On the Macroeconomic and Welfare Effects of Illegal Immigration

Transcription

1 MPRA Munich Personal RePEc Archive On the Macroeconoic and Welfare Effects of Illegal Iigration Xiangbo Liu 28. May 2009 Online at MPRA Paper No. 5469, posted. June :4 UTC

2 On the Macroeconoic and Welfare E ects of Illegal Iigration Xiangbo Liu y University of California, Riverside January 28, 2009 Abstract This paper investigates the acroeconoic and welfare e ects of illegal iigration on the native born within a dynaic general equilibriu fraework with labor arket frictions. A key feature of the odel is that job copetition is allowed for between doestic workers and illegal iigrants. We calibrate the odel to atch soe key statistics of the postwar U.S. econoy. The odel predicts that in the long run illegal iigration is a boon, but the eployent opportunities of doestic workers are strongly negatively a ected. The odel also predicts that the level of doestic consuption has a U-shaped relationship with the share of illegal iigrants. Keywords: Econoic Growth, Iigration, Welfare, Search, Uneployent JEL Classi cation: F22; O4; J64 I a greatly indebted to Jang-Ting Guo and Richard M. H. Suen for their extreely helpful discussions and coents. I also thank Richard Arnott and Todd Sorensen for their insightful suggestions. Of course, any reaining errors are y own. y Departent of Econoics, Sproul Hall, University of California, Riverside, CA Eail: xliu005@ucr.edu.

3 Introduction Illegal iigration is a contentious issue facing ost developed econoies. In the United States, for instance, scholars have heatedly debated the pros and cons of illegal iigration for years. The ain econoic arguent in support of iigration is that it helps increase the supply of labor, reduces the cost of production and hence is good for the econoy. Priary opposing arguents include supposed high rates of use of welfare progras, iigrant poverty and job copetition. Much of the discussion is otivated by concerns for the welfare e ects of illegal iigration on the native born. However, ost research applying partial-equilibriu analysis has only addressed slices of this proble through analyzing the e ects of iigration on labor-arket outcoes. There is only a sall set of theoretical studies that address this issue of illegal iigration in a general equilibriu context. These studies have noticeable liitations. Aong the, Ethier (986) and Bond and Chen (987) are carried out within a static context and they pay particular attention to probles and prescriptions for border control. Following the Rasey tradition, subsequent research suppleents the literature by investigating this issue within a one-sector dynaic general equilibriu fraework. These studies include Hazari and Sgro (2003), Moy and Yip (2006), and Palivos (2009). One coon liitation aong all existing studies is that they assue full eployent in doestic labor arket. These odels thus ignore the e ect of illegal iigration on the eployent opportunities of doestic workers. In fact, one coon arguent in general against iigration is that iigrants har the eployent opportunities of native workers. Studies failing to address this issue cannot capture the whole picture of the e ects of illegal iigration. The priary objective of this paper is to develop a dynaic general equilibriu odel that can be used to evaluate the displaceent e ects of illegal iigration on native workers. To the best of our knowledge, so far no such theoretical odel has been developed. To achieve this objective, this study builds upon the contributions of Shi and Wen (997) and odels illegal iigration in a standard dynaic general equilibriu odel with labor arket frictions. One key feature of our odel that di erentiates it fro the previous literature is that we allow both doestic workers and illegal iigrants to search for jobs at the sae tie, which in turn leads to job copetition between the and consequently increases the uneployent rate of

4 native workers. In the odel econoy, each individual has three alternative, utually exclusive uses of one indivisible unit of tie: searching for a job, working for a r, or enjoying leisure. Firs hire both doestic and illegal iigrant workers. The labor arket is subject to search-atching frictions. Once uneployed doestic workers and job vacancies are atched, the ters of eployent contracts are deterined through bilateral bargaining. We assue that rs are able to distinguish illegal iigrants fro doestic workers and face a punishent for hiring the forer if being caught. The wage rate for illegal iigrants is thus equated to the wage rate of doestic workers inus the expected value of the punishent. We characterize the search equilibriu and prove the existence and uniqueness of stationary equilibriu. The odel generates iportant theoretical predictions due to the incorporation of illegal iigrants. Within this dynaic general equilibriu fraework, we are able to show analytically that the long-run level of the uneployent rate for doestic citizens is increasing in the share of illegal iigrants in the total population for the case in which natives and illegal iigrant workers are perfect substitutes in the production process. We also uncover that the entry of illegal iigrants akes doestic workers face tighter labor arkets in the long run. To develop the quantitative iplications of our odel, we nuerically solve and calibrate the odel to atch soe key statistics of the postwar U.S. econoy. Palivos (2009) nds that illegal iigration necessarily lowers the long-run level of per capita consuption and welfare of doestic citizens. In sharp contrast, the quantitative predictions of this study indicate that the long-run level of consuption of doestic citizens has a U-shaped relationship with the share of illegal iigrants. In other words, an increase in the nuber of illegal iigrants rst reduces and then raises the long-run level of per capita doestic consuption. This nding is due to the presence of four e ects at work. () A positive exploitation e ect. When there is an increase in the nuber of illegal iigrants, a greater nuber of uneployed illegal iigrants are searching for jobs. In contrast, the change in the nuber of doestic workers searching for jobs is sall. This leads to a tighter labor arket which in turn leads to ore erce copetition for jobs. To successfully secure a job, both doestic and foreign labor would have to lower their wages. This raises the r s pro ts which are then distributed to doestic households as dividends. This e ect adds to doestic consuption. (2) A negative capital-using-up e ect. This is due to the fact that in the 2

