1.1 Population

The population differ in age (young, i = 0, and retired, i = 1), in skill level (skilled, s = 1, and unskilled, s = 0) and number of previous generations in the economy. Immigrants, m, are introduced to the economy while young and are considered the first generation in the economy, g = 1. Descendants of immigrant dynasties can be of any generation, g = 2, 3, …, and the natives, n, belong to dynasties that are present in the economy at time t = 0.

While the share of skilled among immigrants, λ ∈ (0, 1), is a ‘choice variable’ for the government, the locals (l, descendants of native and immigrant dynasties) are born skilled with some probability: the share of skilled among descendants of native dynasties is θ ∈ (0, 1), and the share of skilled among immigrant dynasties is either λ or θ depending on the assimilation process.Footnote ³ (For notation also γ(s, g) will be used as the probability for generation g agent having skill level s.)

In the labour market, the skill level directly translates into efficiency level, ɛ(s). The skill level also co-determines, together with generational background, the fertility rate φ(s, g) of the agents. Natives, for the sake of simplicity, are assumed to reproduce with unit fertility.Footnote ⁴ Thus, if μ _t(i, s, g) is defined as a measure on type (i, s, g) agents, the introduction of the new generation of (type s) natives can be presented as:

(1)$$ \mu _{t + 1}\lpar {0\comma \;s\comma \;0} \rpar = \gamma \lpar {s\comma \;0} \rpar \sum \limits_{{s}^{\prime}} \mu _t\lpar {0\comma \;{s}^{\prime}\comma \;0} \rpar $$

and the introduction of the immigrant dynasty descendants as:

(2)$$ \mu _{t + 1} (0, s, g + 1) = \gamma (s,g + 1) \sum \limits_{{s}^{\prime}} \mu_{t} (0,{s}^{\prime},g) \cdot \varphi ({s}^{\prime}\comma \;g) $$

The size and quality of the first generation immigrants is a government policy:

(3)$$ \mu _t (0\comma \;s\comma \;1) = \gamma \lpar {s\comma \;1} \rpar \cdot \psi \sum\limits_{g\ne 1} \sum \limits_{{s}^{\prime}} \mu _t\lpar {0\comma \;{s}^{\prime}\comma \;g} \rpar $$

where ψ is the size of the immigrant population compared to the local-born population. The government chooses not only the size of the group but also the share of skilled among the immigrants: λ = γ(1, 1).

Each agent stays in the model for two periods (except the initial retired population, μ ₀(1, s, n), that are present only for the second period of their lives), viz. individual ageing is deterministic:

(4)$$\mu _{t + 1}\lpar {1\comma \;s\comma \;g} \rpar = \mu _t\lpar {0\comma \;s\comma \;g} \rpar $$

i.e., everybody ages, stays in senescence for one period and leaves the model afterwards (also no return migration is allowed).

An individual agent maximises lifetime utility, which is derived from consumption in both periods:

(5)$$U\lpar {c_t^t \comma \;c_{t + 1}^t } \rpar = v\lpar {c_t^t } \rpar + \beta v\lpar {c_{t + 1}^t } \rpar $$

where $c_j^i$ is the consumption of an agent born at time i during time j; U( ⋅ , ⋅ ) is a time-separable utility function with β ∈ (0, 1) being the time-discount coefficient and v( ⋅ ) being a continuous, twice continuously differentiable, strictly increasing, strictly concave function that satisfies the Inada conditions.Footnote ⁵ To finance consumption an agent uses labour income net of taxes and savings in the first period, and in the second period savings and social security benefits are used. Thus, at time t a (0, s, g) type agent faces the following budget constraints:

(6)$$c_t^t + a_t \le w\varepsilon \lpar s \rpar \lpar {1-\tau_t} \rpar $$

(7)$$c_{t + 1}^t \le \rho w\varepsilon \lpar s \rpar + a_t\lpar {1 + r} \rpar $$

where a _t is the savings, τ _t is the tax rate and ρ is the pension replacement rate.Footnote ⁶ The agent's efficiency is ɛ _s = ɛ(1) and ɛ _u = ɛ(0).

