Bilateral Migration and Multinationals - On the Welfare Effects of Firm and Labor Mobility (Appendix)

APPENDIX A: DERIVATION OF GENERAL EQUILIBRIUM The equilibrium is defined by the set of variables oI?H K, p?Hg q, Ij K, Ii K, I3 ,Y K, IS ,Y Kr, where = I1,2K denotes the firm’s original country and
, N = I
1,1,
1,2,
2,1,
2,2K denotes the wage rate for workers who come from country N and work at country . Similarly to the symmetric case, we use the following equilibrium conditions to solve the equilibrium. 1. Zero Cutoff Profits The zero cutoff profits condition can be expressed as a set of equations: x
?H  = 0 x ?H g (29) =0 (30) The profit function and revenue function of establishments are: /.- E 7 1 D 
? = * ∙ ∙ 2 7 − 1 ? :  ∙ 5 1 − =B, = ≠ 7  x
? = ½ 1  
? ∙ − =, = = 7  
? ∙ The index (31) , (32) denotes the country where the establishment located and the index is the country of origin of the establishment. 5 and : are respectively the aggregate expenditure and the price index of country . We substitute (29) and (30) into (57) to derive the solution for ?H : g ?H/,µ g ?Hµ,/ g / D E/µ :/ 5µ = /.= ?H/ ∙ / ∙ µ ∙ * / ∙ B 2 5 = D E/ : (33) / D Eµ/ : µ 5/ = /.= ?Hµ ∙ µ ∙ / ∙ * µ ∙ B 2 , 5 = D Eµ : where D E  is the unit price of labor bundles for establishments from country (34) and located at country , : is the price index at country , 5  is the aggregate expenditure of country . We can derive from (29), (30), and (33) to get the equations for the average profit of firms as: g -./ ?e -./ ?e xH = = ∙ 9b c − 1; + j ∙ = B ∙ Ž¾ g ¿ ?H ?H  − 1– , =f = I1,2K , (35) where ?e is the average productivity of local establishments and ?e g is the average productivity of foreign establishments. 2. Free Entry Free entry of new firms drives the ex-ante expected profits to zero (i.e., y> = 0 =f = I1,2K). According to this, we derive the average profit of firms as: A xH = = > ∙ À u . OÁ?H T  (36) We combine (35) and (57) to derive the solution for the cutoff productivity for local establishment as: / = + j ∙ = B 7−1 u ?H = 9 ∙ ; ∙  =f = I1,2K . A ∙ => 1+w−7 (37) The probabilities of being a multinational firm are: / .u / .u 1 − @?H/ D E/µ :/ 5µ = /.j/ = = Ž / ∙ µ ∙ * / ∙ B2 – 1 − @
?H/  5 = D E/ : g 1 − @?Hµ D E µ/ :µ 5/ = /.jµ = = Ž µ ∙ / ∙ * µ ∙ B2 – 1 − @
?Hµ  5 = D Eµ : g (38) (39) 3. Labor Market Clearing The labor market clearing condition requires that the aggregate labor supply should equal to the aggregate labor demand for all types of labor in all countries. The labor market clearing condition can be written as: E/ F —,/,/ + F—,µ,/ = F Eµ F —,µ,µ + F —,/,µ = F G—,/,/ + G—,µ,/ = GH/ G—,/,/ + G—,µ,/ = GHµ , (40) (41) (42) (43) where F—,, is the aggregate demand in country for high-skilled workers from country , and G—,, is the aggregate demand in country for high-skilled workers from country . The aggregate labor demands are: F—,/, = i/ ∙ F/ + iµ ∙ jµ ∙ Fµ /, /, F—,µ, = iµ ∙ Fµ + i/ ∙ j/ ∙ F/ µ, µ, G—,/, = i/ ∙ G/ + i/ ∙ jµ ∙ Gµ /, /, G—,µ, = iµ ∙ Gµ + i/ ∙ j/ ∙ G/ µ, µ, (44) (45) (46) (47) I normalize S/,/ to 1 and use (46) and (57) to solve for Sµ,µ , i/and iµ. 4. Migration Incentive Compatibility Condition If there is no migration quota workers are assumed to be able to move across countries by paying the moving costs. Therefore, in equilibrium, we must have the real incomes for workers who migrate to foreign countries equal to the real incomes for the same type of workers who stay in their home countries. Otherwise, workers would keep moving to the country where they can earn higher real wages. The incentive compatibility condition can be written as: 3/,µ 3 µ,µ = µ :/ : 3 µ,/ 3/,/ = / :µ : S/,µ Sµ,µ = µ :/ : Sµ,/ S/,/ = / :µ : Wage variables can be solve by (44), (45), and (48)-(57). 