APPENDIX A. be identical and the indirect utility of a representative consumer is given by

(1)

APPENDIX A

This mathematical appendix derives the political equilibrium production and trade policies under cooperation with a global production externality.

Home and foreign country exhibit similar economic and political structures. Variables and parameters for the foreign country are denoted with an asterisk (*). Supply for industry i in the home country is given by Hotelling’s lemma

(A1) X_i(p_i^s)= ′Π_i(p_i^s)

Each individual j maximizes direct utility

(A ) u_j c _j u c_ij( _ij) u ( )E

i n

2 ₀ Ej

1

= + +

∑

= ^,

where c_0j is the consumption of the numeraire good, c_ij is the consumption of good i by

individual j, and u_Ej( ) is the utility that individual j derives from the state of the environment asE determined by the externality E. The externality is global and is generated by production of one or more nonnumeraire goods e in one or both such that ∂

∂ E X X

X^e ^e E

e

X

( , ^*)

= > 0^and

∂

∂ E X X

X^e ^e E

e

X

( , ^*)

* = * > 0^. All u_ij(.) are assumed to be increasing and concave function. Further assume that the externality is negative, that is ∂

∂ u

E^{E j} = ′ <u_{E j} 0 . All consumers are assumed to be identical and the indirect utility of a representative consumer is given by

(A3) v(p ,y,E) = y+s(p ) + u^d ^d _E( )E ,

where p^d=( ,p p₁^d ₂^d,...,p_n^d) is the vector of consumer prices for nonnumeraire goods, and y represents her income. Consumer surplus from all nonnumeraire goods is

(A4) s(p ) = ^d u [d (p )] - _i _i _i^d p d (p_i^d _i _i^d

i=1 n

∑

) .

∑

(2)

Using Roy’s Identity yields individual demand for good i

(A5) d p s p

i i p

d

i

( ) ( d )

= −∂

∂ .

Total demand for good i in the home country is D_i = Nd_i and ∂

∂ D p ⁱ D

i

d = ′ <i 0 . The price equilibrium conditions for supply and demand in the home and foreign country are

(A a) p_i^s ⁱ p

i i

6 =θ w

τ

( ) ^*

*

A b p_i^s ⁱ* p

i i

6 =θ w

τ

(A c6 ) p_i^d =θ_ip_i^w (A d6 ) p_i^d^* =θ_i^*p_i^w.

The net revenue of the government in the home country is generated by domestic and trade policies. Net per-capita transfer is then

(A ) r( , , ^*, ^*) N pi( )X p( ) p ( )[D p( ) X (p )]

s

i i i

s

i

N i

w

i i i

d

i i

s

i

7 τ θ τ θ = ¹

∑

τ −1 + ¹

∑

θ −1 − ^.

The governments cooperate when they set production and trade policies and efficiency requires that they maximize the weighted sum

( ) [ ( , , , ) ( , , , )]

[ ( , , , ) ( , , , )]

* * * * * * *

* * *

*

* * * *

A A G AG A C aW

A C a W

i L i

8 + = +

+ +

∈

∑

τ θ τ θ τ θ τ θ τ θ τ θ τ θ τ θ

where A=(α_L +a)and A^* =(α^*_L +a^*). The first order condition for the cooperative equilibrium is then

( )

( , , , ) ( , , , )

( , , , ) ( , , , ) ,

* * * * *

*

* * * * * * *

A

A W a W

i L i

9

0

∇ + ∇ +

∇ + ∇ =

∈

∑

β β

τ θ τ θ τ θ τ θ

where β τ τ θ θ= , *, , *.

(3)

Substituting in for the partial derivatives in (A9) yields

( )

( { ( ) [ ( , , , ) ( ) ( , )]}

{ ( ) ( , , , ) ( ) ( , )})

( { ( ) [ ( , , , ) ( ) ( , )]}

* * * *

* * *

* * * * * * *

*

* * *

*

A

A l p N r R s p u X X

a l p Nr R Ns p Nu X X

A l p N r R s p u X X

i i i

s

i N

d

E e e

i L

i i

s

i n

d

E e e

i i i

s

i N i

d

E e e

i L

10

1

∇ + + + + +

+ ∇ + + + + +

+ ∇ + + − + +

+

∈

=

∈

∑

β β

β

τ θ τ θ τ θ τ θ

τ θ τ θ Π

Π Π

a l _i p_i^s N r R N s p N u X X

i n

i i i

d

i E e e

* * * * * * * * * * * * * *

{ ( ) ( , , , ) ( ) ( , )}) .

