The aggregate and redistributive effects of emigration

Abstract

The 2004 EU enlargement has triggered large and rapid migration movements from the new to the old member states. The scale of this outflow was unprecedented in the CEE history and its structure was also different from previous emigration waves as it was more heavily biased towards young and educated people. I exploit this post-accession emigration wave to study the aggregate and redistributive effects of emigration. Using a small open economy general equilibrium model with heterogeneous agents and endogenous migration choice calibrated to Polish data, I show that emigration lowers output per capita and improves the international investment position of the source country. Changes in population structure resulting from population outflows affect the wage distribution between high-skilled and low-skilled workers, thereby increasing economic inequalities. Moreover, I find that lifting labour mobility barriers is beneficial not only for people who move abroad, but also for skilled never-migrants.

Appendices

Appendix

Appendix 1: Definition of competitive equilibrium

Individual state variables are age j, skills s, assets a, individual labour productivity z and migration cost $\lambda$. However, after relocating, the latter no longer affects agent’s decisions. The aggregate state of the economy at time t is completely described by $\Phi ^{stay}_t$ and $\Phi ^{migr}_t$, which are the joint measures of agents residing in the beginning of period t, i.e. before migration decisions are made in, respectively, their home country and abroad, expressed relative to the size of the youngest living generation. $\Phi ^{stay}_t$ is defined over age, skills, assets positions, individual labour productivity and migration costs, and $\Phi ^{migr}_t$ is defined over age, skills, assets positions and individual labour productivity.

Let $j \in \mathcal {J}=\{1,2,...,J\}$, $s\in \mathcal {S}=\{h,l\}$, $a\in \mathcal {A}=[0, \inf ]$, $z\in \mathcal {Z} = \{z_1, z_2, ..., z_N\}$, $\lambda \in \Lambda =[0, \inf ]$, and let ${\textbf {V}} \in \mathcal {J} \times \mathcal {S}\times \mathcal {A} \times \mathcal {Z} \times \Lambda$ and ${\textbf {W}} \in \mathcal {J} \times \mathcal {S}\times \mathcal {A} \times \mathcal {Z}$. Let ${\textbf {B}}(\mathcal {A})$ and ${\textbf {B}}(\Lambda )$ be the Borel $\sigma$-algebra of $\mathcal {A}$ and $\Lambda$, respectively, and P($\mathcal {J}$), P($\mathcal {S}$), P($\mathcal {Z}$) be the power sets of $\mathcal {J}$, $\mathcal {S}$, $\mathcal {Z}$, respectively. Let K be the set of all finite measures over the measurable space (V, ${\textbf {P}}(\mathcal {J}) \times {\textbf {P}}(\mathcal {S}) \times {\textbf {B}}(\mathcal {A}) \times {\textbf {P}}(\mathcal {Z}) \times {\textbf {B}}(\Lambda )$) and M be the set of all finite measures over the measurable space (W, ${\textbf {P}}(\mathcal {J}) \times {\textbf {P}}(\mathcal {S}) \times {\textbf {B}}(\mathcal {A}) \times {\textbf {P}}(\mathcal {Z})$). For further convenience, I also define $\tilde{\Phi }^{stay}_t(\mathcal {J}\times \mathcal {S} \times \mathcal {A}\times \mathcal {Z}) = \int \Phi ^{stay}_t(\mathcal {J}\times \mathcal {S} \times \mathcal {A}\times \mathcal {Z} \times \Lambda )d\lambda$. To ease the notation, in the following I will use $\Phi ^{stay}_t$, $\tilde{\Phi }^{stay}_t$ and $\Phi ^{migr}_t$ instead of $\Phi ^{stay}_t(\mathcal {J}\times \mathcal {S} \times \mathcal {A}\times \mathcal {Z} \times \Lambda )$, $\tilde{\Phi }^{stay}_t(\mathcal {J}\times \mathcal {S} \times \mathcal {A}\times \mathcal {Z})$ and $\Phi ^{migr}_t(\mathcal {J}\times \mathcal {S} \times \mathcal {A}\times \mathcal {Z})$.

Definition

Given initial conditions $b_1$, $\Phi ^{stay}_1$ and $\Phi ^{migr}_1$, a competitive equilibrium is a sequence of functions for the household $\left\{ \right. V_t$, $m_t$: V $\rightarrow$ ${\textbf {R}}_+$ $\left. \right\} _{t=1}^{\infty }$, $\left\{ \right. v^{stay}_t$, $c^{stay}_t$, $a'^{stay}_t$: W $\rightarrow$ ${\textbf {R}}_+\left. \right\} _{t=1}^{\infty }$, $\left\{ \right. \tilde{v}^{migr}_t$, $c^{migr}_t$, $a'^{migr}_t$: W $\rightarrow$ ${\textbf {R}}_+\left. \right\} _{t=1}^{\infty }$, production plans for the firm $\{k_t, h_t, l_t\}_{t=1}^{\infty }$, prices $\{w^{h}_t, w^{l}_t\}_{t=1}^{\infty }$, bequests $\{b_t\}_{t=1}^{\infty }$, growth rates of the youngest generation born in the country $\{n_{1,t}\}_{t=1}^{\infty }$, growth rates of the population living in the country $\{n_{t}\}_{t=1}^{\infty }$, threshold migration costs $\{\lambda ^s_{j,t}\}_{t=1}^{\infty }$, probabilities of emigrating $\{\eta ^s_{j,t}\}_{t=1}^{\infty }$, and measures $\{\Phi ^{stay}_t\}_{t=1}^{\infty }$ and $\{\Phi ^{migr}_t\}_{t=1}^{\infty }$, with $\Phi ^{stay}_t \in {\textbf {K}}$ and $\Phi ^{migr}_t \in {\textbf {M}}$ such that:

