Immigration and the Macroeconomy: An International Real Business Cycle Model

Transcription

1 Immigration and the Macroeconomy: An International Real Business Cycle Model Federico S. Mandelman y Federal Reserve Bank of Atlanta Andrei Zlate z Boston College July 23, 28 (PRELIMINARY - COMMENTS WELCOME) Abstract In contrast to modern international real business cycle literature, which assumes that labor is immobile across countries, we allow for immigration ows and for the corresponding factor payments (remittances). In our model, labor migration depends on the expected di erence between future wages in the destination country and in the country of origin, as well as on the perceived sunk cost of emigration. The sunk cost, which re ects the intensity of border enforcement, partially o sets the incentive to migrate towards the country where labor is more productive, and thus reduces the volatility of the stock of immigrant labor over the business cycle. During economic expansions in the destination economy, established immigrants become relatively scarce while immigrant wages and the resulting remittances increase by more as a result of the sunk cost. Our results show that immigration encourages the accumulation of capital, which in turn increases the productivity of native labor. The overall welfare gains from unskilled immigration for the destination economy (and the damage of tightening the border) increase with the degree of complementarity between skilled and unskilled labor, as well as with the share of the skilled among native labor. The model successfully matches the cyclical dynamics of migration and remittances which we document using data from the U.S. and Mexico. We thank Pedro Silos, Myriam Quispe-Agnoli and brown bag seminar participants at the Federal Reserve Bank of Atlanta for very helpful comments. Gustavo Canavire provided superb research assistance. Part of this project was developed while Andrei Zlate was visiting the FRB of Atlanta and the FRB of Boston, whose hospitality he gratefully acknowledges. The views expressed here are those of the authors and not necessarily those of the FRB of Atlanta or the Federal Reserve System. y Research Department, FRB of Atlanta, GA 339, USA; Federico.Mandelman@atl.frb.org. z Department of Economics, Boston College, Chestnut Hill, MA 2467, USA; zlate@bc.edu.

2 Introduction Labor migration is sizable and has a non-negligible economic impact on the economies involved. The number of foreign-born residents is rising worldwide: As much as 2.5 percent of the total U.S. population in 27 was foreign born, as compared to less than 6 percent in 98, a pattern which is also visible in several other OECD countries (Grogger and Hanson, 28). Labor migration also varies over the business cycle. Jeromy (926) was the rst to document the procyclical pattern of European immigration into the United States, showing that recessions have been associated with drastic declines in immigration ows, while relatively larger in ows occurred during the recovery years after recessions. In Figure we plot the number of apprehensions at the U.S.-Mexico border, which the existing literature uses as a proxy for attempted undocumented immigration into the U.S., 2 along with the U.S./Mexico ratio of real GDP per capita. 3 The chart shows that periods in which the U.S. economy outperformed Mexico s (such as in the early 98s and during the Tequila Crisis in 995) were accompanied by an increase in border apprehensions. The correlations in Figure 2 con rm this pattern. Evidence of procyclical immigration also exists for Canada (Sweetman, 24), the United Kingdom (Gordon et al., 27) and Australia (RBA, 27), among other countries. Nevertheless, labor migration is countercylical with output in the country of origin. For instance, Hanson and Spilimbergo (999) nd that a percent relative decline in the Mexican real wage has been associated with a 6-8 percent increase in U.S. border apprehensions, with this e ect being fully realized within 3 months. Immigrants send remittances home on a regular basis. Conservative estimates indicate that the remittances sent by immigrants from developing economies back to their countries of origin reached $24 billion in 27, which was more than double the amount of In 27, the recorded remittances represented more than 2 percent of the GDP of several receiving countries, 5 while globally For instance, the number of arrivals into the United declined by 39. percent in the depression year of 98. The same was observed during the depression years of , 894 and 922. During these years there were few restrictions on European immigration and most of the arrivals into the U.S. were properly documented (see O Rourke and Williamson, 999). 2 See Hanson (26) for references. Today s legal immigration involves complicated and long administrative processes which are arguably less related to economic considerations (see Hanson, 27). 3 We apply the Baxter-King bandpass lter to the variables in natural logs in order to lter out uctuations with periodicity lower than 8 months and greater than eight years. 4 Due to unrecorded ows through formal and informal channels, the actual numbers are believed to be signi cantly larger than the reported numbers. 5 Examples include Moldova (36.2%), Honduras (25.6%), Guyana (24.3%) and Jordan (2.3%), among many others. Remittances account for roughly 2.5 percent of Mexico s GDP (World Bank, 28). 2

3 they represented the equivalent of two-thirds of the amount of foreign direct investment received by developing economies, thus becoming a principal component of their total nancial in ows (Figure 3). Although remittances have been regarded as less volatile than other capital ows, recent developments have highlighted their cyclical nature. For instance, Figure 4 shows the pattern of remittances from the U.S. to Mexico vis-a-vis the relative performance of these economies. 6 The correlations in Figure 5 indicate that periods with faster U.S. economic growth have been associated with larger out ows of remittances to Mexico. To add to our argument, the evidence in Figure 5(b) shows that the ongoing housing crisis in the U.S. has dampened remittances. 7 Despite this evidence, the workhorse models of international macroeconomics assume that labor is immobile across countries, in contrast to other factors of production such as capital. Instead, immigration is generally analyzed within formal setups limited to comparisons of long-rung positions or to the study of growth dynamics after permanent changes in immigration variables. These models are not suitable for the analysis of immigration dynamics at business cycle frequencies, as they neglect the standard macroeconomic dynamics within a general equilibrium context. This paper aims to bridge the gap between modern international macroeconomic literature and immigration theory. We use a standard dynamic stochastic general equilibrium (DSGE), two-country, real business cycle model along the guidelines of Backus, Kehoe and Kydland (994) in which we allow for labor migration ows. Beginning with Sjaastad (962), economists have regarded migration as an investment decision; thus, we construct a microfounded model of immigration that follows this principle. In our model, the incentive to emigrate depends on the expectation of future earnings at the destination relative to the country of origin, on the perceived sunk costs of emigration, as well as on the exit rate of immigrant labor. The sunk cost of emigration varies in nature, as it may include the cost of searching for employment, the cost of adjusting to a new lifestyle (learning a new language, integration into a new community, housing arrangements, etc.), transportation expenditures, working visa procedures, and in the case of undocumented immigration, the need to hire human smugglers (also known as coyotes) as well as the physical risk and legal implications of illegally crossing the border. The exit rate a ecting the established immigrants has a non-trivial role, as about 7 percent of undocumented Mexican immigrants in the U.S. tend to return to their countries within ten years after their arrival (Reyes, 997). 6 We apply the Baxter-King bandpass lter to the variables in natural logs. 7 Approximately 7 percent of the Mexican immigrant labor in the U.S. works in the construction industry (IADB, 28). 3

4 In our baseline model, we assume that labor can migrate from the foreign to the home economy, and also faces the possibility to return to the country of origin every period. The relative di erence in productivity between the home and foreign economies ensures that the stock of immigrant labor is always positive, despite the sunk immigration cost. Firms in the home economy employ a composite of immigrant and native labor as imperfect substitutes, along with capital. Households in each economy consume both domestic and imported goods, while foreign consumption can be nanced from the income contributed by both immigrant and resident labor. Following a positive productivity shock in the home economy, the demand for both immigrant and native labor increases, along with wages. In turn, the appreciation of immigrant wages stimulates the in ow of new immigrant labor, which enhances the productivity of capital and stimulates capital accumulation over the business cycle. In the foreign economy, the out ow of immigrants and the resulting decline in the pool of resident labor cause wages to rise, which dampens the accumulation of capital and output. However, foreign consumption remains strong due to the increase in remittances. The relative scarcity of the foreign intermediate good increases its relative price by even more in the presence of labor migration out ows, and thus leads to a real exchange rate appreciation for the foreign economy. Thus, our parsimonious model is consistent with the vast empirical evidence showing that the in ows of remittances are associated with an appreciation of the real exchange for the receiving country (Amuedo Dorantes and Pozo, 24; Lopez et al., 28; Lartey et al., 28) as well as with a decline in labor supply (Hanson, 27; Acosta, 26). Stricter border enforcement reduces the incentive for foreign labor to immigrate. Clearly, immigration policies work as a result of the burden of migration costs faced by potential immigrants. In the model with sunk costs, a temporary economic expansion in the destination economy leads to an increase in the immigrant wage; however, the greater incentive for labor migration is partially o set by an increase in the sunk cost. 8 During economic expansions, the immigrants already established in the home economy become relatively scarce, as the increase in the stock of immigrants labor does not keep up with the increase in labor demand; they enjoy relatively higher wages and send larger remittances to the foreign economy. The opposite occurs during recessions, when immigrant labor becomes relatively more abundant and the immigrant wage declines. In the presence of high comple- 8 Without the sunk cost, the appreciation of the immigrant wage would trigger an immediate in ow of immigrants that would erode the wage gap. Following a positive technology shock in Home, the immigration incentive created by the temporary increase in the immigrant wage is partially o set by the higher sunk cost faced by prospective immigrants. The degree of border enforcement is re ected by the relative magnitude of the sunk cost. 4

