Online Appendix or Lerner Symmetry: A Modern Treatment Arnaud Costinot MIT Iván Werning MIT May 2018 Abstract Tis Appendix provides te proos o Teorem 1, Teorem 2, and Proposition 1.
1 Perect Competition For convenience, we irst repeat te deinition o a competitive equilibrium as well as assumptions A1-A3. We ten oer a ormal proo o Teorem 1. 1.1 Equilibrium A competitive equilibrium wit taxes, t {tij k (n)}, subsidies, s {sk ij (n)}, and lump-sum transers, τ {τ()} and T {T ij }, corresponds to quantities c {c()}, l {l()}, m {m( )}, y {y( )}, and prices p {pij k } suc tat: (i) (c(), l()) solves max (ĉ(),ˆl()) Γ() u(ĉ(), ˆl(); ) p(1 + t()) ĉ() = p(1 + s()) ˆl() + π θ() + τ(), or all ; (ii) (m( ), y( )) solves π( ) max p(1 + s( )) ŷ( ) p(1 + t( )) ˆm( ), or all ; ( ˆm( ),ŷ( )) Ω( ) (iii) markets clear: y( ) + l() = c() + m( ); (iv) government budget constraints old: p k ji ( = t k ji ()ck ji () + t k ji ( )mk ji ( )) + T ji p k ij ( sij k ()lk ij () + sij k ( )yk ij ( )) + τ() + T ij, or all i; H i 1.2 Assumptions A1. For any irm, production sets can be separated into Ω( ) = Ω i0 ( ) Ω i0 ( ), were Ω i0 ( ) denotes te set o easible production plans, {m k ji 0 ( ), y k i 0 j ( )}, in country i 0 and Ω i0 ( ) denotes te set o easible plans, {m k ji ( ), yk ij ( )} i =i 0, in oter countries. 1
A2. For any ouseold, consumption sets can be separated into Γ() = Γ i0 () Γ i0 (), were Γ i0 () denotes te set o easible consumption plans, {c k ji 0 ( ), l k i 0 j ( )}, in country i 0; Γ i0 () denotes te set o easible plans, {c k ji ( ), lk ij ( )} i =i 0, in oter countries; and Γ i0 () and Γ i0 () are suc tat H i0 Γ i0 () = {0} and / H i0 Γ i0 () = {0}. A3. For any oreign country j = i 0, te total value o assets eld in country i 0 prior to te tax reorm is zero, π i0 Hj θ() = 0. 1.3 Lerner Symmetry Teorem 1 (Perect Competition). Consider a reorm o trade taxes in country i 0 satisying 1 + t k ji 0 (n) 1 + t k ji 0 (n) = 1 + sk i0 j (n) 1 + si k 0 j (n) = η or all j = i 0, k, and n, or some η > 0; all oter taxes are uncanged. I A1 and A2 old, ten E(t, s) = E( t, s); i A1, A2, and A3 old, ten E(t, s, T) = E( t, s, T). Proo. (E(t, s) = E( t, s)). It suices to establis tat E(t, s) E( t, s), since ten, reversing te notation, one also as E( t, s) E(t, s), yielding te desired equality. For any (c, l, m, y) E(t, s) wit associated (p, τ, T), we sow tat (c, l, m, y) E( t, s) by constructing a new ( p, τ, T) to veriy te equilibrium conditions (i)-(iv). For all, i, j, and k set p ij k = pij k η i i = j = i 0, pij k oterwise, (1.1) τ() = p(1 + t()) c() p(1 + s()) l() π θ(), (1.2) T ij = T ij + [π i π i ] H j θ(), (1.3) wit π { π( )} te vector o irms total proits under te new tax scedule and π i 2