∇ + + − + + =

∑

=

β Π τ θ τ θ

1

0

Adding and subtracting A^*α Π_L _i and Aα^*_LΠ_i^*, and noting that labor supply is unchanged, yields

( ) { ( ) ( ) [ ( ) ( , , , ) ( ) ( )]}

{ ( ) ( ) [ ( ) ( , , , ) ( ) ( )]} .

* * * *

,

*

* * * * * * * * * * * * * * *

,

A A I p A A p Nr R Ns p Nu X X *

A I p AA p N r R N s p N u X X

iL L i i

s

i i

s d

E e e

iL L i i

s

i i

s d

e e e

11 0

∇ − + + + + + +

∇ − + + − + + =

β β

α τ θ τ θ

Π Π

Note that an international transfer R drops out of the maximization problem because it does not affect efficiency. First consider the first-order condition for the home country’s production policy for good i, that is for β = τi.Denoting M_i =D_i −X_i, substituting (A7) into (A10), and then using (A1) and (A5) gives

(4)

( )

[ ( ) ( ) ]

[ ( ) ( )( )]

[ ) ]

(

*

A

A I X p

A A X p

X p

p X p X p

D p

A A p

M p D p

X p

A A N u E X p

N u E X p

A I

iL L i

i s

i

i i

s

i

i i

i s

i i

s

i i

s

i i

i s

i i

i d

i

i w

i

i i i

w

i i

i d

i i

i s

i

E X i

i s

i

E X i

i s

i

12

1 1

−

+ + − + + − ′ −

+ − + − ′ − ′

+ ′ ′ ′ + ′ ′ ′

+

α ∂

∂τ

∂

∂τ τ ∂

∂τ

∂

∂τ

∂

∂τ θ θ ∂

∂τ

∂

∂τ

∂

∂τ

∂

∂τ

iL L i

i s

i

i i

s

i

i i

i s

i i s

i i

i s

i i

i d

i

i w

i

i i i

w

i i

i d

i i

i s

i

E X i

i s

i

E X i

X p

AA X p

X p

p X p

D p

AA p

M p D p

X p

AA N u E X p

N u E X

*

* * *

*

* *

*

* *

*

* * *

*

* * * * *

*

* * * * *

*

* *

)

[ ( ) ( ) ]

[ ( ) ( )( )]

[ )

−

+ + − + − ′ −

+ − + − ′ − ′

+ ′ ′ ′ + ′ ′ ′

α ∂

∂τ

∂

∂τ τ ∂

∂τ

∂

∂τ

∂

∂τ θ θ ∂

∂τ

∂

∂τ

∂

∂τ

∂

1 1

p

A I X p

A A X p

p A A p X X p

X p

A A p D p

X p

A A N u E X p

N u E X p

AA I X

i s

i

iL L i

i s

i

i i

i s

i i

w i

i i

w

i i i

i s

i

i i

i w

i

i w

i i

i d

i i

i s

i

E X i

i s

i

E X i

i s

i

iL L i

∂τ

α ∂

∂τ

τ ∂

∂τ θ τ

θ

τ τ ∂

∂τ θ ∂

∂τ

θ ∂

∂τ

∂

∂τ

∂

∂τ

∂

∂τ α

]

( )

[( ) ] [ ]

[ ( )( )]

[ ) ]

( )

*

* *

*

* *

=

−

+ − ′ + + −

+ − ′ − ′

+ ′ ′ ′ + ′ ′ ′

+ −

1

*

* * *

*

* * * *

* *

*

* * *

*

* * * * *

*

* *

[ ( ) ]

[ ( )( )]

[ ) ] .