5 mentarity between native and immigrant labor, this result helps us to explain the paradoxical evidence that the decrease in immigration in ows caused by the increase in surveillance at the Mexican border during the late 99s coincided with an "explosion" of remittance out ows to Mexico, as documented in Rodriguez-Zamora (28). Undocumented immigrants usually account for most of the unskilled immigrant arrivals and are subject to public scrutiny. In the U.S., they represent 3 to 5 percent of the total new immigrant arrivals (Hanson, 26). In order to take this phenomenon into account, we extend the baseline model by introducing two types of labor in the home economy (skilled and unskilled) while assuming that capital and skilled labor are relative complements as in Krusell et al. (2), and that the native unskilled and immigrant labor are perfect substitutes as in Borjas et al. (28). We calibrate the model to match the empirical regularities of labor migration between Mexico and the U.S., such as the share of the Mexican labor force residing in the U.S., the ratio between the native and immigrant wages in the U.S., and the ratio between the immigrant wage in the U.S. and the resident labor wage in Mexico adjusted for purchasing power parity, while controlling for socio-demographic features such as age and educational attainment. Although the macroeconomic dynamics of the extended model remain unchanged at the aggregate level relative to the baseline, immigration has an asymmetric e ect on the skilled and unskilled labor, bene ting the former and harming the latter. We also explore the e ects of an alternative immigration policy in which sunk costs are set to zero (to re ect lower border enforcement) and a countercyclical tax in imposed on the immigrant wage. The countercylical immigration tax increases the procyclicality of the stock of immigrant labor and improves overall welfare in the home economy. In particular, it improves the stance of native unskilled workers during recessions, when their employment and wages decline by less due to the lower stock of immigrant labor. When computing the welfare e ects of di erent enforcement policies, we account for the presence of both temporary aggregate stochastic shocks (i.e. to account for cyclical considerations) as well as for anticipated deterministic shocks with permanent e ects on the balanced growth path. The results indicate that tightening the border to constrain the in ow of unskilled labor has a negative impact on welfare in the destination economy, particularly when the complementarity between skilled and unskilled labor is relatively higher, and when the share of the skilled labor in total native labor converges to a relatively higher steady-state level. We also extend the baseline model to allow for nancial integration between the home and foreign economies through international trade in bonds. In steady state, as predicted by Lucas (99), nancial 5

6 integration allows capital to migrate towards the economy with a relatively higher rate of return (i.e. in our model, the foreign economy), where the resident labor becomes relatively more productive, receives a higher wage, and has a lower incentive to emigrate. Following a positive technology shock in the home economy, foreign households have the option to lend o shore as an alternative to investing in emigration; thus, nancial integration reduces the cyclicality of labor migration. This paper is related to existing literature that quanti es the e ect of migration in both static (Borjas, 995; Hamilton and Whalley, 984; Moses and Letnes, 24; Walmsley and Winters, 23) and dynamic frameworks (Djacic, 989). Our paper is closely related to Klein and Ventura (27) and Urrutia (998), who use growth models with endogenous labor movement to assess the welfare e ects of removing barriers to labor migration. In the context of DSGE models of international business cycles, our paper is also related to Acosta et al. (27), Chami et al. (26) and Durdu and Sayan (28), who include remittance endowment shocks; to Ghironi and Melitz (25) and Bilbiie et al. (26), who introduce an endogenous rm entry mechanism subject to sunk costs; and to Polgreen and Silos (26), who use skill heterogeneity and capital-skill complementarity with two representative households. The rest of the paper is organized as follows: Section 2 introduces the benchmark and extended model; Section 3 presents the calibration; Section 4 describes the dynamic properties of the model, providing the impulse responses and the second moments of the theoretical economy; Section 5 performs a welfare analysis in the presence of deterministic shocks to the magnitude of sunk cost and to the share of native skilled labor; Section 6 concludes. 2 The Model: Immigration with Sunk Costs The model is representative of a standard two-country setup along the guidelines of Backus, Kehoe, Kydland (994, henceforth BKK). Our setup di ers from that of BKK in that we use for simplicity log-crra preferences and abstract from government purchases and time-to-build in capital formation. In our baseline speci cation, we assume nancial autarky. Each country specializes in the production of a single (intermediate) good. The nal good is a composite of foreign and domestic goods, and can be either consumed or invested. The novel characteristic of our setup is the presence of labor mobility, that in this case occurs from the foreign economy to the home one. In the baseline model, native and immigrant labor form a CES aggregate which enter symmetrically as a single input, along with capital, in a Cobb-Douglas 6

7 production function in the home economy. In the model with an alternative production speci cation, we explore the implications of capital-skill complementarity by introducing two types of labor in the home economy (skilled and unskilled) as in Krusell et al. (997), while assuming that the native unskilled and immigrant labor are perfect substitutes, following the ndings in Borjas et al. (28). 2. The Home Economy Supply of native labor The representative Home household supplies L n;t hours of labor, consumes C t units of the home composite basket, and invests in physical capital K t. It maximizes the inter-temporal utility: max fc t;l n;t;k t+ g E t " X # s t U(C s ; L n;s ) ; () s=t where the period utility function takes the form subject to the constraint: U(C t ; L n;t ) = ln C t (L n;t) + + ; > (2) w n;t L n;t + ( + r t )K t > C t + K t+ : Parameter = > is the Frisch elasticity of labor supply and the inter-temporal elasticity of substitution in labor supply. Following King et al. (998), we use separable preferences and log-utility from consumption in order to obtain balanced growth path in steady state, i.e. the income and substitution e ects of changes in the real wage on hours worked cancel out and generate constant steady-state labor e ort. Variable w n;t is the domestic wage and r t denotes the return on capital net of depreciation, all expressed in units of the home composite good. The usual rst-order conditions with respect to consumption and labor follow: = E t ( + r t+ ) C t ; C t+ (3) w n;t = (L n;t ) : C t (4) Production of the Home Intermediate Good In our baseline model speci cation, we assume that production is a Cobb-Douglas function of capital and a CES aggregate of immigrant and native labor: h i Y h;t = A t (K t ) (Li;t ) + ( ) (Ln;t ) ( ) ; (5) 7

8 where L i;t and L n;t denote immigrant and native labor; is the share of immigrant labor income in Home s total labor income; is a parameter that re ects the productivity of native labor relative to that of immigrant labor in steady state; and is the share of capital in GDP. Thus, the elasticity of substitution between native labor and capital is the same as that between immigrant labor and capital. The supply of immigrant labor is a decision of the foreign household and will be described later. The country-speci c good is used both domestically and o shore: Y h;t = Y h;t + Y h2;t ; (6) where quantity Y h;t denotes total intermediate output produced in the home economy, Y h;t denotes the domestic use of the home-speci c good, and Y h2;t denotes the exports of the home intermediate good to the foreign economy. Consumption and investment are composites of the home and foreign-speci c goods: Y t = i h! (Y h;t ) + (!) (Y f;t ) ; (7) where Y f;t denotes the imports of Home from Foreign. foreign-speci c goods are: The demand functions for the home and Y h;t =! (p h;t ) Y t ; (8) Y f;t = (!) (p f;t Q t ) Y t ; (9) where p h;t is the price of the home-speci c good in units of the Home composite good, p f;t is the price of the foreign good in units of the Foreign composite good, and Q t is the real exchange rate. At the aggregate level, the resource constraint is: and the rule of motion for the capital stock is: Y t = C t + I t ; K t+ = ( ) K t + I t : () Optimality Conditions Firms maximize pro ts: max = A tkt L t fk t;l tg rt k + K t w t L t () 8