5. Migration Quota (48) (49) (50) (51) Notice that the fourth equilibrium condition holds true only if we do not have an effective migration quota in presence. If there are migration quotas, the countries that implement effective migration quotas would have excess demands for foreign workers. In this case, we have: 3 , 3 , > : : For workers who are form country = kƒ < 1 , S, S , > : : =k , (52) <1, (53) that are effectively regulated by migration quota in country . By our notation of the migration quota, the supplies of foreign workers are: F †,, = kƒ ∙ F>,, G†,, = k , , (54) ∙ G>,, , where F>,, and G>,, are amounts of immigration from country (55) to in the unconstraint equilibrium solved by using equilibrium condition 4. We can rewrite (40)-(43) as: F —,, = F †,, G—,, = G†,, , (56) (57) where , = I1,2K. We can then use (56) and (57) to solve equilibrium wages when effective immigration quotas are in presence. APPENDIX B: SIMULATION RESULTS Figure 8: Labor market outcomes as moving costs changing for all migrants 10a. 10b. 5 1200 Low-skilled, USA Low-skilled, Canada High-skilled, USA High-skilled, Canada % change in number of migrants % change in real wage 4 3 2 1 0 -1 -50 -40 -30 -20 -10 0 10 20 30 40 50 10c. 1400 Net Low-skilled Migrants to USA Net High-skilled Migrants to USA 1200 % change in net immigrants 800 600 400 200 0 % change in moving costs for high-skilled migration from both countries 1000 800 600 400 200 0 -200 -50 Low-skilled Migrants from USA to Canada Low-skilled Migrants from Canada to USA High-skilled Migrants from USA to Canada High-skilled Mmigrants from Canada to USA 1000 -40 -30 -20 -10 0 10 20 % change in moving costs 30 40 50 -200 -50 -40 -30 -20 -10 0 10 20 % change in moving costs 30 40 50 Figure 9: Mass of firms and intra-industry reallocation as moving costs changing for all migrants 11a. 11b. 6 0.15 % change in mass of firms 5 of Local Firms, USA of Foreign Firms, USA of Local Firms, Canada of Foreign Firms, Canada 4 3 2 1 0 -0.05 -0.1 -0.15 -40 -30 -20 -10 0 10 20 30 40 -0.2 -50 50 -40 -30 -20 -10 0 10 20 30 40 50 % change in moving costs % change in moving costs 11c. 11d. 0.25 5 Aggregate Productivity in USA Aggregate Productivity in Canada Real GDP per capita, USA Real GDP per capita, Canada % chagne in real GDP per capita 0.2 % chagne in productivity level for Local Firms, USA for Foreign Firms, USA for Local Firms, Canada for Foreign Firms, Canada 0.05 0 -1 -50 Cutoff Productivity Cutoff Productivity Cutoff Productivity Cutoff Productivity 0.1 % change in productivity level Mass Mass Mass Mass 0.15 0.1 0.05 0 -0.05 -0.1 -50 -40 -30 -20 -10 0 10 20 % change in moving costs 30 40 50 4 3 2 1 0 -1 -50 -40 -30 -20 -10 0 10 20 % change in moving costs 30 40 50 Figure 10: Simulation result for moving costs changes for high-skilled migrants 12a. 12b. 4 800 Low-skilled, USA Low-skilled, Canada High-skilled, USA High-skilled, Canada 2 1 0 -1 -2 -3 -50 -40 -30 -20 -10 0 10 20 30 40 50 12c. 400 Net Low-skilled Migrants to USA Net High-skilled Migrants to USA 350 % change in net immigrants 600 500 400 300 200 100 0 % change in moving costs for high-skilled migration from both countries 300 250 200 150 100 50 0 -50 -100 -50 Low-skilled Migrants from USA to Canada Low-skilled Migrants from Canada to USA High-skilled Migrants from USA to Canada High-skilled Mmigrants from Canada to USA 700 % change in number of migrants % change in real wage 3 -40 -30 -20 -10 0 10 20 % change in moving costs 30 40 50 -100 -50 -40 -30 -20 -10 0 10 20 % change in moving costs 30 40 50 Figure 11: Simulation result for moving costs changes for low-skilled migrants 13a. 13b. 5 1200 Low-skilled, USA Low-skilled, Canada High-skilled, USA High-skilled, Canada % change in number of migrants % change in real wage 4 3 2 1 0 -1 -50 -40 -30 -20 -10 0 10 20 30 40 50 13c. 350 Net Low-skilled Migrants to USA Net High-skilled