Part 1 - AL-Tax

Chapter 4The Lagrangian of the problem:⎡ ⎛ I ⎞ ⎛ I ⎞ ⎤B1B1L UB1δ2 UB1ννUB2⎝⎜ω1⎠⎟⎝⎜ω2⎠⎟⎣⎢⎦⎥⎡ ⎛ ⎞I U ⎟⎛ ⎞ ⎤B2δ1 B2ν ν I B2UB1⎝⎜ω2⎠⎟⎝⎜⎠⎟⎣⎢ω1⎦⎥⎛⎡ ⎛ I ⎞ ⎤B1pP 1 1( UA1, UB1)UB1ν IB1⎝⎜ω 1⎠⎟⎝⎜⎣⎢⎥⎦⎡ ⎛I ⎞ ⎤⎞B2 pP 2 2( UA2, UB2)UB2ν I ⎝⎜ω 2⎠⎟ B2 ,⎣⎢⎦⎥⎠⎟where λ, δ 1 , and δ 2 are the multipliers associated with the constraints. The generalmethodology is the following. The first-order conditions with respect to(I B1 ,U B1 ,I B2 ,U B2 ) characterize the best reply of government B to government A’spolicy. We set U B1 U A1 and U B2 U A2 . The first-order conditions are then:λ ⎡ I B11δδλ2 1 p1p1IB1ν ⎛ ⎞ ⎤UB1σ1 ⎝⎜ω1⎠⎟ 0,⎣⎢⎦⎥λ ⎡ ⎛ ⎜I ⎞ ⎤2δδλ1 2 p2 p B2 IB2ν⎜σU2 ⎝ ⎜B2ω2⎠⎟ 0,⎣⎢⎦⎥⎡ 1 ⎛ ⎜I ⎞ ⎛ ⎞B2δ1 ⎤1 ν⎜ 2 νI ⎡B 1 Iω2⎝ ⎜ ω2⎠⎟⎝⎜⎠⎟λp B 22 1 ν ⎛ ⎞⎤ 0,⎣⎢ω1ω 1 ⎦⎥ω2⎝⎜ω2⎠⎟⎣⎢⎦⎥⎡ 1 ⎛ I ⎞1 1 ⎛ I ⎞⎤⎡1δ2 ν BB ν1 ⎛I ⎞⎤ω1⎝⎜ω1⎠⎟ω2⎝⎜ω2⎠⎟λp B 11 1 ν ⎣⎢⎦⎥⎝⎜ω1ω1⎠⎟ 0.⎣⎢⎦⎥The budget constraint becomes:⎡p U⎣⎢1 B1⎛ I ⎞ ⎤ ⎡B1ν IB1 p 2 U⎝⎜ω 1⎠⎟⎦⎥⎣⎢The incentive constraints remain unchanged. We consider solutions to the latterconditions, such that δ 1 0, δ 2 0 and λ 0. If we find a solution, by constructionit will be a symmetric equilibrium of the game. As we consider symmetricequilibria, we use the following notation: U A1 U B1 U 1 , U A2 U B2 U 2 , andI A1 I B1 I 1 , and I A2 I B2 I 2 .B2⎛2ν I ⎞ ⎤BIB2 0.⎝⎜ω2⎠⎟⎦⎥91