9 Thus, the rental rate of capital (plus depreciation) and the real wages are equal to the marginal products of capital, immigrant and native labor, h;t = Y h;t K t = r t + ; (2) = ( ) (Yh;t ) ( ) (A t Kt ) ( ) (L i;t ) = wi;t i;t h;t = ( ) ( ) (Yh;t ) ( ) (A t Kt ) ( ) () (L n;t ) = wn;t n;t (4) 2.2 The Foreign Economy We model labor migration from Foreign to Home. We introduce cross-country labor mobility with sunk immigration costs: Foreign households have the option to work in the home economy, where wages are higher. However, labor migration from Foreign to Home requires a sunk cost per unit of emigrant labor, a cost which in equilibrium equals the present discounted value of the di erence between the future stream of wages obtained as an immigrant in the home economy and the stream of wages obtained in the country of origin. Location of Labor The foreign household supplies L t units of labor every period. They can either emigrate and work in the home economy (L i;t ) or work domestically in the foreign economy (L f;t ): L t = L i;t + L f;t : (5) As will be discussed later, we calibrate the sunk migration cost so that the stock of emigrant labor is always lower than the total labor supply in Foreign in any period t, i.e. L i;t < L t : Similarly, the calibration assures that immigrant wages in the home economy are signi cantly higher, so that the number of immigrants in the home economy is always positive (i.e. L i;t < L t ). For simplicity, we do not allow for labor ow from the home to the foreign economy. Every period foreign workers have the option to emigrate to the home economy. The time-to-build assumption implies that recent immigrants start working one period after arriving at the destination. They continue working in the home economy in all subsequent periods, until an exogenous exit-inducing shock, which hits them with probability l every period, sends them back to their country of origin (i.e. the foreign economy). This shock occurs at the end of every time period, and may be linked to issues such as the likelihood of deportation, the impossibility of nding employment in the home economy, or the lack of adaptation to the new country of residence, etc. 9

10 Thus, the rule of motion for the stock of immigrant labor in Home is: L i;t = ( l )(L i;t + L e;t ); (6) where L e;t is the number of new foreign workers that emigrate to Home every period (i.e. a ow variable), and L i;t is the number of immigrant workers located in Home every period (i.e. a stock variable). Household s Problem The representative foreign household has preferences over real consumption and labor e ort. It maximizes the inter-temporal utility with respect to total labor L t, emigrant labor L e;t and capital K t+ : max fc t ;L t ;Le;t;K t+g E t Utility takes the same form as in (2), and the budget constraint is: " X # ( ) s t U(Cs ; L s) : (7) s=t w t (L t L i;t ) + w i;t Q t L i;t + ( + r t )K t > C t + f e w i;t Q t L e;t + K t+; (8) where w t is the wage in the foreign economy and w t (L t L i;t ) denotes the income from hours worked domestically. We de ne w i;t as the immigrant wage earned in the home economy, so that the immigrants total labor income expressed in units of the foreign composite good is w i;t Qt L i;t. Emigration requires a sunk cost of f e units of immigrant labor, equal to f e w i;t Q t : Finally, r t is the return on foreign capital net of depreciation. Potential emigrants face a trade-o between the sunk migration cost (f e w i;t Qt ) and the present discounted value of the di erence between the streams of future wages at the destination (w i;t Q t ) and in the country of origin (w t ), expressed in units of the foreign composite good. It is useful to re-write the constraint as: w t L t + d t L i;t + ( + r t )K t > C t + f e w i;t Q t L e;t + K t+; (9) where d t is the di erence between the immigrant wage in Home and the wage in the country of origin at time t, expressed in units of the foreign consumption basket: d t = w i;t Q t w t : (2) Using the new budget constraint and the law of motion for the stock of immigrant labor, L i;t = ( l )(L i;t +L e;t ), the optimization with respect to new emigrant labor L e;t every period implies:

11 f e w i;t Q t = X s=t+ [ ( l )] s t C E t t Cs d s ; (2) which shows that, in equilibrium, the sunk emigration cost equals the present discounted gain from emigration, measured as the di erence between the future expected wages at the destination and in the country of origin, expressed in units of the foreign composite good. Production of the Foreign Intermediate Good Foreign production is a Cobb-Douglas function of non-immigrant labor, L f;t ; and capital, K t. Following BKK, the resulting foreign-speci c intermediate good (Y f;t ) can be either used domestically (Y f2;t ) or exported to the home economy (Y f;t ): Y f;t = A t (K t ) L f;t ; (22) Y f;t = Y f;t + Y f2;t : (23) The foreign composite good, Y t ; incorporates amounts of both the foreign-speci c intermediate good (Y f2;t ) and the home-speci c imported good (Y h2;t ): i Yt = h! (Y f2;t ) + (! ) (Y h2;t ) : (24) This nal good composite can be consumed, invested in physical capital, or used for investment in immigration (i.e. used to cover the sunk costs required to send immigrant labor abroad). Finally, capital accumulation is described by: Y t = C t + I t + f e w i;t Q t L e;t : (25) K t+ = ( ) K t + I t : (26) Optimality Conditions and pins down the total labor e ort: Households optimization problem delivers a typical Euler equation = E t ( + rt+) C t Ct+ ; (27) wt Ct = (L t ) ; (28)

12 The demand functions for the home and foreign-speci c goods are: Y f2;t =! (p f;t ) Yt ; (29) Y h2;t = (! ph;t ) Yt ; (3) where p f;t and p h;t Q t ; respectively, are the price of the foreign-speci c and home-speci c good, both expressed in units of the foreign consumption basket. In turn, the net return on capital and local wages are respectively determined by the marginal product of capital and labor: r t = Y f;t K t w t = ( Q t ; (3) ) Y f;t L : (32) f;t 2.3 Alternative Speci cation: Capital-Skill Complementarity in Home In the alternative speci cation, we assume that foreign labor is relatively unskilled and can migrate to Home, where it becomes a perfect substitute for the native home unskilled labor, as in Borjas et al. (28). Capital and native skilled labor are relative complements, whereas capital and unskilled labor (i.e. immigrant and native) are relative substitutes, as in Krusell et al. (2). Optimization with Two Representative Households While the description of the foreign economy remains identical, the home economy now includes a continuum of two types of in nitelylived households that supply units of skilled and unskilled labor, as in Polgreen and Silos (26). Every period t, each of the two representative households consumes c j;t units the home consumption basket and supplies l j;t units of labor, where subscript j 2 fs; ug denotes skilled and unskilled labor, respectively. Thus, the planner maximizes the weighted sum of utilities for the two representative households: max fc s;t;l s;t;c u;t;l u;t;k t+ g t= X s t fsu (c s;t ; l s;t ) + ( ) ( s) U (c u;t ; l u;t )g ; (33) where utility takes the standard form as in (2), and the constraint is: w s;t L s;t + w u;t L u;t + ( + r t )K t > C s;t + C u;t + K t+ ; (34) where s denotes the fraction of skilled households and s is the fraction of unskilled households in the total population; and are the weights of the utility of skilled and unskilled households, 2

13 respectively, in the objective function of the planner. L s;t = sl s;t and L u;t = ( s) l u;t are the aggregate amounts of skilled and unskilled labor which rms hire at the equilibrium wages w s;t and w u;t, respectively. C s;t = sc s;t and C u;t = ( and unskilled households. The maximization problem for the two representative agents generates the usual rst-order conditions: s) c u;t are the aggregate consumptions of the skilled = c s;t = t ; c u;t = E t ( + rt+) t t+ w s;t c s;t = s (l s;t ) s ; w u;t c u;t = u (l u;t ) u : where j; j ; j fs; ug represent weights in the utility function and the inverse of the Frisch elasticity of skilled and unskilled labor supply. ; Production of the Home Intermediate Good function is a nested CES aggregate: n o Y h;t = A t (;t ) + ( ) (2;t ) ; In the alternative speci cation, production of the following components: ;t = L i;t + L u;t ; (35) i 2;t = h (K t ) + ( ) (L s;t ) ; (36) where ;t is a function in which the unskilled immigrant and native labor enter as perfect substitutes; 2;t is a CES function of capital and skilled native labor; is the fraction of unskilled labor in output; =( ) is the share of capital in output. Finally, > governs the elasticity of substitution between skilled and unskilled labor, which is the same as the elasticity of substitution between capital and unskilled labor; > is the elasticity of substitution between capital and skilled labor. Following Krusell et al. (997), we restrict > under the assumption of capital-skill complementarity. 3