5 doestic econoy soe capital has to be used to produce output for the consuption of illegal igrants. This e ect reduces current output which could have been used for doestic consuption and investent. (3) A negative wage depressing e ect. As entioned above, when ore illegal iigrants enter into the econoy, the copetition for jobs becoes ore severe. Thus, the wages for doestic labor are pushed down. (4) A negative displaceent e ect. As uneployed doestic labor and igrants copete for jobs, the chance for uneployed doestic workers to nd a job is reduced. This e ect reduces doestic consuption. Epirical studies on this topic often focus on (3) and (4) (for instance, see Hotchkiss and Myria 2008). These four e ects work together to deterine the relationship between the long-run level of consuption of doestic citizens and the share of illegal iigrants. Under the baseline paraeterization, the negative e ects doinate when the population fraction of illegal iigrants is sall. Thus, an increase in illegal iigration would lead to a drop in consuption. However, as the share of illegal iigrants continues to increase, the long-run level of doestic consuption would rise as the positive e ect eventually doinates. This gives rise to the U-shaped relationship between the long-run level of consuption of doestic citizens and the share of illegal iigrants. In order to shed soe light on the welfare e ects of illegal iigration, we copute the consuption-equivalent level of utility of doestic households and nd that illegal iigration has a positive welfare e ect. In particular, we copare two scenarios: () the econoy stays at the steady state with no illegal iigrants forever; and (2) at t = 0, the host country adits a certain fraction of illegal iigrants and the econoy gradually converges to the new steady state. The welfare easure of illegal iigration is calculated for a wide variety of cobinations of labor supply elasticity and population share of illegal iigrants. For instance, we nd that the doestic households would require a :746-percent increase in their consuption under scenario () in every period when the labor supply elasticity is 0:4 and when there is an increase in the population share of illegal iigrants fro zero to 5 percent. The odel also generates a prediction on eployent opportunities of doestic workers. It predicts that eployent opportunities of doestic workers are strongly negatively a ected in the long run. Speci cally, a greater nuber of doestic workers will leave the labor force when there is an increase in illegal iigration. In contrast, the labor force participation rate for illegal iigrants experiences a slight decrease. This result turns out to be qualitatively supported by the existing epirical evidence (for instance, see Borjas et al., 2007). 3

6 The reainder of the paper is structured as follows. Section 2 presents the search-theoretic odel of uneployent and analyzes the search equilibriu. Section 3 studies the welfare e ect of illegal iigration on doestic citizens and discusses the quantitative iplications of the odel. Finally, Section 4 o ers soe concluding rearks. 2 The Model Consider an econoy that is inhabited by two types of households, i.e., doestic (D) and illegal iigrant (M) households. The nuber of each type of households is noralized to one. Each household consists of any in nitely lived agents. We use L(t) and M(t) to denote the size of each doestic and iigrant household at any tie t 0, respectively. We call N(t) = L(t) + M(t) the total population. Both L(t) and M(t) are assued to be growing at the sae constant rate g > 0. 2 The share of illegal iigrants in the total population is constant over tie = M(t) N(t). Each household eber at each point in tie is endowed with one indivisible unit of tie that has three alternative, utually exclusive uses: searching for a job, working for a r, or enjoying leisure. Throughout we use a superscript i 2 fd; M g to indicate these two types of households. The variable s i (t) is the fraction of the household s tie in work, and si 2 (t) is the fraction of the household s tie in search. The variable s i (t) is also referred to as the search e ort. Accordingly, at the aggregate level, a representative doestic household spends L (t) = s D (t)l(t) of its total aount of tie in search, and L 2 (t) = s D 2 (t)l(t) in work. Siilarly, de ne M (t) = s M (t)m(t) and M 2(t) = s M 2 (t)m(t) as the respective aggregate aount of tie in search and in work for a representative illegal iigrant household. The doestic labor participation rate and uneployent rate can be tered as s D (t) + sd s 2 (t) and D (t) s D (t)+sd 2 (t); respectively. Aggregate output Y (t) is produced according to the Cobb-Douglas production technology that takes as inputs aggregate capital K(t) and aggregate labor L 2 (t) + M 2 (t), Y (t) = [K(t)] [L 2 (t) + M 2 (t)], 2 (0; ); Although both legal and illegal iigrants act as a factor substitute for native labor of siilar skill, in this odel, we only consider illegal iigrants because they work as a cheaper production substitute for doestic workers of the sae level of labor productivity. 2 Iposing this assuption is to ensure balanced growth properties of the odel. 4

7 where is the capital share of national incoe. Doestic labor L 2 (t) and illegal igrants M 2 (t) are assued to be perfect substitutes in production Doestic Household s Utility Maxiization Proble In each period, each household eber derives utility fro consuption and disutility fro working and searching for jobs. The oentary utility function of a typical agent is given by u[c(t); s D (t) + s D 2 (t)] = log c(t) [sd (t) + sd 2 (t)]+ + ; > 0; and > 0; () where denotes the inverse of labor supply elasticity, and is a preference paraeter. The household s total discounted utility is described by U = Z 0 e ( g)t u[c(t); s D (t) + s D 2 (t)]dt: (2) The variables C(t) and c(t) = C(t)=L(t) are aggregate and individual consuption of the doestic household, respectively. 4 The paraeter > 0 is the discount rate, and g the e ective discount rate, which is assued to be greater than zero. A worker receives a wage rate w(t) when he enters an eployent relationship. Let r(t) denote the rate of return to capital net of depreciation at tie t; and (t) be the aount of dividends that a household receives by owning the r. Thus, the oentary budget constraint faced by a representative doestic household is _K(t) + C(t) = w(t)l 2 (t) + r(t)k(t) + (t): Dividing it by the size of population N(t) gives the budget constraint in per capita ters as _k(t) + k(t)g + c(t) = w(t)s D 2 (t) + r(t)k(t) + (t); (3) where _%(t) d%(t) dt is the tie derivative of the variable %(t), = L(t)=N(t) is the ratio of doestic 3 The sae assuption is also adopted in Hazari and Sgro (2003), Moy and Yip (2006), and Palivos (2009). 4 Following Merz (995), we assue that there are a large nuber of agents in each household. They pool their incoe and care only about the household s utility. By doing so they provide each other with coplete insurance against variations in labor incoe due to uneployent. 5