1.2 Government

The government regulates immigration (as presented in equation (3)) and implements the fiscal constitution: the fiscal constitution includes taxation, pension benefits and sustainability of public debt (if it exists), and can be presented as:

(8)$$ \sum\limits_{s\comma g} \rho w\varepsilon \lpar s \rpar \mu _t\lpar {1\comma \;s\comma \;g} \rpar + B_t\lpar {1 + r} \rpar = \sum \limits_{s\comma g} \tau _tw\varepsilon \lpar s \rpar \mu _t\lpar {0\comma \;s\comma \;g} \rpar + B_{t + 1}$$

and

(9)$$\mathop {\lim }\limits_{t\to \infty } B_t\cdot \lpar {1 + r} \rpar ^{{-}t} = 0$$

where B _t is the debt at year t (a borrowing from time t − 1 due at time t). Equation (8) is a general dynamic budget constraint that allows for periodically balanced unfunded PAYG (in case B _t = 0 for any t) as well as dynamically balanced social security. If the dynamically balanced system is used (i.e., non-zero public debt is allowed, B _t ≠ 0 for some t), then equation (8) needs to be augmented by the condition presented in (9) for sustainability.

1.3 Equilibrium and welfare measure

The economy starts with an established social security system and no immigration policy. Starting with the first period immigrants are invited in and the government manages the social security budget (taking into consideration the new demographic changes).

This definition allows infinitely many equilibria depending on the combination of tax rate and size of the public debtFootnote ⁷ in each period. Thus, a further refinement (based on the variability of the tax rate) shall be used to restrict the equilibria to only two types:

Definition 2

A single-tax (or non-zero debt) equilibrium is an equilibrium (as given in Definition 1) with a common tax rate set for all cohorts t > 0

(10)$$\tau _t = \bar{\tau }$$

while the public debt varies accordingly to satisfy (8) and (9).

In equilibrium the agents maximise their lifetime utility given the prices, tax rates and pension replacement. Over the types of equilibria none of the mentioned varies but the tax rate. Thus,Footnote ⁸ the product of the pension replacement rate and the reciprocal of the tax rate shall be considered the measure of welfare:

(11)$$W_t = \displaystyle{\rho \over {\tau _t}}$$

Thence, while in the case of the multiple-tax equilibrium the welfare of (each agent in) the different cohorts will differ, in the case of the single-tax equilibrium one measure W _STE represents all the agents at once (with the exception of the initial (time t = 0) retired agents whose welfare is unaltered in any discussed equilibria).

1.4 Assimilation dynamics

In equilibrium the government sets a tax rate, based on the size of the effective labour force (the base for the contributions and the next period pension claims), to balance the budget (8) and (9). The size of the effective labour force depends on the absolute size of the population and their average efficiency:

(12)$$ N_t = \sum \limits_{s\comma g} \varepsilon \lpar s \rpar \mu _t\lpar {0\comma \;s\comma \;g} \rpar $$

At the start of the economy half of the native population is young (age i = 0), and the other half in senescence (age i = 1). Thus the effective labour force at time t = 0 is

(13)$$N_o = {\cal L}\lpar {\theta \varepsilon_s + \lpar {1-\theta } \rpar \varepsilon_u} \rpar = {\cal L}E_\theta $$

where ${\cal L}$ is exactly half of the total population in the country, and $E_\theta$ is the average efficiency of a native worker. In period t = 1 the government allows migrants to enter. Migrants immediately start participating in the local labour market and the social security system (by contributions first and then pension claims one period later). The government chooses the size of the immigrant group (relative to the local-born population, ψ) and the size of the skilled among migrants, λ, so that the average efficiency among immigrants is

(14)$$E_\lambda = \lambda \varepsilon _s + \lpar {1-\lambda } \rpar \varepsilon _u$$

and the working age population at time t = 1 is

(15)$$N_1 = {\cal L}\lpar {E_\theta + \psi E_\lambda } \rpar $$

From the second period onwards, the descendants of the immigrants enter the labour force and the effective labour force changes depending on their assimilation pattern. Thus the three cases are discussed.