• 1. given home and foreign prices, bequests and initial conditions, for each t:

  • (a) $V_t$ solves the functional equation (with $m_t$ as associated policy function):

    $$\begin{aligned} V_{t}(j,s,a,z, \lambda ) = \max _{m\in \{0,1\}} \quad \{ m( \tilde{v}_{t}^{migr}(j,s,a,{z^{Fs}_{min}})-\lambda \Psi ^s_j) + (1-m) v_{t}^{stay}(j,s,a,z) \} \end{aligned}$$

    (A.1)

  • (b) $v^{stay}_t$ solves the functional equation (with $c^{stay}_t$, $a'^{stay}_t$ as associated policy functions):

    $$\begin{aligned} \begin{aligned}&v_{t}^{stay}(j,s,a,z) =\\&= \max _{c, a'\ge 0} \quad \bigg \{ \bigg . u(c) +\beta \psi ^{s}_{j} \sum _{z'} \pi ^{s}(z'|z) \bigg [ \bigg . \bigg (1-\eta ^s_{j+1,t+1}(a',z')\bigg )v_{t+1}^{stay} (j+1, s, a',z') \\&+ \eta ^s_{j+1,t+1}(a',z')\bigg (\tilde{v}_{t+1}^{migr} (j+1, s, a', {z^{Fs}_{min}}) - \Psi _{j+1} \mathbb {E}(\lambda '|\lambda '<\lambda ^s_{j+1,t+1}(a',z')) \bigg )\bigg . \bigg ]\bigg . \bigg \} \end{aligned} \end{aligned}$$

    (A.2)

    subject to:

    $$\begin{aligned} c+a'=(1+r)(a+ b_t)+w_t^{s}e^{s}_{j}z \end{aligned}$$

    (A.3)

  • (c) $\tilde{v}^{migr}_t$ solves the functional equation (with $c^{migr}_t$, $a'^{migr}_t$ as associated policy functions):

    $$\begin{aligned} \tilde{v}_{t}^{migr}(j,s,a,z)=\max _{c, a'\ge 0} \quad \{ u(c)+\beta \psi ^{Fs}_{j} \sum _{z'} \pi ^{Fs}(z'|z) \tilde{v}_{t+1}^{migr} (j+1,s, a',z')\} \end{aligned}$$

    (A.4)

    subject to:

    $$\begin{aligned} c+a'=(1+r)(a + b_t) +w^{Fs}e^{Fs}_{j}z \end{aligned}$$

    (A.5)

• 2. Marginal product of capital and wages $w^{h}_t$ and $w^{l}_t$ satisfy:

  $$\begin{aligned} A F_{k}(k_t, h_t, l_t) = r + \delta \end{aligned}$$

  (A.6)
  $$\begin{aligned} \begin{aligned}&w^{h}_t = A F_{h}(k_t, h_t, l_t)\\&w^{l}_t = A F_{l}(k_t, h_t, l_t) \end{aligned} \end{aligned}$$

  (A.7)

• 3. Bequests are given by:

  $$\begin{aligned} \begin{aligned} b_t&= \frac{\int (1-\psi ^{s}_{j}) {a'}_{t-1}^{stay,s}(j,s,a,z) (1-\eta ^s_{j,t-1}(a,z)) d\tilde{\Phi }_{t-1}^{stay}}{(1+n_{1,t})(\int d\tilde{\Phi }_t^{stay}+ \int d\Phi _t^{migr})}\\&+ \frac{\int (1-\psi ^{Fs}_{j}) {a'}_{t-1}^{migr}(j,s,a,z^{Fs}_{min}) \eta ^s_{j,t-1}(a,z) d\tilde{\Phi }_{t-1}^{stay}}{(1+n_{1,t})(\int d\tilde{\Phi }_t^{stay}+ \int d\Phi _t^{migr})}\\&+ \frac{\int (1-\psi ^{Fs}_{j}) {a'}_{t-1}^{migr}(j,s,a,z) d\Phi _{t-1}^{migr}}{(1+n_{1,t})(\int d\tilde{\Phi }_t^{stay}+ \int d\Phi _t^{migr})}\\ \end{aligned} \end{aligned}$$

  (A.8)

• 4. Growth rate of the youngest generation born in the country is given by:

  $$\begin{aligned} n_{1,t} = \left( \int \frac{f_{j-23} (1-\eta ^s_{j,t-1}(a,z))\Omega ^s_{-23,1} }{\Omega ^s_{j-23,j}} d\tilde{\Phi }^{stay}_{t-1}\right) -1 \end{aligned}$$

  (A.9)

• 5. Growth rate of the population living in the country is given by:

  $$\begin{aligned} n_{t} = \frac{(1+n_{1,t})\int (1-\eta ^s_{j,t}(a,z))d\tilde{\Phi _t}^{stay}}{\int (1-\eta ^s_{j,t-1}(a,z))d\tilde{\Phi }_{t-1}^{stay}} -1 \end{aligned}$$

  (A.10)

• 6. Threshold migration cost is given by:

  $$\begin{aligned} \lambda ^s_{j,t}(a,z) = \frac{\tilde{v}_{t}^{migr}(j,s, a,z^F_{min})-v_{t}^{stay}(j,s,a,z)}{\Psi ^s_j} \end{aligned}$$