8 to total population, (t) = (t)=n(t) is dividend per capita and k(t) = K(t)=N(t) is capital per capita. 5 The nuber of eployed doestic workers evolves according to _L 2 (t) = (t)l (t) L 2 (t); (4) where (t) is the rate at which uneployed workers nd jobs and > 0 is the job destruction rate. In equilibriu, (t) is deterined by the aggregate nubers of job vacancies and uneployed workers. In the utility axiization proble, however, (t) is taken as given by a representative household. Upon dividing by N(t), an individual s eployent evolves according to the law of otion: _s D 2 (t) = (t)s D (t) s D 2 (t) gs D 2 (t): (5) The representative doestic household s optiization proble is to choose a set of tie paths c(t), s D (t), k(t), s D 2 (t) so as to axiize (2) subject to (3), (5) and two initial conditions: k(0) > 0; > s D 2 (0) > 0: Let (t) and (t) be the costate variables. They denote the shadow prices of household s eployent and capital, respectively. The rst-order conditions of the representative household s optiization proble with respect to c(t), s D (t), k(t), sd 2 (t) and the associated transversality conditions (TVC) are given by u 0 s D u 0 c(t) = (t); (6) (t) = (t)(t); (7) _(t) = ( + )(t) [ (t)w(t) + u 0 (t)]; (8) s D _ (t) (t) = r(t); (9) li e ( g)t (t)s D 2 (t) = 0; (0) t! li e ( g)t (t)k(t) = 0: () t! Equation (7) states the rule for the household to decide how uch e ort it should put into search. It requires the arginal cost of search to be equal to the arginal bene t of search. 5 As de ned earlier, = M(t) N(t) : Thus, = holds true for each tie period. 6

9 Given the separable utility function for in (), cobining (6) and (9) and rearranging ters yield a siple expression for the Euler equation: _c(t) = r(t) ; (2) c(t) where the elasticity of interteporal substitution of consuption is. This condition describes the evolution of individual s consuption. In other words, it states that if r exceeds ; then individual consuption will expand over tie. By using (6), (7), and (8), we obtain _(t) = ( + )(t) + [w(t) u0 c(t) + ](t)(t): (3) u 0 (t) s D An iportant iplication of (3) is that in order to copensate for the search cost the wage rate has to be set above the arginal rate of substitution between leisure and consuption Iigrant Household s Utility Maxiization Proble Siilar to the doestic households, in each period each iigrant household eber derives utility fro consuption and disutility fro working and searching for jobs. The oentary utility function of a typical iigrant agent is given by u[c M (t); s M (t) + s M 2 (t)] = log c M (t) [sm (t) + sm 2 (t)]+ + ; > 0; and > 0: (4) The igrant household s total discounted utility is characterized by U = Z 0 e ( g)t u[c M (t); s M (t) + s M 2 (t)]dt: (5) The variables C M (t) and c M (t) = C M (t)=m(t) are aggregate and individual consuption of iigrant household, respectively. Under the conventional assuptions in the literature, illegal igrants are paid at the wage rate w M (t) which is distinct fro that paid to doestic labor, w(t). 7 This is due to the fact that in ost developed countries, rs have to pay a ne once they are caught 6 It can be shown that if w = u 0 s D (t) u 0 c (t) as in a typical neoclassical odel; the shadow price of eployent (t) can grow without bound. 7 See also Hazari and Sgro (2003), Moy and Yip (2006), and Palivos (2009). 7

10 hiring illegal igrants. In a copetitive arket, the wage rate w M (t) is equated to the arginal product of the illegal iigrants inus the expected value of the punishent. In this study, as described later, the wage rate for doestic workers is endogenously deterined through a Nash bargaining process. Given the assuption that rs are able to distinguish illegal iigrants fro doestic workers, the wage rate for illegal iigrants is therefore equated to the wage rate of doestic workers inus the expected value of the punishent. Moreover, it s assued that illegal igrants do not accuulate capital in the host country. This can be justi ed by the fact that in ost developed countries illegal iigrants nd no way to legally establish credit and own assets. 8 The budget constraint that a representative igrant household faces is therefore C M (t) = w M (t)m 2 (t): (6) Dividing it by N(t) gives the budget constraint in per capita ters as c M (t) = w M (t)s M 2 (t): (7) Analogous to (4), the nuber of eployed illegal iigrants evolves according to _M 2 (t) = (t)m (t) M 2 (t): (8) Upon dividing by N(t), an individual s eployent evolves as follows: _s M 2 (t) = (t)s M (t) ( + g)s M 2 (t): (9) Let ~ (t) be the costate variable of illegal iigrant household s eployent. The axiization conditions for the representative iigrant household with respect to s M (t), sm 2 (t) and the 8 This assuption is also ade in Hazari and Sgro (2003), Moy and Yip (2006), and Palivos (2009). In Hazari and Sgro (2003) and Moy and Yip (2006), it s assued that iigrants do not save and hence their consuption is equal to their incoe. Palivos (2009) assues that iigrants do save but they channel all their savings abroad. The capital accuulation process in the host country is not a ected by the illegal iigrants consuption-saving decisions in either way. Therefore, it doesn t atter how illegal iigrant households split their incoe. 8