1.4.1 Full assimilation: uninherited fertility and uninherited skills

In the case of full assimilation, the second period starts with the natives creating young natives equal to themselves in number and average efficiency, while the immigrants father, with their respective fertility rates φ _s = φ(s, 1), second generation immigrants that have the skill distribution of the natives. Next, the government allows a new cohort of young immigrants to enter the country according to (3), so that in the second period the efficient labour force is

(16)$$N_2 = {\cal L}\lpar {E_\theta + \psi E_\lambda } \rpar \lpar {1 + \psi \Phi_\lambda } \rpar $$

where $\Phi _\lambda$ is the average fertility of the immigrants:

(17)$$\Phi _\lambda = \lambda \varphi _1 + \lpar {1-\lambda } \rpar \varphi _o$$

From the second period onwards, the natives reproduce, the immigrant descendants behave identically to the natives, and the immigrants enter according to (3), so that the population at time t ≥ 2 is

(18)$$N_t^{\,fa} = {\cal L}\lpar {E_\theta + \psi E_\lambda } \rpar \lpar {1 + \psi \Phi_\lambda } \rpar ^{t-1}$$

i.e., increasing proportionally with the weighted sum of the local and immigrant fertility rates.

1.4.2 Partial assimilation: inherited fertility and uninherited skills

In the case of partial assimilation, the labour force dynamics are identical to the previous case up to the second period and can be presented by (13)–(16). However, from the beginning of the third period (while the natives reproduce, the immigrants father according to their fertility, $\Phi _\lambda$ and the government allows young immigrants to enter according to policy rule (3)) the second generation immigrants, having inherited the fertility level of their ancestors, produce third-generation immigrants with the skill-dependent fertility rate φ _s (as the first-generation immigrants do).

The population at time t ≥ 2 is

(19)$$N_t^{\,pa} = {\cal L}\lpar {E_\theta + \psi E_\lambda } \rpar \left({\displaystyle{{\psi \Phi_\lambda } \over {\Phi_\theta + \psi \Phi_\lambda -1}}{\lpar {\Phi_\theta + \psi \Phi_\lambda } \rpar }^{t-1} + \displaystyle{{\Phi_\theta -1} \over {\Phi_\theta + \psi \Phi_\lambda -1}}} \right)$$

where

(20)$$\Phi _\theta = \theta \varphi _1 + \lpar {1-\theta } \rpar \varphi _o$$

is the average fertility level of the immigrant descendants.

Hence, the immigrant generations differ in their average fertility both from the natives (unit fertility) and the immigrants (on average $\Phi _\lambda$). Furthermore, when the share of the skilled among the immigrant population is higher than the natives, the average fertility rate of the descendants is higher than that of the immigrants. Similar to the previous case of full assimilation, the labour force growth over time is driven by the weighted sum of the fertility rates, $\Phi _\theta + \psi \Phi _\lambda \semicolon \;$ however, the labour force growth rate reaches that level only infinitely many periods later.

1.4.3 No assimilation: inherited fertility and inherited skills

In the extreme case of immigrant descendants being identical (in labour-market and reproductive qualities) to their ancestors, i.e., in the case of no assimilation, the size of the population is again the same as the previous two cases for the first two periods. However, while in the first period the labour force is again identical (and is given by (13)–(15)), the immigrant descendants are different in their skill distribution: The efficient labour force in the second period in case of no assimilation is:

(21)$$N_2^{na} = {\cal L}\lpar {E_\theta + \psi E_\lambda \Phi_\lambda + \psi E_\lambda \lpar {1 + \psi \Phi_\lambda } \rpar } \rpar $$

which further changes both the population and the labour force size:

(22)$$N_t^{na} = {\cal L}\left({E_\theta + \psi E_\lambda \displaystyle{{\Phi_\lambda^t {\lpar {1 + \psi } \rpar }^t-1} \over {\Phi_\lambda \lpar {1 + \psi } \rpar -1}}} \right)$$

for t ≥ 1. Again the weighted sum of the immigrant dynasty average fertility, $\Phi _\lambda \lpar {1 + \psi } \rpar \comma \;$ is the driving force for the labour force growth as well as is the limit value that the labour force (and population) growth rate approaches over time.