  (A.11)

• 7. Probability of emigrating satisfies:

  $$\begin{aligned} \eta ^s_{j,t}(a,z) =F(\lambda ^s_{j,t}(a,z)) \end{aligned}$$

  (A.12)

• 8. Markets clear:

  $$\begin{aligned} \begin{aligned}&h_t = \frac{\int e_j^{h} z (1-\eta ^h_{j,t}(a,z)) d\tilde{\Phi }_t^{stay} }{\int (1-\eta ^s_{j,t}(a,z)) d\tilde{\Phi _t}^{stay}}\\&l_t = \frac{\int e_j^{l} z (1-\eta ^l_{j,t}(a,z)) d\tilde{\Phi }_t^{stay} }{\int (1-\eta ^s_{j,t}(a,z)) d\tilde{\Phi }_t^{stay}} \end{aligned} \end{aligned}$$

  (A.13)
  $$\begin{aligned} \begin{aligned}&A\left( \alpha (l_t)^{\gamma }+(1-\alpha )\left[ \rho (k_t)^{\eta }+(1-\rho )(h_t)^{\eta }\right] ^{\frac{\gamma }{\eta }}\right) ^{\frac{1}{\gamma }} = \\&= c_t + i_t + (1+n_{t+1}){nfa}_{t+1} - (1+r){nfa}_t \end{aligned} \end{aligned}$$

  (A.14)

  where

  $$\begin{aligned} c_t=\frac{\int c_{t}^{stay}(j,s,a,z) (1-\eta ^s_{j,t}(a,z)) d\tilde{\Phi }_t^{stay} }{\int (1-\eta ^s_{j,t}(a,z)) d\tilde{\Phi }_t^{stay}} \end{aligned}$$

  (A.15)
  $$\begin{aligned} i_t=k_{t+1}(1+n_{t+1})-(1-\delta )k_t \end{aligned}$$

  (A.16)
  $$\begin{aligned} nfa_t = a_t -k_t \end{aligned}$$

  (A.17)
  $$\begin{aligned} a_t = \frac{\int a(1-\eta ^s_{j,t}(a,z)) d\tilde{\Phi }_{t}^{stay}}{\int (1-\eta ^s_{j,t}(a,z)) d\tilde{\Phi }_t^{stay}} + b_t \end{aligned}$$

  (A.18)

• 9. Law of motion of $\Phi ^{stay}_t$:

  For any subset $C=C_{\mathcal {J}} \times C_{\mathcal {S}} \times C_{\mathcal {A}}\times C_{\mathcal {Z}} \times C_{\Lambda } \subseteq \{2,..., J\}\times \mathcal {S}\times \mathcal {A}\times \mathcal {Z} \times \Lambda$,

  $$\begin{aligned} \Phi ^{stay}_{t+1}(C) = \frac{1}{1+n_{1,t+1}}\int (1-m_t(j,s,a,z,\lambda ))\psi ^{s}_{j} P_t^{stay}\left( (j,s,a,z,\lambda ),C\right) d\Phi ^{stay}_t \end{aligned}$$

  (A.19)

  where

  $$\begin{aligned} P_t^{stay}\left( (j,s,a,z,\lambda ),C\right) = {\left\{ \begin{array}{ll} \pi ^{s}(z'|z)\int _{C_\Lambda }f(\lambda )d\lambda &{} \text {if }j+1\in C_{\mathcal {J}}, s\in C_{\mathcal {S}},\\ &{} a'^{stay}_t(j,a,s,z) \in C_{\mathcal {A}}, z'\in C_{\mathcal {Z}}\\ 0 &{} \text {otherwise} \end{array}\right. } \end{aligned}$$

  (A.20)

  Hereby, f denotes the probability distribution function of $\lambda$.

  For any $C=C_{\mathcal {J}} \times C_{\mathcal {S}} \times C_{\mathcal {A}}\times C_{\mathcal {Z}} \times C_{\Lambda }\subseteq \{1\}\times \mathcal {S}\times \mathcal {A}\times \mathcal {Z} \times \Lambda$,

  $$\begin{aligned} \Phi _{t+1}^{stay}(C) = {\left\{ \begin{array}{ll} \sum \limits _{s\in C_{\mathcal {S}}} \sum \limits _{z\in C_{\mathcal {Z}}} \nu _1^{sz} \int _{C_\Lambda }f(\lambda )d\lambda &{} \text {if }0\in C_{\mathcal {A}}\\ 0 &{} \text {otherwise} \end{array}\right. } \end{aligned}$$

  (A.21)

  where $\nu _1^{sz}$ is the share of workers with skill level s and labour productivity z in the cohort of newborn agents.

• 10. Law of motion of $\Phi ^{migr}_t$:

  For any subset $C=C_{\mathcal {J}} \times C_{\mathcal {S}} \times C_{\mathcal {A}}\times C_{\mathcal {Z}} \subseteq \{2,..., J\}\times \mathcal {S}\times \mathcal {A}\times \mathcal {Z}$,

  $$\begin{aligned} \begin{aligned} \Phi ^{migr}_{t+1}(C) = \frac{1}{1+n_{1,t+1}}\bigg [&\int \psi ^{Fs}_{j} P_t^{migr}\left( (j,s,a,z),C\right) d\Phi ^{migr}_t+\bigg .\\&+\int \eta ^s_{j,t}(a,z)\psi ^{Fs}_{j} Q_t^{migr}\left( (j,s,a,z),C\right) d\tilde{\Phi }^{stay}_t\bigg .\bigg ] \end{aligned} \end{aligned}$$