Migrants to USA 300 % change in net immigrants 800 600 400 200 0 % change in moving costs for high-skilled migration from both countries 250 200 150 100 50 0 -50 -100 -50 Low-skilled Migrants from USA to Canada Low-skilled Migrants from Canada to USA High-skilled Migrants from USA to Canada High-skilled Mmigrants from Canada to USA 1000 -40 -30 -20 -10 0 10 20 % change in moving costs 30 40 50 -200 -50 -40 -30 -20 -10 0 10 20 % change in moving costs 30 40 50 Figure 12: Income changes due to bilateral quotas for all migrants (Solid line represents real wage of native workers and dashed line represents real wage of foreign workers) Low-Skilled Real Wage in USA Low-Skilled Real Wage in Canada 2 % change in real wage % change in real wage 2 1.5 1 0.5 0 -0.5 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 1.5 1 0.5 0 -0.5 -20 0 -18 -16 Migration quota (%) High-Skilled Real Wage in USA -10 -8 -6 -4 -2 0 High-Skilled Real Wage in Canada 0.6 0.6 0.5 % change in real wage % change in real wage -12 Migration quota (%) 0.7 0.5 0.4 0.3 0.2 0.1 0 -0.1 -20 -14 0.4 0.3 0.2 0.1 0 -0.1 -18 -16 -14 -12 -10 -8 -6 Migration quota (%) -4 -2 0 -0.2 -20 -18 -16 -14 -12 -10 -8 -6 Migration quota (%) -4 -2 0 Figure 13: Income changes due to bilateral quotas for high-skilled migrants (Solid line represents real wage of native workers and dashed line represents real wage of foreign workers) Low-Skilled Real Wage in Canada 0.25 0.2 0.2 % change in real wage % change in real wage Low-Skilled Real Wage in USA 0.25 0.15 0.1 0.05 0 0.1 0.05 0 -0.05 -0.05 -0.1 -20 0.15 -18 -16 -14 -12 -10 -8 -6 -4 -2 -0.1 -20 0 -18 -16 -12 -10 -8 -6 -4 -2 0 High-Skilled Real Wage in Canada 0.6 0.6 0.5 % change in real wage % change in real wage High-Skilled Real Wage in USA 0.7 0.5 0.4 0.3 0.2 0.1 0 -0.1 -20 -14 Migration quota (%) Migration quota (%) 0.4 0.3 0.2 0.1 0 -0.1 -18 -16 -14 -12 -10 -8 -6 Migration quota (%) -4 -2 0 -0.2 -20 -18 -16 -14 -12 -10 -8 -6 Migration quota (%) -4 -2 0 Figure 14: Income Changes due to bilateral quotas for low-skilled migrants (Solid line represents real wage of native workers and dashed line represents real wage of foreign workers) Low-Skilled Real Wage in USA Low-Skilled Real Wage in Canada 2 2 % change in real wage % change in real wage 2.5 1.5 1 0.5 0 -0.5 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 1.5 1 0.5 0 -0.5 -20 0 -18 -16 Migration quota (%) High-Skilled Real Wage in USA -10 -8 -6 -4 -2 0 High-Skilled Real Wage in Canada 0.01 0 0 % change in real wage % change in real wage -12 Migration quota (%) 0.01 -0.01 -0.02 -0.03 -0.04 -0.05 -0.06 -0.07 -20 -14 -0.01 -0.02 -0.03 -0.04 -0.05 -0.06 -18 -16 -14 -12 -10 -8 -6 Migration quota (%) -4 -2 0 -0.07 -20 -18 -16 -14 -12 -10 -8 -6 Migration quota (%) -4 -2 0 Figure Row 1 Row 2 Row 3 15: Change in real per capita GDP due to migration quotas – Bilateral migration quotas are implemented for all migrants – Bilateral migration quotas are implemented only for high-skilled migrants – Bilateral migration quotas are implemented only for low-skilled migrants Real GDP per capita, Canada -0.02 -0.04 -0.06 -0.08 -0.1 -0.12 -20 -10 0 % chagne in real GDP per capita % chagne in real GDP per capita Real GDP per capita, USA 0 0 -0.02 -0.04 -0.06 -0.08 -0.1 -0.12 -20 Migration quota (%) -0.01 -0.02 -0.03 -0.04 -0.05 -0.06 -10 0 0.12 0.1 0.08 0.06 0.04 0.02 0 -20 Migration quota (%) -0.01 -0.02 -0.03 -0.04 -0.05 -0.06 -10 0 Real GDP per capita, Canada 0 % chagne in real GDP per capita % chagne in real GDP per capita Real GDP per capita, USA Migration quota (%) -10 Migration quota (%) 0 -0.07 -20 0 Real GDP per capita, Canada % chagne in real GDP per capita % chagne in real GDP per capita Real GDP per capita, USA 0 -0.07 -20 -10 Migration quota (%) 0 -0.05 -0.1 -0.15 -0.2 -0.25 -20 -10 Migration quota (%) 0