2.1 Single-tax equilibrium

The measure for welfare under the single-tax equilibrium is obtained from solving (8) and (9) for tax rate $\bar{\tau }$ and substituting in (11):

(23)$$W_{STE} = \lpar {1 + r} \rpar \left({1 + \displaystyle{{N_o} \over {\sum\nolimits_{t = 1}^\infty {{\textstyle{{N_t} \over {{\lpar {1 + r} \rpar }^t}}}} }}} \right)^{{-}1}$$

It is apparent that welfare depends on the relation of the effective population and the real interest rate. Holding the working assumption that immigration never covers the gap between the internal and external returns of participating in an unfunded social security system (i.e., the population growth rate is lower than the real interest rate), it is possible to analyse the welfare measure under different cases of assimilation.

Lemma 1

In the single-tax equilibrium the welfare measure in the case of full, partial and no assimilation, respectively, reads:

(24)$$W_{STE}^{FA} = \displaystyle{{E_\theta + E_\lambda \psi } \over {E_\lambda \psi + E_\theta \lpar {1 + r} \rpar -\Phi _\lambda E_\theta \psi }}\lpar {1 + r} \rpar $$

(25)$$W_{STE}^{PA} = \displaystyle{{E_\theta + E_\lambda \psi } \over {E_\lambda \psi + E_\theta \lpar {1 + r} \rpar -\Phi _\lambda E_\theta \psi \lpar r/\lpar {1 + r-\Phi_\theta } \rpar \rpar }}\lpar {1 + r} \rpar $$

(26)$$W_{STE}^{NA} = \displaystyle{{E_\lambda \psi r} \over {E_\lambda \psi + E_\theta \lpar {1 + r} \rpar -\Phi _\lambda E_\theta \lpar {1 + \psi } \rpar }} + 1$$

Proof. Follows directly from rearranging equations (13), (15), (16), (18), (19), (21), (22), and (23). ■

Interpretation of Proposition 1 is based on the Breyer (Reference Breyer1989) equivalence result. Thus, the single-tax equilibrium is equal to cancelling the existing pension system and turning the implicit pension debt (the due benefits) into an explicit public debt, and then distributing the debt equally over the future cohorts. Thus immigration (even unskilled) can be understood as an increase in the tax base (for financing a specific sum).

Proposition 2

In the single-tax equilibrium, in the case of full assimilation, welfare decreases with an increase in the share of skilled among immigrants if

(27)$$\displaystyle{{\varepsilon _s-\varepsilon _u} \over {\varphi _u-\varphi _s}} \lt \displaystyle{{E_\theta + \psi E_\lambda } \over {r-\psi \Phi _\lambda }}$$

Proof. The derivative of (24) is

(28)$$\displaystyle{{\partial W_{STE}^{FA} } \over {\partial \lambda }} = P_{FA}\cdot \left({\displaystyle{{\partial E_\lambda } \over {\partial \lambda }}r + \displaystyle{{\partial \Phi_\lambda } \over {\partial \lambda }}E_\theta -\psi \left[{\displaystyle{{\partial E_\lambda } \over {\partial \lambda }}\Phi_\lambda -\displaystyle{{\partial \Phi_\lambda } \over {\partial \lambda }}E_\lambda } \right]} \right)$$

where P _FA is a strictly positive function of model parameters, $\partial E_\lambda /\partial \lambda =$ ɛ _s − ɛ _u > 0 and $\partial \Phi _\lambda /\partial \lambda =$ φ _s − φ _u < 0. Hence, (28) is negative if (27) holds. ■

The proposition claims that under rather permissive assumptions,Footnote ⁹ namely, that the difference in efficiency rate is not disproportionately large compared to the difference in fertility rates, unskilled immigration is preferred. The intuition behind condition (27) is that if the losses (size of the pension system losses $\lpar {r-\psi \Phi_\lambda } \rpar$ multiplied by the losses in efficiency (ɛ _s − ɛ _u)) due to unskilled immigration are smaller than the gains (average efficiency $\lpar {E_\theta + \psi E_\lambda } \rpar$ multiplied by the gain in fertility (φ _u − φ _s)) due to unskilled immigration, then unskilled immigration is preferred.