11 associated TVC are u s M (t) = ~ (t)(t); (20) ~(t) = ( + ) ~ (t) [u c M (t)w M (t) + U s M 2 (t)]; (2) li e ( g)t (t)s ~ M 2 (t) = 0: (22) t! In particular, (20) governs illegal iigrant household s optial decision on the search e ort. 2.3 Production In this econoy, there are a large nuber of identical rs. Firs hire both doestic and foreign labor fro the labor arket to produce output. In order to hire labor, the r has to post job vacancies V (t). Each vacancy costs d > 0 units of output. The probability that a r nds an uneployed worker is (t): Siilar to (t), (t) is deterined by the aggregate nubers of job vacancies and uneployed workers in equilibriu. However, in the pro t axiization proble, (t) is taken as given by a representative r. The law of otion of a r s eployent is given by: _L 2 (t) + M _ 2 (t) = (t)v (t) [L 2 (t) + M 2 (t)]: (23) Taking the factor prices as given, a representative r chooses a set of tie paths fk(t), L 2 (t), M 2 (t), V (t)g so as to axiize its present value of the future pro t streas. Forally, this is given by Max = Z 0 e R t 0 r()d (t)dt; subject to (23), and (t) = F [K(t); L 2 (t) + M 2 (t)] [r(t) + ]K(t) w(t)l 2 (t) w M (t)m 2 (t) pm 2 (t) dv (t): (24) The paraeter is the rate of capital depreciation, and p 2 (0; ) the probability that a r which eploys illegal igrants gets detected. 9 The ne for eploying illegal igrants is noralized to one per illegal iigrant worker. Let (t) and (t) be the costate variables of r s eployent 9 This probability can surely be a ected by a country s enforceent budget. In the present odel, it s assued to be constant. 9

12 of doestic and foreign labor, respectively. Interior solutions of the above axiization proble are characterized by the rst-order conditions F 0 k (t) = r(t) + ; (25) (t) = d (t) ; (26) (t) = (t); (27) _(t) = [r(t) + ](t) + w(t) F L2 (t); (28) _(t) = [r(t) + ](t) + w M (t) + p F M2 (t): (29) Equation (25) is the usual condition which states that the rental rate on capital is equated to the arginal product of capital. Equation (26) governs the r s optial vacancy decisions. The arginal cost of vacancy d equals the arginal bene t of vacancy (t)(t). Equation (28) deonstrates that if there is no vacancy aintaining cost for the r i.e., d = 0, we would obtain the standard neoclassical productivity condition for labor w(t) = F L2 (t). In that case, rs would post an in nite nuber of vacancies and there will not any search frictions in the labor arket. With positive d; however, the wage rate for doestic workers w(t) is less than the arginal product of labor F L2 (t) in this odel. The relationship between the wage rates paid to doestic worker w(t) and illegal iigrant w M (t) is given by w M (t) = w(t) p: (30) The wage rate w(t) for doestic workers is deterined through a Nash bargaining process which will becoe clear later on. As p is positive, it follows that wage rate w M (t) paid to illegal igrants is strictly lower than that paid to doestic labor. Notice that the above condition also indicates that the punishent of hiring illegal iigrants is copletely borne by the illegal iigrants theselves. Firs therefore do not su er directly fro eploying illegal iigrant workers. 2.4 Matching and Wage Deterination The labor arket is subject to search-atching frictions. Vacant jobs and uneployed workers are brought together in a pair-wise fashion by a stochastic search-atching process. The search 0

13 part follows fro the fact that both doestic workers and iigrants invest soe tie and e ort in searching for jobs. Meanwhile, rs seek workers to ll vacant job positions. The atching part of the process is derived fro a atching function which pairs the uneployed workers with vacancies. For analytical convenience, we eploy a Cobb-Douglas atching function with constant returns-to-scale. 0 The nuber of successful job atches is deterined by the following atching function: [V (t); L (t) + M (t)] = 0 [V (t)] [L (t) + M (t)] ; 2 (0; ) where V (t) is the nuber of vacancies, L (t)+m (t) is the nuber of uneployed workers searching for jobs, is the elasticity of vacancy in job atches, and 0 > 0 is assued to be constant over tie. Given the Cobb-Douglas atching function, the vacancy-atching rate (t) and the job- nding rate (t) are obtained as follows: (t) = (t) V (t) = 0[x(t)] ; (3) (t) = (t) L (t) + M (t) = 0[x(t)] ; (32) ) (t) = (t) x(t) ; where the ratio between the vacancies and the uneployed workers, x(t) = V (t) L (t)+m (t); is conventionally labeled as the tightness of the labor arket. Intuitively, it captures the pressure that uneployed workers and rs face in the labor arket. Speci cally, workers and eployers face a tighter labor arket when x(t) is saller. The above expressions ake clear the dependence of the rates (t) and (t) on the tightness of the labor arket x(t): In particular, (t) falls with x(t) and (t) rises with x(t): With the use of this atching function, the equilibriu outcoes are not Pareto optial. This is due to the presence of search externalities inherent in the odel. The intuition is as follows. On the one hand, with ore uneployed workers participating in search, rs will be bene cial since vacancies are ore likely to be lled. However, uneployed workers will su er as their chance to 0 The Cobb-Douglas atching function is also epirically veri ed. See, for instance, Blanchard and Diaond (989).

