Referendum-led Immigration Policy in te Welfare State YUJI TAMURA Department of Economics, University of Warwick, UK First version: 12 December 2003 Updated: 16 Marc 2004 Abstract Preferences of eterogeneous citizens over te different levels of low-skilled immigration are examined in a two-period overlapping generation model in wic an inflow of low-skilled immigrants affects te ost economy via tree cannels te labour market, te income support programme and te pay-as-you-go pension system. In most of te cases, te model can predict unique immigration policy determined by majority voting. However, a voting cycle can also arise in certain circumstances, subjecting a referendum outcome to manipulation. Key words: international migration, majority voting, skill acquisition, welfare state JEL classification: D70, F22, I38 Email: y.tamura@warwick.ac.uk Acknowledgement: Tis article is a derivative of te autor s PD dissertation at te university. He as benefitted from generous supervision provided by Professor Benjamin Lockwood.... All remaining errors are mine. 1
1 Introduction Immigration is a politically sensitive subject, for a certain level of a certain type of immigration is likely to ave different impacts on different individuals in te ost country. We restrict our attention to te economic impacts of low-skilled immigrant workers in te welfare state and teir implications to te referendum-led policy formation wit respect to te level of low-skilled labour immigration. We consider tree cannels via wic te impacts of immigration become manifest in te ost country. Tese are te labour market, te intragenerational transfer programme and te intergenerational transfer system. Tis work is motivated by te fact tat te existing literature on te political economy of immigration policymaking in te welfare state is rougly split into two. One studies te subject in te static context of intragenerational transfer, and te oter in te dynamic context of intergenerational transfer. 1 However, intra- and intergenerational transfers often coexist in te welfare state. Terefore, we provide a framework wic contains bot types oftransfersandenablesustoexamineteinteractiveeffect of low-skilled immigration troug tese redistribution programmes. We extend te frameworks provided by Razin & Sadka (2000) and Casarico & Devillanova (2003) in order to derive referendum outcomes. Tese studies examined te impact of an inflow of low-skilled immigrant workers in one period on te ost country wic is inabited by eterogeneous agents of overlapping generations under a pay-as-you-go pension sceme. Razin & Sadka (2000) found tat, if te production factor prices were fixed, te only cange would be a gain by current pensioners te result is driven by te specification of a fixed payroll tax rate. If te factor prices were flexible, owever, te effect would be not only a gain by current pensioners but also a loss to all current workers and te subsequent generations. Immigration 1 Kemnitz (2002), for instance, deals wit an unemployment insurance sceme, altoug a ric model is developed for te static analysis. On te oter and, Scolten & Tum (1996) and Haupt & Peters (1998) focus on a pay-as-you-go pension sceme. 2
increases te supply of labour relative to capital, wic reduces te wage rate. Since immigrant workers are assumed to enter in one period only, te wage rate, after a drop, starts rising to te pre-immigration stady state level over periods. Until ten, te size of te per capita pension benefit is less tan it would ave been witout immigration, as te tax base is smaller from te post-immigration period onwards tan it would ave been wit no immigrants. 2 Casarico & Devillanova (2003) provide a ricer model in terms of te labour market as a transmission mecanism. First, tey assume tat igskilled and low-skilled labour are imperfect substitutes of one anoter. Accordingly, two separate wage rates exist in te labour market. 3 Tis way of modelling te labour market seems to be sensible for our analysis because tereissomeevidencetatlow-wageearnersaremorelikelytopreferreduced immigration tan ig-wage earners. 4 Second, tey endogenise te 2 A gain to current pensioners arises because, altoug te wage rate falls, immigration also increases te number of contributors in te period of entry. 3 Razin & Sadka s (2000) model assumes ig-skilled and low-skilled labour wic are perfect substitutes of eac oter. Terefore, tere is a single wage rate in te labour market. Te difference between ig-skilled and low-skilled workers is ten te quantity of substitutable labour being endowed. 4 See Sceve & Slaugter (2001, Table 2, p. 140) wic examined te data from te National Election Studies surveys in te United States in te 1990s. Tey also found tat tose wit low educational attainment are more likely to oppose immigration tan more educated individuals. Since te type of immigrants was not specified in te survey questionnaire, teir findings do not necessarily conform wit te typical economic teoretical argument tat an increase in te size of te labour force of one type would damage te labour market opportunities for workers wo are already in te ost country and belong to te same labour type. Neverteless, if we assume tat te survey respondents ad a biased view tat immigrants are typically poorly educated and are likely to work in lowwage sectors once arrived in te ost country, te preferences of te respondents examined by Sceve & Slaugter (2001) may well reflect te fact tat people anticipate te impacts of immigration just as te standard economic teoretical model would predict tem. As people s perception matters in voting, weter influenced by media or learned troug actual job market experiences, models wic incorporate adverse labour market effects of immigration would be good predictors of majority voting outcomes on immigration policy. 3