∂

∂τ

τ ∂

∂τ θ ∂

∂τ τ ∂

∂τ θ τ

θ ∂

∂τ

∂

∂τ

∂ ∂

p

AA X p

X p

p

AA p D p

X p

AA N u E X p

N u E X p

i s

i

i i

i s

i

i i

i w

i

i i

i s

i i

i i w

i w

i i

i d

i i

i s

i

i s

+ − + − ′

+ − ′ − ′

+ ′ ′ ′ + ′ ′ ′ =

1

0

(5)

Differentiating the price equilibrium conditions (A6) with respect to τi yields

(A a) p_i^s p p

i i

w

i i

w i

i

13 ∂ ₂

∂τ θ τ

∂

∂τ

θ

= − τ⋅

( )

* *

A b p_i^s * p

i i

i i w

i

13 ∂

∂τ θ τ

∂

= ∂τ

(A c) p_i^d p

i i

i w

i

13 ∂

∂τ θ ∂

= ∂τ

( )

*

A d p_i^d * p

i

i w

i

13 ∂ ii

∂τ θ ∂

= ∂τ .

Using equations (A13), and collecting terms, the first order condition for the production policy (A12) can be written as

( )

( ) ( )

[( ) ( )]

[ ( )( ( ))]

[ ( ))

*

A

A I X p p

A A X p p p

A A p D p

X p p

A A Nu E X p p

Nu E X

iL L i

i

i i

w

i i

w i

i

i i

i

i i

w i

i i

w

i i

w i

i

i w

i i

i w

i

i i

i

i i

w

i i

w i

i

E X i

i

i i

w

i i

w i

i

E X i

i

14

1

2

− − +

− ′ − +

− ′ − ′ −

+ ′ ′ − + ′ ′

α θ

τ

∂

∂τ

θ τ

τ θ

τ

θ τ

∂

∂τ

θ τ

θ ∂

∂τ θ θ

τ

∂

∂τ

θ τ θ

τ

∂

∂τ

θ τ

∂

∂τ

α θ

τ

∂

∂τ

τ θ

τ

∂

∂τ θ τ

θ θ ∂

∂τ θ τ

∂

∂τ

θ τ

∂

∂τ

θ τ p

A I X p

AA X p

p

AA p M p

AA N u E X p

N u E X p p

i w

i

iL L i

i

i i

w

i

i i

i

i i

w

i i

w

i w

i i i

i w

i

E X i

i

i i

w

i

E X i

i

i i

w

i i

w i

i

]

( )

[( ) ]

[ ( ) ]

[ ( )]

.

*

* * *

*

* * *

*

* * * *

* * *

*

* *

+

−

+ − ′

+ ′ ′ + ′ ′ −

=

1

0

2

(6)

To get ^∂

∂τ p_i^w

i

totally differentiate the world market equilibrium condition

D p X p

i i i

w i

i i

w

i

i i i

w i

i i

w

i

( ) ( ) [ ^*( ^* ) ^*( )]

*

θ θ *

τ θ θ

− = − − τ , where the asterisk stands for the variables

in the foreign country. This yields

( ) ( )

( ) ( ) ( )

* * * * *

*

A p X p

D D

X p

M M

i w

i

i i

w

i i

X

i i

X

i i

w

i i i i

i i

i

15 ∂ ² i²

∂τ θ θ θ θ

θ τ

τ τ

θ τ

= − ′

′ − + ′ − = − ′

′ + ′

′ ′ ,

whereM_i′ = ′ −D_i ^Xⁱ^′

τi and M_i′ = ′ −D_i ^Xⁱ^′

i

* * ^*

τ*. Differentiating the price equilibrium conditions, p_i^s ⁱ p

i i

=θ w

τ and p_i^d =θ_ip_i^w, and using equation (15) yields

( ) ( )

( )

* *

1 6a p p₂ D M

M M

i s

i

i w

i i i i

i i i i i

∂

∂τ

θ θ θ

τ θ θ

= − ′ + ′

′ + ′

( ) ( )

* *

1 6b p X ² p

M M

i d

i

i i i

w

i i i i

i

∂ i

∂τ

θ

θ θ

θ τ

= − ′

′ + ′

( )

* *

16c p M p

M M

i s

i

i i i

w

i i i i i

∂

∂θ

θ

τ θ θ

= ′

′ + ′

( ) .