14 atch theselves with a job is reduced. On the other hand, with ore open vacancies, uneployed workers win while rs searching for workers lose. Thus, the decentralized outcoe is not e cient because workers and rs do not take into consideration the costs that they ipose on others. The activities that generate a negative externality are carried out to a greater extent than are socially desirable. Uneployed doestic workers and vacant jobs eet in pairs. A successful job atch generates a surplus for both uneployed doestic workers and eployers. How is this surplus shared between the? It is a atter of bargaining. The standard search and atching odel assues that by choosing a proper wage rate this surplus is axiized according to the Nash solution to a bargaining proble. In particular, if a r hires a doestic worker, then the surplus to the r fro eploying hi is f 0 (t) w(t): On the other hand, if a doestic worker chooses to work for a r, then s D 2 the gain to hi fro accepting the job is w(t) [ a utually advantageous deal. Let 2 (0; ) and u 0 s D (t) 2 u 0 c(t) ]:2 Hence, there exists a possibility of represent the relative bargaining powers of doestic labor and rs, respectively. A doestic worker and a r jointly deterine the eployent contract under the assuption that each r-worker pair takes the behavior of other such pairings as given. The optial wage contract under Nash bargaining is derived by solving Max w(t) f( ) log[f 0 (t) w(t)] + log[w(t) ( s D 2 u 0 (t) s D 2 u 0 c(t) )]g: The solution of this is given by w(t) = f 0 (t) + ( s D 2 )[ u 0 s D 2 (t) ]: (33) u 0 c(t) The optial wage is a weighted average of the worker s arginal product of labor and reservation wage, which is the arginal rate of substitution between consuption and leisure. If doestic workers have relative stronger bargaining strength, i.e., is closer to one, then the optial wage is closer to the arginal product of labor. In this odel, illegal iigrants have no bargaining As denoted above, k(t) = K(t)=N(t); the Cobb-Douglas production function in per capita ters can be written as f[k; s D 2 + ( )s M 2 ] = k [s D 2 + ( )s M 2 ] : 2 The derivative of u with respect to s D 2 (t) is negative i.e., u 0 u 0 < 0: The expression [ s D s D 2 (t) ] represents the 2 (t) doestic workers endogenized reservation wage. u 0 c(t) 2

15 power, i.e., they are not allowed to bargain over the wage with the rs. Rather, as entioned above, their wage rate w M (t) is deterined by (30). 2.5 Market Equilibriu In this subsection, we provide all the necessary ingredients of this odel as follows: De nition A search equilibriu consists of a set of tie paths fc(t); c M (t); k(t); v(t); s D (t); sd 2 (t); sm (t); sm 2 (t) j t 0g; prices fr(t); w(t); w M(t) j t 0g; pro t f(t) j t 0g; and atching rates f(t); (t) j t 0g such that. Given fr(t); w(t); (t); (t) j t 0g, fc(t); k(t); s D (t); sd 2 (t) j t 0g solves the doestic household s proble. 2. Given fw M (t); (t) j t 0g, fc M (t); s M (t); sm 2 (t) j t 0g solves the iigrant household s proble. 3. Given fr(t); w(t); w M (t); (t) j t 0g, fk(t); v(t); s D 2 (t); sm 2 (t) j t 0g solves the r s proble. 4. The wage rate w(t) is deterined by (33). 5. The atching rates are given by (3) and (32): 6. All arkets clear. (a) The goods arket clears at every t 0, i.e., C(t) + _ K(t) + K(t) = F [K(t); L 2 (t) + M 2 (t)] C M (t) pm 2 (t) dv (t) for all t 0: (b) In the labor arket, in equilibriu, the ows of workers into eployent ust equal the ows of vacancies atched with uneployed agents, i.e., (t)[l (t) + M (t)] = (t)v (t): (4), (8), and (23) indicate that the total supply for labor equals the deand for labor at every t 0. 3

16 2.6 Characterization of Equilibriu After siple anipulations and substitutions, the equilibriu de ned above is suarized by the following seven di erential equations which together deterine the dynaic properties of [c(t); k(t); x(t); s D (t); sd 2 (t); sm (t); sm 2 (t)]t : _c(t) c(t) = f 0 k (t) ; _k(t) = f(t) k(t)g k(t) c(t) [f 0 (t) + ( )u 0 (t) s D s D 2 2 u 0 c(t) ]sm 2 (t)( ) dv(t); (35) _x(t) = x(t) 0 ( + ) (t)( ) u (t) d( ) [f s D(t) s D 2 ]; (36) 2 u 0 c(t) u 0 (t) _s D s (t) = ( + ) D u 00 (t)s D s D (t) + u 0 0 c(t)w(t)(t) u (t)(t) u 00 (t)s D s D (t) + s D 2 u 00 (t)s D (37) s D (t); _s D 2 (t) = (t)s D (t) ( + g)s D 2 (t); (38) _s M (t) = ( + ) u 00 s M (t) (t)s M (t) + u 0 (t)w c M M (t)(t) (t)s M (t) + u 0 s M u 00 s M u 0 s M 2 u 00 s M (t)(t) (t)s M (34) (t); (39) _s M 2 (t) = (t)s M (t) ( + g)s M 2 (t): (40) The three initial conditions of this syste are k(0); s D 2 (0); and sm 2 (0):3 Proposition : A unique steady state exists. All proofs can be found in the Appendix. Notice that this unique steady state is in per capita ters. All aggregate variables, such as K(t); C(t); are still growing at rate g > 0: 3 Quantitative Analysis In this section, we develop the quantitative iplications of our odel. Thus, we rst nuerically solve and calibrate the odel to atch soe key statistics of the postwar U.S. econoy. Then we discuss those quantitative predictions in order. Speci cally, we answer the following three questions:. Will the long-run level of consuption of doestic citizens decrease in the population share of illegal iigrants? 3 The atheatical derivations of these di erential equations are available fro the author upon request. 4

17 2. How doestic workers eployent opportunities are a ected by illegal iigration ows in the long run? 3. What is the welfare e ect of illegal iigration on the host country? 3. Paraeterization This subsection presents the procedure used to paraeterize the odel. The speci c nuerical values to the paraeters of the odel are assigned so that the odel can atch as closely as possible soe key statistics for the U.S. econoy for the postwar period. In particular, the odel ais to atch U.S. facts on the labor participation rate, the uneployent rate, the average capital-output ratio, and the real interest rate. There are eleven paraeters which need to be assigned in this odel: the preference paraeters,, and ; the production paraeters and ; the search-atching paraeters 0 ; ;, the rate of population growth g; the bargaining power of doestic labor and the unit cost of vacancy d: As a tie period is noralized to be one quarter, each paraeter is interpreted quarterly. The preference paraeter is set equal to 3:7939 to atch the steady-state labor force participation rate of 0:68. This is consistent with the U.S. labor force participation rate for the population aged 6 years old and over in the postwar period. 4 We choose the depreciation rate on capital = 0:008 so that the quarterly capital-output ratio in the steady state is equal to 2; which roughly atches the average capital-output ratio in postwar U.S. data (Cooley et al., 995). The unit cost of vacancy d is set at 2:064 to achieve the steady-state uneployent rate of 0:06, which atches the U.S. quarterly average uneployent rate in the postwar period. 5 We use the discount rate = 0:0 so that the steady-state annual interest rate is roughly 4% (Siegel 2002). 6 The value of the capital s share of national incoe is set to 0:25, which falls in the range in Gollin (2002). 7 We set the value of = 2:5 to obtain a labor supply elasticity of { = 0:4 (Killingsworth 983). We also allow { to take di erent values 0:2, 0:7 and. The paraeter 0 is coonly noralized to one. As indicated in Blanchard and Diaond (989), the paraeter for 4 Source: U.S Bureau of Labor Statistics < 5 U.S. Bureau of Labor Statistics has docuented the annual average uneployent rate fro 948 to the present. 6 Siegel (2002) suggests the average of the real return to stock and long-ter bonds over the period is Gollin (2002) indicates that the labor shares for ost countries fall in the range of

18 the elasticity of vacancy in job atches is 0:6, hence = 0:6: We use the exogenous job destruction rate = 0:05; which resebles the quarterly eployent-uneployent transition probability (Shi and Wen 999). The value of bargaining power of labor is set to 0:5, a value coonly used in the literature. We use the rate of population growth g = 0:0027 as the annual population growth rate in the postwar US is roughly %: The baseline paraeterization is suarized in Table. Table : Baseline Paraeter Values Preferences = 0:0; = 2:5; = 3:7939: Production = 0:25; = 0:008: Matching 0 = ; = 0:6; = 0:05: Others g = 0:0027; = 0:5; d = 2:064: 3.2 Local Dynaics We next exaine local dynaics by linearizing the syste of di erential equations in the neighborhood of the steady state. As stated in Proposition, the nonlinear dynaic syste has a unique steady state at (c ; k ; x ; s D ; sd 2 ; sm ; s M 2 ) T. Let J be the 7 7 Jacobian atrix evaluated at the steady state. The dynaic properties of the linearized syste is deterined by the eigenvalues of the Jacobian atrix J: The predeterined variables are k(t); s D 2 (t); and sm 2 (t): Saddle-path stability requires that the nuber of stable eigenvalues be exactly the sae as the nuber of predeterined variables. Therefore, the atrix J needs to have three stable eigenvalues and four unstable eigenvalues in order to ensure the existence of a unique transition path. In the quantitative exercise, we allow the population share of illegal iigrants to vary between 0 and 0:5. 8 In all of these experients we obtain three stable eigenvalues and four unstable eigenvalues. By allowing { to take values in f0:2; 0:4; 0:7; g, we nd that the above result is robust with respect to changes in the labor supply elasticity. The values of the stable eigenvalues are reported in Table 2. Thus, the unique steady state is saddle-path stable under the baseline paraeterization. 8 In this study, we only consider the case in which the nuber of illegal iigrants is less than that of doestic citizens. Therefore, we allow to vary fro 0 through 0:5. 6

19 3.3 Macroeconoic E ects In this subsection, we develop the quantitative iplications of the odel. In particular, we focus on the steady-state e ects of illegal iigration. In order to deonstrate the econoic ipact of illegal iigration on doestic residents, we now perfor soe coparative static experients. In the rst coparative static experient we are concerned with the e ects on the long-run level of doestic consuption when there is an increase in the share of illegal iigrants in the population. Speci cally, by allowing the fraction of iigration to take values fro 0 through 0:5; we copute a series of steady states to capture the response of doestic consuption to an in ux of illegal iigrants. The quantitative prediction of the present odel is that the long-run level of consuption of doestic citizens has a U-shaped relationship with the share of illegal iigrants (see Figure 2). In other words, an increase in the nuber of illegal iigrants rst reduces and then raises the long-run consuption of the doestic citizens. 9 The intuitions of these results are as follows. The presence of illegal igration has four e ects. The rst one is the exploitation e ect. As shown in Figure 2, when there is an increase in the nuber of illegal iigrants, a greater nuber of uneployed illegal iigrants are searching for jobs. In contrast, the change in the nuber of doestic workers searching for jobs is sall. This leads to a tighter labor arket which in turn leads to ore erce copetition for jobs. To successfully secure a job, both doestic and foreign labor would have to lower their wages. The rs therefore ake ore pro ts. In turn, doestic citizens receive ore dividends which can be used for consuption and investent. This e ect adds to doestic consuption. Second, the capital-using-up e ect. This is due to the fact that the illegal iigrants do not save in the doestic econoy. Soe capital has to be used to produce output for the consuption of illegal igrants. This e ect reduces current output which could have been used for doestic consuption and investent. Third, the displaceent e ect. As uneployed doestic labor and igrants copete for jobs, the chance for uneployed doestic workers to nd a job is reduced. This e ect lowers their consuption. Fourth, the wage depressing e ect of illegal iigrants. 20 As ore 9 As the nuber of illegal iigrants increases, the variable decreases and vice versa. 20 This wage depressing e ect of illegal iigrant workers has been docuented in Hotchkiss and Myria (2008). Borjas (2003) also concludes that a 0-percent increase in labor supply could reduce wages by 3-4 percent. 7