  (A.22)

  where

  $$\begin{aligned} P_t^{migr}\left( (j,s,a,z),C\right) = {\left\{ \begin{array}{ll} \pi ^{Fs}(z'|z) &{} \text {if }(j+1, s, a'^{migr}_t(j,a,s,z), z')\in C\\ 0 &{} \text {otherwise} \end{array}\right. } \end{aligned}$$

  (A.23)

  and

  $$\begin{aligned} Q_t^{migr}\left( (j,s,a,z),C\right) = {\left\{ \begin{array}{ll} \pi ^{Fs}(z'|z_{min}^{Fs}) &{} \text {if }(j+1, s, a'^{migr}_t(j,a,s,z_{min}^{Fs}), z')\in C\\ 0 &{} \text {otherwise} \end{array}\right. } \end{aligned}$$

  (A.24)

  For any $C=C_{\mathcal {J}} \times C_{\mathcal {S}} \times C_{\mathcal {A}}\times C_{\mathcal {Z}}\subseteq \{1\}\times \mathcal {S}\times \mathcal {A}\times \mathcal {Z}$,

  $$\begin{aligned} \Phi _{t+1}^{migr}(C) = 0 \end{aligned}$$

  (A.25)

Definition

A stationary equilibrium is a competitive equilibrium in which per capita variables and functions as well as prices are constant.

Appendix 2: Solution algorithm

In order to solve the model I need to discretize the values of individual savings and idiosyncratic productivity. For individual savings I use the log-space grid with 500 elements. To approximate the distribution of the individual productivities, I use six-state Markov chains.

For each discrete value of the state space, I calculate the optimal decision to the agents’ optimization problems. Since these problems cannot be solved analytically, I use the numerical maximization methods to obtain the decision rules.

Stationary equilibrium

The algorithm used to solve the stationary equilibrium of the model can be summarized in the following steps:

• 1. Guess labour supplies per capita h and l, growth rate of the first generation $n_{1}$ and bequests b.

• 2. Use the world interest rate r to calculate capital per capita k using Eq. (A.6).

• 3. Calculate wages $w^{h}$ and $w^{l}$ from Eqs. (A.7).

• 4. Compute agents’ decisions at the last possible age J. Note that in next period all agents die for sure, so in the last period of life, they consume all their resources. Use backward induction to solve decision problems of younger agents.

• 5. Use computed decision rules to calculate new values of h and l using Eq. (A.13), $n_{1}$ using Eq. (A.9) and b using Eq. (A.8), and compare them with the initial guesses. If they are not sufficiently close to each other, update your guess and repeat the algorithm from step 2 on.

Transiton path

Let $t\in \{0,1,...\}$ denote time along transition, with $t=0$ representing the initial non-migration stationary equilibrium, i.e. equilibrium in which emigration probabilities $\eta ^s_j(a,z)$ are not determined endogenously by A.12 but they are exogenously set to zero. Let $t=1$ represent the period of opening the borders.

The transition path is computed using the following algorithm:

• 1. Use algorithm presented above to calculate the stationary equilibria before and after opening the borders.

• 2. Note that the growth rate of the first generation in period t $n_{1,t}$ is determined by the population structure from period $t-1$, see Eq. (A.9). Hence, in period $t=1$ it is the same as in the initial stationary equilibrium, i.e. $n_{1,1} = n_{1,0}$.

• 3. Assume that the economy moves from the non-migration to migration stationary equilibrium in the arbitrary, big enough, number of periods $\mathcal {T}$. In other words, assume that in period $t=\mathcal {T}$ the economy is in the migration stationary equilibrium.

• 4. Guess the path of labour supplies per capita $\{h_t\}_{t=1}^{\mathcal {T}-1}$ and $\{l_t\}_{t=1}^{\mathcal {T}-1}$ and bequests $\{b_t\}_{t=1}^{\mathcal {T}-1}$.

• 5. Use those paths and the world interest rate r to calculate the sequence of capital stock per capita $\{k_t\}_{t=1}^{\mathcal {T}-1}$ from Eq. (A.6).

• 6. Calculate the sequence of wages $\{w^{h}_t\}_{t=1}^{\mathcal {T}-1}$ and $\{w^{l}_t\}_{t=1}^{\mathcal {T}-1}$ using Eqs. (A.7).

• 7. Starting from $t=\mathcal {T}$, use backward induction to compute the sequence of agents’ optimal decision rules, taking as given the path of prices calculated in the previous step.

• 8. Starting from $t=1$, for each period use agents’ decision rules to calculate new values of labour supplies $h_t$ and $l_t$ (Eqs. (A.13)), and bequests $b_t$ (Eq. (A.8)), and next period growth rate of the first generation $n_{1,t+1}$ (Eq. (A.9)).

• 9. Compare the new sequence $\{h_t\}_{t=1}^{\mathcal {T}-1}$, $\{l_t\}_{t=1}^{\mathcal {T}-1}$ and $\{b_t\}_{t=1}^{\mathcal {T}-1}$ computed in the previous step with the initial guesses. If they are not sufficiently close to each other, update your guess and repeat from step 5 on.

• 10. Check whether in period $t=\mathcal {T}-1$ the economy is sufficiently similar to the economy in period $t=\mathcal {T}$. If not, increase $\mathcal {T}$ and repeat from step 4 on.

Appendix 3: Foreign economy

This Appendix provides the description of the foreign economy. More precisely, I present the definition of competitive equilibrium and provide calibration details.

