Part 1 - AL-Tax

two incentive constraints similar to (4.1) and (4.2), and the budget constraintbecomes:The government still maximizes the social welfare function with respect to(I B1 ,U B1 ,I B2 ,U B2 ). The program of the government B is similar to the autarky case,except that the number of skilled and unskilled workers in country B is nowendogenous. In mathematical terms:s.t.⎡ ⎛ I ⎞B1pPU 1 1( A1, UB1) UB1ν I⎢ ⎝⎜ω 1⎠⎟⎣B1max U B1 ,⎛ I ⎞ ⎛B I ⎞1 B1UB1ννUB2 0,⎝⎜ω1⎠⎟⎝⎜ω2⎠⎟ ⎛ I ⎞ ⎛B2I ⎞B2UB2ννUB1 0,⎝⎜ω2⎠⎟⎝⎜ω⎟1⎠⎡ ⎛ ⎞pP U U U I ⎤B11 1( A1, B1)B1ν ⎝⎜ω ⎟⎣⎢1⎠ ⎡⎛ I BIB1 p2P2( UA2, UB2)UB2ν ⎦⎥⎝⎜⎣⎢ωWe assume that the best reply of government B is fully characterized by the firstorderconditions of this program: This problem has an interior solution and thesecond-order conditions are always satisfied.Given this, it is convenient to define a threshold value for σ 2 :⎡ Iɶσ p1 I2 fb ν ⎛ ⎞⎤2fb p⎝⎜ω2⎠⎟⎣⎢⎦⎥⎤ ⎡ ⎛I ⎞ ⎤B2pPU 2 2( A2, UB2) UB2ν I ⎦⎥⎝⎜ω ⎠⎟B2 0.⎣⎢2 ⎦⎥(4.4)Given the properties of the function ν, it is easy to see that σ ~ > 0. One can interpretσ ~ as the maximum difference between the income tax paid by the skilledworkers and the income tax paid by the unskilled, such that skilled workers haveno incentive to misreport their type. The quasi-linearity of the utility functionallows us to define easily such a value. 5As we consider similar countries, we restrict the analysis to symmetric equilibriain which there is no migration:⎡I⎣⎢P( U , U ) P ( U , U ) 1.1 A1 B1 2 A2 B21⎛ν ⎜I ω1 fb 1fb ⎝⎜2⎞⎤⎠⎟.⎦⎥22Chapter 4⎞ ⎤I ⎠⎟B2 0.⎦⎥83