* *

16d p M p

M M

i d

i

i i i

w

i i i i

∂

∂θ

θ

θ θ

= ′

′ + ′

Using equations (A15) and (A16) and solving implicitly for the home production policies yields

(7)

( ) ( ) ( )

( )

( )[ ( ) ( )

( ) ]

( )

( ) ( ) ( )

* *

*

* *

*

* *

*

A I

a

X p X

p M p X X p p M M

X p p D M

I a

X

p D M

X

i

iL L

L

i

i s

i

i i

w

i i i

w

i i i

w

i i

w

i i i i

i i

w

i i

w

i i i i

iL L

L

i

i s

i i i i

i

i i

17 1

1

2

τ α

α

θ τ θ θ θ θ

θ θ θ

α

α θ θ τ

θτ θτ θ

τ

− = − −

+ ′

− − ′ − ′ + ′ ′ + ′

− ′ ′ + ′

− −

+ ′ + ′ − − ′^*

* *

*

* *

*

* *

*

* *

*2

*

( )

( )[

( )]

( ) ( )

( )

θ τ θτ

θτ

θ τ

θ θ

i i

i i i i

i

i i

i i i i

E E X

i s

E X E X i

i s

i i i i

D M

M

D M

Nu N u E p

Nu E N u E X

p D M

′ + ′

− − ′

′ + ′

− ′ + ′ − ′ + ′ ′

′ + ′ 1

( ) ( ) ( )

( )

( ) ( )

( )

[( ) ( )][

( )]

( )

*

* *

*

* *

*

* *

*

* *

*

*2

A I

a

X p X

I a

X

p D M

p X

p D M

p M

p D M

Nu N u E p

i

iL L

L

i

i s

i

iL L

L

i

i s

i i i i

i

i w

i

i s

i i i i

i i

i w

i i

i s

i i i i

E E X

i s

i i

18 1

1

1 1

τ α

α

α θ θ

τ θ θ

θ θ θ

θ θ

θ τ

− = − −

+ ′ − −

+ ′ + ′

− − ′

′ + ′

+ − − − ′

′ + ′

− ′ + ′ − ′ + ′ ′

′ + ′

( )

( ) .

* *

*

* *

*

Nu N u E X *

p D M

E E X i

i s

i i i i

i i

θ τ

θ θ

The foreign production policy can be derived in a similar manner as the home production policy

( ) ( ) ( )

( )

( ) ( )

( )

[( ) ( )][

( )]

( )

* *

*

* * * * *

* * *

*

* * *

* *

*

A I

a

X p X

I a

X

p D M

p X

p D M

p M

p D M

Nu N u E p

i

iL L

L

i

i s

i

iL L

L

i

i s

i i i i

i

i w

i

i s

i i i i

i i

i w

i i

i s

i i i i

E E X

i i

19 1

1

1 1

2 2

τ α

α

α θ θ

τ θ θ

θ θ θ

θ θ

θ τ

− = − −

+ ′ − −

+

′

′ + ′

− − ′

′ + ′

+ − − − ′

′ + ′

− ′ + ′

i s

E E X i

i s

i i i i

Nu N u E X

p D M

i i

*

* *

* * *

( )

− ′ + ′ ′

′ + ′

θ τ

θ θ

(8)

Equations (A18) and (A19) are the same as equations (20a) and (20b) in chapter 4. Now consider the first-order condition for the trade policy for industry i, that is for β θ= _i in (A10).

Substituting (A7) into (A10), and using (A1) and (A5) gives

( )

[ ( ) ( ) ]

[ ( ) ( )( )]

[

*

A

A I X p

A A X p

X p

p X p

D p

A A p

M p M p D p

X p

A A N u E X p

iL L i

i s

i

i i

s

i

i i

i s

i i s

i i

i s

i i

i d

i

i w

i

i i i

w

i i

w

i i

i d

i i

i s

i

E X i

i s

20

1 1

−

+ + − + − ′ −

+ − + + − ′ − ′

+ ′ ′ ′

α ∂

∂θ

∂

∂θ τ ∂

∂θ

∂

∂θ

∂

∂θ θ θ ∂

∂θ

∂

∂θ

∂

∂θ

∂

∂θ

α ∂

∂θ

∂

∂θ τ ∂

∂θ

∂

∂θ

∂

∂θ θ θ ∂

i

E X i

i s

i

iL L i

i s

i

i i

s

i

i i

i s

i i

s

i i

i s

i i

i d

i

i w

i

i i i

w

i i

i d

N u E X p

A I X p

AA X p

X p

p X p

D p

AA p

M p D p

) ]