Competitive equilibrium

Individual state variables are age j, skills s, assets a and individual labour productivity z. The aggregate state of the economy at time t is completely described by $\Phi ^F_t$, which is the joint measure of agents born in the foreign country defined over age, skills, assets positions and individual labour productivity, expressed relative to the size of the youngest generation.

Let $j \in \mathcal {J}=\{1,2,...,J\}$, $s\in \mathcal {S}=\{h,l\}$, $a\in \mathcal {A}=[0, \inf ]$, $z\in \mathcal {Z} = \{z_1, z_2, ..., z_N\}$, and let ${\textbf {W}} \in \mathcal {J} \times \mathcal {S}\times \mathcal {A} \times \mathcal {Z}$. Let ${\textbf {B}}(\mathcal {A})$ be the Borel $\sigma$-algebra of $\mathcal {A}$ and P($\mathcal {J}$), P($\mathcal {S}$), P($\mathcal {Z}$) be the power sets of $\mathcal {J}$, $\mathcal {S}$, $\mathcal {Z}$, respectively. Let M be the set of all finite measures over the measurable space (W, ${\textbf {P}}(\mathcal {J}) \times {\textbf {P}}(\mathcal {S}) \times {\textbf {B}}(\mathcal {A}) \times {\textbf {P}}(\mathcal {Z})$). To ease the notation, in the following I will use $\Phi ^F_t$ instead of $\Phi ^F_t(\mathcal {J}\times \mathcal {S} \times \mathcal {A}\times \mathcal {Z})$.

Definition

Given initial conditions $b_1^F$ and $\Phi ^F_1$, a competitive equilibrium is a sequence of functions for the household $\left\{ \right. V^F_t$, $c^F_t$, $a'^F_t$: W $\rightarrow$ ${\textbf {R}}_+$ $\left. \right\} _{t=1}^{\infty }$, production plans for the firm $\{k^F_t, h^F_t, l^F_t\}_{t=1}^{\infty }$, prices $\{r_t, w^{Fh}_t, w^{Fl}_t\}_{t=1}^{\infty }$, bequests $\{b^F_t\}_{t=1}^{\infty }$, population growth rates $\{n^F_{t}\}_{t=1}^{\infty }$, and measures $\{\Phi ^F_t\}_{t=1}^{\infty }$, with $\Phi ^F_t \in {\textbf {M}}$ such that:

• 1. given prices, bequests and initial conditions, for each t, $V^F_t$ solves the functional equation (with $c^F_t$, $a'^F_t$ as associated policy functions):

  $$\begin{aligned} V^F_t(j,s,a,z) = \max _{c, a'\ge 0} \quad \{ u(c)+\beta ^F \psi ^{Fs}_{j} \sum _{z'} \pi ^{Fs}(z'|z) V_{t+1}^{F} (j+1,s, a',z')\} \end{aligned}$$

  (C.1)

  subject to:

  $$\begin{aligned} c+a'=(1+r_t)(a + b_t^F) +w_t^{Fs} e_{j}^{Fs}z \end{aligned}$$

  (C.2)

• 2. Prices $r_t$, $w^{Fh}_t$, $w^{Fl}_t$ satisfy:

  $$\begin{aligned} \begin{aligned} r_t&= A^F F^F_{k}(k^F_t, h^F_t, l^F_t) -\delta ^F\\ w^{Fh}_t&= A^F F^F_{h}(k^F_t, h^F_t, l^F_t)\\ w^{Fl}_t&= A^F F^F_{l}(k^F_t, h^F_t, l^F_t) \end{aligned} \end{aligned}$$

  (C.3)

• 3. Bequests are given by:

  $$\begin{aligned} b^F_t = \frac{\int (1-\psi ^{Fs}_{j}) {a'}_{t-1}^{Fs}(j,s,a,z) d\Phi _{t-1}^{F}}{\int (1+n^F_{1}) d\Phi _t^{F}} \end{aligned}$$

  (C.4)

• 4. Population growth rate is given by:

  $$\begin{aligned} n^F_{t} = \frac{(1+n^F_{1})\int d\Phi ^F_t}{\int d\Phi ^F_{t-1}} -1 \end{aligned}$$

  (C.5)

• 5. Markets clear:

  $$\begin{aligned} k^F_{t} = \frac{\int a d\Phi _{t}^{F}}{\int d\Phi _t^{F}} + b^F_t \end{aligned}$$

  (C.6)
  $$\begin{aligned} \begin{aligned} h^F_t&= \frac{\int e_j^{Fh} z d\Phi _t^{F}}{\int d\Phi _t^{F}}\\ l^F_t&= \frac{\int e_j^{Fl}z d\Phi _t^{F}}{\int d\Phi _t^{F} } \end{aligned} \end{aligned}$$

  (C.7)
  $$\begin{aligned} A^F \left( \alpha ^F (l_t^F)^{\gamma }+(1-\alpha ^F)\left[ \rho ^F (k^F_t)^{\eta }+(1-\rho ^F)(h_t^F)^{\eta }\right] ^{\frac{\gamma }{\eta }}\right) ^{\frac{1}{\gamma }}= c^F_t + i^F_t \end{aligned}$$

  (C.8)

  where

  $$\begin{aligned} c^F_t= \frac{\int c_{t}^{F}(j,s,a,z) d\Phi _t^{F}}{\int d\Phi _t^{F}} \end{aligned}$$

  (C.9)

  and

  $$\begin{aligned} i^F_t=k^F_{t+1}(1+n^F_{t+1})-(1-\delta ^F)k^F_t \end{aligned}$$

