w i A i W 1 \u03c3 X j\u03c4 1 \u03c3 ji wj Tj 1 \u03c3I Labor market clearing X i L i L ECON280D

# W i a i w 1 σ x jτ 1 σ ji wj tj 1 σi labor market

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w i A i W 1 - σ = X j τ 1 - σ ji × w j T j 1 - σ I “Labor market clearing”: X i L i = ¯ L ECON280D. Spring 2018. C. Gaubert Lecture 3.2 Equilibrium, Estimation, Counterfactuals 7 / 39
Equilibrium without spillovers I Suppose A i = ¯ A i and T i = ¯ T i exogenous. I Rearranging “Market clearing” + “welfare equalization” W σ - 1 × w σ i L i = X j τ 1 - σ ij × T σ - 1 i × A σ - 1 j × w σ j L j I Note that this system of N equations can be written as: λ x = K x , I K is a known N × N matrix with i th row and j th column τ ij T i A j 1 - σ I x is an unknown N × 1 vector with i th row w σ i L i I λ = W σ - 1 is an unknown scalar. ECON280D. Spring 2018. C. Gaubert Lecture 3.2 Equilibrium, Estimation, Counterfactuals 8 / 39
Equilibrium without spillovers I Rearranging “Balanced trade” + “welfare equalization”: W σ - 1 × w i 1 - σ = X j τ 1 - σ ji × T σ - 1 j × A σ - 1 i × w j 1 - σ I This system of N equations can be written as: λ y = K T y , I with the same matrix K and scalar λ = W σ - 1 ECON280D. Spring 2018. C. Gaubert Lecture 3.2 Equilibrium, Estimation, Counterfactuals 9 / 39
ECON280D. Spring 2018. C. Gaubert Lecture 3.2 Equilibrium, Estimation, Counterfactuals 10 / 39
With spillovers I With spillovers, things are more complicated. One cannot separate the system into two independent systems (in w , and w σ L ). I Plugging into equilibrium system: A i = ¯ A i L β i T i = ¯ T i L α i I we get W σ - 1 × w σ i L 1 - α ( σ - 1) i = X j τ 1 - σ ij × ¯ T i σ - 1 × ¯ A σ - 1 j × w σ j L 1 - β (1 - σ ) j W σ - 1 × w 1 - σ i L β (1 - σ ) i = X j τ 1 - σ ji × ¯ T σ - 1 j × ¯ A σ - 1 i × w 1 - σ j L α ( σ - 1) j I AA(2014) prove Proposition 2 on existence and uniqueness of spatial equilibria in this case. ECON280D. Spring 2018. C. Gaubert Lecture 3.2 Equilibrium, Estimation, Counterfactuals 11 / 39
Proposition 2: Uniqueness, Existence with spillovers I Assume symmetric trade costs τ ij = τ ji I Define γ 1 = 1 - α ( σ - 1) - βσ γ 2 = 1 + ασ + + ( σ - 1) β I Proposition 2 : if γ 1 6