -6 Similarly {X₂, P2 = -2 + I . ( . E) 32 (1H2) Now VCP, P2,T, T2) =107) [r (47)]htetas Con (147 -H + CI+TO)! P!-€ +(1+T2') in palcon

> -(1+)3€+(1-e) p!-((1+73€ = ple (1+TiJ€ (1-28) » - R? (147)” (1-21) -E- = pt. (E) (1+Tv) 3 dritti) EN (HT)-2-1 - P2 d (1+T2) ► di To) - [8; +Tisa pro =0 > pilt (147) € (1-2E) - 4 [+*+t, Pit(e) (147)]=0 >> P.144 (147,5€ (-26) = EXT+Tipifce)+75€ >ple (+7)= (1-26) = M (P, (471)) * (Ittife) (1:1) » Pi (+26) = M (1-6Ti (1+T05)

> H = - tiv P, (1-26) I-ETICHT) and = P2 (1-21) Antall HT - using (GV) and (v) P, (1-26) = 12 (1-20) LETI (1+1) InTz (1+T2)' Thall+TIT 6D-1) = 1x* = *** dll+T) → xi* = 1-4 (+70) 4 (1-28) = XTP, (1-28) - P(1-2) = similarly 3 (1-21) = 1 HE TI (1+774 - 1. ARTz (1+T25 -); ETI = noz (1+TO) CHR)