skill acquisition decision of native workers. Since immigration affects te wage rates, te profitability of skill acquisition also canges according to te level of immigration. 5 As a result of an endogenised flow of native workers from te low-skilled to ig-skilled labour pool, te impact of low-skilled immigration on te two wage rates is moderated. 6 Teir analysis reveals tat currently low-skilled workers can be divided into tree groups for a given level of low-skilled immigration: tose wo would remain low-skilled, tose wo would become ig-skilled but would be better off witout te immigration and tose wo would become igskilled and would be better off wit it. Te second group is pused out of te low-skilled labour force because, altoug becoming ig-skilled wit te immigration does not yield as ig income as remaining low-skilled witout it, remaining low-skilled wit te immigation gives tem even lower income tan becoming ig-skilled wit it. Tey prefer no immigration to tis level of immigration. On te oter and, te tird group is pulled to te ig-skilled labour force because becoming ig-skilled wit te immigration yeilds iger income tan remaining low-skilled not only wit it but also witout it. Tey prefer tis level of immigration to te status quo. Since te interest group division is important to te determination of majority voting outcomes, we follow Casarico & Devillanova (2003) and endogenise te skill acquisition decision. Tese two studies offer a platform for our study of te formation of referendum-led immigration policy. We include an explicit intragenerational transfer wen we describe our model in section 2. 7 After constructing 5 See, for example, Ciswick (1989) for suc a model. 6 Tis is one potential reason wy empirical evidence for te impact of immigrant workers on te ost country s labour market is mixed. Endogenous skill acquisition decisions of native workers may lessen te labour market impact of immigration. LaLonde & Topel (1997), for instance, found tat immigration would ave a small impact on te labour market at destination. 7 Te two studies mentioned do include suc a transfer but implicitly because teir pay-as-you-go pension scemes are made of a single tax rate on wage earnings and a flat lump sum per capita benefit. 4
a model, we examine te preferences of individual voters so as to observe te impacts of low-skilled labour immigration on natives in section 3. We ten derive majority voting outcomes determining quantitative immigration policy. 8 In addition to immigration occuring in one period only, we also examine te cases of immigration tat takes place in every period. Section 4 concludes. 2 Model Consider overlapping generations of agents wo live for two periods in a country. In te first period, eac agent supplies labour to earn wage income, saves a fraction of te disposable income for te second period and consumes te rest. In te second period, te agent does not work, receives apensionbenefit and witdraws te savings wic ave earned interest over one period. Se/e consumes all te income in te second period, i.e., no bequest. Te pension sceme is balanced pay-as-you-go (PAYG). Tat is, te sum of pension benefits received by current pensioners equals te sum of contributions paid by current workers. In addition to te unfunded pension sceme wic is intergenerationally redistributive, an income support programme provides eac low-wage earner wit a flat lump sum benefit. Te programme is financed by a linear tax on te gross wage of all workers, and ence intragenerational redistribution takes place from te ric to te poor at te same time as young agents support te elderly. 2.1 Population Agents are categorised into two groups natives and immigrants. We draw a line between tese two groups by te voting rigt endowment: natives can 8 Casarico & Devillanova (2003) revealed an interesting group division but did not examine politico-economic outcomes. 5
vote, but immigrants cannot. We assume tat all natives exercise teir voting rigts in a referendum on quantitative immigration policy. On te oter and, all agents including immigrants are entitled to economic rigts in te ost country. Hence natives and immigrants are equally entitled to benefits of te PAYG pension sceme and te income support programme. Assume for simplicity tat tere as been no immigration in te past. Te total number of working natives in period t is denoted by N t. Te growt rate of te native population is assumed to be a positive constant, N i.e., δ > 0. Accordingly, t 1 N t = 1 were N 1+δ t 1 is te total number of pensioners in period t. Tis ratio expresses te PAYG system dependency ratio in period t witout immigration. We assume tat native agent i is born low-skilled wit parameter e i wic indicates an idiosyncratic pecuniary cost to become ig-skilled. Te smaller te value of e i is, te less costly it is for worker i to become ig-skilled. We assume tat e i [0, e] weree is te igest cost of skill acquisition among te native workers. We furter assume for ease of exposition tat te cost parameter is uniformly distributed among young native workers. 9 Te existence of te idiosyncratic cost of skill acquisition implies tat, wile some native workers can afford to become ig-skilled, skill acquisition is too costly for te oters. Let e t be te tresold level of te skill acquisition cost in period t. Young native i wit e i > e t remains low-skilled in period t. Since we assume a uniform distribution for e i,wecanwrite te proportion of ig-skilled workers in te native workforce by using its cumulative distribution function as t := e t [0, 1] (1) e were e t is later defined as a function of immigration. 10 Consider now tat immigrants enter te country. We assume tat tey are always low-skilled workers at te entry and are fully employed. Once 9 It is possible to assume oter, peraps more realistic, distributions for e i,tougitis ten likely tat we need to resort to numerical simulation. 10 See section 2.7 below. 6
entered, tey remain in te ost country for two periods of teir lifetime. Low-skilled immigrants wo enter in period t amount to M t. We assume tat immigrant workers do not bring in dependants wen tey arrive and ave te same reproductive beaviour as natives during teir working life. Furtermore, all cildren of immigrants wo are born in te ost country are given voting rigts. Tat is, tey are classified as natives in our model. Definition 1. By immigration, we mean low-skilled immigrant workers wo are in te first period of teir lifetime. If suc immigration takes place in period t only, te PAYG system dependency ratio is t 1 N N t +M t in period t and is N t+m t N t+1 in period t +1 were N t+1 =(1+δ)(N t + M t ). Tat is, immigrants reduce te pensioner-worker N ratio in teir working period, i.e., t 1 N t +M t < 1, and te ratio returns to te 1+δ pre-immigration level wen tey retire, i.e., N t+m t N t+1 = 1, if immigration is 1+δ temporary. Definition 2. By temporary immigration, we mean tat low-skilled immigrant workers enter te ost country in one period only. We denote by m te growt rate of te low-skilled workforce due to immigration in period t, i.e., M t m t :=. (2) (1 t )(1+δ) N t 1 Te total supply of low-skilled labour in period t is L t := (1 t )(1+δ) N t 1 + M t, assuming tat eac working agent provides one unit of labour. Since we assume tat immigrant workers are always low-skilled, te total supply of ig-skilled workers in period t is H t := t (1 + δ) N t 1. 2.2 Production Te production in te ost country, Y, is caracterised by te following Cobb-Douglas function wic exibits constant returns to scale: Y t (K t,h t,l t ):=K γ t H ϕ t L % t (3 ) 7
