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

∂ tx

-   g ʃ dz I

1' 1 ∂ qx

ug d x

-4-—+

ug d qx

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

L

d ty

- g ʃ d z .
ui Idqy +

ug d x

—--+

u1d qy

u3 ⅛L f + γ
uι dqy )

{y+txdx+ty⅜} =0 (30b)

L
∂ T

= - '½ dm

ug d x

-I—-—
ug d m

+ u3 y y ¾ -
u
g 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 ∂ q
x

ug ∂b
+ ug dqx

+ u| lb) + 7 (tx f- + ty Ib) = 0
u
i ∂qx J      [ ∂ qx     ∂qx J

½ ∂b
Id qy

ug ∂b

+ ug dqy

+u Я+γ ½txdb+ty b ¾ = 0
u
i 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

1

g
u2

O
uɪ

O
u3
uɪɪɪ

— qx

1

0

qy
0

1

Ox
Oqx
∂y
Oqx

Ox
∂qχ
Oy
dqy _

= Y [ tx

ty ]

Ox
Oqx
Oy
Oqx

Ox

Oqy
Oy

Oqy _

.    (32)

OS.

Xx

Because the substitution matrix (Oqx O qy ) is negative definite, (32) reduces to

Oq x Oqy

u.           U2     A /           ug                      /99 A

7tχ = qx--g , and γty = qy--y.                   (33a)

u1                  u1

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

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

Derivationof ug(z,x,y)

By definition of F()

ГУ

z +   μ(χ)=

JyD


f x,y,u(^z + У μ(χ),χ,yj


(34)


f χ,y,u^z + У μ(χ)dχ,χ,y


- У μ(χ)dχ


(35)


12




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