  (C.10)

• 6. Law of motion: For any subset $C=C_{\mathcal {J}} \times C_{\mathcal {S}} \times C_{\mathcal {A}}\times C_{\mathcal {Z}} \subseteq \{2,..., J\}\times \mathcal {S}\times \mathcal {A}\times \mathcal {Z}$,

  $$\begin{aligned} \Phi ^F_{t+1}(C) = \frac{1}{1+n^F_{1}}\int \psi ^{Fs}_{j} P_t^F\left( (j,s,a,z),C\right) d\Phi ^F_t \end{aligned}$$

  (C.11)

  where

  $$\begin{aligned} P_t^F\left( (j,s,a,z),C\right) = {\left\{ \begin{array}{ll} \pi ^{Fs}(z'|z) &{} \text {if }(j+1, s, a'^F_t(j,a,s,z), z')\in C\\ 0 &{} \text {otherwise} \end{array}\right. } \end{aligned}$$

  (C.12)

  For any $C=C_{\mathcal {J}} \times C_{\mathcal {S}} \times C_{\mathcal {A}}\times C_{\mathcal {Z}} \subseteq \{1\}\times \mathcal {S}\times \mathcal {A}\times \mathcal {Z}$,

  $$\begin{aligned} \Phi _{t+1}^F(C) = {\left\{ \begin{array}{ll} \sum \limits _{s\in C_{\mathcal {S}}} \sum \limits _{z\in C_{\mathcal {Z}}} \nu _1^{Fsz}&{} \text {if }0\in C_{\mathcal {A}}\\ 0 &{} \text {otherwise} \end{array}\right. } \end{aligned}$$

  (C.13)

  where $\nu _1^{Fsz}$ is the share of workers with skill level s and labour productivity z in the cohort of newborn agents.

Calibration

Table 6 presents calibrated model parameters. Calibration targets are depicted in Table 7.

Table 6 Foreign economy: calibrated model parameters

Table 7 Foreign economy: target moments

Appendix 4: Remittances scenario

Outward population movements are usually accompanied by increasing inflow of remittances. These transfers from abroad may affect both the well-being of those staying behind and the macroeconomic aggregates of remittance-receiving countries (see, e.g. Rapoport and Docquier, 2005; OECD, 2007; di Giovanni et al., 2015). Taking into account their potentially important role, in this robustness exercise I augment my model with remittances. In this scenario, I recalibrate the migration costs and set $\mu _h$ equal to 0.855 and $\mu _l$ equal to 1.192, to ensure that the size and skill structure of emigration in the stationary equilibrium are the same as in the baseline migration steady state. I also modify the budget constraint of the emigrant given in the baseline model by Eq. (2) to:

$$\begin{aligned} c+a'=(1+r)(a + b_t) + (1-\kappa )w^{F}e^F_{j}z \end{aligned}$$

(D.1)

with $\kappa$ being the fraction of labour income sent to the home country.

To calibrate $\kappa$, I use the survey data on emigrants from Poland collected by Polish central bank. I use data from 2018, which is the last vintage available. In that year, the survey was carried out in four countries on a sample of 1500 individuals in Great Britain, 1500 in Germany, 1000 in Norway and 700 in the Netherlands. The interviews were conducted among Polish citizens aged between 18 and 65 who had left Poland for work purposes. More detailed information on the survey and its methodology can be found in, e.g. Chmielewska et al. (2019).

First, I took a look at the questions about the net monthly wage and the amount of money transferred to Poland in the last 12 months. Since in this type of questions, respondents usually prefer not to mention exact numbers, they were also asked to choose one of the provided wage/transfer ranges. If the respondent chose to pinpoint only the wage/transfer range, I assigned him the middle point in the given category. The respondents choosing the last, open-ended wage/transfer category were assigned the value of average wage/transfer calculated for the interviewees belonging to the respective wage/transfer category who also provided the exact answer. Since no emigrant residing in Norway and pointing the last open-ended income range provided the exact value for his wage, it was impossible to evaluate the income of the highest labour income earners in this emigration country. Hence, finally I decided to exclude the respondents from Norway from my analysis. To convert the answers provided by emigrants in Great Britain, Germany and the Netherland into one currency, I used the average exchange rates from 2018.

The ratios of average money transfer remitted to Poland by individuals sending remittances (i.e. declaring transfer greater than zero) to average net annual wage (twelve times the average net monthly wage) calculated for different age and skill groups are provided in Table 8. The ratio turns out to be pretty constant across the groups and amounts to around 0.2, with a little bit lower value for young skilled agents. Table 9 presents the fractions of emigrants sending money to Poland. Again, this fraction does not vary significantly across different groups and is equal to around 0.35. As the numbers of observations in particular age-skill categories are quite small, in my calibration I decided to differentiate the fraction of remitted income only with respect to skills. Multiplying the respective numbers from the last columns of Tables 8 and 9 gives us the value of $\kappa$ equal to 0.06 for skilled agents and equal to 0.08 for the unskilled ones.

Table 8 Average fraction of income remitted to Poland by individuals sending remittances.

Table 9 Ratio of emigrants sending money to Poland to all emigrants from Poland.

