An alternative way to model merit good arguments

Appendix

• Derivation of the first best tax rules.

The first order conditions for an interior maximum are given by:

∂ L

∂ t_x

- g ʃ ^dz I

1' 1 ∂ q_x

u^g d x

-4-—+

u^g d qx

^u3 ∂^y-1 + γ (x + tx 7 + ty |y.) =0 (30a)
ug ∂ qxj I ∂ qx ^y dqxj

∂ L

^{d t}y

- g ʃ ^d z .
^uⁱ Idqy ⁺

u^g d x

—--+

^u1^d qy

u3 ⅛L f + γ
^uι ^dqy )

{^y+^t^xdx⁺^t^y⅜} =⁰^(30b)

∂L
∂ T

= ^- '½ dm

u^g d x

-I—-—
ug d m

₊ u3 ^y y ¾ _-ug d m J

γ {1+tx dm+ty 1}

Performing ∣L — ^dLx and ∣L — ^dLy gives:
'ɔ Otx OT Oty ∂T & 'ɔ

r db
t ∂ qx

ug ∂b
⁺ u^g dqx

+ u| l^b) + 7 (tx f- + ty I^b) = 0
ui ∂qx J [ ∂ qx ∂qx J

½ ∂b
I^d qy

ug ∂b

⁺ u^g dqy

+u∣ Я+γ ½tx^db+ty ∣^b ¾ = 0
ui ∂ qy) L ∂qy ∂qy J

(31a)

(31b)

where a^denotes a compensated price effect and 7 =^f ~. Using the homogeneity
condition on the compensated price effects, (31) may be rewritten as

g
^u2

O
uɪ

O
^u3
uɪɪɪ

^{— q}x

—qy
0

Ox
Oqx
∂y
Oqx

Ox
∂q_χOy
dqy _

= Y [ tx

^ty ]

Ox
Oqx
Oy
Oqx

^Oqy
Oy

^Oqy _

. (32)

OS.

Because the substitution matrix (Oqx O q^y ) is negative definite, (32) reduces to

Oq x Oq_y

^u. ^U2 A / ^ug /99 A

7tχ = qx--g , and γty = q_y--_y. (33a)

^u1 ^u1

Inserting these conditions back in the FOC for T, shows that 7 = 1.

Since q_x = u² and q_y = uɪɪ, the first best tax rates are as in the text.

• Derivationof u^g(z,x,y)

By definition of F(∙)

ГУ

^{z +} μ(χ)dχ =

JyD

^{f x,}y^,u(^z ⁺ У μ(χ)dχ^,χ,yj

(34)

^{f χ,}y^,u^z ⁺ У μ(^χ)^dχ^,χ,y

^- У μ(χ)^dχ

(35)

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An alternative way to model merit good arguments

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