{ π i ( )} te vector o proits derived rom transactions in country i, π( ) = [ p ij k (1 + sk ij ( ))yk ij ( ) pk ji (1 + t k ji ( ))mk ji ( )], i, π i ( ) = [ p ij k (1 + sk ij ( ))yk ij ( ) pk ji (1 + t k ji ( ))mk ji ( )]. Given te cange in taxes rom (t, s) to ( t, s) tat we consider, equation (1.1) implies tat all ater-tax prices aced by buyers and sellers rom country i 0 are multiplied by η, wile oter ater-tax prices remain uncanged, p k ji 0 (1 + t k ji 0 (n)) = ηp k ji 0 (1 + t k ji 0 (n)), (1.4) p i k 0 j (1 + sk i 0 j (n)) = ηpk i 0 j (1 + sk i 0 j (n)), (1.5) (1 + t k ji (n)) pk ji = (1 + tk ji (n))pk ji, (1.6) (1 + s k ij (n)) pk ij = (1 + sk ij (n))pk ij, (1.7) i i = i 0. In turn, proits in te proposed equilibrium satisy π i η i i = i 0, π i = oterwise. π i (1.8) First, consider condition (i). Equation (1.2) implies tat te ouseold budget constraint still olds at te original allocation (c(), l()) given te new prices, p, taxes, t and s, and transers, τ. Under A2, equations (1.4) and (1.5) are tereore suicient or condition (i) to old in country i 0, wereas equations (1.6) and (1.7) are suicient or it to old in countries i = i 0. Next, consider condition (ii). Under A1, equations (1.4) and (1.5) are again suicient or condition (ii) to old in country i 0, wereas equations (1.6) and (1.7) are suicient or it to old in countries i = i 0. Since te allocation (c, l, m, y) is uncanged in te proposed equilibrium, te good market clearing condition (iii) continues to old. Finally, we veriy te government budget balance condition (iv). Let R i and R i denote te net revenues o country i s government at te original and proposed equilibria, R i p k ji ( t k ji ()ck ji () + t k ji ( )mk ji ( )) + T ji p k ij ( sij k ()lk ij () + sij k ( )yk ij ( )) τ() T ij, H i 3
R i p k ji ( t k ji ()ck ji () + t k ji ( )mk ji ( )) + p k ij ( s k ij ()lk ij () + In any country i = i 0, equations (1.1) (1.3) imply T ji s k ij ( )yk ij ( )) H i τ() R i = R i + [π j π j ] θ() + [ π π] θ() [π i π i ] θ(). H i H i H j Using te government budget constraint in country i at te original equilibrium, R i = 0, and noting tat [π j π j ] θ() = H i we tereore arrive at [π π] θ() [π i π i ] θ(), H i H i R i = [π i π i ] j H j θ(). T ij. Togeter wit equation (1.8), tis implies government budget balance, R i i = i 0. Let us now turn to country i 0. Equation (1.2) and A2 imply = 0, or all R i0 = p k ji 0 ( c k ji 0 ()) + p i k 0 j ( li k 0 j ()) + π θ() H i0 [ p i k 0 j sk i 0 j ( )yk i 0 j ( ) pk ji 0 t k ji 0 ( )m k ji 0 ( )] +, By equation (1.3), tis is equivalent to R i0 = p k ji 0 ( c k ji 0 ()) + p i k 0 j ( li k 0 j ()) + π θ() H i0 T ji0 [ p i k 0 j sk i 0 j ( )yk i 0 j ( ) pk ji 0 t k ji 0 ( )m k ji 0 ( )] + [T ji0 + [π j π j ] θ()], 0 H i0 0 [T i0 j + [π i0 π i0 ] H j θ()]. Togeter wit te ouseolds budget constraints, te government budget constraint in T i0 j. 4
country i 0 in te original equilibrium implies p k ji 0 ( c k ji 0 ()) + T i0 j = 0 Combining te two previous observations, we get R i0 = ( p k ji 0 p k ji 0 )( c k ji 0 ()) + pi k 0 j ( li k 0 j ()) + π θ() + T ji0. H i0 0 ( p i k 0 j pk i 0 j )( li k 0 j ()) [ p i k 0 j sk i 0 j ( )yk i 0 j ( ) pk ji 0 t k ji 0 ( )m k ji 0 ( )] + [pi k 0 j sk i 0 j ( )yk i 0 j ( ) pk ji 0 t k ji 0 ( )m k ji 0 ( )],, + [ π i0 π i0 ] j H j θ(). Using equation (1.1) and te deinitions o π i0 and π i0, tis simpliies into R i0 =(1 η) k p k [ c k () + m k ( ) li k 0 i 0 () yi k 0 i 0 ( )]. Togeter wit te good market clearing condition (iii), tis proves government budget balance R i0 = 0. Tis concludes te proo tat (c, l, m, y) E( t, s). (E(t, s, T) = E( t, s, T)). As beore, it suices to establis E(t, s, T) E( t, s, T). Equations (1.3) and (1.8) imply T ij i i = i 0 and j = i, T ij = T ij + (1 η)π i Hj θ() i i = i 0 and j = i 0. Under A3, tis simpliies into T ij = T ij or all i = j. Togeter wit te irst part o our proo, tis establises tat (c, l, m, y) E( t, s, T). 