14 The pro t maximization problem of rms generates the following optimality h;t = (A t ) (Y h;t ) (2;t ) (K t ) = r t + ; h;t h;t = (A t A u;t ) Y h;t = wu;t ; u;t L i;t + L h;t = 2 (A t ) (Y h;t ) (2;t ) () (L s;t ) = w s;t ; s;t where = ( ) and 2 = ( ) ( ) : The rest of the economy is described by the equations of the baseline speci cation model outlined in the previous section. The only exception is the resource constraint in the home economy, which becomes: Y t = C s;t + C u;t + I t : (4) 2.4 Trade Balance and Remittances The current account balance, measured in units of the Home composite good, is: CA t = p h;t Y h2;t p f;t Q t Y f;t w i;t L i;t : (4) Under nancial autarky, the current account balance is zero (CA t = ): The trade balance (T B t = p h;t Y h2;t p f;t Q t Y f;t ) equals the immigrant labor income (w i;t L i;t ). It is useful to show that, using the resource constraint Y t = p h;t Y h;t + p f;t Q t Y f;t = C t + I t ; we can re-write the home GDP expressed in units of the home-speci c good as: p h;t Y h;t = C t + I t + T B t : (42) Similarly, using that Yt = p h;t Qt Y h2;t + p f;t Y f2;t = Ct + It + f e w i;t Qt L e;t, we can write the foreign GDP expressed in units of the foreign-speci c good as: p f;t Y f;t = C t + I t + f e w i;t Q t L e;t Q t T B t : (43) We use the immigrant labor income w i;t L i;t as a proxy for remittances. From a theoretical standpoint, workers remittances, t ; could also be proxied as the di erence between () the immigrant labor income and (2) the immigrant labor s share in foreign consumption and investment in both physical capital and emigration net of the return on foreign capital. Using the budget constraint of the foreign household in (9), we write the alternative proxy for remittances as: t = w i;t Qt L i;t L i;t C L t + f e w i;t Qt L e;t + Kt+ ( + rt )K t : (44) t 4

15 3 Calibration We introduce an asymmetric steady state across countries using uneven discount factors, >. 9 Thus, the relatively larger capital accumulation in Home, where households are more patient, provides a wage incentive for immigrant foreign labor. We use the standard quarterly calibration from BKK: = :5 is the elasticity of substitution between the Home and Foreign-speci c goods in the composite basket of both countries; = :33 is the share of capital in GDP; = :25 is the depreciation rate of the capital stock;! = :85 re ects the degree of home bias in Home and! = :75 shows home bias in Foreign; we set! >! in order to account for the relatively greater trade openness in Mexico relative to the U.S. The inverse of the elasticity of labor supply to labor is = :33. We also set = :66; following the nding in Hotchkiss and Quispe-Agnoli (28) that the labor supply elasticity of undocumented immigrants is half the value of the labor supply elasticity of U.S. workers. We also calibrate the weights on the disutility from labor and so that the total labor supply in steady state is normalized to.5 in each country. We set the quarterly exit rate of immigrant labor l = :7, using the nding of Reyes (997) that approximately 5 percent of the undocumented Mexican immigrants return to their country of origin within two years after their arrival in the U.S. (which corresponds to a quarterly exit rate of.635), and that 65 percent of them return within four years after their arrival (i.e. quarterly exit rate of.83). 2 Baseline Model Calibration For the baseline model with symmetric elasticity of substitution between capital and each type of labor (native and immigrant), the calibration parameters are described in Table 4.. We choose parameter values for ; ; and f e so that the model matches the following two empirical ratios in steady state: () The share of Mexico s labor force residing in the 9 The calibration = :99 and = :98 re ects a larger quarterly interest rate in Foreign (where capital is scarce) relative to Home in steady state (r = :2 and r = :, respectively). This calibration also implies that Foreign immigrants living in the Home economy will have a preference bias towards the Home rather than the Foreign goods. One caveat is that the labor supply elasticity of immigrant labor originating in Foreign is not necessarily equal to the labor supply elasticity of the foreign labor that resides in Foreign. However, the results are similar when assuming that the elasticity of labor supply is the same for Foreign immigrant and resident workers, as we do in this paper. The alternative results, not reported here, are available upon request. 2 Using the information that 35 percent of the undocumented Mexican immigrants are still in the U.S. four years after their arrival we compute the quarterly exit rate as ( l;4y ) 6 = :35: 5

16 U.S., L i L U.S., w w i = : (Hanson, 26); (2) The ratio between the wages of native and immigrant labor in the = 2:. 3 In order to achieve this, we set = :8 (the share of immigrant labor in total labor income), = 5:4 (the relative productivity of native vs. immigrant labor), f e = 4 (the sunk cost of labor migration) and = :55 (the elasticity of substitution between native and immigrant labor). 4 Given the key role of the degree of complementarity between native and immigrant labor, we perform robustness checks with low and high substitutability between immigrant and native workers, = :5 and = 2:5. Table 4. Baseline Model Calibration = :8 = 5:4 = :55 f e = 4 Share of immigrant labor in total labor income Relative productivity of native vs. immigrant labor Elasticity of substitution between native and immigrant labor Sunk cost of labor migration Alternative Model Calibration For the alternative model with two types of native labor in Home (skilled and unskilled), in which native unskilled and immigrant labor are perfect substitutes, the calibration is summarized in Table 4.2. We de ne the pool of native unskilled labor to include the adult population without a high school degree; using data from the U.S. Census Bureau, we set the share of unskilled labor at ( s) = :: We choose values for parameters e; e ; e; e and e f e so that the model generates a set of three steady state-ratios that match the empirical evidence from the U.S. and Mexico: () The share of Mexico s labor force residing in the U.S is L i L = :, as discussed above (Hanson, 26). (2) The ratio between the wages of the native skilled and unskilled labor in the U.S. is ws w u = 2:2. 5 (3) The ratio between the immigrant hourly wage in Home and the wage of foreign labor in the country of origin, expressed 3 For the immigrant wage we use the average hourly wages for immigrant Mexican males in the U.S. (28 to 32 years of age, with 9 to years of schooling completed) provided by Hanson (26); we also compute the weighted average hourly wage of the U.S. native labor using data from the U.S. Census Bureau (27). 4 We take the estimate of the elasticity of substitution between skilled and unskilled labor (:26) under the symmetric model setup in Krusell et al. (997) as a benchmark for the value of in our baseline model. 5 We take the weighted average of hourly earnings for the U.S. skilled labor (i.e. high school degree or more), as well as for the U.S. unskilled labor (i.e. without a high school degree) using data provided by the U.S. Census Bureau (26, 27). We divide the sample into four groups: (a) no high school degree; (b) completed high school; (c) some college or associate s degree; and (d) bachelor s degree or higher. Then we take the average of the respective earnings weighted by their share in the total population. 6

17 in units of the same consumption basket, is w i Qw = 3:64. 6 Using a broader de nition for unskilled labor (i.e. population without college degree), Krusell et al. (2) estimate the elasticity of substitution between capital and unskilled labor at.67. Since our de nition of unskilled labor (i.e. adult population without high school degree) is narrower than that in Krusell et al. (2), we regard their estimate as an upper bound for the range of elasticities e in our model. Krusell et al (2) also nd that the elasticity of substitution between capital and skilled labor is.67; we take this value as a lower bound for the range of elasticities e in our model, since we use a broader de nition of skilled labor than Krusell et al. (997). Thus, we choose e = :25, e = :, e = 4:26, e f e = 7:5 and e = :, parameter values which allow the model in steady state to match the empirical ratios described above. Finally, Krueger and Perri (27) and Attanasion and Davis (996) nd that di erences in the consumption of population groups with di erent levels of educational attainment (e.g. skilled and unskilled) closely re ect the income di erences between the respective groups. Therefore, we set the weight on the utility of representative skilled household = :688, so that the consumption ratio for the Home representative skilled and unskilled households matches the corresponding wage ratio, c s c u = ws w u = 2:2: Table 4.2 Extended Model Calibration s = :9 Share of Home skilled in total households e = : Share of native + immigrant unskilled in GDP e = =( e) Share of capital in GDP e = :25 e = : e = 4:26 ef e = 7:5 = :688 Elasticity of substitution, capital vs. unskilled labor Elasticity of substitution, capital vs. skilled labor Relative productivity of native vs. immigrant labor Sunk cost of labor migration Weight on the utility of skilled labor 6 In order to build this ratio, we use the wage data provided in Hanson (26) for () the hourly wage of the recent Mexican immigrants in the U.S., and (2) the hourly wage of those of similar age and educational attainement that reside in Mexico (i.e. males between years of age with 9 to years of schooling), adjusted for purchasing power parity. The wage ratios for other age and educational attainment groups are similar (see Hanson, 26). 7

