Part 1 - AL-Tax

Chapter 4We keep the same methodology. The first-order conditions give the bestreply of government B to the government A policy. We set U B1 U A1 and U B2 U A2 ,and we take arbitrary values for U B1 and U B2 . The first-order conditions become:αδ λδλ ⎡σ I B12 1 p1 p1IB1ν ⎛ ⎞ω⎤UB11 ⎝⎜1⎠⎟ 0,⎣⎢⎦⎥λ ⎡ ⎛ I B21δδλ1 2 p2p2IB2ν ⎜ ⎞ ⎤⎜⎜σUB22 ⎝ω2⎠⎟ 0,⎣⎢⎦⎥⎡ 1 ⎛ I B2δ1ν⎜ ⎞ ⎛ ⎞1 ⎤⎜⎜2 νI Bω2⎝ω2⎠⎟⎝⎜⎠⎟⎣⎢ω1ω ⎡ ⎛ ⎞ 1 I ⎤λpB 22 1 ν1 ⎥⎦⎝⎜⎠⎟ 0,⎣⎢ω2ω2⎦⎥⎡ 1 ⎛ I ⎞1 1 ⎛ I ⎞⎤⎡1δ2 ν BB ν1 ⎛I ⎞⎤ω1⎝⎜ω1⎠⎟ω2⎝⎜ω2⎠⎟λp B 11 1 ν ⎣⎢⎦⎥ω ⎝⎜1 ω1⎠⎟ 0.⎣⎢⎦⎥The budget constraint becomes:⎡p U⎣⎢1 B1⎛ I ⎞ ⎤ ⎡B1ν IB1 p 2 U⎝⎜ω 1⎠⎟⎦⎥⎣⎢B2⎛2ν I ⎞ ⎤BIB2 0.⎝⎜ω2⎠⎟⎦⎥The incentive constraints remain unchanged. We consider symmetric solutionsto the latter conditions. We keep the same notation.Again, tedious but straightforward computation shows that:● If σ 2 ∉ S, ɶ the equilibrium policy involves:⎛ p ⎞21 αpI2 I2 fb I1 I1 fb U2 I2 fb ⎝⎜1⎠⎟ σ , , p2p2σαp p σ11112Iν ⎛ ⎝⎜ω2 fb 2⎞and⎠⎟ ,U Ip1 1 fb 2 p1⎛ p ⎞21 α σ⎝⎜p1⎠⎟p2p2σαp p σ11112⎛ν I 1 fb ⎞⎝⎜ω1⎠⎟ .93