Lecture3.2_2019_handout.pdf - ECON280D Lecture 3.2 Quantitative Spatial Equilibrium Models Equilibrium Estimation Counterfactuals Cecile Gaubert U.C

# Lecture3.2_2019_handout.pdf - ECON280D Lecture 3.2...

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ECON280D Lecture 3.2: Quantitative Spatial Equilibrium Models: Equilibrium, Estimation, Counterfactuals Cecile Gaubert U.C. Berkeley Thanks to Treb Allen for sharing his slides ECON280D. Spring 2018. C. Gaubert Lecture 3.2 Equilibrium, Estimation, Counterfactuals 1 / 39
Allen and Arkolakis (2014) I N different regions of a country Workers I Perfectly mobile between regions of a country I Supply 1 unit of labor inelastically and earn wage w i I CES preferences over different varieties of the traded good, elasticity of substitution σ I We can write the indirect utility function of living in location i as: W i = w i P i × A i , where A i is the amenity of living in location i . I Welfare equalization implies: W = w i P i A i ⇐⇒ P i = w i A i W , ECON280D. Spring 2018. C. Gaubert Lecture 3.2 Equilibrium, Estimation, Counterfactuals 2 / 39
Production side: an Armington trade model I Armington assumption : I each region produces its own differentiated variety of the good I within region, all firms are perfectly competitive and produce this same variety I Labor is the only factor of production ( L i workers in i ) I Productivity of worker in i in T i I Geography: trade from i to j subject to an iceberg trade cost τ ij 1. I Price of good from i in j is therefore: w i T i τ ij ECON280D. Spring 2018. C. Gaubert Lecture 3.2 Equilibrium, Estimation, Counterfactuals 3 / 39
Gravity for trade flows I Define: I Y j total income in region j I X ij expenditure of region j spent on goods from region i I π ij = X ij N k =1 X kj share of expenditure of region j spent on goods from region i I Given CES demand: π ij = τ ij w i T i 1 - σ P 1 - σ j I We get gravity in trade flows: X ij = τ 1 - σ ij × w i T i 1 - σ × Y j P 1 - σ j ! where: I P 1 - σ j k τ kj w k T k 1 - σ I Y j = w j L j . ECON280D. Spring 2018. C. Gaubert Lecture 3.2 Equilibrium, Estimation, Counterfactuals 4 / 39
Equilibrium Conditions from Armington trade model I Equilibrium condition #1: Income is equal to total sales (“Market clearing”): equality of income and expenditures Y i = X j X ij ⇐⇒ w i L i = X j τ 1 - σ ij × w i T i 1 - σ × w j L j P 1 - σ j ! I Equilibrium condition #2: Income is equal to total purchases (“Balanced trade”): Y i = X j X ji ⇐⇒ w i L i = X j τ 1 - σ ji × w j T j 1 - σ × w i L i P 1 - σ i ! P 1 - σ i = X j τ 1 - σ ji × w j T j 1 - σ I Notes: I Balanced trade = definition of price index I Market clearing + balanced trade: Can solve for wages { w i } given trade costs, productivities, and population { τ ij , T i , L i } . ECON280D. Spring 2018. C. Gaubert Lecture 3.2 Equilibrium, Estimation, Counterfactuals 5 / 39
Model equilibrium in Allen Arkolakis I “Market clearing” (note: same as Armington trade model) w i L i = X j τ 1 - σ ij × w i T i 1 - σ × w j L j P 1 - σ j ! I “Balanced trade” (note: same as Armington trade model) P 1 - σ i = X j τ 1 - σ ji × w j T j 1 - σ I “Welfare equalization” (only in economic geography): P i = w i A i W I “Aggregate Labor market clearing” (only relevant in economic geography): X i L i = ¯ L ECON280D. Spring 2018. C. Gaubert Lecture 3.2 Equilibrium, Estimation, Counterfactuals 6 / 39
Model equilibrium (ctd.) I “Market clearing” + “welfare equalization”: w i L i = X j τ 1 - σ ij × w i T i 1 - σ × w j L j w j A j W 1 - σ I “Balanced trade” + “welfare equalization”: