SIMON FRASER UNIVERSITY. Department of Economics

SMO FRASER UVERSTY Department o Economics Discssion Paper Series Discssion Paper #00-19 A political economy model o immigration qotas J. Ats Amegashie September, 2000 OU S SOM ME P S R E S T Mailing Address: 8888 University Drive, Brnaby, B.C., V5A 1S6, Canada http://www.s.ca/economics

A political economy model o immigration qotas* J. Ats Amegashie Department o Economics Simon Fraser University Brnaby, British Colmbia Canada V5A 1S6 September 19, 2000 E-mail: jamegash@s.ca Abstract The paper examines a model in which the nmber o immigrants allowed into a contry is the otcome o a costly political lobbying process between a irm and a nion. The nion and the irm bargain over the wage o natives ater the nmber o immigrants that will be permitted is known. consider two contest sccess nctions: one in which the lobbyist with the higher eort is not necessarily the winner and another in which the lobbyist with the higher eort wins with certainty (i.e., the all-pay action). Comparative statics reslts are derived to show how the reservation wage o immigrants, the size o the nion, the sensitivity o the legislatre to lobbying, the reservation wage o natives, the price o the irm s prodct and the irm s bargaining power aect immigration qotas and the post-immigration wage o natives. also discss some limitations o my reslts. Keywords: all-pay action, immigration, lobbying, mixed strategy, ash bargaining. JEL Classiication: D72, D73, J5, J61. *This paper is a chapter o my on-going Phd thesis at Simon Fraser University. My thanks are de to my spervisors, Dog Allen and Gordon Myers, or their perceptive comments which have signiicantly improved this paper. An earlier version o this paper was circlated as RM working paper #00-01, Simon Fraser University, Canada. My thanks are de to Don Devoretz or comments. The sal disclaimer applies.

1 1. ntrodction There are several actors which inlence the nmber o immigrants allowed into a contry. Some o these inclde excess demand or labor, competing opportnities in other contries [Devoretz and Maki (1983)], hmanitarian reasons, and amily re-niication as is the case in Canada s immigration policy [see also Akbar and Devoretz (1993) and Borjas (1994)]. t is widely recognized that a political lobbying process also inlences the nmber o immigrants into a contry [see, or example, Benhabib (1996), Borjas (1994), Devoretz (1996), Goldin (1994), Green and Green (1996)]. For example Goldin (1994, p. 223) writes: Becase the story o immigration restriction is a legislative one, its main players will be representatives, senators, and presidents. Bt behind the legislative tale are the shiting interests o varios grops. The irst is organized labor and norganized labor. Owners o capital are the second and immigrants both new and old are the third. The interests o these grops will dier becase immigration has dierent eects on them. For example, immigration may depress the wages o nskilled workers reslting in an increase in the share o income accring to skilled workers and capital owners. Borjas (1995) calls this gain the immigration srpls. ndeed, some empirical papers have shown that immigration cold depress the wages o the native born [see, or example, Borjas (1994) and the reerences cited therein]. However, in theory, this depends on one s assmptions abot the hman capital content o immigrants and whether they are sbstittes or complements to native labor. To the best o my knowledge, am not aware o any work which ormally shows how lobbying may inlence the nmber o immigrants. Benhabib (1996) ses a median-voter

2 model to determine the qality o immigrants bt not the nmber o immigrants allowed into a contry. 1 He sggests that examining the latter isse will be an interesting research endeavor. ndeed, on a more general note, Borjas (1994, p. 1693) observes: rther research on the political economy o immigration might greatly improve or nderstanding o the properties o the eqilibrim in the immigration market. n this paper, model the lobbying contest and examine how the reservation wage o immigrants, the size o the nion, the sensitivity o the legislatre to lobbying, the price o the irm s prodct and irm s bargaining power aect the post-immigration wage o natives and the permissible nmber o immigrants o homogeneos qality. By permissible nmber o immigrants, simply mean the nmber o immigrants that wold be allowed into a contry (i.e., the immigration qota). One may think o this as the annal target level o immigration in Canada, which is tabled by the minister nder the 1976 mmigration Act or as the annal qota o H1B visas issed in the USA or the qota nder the US visa lottery scheme. 2 imagine native or indigenos workers as being represented by a nion. ote, however, that this jst or the sake o exposition. What matters is that there is a lobbying grop or native workers. The paper is organized as ollows: in the next section, present a lobbying contest based on the model in Amegashie (1999a, 2000b). postlate a relationship between the post-immigration wage o natives and the nmber o immigrants. then model the lobbying contest nder two alternative assmptions: in one model the contestant who 1 See also Hillman and Weiss (1999) who se a median-voter model to examine permissible illegal immigration. This paper ocses on legal immigrants. 2 For recent analyses o immigration qotas, see Myers and Papageorgio (2000a, 2000b).

3 exerts the higher lobbying eort is not necessarily the winner. n the other model, the contestant with the higher eort is the winner. n section 3, prse similar analysis bt derive the relationship between the post-immigration wage o immigrants and the nmber o immigrants rom a bargaining model. Section 4 concldes the paper. 2. The immigration lobbying contest As discssed above, there are actors other than lobbying which determine the nmber o permissible immigrants. However, some o these actors may well be related to lobbying. For example, even i the nmber o immigrants is partly determined by excess demand in the labor market as in Devoretz and Maki (1983), this inormation cold be spplied by irms as part o their lobbying eorts. Hence it is not nreasonable to assme that immigration qotas are the otcome o a political lobbying process. Let the crrent wage be w. All the nionized native workers will receive w in the absence o immigrants. Let w < w be the post-immigration wage o the natives. Let the nmber o native workers be ixed at L. Sppose that the irm has the prodction nction, (L) = L σ, where L = L + L is the total nmber o workers and 0 < σ 1. also assme that all immigrants are paid w (< w ), their reservation wage. 2.1 mperectly discriminating contest (the lottery model) Let y and x be the irm s and nion s lobbying expenditres respectively. The expenditres o both lobbyists are snk, whether they win or lose the contest. Following Tllock (1980), let the probability that the irm wins the lobbying contest be

4 θ(x, y) = τ y τ τ x + y, where τ > 0 is a measre o the sensitivity o the politician or legislative body to lobbying expenditres. 3 The nion wins the contest with probability (1- θ). A lobbyist wins the contest i its preerred nmber o immigrants is approved by the legislative body. ote that or any inite τ, the lobbyist with higher expenditre does not necessarily win the contest. This may be the case or two reasons: (i) i lobbying expenditres are not in terms o money, then the legislative body may not be able to perectly discriminate between the expenditres o the lobbyists, and (ii) there may be other actors, apart rom lobbying expenditres, which determine the otcome o the contest [see Clark and Riis (1996) or a ormal proo; or an axiomization o this contest sccess nction, see Skaperdas (1996) and Kooreman and Schoonbeek (1997)]. 4 lobbying expenditres are in terms o money (e.g., campaign contribtions), then this contest sccess nction cold be seen as a redced orm o the objective nction in a lobbying model like in Grossman and Helpman (1994). n their model, the government or politician cares abot campaign contribtions bt also cares abot other actors (e.g., the welare o the lobbyists and other agents in the economy). n the next section consider the all-pay action, where the lobbyist with the higher expenditre wins the contest with certainty. 5 Baye et al. (1993) and Che and Gale (1998) arge that a politician might 3 See Amegashie (1998, 1999a, 1999b), Glazer and Konrad (1999), Konrad (1999), Ellingsen (1991), and Rama (1997) or extensions o Tllock s model. For an interesting srvey, see itzan (1994). 4 ote that there is an aspect o the sccess probability in this contest that do not model. the legislative body determines the winner throgh majority voting, then it not only the expenditre o a contestant which matters bt also how this expenditre is allocated among the members o the legislatre. Sch a game is modeled in Congleton (1984) in his analyses o the eect o committees on rent-seeking eort. To simpliy the game, ignore this problem by assming that the otcome o the contest is determined by only single politician. This assmption is implicit in other lobbying models (e.g., Grossman and Helpman (1994)). 5 Baye et al. (1993) Che and Gale (1998), Glazer and Konrad (1999) and Konrad (2000) have sed the allpay action to examine political corrption, caps on campaign contribtions, taxation, and strategic trade policy.

5 choose this contest sccess nction i his objective is to maximize his campaign contribtions or his income rom bribery. Assme that i the irm wins the contest, then the immigration qota is given by L = g(y), where g (y) > 0. For simplicity, assme that g(y) = by, with b > 0. Assme that the irm sells its inal prodct at a constant price, p > w. 6 se an imperectly competitive model o both the prodct and labor markets. t wold seem natral to se a perectly competitive model to examine this isse since that prodces a conlict between labor and the owners o capital with respect to their preerences or immigration. do not se a perectly competitive model becase it gives a corner soltion or the immigration qota chosen by the irm. This does not make it possible to obtain interesting comparative statics reslts (see ootnote 21). ote that the model here is similar to Amegashie (1999a, 2000b), except that the contestants are non-identical. 7 The irm s lobbying expenditre aects the size o the rent it gets rom employing immigrants and indeed, L, is an increasing nction o its lobbying eort. 6 assme that the irm is a monopolist, so have to explain why it sells at a constant price. make this assmption to simpliy the analysis. However, one cold jstiy it as ollows. Sppose that given the monopolist s crrent otpt and zero immigration, it has excess demand at its crrent price. However, it may not increase its price i individal demand is sbject to a positive consmption externality (i.e., bandwagon eect) where the demand o a consmer is increasing in the demand o other consmers (see Becker, 1991). Hence the monopolist can increase spply and redce the excess demand gap withot changing its price. ndeed, a positive consmption externality is not reqired or this reslt to go throgh. Consider a drable good or which consmers by only a nit at a time. Assme that at the crrent qantity, there is excess demand sch that the consmers who are withot the good are willing to pay a maximm price eqal to the monopolist s crrent price. Then the monopolist can sell to these consmers withot changing its price. 7 See the exchange between Amegashie (2000a) and Clark (2000) on this model. ote, however, that Clark s critiqe is not relevant here becase am not interested in the relationship between aggregate lobbying expenditres and the nmber o lobbyists.

6 n section 3, present a model (hereater model 2) with bargaining and lobbying. Unlike the model (hereater model 1) in this section, the nmber o immigrants in model 2 is independent o y. n model 2, ind that the soltion in the lobbying sb-game gives a positive relationship between the irm s lobbying eort and its preerred nmber o immigrants, or example, y = y( L ). t is important to note that model 2 is ndamentally dierent rom model 1. n model 2, y is increasing in L de to strategic considerations. n this model (i.e., model 2), i the nion does not lobby, then the legislative body or politician will approve the irm s proposal o L > 0 immigrants even i the irm does not lobby. ndeed, in model 2, the irm s lobbying eort determines the probability that it wins the contest bt not the nmber o immigrants. n model 1, the irm cannot get L > 0 withot lobbying, even i the nion does not lobby. Withot any lobbying, L = 0. So nlike model 2, the positive relationship between y and L is not de to strategic eects. t is based on the assmption that the irm s lobbying eort determines the immigration qota, i it wins the contest. That is, the irm knows that i it wants L immigrants, it has to spend y = ( L /b) dollars in lobbying and it will be sccessl with probability, θ(x, y). The nion lobbies or zero immigration. Ths the irm s lobbying eort determines two things: (i) the nmber o immigrants, and (ii) the probability that it will be sccessl. consider both models becase think each captres some aspect o the real world. The dierent models also enable me to check the robstness o my reslts. Finally, a technical problem acconts or the switch rom model 1 to model 2. Assming that σ =1, 8 the irm s expected proit cold be written as 8 consider the case o 0 < σ < 1 in section 3.

7 π = θ[p( L + L ) w L - w which cold be simpliied, noting that L )] + (1 - θ)[ p L w L = by, to obtain L ] y, π = θ[(p- w )by + ( w - w ) L ] + (p w ) L y. (1) Assming that the nion seeks to maximize the total wage o its members, its payo can be written as π = (1 - θ)w L + θ w which can be rewritten as π = (1- θ)[( w - w ) L x, L ] + w L x. (2) ote that V (w - w ) L and V (p- w )by + (w - w ) L are the nion s and irm s valations (respectively) o winning the contest. The irm has a higher valation than the nion, i y > 0. ndeed, this is the case in eqilibrim. 9 now model how the nmber o immigrants aects the post-immigration wage o natives. postlate that the higher the nmber o immigrants, the lower is the postimmigration wage o natives. That is w = w ( L ), where w is decreasing inl. α Speciically, assme that w = w - β ( L ) > w, 10 where 0 < α < 1 and β > 0. When L = 0, then w = w. have in mind a bargaining process where the nion and the irm bargain over the wage bt not over employment (i.e., the crrent employment level o 9 This is becase x = 0, y = 0, cannot be an eqilibrim, since then each contestant has the incentive to spend a small bt positive amont on lobbying and win the contest with certainty. O corse, this argment does not apply to the eqilibrim with x = 0 and y > 0. 10 n the nmerical soltion to the game, choose the parameters carelly to satisy this restriction. ote that i natives were paid w, then that cannot be consistent with a lobbying eqilibrim. The natives can still obtain the wage o w withot lobbying. One might arge that it is against the law to discriminate against immigrants by paying them less than natives. cold jstiy this by arging the natives are initially more prodctive than immigrants. This can be incorporated into the model, by writing the prodction nction as (L) = ( L + λl ) σ, where 0 < λ < 1. This speciication will not aect my reslts.

8 natives remains nchanged). 11 The possibility o being able to lobby or immigrants pts the irm in a stronger bargaining position than wold be the case i this possibility did not exist. That is, the higher is the nmber o immigrants, the higher is the irm s otside option (i.e., the irm s payo when there is no agreement between the nion and the irm). t is the otcome o this bargaining process that is captred in the relationship, α w = w - β ( L ). n section 3, derive a similar relationship by assming that the postimmigration wage is the otcome o a ash bargaining between the irm and the nion. Ptting w = w - β ) α ( L into (1) and (2) and noting that L = by, we get π = θ[(p- w )by + ϕ L y α ] + (p w ) L y. (1a) and π = (1- θ)[ϕ L y α ] + ( w - ϕy α ) L x, (2a) where ϕ βb α. Assming that τ = 1, 12 the optimal vales o x and y mst satisy the pair o Khn-Tcker conditions below in a Cornot-ash eqilibrim: π y and 1 y = V 2 x + y (x + y) + y x + [bˆ y + αϕl y α 1 π ] 1 0, y 0, y = 0 y (3) 11 Since the nion wants to maximize its total wage receipts, one cold instead assme that the wage o natives is held constant bt the nion and the irm bargain over employment. So long as immigration redces the employment o natives or redces their wage, whether the nion and irm bargain over only the wage or employment will not aect the analysis. they bargain simltaneosly over the wage and employment, that will complicate the model withot adding any additional insight. Besides, it is debatable whether irms and nions bargain over wages and employment simltaneosly (see, or example, Booth, 1995, p. 128). 12 This assmption is made or simplicity.

9 π x 1 x π = V 1 0 2 x y (x y), x 0, x = 0 + + x (4) where V ( w - w ) L, V (p- w )by + ( w - w ) L and bˆ (p- w )b. ow x* = 0 and y* = [ 1/(1 α) αϕ L /(1 > 0 satisy the conditions in (3) and (4), bˆ )] i bˆ + α > 1. To see, note that when x = 0 and y > 0, then (3) becomes bˆ + αϕl y α-1 1 = 0, which gives y* = [ αϕ L /(1 bˆ )] 1/(1 α). t remains to show that given y*, π / x < 0 at x = 0, i bˆ + α > 1. ow at x = 0, π / x = βb α y α L /y 1. Given y = y*, this derivative is negative, i bˆ + α > 1. Hence x* = 0 and y* = [ αϕ L /(1 bˆ )] 1/(1 α) The immigration qota is > 0 is a ash eqilibrim. * L = by* = b[ αϕl /(1 bˆ )] 1/(1 α) (5) ow we want to perorm some comparative statics on the immigration qota. However, it may not be very interesting to do so given that the soltion or one o the lobbyists is zero. n what ollows, look or an interior soltion (i.e., x > 0, and y > 0), given bˆ + α < 1 and compare its comparative statics properties with those o the soltion given by eqation (5). ow at an interior soltion, the ollowing pair o eqations mst be satisied: π y 1 y = V 2 x + y (x + y) y + x + [bˆ y + αϕl y α 1 ] 1 = 0 (3a)

10 and π x 1 x = V 2 x + y (x + y) 1 = 0 (4a) Eqations (3a) and (4a) cold be simpliied to obtain x(v + bˆ y) + y(x + y) [bˆ + αϕ L ] = (x + y) 2 (3b) y α 1 and yv = (x + y) 2 (4b) where in (3b), have made se o the act that V = V + bˆ y. now need to solve (3b) and (4b) simltaneosly or x and y. proceed as ollows: Let T x + y be aggregate lobbying expenditres. Let {x**, y**} be the optimal soltion. Then T** = x** + y**. Having ond x** and y**, one can write y** = γt** and x** = (1-γ)T**, where 0 < γ < (1+α) 1/(1 α) 1. Ptting these expressions into (4b) and simpliying gives T** = [ ϕl γ ]. oting that y** = γt**, x** = (1-γ)T** and T** = x** + y** = [ (1+α) 1/(1 α) ϕ L γ ], pt these expressions into (3b) to get (1-γ)[bˆ (γt**) + ϕ L (γt**) α ] + γt**[ bˆ + which can be simpliied rther to obtain α 1 αϕ L ( γt * ] - T** = 0 *) bˆ γ + ϕl γ α (T**) α-1 [1 - γ + α] - 1 = 0 (6) oting that (T**) α-1 (1+α) = [ ϕ L γ ] 1, eqation (6) can be written as bˆ γ 2-2γ + (1 + α) = 0 (6a)

11 ote that γ can be solved rom eqation (6a). Once its vale is obtained, it ollows that x** and y** have been obtained. The soltion to (6a) is given by 1± γ* = 1 bˆ(1 + α) bˆ For a real root, we reqire bˆ (1 + α) 1. For an interior soltion, we reqire 0 < γ* < 1. The larger root does not satisy this condition since it reqires bˆ + α > 1 or γ* < 1. Bt as shown above γ* = 1 i bˆ + α > 1. The smaller root satisies γ* < 1 i bˆ + α < 1. Hence the soltion is 1 γ* = 1 bˆ(1 + α) bˆ (6b) ote that γ* > 0.5. This means that the irm has a higher probability o winning the contest. This is de to the act that the irm has a higher valation than the nion (i.e., V > V ) in eqilibrim, since y** > 0. 13 check that the second-order conditions or a local maximm hold. 14 Since γ θ, the expected immigration qota is e L = γ*(by**) = b(γ*) 2 T** = b(ϕ L ) 1/(1-α) (γ*) (3-α)/(1-α) (7) From (6b), ( γ*/ bˆ ) > 0. oting that bˆ (p- w )b, it ollows that ( γ*/ w ) < 0 and ( γ*/ p) > 0. Thereore L e / w = b(ϕ L ) 1/(1-α) [(3-α)/(1-α)] (γ*) (3-α)/(1-α)-1 ( γ*/ w ) < 0. This gives the ollowing proposition: 13 ote that the irm will still have a higher valation than the nion even i L is independent o y. Any positive nmber o immigrants chosen by the irm will give this reslt, given that σ = 1. 14 t is easy to conirm this or the nion. For the irm, the condition reqired is 2(1-γ)( bˆ γ 2 (1-γ) + α - γ) - α(1 - α) < 0.

12 Proposition 1: The lower is the reservation wage o immigrants, the higher is the expected immigration qota. The intition behind proposition 1 is simple. A all in w increases the valation o the irm withot changing the valation o the nion. This increases the irm s lobbying eort, its probability o sccess and hence the expected immigration qota. This proposition and all the others which ollow is conirmed by the closed- orm soltion given by eqation (5). Sppose there is an increase the price o the irm s prodct. Then L e p γ = k p where k b[(3-α)/(1-α)](ϕ L ) 1/(1-α) (γ*) (3-α)/(1-α)-1. Since γ*/ p > 0, we get the ollowing proposition. Proposition 2: An increase in the price o the irm s inal prodct reslts in an increase in the expected immigration qota. The intition or this proposition is similar to that or proposition 1. Since my model is a partial eqilibrim model, proposition 2 (and probably all my propositions) will only make sense i we consider the immigration qota or a particlar indstry. Proposition 2 accords very well with intition or it implies that the prospect o bigger proits will indce irms to lobby more or immigrants and increase the immigration qota. am

13 inclined to believe that this is one o the most robst predictions o the model or it sggests that immigration qotas will increase when the economy is booming and will decrease when the economy is slowing down. ndeed, Shghart et al. (1986) ind that immigration restrictions are tighter dring recessions and looser dring expansions. ote that we can rewrite the prodction nction as (L) = AL σ, where A > 0 represents the state o technology. Then A is analytically eqivalent to p. Ths the ollowing proposition ollows. Proposition 2a: The more eicient is the irm s technology, the higher is the expected immigration qota. n what ollows, revert to the assmption that A = 1. Since w =w - ϕ(γ*t**) α, we obtain w / w = -ϕ{ [(γ*t**) α ]/ γ*}( γ*/ w ). Given that given T** is increasing in γ* and ( γ*/ w ) < 0, it ollows that w / w > 0. This gives the ollowing proposition: Proposition 3: The lower is the reservation wage o immigrants, the lower is the expected post-immigration wage o natives. ow L e / L = k( γ*/ L ) + {bϕ 1/(1-α) ( L ) 1/(1-α)-1 (γ*) (3-α)/(1-α) /(1-α)}. Since ( γ*/ L ) = 0, it ollows that L e / L > 0. This gives the ollowing proposition:

14 Proposition 4: An increase in the size o the nion leads to an increase in the expected immigration qota. An increase in the size o the nion increases the valation o both the nion and the irm. Since there is no change in γ*, it mst be the case that x** and y** increase by the same proportion. However, the increase in y** means that the immigration qota increases. ote that an increase in the size o the nion increases the irm s valation becase the all in the total wage bill or natives as a reslt o immigration is bigger. 2.2 The all-pay action (perectly discriminating contest) n the section, model the contest nder the alternative assmption that the lobbyist with the higher eort wins with certainty. This corresponds to the case where τ is ininitely large. That is, the probability that the irm wins is now 1 θˆ(x, y) = 1/ 2 0 i i i y > x y = x y < x shall draw on reslts in the all-pay action literatre by Hillman and Riley (1989) Ellingsen (1991) and Baye, Kovenock and de Vries (1993, 1996) to solve the game. However, their reslts have to be applied to the model in this paper with cation. This is becase while the literatre on all-pay actions ocses on contestants with exogenos valations, my model has contestants whose valations depend on lobbying expenditres. 15 15 To the best o my knowledge, the only papers which examine all-pay actions with endogenos valations are Amegashie (2000b) and Konrad (2000). However, in Konrad (2000) the valations do not depend on the expenditres o the contestants nor on any choice variable by the contestants.

15 t is a standard reslt that there is no eqilibrim in pre strategies in all-pay actions. To see this, consider the standard case where V and V are ixed (i.e., do not depend on y). Sppose the nion bids 0 < x V. Then the irm s optimal response is y = x + ε < V (i.e., marginally higher than x). Bt then x > 0 cannot be an optimal response to y = x + ε. Also it is obvios that x = y = 0 cannot be an eqilibrim. Hence, there is no eqilibrim in pre strategies. However, in this model, there may be an eqilibrim in pre strategies. To see this, note that when the nion bids zero, the irm will choose y to maximize σ ˆπ = [ p(l + L) wl wl ] y. Sppose σ = 1, then the optimal soltion y* = [ αϕ L /(1 bˆ )] 1/(1 α), which is the soltion obtain in lottery model with bˆ + α > 1. Let V = V * given that y = y* and assme that y* V *. Then when the irm bids y*, the nion s optimal reply is zero and the irm wins the contest with certainty. This is an eqilibrim becase the irm has no incentive to bid less than y*, given that the nion bids zero. Ths there is an eqilibrim in pre strategies, i y* holds i bˆ + α 1. Hence the expected immigration qota is V *. 16 This condition e L t L * = b y*= b [ 1/(1 α) αϕ L /(1. bˆ )] Then the propositions derived above immediately ollow. 16 See Amegashie (2000b) or a more detailed analysis o an all-pay action with a pre-strategy eqilibrim.

16 The dierence between the expected immigration qotas nder the all-pay action and the imperectly discriminating contest, when bˆ + α > 1, is e * L t L = 0 Ths we get the ollowing proposition: Proposition 5: The sensitivity o the legislatre to lobbying may have no eect on the expected immigration qota. do not have an intition or this proposition. However, it has the ollowing implication. Sppose, as in Che and Gale (1997), we interpret the sensitivity parameter (o the legislatre), τ, as a relection o a contry s political cltre. Proposition 5 implies that, a priori, one shold not necessarily expect any dierence between the immigration lows o two democratic contries which are similar in several respects except that they dier in their tolerance or the sale o political avors. From inspection o eqation 7, it is easy to see that the expected immigration qota is increasing in ϕ. Given that ϕ βb α, it ollows that the expected immigration qota is increasing in β. oting that w = w - β ) α ( L, it is obvios that or a given nmber o immigrants, the all in the post-immigration wage o natives is higher, the higher is β. This leads s to the ollowing proposition. Proposition 6: The bigger is the negative eect o immigration on the wage o natives, the higher is the expected immigration qota.