18 4 Results 4. Impulse Responses ) Baseline model, high vs. low barriers to immigration (positive technology shock in Home) Technology follows an autoregressive process of the form log A t = log A t + " t, where the persistence parameter is = :95. In Figure 6, following a positive technology shock in Home, the increase in the immigrant wage premium encourages the entry of immigrants. However, due to the complementarity between capital and immigrant labor, the higher sunk cost of immigration (f e = 6) dampens the accumulation of capital relative to the model with low sunk cost (f e = ): Over the business cycle, the weaker capital accumulation hurts labor productivity and generates a lower increase in the wage of native labor, with a corresponding e ect on home consumption. Higher sunk costs discourage emigration from Foreign to Home, which results in a higher wage for the immigrant labor already established in Home. Overall, foreign consumption increases by less under the scenario with high sunk costs (i.e. due to the lower income from immigrant labor), whereas foreign output increases by more (i.e. due to the larger amount of resident labor and greater capital accumulation). The relatively greater increase in foreign output along with the smaller increase in consumption release pressure on the real exchange rate for Foreign, which depreciates by less in the presence of high immigration costs in Home. 8

19 35 Consumption, Home (c) 4 Consumption, Foreign (c2) 8 Immigrant wage premium (d) 4 Immigrant labor entry (he) Immigrant labor stock (hi) 6 Native labor (hn) 8 Capital stock, Home (k) 4 Capital stock, Foreign (k2) Real exchange rate (q) 8 Wage of Home natives (wn) Wage of immigrant labor (wi) 25 Wage in Foreign (w2) GDP, Home (yh) GDP, Foreign (yf) 5 Exports of Home (yh2) 8 Imports of Home (yf) Low sunk costs (fe=) High sunk costs (fe=6) -2 Figure 6. Baseline model, low vs. high sunk costs (positive technology shock in Home) 9

20 2) Baseline model, high complementarity between native and immigrant labor (positive technology shock in Home) The impulse responses in Figure 7 show that, relative to the baseline calibration, high complementarity ( = :5) makes border enforcement more harmful for the home economy: Following a positive technology shock in North ( = :95), higher complementarity dampens the increase in the demand for native labor and capital accumulation, which results in a lower increase in the wage of native labor and Home consumption. One notable qualitative di erence from the baseline model with = :5 is that high complementarity reverses the negative impact of high barriers to immigration on foreign consumption: Despite the high sunk costs, the stronger increase in the wage of immigrant labor o sets the e ect of a lower stock of immigrant labor on remittances. 3 Consumption, Home (c) 6 Consumption, Foreign (c2) 2 Immigrant wage premium (d) 6 Immigrant labor entry (he) Immigrant labor stock (hi) 6 Native labor (hn) 8 Capital stock, Home (k) Capital stock, Foreign (k2) Real exchange rate (q) 8 Wage of Home natives (wn) 2 Wage of immigrant labor (wi) 4 Wage in Foreign (w2) GDP, Home (yh) GDP, Foreign (yf) 3 Exports of Home (yh2) 5 Imports of Home (yf) Low sunk costs (fe=) High sunk costs (fe=6) -5 Figure 7. Baseline model, low elasticity of substitution between native and immigrant labor (positive technology shock in Home) 2

21 3) Alternative model, countercyclical immigration tax vs. sunk cost (negative technology shock in Home) In the model with skilled and unskilled labor in Home and capital-skill complementarity, we now compare the e ects of two immigration policies: () immigrant entry with sunk costs, as described above, and (2) while setting the sunk cost equal to zero, we introduce a countercyclical tax on the immigrant wage, payable every period: ( + t )w i;t = MP L i;t where the countercyclical tax is: t = Yh;t Y h ; < : We set the sunk cost of immigration f e = 3:23 under policy () and the tax parameter = :5 under policy (2) so that the two calibrations generate the same stock of immigrant labor in steady state. The countercyclical tax on immigration generates an income transfer from the foreign to the home households, a transfer which we illustrate by comparing the steady-state levels of the relevant variables in Table 4.3: C() > C(2) and C () < C (2), where indices () and (2) refer to the two policies described above. Table 4.3 Steady-State Levels Counter-cyclical tax Sunk cost f e = ; = :5 f e = 3:23; = L i :225 :225 w i 7:372 6:62 C 4:866 4:837 C :849 :944 Q 3:6 3:24 The impulse responses are described in Figure 8. Following a negative technology shock in Home ( = :95), the resulting recession lowers the productivity of labor. Therefore, the wages of both skilled and unskilled labor decline. Under policy (2), the countercyclical tax on immigrant labor acts as an extra deterrent to immigrant entry, in addition to the decline in wages caused by the lower productivity. On impact, immigrant entry declines as the forward-looking foreign household re-optimizes the stock of immigrant labor to be signi cantly lower during the recession. With the tax in place, the native 2

22 unskilled labor bene ts from the sharp decline in the entry of immigrants. As a consequence, the unskilled wages and the labor supply do not fall by as much under policy () as under policy (2). The foreign economy su ers from the countercyclical tax on emigration imposed by Home. The lower stock of immigrant labor and the lower immigrant wages (both due to the tax) negatively a ect the amount of remittances, which leads to lower foreign consumption relative to the policy with sunk emigration costs. However, the larger amount of labor available in Foreign encourages capital accumulation and output in Foreign. Under policy () with sunk costs, the barrier to immigration declines along with the immigrant wage during the recession. Therefore, the decline in entry is dampened relative to policy (2), despite the lower gap between the immigrant and foreign wage. -2 Consumption, Home skilled (cs) Consumption, Foreign (c2) 2 Immigrant wage premium (d) Immigrant labor entry (he) Immigrant labor stock (hi) 2 Home skilled labor (hs) x 4 Capital stock, Home (k) 4 Capital stock, Foreign (k2) Real exchange rate (q) -8 Wage of Home skilled (ws) -2 Wage of immigrant labor (wi) Wage in Foreign (w2) GDP, Home (yh) 8 GDP, Foreign (yf) Home unskilled labor (hu) Wage of Home unskilled (wu) Sunk cost Countercyclical immigration tax - Figure 8. Alternative model, countercyclical immigration tax vs. sunk cost (negative technology shock in Home) 22

23 4.2 Theoretical Moments Baseline model with trade in bonds We study the theoretical moments of the baseline model with labor migration while allowing for trade in international bonds across countries. Following Ghironi and Melitz (25), we assume that: () International asset markets are incomplete, as households in each country issue risk-free bonds denominated in their own currency. (2) Nominal returns are indexed to in ation in each economy, so that each type of bond provides a real return denominated in units of that country s consumption basket. (3) In order to avoid the indeterminacy of steady-state net foreign assets and non-stationarity, we introduce quadratic costs of adjustment for bond holdings, a tool which allows us to pin down the steady state and also to ensure stationarity in the presence of temporary shocks. The in nitely-lived representative agent maximizes the inter-temporal utility subject to the constraint: where r k t w t L t + + rt k K t + + rt b B h;t + + rt b Q t B f;t + T t (45) > C t + K t+ + B h;t+ + 2 (B h;t+) 2 + Q t B f;t+ + 2 Q t (B f;t+ ) 2 ; is the rental rate of capital in Home; r b t and r b t are the rates of return of the home and foreign bonds; ( + r b t)b h;t and ( + r b t )Q t B f;t are the principal and interest income from holdings of the home and foreign bonds; 2 (B h;t+) 2 and 2 Q t (B f;t+ ) 2 are the cost of adjusting holdings of the home and foreign bonds, respectively; T t is is the fee rebate. We set = :25. We add the two Euler equations for bonds to the baseline model: + B h;t+ = E t ( + rt+) b C t C t+ Qt+ + B f;t+ = E t ( + r Q t+) b C t t C t+ With trade in bonds, the budget constraint of the foreign household becomes: wt (L t L i;t ) + w i;t Qt L i;t + + r k t Kt + + rt b ; (46) : (47) Qt Bh;t + + r b Bf;t + T t (48) > C t + f e w i;t Q t L e;t + K t+ + Q t B h;t+ + 2 Q t B h;t+ 2 + B f;t+ + 2 B f;t+ 2 ; and the corresponding Euler equations for bonds are: + B h;t+ = E t Qt Q t+ ( + r b t+) C t + B f;t+ = E t ( + r b t+) C t C t+ C t+ t ; (49) : (5) 23