2 Imperect Competition For convenience, we repeat te deinition o an equilibrium under imperect competition as well as assumption A1. We ten oer a ormal proo o Teorem 2. 2.1 Equilibrium An equilibrium requires ouseolds to maximize utility subject to budget constraint taking prices and taxes as given (condition i), markets to clear (condition iii), and govern- 5
ment budget constraints to old (condition iv), but it no longer requires irms to be pricetakers. In place o condition (ii), eac irm cooses a correspondence σ( ) tat describes te set o quantities (y( ), m( )) Ω( ) tat it is willing to supply and demand at every price vector p. Te correspondence σ( ) must belong to a easible set Σ( ). For eac strategy proile σ {σ( )}, an auctioneer ten selects a price vector P(σ) and an allocation C(σ) {C(σ, )}, L(σ) {L(σ, )}, M(σ) {M(σ, )}, and Y(σ) {Y(σ, )} suc tat te equilibrium conditions (i), (iii), and (iv) old. Firm solves max P(σ)(1 + s( )) Y(σ, ) P(σ)(1 + t( )) M(σ, ), (2.1) σ( ) Σ( ) taking te correspondences o oter irms {σ( )} = as given. 2.2 Assumptions A1. For any irm, production sets can be separated into Ω( ) = Ω i0 ( ) Ω i0 ( ), were Ω i0 ( ) and Ω i0 ( ) are suc tat eiter Ω i0 ( ) = {0} or Ω i0 ( ) = {0}. In line wit te proo o Teorem (1), we deine te unction ρ η mapping p into p using (1.1), tat is, ρ η (pij k ) = pij k η i i = j = i 0, pij k oterwise. (2.2) Its inverse ρ 1 η is given by ρ 1 η (p k ij ) = pij k /η i i = j = i 0, p k ij oterwise. For any η > 0, we assume tat i σ( ) Σ( ), ten σ( ) = σ( ) ρ 1 η Σ( ). 2.3 Lerner Symmetry Teorem 2 (Imperect Competition). Consider te tax reorm o Teorem 1. I A1 and A2 old, ten E(t, s) = E( t, s); i A1, A2, and A3 old, ten E(t, s, T) = E( t, s, T). Proo. Fix an equilibrium wit strategy proile σ, taxes (t, s), auctioneer s coices P(σ ), 6
C(σ ), L(σ ), M(σ ) and Y(σ ), and realized prices p = P(σ). Deine a new strategy proile σ = σ ρ 1 η. We sow tat σ is an equilibrium strategy, wit taxes ( t, s) and auctioneer coices, P( σ ) = ρ η (P( σ ρ η )), C( σ ) = C( σ ρ η ), L( σ ) = L( σ ρ η ), M( σ ) = M( σ ρ η ), Ỹ( σ ) = Y( σ ρ η ), and realized prices p = P( σ) = ρ η (p). We ocus on te proit maximization problem o a given irm ; te rest o te proo is identical to te perect competition case. Deine te set o easible deviation strategies or irm at te original and proposed equilibria D,σ = {σ (σ ( ), σ( )) or all σ ( ) Σ( )}, D, σ = { σ ( σ ( ), σ( )) or all σ ( ) Σ( )}, were σ( ) = {σ( )} = Π = Σ( ) and σ( ) = { σ( )} = Π = Σ( ). By assumption, σ( ) = σ( ) ρ 1 η Σ( ). We tereore need to prove tat P( σ)(1 + s( )) Ỹ( σ, ) P( σ)(1 + t( )) M( σ, ) P( σ )(1 + s( )) Ỹ( σ, ) P( σ )(1 + t( )) M( σ, ), (2.3) or all σ D, σ. By condition (2.1), σ satisies P(σ)(1 + s( )) Y(σ, ) P(σ)(1 + t( )) M(σ, ) P(σ )(1 + s( )) Y(σ, ) P(σ )(1 + t( )) M(σ, ), (2.4) or all σ D,σ. Decompose (M(σ, ), Y(σ, )) = (M i0 (σ, ), M i0 (σ, ), Y i0 (σ, ), Y i0 (σ, )) so tat (M i0 (σ, ), Y i0 (σ, )) Ω i0 ( ) and (M i0 (σ, ), Y i0 (σ, )) Ω i0 ( ). Decompose P(σ ), t( ) and s( ) in te same manner. Wit tis notation, A1 and (2.4) imply P i0 (σ)(1 + s i0 ( )) Y i0 (σ, ) P i0 (σ)(1 + t i0 ( )) M i0 (σ, ) P i0 (σ )(1 + s i0 ( )) Y i0 (σ, ) P i0 (σ )(1 + t i0 ( )) M i0 (σ, ) (2.5) 7
and P i0 (σ)(1 + s i0 ( )) Y i0 (σ, ) P i0 (σ)(1 + t i0 ( )) M i0 (σ, ) P i0 (σ )(1 + s i0 ( )) Y i0 (σ, ) P i0 (σ )(1 + t i0 ( )) M i0 (σ, ), (2.6) as one o te two inequalities olds trivially as an equality wit zero on bot sides. For any σ Π Σ( ) and σ = σ ρ η Π Σ( ), te new auctioneer s coices imply P( σ )(1 + s( )) Ỹ( σ, ) P( σ )(1 + t( )) M( σ, ) = ρ η (P( σ ρ η ))(1 + s( )) Y( σ ρ η, ) ρ η (P( σ ρ η ))(1 + t( )) M( σ ρ η, ) Equation (2.2) urter implies, = ρ η (P(σ ))(1 + s( )) Y(σ, ) ρ η (P(σ ))(1 + t( )) M(σ, ) ρ η (Pij k(σ ))(1 + s ij k ( )) = ηpij k(σ )(1 + sij k ( )) or all j and k i i = i 0, Pij k(σ )(1 + sij k ( )) or all j and k i i = i 0, ρ η (Pji k(σ ))(1 + t k ji ( )) = ηpji k(σ )(1 + t k ji ( )) or all j and k i i = i 0, Pji k(σ )(1 + t k ji ( )) or all j and k i i = i 0. Tus, it ollows tat and P i0 ( σ )(1 + s i0 ( )) Ỹ i0 ( σ, ) P i0 ( σ )(1 + t i0 ( )) M i0 ( σ, ) = η ( P i0 (σ )(1 + s i0 ( )) Y i0 (σ, ) P i0 (σ )(1 + t i0 ( )) M i0 (σ, ) ), (2.7) P i0 ( σ )(1 + s i0 ( )) Ỹ i0 ( σ, ) P i0 ( σ )(1 + t i0 ( )) M i0 ( σ, ) = P i0 (σ )(1 + s i0 ( )) Y i0 (σ, ) P i0 (σ )(1 + t i0 ( )) M i0 (σ, ). (2.8) Since or any σ D, σ, we ave σ = σ ρ η D,σ, (2.5)-(2.8) imply P i0 ( σ)(1 + s i0 ( )) Ỹ i0 ( σ, ) P i0 ( σ)(1 + t i0 ( )) M i0 ( σ, ) P i0 ( σ )(1 + s i0 ( )) Ỹ i0 ( σ, ) P i0 ( σ )(1 + t i0 ( )) M i0 ( σ, ), 8
and P i0 ( σ)(1 + s i0 ( )) Ỹ i0 ( σ, ) P i0 ( σ)(1 + t i0 ( )) M i0 ( σ, ) P i0 ( σ )(1 + s i0 ( )) Ỹ i0 ( σ, ) P i0 ( σ )(1 + t i0 ( )) M i0 ( σ, ), or all σ D, σ. Adding up tese last two inequalities gives (2.3). 3 Nominal Rigidities For convenience, we repeat te adjustment in prices beore taxes, p ij k pij k η i i = j = i 0, = 1 oterwise. (3.1) For parts o te proo o Proposition 1, we will use te act tat given te tax reorm o Teorem 1, equation (3.1) is equivalent to p k ij (1 + sk ij (n)) p k ij (1 + sk ij (n)) = pk ji (1 + tk ji (n)) p k ji (1 + tk ji (n)) = η or all j and k, i i = i 0, 1 or all j and k, i i = i 0. Proposition 1. Consider te tax reorm o Teorem 1 wit η = 1. Suppose p P(t, s) and p satisies (3.1). Ten p P( t, s) olds i prices are rigid in te origin country s currency ater sellers taxes or te destination country s currency ater buyers taxes, but not i tey are rigid beore taxes. Likewise, p P( t, s) olds i prices are rigid in a dominant currency beore taxes and country i 0 = i D, but not i i 0 = i D. Proo. We irst consider te tree cases or wic p P( t, s). Case 1: Prices are rigid in te origin country s currency ater sellers taxes, (3.2) P(t, s) = {{pij k } {e l} suc tat pij k (1 + sk ij (n)) = pk,i ij (1 + sk ij (n))/e i or all i, j, k,n}. Consider p P(t, s). Let us guess ẽ i0 /e i0 = 1/η and ẽ i /e i = 1 i i = i 0. For any j, k, consider irst i = i 0. From (3.2), we ave p ij k (1 + sk ij (n)) = pk ij (1 + sk ij (n)) = pk,i ij (1 + sk ij (n))/e i = p k,i ij (1 + sk ij (n))/ẽ i. 9