17 This reslt seems conter-intitive. However, it is de to the act that the bigger is the negative eect o immigration on the wage o natives, the higher is the valation o the irm reslting in a siciently big increase in its lobbying eort. t is important to note that all my propositions crcially depend on the extent to which immigration is inlenced by lobbying. The analysis so ar has been based on the assmption that y* V *. This made it possible to ind an eqilibrim in pre strategies. Sppose instead that y* < V * (i.e., bˆ + α < 1). Then there is no eqilibrim in pre strategies. To see, note that i y* < V *, bidding y* cannot be an eqilibrim or the irm since the nion can marginally bid higher than y* and win the contest with certainty. Sppose the irm chooses ŷ y*, sch that ŷ Vˆ, where Vˆ is the vale o V when y = ŷ. Then i the irm bids ŷ, the nion will bid zero. Bt then given that nion bids zero, the irm shold bid y* instead o ŷ. Thereore, there is no eqilibrim in pre strategies. There may be an eqilibrim in mixed strategies, when y* < V *. However, am nable to characterize the eqilibrim. The soltion (or the lack thereo) is the sbject o tre research. 17 n the next section, ind a mixed-strategy eqilibrim or the case where L (and or that matter the valations) are independent o y. 17 The problem arises becase i a mixed-strategy eqilibrim exists, it mst be the case that the nion does not know the actal bid o the irm, bt only its expected vale. Since the nion s valation depends on the irm s bid, it ollows that the nion does not know its (own) actal valation bt only its expected vale. t is, however, diiclt to ind a reasonable expectation o the nion s valation that gives valid eqilibrim cmlative distribtion nctions. A detailed explanation is available on reqest.

18 3. The immigration contest and ash bargaining n section 2, postlated that the post-immigration wage is inversely related to the nmber o immigrants. n this section derive a similar relationship sing the ash bargaining soltion. As beore, assme that the nion and the irm bargain over the wage holding the employment level o native workers constant. ote that bargaining over the wage takes place ater the otcome o the lobbying game is known. solve the game backwards. irst solve the bargaining sb-game and then solve the lobbying sb-game. Let w ~ be the wage o the native workers when there is no agreement between the irm and the nion. Call this the reservation wage o natives. n the event o no agreement, the irm will only employ immigrants. 18 Hence the ash bargaining prodct may be written as w ~ σ σ Ω = ( wl L )[p(l + L) wl wl (pl wl)], (8) where 0 < σ < 1 and L 0. The maximand in (8) can be rewritten as w ~ σ σ Ω = log( wl L ) + log[ p(l + L ) wl wl (pl wl )] (8a) Maximizing (8a) with respect to w gives p(l w* = σ + L) pl 2L σ + w ~ L (9) t ollows rom (9) that w */ L < 0, given that 0 < σ < 1. ote that (9) shows that w * may not be a nction o only L bt may also be a nction o p, L, and w ~. This implies that it may not be correct to treat β and w (in w = w - β ) α ( L ) as constants in 18 The analysis still goes throgh i the irm employs only a raction o the native labor orce in the event o a disagreement.

19 the comparative statics exercises involving p and L in section 2. t trns ot, however, that this new speciication does not necessarily aect my reslts. (i.e., L = 0), is From (9), w *, the wage o the natives when the nion wins the lobbying game σ p(l ) + w ~ L w* = 2L (10) solve the lobbying sb-game by assming that the lobbyist with the higher eort wins with certainty (i.e., all-pay action). The contestants valations are V = [p(l + L ) σ wl w * L (pl σ w * L )] > 0 and V = (w * L w *L ) > 0. However, modiy the model since as noted above am nable to ind an eqilibrim in mixed-strategies when the contestants valations are a nction o expenditres. The main modiication is that L is no longer a nction o y. The irm chooses L and given the L chosen, both contestants choose their lobbying expenditres, y and x. Since L is known and independent o y at the lobbying stage, it ollows that the contestants valations are independent o y. Hence there is no eqilibrim in pre strategies in the lobbying sb-game. look or a mixed-strategy eqilibrim. ote that the contestants valations are still endogenos since they depend on L. Sppose that V > V σ σ. This condition holds i p[(l + L ) L ] wl 0. > Unlike beore, V is not necessarily greater than V, given 0 < σ < 1. To ensre that V > V assme that w is siciently small and/or p is siciently high and/or σ siciently close to 1. Sppose the irm lobbies or Lˆ immigrants. Let Vˆ and the nion and the irm (respectively) when L = Lˆ and assme that Vˆ be the valations o Vˆ < Vˆ.

20 A crcial point is that, in eqilibrim, no lobbyist will bid a positive amont with positive probability. This means that eqilibrim strategies are continos mixed strategies. To see this, sppose (withot any loss o generality) that the nion bids x = ~ x > 0 with positive probability. Then the probability that the irm wins rises discontinosly as a nction o x at x = ~ x. Hence there is some ε > 0 sch that the irm will bid on the interval [ ~ x - ε, ~ x ] with zero probability. Bt then the nion is better o spending ~ x - ε instead o ~ x since its probability o winning is the same. This contradicts the hypothesis that x = ~ x is an eqilibrim strategy. Hence apart rom the bid o zero which may occr with positive probability, no other bid will be played with positive probability [see Hillman and Riley, 1989]. Given that eqilibrim strategies are continos mixed strategies, it ollows that the probability o a tie is zero. Thereore when the irm bids Vˆ, it mst win with probability 1 in a mixed-strategy eqilibrim. So the pper bond o the spport o its eqilibrim mixed strategy is Vˆ. Similarly, the pper bond o the spport o the nion s eqilibrim mixed strategy is Vˆ. The lower bond o the spport o the irm s eqilibrim mixed strategy is zero. To see this, sppose to the contrary that, in eqilibrim, the irm bids less than y with zero probability, where 0 < y< Vˆ. Given this strategy by the irm, it is never optimal or the nion to bid any x sch that 0 < x < y. Bt then the irm cold redce its bid below y withot changing its probability o winning, contradicting the hypothesis that, in

21 eqilibrim, the irm cannot do better than bidding a minimm o y. 19 Hence the lower bond o the spport o the irm s eqilibrim mixed strategy is zero. t ollows that the spport o the irm s mixed strategy eqilibrim is [0, Vˆ ]. A similar reasoning establishes that the nion has the same eqilibrim spport [see Hillman and Riley, 1989]. As arged above the irm wins the contest with probability 1 i it bids y = Vˆ in a mixed-strategy eqilibrim. This gives a payo o Vˆ - Vˆ > 0. Since every bid in the spport o the irm s eqilibrim mixed-strategy mst yield the same expected payo, it ollows that the irm s expected payo in a mixed-strategy eqilibrim is Vˆ - Vˆ. A similar argment also establishes that the nion s expected payo, in a mixed-strategy eqilibrim, is zero. Let G (y) and G (x) be the cmlative density nctions (c.d. s) associated with the players mixed strategies. Then in the mixed-strategy eqilibrim, 20 the ollowing eqations mst hold: G (y) Vˆ y = Vˆ Vˆ (11) and G (x) Vˆ x = 0 (12) Solving eqations (11) and (12) gives the eqilibrim c.d. s, which cold be written as 19 ote that the nion never bids less than y and the probability o a tie is zero. Then it ollows that when the irm bids y, it expects the nion to bid greater than y, which means it (i.e., the irm) loses with certainty and its payo is negative. Thereore the irm is better o bidding zero instead o y > 0. 20 n the all-pay action, the mixed-strategy eqilibrim is niqe when there are only two contestants [see Baye, Kovenock and devries (1993, 1996) and Hillman and Riley (1989)].

22 y G (y) = or y [0, Vˆ and Vˆ ] Vˆ x G (x) = 1 + or x [0, Vˆ Vˆ Vˆ ]. The eqilibrim c.d. s show that the irm bids niormly on [0, Vˆ ], while the nion pts a probability mass eqal to (1 Vˆ / Vˆ ) on x = 0. The expected lobbying expenditres are Vˆ Vˆ 2 Vˆ (Vˆ ) Ey* = ydg (y) = and Ex* = xdg (x). 2 = 2Vˆ 0 0 The probability that the irm wins the contest is eqal to the probability that it bids more than the nion s expected bid. This eqals [1 G (Ex*)] = [1 Ex*/ Vˆ ] = (1 - Vˆ /2Vˆ ). Then the probability that the nion wins is Vˆ /2Vˆ. ote that there is nothing novel abot my soltion. Given Vˆ and Vˆ, have simply ollowed the soltion method in Hillman and Riley (1989), Ellingsen (1991) and Baye et al. (1996). now determine the irm s choice o the nmber o immigrants. Since the irm s payo in a mixed-strategy eqilibrim is Vˆ, it will choose Vˆ Lˆ to maximize σ σ Vˆ = [ p(l + ) pl] Vˆ Lˆ wlˆ This gives Lˆ * σp = w 1/(1 σ) L (13)

23 assme that * Lˆ > 0. 21 also assme that Vˆ > 0, given Vˆ * Lˆ. 22 oting that V = (w * L w *L ), it ollows that Vˆ is increasing in Lˆ, since w*/ Lˆ < 0. Hence Ey* is increasing in Lˆ. However, a change in Lˆ has an ambigos eect on Ex*. This is becase while an increase in Lˆ increases Vˆ, it also leads to an increase in Vˆ or Lˆ < * Lˆ. ote that i there were no lobbying the irm will choose the nmber o immigrants σ to maximize p(l + Lˆ ) wlˆ w *L ]. Let the optimal soltion be [ ** Lˆ. Then ** Lˆ > Lˆ since w */ Lˆ < 0. This gives the ollowing straightorward proposition: * Proposition 7: The irm proposes a smaller nmber o immigrants when it has to lobby or its proposal compared to the nmber o immigrants it wold have proposed in the absence o lobbying. 21 ote that the analysis can be prsed within a perectly competitive model. t is well known that, with diminishing retrns and complementarity between capital and labor, a competitive model gives conlicting interests o capital owners and labor with respect to their preerences or immigration. Sppose the prodction is (L) = K µ L σ, where σ + µ = 1; K is capital and L = L + L. Let K be ixed at K. n a competitive eqilibrim, the wage rate will be w = pσ(l + L ) σ-1 K µ. t ollows that the wage rate is decreasing in the nmber o immigrants i 0 < σ < 1. The retrn on capital is pµ(l + L ) σ K µ-1 which is increasing in the nmber o immigrants i 0 < µ < 1. This speciication gives a corner soltion or the immigration qota chosen by the irm, since (V V )/ L > 0. This makes it impossible to obtain interesting comparative statics reslts. t also does not permit an analysis o the kind in section 2.1. To get the soltion in section 2.1, assmed that σ = 1. σ < 1, then the eqivalent o eqation (5), sing w = p σ(l + L ) σ-1 K µ, is very non-linear making it impossible to solve. On the other hand i σ = 1, then w is independent o L. This latter criticism also applies to the bargaining model in this section. 22 One might arge that lobbying is not in the interest o both parties. They cold maximize the srpls rom employing immigrants and share it. The nion will preer this so long it gets the existing native wage o w and a proportion o the srpls. Bt the irm might not like this. With lobbying, the irm does not have to pay w to the natives, i it wins the contest. The irm only pays w < w, or L > 0. Also i the irm's share o the srpls is siciently low, it will preer the lobbying eqilibrim to the zero-lobbying cm sharing eqilibrim.

24 The expected immigration qota is Lˆ e 0.5Vˆ = (1 Vˆ ) Lˆ * (14) ote that Vˆ is not a nction o w. Since Vˆ and * Lˆ are both decreasing in w, it ollows that Lˆe / w 0. This conirms proposition 1. Since the post-immigration < wage o natives is decreasing in the nmber o immigrants, it ollows that the lower is the reservation wage o natives, the lower is the expected post-immigration wage o natives (i.e., proposition 3). Proposition (2) is easy to veriy. From eqations (13) and (14), we note that Lˆ e / L has an ambigos sign (i.e., the size o nion has an ambigos eect on the immigration qota). This reslt diers rom proposition 4; it sggests that proposition 4 is not robst. To veriy proposition 5, need to compare the model in this section (model 2) with the imperectly discriminating contest version o model 1. However, this comparison is not possible becase it is not only the sensitivity parameter which diers in the two models bt also the act in model 1 L depends on y bt in model 2, it does not. ote that ( w */ Lˆ )/ p < 0. That is, or a given nmber o immigrants, the all in the post-immigration wage o natives is bigger the higher is the price o the irm s prodct. That is, the higher is the price the bigger is the negative eect o immigration on the wage o natives. This is becase a higher price increases the irm s option vale in the bargaining game and hence increases its bargaining power. Given that in this model, the expected immigration qota is increasing in p and noting that a higher p increases the

25 negative eect o immigration on the wage o natives, proposition 6 can be re-stated as ollows. Proposition 6a: The stronger is the irm s bargaining power, the higher is the expected immigration qota. ote that i we sbtract eqation (9) rom eqation (10), the reservation wage o natives drops ot. Hence the nion s valation is independent o the reservation wage o natives. This is becase the native workers are garanteed this wage, whether or not there is immigration. 23 Since the irm s valation is also independent o the reservation wage o natives, this gives the ollowing proposition: Proposition 8: The expected immigration qota is independent o the reservation wage o native workers. Sppose that at * Lˆ, Vˆ 0. Then the payo o the irm in the all-pay action is Vˆ zero, regardless o the nmber o immigrants chosen. 24 the irm sets Lˆ = 0, then the valations o both contestants is zero and there is no lobbying. Bt this is not a ash eqilibrim. Since the irm s payo is zero, regardless o the nmber o immigrants chosen, we cold assme that it chooses L > 0. Withot any loss o generality, assme that the irm chooses L = Lˆ *. Then the expected immigration qota is now 23 This might not be the case in a general eqilibrim model. 24 * assme, however, that Vˆ > 0 at Lˆ.

26 e Vˆ * L ~ = Lˆ (15) 2Vˆ t is straightorward to conirm the comparative statics reslts above. Ths my reslts hold even i the nion has a higher valation. 4. Conclsion The propositions in this paper oer some potentially testable hypotheses, which may be the basis or tre empirical work on immigration. However, sch an analysis is not withot problems. As noted in section 1, several actors apart rom those in my model, determine the nmber o immigrants. Thereore, it may not be easy to isolate these actors in empirical work. Also while my propositions are with reerence to the expected nmber o immigrants, any empirical work wold se the actal nmber o immigrants. Since what am interested in is how the nmber o immigrants responds to certain variables, this may not be problem i the expected immigration qota is positively correlated with the actal immigration qota. One might arge that the reslts in this paper are obvios. Bt they are probably not. For example, consider proposition 6. the negative eect o immigration on the wage o natives increases, then this increases the valations o both the irm and the nion. A priori, it is not obvios, in a lobbying model that the nmber o immigrants will increase given that there is an increase in the valations o both contestants. The paper shows that the direction is an increase in the expected immigration qota. ote that or a given change in any exogenos variable there are three possibilities: (a) the change reslts in an

27 increase 25 in the valations o both contestants (b) the change increases the irm s valation bt not the nion s valation, and (c) the change increases the nion s valations bt not the irm s valation. 26 The paper has shown how immigration qotas are adjsted in these three cases. Admittedly, my model is a very simple one and recognize its limitations. Given that immigration is a dynamic isse, my static ramework may ignore some very important isses. For example, immigrant workers may join the nion, increasing its size over time or may orm a coalition with the irm to lobby, i they care abot other prospective immigrants. n section 2, cold have assmed that L = h(y, x) 0, where h/ y > 0 and h/ x < 0; that is, the immigration qota is decreasing in the nion s lobbying eort. nstead assmed that the nion s eort redces the probability that the irm will be sccessl bt does not aect the nmber o immigrants, when the irm wins. When the nion wins the contest, there is no immigration. Ths, in my ormlation, h(y, x) = 0, when the nion wins and eqals h(y, x) > 0, when the irm wins. The reslts o this paper probably do not hinge on this ormlation, since most o them still hold in section 3 where the nmber o immigrants is independent o y. n spite o the paper s shortcomings, hope that have oered some insight into the natre o the politico-economic eqilibrim in the immigration market. 25 ocs on the case o an increase or the sake o exposition. 26 ote, however, that i the nion cares abot only total wage receipts, then case (c) is not possible.

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