24 The market clearing conditions for bonds are: B h;t+ + Bh;t+ = ; (5) B f;t+ + Bf;t+ = : (52) Under nancial integration, we replace the balanced current account condition (T B t w i;t L i;t = ) from the model with nancial autarky with the expression for the balance of international payments: p h;t Y h2;t p f;t Q t Y f;t {z } Trade balance w i;t L i;t + r {z } tb b h;t + rt b Q t B f;t = (B h;t+ B h;t ) Q t (B f;t+ B f;t ) {z } {z } Immigrant labor income Investment income Change in bond holdings (53) which shows that the current account balance (i.e. the trade balance plus investment income minus remittances) must equal the negative of the nancial account balance (i.e. the change in bond holdings). Thus, nancial integration through trade in country-speci c bonds adds 6 variables (B h;t ; B f;t ; B h;t ; B f;t ; rb t and r b t ) and 6 equations (46, 47, 49, 5, 5 and 52) to the baseline model with nancial autarky. Theoretical moments Next we compute the second moments of the theoretical economy described by the baseline model with international trade in bonds. We assume that technology follows the bivariate process below: 2 4 log A t log A t 3 5 = 2 4 A AA A A A log A t log A t t where the persistence parameters are A = A = :979 as in King and Rebelo (2) and the spillover parameters are AA = A A =. The variances of innovations are.73 percent and the covariance is t 3 5 ;.9 percent. Table 4.4 Empirical Moments: Corr X t ; GDP US GDP t+j MEX j X Immigrant entry Remittances Trade balance

25 Table 4.5 Model-Generated Moments (A) Baseline Model, Financial Integration, Corr(X t ; GDP t+j ) j X Immigrant entry (Le) Remittances (wi*li) Remittances 2 ( ) Trade balance (B) Baseline Model, Financial Integration, Corr(X t ; GDP t+j) j X Immigrant entry (Le) Remittances (wi*li) Remittances 2 ( ) Trade balance (C) Baseline Model, Financial Autarky, Corr(X t ; GDP t+j ) j X Immigrant entry (Le) Remittances (wi*li) Remittances 2 ( ) Trade balance (D) Model without Immigration (BKK94), Financial Integration, Corr(X t ; GDP t+j ) j X Trade balance (E) Cross-Country Correlations, GDP and Consumption Baseline Model, Baseline Model, No Immigration Trade in Bonds Financial Autarky Trade in Bonds Corr (GDP, GDP*)..2.2 Corr (C, C*) The empirical moments of labor migration ows, remittances, and the U.S.-Mexico trade balance are summarized in Table 4.4; we report the correlations of each of the three indicators with the ratio of U.S.-Mexico GDP per capita measured at various lags and leads. 7 The theoretical moments for the corresponding variables (the ow immigrant labor L e;t, remittances, and the trade balance 7 We apply the Baxter-King bandpass lter to the variables in natural logs in order to lter out uctuations with periodicity lower than 8 months and greater than eight years. 25

26 T B e;t ) are summarized in Table 4.5. We provide moments for the two measures of remittances used in our model: () the immigrant labor income (w i;t L i;t ), and (2) the di erence between immigrant labor income and the immigrant share in the foreign households consumption and investment net of the return on capital, ( t ), as described in (44). Thus, panels (A) and (B) in Table 4.5 show the correlations with home and foreign GDP for the baseline model with nancial integration; panel (C) shows the correlations with home GDP for the baseline model with nancial autarky; panel (D) shows the correlations with home GDP for the model without labor migration which allows for trade in bonds. The moments of the baseline model with nancial integration are consistent with those in the data. The entry of new immigrant labor in Home is procylical with home GDP and countercylical with foreign GDP. Remittances are strongly procylical with the output of the country that hosts the immigrant labor, a result which is robust across the two measures of remittances. The trade balance is countercylical with the home GDP in the baseline model, but the correlation is less negative than in the model without immigration. This result is due to the fact that in the baseline model with labor mobility, foreign households have the option of investing in emigration at the same time as lending to the home households. Thus, immigration acts as a substitute for foreign investment in the home economy, and causes the trade balance to be less countercylical than in the model without labor mobility. Nonetheless, the cross-country correlations of national income and consumption in panel (E) of Table 4.5 show that labor mobility enhances the co-movement of consumption and decreases the co-movement of output relative to the model without migration: The correlation of consumption increases due to remittances, whereas the co-movement of output declines due to the labor out ow from Home to Foreign. 26

27 5 Welfare Analysis 5. Baseline Model: Permanent Increase in Border Enforcement In this section we analyze the welfare e ects of a sudden and permanent increase in the sunk immigration cost (from f e = 4 to f e = 5). The transition paths to a new steady state in Figure 9 show that the declining availability of immigrant labor makes capital less productive and therefore dampens investment, which leads to a decline in the capital stock. Due to the higher entry barriers, rms initially substitute the immigrant with native labor. Despite the lack of increase in native wages, the inter-temporal optimization determines native households to commit more hours in the present, when wages and the rental rate of capital are relatively higher than in the future. However, over the transition path, the declining stock of capital causes the demand and wage of native labor to fall, which explains the decline in hours worked by native households. We study the welfare e ect of the permanent increase in the sunk cost over a wide range of values for the elasticity of substitution between immigrant and native labor in the baseline model, i.e. 2 [:5; 2:5]. Thus, we compare the home welfare in the initial steady state: V = U C f e=4; L fe=4 (54) with home welfare in the new steady state to which the economy converges after the increase in the sunk cost of immigration: 8 V = U C f e=5; L fe=5 : (55) Next we de ne the constants C and C to denote the permanent streams of aggregate consumption that would generate the welfare values V and V : V = ln(c ); (56) V = ln(c ); (57) and compute the consumption-equivalent welfare gain ( > ) or loss ( < ) associated with the new steady state to which the economy converges after the permanent increase of the barriers to 8 We solve the model using a second order aproximation around the steady state. We use the FORECAST Dynare instruction to de ne the initial and the end values that the sunk cost. Nevertheless, we de ne welfare as the present discounted value of the stream of expected future utilities, V = U + E [V (+)]; thus, we compare the original steady state welfare (available in the ys_ output matrix provided by Dynare) with the new steady state welfare to which the economy converges after the sunk cost increase (available in the dr_:ys ouput matrix). 27

28 immigration: = C : (58) C The results in Figure show that the home economy experiences a consumption-equivalent welfare loss for the entire range of values 2 [:5; 2:5] of the elasticity of substitution between immigrant and native labor. Moreover, the loss increases with the degree of complementarity between capital and immigrant labor. 5.4 Consumption, Home (c).43 Consumption, Foreign (c2) 4 Immigrant wage premium (d) 2.5 x -3 Immigrant labor entry (he) Immigrant labor stock (hi).492 Home native labor (hn) 75 Capital stock, Home (k) 9.2 Capital stock, Foreign (k2) Real exchange rate (q) 5.75 Wage of Home natives (wn) 7 Wage of immigrant labor (wi).79 Wage in Foreign (w2) GDP, Home (yh).26 GDP, Foreign (yf). Exports of Home (yh2).35 Imports of Home (yf) New steady state Transition path Figure 9. Baseline model, permanent increase in border enforcement (sudden increase in f e from 4 to 5) 28

29 Consumption-equivalent welfare gain (λ), % Elasticity of substitution between native and immigrant labor (θ) Figure. Baseline model, consumption-equivalent welfare gain/loss, permanent increase in border enforcement (sudden increase in f e from 4 to 5) 29