Proposition 3

In the single-tax equilibrium welfare decreases with increase in the share of skilled among immigrants if

1. (1) in the case of partial assimilation

   (29)$$\displaystyle{{\varepsilon _s-\varepsilon _u} \over {\varphi _u-\varphi _s}} \lt \displaystyle{{E_\theta + \psi E_\lambda } \over {r-\psi \Phi _\lambda + 1-\Phi _\theta }}$$

2. (2) in the case of no assimilation

   (30)$$\displaystyle{{\varepsilon _s-\varepsilon _u} \over {\varphi _u-\varphi _s}} \lt \displaystyle{{1 + \psi } \over {1 + r-\Phi _\lambda \lpar {1 + \psi } \rpar }}$$

Proof. The derivatives of (25) and (26):

(31)$$\displaystyle{{\partial W_{STE}^{PA} } \over {\partial \lambda }} = P_{PA}\cdot \left[{\displaystyle{{\partial E_\lambda } \over {\partial \lambda }}\lpar {1 + r-\lpar {\Phi_\theta + \psi \Phi_\lambda } \rpar } \rpar + \displaystyle{{\partial \Phi_\lambda } \over {\partial \lambda }}\lpar {E_\theta + \psi E_\lambda } \rpar } \right]$$

(32)$$\displaystyle{{\partial W_{STE}^{NA} } \over {\partial \lambda }} = P_{NA}\cdot \left[{\displaystyle{{\partial E_\lambda } \over {\partial \lambda }}\lpar {1 + r-\Phi_\lambda \lpar {1 + \psi } \rpar } \rpar + \displaystyle{{\partial \Phi_\lambda } \over {\partial \lambda }}\lpar {1 + \psi } \rpar } \right]$$

(where P _PA and P _NA are positive) are negative if, respectively, (29) and (30) hold. ■

Again under rather permissive assumptions, unskilled immigration is preferred. The economic intuition is also similar to the previous case.

2.2 Multiple-tax equilibrium

In the multiple-tax equilibrium the welfare measure collapses to simple efficient population change, and the welfare dynamics are effectively the population dynamics:

(33)$$W_t = \displaystyle{{N_t} \over {N_{t-1}}}$$

Lemma 2

The young of the first period (irrespective of the assimilation scenario) have the welfare:

(34)$$W_1^{FA} = W_1^{PA} = W_1^{NA} = \displaystyle{{E_\theta {\rm} + E_\lambda \psi } \over {E_\theta }} = 1 + \displaystyle{{E_\lambda } \over {E_\theta }}\psi $$

Proof. Follows directly from (13), (15) and (33). ■

Palpably, the voting equilibrium without dynastic altruism will result in a selective immigration policy where only skilled immigrants are allowed in: the welfare of the retired group is unaltered while the young are strictly better off.

Lemma 3

If the immigrant descendants fully assimilate, the welfare at period t ≥ 2 is:

(35)$$W_{t \ge 2}^{FA} = 1 + \psi \Phi _\lambda $$

Proof. Follows directly from (15), (18) and (33). ■

Propositions 4 and 5 suggest that there is a strong conflict of interest between the first and subsequent cohorts of the population: While the first cohort strictly prefers skilled immigrants, all the other cohorts enjoy higher welfare if the policy does not favour the skilled. The lower the share of skilled among immigrants the higher the welfare of the later cohorts is. However, it is worth noting that, if the share of skilled among immigrants is very small, the initial cohort may have lower welfare than their descendants.

Proposition 6

In the case of full assimilation, if

(36)$$E_\lambda \lt E_\theta \Phi _\lambda $$

then the welfare of the first (policy-setting) cohort is lower than the welfare of all the other cohorts.