( )

[ ( ) ( ) ]

[ ( ) ( )(

*

* * *

*

* * * *

*

* * *

*

* * * * *

*

+ ′ ′ ′

+ −

+ + − + − ′ −

+ − + − ′

1 1

∂θ

∂

∂θ

∂

∂θ

∂

∂θ

i i

i s

i

E X i

i s

i

E X i

i s

i

X p

AA N u E X p

N u E X p

− ′

+ ′ ′ ′ + ′ ′ ′ =

*

* * * * *

*

* *

)]

[ ) ] 0.

Differentiating the price equilibrium conditions (A6) with respect to θi yields

(A a) p ( p )

i p

s

i i

w

i i

21 ∂ 1 w

∂θ τ θ ∂

= ∂θ +

( )

* *

A b p_i^s * p

i i

w

i

21 ∂

∂θ θ τ

∂

= ∂θ

(A c) p p

i p

d

i i

i w

i i

21 ∂ w

∂θ θ ∂

= ∂θ +

( )

*

A d p_i^d * p

i i

i w

i

21 ∂

∂θ θ ∂

= ∂θ

Substituting (A21) into (A20) and collecting terms yields

(9)

( )

( ) ( )

[ ( ) ]

[( ) ( )]

[ ( )( )(

*

A

A I X p

p

A A X p

p p X X p

A A D p

p p D D p

A A X p p

p

A A p D X

iL L i

i i

i w

i i

w

i i

i w

i i

w i

w

i i i

i w

i

i i

i w

i i w

i w

i i i

i w

i

i i

i

i i

w

i i

i w

i i

w

i w

i i

i

22

1

1 1

1

− +

+ + − −

+ − + + +

+ − ′ +

+ − ′− ′

α τ θ ∂

∂θ τ τ θ ∂

∂θ θ ∂

∂θ

θ ∂∂θ θ ∂

∂θ

τ θ

τ τ θ ∂

∂θ

θ τ θ ∂p

p

A A N u E X p

p N u E X p

A I X p

AA X p

X p

AA D p

D p

AA p X

i w

i i

w

E X i

i i

i w

i i

w

E X i

i

i i

w

i

iL L i

i

i i

w

i

i i

i

i i

w

i

i i

i w

i

i i

i w

i

i i

i w

i

i i

w i

∂θ

τ θ ∂

∂θ

θ τ

∂

∂θ

α θ

τ

∂

∂θ

τ θ

τ

∂

∂θ θ ∂

∂θ θ ∂∂θ θ ∂

∂θ

τ θ

+

+ ′ ′ ′ + + ′ ′ ′

+ −

+ − +

+ − ′

)]

[ ( )) ]

( )

[ ]

[( ) (

*

* * *

*

* * *

*

* *

* * * * *

* * *

1

1 ⁱ

i i

w

i

i w

i i i

i i

i w

i

E X i

i

i i

w

i

E X i

i i

i w

i i

w

p

AA p D X p

AA N u E X p

N u E X p

p

*

* * * *

*

* * * * *

*

* *

) ]

[ ( ) ( ) ]

[ ) ( )]

.

τ

∂

∂θ

θ θ τ ∂

∂θ θ

τ

∂

∂θ τ θ ∂

∂θ

2

1

1 0

+ − ′ − ′

+ ′ ′ ′ + ′ ′ ′ +

=

Total differentiation of the market equilibrium condition yields

( ) ( )

* * ^* * *

*

A p D p

D D

M p

M M

i w

i

i X

i w

i i

X

i i

X

i i

w

i i i i

i i

23 ∂

∂θ θ θ θ θ

τ

τ τ

= − ′ −

′ − + ′ − = − ′

′ + ′

′

′ ′

Substituting equation (A23) into equation (A22) and solving for the difference in trade policies yields