were te output sare parameters γ, ϕ and % are all on te interval (0, 1) and γ + ϕ + % = 1. Te marginal product of capital, K, is te interest rate, r t := Y t K t = γk γ 1 t H ϕ t L % t. We assume international perfect mobility of capital, and te interest rate, r, is exogenously given. Accordingly, for a fixed interest rate, K t = 1 ϕ % γ 1 γ 1 γ 1 γ H r t Lt. By substituting tis expression back into te production function above, we get Y t (H t,l t )=AH α t L 1 α t were A := 1 γ 1 γ and α := ϕ (0, 1). Tus, capital exists but does not r ϕ+% explicitly enter te production function. Te amount of capital perfectly adjusts to te interest rate wic is exogenous. For ease of exposition, we normalise A = 1. Terefore, our production function reduces to Y t (H t,l t )=H α t L 1 α t. (3) Under perfect competition, firms make zero profit. Wages perfectly adjust for full employment. By differentiating te production function (3) wit respect to H and L respectively, we obtain te marginal product of labour of eac skill type, i.e., µ 1 α Y t Lt = α (4 ) H t H t and µ α Y t Lt =(1 α). (5 ) L t H t Tese are te gross wages, w H and w L, for one unit of ig-skilled and low-skilled labour respectively. Using equation (2), we rewrite tese marginal products of labour as follows: w H t (1 t (m t )) (1 + m t ) (m t, t (m t )) := α t (m t ) 1 α (4) and (1 wt L t (m t )) (1 + m t ) (m t, t (m t )) := (1 α) t (m t ) α (5) 8
were it is implied tat t is a function of m t. As mentioned wen we defined t in (1) above, we later define e t as a function of m t. We will ten comment on te sapes of tese wage functions. 11 2.3 Income support Our economy operates an income support programmeforlow-wageearners. We simply assume tat all low-skilled workers receive suc support wic is flat lump sum, θ. Te programme is financed troug a programme-specific tax, µ. Te budget constraint in period t is ten L t θ t = µ t Y t (6 ) were Y t = wt H H t + wt L L t. Te tax is imposed on all workers, and te revenue is sared by only low-skilled workers. Hence pensioners are not affected by te programme, wile ig-skilled workers redistribute to lowskilled workers. We fix te per capita support exogenously. Terefore, te tax rate is residually determined by immigration policy, making te policy coice unidimensional. We ten rewrite te constraint (6 ) by substituting (2), (4) and (5) into it as 12 (1 t (m t )) (1 + m t ) µ t (m t, t (m t )) = θ. (6) t (m t ) 2.4 Pension Te pension sceme in te economy is balanced PAYG. We assume tat te per capita pension benefit, b, isaflat lump sum payment for all pensioners. Accordingly, te following budget constraint must old in any one period: (N t 1 + M t 1 ) b t = τ t Y t (7 ) 11 See lemmata 3 and 4 below. 12 See lemma 5 below for te beaviour of te programme-specific taxrate. 9
were τ is te payroll tax rate common to all workers. Te left and side of te equation represents te total amount of pension benefits to be paid to te pensioners in period t, and te rigt and side te total amount of contributions to be collected from te workers in tat period. Let us assume tat te payroll tax rate is exogenously given. Accordingly, te amount of te per capita benefit adjusts residually to te level of immigration, again making te policy coice unidimensional. We continue to assume tat no immigrant entered in period t 1, so we can rewrite te budget constraint (7 ) by substituting (2), (4) and (5) into it as 13 1 α (1 t (m t )) (1 + m t ) b t (m t, t (m t )) = τ (1 + δ) t (m t ). (7) t (m t ) Note tat our flat lump sum pension sceme implies redistribution among pensioners of te same generation. Tat is, all agents receive te same amount of pension in te post-retirement period, wereas ig-skilled ones contribute more to te pension system tan te low-skilled during te working period. Hence our PAYG pension sceme is bot inter- and intragenerationally redistributive. If immigrants continue to enter te ost country in te next period, te same expression (7) is applicable by canging te time subscript to t +1. M Te immigration rate is ten m t+1 = t+1. (1 t (m t ))(1+δ)[(1+δ)N t 1 +M t On te ] oter and, if immigration is temporary, 14 te per capita pension benefit in period t +1becomes µ 1 α 1 t+1 (0) b t+1 (0, t+1 (0)) = τ (1 + δ) t+1 (0). t+1 (0) Tus, wen immigration is one-off, tepercapitabenefit returns to te preimmigration level in te next period. Tat is, today s immigrants are not extra burdens in te next period under our PAYG pension system. Tis observation results from our assumption wit regard to te fertility of immigrants. 13 See lemmata 6 and 7 below for te beaviour of te per capita pension benefit. 14 See definition 2 above. 10
2.5 Consumption Worker i maximise teir utility over te lifetime consumption, c. In te first period of life, te budget constraint is ( (1 s) (1 τ c i µt ) wt H e i if ig-skilled; t (m t, t (m t )) := (1 s) (1 τ µ t ) wt L + θ oterwise, (8) were s is te saving rate common to all workers. life, te budget constraint is In te second period of : = c i t+1 (m t, t (m t ),m t+1, t+1 (m t+1 )) ( (1 + r) s (1 τ µt ) wt H e i + b t+1 if ig-skilled; (1 + r) s (1 τ µ t ) w L t + θ + b t+1 oterwise. (9) By substituting into te saving term, we combine te constraints for two periods respectively, i.e., ( c i t + ci t+1 (1 τ 1+r = µt ) wt H e i + b t+1 if ig-skilled; 1+r (10 ) (1 τ µ t ) wt L + θ + b t+1 oterwise. 1+r Te left and side of te equality sign expresses te lifetime consumption, wile te rigt and side expresses te lifetime income. Let us define te lifetimeincomeas ( zt i w H (1 τ t,wt L µt ) w H,µ t,b t+1 := t e i + b t+1 if ig-skilled; 1+r (1 τ µ t ) wt L + θ + b t+1 oterwise, 1+r (10) were wt H, wt L, µ t and b t+1 alldependonimmigration. 15 In section 2.7 below, we model te skill acquisition decision of agent i. 2.6 Utility We assume tat all workers ave an identical Cobb-Douglas utility function, i.e., u i t c i t,ct+1 i := c i β t c i 1 β t+1 (11) 15 Seelemmata3to7below. 11