30 5.2 Alternative Model: Gradual Increase in the Share of Native Skilled This section explores the impact of immigration barriers on welfare in the presence of a gradual and permanent increase in the share of skilled native labor in Home. In the extended model with two types of native labor (skilled and unskilled), we introduce a deterministic growth path in the share of skilled native labor, allowing it to increase from.9 to.97 over 2 years. We assume that households take into account the expected growth path of the share of skilled labor when solving their inter-temporal optimization problem, and compute the consumption-equivalent welfare gain (or loss) associated with the increasing share of skilled labor relative to the initial steady state. To this end, we compare the Home welfare in the initial steady state: V = su cs ; l s + ( ) ( s) U cu ; l u with Home welfare as of period t when households learn about the growth path of the share of skilled labor: 9 Vt = Et X (59) v=t v fs v U (c s;v ; l s;v ) + ( ) ( s v ) U (c u;v ; l u;v )g : (6) Next we de ne the constants C and C to denote the permanent streams of aggregate consumption that would generate the welfare values V and V t : V = ln(c ); (6) V t = ln(c ); (62) and compute the consumption-equivalent welfare gain ( > ) or loss ( < ) associated with the deterministic increase in the share of skilled labor relative to the initial steady state in the presence of barriers to immigration: = C : (63) C The results in Figure show that the welfare loss increases with the magnitude of barriers to immigration and with the degree of complementarity between capital and immigrant labor. Although the immigrant and native unskilled labor are perfect substitutes, the welfare loss su ered by the home unskilled households is o set by the larger accumulation of capital which enhances the productivity of home skilled labor in the presence of immigration. 9 Was previously described, we use the FORECAST Dynare instruction to de ne the list of values that the share of skilled labor takes over the transition path. Thus, we compare the steady state welfare (available in the y_ output matrix provided by Dynare) with the mean welfare as of the period when households learn about the deterministic path of the share of skilled labor (available in the oo_:forecast:means ouput matrix). 3

31 Using the extended model with two types of native labor (skilled and unskilled), we repeat the welfare analysis with the share of skilled native labor increasing deterministically from a lower initial level (i.e. from.5 to.57 over 2 years). As shown in Figure 2, in contrast to the previous exercise, we nd that the welfare gain increases with the size of immigration sunk costs (rather than decreases) when the share of skilled labor starts increasing from a relatively lower level (.5 vs..9). When a larger fraction of the native labor becomes exposed to competition from the immigrant labor, the welfare loss of the home unskilled exceeds the welfare gains of the home skilled labor that bene ts from the greater accumulation of capital. This leads to an overall welfare loss for the home economy. 6 4 Consumption-equivalent welfare gain (λ), % No barriers to immigration (fe=) Sunk costs (fe=7) Elasticity of substitution between capital and immigrant labor (θ) Figure. Consumption-equivalent welfare gain/loss relative to the initial steady state, increasing share of native skilled labor (from.9 to.97 over 2 years). 3

32 5 4.5 Consumption-equivalent welfare gain (λ), % No barriers to immigration (fe=) Sunk costs (fe=7) Elasticity of substitution between capital and immigrant labor (θ) Figure 2. Consumption-equivalent welfare gain/loss relative to the initial steady state, increasing share of native skilled labor (from.5 to.57 over 2 years). 32

33 6 Conclusion This paper attempts to bridge the gap between modern international macroeconomics and immigration theory. In contrast to the former, we allow for labor mobility across countries; in contrast to the latter, we consider the business cycle dynamics and account for the transmission of aggregate stochastic shocks across countries in the presence of labor migration. In the baseline model, we introduce labor migration ows within a parsimonious two-country international real business cycle setup. The incentive to migrate depends on the di erence between the expected future earnings at the destination and in the country of origin, on the perceived sunk costs of labor migration, as well as on the exogenous rate of return to the country of origin. Immigration stimulates the accumulation of capital in the destination economy, which in turn increases the productivity of native labor. The baseline model successfully matches the cyclical dynamics of labor migration and remittances which we document using U.S. and Mexican data. International borrowing and lending facilitate capital ows and reduce the incentive of foreign labor to emigrate in steady state. Over the business cycle, however, the ow of capital towards the expanding economy reinforces the cyclical pattern of labor migration. In an alternative speci cation, we extend the baseline model to allow for skill heterogeneity among home households in the presence of capital-skill complementarity. The overall welfare gain from unskilled immigration for the destination economy increases with the degree of complementarity between the skilled and unskilled labor, as well as with the share of the skilled in total native labor (and so does the damage of "tightening" the border). At the sectoral level, the in ow of unskilled immigrants harms the welfare of unskilled native workers, but a compensation policy mechanism in the form of a countercyclical tax on the immigrant wage could address this issue. As explained in Hanson and Grogger (28), the sunk immigration cost has a key role in explaining the observed wage di erentials across countries. Thus, the general equilibrium stochastic model presented here allows for the estimation of sunk costs while it overcomes the identi cation issues that prevail in the existing reduced-form models of immigration. Our model does not include the possibility of skilled labor migration, also referred to as the "brain drain" phenomenon. Future research should explore this issue. 33

34 References [] Acosta, Pablo (26) Occupational Choice, Migration and Remittances in El Salvador, mimeo. [2] Acosta, Pablo A. & Emmanuel K.K. Lartey & Federico S. Mandelman (27). "Remittances and the Dutch disease," Working Paper 27-8, Federal Reserve Bank of Atlanta. [3] Amuedo-Dorantes, Catalina & Susan Pozo (24). "On the Use of Di ering Money Transmission Methods by Mexican Immigrants," Economic Working Papers at Centro de Estudios Andaluces E24/6, Centro de Estudios Andaluces. [4] Attanasio, O. and Davis, S. (996), Relative Wage movements and the Distribution of Consumption, Journal of Political Economy, 4, [5] Backus, D. K., P.J. Kehoe & F.E. Kydland, (994). "Dynamics of the Trade Balance and the Terms of Trade: The J-Curve?" The American Economic Review, 84() (March). [6] Bilbie, F.O., F. Ghironi and M.J. Melitz (26). "Endogenous Entry, Product Variety, and Business Cycles," mimeo, Boston College, December 7, 26. [7] Polgreen, Linnea and Pedro Silos (25). "Capital-Skill Complementarity and Inequality: a Sensitivity Analysis," Working Paper 25-2, Federal Reserve Bank of Atlanta. [8] Borjas, George J. (995), Assimilitation and Changes in Cohort Quality Revisited: what happened to Immigrant Earnings in the 98s? Journal of Labor Economics 3(3):2-45. [9] Borjas, George J. & Je rey Grogger & Gordon H. Hanson (28). "Imperfect Substitution between Immigrants and Natives: A Reappraisal," NBER Working Papers 3887, National Bureau of Economic Research, Inc. [] Chami, R.; Cosimano, T. and Gapen, M.(26) Beware of Emigrants Bearing Gifts: optimal Fiscal and Monetary Policy in the Presence of Remitttances, IMF Working Paper 6/6. [] Djacic (987), Illegal Alliens, Unemployment, and Immigration Policy, Journal of Development Economics, 2() pp [2] Durdu, Ceyhun Bora and Sayan, Serdar (28). Emerging Market Business Cycles with Remittance Fluctuations, IMF Sta Papers. 34

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38 A Appendix A. Baseline Model with Financial Autarky, Steady State The foreign economy In steady state, A =. With the classic Cobb-Douglas production function Y f = (K ) L f, it is straightforward to solve for the steady state in the foreign economy: r = ; (64) Y f K = r + ; (65) K = Y f = w = ( Yf K Yf K L f ; (66) K = ) Y f L f r + r = ( + ) L f ; (67) ; (68) I = K : (69) The home economy For the home economy, we solve the steady state numerically using a system of eight non-linear equations (7, 7, 75-8) in eight unknowns (Y h, K, L i, Y h2, Y f, p h, p f, Q), as described below. Step : With A =, output and the marginal product of capital are: h i Y h = K (Li ) + ( ) (Ln ) ( ) ; = Y h K = r + : (7) Step 2: Using the steady-state expression for the present discounted value of the future gains from immigration, f e Q w i = ( l ) ( l ) d, we obtain: Q w i = w + d; (72) = w + ( l ) f e Q w i : (73) ( l ) Thus, the steady state ratio of the immigrant wage and the wage in in the country of origin expressed in units of the same consumption basket is: w i w Q = ( l ) f e ; (74) ( l ) 38