Proof. Follows directly from (34) and (35). ■

That is, if the efficient size of the immigrant population is less than the efficient size of the immigrant descendant population (average efficiency multiplied by size) then the first cohort is worse off compared to every other cohort. The reverse also holds, i.e., if the share of skilled among immigrants is high, then the first generation measure of welfare is higher than that of the other cohorts.Footnote ¹⁰

Lemma 4

If the immigrant descendants partially assimilate, the welfare of the agents in the second period is

(37)$$W_2^{PA} = 1 + \psi \Phi _\lambda $$

then grows over time according to

(38)$$W_{t \gt 2}^{PA} = \displaystyle{{\Phi _\theta -1 + \psi \Phi _\lambda {\lpar {\Phi_\theta + \psi \Phi_\lambda } \rpar }^t} \over {\Phi _\theta -1 + \psi \Phi _\lambda {\lpar {\Phi_\theta + \psi \Phi_\lambda } \rpar }^{t-1}}}$$

and approaches level of

(39)$$W_\infty ^{PA} = \Phi _\theta + \psi \Phi _\lambda $$

in the limit.

Proof. Equations (37) and (38) follow directly from (15), (19), and (33). The derivative of (38) with respect to time t:

(40)$$\displaystyle{\partial \over {\partial t}}W_{t \gt 2}^{PA} = \displaystyle{{\Phi _\lambda \psi \lpar {\Phi_\theta -1} \rpar \lpar {\Phi_\lambda \psi + \Phi_\theta -1} \rpar {\lpar {\Phi_\lambda \psi + \Phi_\theta } \rpar }^{t + 1}} \over {{\lpar {\lpar {\Phi_\lambda \psi + \Phi_\theta } \rpar \lpar {\Phi_\theta -1} \rpar + \Phi_\lambda \psi {\lpar {\Phi_\lambda \psi + \Phi_\theta } \rpar }^t} \rpar }^2}}\ln \lpar {\Phi_\lambda \psi + \Phi_\theta } \rpar $$

is positive given that $\Phi _\theta \gt 1$ by construction. Also, it is straightforward to note that

(41)$$\mathop {\lim }\limits_{t\to \infty } \displaystyle{{\Phi _\theta -1 + \psi \Phi _\lambda {\lpar {\Phi_\theta + \psi \Phi_\lambda } \rpar }^t} \over {\Phi _\theta -1 + \psi \Phi _\lambda {\lpar {\Phi_\theta + \psi \Phi_\lambda } \rpar }^{t-1}}} = \Phi _\theta + \psi \Phi _\lambda $$

thus proving the last part of the lemma. ■

As in the case of full assimilation, the results unequivocally indicate the reverse effect of skilled immigration on the welfare of the first (policy-setting) cohort and all the subsequent cohorts: While the first cohort is better off with skilled immigrants, future cohorts prefer the unskilled. However, as opposed to the case of full assimilation, in the case of partial assimilation there is a possibility that the first cohort will have higher welfare than the cohorts immediately following after, but in future there will be cohorts that will enjoy welfare higher than that of the first generation.

Thus, the dynamics in the case of partial assimilation are richer, allowing different welfare ranking, but again a large share of skilled immigrants allows the first cohort to have the highest welfare (and the higher the share the higher the welfare is), while a small number of skilled among immigrants gives the first cohort the lowest welfare.

Lemma 5

If the immigrant descendants do not assimilate, the welfare of cohorts t ≥ 2 is

(45)$$W_{t \ge 2}^{NA} = \displaystyle{{\lpar {\Phi_\lambda \lpar {1 + \psi } \rpar -1} \rpar E_\theta + \lpar {\Phi_\lambda^{t + 1} {\lpar {1 + \psi } \rpar }^{t + 1}-1} \rpar E_\lambda \psi } \over {\lpar {\Phi_\lambda \lpar {1 + \psi } \rpar -1} \rpar E_\theta + \lpar {\Phi_\lambda^t {\lpar {1 + \psi } \rpar }^t-1} \rpar E_\lambda \psi }}$$

that reaches the limit value of $W_\infty ^{NA} = \Phi _\lambda \lpar {1 + \psi } \rpar \comma \;$ while growing if

(46)$$\Phi _\lambda \lpar {1 + \psi } \rpar \gt 1 + \lpar E_\lambda /E_\theta \rpar \psi $$

and decreasing otherwise.

Proof. From (22) and (33) directly follows (45). The limit value of (45) is also easily obtainable. The derivative of (45) with respect to time is

(47)$$\displaystyle{\partial \over {\partial t}}W_{t \ge 2}^{NA} = \displaystyle{{F_{\lambda \comma \psi }E_\lambda \psi {\lpar {F_{\lambda \comma \psi } + 1} \rpar }^t\ln \lpar {F_{\lambda \comma \psi } + 1} \rpar } \over {{\lpar {F_{\lambda \comma \psi }E_\theta + \lpar {{\lpar {F_{\lambda \comma \psi } + 1} \rpar }^t-1} \rpar E_\lambda \psi } \rpar }^2}}\lpar {F_{\lambda \comma \psi }E_\theta -E_\lambda \psi } \rpar $$

where

(48)$$F_{\lambda \comma \psi } = \Phi _\lambda \lpar {1 + \psi } \rpar -1$$

Expression in (47) is positive if the second multiplier in the product is positive. Equation (46) presents the required condition. ■

Proposition 9

In the case of no assimilation, if condition (46) holds, i.e.,

(49)$$\lambda \lt \displaystyle{{E_\theta \varphi _u-\varepsilon _u} \over {E_\theta \lpar {\varphi_u-\varphi_s} \rpar + \varepsilon _s-\varepsilon _u}}$$

then the first cohort (i.e., the policy-setting cohort) has the lowest welfare that sequentially increases for subsequent cohorts.

If condition (46), or (49), holds with the opposite sign, the first cohort has the highest welfare, while welfare sequentially decreases for the subsequent cohorts.

If the expressions on both sides of (46), or (49), are equal, then the welfare measure for all cohorts t ≥ 1 are the same.

Proof. The welfare of the second cohort is

(50)$$\eqalign{W_2^{NA} = & \displaystyle{{\lpar {\Phi_\lambda \lpar {1 + \psi } \rpar -1} \rpar E_\theta + \lpar {\Phi_\lambda^3 {\lpar {1 + \psi } \rpar }^3-1} \rpar E_\lambda \psi } \over {\lpar {\Phi_\lambda \lpar {1 + \psi } \rpar -1} \rpar E_\theta + \lpar {\Phi_\lambda^2 {\lpar {1 + \psi } \rpar }^2-1} \rpar E_\lambda \psi }} \cr & = 1 + \displaystyle{{\Phi _\lambda ^2 {\lpar {1 + \psi } \rpar }^2E_\lambda \psi } \over {E_\theta + \lpar {\Phi_\lambda \lpar {1 + \psi } \rpar + 1} \rpar E_\lambda \psi }}} $$

which is more than the welfare of the first cohort $\lpar {W_2^{NA} \gt W_1^{NA} } \rpar$ if

(51)$$1 + \displaystyle{{\Phi _\lambda ^2 {\lpar {1 + \psi } \rpar }^2E_\lambda \psi } \over {E_\theta + \lpar {\Phi_\lambda \lpar {1 + \psi } \rpar + 1} \rpar E_\lambda \psi }} \gt 1 + \displaystyle{{E_\lambda } \over {E_\theta }}\psi $$

The expression simplifies into (46), or further into (49). Under the same condition (46), according to Lemma 5, the welfare measure for all subsequent cohorts grows. This concludes the proof for the first part of the proposition. The other two cases are proved similarly. ■

Again it is clear that the first cohort benefits from a high-skilled-favouring immigration policy, and that later cohorts would prefer the share of skilled immigrants to be smaller initially. A highly-skilled immigrant group leaves the first (policy-setting) cohort with high welfare, while the welfare of all subsequent cohorts decreases rapidly. The results are similar to Kudrna et al. (Reference Kudrna, Tran and Woodland2019) where the current generations prefer a type of pension policy that is detrimental to the well-being of the future generations.

Of note is that the model is dynamically inconsistent and any cohort will prefer to increase the share of skilled among immigrants in their own period, thus increasing its own and decreasing the subsequent cohorts' welfare.