were we assume β (0, 1). Form te Lagrangian using (11) wit (10 ). Te first-order condition for utility maximisation suggests c i t c i t+1 = β (1 β)(1+r) were we substitute into te Lagrangian multiplier for L =0and c i t Plug tis into (10 ) to obtain te following demand functions: L c i t+1 =0. c i t = βz i t and c i t+1 =(1 β)(1+r) z i t. By substituting tese demands back into te utility function (11), we obtain te following indirect utility function: vt i z i t := β β (1 β) 1 β (1 + r) 1 β zt i (12) were zt i is definedin(10)wicindicatestatimmigrationaffects te preferences of agents via tree cannels te labour market, te income support programme and te PAYG pension system. Since te relationsip between vt i and zt i is positive linear in expression (12), we focus on zt i to examine te preferences of native workers in te subsequent analysis. 2.7 Skill acquisition Young native i is assumed to become ig-skilled if (1 τ µ t ) w L t + θ + b t+1 1+r (1 τ µ t) w H t e i + b t+1 1+r or e i (1 τ µ t ) w H t wt L θ (13) wit given immigration. Tat is, to become ig-skilled, se/e requires te lifetime income for ig-skilled labour to be at least as ig as tat for 12
low-skilled labour. 16 Note tat we need to compare only te first-period incomes because bot ig-skilled and low-skilled workers receive te same amount of pension in te post-retirement period. Te worker wo pays te most to become ig-skilled in period t is ten born wit te idiosyncratic cost of e t w H t,wt L,µ t := (1 τ µt ) wt H wt L θ. (14 ) By substituting expressions (4), (5) and (6) into it, we obtain ½ ¾ (1 t (m t )) (1 + m t ) e t (m t, t (m t )) = 1 τ θ t (m t ) ( 1 α (1 t (m t )) (1 + m t ) (1 t (m t )) (1 + m t ) α (1 α) t (m t ) t (m t ) θ. Wit tis last expression, we are now able to determine ow te skill acquisition decision is influenced by immigration and ence, via definition (1), ow te skill composition of te native workforce canges over te range of policy alternatives [0, m] werem is te maximum feasible rate of immigration and is exogenously given. 17 We ave te following two observations from (14). Lemma 1. Suppose no income support for low-skilled workers. Te proportion of ig-skilled workers in te native workforce is ten monotonically increasing in immigration. Furtermore, d t t (0, 1). Proof. See appendix 1. Lemma 2. Suppose tere is income support for low-skilled workers. Te proportion of ig-skilled workers in te native workforce is ten initially 16 Refer to expression (10). 17 Tis can be identified by assuming a given reservation wage for immigrants, w L,below wic tey would not accept to enter te ost country, i.e., from expression (5). µ 1 α m := w L 1 α (m) 1 (m) 1 (14) α ) 13
increasing but subsequently decreasing in immigration, provided tat (A) te size of per capita income support is not excessively large, (B) te maximum feasible rate of immigration is sufficiently ig and (C) te maximum cost of skill acquisition is sufficiently ig. Furtermore, d t t ( 1, 1). Proof. See appendix 2. Lemma 1 implies tat te wage gap between ig-skilled and low-skilled labour monotonically widens wen immigration increases in te ost country. 18 As a result, an increasing number of native workers pay e i to become ig-skilled, as te rate of immigration rises. Lemma 2 suggests tat te wage gap again widens as m increases, but it begins to srink if m continues to increase beyond a certain level. Tat is, too many immigrants require excessive redistribution troug te income support programme. Terefore, over a range of different rates of immigration, tere is unique interior policy wic maximises te proportion of ig-skilled workers in te native workforce. As appendix 2 sows, tis result is subject to tree assumptions (A), (B) and (C). We assume tat tey old trougout our subsequent analyses. Note tat, altoug t decreases wit ig m t wen θ > 0, it does not fall below t (0) because we assume tat currently ig-skilled workers cannot become low-skilled. Tat is, skill acquisition cannot be reversed. Hence we ave a smaller range for t tan defined in (1), i.e., t (m t ) [ t (0), 1] m t [0, m]. Usingtesetwolemmataabout t, we now summarise te signs of dwh t dw L t t, dµ t t and db t t by te following five lemmata. t, Lemma 3. Suppose no income support for low-skilled workers. Te igskilled wage rate is ten monotonically increasing in immigration, and te low-skilled wage rate is monotonically decreasing in it. Proof. See appendix 3. 18 Wen θ = 0, expression (14 ) merely reflects te canging difference between te ig-skilled and low-skilled wages, i.e., e t := (1 τ) w H t wt L. 14
Lemma 4. Suppose tere is income support for low-skilled workers. Suppose assumptions (A), (B) and (C) of lemma 2 old. Te ig-skilled wage rate is ten monotonically increasing in immigration, and te low-skilled wage rate is monotonically decreasing in it. Proof. See appendix 4. Lemma 5. Suppose tere is income support for low-skilled workers. Suppose assumptions (A), (B) and (C) of lemma 2 old. Te tax rate specific to te income support programme for low-skilled workers is ten monotonically increasing in immigration. Proof. See appendix 5. Lemma 6. Suppose no income support for low-skilled workers. Te per capita pay-as-you-go pension benefit is ten monotonically increasing in immigration. Proof. See appendix 6. Lemma 7. Suppose tere is income support for low-skilled workers. Suppose assumptions (A), (B) and (C) of lemma 2 old. Te per capita payas-you-go pension benefit is ten monotonically increasing in immigration. Proof. See appendix 7. Tese lemmata indicate tat, as far as assumptions (A), (B) and (C) of lemma 2 old, we always ave dwh t t > 0, dwl t dµ t < 0, t t > 0and db t t > 0 regardless of weter te income support programme operates or not. We use lemmata 1 to 7 extensively wen we prove te propositions in section 3. 2.8 Interest group division Altoug te skill acquisition decision tresold (14 ) determines te skill composition of te native labour force ifagivenrateofimmigrationactu- ally takes place, it does not indicate wat proportion of te working native 15