20 illegal iigrants enter into the econoy, the copetition for jobs becoes ore severe. Thus, the wages of doestic labor are pushed down. The net ipact of illegal iigration on doestic consuption hinges upon the agnitude of these four e ects. If the exploitation e ect doinates, doestic consuption will rise. Otherwise, it will fall. More precisely, the steady-state equilibriu value of the doestic consuption c is deterined by c = w s D 2 + ( g) k + : (4) To understand the intuition of this coparative static nding, we di erentiate (4) with respect to and obtain dc d dsd 2 = [w d positive {z } the displaceent e ect + s D 2 dw ] d positive {z } the wage depressing e ect + [ ( g)dk + ( {z d} ( g)k ) ] {z } positive negative {z } the capital-using-up e ect + [ d {z d} negative + ( ) ] {z } negative {z } the exploitation e ect (42) Equation (42) iplies that there are three results generated by an increase in the population share of illegal iigrants (a decrease in ) in the U.S. The rst result is that doestic labor incoe falls. This is due to the displaceent and negative wage depressing e ects, which are captured by the two ters in the rst square bracket of (42), respectively. The second square bracket re ects the positive exploitation e ect. The reason is that as ore illegal igrants are in the U.S., doestic households receive ore dividends which can be used for consuption and investent. The last square bracket shows how the doestic capital incoe is a ected by the in ows of illegal igrants. When the share of illegal iigrants goes up, the capital per worker k declines while the capital per doestic citizen k rises, which generates additional incoe for doestic households.2 Figure 2 suarizes the responses of the key variables in this odel (c ; x ; k ; w ; s D + s D 2 ; sm + s M 2 ; s D s D +sd 2 ; s M s M +s M 2 ) to a gradual increase in illegal iigration. 22 When it goes up, the odel predicts that workers and eployers face a tighter labor arket, i.e., x goes down, 2 In the steady state, r = : Therefore, the capital incoe is solely deterined by the quantity of capital. 22 For di erent values of { (e.g. { = 0:2; 0:7; and ); see corresponding Figure ; 3;and 4 in the appendix. 8

21 the capital per worker k reduces, the uneployent rate for doestic labor s D s D +sd 2 rises, the wage rate w for doestic labor drops, the fraction of doestic residents in search s D rises rst and then declines, and the labor force participation rate of doestic residents s D + s D 2 falls. Notice that coparing with doestic residents, we observe that there is only a sall reduction in the labor force participation rate for illegal iigrants s M + s M 2. The reason for this is that with ore illegal iigrants in search, doestic workers nd the opportunity cost of searching for jobs becoes higher so that it s optial to withdraw fro supplying labor and to enjoy leisure instead. 23 This result turns out to be consistent with the existing epirical evidence. Borjas et al. (2007) report that a 0-percent iigrant-induced increase in the supply of a particular skill group is associated with a reduction in the black eployent rate of 3:5 percentage points, and a :6 percentage point reduction in the eployent rate of white en. Our results are in sharp contrast with those obtained in previous studies. Analyzing the issue of illegal iigration under the full eployent assuption, Hazari and Sgro (2003) conclude that illegal iigration necessarily lowers the long-run per capita doestic consuption. Palivos (2009) obtains an unabiguous positive e ect of illegal iigration. It raises the consuption and welfare of doestic workers. Palivos (2009) also considers a case in which a binding iniu wage only applies to unskilled workers. His nding is that illegal iigration decreases doestic consuption. 3.4 Welfare E ects In order to answer the question of how illegal iigration a ects doestic welfare, we copute and copare, using a consuption-equivalent easure as in Lucas (987), the level of utility of doestic households under two scenarios. Let c(t; ); s D (t; ); sd 2 (t; ) denote the equilibriu tie paths when the population share of illegal iigrants is : The lifetie utility of the representative doestic household is given by U() = Z 0 flog c(t; ) (t; )]+ [sd (t; ) + sd 2 ge ( g)t dt: + 23 The gap between doestic worker s wage and reservation wage is narrowed with ore iigrants owing into the country. 9