Since the available dataset lacks the information on remittance recipients, I assume that the money sent to the home economy by emigrants is distributed equally among all residents of the country. Hence, the budget constraint of the agent who decides to stay in the current period in his home country, i.e. Eq. (5) from the baseline model, is now given by:

$$\begin{aligned} c+a'=(1+r)(a+ b_t)+w_t e_{j}z + \Xi _t \end{aligned}$$

(D.2)

with $\Xi _t$ denoting the transfer received by home country residents from abroad and calculated as:

$$\begin{aligned} \Xi _t = \frac{ \int \kappa ^s w^{Fs}e_j^{Fs}z d\Phi _t^{migr}(\mathcal {J}\times \mathcal {S} \times \mathcal {A}\times \mathcal {Z})+\int \kappa ^s w^{Fs}e_j^{Fs}z\eta ^s_{j,t}(a,z) d\tilde{\Phi }_t^{stay}(\mathcal {J}\times \mathcal {S} \times \mathcal {A}\times \mathcal {Z})}{\int (1-\eta ^s_{j,t}(a,z)) d\tilde{\Phi }_t^{stay}(\mathcal {J}\times \mathcal {S} \times \mathcal {A}\times \mathcal {Z}) } \end{aligned}$$

(D.3)

where $\tilde{\Phi }_t^{stay}$ and $\Phi _t^{migr}$ denote the measures of agents who reside in the beginning of period t, i.e. before migration decisions are made, in, respectively, home country and abroad, expressed relative to the size of the first generation.

Note that in order to solve this version of the model, I need to additionally guess and update in each iteration the value (or the path in case of solving the transition path) of money transfer $\Xi _t$.

Although the long run changes in the economic aggregates in the analysed scenario are roughly similar to the changes observed in the baseline model, see Table 10, several differences stand out.

Table 10 Comparison of stationary equilibria: remittances scenario

Even though the size of the emigrant flow is the same in both scenarios, the stock of emigrants is a bit higher in the model with remittances. Since in the remittances scenario the incentives to emigrate are lower, in order to ensure the desired size of migration flows, the utility costs of migration have to be also calibrated to the level lower than in the baseline model. This modification affects differently the willingness to relocate of various age cohorts and shifts the age distribution of emigration flows towards younger generations. Since agents decide to change the country of their residence earlier in life, the ratio of emigrants to individuals born in the country in the analysed scenario is higher than in the baseline model, which in turn leads to stronger changes in the macroeconomic aggregates. The only variable which reacts less are the bequests. This is the immediate consequence of the obligation to remit part of the income and hence lower asset accumulation by emigrants.

In the migration equilibrium the sum of money remitted to the home country is equal to 0.98% of its output. This is roughly consistent with the data: according to the World Bank statistics, remittances received by Poles in 2020 constituted around 1.1% of Poland’s GDP.

Table 11 Inequality measures: remittance scenario

The inequality measures presented in Table 11 confirm that emigration raises labour income disproportions. The rise in labour income inequalities is, however, a bit weaker than in the baseline scenario. While the difference between wages per efficiency unit of labour received by skilled and unskilled individuals is greater than in the baseline migration equilibrium, the selectivity in terms of individual productivity is more pronounced. This stronger outflow of people from the lower end of individual productivity distribution limits the within-skill-group inequalities and hence mitigates the overall labour income disproportions. Moreover, in this scenario the skill distribution of population shifts more towards unskilled agents who are relatively less income-differentiated. This also pushes overall labour income inequalities down. This stronger shift results from the higher than in the baseline migration equilibrium stock of emigrants who are more skilled than the stayers.

Table 12 Welfare effects: non-migration equilibrium versus remittance scenario

Interestingly, Gini coefficients based on incomes including remittances, i.e. labour income and remittances received, and total disposable income which additionally includes bequests and capital income, slightly decline. My analysis shows therefore that money transfers from abroad more than offset rising labour income inequalities in the country of origin. It should be noted that impact of remittances on income inequalities crucially depends on who and how much receives in money transfers from abroad. The empirical analysis conducted by Giannetti et al. (2009), which was based on the EU-SILC data, shows that in CEE countries, including Poland, remittances are mainly paid to households from the lower end of the income distribution. Incorporating this finding into my model instead of assuming that remittances are distributed equally among all citizens of the sending country would only strengthen the model conclusions that remittances limit income inequalities.

Table 12 presents the long run welfare effects of opening the borders. Obviously, the gains are higher than in the model without money transfers from abroad. Contrary to the baseline model, in this scenario the unskilled never-migrants benefit from the open borders even after eliminating the positive role of higher bequests. On average, they would require 0.89% increase of lifetime consumption in the non-migration steady state in order to be indifferent to being born in the new migration equilibrium.

In the analysed scenario, the bequests have much smaller impact on the evolution of welfare measures. As we can see in Fig. 7, they affect the level of welfare effects, but, contrary to the baseline scenario, do not change the conclusions which age cohorts benefit more and which less from emigration possibility.

Fig. 7

Welfare effects: remittances scenario. Notes: Fixed bequests: bequests along the transition are fixed at the level from non-migration equilibrium

As far as the future generations are concerned, the cohorts born late after opening up the borders benefit the most. This holds true for both skill groups and results from the rising remittances received by the individual living in the home country. Looking at the situation of skilled current generations let us notice that the agents who are young at the moment of removing migration barriers gain the most. These agents have relatively big chances to emigrate to the richer country. Moreover, even if they do not decide to relocate, they enjoy higher wages in the home country and receive transfers from abroad. The gains of the elder workers are lower and then they again increase for the retirees. The individuals after retirement have no incentives to change the country of their residence, but since their consumption is relatively low, even small rise in disposable income due to received remittances leads to high percentage change in consumption. For the unskilled pensioners, this percentage change in consumption is even higher. The welfare effects for unskilled agents who are young when the borders open are relatively low due to falling wage per unit of efficiency labour.

