2.1. Labor Market Structure

Following the recent immigration literature [see Borjas (Reference Borjas2003), Ottaviano and Peri (Reference Ottaviano and Peri2012), Docquier et al. (Reference Docquier, Özden and Peri2013), Edwards and Ortega (Reference Edwards and Francesc2016)], total labor in each period t is aggregated in a nested CES function. The first nest accounts for imperfect substitution between less- and tertiary-educated workers. The second nest, accounts for imperfect substitution between natives and immigrants. One labor aggregate is considered for less-educated natives and immigrants, and another for highly educated natives and immigrants. Hence, the model allows for a different skill-specific complementarity between natives and foreign-born workers. The third nest accounts for imperfect substitution between legal and undocumented low-educated immigrants.

The total labor aggregate (Q _t ) is a nested CES function of the high- (Q _{h, t} ) and low-educated labor (Q _{l, t} ) expressed in efficiency units:

(1) $$\begin{equation} Q_t=\left[\theta _{h}Q_{h,t}^{\frac{\sigma _{H}-1}{\sigma _{H}}}+(1-\theta _{h}) Q_{l,t}^{\frac{\sigma _{H}-1}{\sigma _{H}}}\right]^{\frac{\sigma _{H}}{\sigma _{H}-1}}, \end{equation}$$

where θ _h is the relative productivity of high-educated labor and σ _H the elasticity of substitution between the two education groups.

The high-educated labor aggregate regroups tertiary-educated natives (N _{h, t} ) and immigrants (M _{h, t} ):

(2) $$\begin{equation} Q_{h,t}=\left[\theta _{e}N_{h,t}^{\frac{\sigma _{E}-1}{\sigma _{E}}}+\left(1-\theta _{e}\right)M_{h,t}^{\frac{\sigma _{E}-1}{\sigma _{E}}}\right]^{\frac{\sigma _{E}}{\sigma _{E}-1}}, \end{equation}$$

where θ _e is the relative productivity of educated native workers and σ _E is the elasticity of substitution between tertiary-educated natives and legal immigrants. Note that undocumented immigrants are not allowed to work as highly educated workers and are thus all assumed to be less educated. Orrenius and Zavodny (Reference Orrenius and Zavodny2004) argue that, although granting legal status might increase the competition between legalized and native workers, the latter keep a certain protection due to their language skills, their higher education level, and their better knowledge of the labor market institutions. The low-educated labor aggregate, Q _{l, t} , can therefore account for imperfect substitution between less-educated natives, N _{l, t} , and the less-educated foreign workers’ labor aggregate, Q _{m, t} :

(3) $$\begin{equation} Q_{l,t}=\left[\theta _{n}N_{l,t}^{\frac{\sigma _{N}-1}{\sigma _{N}}}+(1-\theta _{n})Q_{m,t}^{\frac{\sigma _{N}-1}{\sigma _{N}}}\right]^{\frac{\sigma _{N}}{\sigma _{N}-1}}. \end{equation}$$

The parameter θ _n represents the relative labor productivity level of native workers and σ _N is the elasticity of substitution between the native and the foreign-born workforce. Legal (M _{l, t} ) and undocumented (I _{l, t} ) foreign-born workers are also imperfect substitutes and regrouped in one additional nest:

(4) $$\begin{equation} Q_{m,t}=\left[\theta _{m}M_{l,t}^{\frac{\sigma _{M}-1}{\sigma _{M}}}+\left(1-\theta _{m}\right)\left(I_{l,t}\right)^{\frac{\sigma _{M}-1}{\sigma _{M}}}\right]^{\frac{\sigma _{M}}{\sigma _{M}-1}}. \end{equation}$$

The complementarity between legal and undocumented immigrant workers allows to account for a lower productivity of the latter [Chiswick (Reference Chiswick1988)]. Undocumented workers might moreover be restrained on their mobility due to the lack of proper documentation, lower information about employment possibilities, or networks concentrated in certain sectors [Massey (Reference Massey1987)]. Kossoudji and Cobb-Clark (Reference Kossoudji and Cobb-Clark2002) argue that the IRCA’s amnesty provisions impacted on the wages of legalized workers mainly by improving their labor mobility, allowing them to access better-paid jobs. The CES production function can account for these different explanations.Footnote ⁶

2.2. Profit Maximization

The production is represented by a Cobb–Douglas function, using capital K _t and labor expressed in efficiency units, Q _t . Capital is given by the total savings of the previous period, such that K _{t + 1} = S _t , and full depreciation is assumed:Footnote ⁷

(5) $$\begin{equation} Y_t=AK_{t}^\alpha Q_t^{1-\alpha }. \end{equation}$$

The firm’s problem can be written:

(6) $$\begin{eqnarray} \max _{Z_t, I_t} \pi &=& AK_{t}^\alpha Q_{t}^{1-\alpha }-w_{h,n,t}N_{h,t}-w_{h,m,t}M_{h,t}-w_{l,n,t}N_{l,t}\nonumber\\ &&-\,w_{l,m,t}M_{l,t}-w_{l,i,t} I_t-R_tK_t, \end{eqnarray}$$

where Z _t = N _{h, t} , M _{h, t} , N _{l, t} , M _{l, t} , K _t . The interest factor takes its competitive equilibrium value:

(7) $$\begin{equation} R_t=\frac{\alpha Y_t}{K_t}. \end{equation}$$

2.3. Utility Maximization

Each agent lives for two periods. Individual subscripts are left out for notational simplicity and the superscript (t) denotes the period of birth only when a distinction is necessary. When young, each individual supplies one unit of labor inelastically. Her income is either consumed or saved. The savings are used to consume when she becomes old and no bequests are left.Footnote ⁸ The lifetime utility of a j-type agent, born at time t, is given by

(8) $$\begin{equation} U_j^t=\ln (c_{j,t})+\beta \ln (d_{j,t+1})-V_j, \ \ \ \ \ \text{where}\ {\it j = hn,hm,ln,lm,li}. \end{equation}$$

β is the type-independent discount factor, whereas c _{j, t} and d _{j, t + 1} represent, for an agent of type j, the consumption of the single good at time t and t + 1. V _j is a non-monetary fixed cost that the illegal status imposes on immigrants without proper documentation. Thus, V _i ⩾ 0, whereas V _j = 0 for j = hn, hm, ln, lm.Footnote ⁹ This cost might reflect a variety of restrictions imposed by the illegal status, like the discomfort due to the irregular situation, the fear to be caught or limitations in the daily life that the absence of legal status imposes. I assume that this disutility is so strong that every undocumented immigrant prefers to be legalized and thus applies for an amnesty.Footnote ¹⁰ Given that she lives for two periods, the lifetime budget constraint of a j-type agent can be written:

(9) $$\begin{equation} \psi _{j}^t=c_{j,t}+\frac{d_{j,t+1}}{R_{t+1}}, \end{equation}$$

where R _{t + 1} is the return on savings. The annualized lifetime income of a j-type worker born in period t, ψ ^t _j , is given by

(10) $$\begin{equation} \psi _{j}^t = w_{j,t}(1-\tau _t^b-\tau _t^p)+g_t+\frac{p_{t+1}}{R_{t+1}}\qquad {\it for\ j = hn,hm,lm,ln},\\ \end{equation}$$

(11) $$\begin{equation} \psi _{i}^t = w_{i,t}(1-\tau _t^b\nu ^b-\tau _t^p\nu ^p)+\Theta g_t, \end{equation}$$

where w _{j, t} is the j-type worker’s wage, g _t the constant public transfer provided by the government to legal workers (i.e. when the individual is young) and p _{t + 1} the public pension transfer perceived at retirement. The income tax rate, τ ^b _t , and the pension contribution rate, τ ^p _t , are used to finance the general government budget and pensions, respectively. Undocumented workers are assumed to contribute to the government budget by paying a fraction ν ^b of the income taxes, and a fraction ν ^p of the pension contributions, paid by legal workers.Footnote ¹¹ Undocumented workers are eligible for specific public expenditures (e.g. urgent medical care, public education for their children). Therefore, they impose a cost on the government budget and the fraction of the transfers that an undocumented individual receives is denoted by Θ. However, they are not entitled to transfers from the public pension system when they retire.

Maximizing (8) subject to (9) yields per capita consumption and savings, which given the logarithmic utility function, are a constant fraction of the disposable life-cycle income:

(12) $$\begin{equation} c_{j,t} = \frac{\psi _{j}^t}{1+\beta },\\ \end{equation}$$

(13) $$\begin{equation} s_{j,t} = \frac{\beta \psi _{j}^t}{1+\beta }-\frac{p_{t+1}}{R_{t+1}},\\ \end{equation}$$

(14) $$\begin{equation} d_{j,t+1} = \frac{\beta \psi _{j}^t}{1+\beta }R_{t+1}. \end{equation}$$

Denoting the total number of j-type documented workers at time t with T _{j, t} , the aggregate disposable income is defined as

(15) $$\begin{eqnarray} \Psi _t&=&\sum\limits _{{j\in Z}} T_{j,t}\left(w_{j,t}(1-\tau _t^b-\tau _t^p)+g_t+\frac{p_{t+1}}{R_{t+1}}\right)\nonumber\\ &&+\,I_{l,t}\left(w_{l,i,t}(1-\nu ^b\tau _t^b-\nu ^p\tau _t^p)+\Theta g_t\right), \end{eqnarray}$$

where Z = hn, hm, ln, lm is used for notational convenience. The corresponding consumption and savings aggregates become

(16) $$\begin{equation} C_t=\frac{\Psi _t}{1+\beta },\ \ S_t=\frac{\beta \Psi _t}{1+\beta }-\frac{1}{R_{t+1}}\displaystyle \sum\limits _{{j\in Z}} T_{j,t+1}p_{t+1},\ \ D_t=R_t \frac{\beta }{1+\beta }\Psi _{t-1}. \end{equation}$$

2.4. The Government Budget

Income taxation collected at the rate of τ ^b _t constitutes the government’s revenue, whereas the pension contribution rate, τ ^p _t , finances the pay-as-you-go pension system.Footnote ¹² The split between two distinct government budgets, which are assumed to be balanced at each period t, allows to disentangle different fiscal effects. Public expenditures of the government consist of constant structural spending, G, and per capita transfers, g (which is an intra-generational transfer). The pay-as-you-go pension system collects contributions from workers and redistributes the funds in the form of a constant pension, p, to the retired individuals (and therefore represents an inter-generational transfer). Rearranging the two budget constraints yields the income tax rate:

(17) $$\begin{equation} \tau _t^b=\frac{g \left(N_{h,t}+M_{h,t}+N_{l,t}+M_{l,t}+\Theta I_{l,t}\right)+G}{N_{h,t}w_{h,n,t}+M_{h,t}w_{h,m,t}+N_{l,t} w_{l,n,t}+M_{l,t} w_{l,m,t}+\nu ^b I_{l,t} w_{l,i,t}}, \end{equation}$$

and the pension contribution rate:

(18) $$\begin{equation} \tau _t^p=\frac{p \left(N_{h,t}+M_{h,t}+N_{l,t}+M_{l,t}\right)}{N_{h,t}w_{h,n,t}+M_{h,t}w_{h,m,t}+N_{l,t} w_{l,n,t}+M_{l,t} w_{l,m,t}+\nu ^p I_t w_{l,i,t}}, \end{equation}$$

that balance the respective budget. The presence of undocumented workers in the economy is not neutral to the government budget. In the United States, there is evidence that some employers report and pay taxes on the wages paid to undocumented workers [either knowingly or unknowingly; see Becerra et al. (Reference Becerra, Androff, Ayón and Castillo2012) and the Congressional Budget Office (2007) report on the fiscal impact of undocumented immigrants and the studies reviewed therein]. Using firms’ Unemployment Insurance wage reports for the State of Georgia, Hotchkiss and Quispe-Agnoli (Reference Hotchkiss and Quispe-Agnoli2008) check the validity of the workers’ SSN and find that 0.39% of them (or the equivalent of “just over one million workers”) are invalid and used by undocumented immigrants. Thus, although they do not benefit from the public pension, undocumented workers partially pay income taxes (at compliance rate ν ^b ) and pension contributions (at compliance rate ν ^p ), and are entitled to a fraction (Θ) of the public transfer, g. Moreover, they contribute to the structural spending, G.

2.5. Population Dynamics

Population growth rates may differ among different agent groups. For simplicity, I assume exogenous population growth and education type. Children born in the United States have the U.S. citizenship, independent of their parents’ national and legal status. However, for notational simplicity, I assume here that each group of agents grows at a specific rate (i.e. immigrant children are implicitly accounted for in the growth rate of natives, whereas the growth rate of immigrants can be thought of as the inflow of new immigrants). Thus, high-skilled individuals give birth to high-skilled children and low-skilled individuals give birth to low-skilled children. Moreover, population growth rates can differ for legal and undocumented immigrants. Hence, population dynamics are written:

(19) $$\begin{equation} N_{h,t+1} = N_{h,t}n_{h,n,t}, \\ \end{equation}$$

(20) $$\begin{equation} M_{h,t+1} = M_{h,t}n_{h,m,t}, \\ \end{equation}$$

(21) $$\begin{equation} N_{l,t+1} = N_{l,t}n_{l,n,t}, \\ \end{equation}$$

(22) $$\begin{equation} M_{l,t+1} = M_{l,t}n_{l,m,t}, \\ \end{equation}$$

(23) $$\begin{equation} I_{l,t+1} = I_{l,t}n_{l,i,t}. \end{equation}$$

In the benchmark, the total population size and structure are assumed to remain constant (with the skill-origin-status specific population growth rate n _{e, s, t} = 1 ∀e = h, l and s = n, m, i). This assumption is relaxed in Section 6.1.

2.6. Definition of the Equilibrium

3.1. Short-Run Effects of a Policy Shock

In the short run, the capital stock K _T is determined by the agents’ savings of the previous period. Given the assumptions of the model (i.e. that the pension transfer amount is constant), the retired agents in period T (born in T − 1) are only affected by the policy shock through the impact that it has on their return on savings:

(28) $$\begin{equation} \Delta R_T=\alpha \left(\frac{\Delta {Q_{T}}}{K_T}\right)^{1-\alpha }. \end{equation}$$

The numerator in equation (28) captures the change in the composition of the workforce, which depends on the evolution of the foreign-born labor aggregate ΔQ _{m, T} = Q _{m, T} − Q _{m, s} :Footnote ¹⁸

(29) $$\begin{eqnarray} \begin{array}{rcl} \Delta Q_{m,T}&>&0 \displaystyle\Leftrightarrow (M_{l,s}+\varsigma \eta I_{l,s})^{\frac{\sigma _{M}-1}{\sigma _{M}}}-M_{l,s}^{\frac{\sigma _{M}-1}{\sigma _{M}}}\\[6pt] &>& \displaystyle\frac{1-\theta _m}{\theta _m} \left(I_{l,s}^{\frac{\sigma _{M}-1}{\sigma _{M}}}-(I_{l,s}(1-\eta -\delta ))^{\frac{\sigma _{M}-1}{\sigma _{M}}}\right). \end{array} \end{eqnarray}$$

The fraction η of immigrants whose situation is regularized receives a higher wage and the same transfers as the legal immigrants (instead of the fraction Θ, they received as undocumented workers). The model implies, by construction, that the fiscal channel affects workers only and not the retired agents: the workers pay an income tax and the contribution to the pension system and receive public transfers. The retired generation benefits from the provision of the constant pension, to which legalized workers contribute to. Hence, retired agents (i.e. individuals born in T − 1) are only affected through the change in their savings’ return. For agents born in T, the policy impacts their net income when they are young in addition to the effect on the interest factor when they retire:

(30) $$\begin{eqnarray} \begin{array}{ll} \Delta {\psi _{j,T}}= &(1-\tau _{T}^b-\tau _T^p)\Delta w_{j,T}-w_{j,T}(\Delta {\tau _T^b}+\Delta {\tau _T^p})+\frac{p}{R_{s}}\frac{\Delta R_{T+1}}{R_{T+1}} \ \ \ \text{for} \ j \ne i^{\ell } \\[4pt] \Delta {\psi _{i,T}^{\ell }}= &(1-\tau _T^b-\tau _T^p)\Delta w_{i,T}- w_{i,s}(\Delta \tau _T^b+\Delta \tau _T^p+\tau _s^b(1-\nu ^b)\\[4pt] &\quad+\,\tau _s^p(1-\nu ^p))+(1-\Theta )g+\frac{p}{R_{T+1}}, \end{array} \end{eqnarray}$$

where Δw _{j, T} = w _{j, T} − w _{j, s} represents the change in the j-type agents’ wage rate implied by a change in the population structure with a fixed capital stock. The following assumption is made on the values of the elasticities of substitution:

This ensures that low-educated native workers’ wage rate (w _{l, n, t} ) decreases with the size of the complementary low-educated foreign workforce (Q _{m, t} ), whereas the high-educated native wage rate (w _{h, n, t} ) increases (see Appendix B.4):

(31) $$\begin{equation} \frac{\partial w_{l,n,t} }{\partial Q_{m,t}}<0 ; \ \frac{\partial w_{h,n,t}}{\partial Q_{m,t}}>0. \end{equation}$$

A j-type worker’s net income is therefore affected by the policy’s impact on her gross wage rate and the net fiscal effect of the migration policy, reflected in the change of the income tax rate, Δτ ^b _T , and the pension contribution rate, Δτ ^p _T . Using the general notation, a migration policy affects these two rates as follows:

(32) $$\begin{equation} \Delta {\tau _T^b} = \frac{g(\eta I_{s}-(\delta +\eta )\Theta I_{s})-\tau _T^b\Delta W_T^b}{W_{s}}, \\ \end{equation}$$

(33) $$\begin{equation} \Delta {\tau _T^p} = \frac{p\eta I_{s}-\tau _T^p\Delta W_T^p}{W_{s}}, \end{equation}$$

where W^b _s = N _{h, s} w _{h, n, s} + M _{h, s} w _{h, m, s} + N _{l, s} w _{l, n, s} + M _{l, s} w _{l, m, s} + ν ^b I _{l, s} w _{l, i, s} is the steady state taxable income base for the government budget and W^p _s = N _{h, s} w _{h, n, s} + M _{h, s} w _{h, m, s} + N _{l, s} w _{l, n, s} + M _{l, s} w _{l, m, s} + ν ^p I _{l, s} w _{l, i, s} is the steady state taxable income base for the pension system. The first term in the numerator of equations (32) and (33) captures the change in the composition of the population entitled to the public transfer and the pension, respectively. The lower is the access of undocumented workers to public transfers (i.e. the lower is Θ), the higher is the regularization’s effect on the income tax rates. The second term in the numerator of equations (32) and (33) accounts for the change in the taxable income base due to the evolution of the population structure. The taxable income base is affected by the change in the wage rates of all documented workers and the change in the taxable income of the undocumented workers (i.e. the second line in equations (34) and (35)):

(34) $$\begin{eqnarray} \begin{array}{ll} \Delta W_T^b= &N_{h,s}\Delta {w_{h,n,T}}+M_{h,s}\Delta {w_{h,m,T}}+N_{l,s}\Delta w_{l,n,T}+M_{l,s}\Delta {w_{l,m,T}}\\ &+\,I_{l,s}(w_{\ell ,T}\eta +(1-\eta -\delta )\nu ^b w_{l,i,T}-(\eta +\delta ) \nu ^b w_{l,i,s}) \end{array} \end{eqnarray}$$

and

(35) $$\begin{eqnarray} \begin{array}{ll} \Delta W_T^p= &N_{h,s}\Delta {w_{h,n,T}}+M_{h,s}\Delta {w_{h,m,T}}+N_{l,s}\Delta w_{l,n,T}+M_{l,s}\Delta {w_{l,m,T}}\\ &+\,I_{l,s}(w_{\ell ,T}\eta +(1-\eta -\delta )\nu ^p w_{l,i,T}-(\eta +\delta ) \nu ^p w_{l,i,s}) . \end{array} \end{eqnarray}$$

The taxable income of the undocumented workers (i.e. the second line in equations (34) and (35)) is affected by three elements. The first captures the increase in the taxable income base due to the legalized workers (who receive wage w _{ℓ, T} ). The second term represents the taxable income of the remaining undocumented workers, whereas the third term captures the loss of taxable income due to an amnesty or a deportation of undocumented workers. The fiscal impact of a migration policy results from the effects implied by the change in the population structure on the taxable income base and the net fiscal contribution of legalized agents (i.e. whether the additional taxes that they pay compensate for the additional transfers that they perceive, see equations (32) and (33)). A larger taxable income base (with ΔW^x _T > 0 for x = b, p) decreases the income tax rate, τ ^b _T , and the pension contribution rate, τ ^p _T , respectively.

3.2. Long-Run Effects

Each policy considered in this paper is by assumption a one-time shock (i.e. a “one-shot” policy) and no further changes in the population structure occur after the shock.Footnote ¹⁹ Under a constant population structure, the long-run effects in any of the two policy scenarios depend entirely on the capital accumulation dynamics that it generates. The difference in the inter-temporal welfare of two subsequent generations (not directly affected by the shock) is therefore exclusively due to the different capital levels and its implications (on income and taxation). In the post-shock periods, the disposable income of a legal j-type agent adjusts compared to the one of the previous generation:

(36) $$\begin{eqnarray} \Delta \psi _{j,T+z}&=&(1-\tau _{T+z}^b-\tau _{T+z}^p)\Delta w_{j,T+z}-(\Delta \tau _{T+z}^b+\Delta \tau _{T+z}^p)w_{j,T+z-1}\nonumber\\ &&+\,\frac{p}{R_{t+z}}\frac{\Delta R_{T+z}}{R_{T+z}} \ \ \ \text{with} \ z\ge 2. \end{eqnarray}$$

Under a constant population, the wage rates (the interest factor) increase (decreases) with the capital level. A higher capital accumulation implied by a policy shock thus leads to higher gross wages and a lower interest factor for the following generations. The tax rates decrease with the taxable income base:

(37) $$\begin{equation} \Delta \tau _{T+z}^x=-\frac{\tau _{T+z-1}^x \Delta {W_{T+z}}}{W_{T+z}} \ \ \ \text{with} \ z\ge 2 \ {\it {\rm and}\ x = b, p}. \end{equation}$$

The evolution of a j-type agent’s disposable income (given by equation (36)) is thus linked to the change in the total labor aggregate.

3.3. The Case of an Amnesty

An amnesty implies a change in the regularized workers’ gross wage (which is likely to increase) and provides them a full access to public transfers. Simultaneously, legalized immigrants must comply to the payment of all income taxes and pension contributions.

The impact of a regularization on the labor aggregate is not clear (see Proposition 1). If the latter increases, the capital–labor ratio decreases in the short run (i.e. at constant capital stock) which leads to higher returns on the retired individuals’ savings. The less-educated workers earn lower gross wages, whereas the highly educated workers benefit from higher wages. The fiscal impact of the amnesty depends on the change in the net contribution of undocumented workers and the taxable income of the legal workers. Income taxes used for the public transfer can decrease if the additional taxes levied on the legalized workers compensate for their additional transfers. Undocumented workers are net contributors to the pension system (from which they cannot benefit). When the amnesty occurs in T, the legalized workers contribute to finance the pensions of the retired individuals and hence τ ^p _T is certain to decrease (which benefits legal workers). However, legalized individuals benefit from a pension when they are old, which weights on the next generation of workers and impacts the contribution rate that the latter have to pay (τ ^p _{T + 1}).

The generation born after the period in which the shock occurs, (T + 1) faces a capital stock which was built up by the previous generation’s savings. These increase if the additional income received by the legalized and the highly educated workers compensates the decrease of the less-educated workers' income. A higher capital accumulation implies an increase in wages which can partially counterbalance the impact of the change in the population structure. The net effect of an amnesty depends on the combination of these different effects.

3.4. The Case of a Deportation

The deportation of a fraction δ of the undocumented workers decreases labor in efficiency units (see Proposition 1).Footnote ²⁰ Thus, the capital–labor ratio decreases and reduces the return on the retired individuals’ savings. The latter are therefore certain to lose from the deportation of undocumented workers. The lower workforce implies, under Assumption (1), a decrease (increase) in the high-educated (low-educated) workers’ wages. The fiscal impact of a deportation depends on the net contribution of the deported workers and their impact on the taxable income of remaining workers. If the deported workers were net contributors to the government budget, a deportation is likely to increase the income tax rate τ ^b _T . The contribution rate to the pension system, τ ^p _T , is certain to increase as undocumented workers are net contributors. The deportation of these net contributors is not compensated by an increase in the taxable income base.

The generation born after the period in which the shock occurs, (T + 1) faces a capital stock which was built up by the previous generation’s savings. Although less-educated workers earn higher wages, these cannot compensate the loss of the undocumented workforce and the lower wages of highly educated workers. The lower capital accumulation that follows decreases wages in the long run. The net effect of a deportation depends on the combination of these different effects.

4.1. The Impact of an Amnesty – Benchmark Results

In the benchmark amnesty scenario, it is assumed that 100% of the undocumented workers are regularized (η = 1, δ = 0). Even though a 100% legalization rate is never observed in reality [Levinson (Reference Levinson2005)], this scenario allows to highlight the potential upper-bound effects of an amnesty. Figure 1 shows the impact of a migration policy on the lifetime utility of a low- and high-skilled native, born in period t, respectively. The average native’s welfare, accounting for the skill structure of the native population, is also provided. Each curve represents the deviation of the lifetime utility of an individual born in t relative to the baseline scenario where no shock occurs, as shown in equation (26). The shock occurs in period T.

Figure 1. Effects of a migration policy: amnesty and deportation. (a) Amnesty. (b) Deportation. Effect of a migration policy (an amnesty in Figure 1a and a deportation in Figure 1b) on the lifetime utility of a low-educated, high-educated, and average native individual born in t (horizontal axis). The policy shock occurs at time T and the reference steady state utility belongs to the generation born in period T − 2 (i.e. that is not affected by the policy shock).

Figure 1a shows that the impact of a complete amnesty on natives’ lifetime utility is generally limited, with the change lying between −0.07% and +0.03%. The retired agents in period T, born in period T − 1, benefit from a higher return on their savings due to the lower capital–labor ratio implied by the increase in the labor aggregate [see equation (28)]. This translates into a higher lifetime utility compared to the baseline scenario (i.e. without an amnesty). The workers born in period T, in which the shock occurs, are impacted through several channels. The increase of the labor aggregate decreases less-educated workers’ wages by 1%. The lower wages of the less-educated natives are however more than compensated by the higher wages of the highly educated natives (+0.05%) and the regularized individuals. The taxable income base therefore increases, which reduces the income tax rate (−0.054 percentage points). In the period of the shock, legalized individuals contribute to finance the pension of retired individuals without yet imposing a cost on the system (as they perceive pensions only one period later). The pension contribution rate decreases by 0.12 percentage points.

The higher income base implies a higher capital accumulation in the long run. However, the capital–labor ratio remains below the baseline level which translates into a higher interest rate and lower wage rates. In addition, under a constant population structure, the increase in taxable income base leads to a lower income tax rate, τ ^b . On the other hand, the fact that legalized individuals in period T have access to pensions in period T + 1 increases the pension contribution rate τ ^p . The combination of these different channels leads to a slight increase in highly educated natives’ welfare in the long run. The negative impact on wages dominates the other channels for the less-educated natives, such that their lifetime utility decreases compared to the baseline scenario.

4.2. The Impact of a Deportation – Benchmark Results

In the benchmark deportation scenario, it is assumed that 100% of the undocumented workers are expelled (η = 0, δ = 1). This rather extreme assumption allows to compare this policy to a complete amnesty. The change in the total number of workers (and thereby the labor aggregate) at time T constitutes the main difference between a deportation and an amnesty. Note that tracking and expelling undocumented residents is done at no cost such that the impact of a deportation should be seen as a lower bound.

Figure 1b shows that the impact of a deportation on a native agent’s lifetime utility ranges between −0.07% and +0.12%. Decreasing the size of the workforce through a deportation of the whole undocumented workforce decreases the capital–labor ratio. This decreases the interest factor that retired individuals (in period T) receive on their savings. Less-educated native and immigrant workers benefit from an increase of 1.9% and 3.9% in their respective wages. The high-skilled wage decreases by 0.1%. Even though undocumented workers are net contributors to the government budget, the income tax rate τ ^b decreases in the short run (−0.037 percentage points) due to the increase in less-educated workers’ wages. This decrease is however lower than under an amnesty. The pension contribution rate increases in period T, as undocumented workers are net contributors to the pension system.

In the long run, capital accumulates less than under the baseline scenario, implying a lower (higher) increase (decrease) in less-educated (highly educated) workers’ wages. The lower capital–labor ratio also implies a slightly higher interest factor. The higher wages of less-educated workers are however not enough to compensate for the loss of the undocumented workers’ fiscal contribution, such that the income tax rate and the pension contribution rate are higher in the long run. The combination of these different elements leads to an improvement of less-educated natives’ welfare in the long run, whereas highly educated natives experience a welfare loss. The average native’s welfare is also below the baseline scenario (i.e. without a policy shock).

5.1. Structure of the Government Budget

Figure 2 depicts how changing various key parameters of the model affects the impact of an amnesty on the average native’s welfare. This captures the role played individually by each parameter and allows a deeper understanding of the mechanisms at hand. The following results relate to a full liberalization, if not stated otherwise.

Figure 2. Robustness to a change in the structure of public finances. (a) Public consumption (as % of GDP) (b) Share of public funds devoted to pensions. (c) Contribution to the government budget (ν ^b ). Contribution to the pension system (ν ^p ). (e) Discrimination in the access to transfers (Θ). Note: Sensitivity of an average native’s welfare (born in period t) to a change in different parameters defining the structure of the public finances with an amnesty occurring at time T.

Figure 2a highlights the effects of an amnesty for different values of the public consumption (expressed as % of GDP). The 0% line shows the case where no public funds are consumed (and hence all income taxes are transferred to the agents), whereas the 17.6% line captures the case where the whole government budget is consumed (and hence the agents receive no transfers or pensions). In the latter case, the new steady state welfare is slightly above the initial value. Legalizing workers implies a higher income tax base in this case, which serves to finance the infrastructure (i.e. a fixed cost which does not enter the utility function of the representative agent). Legalized workers do not receive any form of transfers but fully contribute to the public expenses (due to full compliance to the payment of income taxes) which reduces the tax burden on native workers. On the other hand, if the whole budget is spent on transfers, average native welfare decreases more than under the benchmark calibration. The reason is that the regularized workers’ net contribution is lower. As undocumented workers, they subsidize the legal workers’ transfers. Once regularized, they have access to the same transfer amount.

Figure 2b highlights the impact of an amnesty under different assumptions for the share of public funds devoted to pensions (i.e. transfers to the retired generation). In the absence of a pension system, the complete fiscal impact of undocumented workers falls on the general government budget, which is composed of a structural expense and transfers to the workers (i.e. a unique intra-generational transfer). The income tax rate increases in the short run as the increase of the legalized individuals’ wages is not enough to compensate for the additional transfers that they perceive and the impact that they have on legal workers’ wages. Individuals born at the period of the shock are particularly harmed. However, capital accumulation implies that the average native worker is better off in the long run. A full dedication of public transfers to the pensions (i.e. a unique inter-generational transfer), substantially decreases the pension contribution rate τ ^p _T used to finance pensions in the period of the shock, which improves the native agent’s welfare beyond the benchmark level. However, once they retire, the regularized individuals have access to the full pension and thus total pension expenses exceed the benchmark level. Hence, average native welfare is below the benchmark level in the long run.

Figure 2c shows the impact of varying the undocumented workers’ compliance to income taxation (ν ^b ) while keeping compliance to the pension contributions (ν ^p ) at 50%. If only 10% of undocumented workers pay income taxes, a complete amnesty has a stronger positive impact on the government budget due to the full compliance to income taxation that it generates. If undocumented workers already fully comply to income taxes, a legalization generates a higher fiscal burden, as the legalized individuals benefit from full access to public transfers while their contribution only increases due to their higher wages. Average welfare is therefore lower in the full compliance scenario. Figure 2d shows the impact of varying the undocumented workers compliance to the pension contributions (ν ^p ) while keeping compliance to income taxation (ν ^b ) at 50%. The pension system differs from the general government budget because undocumented workers contribute to the fund without being able to perceive any transfer from it (i.e. they are net contributors). The lower is the initial compliance rate, the higher is the additional contribution to the system at the period of the shock (as legalized workers only perceive the pension one period later) and the stronger is the decrease in the pension contribution rate. This explains why the impact of an amnesty is less negative under low compliance than in the benchmark. If undocumented workers fully comply to the payment of pension contributions, a legalization affects the pension system only through its impact on wages. The increase of legalized workers’ wages can compensate for the decrease of the legal workers’ wages, thereby leading to a reduction of the taxation rate, which is however smaller than in the benchmark.

Figure 2e shows the importance of undocumented immigrants’ access to the public transfer. The higher is availability of public support for undocumented workers, the lower is the fiscal effect of a regularization given that it generates a lower additional fiscal burden and a higher taxable income base. If undocumented workers receive only 1% of the transfers prior to regularization (Θ= 0.01), the average native loses at most 0.03% of her welfare with regularization. On the other hand, the absence of discrimination in the access to public transfers (Θ = 1) leads to a pure increase in the fiscal base. The decrease in the income tax rate is enough to compensate for the lower wages of less-educated native workers such that native workers’ net income is higher in the long run. Thus, the average native’s welfare level is above the benchmark amnesty and close to the baseline value.Footnote ³⁰

5.2. Number of Undocumented Immigrants

Figure 3 analyzes the impact of changing the number of undocumented workers affected by a policy. Figures 3a and 3b show the effect on average native welfare of regularizing or deporting a lower fraction of the undocumented workers, respectively. Reducing the share of the 7.9 million undocumented individuals affected by the policy mitigates its impact. The average retired and working native benefit less from an amnesty, whereas the long-term impact is less negative than under a full amnesty. In the case of a deportation, the average retired individual (i.e. born in T − 1) experiences a lower decrease in welfare, whereas the average native worker benefits less from it. The long-run average welfare level is also above the benchmark one. In Figures 3c and 3d, the two shocks are simulated in the benchmark economy with a lower (5 million) or a higher (12 million) number of undocumented workers. The results for both scenarios with a lower number of undocumented immigrants follow the same rationale as the scenarios presented in Figures 3a and 3b (i.e. a mitigation of the policy’s impact). A higher number of undocumented workers amplifies the effects observed in the benchmark. A legalization of 12 million undocumented workers increases benefits to retired individuals but decreases benefits to the workers and the long-run average native welfare level. A deportation of the same number of individuals, decreases retired individuals’ welfare but increases that of the native workers. Reduced capital accumulation however implies a long-run welfare level below the benchmark one.

Figure 3. Robustness to a change in the number of undocumented workers. (a) Proportion of regularized. (b) Proportion of deported. (c) Amnesty – Number of undocumented workers. (d) Deportation – Number of undocumented workers. Note: Sensitivity of an average native’s welfare (born in period t) to a change in the number of undocumented workers affected by a policy shock occurring at time T.

5.3. Sensitivity to the Elasticity of Substitution

Figure 4 underlines the impact of changing the different elasticities of substitution. In each alternative scenario, the value of one of the four elasticities is varied while the remaining three elasticities are kept at their respective benchmark level (which are σ _H = 3, σ _E = σ _N = 20, and σ _M = 1, 000). Note that all the combinations of the elasticities of substitution respect Assumption (1). The values chosen are in general quite extreme and do not necessarily rely on estimations from the empirical literature. The purpose of this section is to highlight to what extent varying the values of these elasticities can change the results.

Figure 4. Robustness to a change in the elasticities of substitution. (a) Change in σ _H . (b) Change in σ _E . (c) Change in σ _N . (d) Change in σ _M . Sensitivity of an average native’s welfare (born in period t) to the change in the elasticities of substitution with an amnesty occurring at time T.

Figure 4a shows that a higher substitutability between less and highly educated workers (i.e. a higher σ _H ) reduces the negative impact of a policy affecting the composition of the less-educated immigrant labor aggregate. In this case, the labor aggregate increases more than in the benchmark. High-skilled wages increase less, whereas less-educated wages decrease less. However, the interest rate increases and the income tax rate decreases slightly more relative to the benchmark simulation. Figure 4b highlights that the elasticity of substitution between highly educated natives and immigrants does not impact the results because the high-skill labor aggregate is not directly affected by a migration policy focusing on less-educated immigrants. The elasticity between less-educated natives and immigrants is of particular importance (see Figure 4c). A low value of σ _N implies a high complementarity between less-educated natives and immigrants, which mitigates the decrease of less-educated natives’ wages [as it is the case for σ _N = 6, which is the lowest value estimated in the literature, see Manacorda et al. (Reference Manacorda, Manning and Wadsworth2008)] and their long-run welfare loss.

The most disputable calibration choice is the value for the elasticity of substitution between legal and undocumented foreign-born workers, which has not been estimated in the literature so far. I follow the strategy of Edwards and Ortega (Reference Edwards and Francesc2016) and set a high initial value for this parameter (σ _M = 1, 000). This value is already so high that increasing it further barely affects natives’ welfare. However, if a higher complementarity is assumed, with σ _M = 20, the less-educated natives’ wages decrease less, whereas the highly educated natives’ wages increase more than in the benchmark scenario. Capital accumulation therefore increases which leads to a higher long-run average welfare level.

6.1. Population Dynamics

The benchmark calibration assumed a constant population. In this section, the population size changes using two different approaches. The first set of simulations allows for different population growth rates n _{s, t} , for s = hn, hm, ln, lm, from period T on (i.e. retired individuals in T are not affected). Note that the undocumented workforce is kept constant (and equal to 0 after a regularization). Three different scenarios, detailed in Table 2, are compared to the benchmark. “Pop. Scenario 1” uses a skill-specific population growth rate, calibrated on the population projections of the U.S. Census (2014) for the period 2015–2040. The native and foreign-born populations are projected to grow by 12.5% and 46.8%, respectively. In order to account for skill-specific population growth rates and for the increasing trend in educational attainment, the native and foreign-born tertiary-educated population is assumed to grow 40% and 15% faster than the respective less-educated population group. This allows to consider a catch-up of the immigrant population in terms of educational attainment. “Pop. Scenario 2” assumes a constant highly educated population and uses the growth rates for the less-educated population from “Pop. Scenario 1”. Hence, the fraction of highly educated individuals among the total population decreases in this scenario. “Pop. Scenario 3” assumes a constant less-educated population and uses the growth rates for the highly educated population from “Pop. Scenario 1”. Hence, the fraction of highly educated individuals among the total population increases.

Table 2. Population Dynamics – different population growth rates

These scenarios are shown in Figure 5c.

Figures 5a and 5b show the impact of an amnesty under the three different scenarios on a highly educated and a less-educated native’s welfare, respectively. The population size is only affected from period T + 1. Hence, retired individuals in period T (born in T − 1) are not impacted by the change in the population growth. Native workers, born in T, are only affected by a change in the return on savings when they are old. The lung-run impact of a policy shock in the presence of population growth is affected by two main channels. First, the effect of a policy that changes the structure of the population dissipates with an increasing population size: e.g. changing the status of 5% of the total population size at period T has a different impact on individuals living in period T + 3 if the population size is constant or doubles over that time period. Second, the effect of a change in the population’s skill-composition varies with the education level of the individual. The differential population growth rates affect the wages, the income tax rates and the pension contribution rate of generations born after period T. The fiscal burden increases with the population size, whereas income taxation tends to decrease with the share of highly educated workers in the economy (because it is proportional to the wages).

Figure 5. Impact of population dynamics. (a) Population growth – highly educated native. (b) Population growth – less-educated native. (c) Population growth – average native. (d) Number of undocumented workers. Note: Sensitivity of a native’s welfare (born in period t) to a change in the population growth rates. Figures 5a and 5b show the impact of differential population growth rates on the welfare of a highly educated and less-educated native, respectively. Figure 5c shows the impact of differential population growth rates on the average native’s welfare. Figure 5d shows how a change in the number of undocumented workers after an amnesty in period T (i.e. an “attraction effect”) affects the average native’s welfare. The different scenarios are shown in Tables 2 and 3.

The number of less and highly educated individuals increases in “Pop. Scenario 1”, although the share of the latter in the total population increases over time. As all the labor aggregates increase, the impact of an amnesty dissipates over time relative to the benchmark scenario. The long-run increase (decrease) in highly educated (less-educated) wages is mitigated. The capital–labor ratio increases over time. Although this has a negative impact on the returns to savings, the lower decrease in less-educated native workers’ wages improves their welfare relative to the benchmark scenario. The positive long-run impact of an amnesty on highly educated natives’ welfare is however mitigated by its lower impact on wages. In “Pop. Scenario 2,” the share of less-educated natives in the population progressively increases, whereas the number of highly educated workers remain constant. As the less-educated labor aggregate increases, the negative long-term effect of the amnesty on the less-educated wage rates dissipates. This effect dominates the other channels and leads to a lower long-run welfare decrease for the less-educated workers. The decrease in the labor aggregate reduces capital accumulation and increases income taxation. The combination of these channels decreases the long-run welfare of highly educated native workers below the benchmark level. “Pop. Scenario 3” implies an increase in the share of highly educated workers in the economy, whereas the number of less-educated workers remains constant. The less-educated labor aggregate does not change relative to the benchmark case, and thus the negative impact of an amnesty on the less-educated native workers’ wages remains important. The higher share of high-wage earners leads to a stronger capital accumulation and a lower interest factor. The income tax rate slightly decreases at the same time. The combination of these channels, and the fact that an amnesty mainly affects less-educated workers, implies that its welfare effect does not change much in the alternative scenario in which the less-educated population is constant. The impact of a migration policy on the average native welfare level, depicted in Figure 5c, is a combination of the channels described above and a population composition effect. For example, although the effect of an amnesty under “Pop. Scenario 3” is close to the benchmark for both groups of natives, the average native welfare level shows an increasing trend. This is due to a pure population composition effect: the average native is increasingly educated, and thus less affected by an amnesty.

In the second set of robustness exercises, an amnesty can serve as a signal for new prospective immigrants, who hope that such a procedure is repeated in the future (see Table 3 for details). Thus, although intended as a means to manage existing stocks of undocumented workers, an amnesty can motivate new entries through this “attraction effect.” Footnote ³¹ “Ileg. Scenario 1” assumes that the regularized population is replaced at a 50% rate by new undocumented workers. Hence, while 7.9 million undocumented workers are regularized, the undocumented workforce in period T increases by 3.95 million new undocumented individuals. “Ileg. Scenario 2” assumes that the “attraction effect” materializes one period after the amnesty. Hence, in period T, the full undocumented population is regularized while 3.95 million new undocumented individuals arrive at the beginning of period T + 1. Note that after these different shocks, the undocumented population is assumed to remain constant in all the scenarios.

Table 3. Population Dynamics – nb. of undoc. workers

These scenarios are shown in Figure 5d.

Figure 5d compares these two scenarios with the benchmark. “Ileg. Scenario 1” increases the undocumented workforce in the period of the amnesty, T, thereby decreasing the capital–labor ratio, which benefits the retired agents (born in T − 1). At the same time, the higher labor aggregate decreases wages further, which is not offset by the positive contribution of these new undocumented workers to the government budget. Hence, the average native is worse off in the period of the shock. These mechanisms are delayed in “Ileg. Scenario 2,” where the inflow of new undocumented workers arrives one period later. Hence, the impact of these undocumented immigrants only occurs through the interest rate that natives perceive in period T + 1 (i.e. wages in T are not affected). As in the previous scenario, new arrivals are particularly detrimental for the workers of that period (here, T + 1) who receive lower wages. The long-run population is higher than the benchmark level and identical in both alternative scenarios. Capital accumulation is therefore higher, which implies an equilibrium above the benchmark one.

6.2. Lower Productivity of Legalized Workers

In this section, regularized workers are assumed to have a lower productivity than documented immigrant workers [ς<1 in equation (25)]. This allows to account for a negative selection of undocumented immigrants into the illegal status (i.e. assuming that undocumented immigrants are, on average, inherently less productive). Amuedo-Dorantes and Bansak (Reference Amuedo-Dorantes and Bansak2011) find evidence on occupational mobility among individuals legalized under the IRCA, though the impact on their labor market participation is mixed [see also Casarico et al. (Reference Casarico, Facchini and Frattini2016) for a theoretical framework]. In a recent study using New Immigrant Survey data, Lofstrom et al. (Reference Lofstrom, Hill and Hayes2013) analyze how acquiring Legal Permanent Resident status influences labor market outcomes for previously undocumented workers. They do not find evidence of improved employment outcomes attributable to legal status for the low skilled.

Setting the productivity parameter ς<1 leads to a lower increase in the labor aggregate, which reduces the return on savings and the welfare of retired individuals in period T (see Figure 6). If the relative productivity parameter is sufficiently low, legalized workers are less productive than they were as undocumented workers. The labor aggregate decreases in that case, which implies a lower return rate on the retired agents’ savings (as it is the case in the scenario with ς = 0.5). The impact on the young generation is more mitigated. The lower increase in the less-educated labor aggregate leads to a lower decrease of the less-educated wage rates, whereas the highly-educated workers’ wages increase less. A lower ς implies lower wages for the regularized workers, who therefore contribute less to the government budget and the pension system (while they perceive the same transfers). Hence, the lower are the legalized workers’ wage rates, the higher are income tax rates. The lower net incomes lead to lower capital accumulation and thus to long-run average native welfare levels below those observed in the benchmark.

Figure 6. Different productivity of legalized workers. Note: Sensitivity of an average native’s welfare (born in period t) to a change in the relative productivity of a legalized worker with an amnesty occurring at time T.

6.3. Discrimination

In this section, I explore alternative explanations for the lower wages of undocumented migrants and address the consequences of an immigration policy reform under these assumptions. Rivera-Batiz (Reference Rivera-Batiz1999) argues that only half of the 41.8% wage premium that legal immigrants have over undocumented immigrants can be explained by observable characteristics. In the model, this traduces in employers being able to pay undocumented workers below their marginal productivity. Palivos (Reference Palivos2009) provides an alternative explanation based on a Nash bargaining model for the wages, where an undocumented immigrant has an outside option far below the legal migrants’ wage rate and therefore accepts to work for a discounted wage. In the presence of labor market discrimination, the representative firm extracts a rent by paying undocumented workers a fraction γ<1 of their marginal product (w _{l, i, t} ). The firm’s problem can be written as

(40) $$\begin{eqnarray} \max _{Z_t, I_{l,t}} \pi &=& AK_{t}^\alpha Q_{t}^{1-\alpha }-w_{h,n,t}N_{h,t}-w_{h,m,t}M_{h,t}-w_{l,n,t}N_{l,t}-w_{l,m,t}M_{l,t}\nonumber\\ &&-\,\gamma w_{l,i,t} I_{l,t}-R_t K_t , \end{eqnarray}$$

where Z _t = N _{h, t} , M _{h, t} , N _{l, t} , M _{l, t} , K _t and γ represent the fraction of her marginal productivity that an undocumented immigrant receives (and hence 1−γ represents the extent of discrimination on the labor market). In the absence of controls and sanctions for hiring undocumented immigrants, the firm thus hires the complete fixed stock of undocumented workers.

Alternatively, it can be assumed that the lower wages are due to the risk of financial sanctions for hiring undocumented workers passed on to them by the firm [see Chau (Reference Chau2001)]. In that case, the representative firm maximizes profits as follows:

(41) $$\begin{eqnarray} \max _{Z_t, I_{l,t}} \pi &=& AK_{t}^\alpha Q_{t}^{1-\alpha }-w_{h,n,t}N_{h,t}-w_{h,m,t}M_{h,t}-w_{l,n,t}N_{l,t}-w_{l,m,t}M_{l,t}\nonumber\\ &&-\,w_{l,i,t}I_{l,t}-\varsigma I_{l,t}-R_tK_t, \end{eqnarray}$$

where ς represents the cost for hiring one undocumented worker (e.g. a sanction). Both interpretations lead to a wage below marginal productivity for undocumented immigrants, which is equivalent when ς = w _{l, i, t} (γ − 1). The difference between the two approaches is that the funds collected through sanctions would increase the government budget, whereas the rent is distributed directly to the legal capital owners (i.e. retired individuals).Footnote ³² While the micro foundation might be clearer in the case of a fine, it is hard to argue that an amnesty would lead to a significant decrease in the public revenues through the lower amount of sanctions collected, given that these are rarely imposed in practice.Footnote ³³

If the firms pay undocumented immigrants below their marginal productivity, the dividend to be distributed is equal to the total rent captured by the capital holders divided by the number of retired legal agents (i.e. born in T − 1):

(42) $$\begin{equation} \Gamma _t=\frac{(1-\gamma )I_{l,t}^t w_{l,i,t}}{N_{h,t}^{t-1}+M_{h,t}^{t-1}+N_{l,t}^{t-1}+M_{l,t}^{t-1}}. \end{equation}$$

The per capita rent received by legal workers is added to their lifetime budget constraint:

(43) $$\begin{equation} \psi _{j}^t=w_{j,t}(1-\tau _t^b-\tau _t^p)+g+\frac{p+\Gamma _{t+1}}{R_{t+1}} \ \ \ \ \ {\it for\ j = hn,hm,ln,lm}.\ \end{equation}$$

Hence, legal workers’ welfare is additionally affected by the migration policy’s impact on the rent:

(44) $$\begin{eqnarray} \Delta \Gamma _T&=&\frac{(1-\delta )(1-\gamma )I_{l,t}^tw_{l,i,T}}{L_T^{t-1}}-\frac{(1-\gamma )I_{l,t}^tw_{l,i,t}}{L_t^{t-1}}\nonumber\\ &=&\frac{(1-\gamma )I_{l,t}^t\Delta w_{l,i,T}-\delta (1-\gamma )I_{l,t}^tw_{l,i,T}}{L_t^{t-1}}, \end{eqnarray}$$

where L ^{t − 1} _t = L_T ^{T − 1} = N ^{T − 1} _{h, T} + M _{h, T} ^{T − 1} + N ^{T − 1} _{l, T} + M _{l, T} ^{T − 1} denotes the number of legal retirees in period T (born in T − 1). The first term in the numerator of equation (44) captures the effect of a regularization on the rent extracted among the remaining undocumented workers. This term increases if the legalization increases the wage paid to the remaining undocumented workers. The second term captures the loss due to the decrease in the number of undocumented workers. Note that for the rent, the impact of a deportation is similar to the one of regularization: the number of undocumented workers on which a rent is captured decreases. The only difference between both policies comes from the different impact on the wage rate of the remaining undocumented workers, Δw _{l, i, T} . The latter decreases less under a deportation, as the labor aggregate decreases. In the absence of labor market discrimination, with γ = 1, the rent disappears and the benchmark framework is recovered.

I simulate the model including the discrimination on the labor market. I first set γ = 0.75 (i.e. undocumented workers receive 75% of their marginal productivity), based on the rationale provided by Rivera-Batiz (Reference Rivera-Batiz1999). Palivos (Reference Palivos2009) sets this value to 0.71, assuming that migrants and natives earn the same wage. The model is also simulated with γ = 0.5. The loss of the rent, which increases with the degree of discrimination, acts differently on native agents, depending on their age. Retired individuals, who perceive the rent extracted on the undocumented workers, suffer a welfare loss which increases with the extent of the discrimination (see Figure 7a). Wages of working natives are not additionally affected by the discrimination of undocumented workers but the policy shock impacts their tax and pension contribution rates. The higher is the discrimination that regularized workers face, the higher is the increase in their gross wages and their fiscal contribution. However, the more their wages increase, the more regularized individuals save, which increases capital accumulation and decreases the interest factor of the next generation. Hence, the lower return on their savings and the loss of the rent explain why native workers are particularly harmed by the regularization. The capital accumulation dynamics impact the trajectory of the wages and the return rate on savings. Figures 7c and 7d show that both less and highly educated natives are better off after a regularization in the long run if there is some discrimination on the labor market.

Figure 7. Impact of labor market discrimination of undocumented immigrants. (a) Amnesty with discrimination. (b) Deportation with discrimination. (c) Amnesty with discrimination – low-skilled native (d) Amnesty with discrimination – high-skilled native Note: Sensitivity of an native’s welfare (born in period t) to the inclusion of labor market discrimination of undocumented workers with an amnesty occurring at time T.

A deportation in the presence of discrimination is particularly harmful to capital owners as in addition to the loss of the rent, the return rate that they perceive on their savings decreases with the reduction of the labor aggregate (see Figure 7b). Less-educated native workers’ wages increase, but this cannot compensate the loss of the rent and the lower return rates that they receive in the next period. Average native welfare is thus below the benchmark level. In addition, the higher is discrimination, the lower is the contribution of undocumented workers to the budget because taxation is proportional to wages. Thus, deportation under strong discrimination can have a positive impact on the government budget, which leads to a long-run average welfare level above the benchmark one.