22 The consuption-equivalent easure () is de ned by Z 0 flog[ + ()]c 0 (t; 0) [sd0 (t; 0) + sd0 2 (t; 0)]+ ge ( g)t dt = U() + log[ + ()] + U(0) = U() g If () > 0; then U(0) < U() which eans that the doestic households are better o in the presence of illegal iigrants. In particular, the doestic households would require a ()-percent increase in c 0 (t; 0) in every period so as to ake theselves indi erent between = 0 and > 0: Hence, illegal iigrants create a welfare gain to the host country s econoy. On the contrary, if () < 0; then U(0) > U(): The doestic household are now willing to surrender ()-percent of c 0 (t; 0) in every period so as to expel the illegal iigrants. This eans that illegal iigrants lead to a welfare loss to the host country. Suppose the econoy starts at the steady state with = 0: The two scenarios that we consider are as follows:. The econoy stays at the steady state with = 0 forever. 2. At t = 0, the host country adits > 0 fraction of illegal iigrants and the econoy gradually converges to the new steady state. Hence, U() is coputed based upon the transition path. To account for the transition path, the procedure described in Cooley and Ohanian (997) is carried out. Given a speci c nuerical value of, we can siply calculate the corresponding (): 24 Table 3 shows the welfare easure of illegal iigration. Three results can be drawn fro Table 3: First, it is instructive to note that illegal iigration induces iportant net gains aong doestic citizens for any values of (; {). For instance, when { = 0:4 and when there is an increase in fro zero to 5 percent in the US, the doestic households would require a :746-percent increase in c 0 (t; 0) in every period. Second, these gains increase in the share of illegal iigrants in the population for each xed { that we consider. Third, () clearly depends upon the agnitude 24 We further restrict our attention to the case in which can alter only fro 0 through 20%. This is due to the fact that in the traditional host countries, nearly 24:6% of the population in Australia, 22:5% in New Zealand, 8:9% in Canada, and 2:3% in the United States is foreign-born (United Nations 2004). Aong the foreign-born, only a fraction of the are illegal iigrants. 20

23 of the labor supply elasticity. In particular, for each xed > 0, () decreases with the labor supply elasticity {: Table 3: Welfare easure of illegal iigration a { = 0:2 { = 0:4 { = 0:7 { = () () () () 5% 0:756% 0:746% 0:73% 0:727% 0% 2:02% :978% :833% :78% 20% 6:684% 6:34% 5:98% 5:75% b a Values of other paraeters reain the sae as in Table. In order to shed soe light on the above coputational results, we di erentiate the doestic household s utility with respect to and obtain the following expression (43). As we require the econoy to ove fro zero to an arbitrary aount of illegal iigration, we evaluate (43) at = 0. Equation (43) reveals that the e ect of illegal iigration on doestic welfare depends on two factors: () the change in the level of per capita consuption of doestic citizens, and (2) the change in the doestic labor participation rate. du[c(t); s D (t) + sd 2 (t)] d = u c (t) dc(t) d j =0 + u s D +s D(t)d[sD (t) + sd 2 (t)] j 2 =0 : (43) d These two factors jointly deterine the welfare e ect of illegal iigration. In general, it s not possible to obtain de nite results analytically. We thus resort to nuerical exercises. We focus on one particular exaple and exaine the transition paths of consuption and leisure. The exaple that we consider here is when { = 0:4 and when there is an increase in fro zero to 5 percent. Under the baseline paraeterization, illegal iigration lowers doestic consuption level throughout the entire transition. It rst reduces and then raises the labor force participation rate during the transition (see Figure 5). By (43), we know that the overall welfare e ect is abiguous as these two changes tend to ove doestic welfare in opposite directions. Nevertheless, 2

24 according to our siulation, the positive welfare e ect doinates. Thus, illegal iigration induces a welfare gain to the host country s econoy. This welfare gain coes fro an increase in leisure. 4 Concluding Rearks This paper contributes to the existing literature on welfare e ect of illegal iigration on doestic workers by introducing illegal iigration into a standard dynaic general equilibriu fraework with labor arket frictions. We therefore construct and calibrate a search-theoretic odel. In the odel econoy, illegal iigrants enter doestic production as perfect substitutes for doestic workers. They are allowed to spend their one indivisible unit of tie in searching for a job, working for a r, or enjoying leisure in each period. Firs hire both doestic and illegal iigrant workers. Once uneployed doestic workers and vacant jobs are paired with each other, they jointly deterine the wage rate through bilateral Nash bargaining. As we assue that rs are able to distinguish illegal iigrants fro doestic workers and face a punishent for hiring the forer if being caught. The wage rate for illegal iigrants is thus equated to the wage rate of doestic workers inus the expected value of the punishent. We characterize the search equilibriu and prove the existence and uniqueness of stationary equilibriu. In contrast to the previous studies, our analysis reveals three striking results. First, although illegal iigration is indeed a boon to the United States, it signi cantly hars the eployent opportunities of doestic workers. Naely, it increases the uneployent rate for doestic workers. Furtherore, it forces the to face a tighter labor arket and even to leave the labor force. Second, we quantitatively prove that the long-run level of consuption of doestic citizens has a U-shaped relationship with the share of illegal iigrants. Third, illegal iigration s negative ipact on native wages has been found in this fraework. This result turns out to be qualitatively consistent with the epirical evidence. To close the paper, we like to point out one line of future research. In this study, we assue that doestic workers and illegal iigrants are perfect substitutes. However, epirical evidence docuents that even with the sae level of education, they are not perfect substitutes. 25 Therefore, the analysis will becoe ore interesting if illegal iigrants can be odeled as a separate factor 25 For a related dicussion, see, aong others, Borjas (2003) and Card and Leieux (200). 22

25 of production. Moreover, in real life, the debate over illegal iigration has also concerned with its distributional e ects. Assuing that doestic and foreign labor di er in ters of their production skills, the distributional ipact of illegal iigration on doestic workers can be analyzed in a search-theoretic fraework. Nevertheless, this extension would not be trivial. We have to consider a two-sector version of the search odel. This could signi cantly increase the diension of the dynaic syste. 23

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