population prefer te given rate to te status quo. Te reason is tat te satisfaction of condition (13) does not necessarily mean (1 τ µ t (0)) wt L (0)+θ+ b t+1 (0) 1+r (1 τ µ t (m t )) wt H (m t ) e i + b t+1 (m t+1 ) 1+r or e i (1 τ µ t (m t )) wt H (m t ) (1 τ µ t (0)) wt L (0) θ + 1 1+r (b t+1 (m t+1 ) b t+1 (0)) (15) for given m t,m t+1 > 0. In oter words, worker i requires er/is lifetime income wit given immigration to be at least as ig as it would be witout it, if te given immigration is to be preferred to te status quo. Wen condition (15) does not old, se/e prefers te status quo to te given immigration, even if condition (13) olds. Tis point is made by Casarico & Devillanova (2003), and we formalise te idea witin our model in order to examine majority voting outcomes. Let us define e t as follows: 19 e t (m t, t (m t ),m t+1, t+1 (m t+1 )) : = [1 τ µ t (m t, t (m t ))] w H t (m t, t (m t )) (16 ) [1 τ µ t (0, t (0))] wt L (0, t (0)) θ + 1 1+r [b t+1 (m t+1, t+1 (m t+1 )) b t+1 (0, t+1 (0))]. By substituting (4), (5), (6) and (7) into it, we obtain e t (m t, t (m t ),m t+1, t+1 (m t+1 )) (16) ½ ¾ 1 α (1 t (m t )) (1 + m t ) (1 t (m t )) (1 + m t ) : = 1 τ θ α t (m t ) t (m t ) µ α µ α 1 t (0) 1 t (0) 1 τ θ (1 α) θ t (0) t (0) ( +τ 1+δ 1 α µ ) 1 α (1 t+1 (m t+1 )) (1 + m t+1 ) 1 t+1 (0) t+1 (m t+1 ) t+1 (0). 1+r t+1 (m t+1 ) t+1 (0) 19 Definition (16 ) assumes tat 0 gives te igest z i if skill acquisition does not take place and ence becomes irrelevant in proposition 8 towards te end of section 3 were a possibility of z i (0) <z i (m) arisestoinfluence te majority voting outcome. 16
Tenativeagentwoisbornwite t would experience no cange in er/is lifetime income, weter given immigration takes place or not. Agents born wit e i [0,e t ] would ten become ig-skilled and would be better off wit te given immigration. Agents born wit e i (e t, e t ] would also become ig-skilled but prefer no immigration to te given immigration because se/e enjoys a iger level of lifetime income witout it by remaining low-skilled. Let t := e t e [ t (0), 1] (17) be te proportion of ig-skilled workers wo do not lose any in te lifetime income wit given immigration. 20 Togeter wit te skill acquisition rule (13), native workers can ten be divided into tree groups at eac potential rate of immigration as follows: e e t (m t ) e 1 t (m t ); e t (m t ) e t (m t,m t+1 ) t (m t ) t (m t,m t+1 ); e e t (m t,m t+1 ) t (m t,m t+1 ). e Given immigration, te first group consists of tose wo remain low-skilled. Te second group consists of tose wo become ig-skilled wit tat immigration but ten ave lower lifetime income tan wen tey remain lowskilled witout it. Tis second group is, in oter words, pused to become ig-skilled by te immigration. Tese two groups prefer no immigration to tis given immigration policy. For te same given immigration, te tird group consists of tose wo become ig-skilled and ave at least as muc lifetime income as witout tat immigration. Tis group includes tose wo are pulled to become ig-skilled by te immigration. Tey prefer tis immigration policy to te status quo. 20 Note tat, wen m t = m t+1 = 0, expression (16 ) is te same as expression (14 ) at m = 0. Hence t (0) = t (0). 17
Definitions (1) and (17) mean tat tese tree groups cange in teir relative size wit eac possible immigration policy because expressions (14) and (16) define bot tresold costs of skill acquisition, e t and e t, as functions of immigration. We ave already summarised te beaviour of t in lemmata 1 and 2 above. We now summarise te beaviour of defined in (17). Notice tat, wile expression (14) is determined by one rate of immigration, namely m t, two rates of immigration are present in (16), i.e., m t and m t+1. For ease of exposition, we consider cases were m t = m t+1. Definition 3. By permanent immigration, we mean tat te same rate of immigration occurs in every period. In te case of permanent immigration, we assume tat, once a decision is taken, te policy becomes effective from period t onwards witout fear of policy cange in te future. Te optimal coice of immigration is ten te steady state solution, i.e., m = m t+j j 0. Note tat, if te cosen rate of immigration is positive in period t, te ratio of low-skilled to igskilled labour canges permanently from L H t 1 = 1 t 1(0) t 1 to L = (0) H t+j (1 t+j (m ))(1+m ) t+j j 0. Hereafter, we drop all te time subscripts because (m ) tey are unnecessary in eiter temporary or permanent scenarios. 21 Lemma 8. Consider temporary immigration. Suppose no income support for low-skilled workers. Te proportion of workers wo prefer immigration to te status quo is ten monotonically increasing in immigration. Proof. No income support for low-skilled workers means θ = 0. Ten, wen immigration policy is temporary, (16 ) reduces to e (m, (m)) := (1 τ) w H (m, (m)) w L (0,(0)). If θ =0,ten dwh > 0bylemma3 and ence de > 0. d Since > 0 from (17), d > 0. de Lemma 9. Consider permanent immigration. Suppose no income support for low-skilled workers. Te proportion of workers wo prefer immigration to te status quo is ten monotonically increasing in immigration. 21 See definition 2 above for temporary immigration. 18
Proof. No income support for low-skilled workers means θ = 0. Ten, wen immigration policy is permanent, (16 ) reduces to e (m, (m)) := (1 τ) w H (m, (m)) w L (0,(0)) + 1 [b (m, (m)) b (0,(0))]. If 1+r θ =0,ten dwh > 0 by lemma 3, db de > 0bylemma6andence > 0. Since d de > 0 from (17), d > 0. Lemma 10. Consider temporary immigration. Suppose tere is income support for low-skilled workers. Suppose assumptions (A), (B) and (C) of lemma 2 old. Te proportion of workers wo prefer immigration to te status quo is ten initially increasing but subsequently decreasing in immigration, provided tat condition (A.2.1 ) for assumption (A) is replaced by (A.8.1 ). Furtermore, tis proportion is at its maximum before te proportion of ig-skilled workers reaces its peak. Proof. See appendix 8. Lemma 11. Consider permanent immigration. Suppose tere is income support for low-skilled workers. Suppose assumptions (A), (B) and (C) of lemma 2 old. Te proportion of workers wo prefer immigration to te status quo is ten initially increasing but subsequently decreasing in immigration, provided tat condition (A.2.1 ) for assumption (A) is replaced by (A.9.1 ). Furtermore, tis proportion is at its maximum before te proportion of ig-skilled workers reaces its peak. Proof. See appendix 9. Tese four lemmata indicate tat, regardless of weter immigration is temporary or permanent, if te income support programme is absent, te proportion of workers wo prefer immigration to te status quo monotonically increases in m. If te income support programme is in operation, it as a peak over (0,m )werem maximises, i.e., te solution for (A.2.1 ) in appendix 2. We use tese four lemmata wen we prove te propositions we state below. 19