To sum up, accounting for remittances does not change the general conclusions about the impact of emigration on macroeconomic aggregates. More interestingly, it turns out that money transfers from abroad more than offset the effects of rising wage differential and lead to lower inequalities in disposable income. In the presence of remittances, the welfare gains are higher and, contrary to the baseline scenario, even the unskilled never-migrants benefit from open borders.

Appendix 5: Convergence scenario

The real convergence, i.e. the process whereby the GDP per capita of lower-income regions grows faster than that of higher-income regions, supports income equalisation, limiting thereby the incentives to emigrate. In order to reflect it, in this scenario I modify the baseline model and assume that the total factor productivity in the home country is no longer constant but it grows exogenously and finally reaches the level observed abroad. The speed of convergence is determined by estimating the following equation:

$$\begin{aligned} gdp_t - gdp_t^* = \alpha _0 + \alpha _1(gdp_{t-1}-gdp^*_{t-1})+\epsilon _t \end{aligned}$$

(E.1)

where $gdp_t$ denotes log of purchasing power adjusted GDP per capita in period t in Poland and $gdp_t^*$ denotes log of purchasing power adjusted GDP per capita in period t in the EU15 countries. The equation is estimated on annual data from the post-communist period (1990-2017) that is taken from Penn World Table database (Feenstra et al., 2015). The estimated speed of convergence, $1-\hat{\alpha }_1 = 0.0165$, is roughly in line with the estimates for CEE region obtained in other studies (Matkowski and Próchniak, 2004) and with the general result prevailing in the economic literature that the convergence rate amounts to around 2% (Barro, 2012).

TFP convergence and the resulting narrowing wage gap between sending and destination economies have a major impact on agents’ migration decisions. In the stationary equilibrium in which TFP in the sending country reaches the level observed abroad, individuals do not want to relocate, i.e. migration flows are equal to zero. This holds true even if the utility costs of migration are set to zero. As the calibration of the utility migration costs does not affect the stationary equilibrium, in this scenario I set them to target the moments along the transition. More specifically, the upper limits of the uniform distributions of the migration utility costs $\lambda$ are calibrated to $\mu _h = 0.73$ and $\mu _l =2.66$ and ensure that in the first 10 years of transition the average ratio of emigration flows to total population born in the country and the average share of skilled emigrants in total emigrant stock are equal to the respective averages from the baseline transition path.

Since in the steady state with full TFP convergence agents do not emigrate, all the differences compared to the initial non-migration equilibrium follow from different TFP levels. In the short run, however, part of the agents born in the home country decide to change the country of their residence. Changes observed in the economy along the transition are presented in Fig. 8.

Fig. 8

Macro variables along the transition: convergence scenario. Notes: Figures e–i: values in the non-migration equilibrium normalized to 100

Just after opening up the borders around 0.38% of country’s population moves abroad. In later periods, the outflow declines and finally, after around 40 years, completely vanishes. At about this time, ratio of number of emigrants to total population reaches its peak: circa 4.31% of agents born in the sending country reside abroad. Afterwards, since there are no new migrants, we observe the gradual decline in emigrant stock. In around 100 years after allowing for labour mobility, the emigrants’ population entirely disappears. Hence, in this scenario migration turns out to be a transitory phenomenon, although lasting for many years.

In the non-migration equilibrium, wages per efficiency unit of labour received by skilled and unskilled workers are equal. The convergence of TFP changes this situation and hence affects the skill premium. Skilled workers gain not only because of the rising productivity of the economy, but also due to increasing capital supply and the complementarity of skilled labour and capital in the aggregate production. As a result, wage per efficiency unit of skilled labour rises much more strongly than the wage of unskilled workers.

Since in the short run the population gets older and there are more people in the retirement age, the labour supplies per capita drop. My calibration of utility costs of migration ensures, that in the first transition periods there are relatively more skilled agents among the emigrants than among the stayers. However, as the rise in wages of skilled workers is from the very beginning more pronounced than the rise in wages of unskilled individuals, crossing the borders sooner becomes unattractive for skilled workers. Hence, the fraction of skilled individuals rises and the rebound in the skilled labour is quicker and stronger than in case of the unskilled labour. In around 45th transition period, the skilled labour supply per capita reaches its peak and is about 1.4% higher than in the non-migration stationary equilibrium.

Interestingly, the evolution of net foreign assets to output ratio along the transition is not monotonic. After the initial decline, the ratio rises and finally stabilizes at the level higher than in the non-migration equilibrium. Very similar behaviour is observed in case of the total assets to output ratio. Clearly, the drop observed in the short run is a standard implication of TFP convergence: as the agents expect to earn more in the future, they decrease their savings in order to smooth their consumption. On the other hand, the capital stock to output ratio falls along the transition. This suggests that the technological progress in the assumed production function is not neutral for the factor shares.

To sum up, accounting for TFP convergence limits the incentives to relocate, and hence we observe no emigration flows in the long run. The analysis of the transition path reveals, however, that in the short run part of the agents born in the home country decide to change the country of their residence. Hence, in this scenario migration turns out to be a transitory phenomenon, although lasting for several decades. The productivity convergence affects the skill premium as wages of skilled workers increase not only because of the rising efficiency of the economy, but also due to rising capital supply and the complementarity of skilled labour and capital in the aggregate production. This redistribution effect is in the short run additionally strengthened by the changes in the population education structure since emigration flows are from the very beginning biased towards skilled agents.