Next consider i = i 0. From (3.2), we ave p k i 0 j (1 + sk i 0 j (n)) = ηpk i 0 j (1 + sk i 0 j (n)) = η pk,i 0 i 0 j (1 + sk i 0 j (n))/e i 0 = p k,i 0 i 0 j (1 + sk i 0 j (n))/ẽ i 0. Tis establises tat p P( t, s). Case 2: Prices are rigid in te destination country s currency ater buyers taxes, P(t, s) = {{pij k } {e l} suc tat pij k (1 + tk ij (n)) = pk,j ij (1 + t ij k (n))/e j or all i, j, k,n}. Consider p P(t, s). Let us guess ẽ i0 /e i0 = 1/η and ẽ i /e i = 1 i i = i 0. For any i, k, consider irst j = i 0. From (3.2), we ave p ij k (1 + t ij k (n)) = pk ij (1 + tk ij (n)) = pk,j ij (1 + t ij k (n))/e j = p k,j ij (1 + t ij k (n))/ẽ j. Next consider j = i 0. From (3.2), we ave p k ii 0 (1 + t k ii 0 (n)) = ηp k ii 0 (1 + t k ii 0 (n)) = η p k,i 0 ii 0 (1 + t k ii 0 (n))/e i0 = p k,i 0 ij (1 + t k ii 0 (n))/ẽ i0. Tis establises tat p P( t, s). Case 3: Prices are rigid in a dominant currency beore taxes are imposed, and i 0 = i D, P(t, s) = {{pij k } {e l} suc tat pij k = pk,i D ij /e id or all i = j, k and pii k = pk,i ii /e i or all k}. Consider p P(t, s). Let us guess ẽ i0 /e i0 = 1/η and ẽ i /e i = 1 i i = i 0, including ẽ id /e id = 1 since i 0 = i D. For any k, j, consider irst i = j. From (3.1), we ave p ij k = pk ij = pk,i D ij /e id = p k,i D ij /ẽ id. Next consider i = j = i 0. From (3.1), we ave p k ii = pk ii = pk,i ii /e i = p k,i ii /ẽ i. Finally, consider i = j = i 0. From (3.1), we ave Tis establises tat p P( t, s). p k = ηp k = η p k,i 0 /e i0 = p k,i 0 /ẽ i0. We now turn to te tree cases or wic p / P( t, s). 10
Case 4: Prices are rigid in te origin country s currency beore sellers s taxes, P(t, s) = {{p k ij } {e l} suc tat p k ij = pk,i ij /e i or all i, j, k,n}. Consider p P(t, s). Suppose p P( t, s). From (3.1), we ave p k i 0 j = pk i 0 j = pk,i 0 i 0 j /e i 0 = p k,i 0 i 0 j /ẽ i 0 i j = i 0, p k = ηp k = η p k,i 0 /e i0 = p k,i 0 /ẽ i0 oterwise. Te irst equation gives ẽ i0 /e i0 = 1; te second gives ẽ i0 /e i0 = 1/η. A contradiction. Case 5: Prices are rigid in te destination country s currency beore buyers taxes, P(t, s) = {{p k ij } {e l} suc tat p k ij = pk,j ij /e j or all i, j, k,n}. Start wit p P(t, s). Suppose p P( t, s). From (3.1), we ave p k ii 0 = p k ii 0 = p k,i 0 ii 0 /e i0 = p k,i 0 ii 0 /ẽ i0 i i = i 0, p k = ηp k = η p k,i 0 /e i0 = p k,i 0 /ẽ i0 oterwise. Te irst equation gives ẽ i0 /e i0 = 1; te second gives ẽ i0 /e i0 = 1/η. A contradiction. Case 6: Prices are rigid in a dominant currency beore taxes are imposed, and i 0 = i D, P(t, s) = {{pij k } {e l} suc tat pij k = pk,i 0 ij /e i0 or all i = j, k and pii k = pk,i ii /e i or all k}. Start wit p P(t, s). Suppose p P( t, s). From (3.1), we ave p k i 0 j = pk i 0 j = pk,i 0 i 0 j /e i 0 = p k,i 0 i 0 j /ẽ i 0 i j = i 0, p k = ηp k = η p k,i 0 /e i0 = p k,i 0 /ẽ i0 oterwise. Te irst equation gives ẽ i0 /e i0 = 1; te second gives ẽ i0 /e i0 = 1/η. A contradiction. 11