3 Results In te framework described above, we now examine te preferences of individuals over te rate of immigration and derive majority voting outcomes. We assume tat a referendum takes place in te beginning of period t, and all natives rationally vote on te rate of immigration to be permitted into te country. Our focus is on te determination of te variable, m. An infinite number of potential policy alternatives over te interval [0, m] arecompared pairwise were m is te maximum feasible rate of immigration. 22 We examine impacts of bot temporary and permanent immigration on individual preferences. Temporary immigration is defined in definition 2 above and is examined by Razin & Sadka (2000) and Casarico & Devillanova (2003). Permanent immigration is definedindefinition 3 above. Tat is, we assume tat, once a decision is taken, te policy becomes effective from period t onwards witout fear of policy cange in te future. Native workers attempt to optimise teir lifetime consumption/income at te beginning of teir working period wen te referendum takes place. Teobjectiveofworkingnativei is max m vi z i w H (m, (m)),w L (m, (m)),µ(m, (m)),b(m, (m)) s.t. m [0, m] and (m) [ (0), 1]. (18) Since expression (12) indicates tat te utility is a positive monotonic transformation of te lifetime income, we examine te sape of z i. Te first-order total derivative of z i wit respect to permanent immigration is dz i (m, (m)) = [1 τ µ (m, (m))] dwk (m, (m)) w k dµ (m, (m)) (m, (m)) + 1 1+r db (m, (m)) (19) were k = {H, L}. For temporary immigration, te tird term on te rigt and side drops out. Wen te income support programme is absent, te second term disappears, and µ = 0intefirst term. 22 See footnote 17. 20
Finally, te objective of pensioners is max m b (m, (m)) s.t. m [0, m] and (m) [ (0), 1] (20) because immigration cannot affect teir first-period income earned in te previous period in our model. 23 From objective (20), lemmata 6 and 7 indicate tat pensioners always vote for m, for immigration only increases teir income and ence utility. 24 Te preferences of workers are more complicated tan pensioners, and we examine tem in detail below. 3.1 Temporary immigration witout income support In tis case, objective (18) and lemma 3 suggest tat ig-skilled workers vote for m, wile low-skilled workers vote for te status quo. Objective (20) and lemma 6 mean tat pensioners vote for m. Since currently ig-skilled workers and pensioners exibit te same ordinal preference over te interval [0, m], we consider te following two different scenarios regarding (0), i.e., te proportion of ig-skilled workers in te native workforce at te voting 23 If we relax our assumption of te fixed interest rate, immigration is likely to cange te marginal product of capital and tus affect pensioners troug savings as well as te PAYG pension benefit. However, since an increase in labour due to immigration would raise te interest rate, te preference of pensioners is unlikely to be modified even if we introduce flexibility for te interest rate. 24 Tis is a consequence of using an unindexed flat lump sum pension and fixing te PAYG payroll tax rate. If te size of per capita pension is fixed instead, pensioners become indifferent in our model. Scolten & Tum (1996) and Haupt & Peters (1998) index te per capita pension benefit to te prevailing wage rate. In teir model, pensioners are all against immigration tat depresses te single wage rate because age is te only dimension of eterogeneity among agents. In our model, single-mindedness of retirees can be canged by, for instance, linking te per capita pension benefit to te contribution istory, e.g., te benefit for ig-skilled pensioners is indexed to te prevailing wage for ig-skilled labour. In addition, incorporating teir altruism towards te younger generation would make teir preference non-monotonic. 21
stage: Scenario I (1 + δ)(1 (0)) < (1 + δ) (0) + 1 Scenario II (1 + δ)(1 (0)) > (1 + δ) (0) + 1 In scenario I, currently ig-skilled workers and pensioners form te majority wen voting takes place. In scenario II, currently low-skilled workers form te majority. Scenario I is a straigtforward case, as te following proposition states. Proposition 1. Consider temporary immigration policy. Suppose no income support for low-skilled workers. If currently ig-skilled workers and pensioners form te majority at te voting stage, te referendum-led policy is to permit te maximum feasible rate of immigration. Proof. No income support for low-skilled workers means θ = 0. Ten, d > 0 by lemma 1. Under scenario I, d > 0 implies m [0, m], (1 + δ)(1 (m)) < (1 + δ) (m) + 1. Tat is, ig-skilled workers and pensioners continue to form te majority over te interval [0, m]. Wen immigration policy is temporary wit θ = 0, expression (10) for te igskilled reduces to z i := (1 τ) w H e i. Sinceθ =0, dwh > 0bylemma3and ence dzi db > 0 for te ig-skilled. Also > 0 by lemma 6. Objectives (18) and (20) ten imply tat everyone in te majority monotonically increases er/is utility as m increases. Te Condorcet winner is tus m. Under scenario II, tere are two possibilities, as te following proposition states. Proposition 2. Consider temporary immigration policy. Suppose no income support for low-skilled workers. If currently low-skilled workers form te majority at te voting stage, te referendum-led policy is to permit (i) te maximum feasible rate of immigration if tere exists m [0, m] satisfying e ( m, ( m)) ( m, ( m)) e = but (ii) no immigration if tere is no suc m [0, m]. δ e, (21) 2(1+δ) 22
Proof. See appendix 10. According to tis proposition, currently ig-skilled workers and pensioners do not ave to form te majority to acieve teir most preferred policy wen a referendum takes place, if tere exists feasible policy tat induces a sufficient number of currently low-skilled workers to ave teir lifetime income by becoming ig-skilled at least as ig as wen remaining low-skilled. Condition (21) is a rearrangement of (1 + δ)(1 ( m, ( m))) = (1 + δ) ( m, ( m)) + 1. Te left and side of te inequality sign consists of tose wo remain lowskilled and tose wo become ig-skilled but ten ave lower lifetime income at m tan wen m = 0. Te rigt and side is te sum of all retired pensioners, currently ig-skilled workers and tose wo are currently lowskilled but become ig-skilled at m and earn at least as ig lifetime income as in te status quo. low e i z ig e i middle e i 0 m Figure 1. Lifetime Incomes of Workers wit Different Costs of Skill Acquisition over Temporary Immigration Policy wen te Income Support Programme is Absent 23
Satisfaction of condition (21) guarantees tat tere are tose currently low-skilled workers wo join ig-skilled workers and pensioners to form te majority in favour of m against any oter policy alternatives. In figure 1, te lifetime income of a worker wit middle e i implies tat se/e most prefers m because z i (0) <z i (m), altoug tis worker is currently low-skilled. 25 If asufficient number of currently low-skilled workers ave tis type of lifetime income projections wit a kink and z i (0) <z i (m), condition (21) is met, and outcome (i) occurs. An implication of tese two propositions is tat tere migt be a tendency to decide on very liberal policy of temporary immigration if immigrants were not burdens in te welfare state, namely no income support in our model. 26 3.2 Permanent immigration witout income support Wen immigration policy is permanent, te lifetime income for te lowskilled is no longer monotonically decreasing in immigration. Lemma 12. Suppose no income support for low-skilled workers. Te lifetime income for te low-skilled is ten initially decreasing but subsequently increasing in permanent immigration, provided tat (D) te PAYG payroll tax rate is not excessively ig and (B) te maximum feasible rate of immigration is sufficiently ig. Proof. See appendix 11. On te oter and, te preference of currently ig-skilled workers over policy alternatives remains te same as in te case of temporary immigration. Altoug lemmata 3 and 6 indicate tat teir lifetime income is iger tan wen immigration is temporary, teir ordinal preference over policy 25 Te projections in figure 1 are calibrated wit α =.6, τ =.2, e =3an =1. Low e i is set to.6, and middle e i to.73. Since z L does not depend on te idiosyncratic cost of skill acquisition, any specific value is not assumed for ig e i. 26 As we observed in section 2.4 above, immigrants are not burdens via te PAYG pension system in our model. 24
alternatives does not cange and is monotonically increasing in permanent immigration. Since objective (20) and lemma 6 imply tat te utility of pensioners is always increasing in immigration, tey and currently ig-skilled workers again sare a same ordinal preference. Terefore, we continue to examine te majority voting outcomes under te same two scenarios as in te previous section. Te outcome under scenario I is again straigtforward. Proposition 3. Consider permanent immigration policy. Suppose no income support for low-skilled workers. If currently ig-skilled workers and pensioners form te majority at te voting stage, te referendum-led policy is to permit te maximum feasible rate of immigration. Proof. No income support for low-skilled workers means θ = 0. Ten, d > 0 by lemma 1. Under scenario I, d > 0 implies m [0, m], (1 + δ)(1 (m)) < (1 + δ) (m) + 1. Tat is, ig-skilled workers and pensioners continue to form te majority over te interval [0, m]. Wen immigration policy is permanent wit θ = 0, expression (10) for te igskilled reduces to z i := (1 τ) w H e i + b dwh. Sinceθ =0, > 0bylemma 1+r 3, db dzi > 0 by lemma 6 and ence > 0 for te ig-skilled. Objectives (18) and (20) ten imply tat everyone in te majority monotonically increases er/is utility as m increases. Te Condorcet winner is tus m. Tis result is te same as in proposition 1 for temporary immigration witout income support under scenario I, since te preference of te majority over policy alternatives as not canged. Under scenario II, we again ave te same outcome possibilities as for temporary immigration policy in proposition 2, but te conditions for tese possibilities are now sligtly different because we take into consideration te preferences of currently low-skilled workers tat are not te same as in te case of temporary immigration. Proposition 4. Consider permanent immigration policy. Suppose no income support for low-skilled workers. If currently low-skilled workers form te majority at te voting stage, te referendum-led policy is to permit (i) te 25
maximum feasible rate of immigration if eiter (1 τ) w L (0) + b (0) 1+r (1 τ) wl (m)+ b (m) 1+r (22) old or condition (22) does not old and tere exists m [0, m] satisfying condition (21), but (ii) no immigration if condition (22) does not old and tere is no m [0, m] satisfying condition (21). Proof. See appendix 12. Wat proposition 4 indicates is tat te tendency to decide on very liberal policy of immigration is even stronger wen policy is permanent tan wen it is temporary. First, all currently low-skilled workers may most prefer m wen condition (22) is met, and everyone agrees on m. Second, by comparing e in te proofs for lemmata 8 and 9, we observe tat condition (21) is more easily met for permanent policy tan for temporary policy even if condition (22) does not old. Tis result is intuitive because immigration in te next period is beneficial for all current workers during teir post-retirement period via te PAYG pension system. 3.3 Temporary immigration wit income support We now introduce te income support programme for low-skilled workers. As explained above, a common tax rate is applied to te earning of every worker as te source of te transfer fund, and every low-skilled worker, weter native or immigrant, receives a fixed lump sum benefit. Te preference of pensioners is not influenced via tis welfare programme, objective (20) and lemma 7 imply tat teir utility is monotonically increasing in m. Lemma 13. Suppose tere is income support for low-skilled workers. Suppose assumptions (A), (B) and (C) of lemma 2 olding. Te lifetime income for te ig-skilled is ten initially increasing but subsequently decreasing in temporary immigration, provided tat condition (A.2.1 ) for assumption (A) 26
is replaced by (A.8.1 ). Te lifetime income for te low-skilled is monotonically decreasing in it. Proof. See appendix 13. Remember tat, as stated in lemma 10, initially increases but subsequently decreases in m. Note tat we observe in appendices 8 and 13 tat z i for te ig-skilled and experience teir peaks wit te same rate of immigration. 0 low e i m middle e i ig e i z very ig e i Figure 2. Lifetime Incomes of Workers wit Different Costs of Skill Acquisition over Temporary Immigration Policy wen te Income Support Programme Operates Te workers wo remain low-skilled at any rate of immigration always ave te lifetime income for te low-skilled definedinlemma13. However, telifetimeincomeofworkeri among te rest of te currently low-skilled workers is some combination of z i for te low-skilled and ig-skilled. Wen m is low, condition (13) does not old. Te lifetime income of currently lowskilled worker i is ten identified by z i for te low-skilled. However, as m 27
continues to increase, condition (13) is met. Wen it is satisfied by equality, er/is lifetime income switces from te low-skilled to te ig-skilled witout discontinuity. If condition (13) continues to old until it reaces m, te lifetime income exibits only one kink at te switcing point over [0, m]. If condition (13) is violated wit ig m, te lifetime income exibits two kinks because tere are two switcing points. Figure 2 illustrates two projections wit one kink, i.e., te curves labelled as middle e i and ig e i. 27 Because of tese non-single-peaked preferences, altoug z i for te igskilled and te low-skilled are respectively single-peaked, e.g., te curves labelled as low e i and very ig e i respectively in figure 2, we cannot simply rely on Black s (1958) median voter teorem. Since we now observe tat pensioners and currently ig-skilled workers do not sare a same ordinal preference over te policy interval [0, m], we consider te following two scenarios regarding (0) (0, 1) and δ > 0 wen a referendum takes place. Scenario III (1 + δ)(1 (0)) + 1 < (1 + δ) (0) Scenario IV (1 + δ) (0) + 1 < (1 + δ)(1 (0)) In scenario III, currently ig-skilled workers form te majority. In scenario IV, currently low-skilled workers do so. 28 Proposition 5. Consider temporary immigration policy. Suppose tere is income support for low-skilled workers. Suppose assumptions (A), (B) and (C) of lemma 2 old. If currently ig-skilled workers form te majority at te voting stage, te referendum-led policy is to permit (1 α)(1 τ) ˆm := θ 1 α (ˆm) 1 (0, m), (23) 1 (ˆm) provided tat condition (A.2.1 ) for assumption (A) is replaced by (A.8.1 ). 27 Te figure illustrates te case were α =.6, τ =.2, e =3,m =1andθ =.095. Low e i is set to.46, middle e i to.475 and ig e i to.49. 28 Te case were pensioners dominate te majority, i.e., δ < 1, is trivial, for teir utility monotonically increases in m, according to lemma 10. Te cosen policy is ten m. 28
Proof. Since we assume tat currently ig-skilled workers cannot become low-skilled, tey continue to form te majority over [0, m] under scenario III. By setting dzi = 0 for te ig-skilled in (A.13.1), we solve for m wic maximises z i for te ig-skilled because it is initially increasing but subsequently decreasing in m, as stated by lemma 13. Te Condorcet winner is ten tis solution wic we denote by ˆm. Under scenario III, we find intermediate policy as a unique equiliblium wen te income support programme redistributes from ig-skilled to lowskilled workers including immigrants. Altoug immigration initially increases te utility of te majority, if it is too muc teir utility begins falling after a peak. Tis interior policy, ˆm, appears to be a strong candidate even if currently ig-skilled workers do not dominate te majority, as te following proposition indicates. Proposition 6. Consider temporary immigration policy. Suppose tere is income support for low-skilled workers. Suppose assumptions (A), (B) and (C) of lemma 2 old, were condition (A.8.1 ) replaces (A.2.1 ) for assumption (A). If currently low-skilled workers form te majority at te voting stage, te referendum-led policy is to permit (i) ˆm if tere exists m (0, ˆm) satisfying condition (21), but (ii) no immigration if tere is no suc m (0, ˆm). Proof. See appendix 14. Condition (21) assures tat tose wo prefer te status quo to m do not form te majority, even toug currently low-skilled workers form te majority. If tis condition olds, a referendum leads to ˆm defined in (23), as sown in appendix 14. In sum, propositions 5 and 6 imply tat te currently ig-skilled workers most preferred policy, ˆm, is a strong candidate in a referendum. 29
3.4 Permanent immigration wit income support Finally, we consider te interaction of te tree cannels wic transmit te effects of permanent immigration. As before, te preference of pensioners continues to increase in immigration monotonically, as objective (20) and lemma 7 imply. Lemma 14. Suppose tere is income support for low-skilled workers. Suppose assumptions (A), (B), (C) and (D) of lemmata 2 and 12 olding. Te lifetime income for te ig-skilled is ten initially increasing but subsequently decreasing in permanent immigration, provided tat condition (A.2.1 ) for assumption (A) is replaced by (A.9.1 ). Te lifetime income for te lowskilled is initially decreasing but subsequently increasing in it, provided tat φ is replaced by ψ in condition (A.11.1 ) for assumption (D). Proof. See appendix 15. As in te previous section, currently ig-skilled workers and pensioners do not sare a same ordinal preference over te policy interval [0, m]. Hence we continue to consider te referendum outcomes under scenarios III and IV. Proposition 7. Consider permanent immigration policy. Suppose tere is income support for low-skilled workers. Suppose assumptions (A), (B) and (C) of lemma 2 old. If currently ig-skilled workers form te majority at te voting stage, te referendum-led policy is to permit µ (1 α)(1 τ) 1 α ḿ : = + θ α τ θ µ 1+δ (ḿ) µ 1+r 1 ψ (ḿ) 1+ψ (ḿ) α (ḿ) 1 α 1 α (ḿ) 1 (0, m), (24) 1 (ḿ) provided tat condition (A.2.1 ) for assumption (A) is replaced by (A.9.1 ). Proof. Since we assume tat currently ig-skilled workers cannot become low-skilled, tey continue to form te majority over [0, m] under scenario III. By setting dzi = 0 for te ig-skilled in (A.15.1), we solve for m wic maximises z i for te ig-skilled because it is initially increasing 30