39 where = when f e =, i.e. with zero sunk cost of labor migration, the wage ratio is equal to unit. Next, we insert w i i and w ( ) ( ) K ( ) into the previous equation to obtain: ( ) (Y h ) L i;t {z } w i Step 3: The balanced current account condition implies: r = ( + ) {z } w Q: (75) p h Y h2 = p f QY f + L i w i ; (76) where w i is given above. Step 4: We write the demand ratios for the two intermediate goods in each economy as: Y h Y h2 =! ph ; (77) Y f! p f Q Y f Y f =! pf Q Y h2! : (78) Step 5: The price indexes for the composite good of each country are: p h =! (p h ) + (!)(p f Q) ; (79) =! (p f ) + (! ph ) : (8) Q A.2 Alternative Model with Financial Autarky, Steady State The presence of skill heterogeneity among native labor (skilled and unskilled) in Home requires several modi cations in the calculation of steady state relative to the baseline model. In the system of eight equations in eight unknowns described above, L n becomes L s (i.e. native skilled labor). One must also distinguish between individual vs. aggregate labor supply (i.e. l j vs. L j ) and consumption (i.e. c j vs. C j ) for the representative skilled and unskilled households (where j 2 fs; ug). Thus, equations 7, 75 and 76 are replaced by: i (Y h ) = (Li + L u ) + ( ) h K + ( ) (L s ) ; (8) Y h L i + L u {z } w i r = ( + ) {z Q; (82) } p h Y h2 p f Q = Y f + w p f Q L Y h i L i + L u {z } w i : (83) 39

40 A.3 Baseline Model with International Trade in Bonds, Steady State The presence of quadratic costs of adjustment for bond holdings allows us to pin down their steadystate levels. From + B h = ( + r b ), + Bh = ( + r b ) and B h + Bh = ; it follows that: r b = 2 + ; (84) B h = Bh = ( + rb ) : (85) Similarly, using that + B f = ( + r b ), + B f = ( + r b ) and B f + B f = ; it follows that: r b = 2 + = r b ; (86) B f = Bf = ( + rb ) : (87) Finally, the balanced current account condition (76) is replaced by the expression for the balance of international payments (53) in steady state: p h;t Y h2;t p f;t Q t Y f;t w i;t L i;t + r b tb h;t + r b t Q t B f;t = : (88) The steady state solutions for the remaining variables are as in Appendix A.. A.4 Benchmark Model without Labor Migration In the model without labor migration, each country specializes in the production of a single good, labeled Y h;t for home and Y f;t for foreign, as in Backus, Kehoe, Kydland (994). We use log-crra preferences and abstract from government purchases and time-to-build in capital formation. 4

41 The model with nancial autarky The home economy is characterized by equations in variables (r t ; w t ; C t ; L t ; Y h;t ; Y t ; Y h;t ; Y h2;t ; I t ; K t ; p h;t ): Ct = ( + r t )E t ; C t+ (89) w t = L C t ; t (9) Y h;t = A t K t L t ; (9) Y h;t = Y h;t + Y h2;t ; (92) (Y t ) =! (Y h;t Y h2;t ) + (!) (Y f;t ) ; (93) Y t = C t + I t ; (94) K t = I t + ( ) K t ; (95) Y h;t =! (p h;t ) Y t ; (96) Y f;t = (!) (p f;t Q t ) Y t ; (97) r t = Y h;t+ K t (98) w t = ( ) Y h;t L t (99) All equations for the foreign economy are similar. Note that the price of the home intermediate good expressed in units of the foreign consumption basket is Qt p h;t ; therefore, the demand functions for the home and foreign-speci c good in the foreign economy are: Y f2;t =! (p f;t ) Y t and Y h2;t = (! ) Qt p h;t Y t, respectively. Technology follows the process: log A t = log A t + e t ; log A t = log A t + e t The real exchange rate Q t is pinned down by the trade balance, measured in units of the home composite good: NX t = Y h2;t p h;t {z } exports Under nancial autarky and without remittances, NX t =. Y f;t p f;t Q t : () {z } imports Financial integration, trade in risk-free bonds International trade in risk-free bonds (with quadratic cost of adjustment of bond holdings) adds 6 extra variables (i.e. the rates of return of the 4

42 home and foreign bonds, r b t and r b t ; holdings of the home and foreign bonds by home households, B h;t and B f;t ; holdings of the home and foreign bonds by foreign households, Bh;t and B f;t ) and 6 new equations to the model with nancial autarky: + B h;t+ = E t ( + rt+) b C t ; () C t+ Qt+ + B f;t+ = E t ( + r Q t+) b C t t C t+ + B h;t+ = E t Qt Q t+ ( + r b t+) C t + B f;t+ = E t ( + r b t+) C t C t+ C t+ ; (2) ; (3) ; (4) B h;t+ + Bh;t+ = ; (5) B f;t+ + Bf;t+ = : (6) The expression for the balance of international payments replaces the balanced trade condition from the model with nancial autarky: p h;t Y h2;t p f;t Q t Y f;t + r b tb h;t + r b t Q t B f;t = : (7) A.5 Benchmark Model without Labor Migration, Asymmetric Steady State In steady state, A = A = : In each country, Y h K r = ; r = ; (8) = r! Y h K = r + ; Y f K = r + ; (9) Y h = K L Yh! K = L; K Yf = L ; () Y h = Yh K = K r + K K L; Yf = r + L ; () I = K; I = K : (2) The symmetric case is described by: The solution with symmetric calibration parameters for the two economies p h = p f = Q = : (3) Y h = Y f2 =!Y h : (4) Y h2 = Y f = (!)Y h ; (5) 42

43 where (!) represents the share imports in GDP. Using that Y h =!Y h and Y h2 = (!)Y h, Y = i h! (Y h ) + (!) (Y f ) = Y h ; (6) C = Y I: (7) Asymmetric steady state asymmetries of the type 6= ; 6= ; 6= This section describes the steady-state solution for cross-country and! 6=! :The equations (8)-(2) still hold. We obtain the steady-state solutions numerically using a system of 5 equations in 5 unknowns (Y h ; Y f2 ; p h ; p f ; Q): Y h =! ph ; (8) Y f Y f2! p f Q Y f2 =! pf Q Y h Y h2! ; (9) In nancial autarky, the balanced trade condition is: p h =! (p h ) + (!)(p f Q) ; (2) =! (p f ) + (! ph ) (2) Q Y h2 p h Y f p f Q = : (22) With nancial integration, balanced trade is replaced by the expression for the balance of international payments: p h;t Y h2;t p f;t Q t Y f;t w i;t L i;t + r b tb h;t + r b t Q t B f;t = : (23) 43

44 Figure. U.S.-Mexico border apprehensions vs. the U.S.-Mexico GDP per capita ratio q 983q3 984q 984q3 985q 985q3 986q 986q3 987q 987q3 988q 988q3 989q 989q3 99q 99q3 99q 99q3 992q 992q3 993q 993q3 994q 994q3 995q 995q3 996q 996q3 997q 997q3 998q 998q3 999q 999q3 2q Apprehensions at the U.S.-MExico border (left axis) Source: Hanson (27) and Haver Statistics Note: We apply the Baxter-King bandpass filter to the quarterly data in natural logs. US-Mexico GDP per capita ratio (right axis) Figure 2. Correlations of apprehensions at the U.S.-Mexico border with the j lags and leads of the U.S.-Mexico GDP per capita ratio j 44

45 Figure 3. Remittances, private debt and foreign direct investment in developing countries US$ billion year Remittances Private debt & portfolio equity FDI Sources: Global Economic Prospects 26: Economic Implications of Remittances and Migration (World Bank), World Development Indicators 27, and Global Development Finance 27. Source: Global Economic Prospects, World Bank. 45

46 q 998q2 998q3 Figure 4. Remittances from the U.S. to Mexico vs. the U.S.-Mexico GDP per capita ratio 998q4 999q 999q2 999q3 999q4 2q 2q2 2q3 2q4 2q 2q2 2q3 2q4 22q 22q2 22q3 22q4 23q 23q2 23q3 23q4 24q 24q2 24q3 24q4 25q Remittances from the U.S. to Mexico (left axis) US-Mexico GDP per capita ratio (right axis) Source: Banco de Mexico and Haver Statistics. Note: We apply the Baxter-King bandpass filter to the quarterly data in natural logs. Figure 5. Correlations of the U.S.-Mexico remittances with the j lags and leads of the U.S.-Mexico GDP per capita ratio j 46

47 Figure 5(b). Growth in remittances from the U.S. to Mexico vs. the growth in U.S. housing starts Source: Banco de Mexico and Haver Statistics. 47