Permanent Migrants and Cross-Border Workers: The Effects on the Host Country

Abstract

When we use the word “migrants,” we tend to disregard the variation in their behavior. We can in fact classify migrants according to length of stay, purpose of migration, geographical origin, or historical background. In this paper, I focus on the time interval of migration and distinguish the three types of migrants: permanent migrants, temporary migrants, and cross-border workers.

This paper was originally published by the Journal of Regional Science as an article in its vol. 38 (1999) issue.

Appendices

Appendix 1

With p _N being constant, total differentiation of Eqs. 2.1, 2.2, 2.3, and 2.4 with respect to L _N , L _T , K _N , K _T , and L _M yields

$$ \begin{array}{cc} & {p}_N{N}^{LL}d{L}_N+{p}_N{N}^{LK}d{K}_N-{T}^{LL}d{L}_T-{T}^{LK}d{K}_T=0,\\[6pt] & {p}_N{N}^{KL}d{L}_N+{p}_N{N}^{KK}d{K}_N-{T}^{KL}d{L}_T-{T}^{KK}d{K}_T=0,\\[6pt] & d{L}_T+d{L}_N-d{L}_M=0,\\[6pt] & d{K}_T+d{K}_N=0,\end{array} $$

(A2.1)

where N ^LL ≡ ∂N ^L /∂L _N and N ^LK , N ^KL , N ^KK , T ^LL , T ^LK, and T ^KK are defined in a similar way.

Because N(L _N , K _N ) and T(L _T , K _T ) are linearly homogeneous, we apply Euler’s theorem and obtain,

$$ \begin{array}{l}{N}^{LL}{L}_N+{N}^{LK}{K}_N=0,\\ {N}^{KL}{L}_N+{N}^{KK}{K}_N=0,\\ {T}^{LL}{L}_T+{T}^{LK}{K}_T=0,\\ {T}^{KL}{L}_T+{T}^{KK}{K}_T=0.\end{array} $$

(A2.2)

The first two equations of (A2.1) can be expressed as follows by substituting other equations:

$$ \begin{array}{ll} & \left[\begin{array}{c@{\quad}c} {p}_N{k}_N{N}^{LK}+{k}_T{T}^{LK} & -\left({p}_N{N}^{LK}+{T}^{LK}\right) \\[6pt] -\left({p}_N{N}^{LK}+{T}^{LK}\right) & {p}_N\left(1/{k}_N\right){N}^{LK}+\left(1/{k}_T\right){k}_T{T}^{LK} \end{array}\right]\left[\begin{array}{c} d{L}_T \\[6pt] d{K}_T \end{array}\right]\\[12pt] & \quad =\left[\begin{array}{c} {k}_N{p}_N{N}^{LK} \\[6pt] -{p}_N{N}^{LK} \end{array}\right]d{L}_M.\end{array} $$

(A2.3)

Let Φ stand for the determinant of the two-by-two matrix in (A2.3), then $\Phi =\left[{\left({k}_T-{k}_N\right)}^2/{k}_T{k}_N\right]{p}_N{N}^{LK}{T}^{LK}>0.$ Therefore,

$$ \frac{d{L}_T}{d{L}_M}=\frac{1}{\Phi}{p}_N{N}^{LK}{T}^{LK}\frac{k_N-1}{k_T}=\frac{k_N}{k_N-{k}_T}, $$

$$ \frac{d{K}_T}{d{L}_M}=\frac{1}{\Phi}{p}_N{N}^{LK}{T}^{LK}\left({k}_T-{k}_N\right)=\frac{k_T{k}_N}{k_N-{k}_T}, $$

$$ \frac{d{L}_N}{d{L}_M}=1-\frac{d{L}_T}{d{L}_M}=\frac{-{k}_T}{k_N-{k}_T}, $$

$$ \frac{d{K}_N}{d{L}_M}=-\frac{d{K}_T}{d{L}_M}=\frac{-{k}_T{k}_N}{k_N-{k}_T}. $$

Hence, $d{L}_T/d{L}_M>\left(<\right)0,d{K}_T/d{L}_M>\left(<\right)0,d{L}_N/d{L}_M<\left(>\right)0,$ and $d{K}_N/d{L}_M<\left(>\right)0,$ according to ${k}_T-{k}_N<\left(>\right)0$. Thus, $dT/d{L}_M>\left(<\right)0$ and $dN/d{L}_M<\left(>\right)0$ if ${k}_T-{k}_N<\left(>\right)0$. The sign of dL _T /dL _M is opposite to that of dL _N /dL _M , implying that in view of the third equation of (A2.1), dL _N /dL _M must be greater than unity when $d{L}_N/d{L}_M>0$.

Next, holding L _M constant, total differentiation of Eqs. 2.1, 2.2, 2.3, and 2.4 with respect to p _N , L _N , L _T , K _N , and K _T gives

$$ \begin{array}{cc} & {N}^Ld{p}_N+{p}_N{N}^{LL}d{L}_N+{p}_N{N}^{LK}d{K}_N-{T}^{LL}d{L}_T-{T}^{LK}d{K}_T=0,\\[6pt] & {N}^Kd{p}_N+{p}_N{N}^{KL}d{L}_N+{p}_N{N}^{KK}d{K}_N-{T}^{KL}d{L}_T-{T}^{KK}d{K}_T=0,\\[6pt] & d{L}_T+d{L}_N=0,\\[6pt] & d{K}_T+d{K}_N=0.\end{array} $$

(A2.4)

The first two equations of (A2.4) can be expressed, by substituting the other two equations of (A2.4) and (A2.2), as

$$ \begin{array}{ll} & \left[\begin{array}{c@{\quad}c} {p}_N{k}_N{N}^{LK}+{k}_T{T}^{LK} & -\left({p}_N{N}^{LK}+{T}^{LK}\right) \\[8pt] -\left({p}_N{N}^{LK}+{T}^{LK}\right) & {p}_N\left(1/{k}_N\right){N}^{LK}+\left(1/{k}_T\right){k}_T{T}^{LK} \end{array}\right]\left[\begin{array}{c} d{L}_T \\[8pt] d{K}_T \end{array}\right]\\[10pt] & \quad =\left[\begin{array}{c} -{N}^L \\ -{N}^K \end{array}\right]d{p}_N,\end{array} $$

(A2.5)

and

$$ \displaystyle\frac{d{L}_T}{d{p}_N}=\frac{1}{\Delta}\left|\begin{array}{c@{\quad}c} -{N}^L & -\left({p}_N{N}^{LK}+{T}^{LK}\right) \\[6pt] -{N}^K & {p}_N\left(1/{k}_N\right){N}^{LK}+\left(1/{k}_T\right){k}_T{T}^{LK} \end{array}\right|, $$

$$ \displaystyle \frac{d{K}_T}{d{p}_N}=\frac{1}{\Delta}\left|\begin{array}{c@{\quad}c} {p}_N{k}_N{N}^{LK}+{k}_T{T}^{LK} & -{N}^L \\ -\left({p}_N{N}^{LK}+{T}^{LK}\right) & -{N}^K \end{array}\right|, $$

where Δ is the determinant of the left-hand-side matrix of (A2.5). To obtain the result of Eq. 2.8, it is necessary to use (A2.2) to substitute T ^LL = k _T T ^LK, and then substitute the expressions for dL _T /dp _N and dK _T /dP _N , given above.

Appendix 2

Following Wong (1995), let us define the gross domestic product (GDP) function as

$$ g\left(p,v\right)=\underset{Q}{\mathrm{Max}}\left\{pQ:\Gamma \left(Q,v\right)\le 0\right\}, $$

where v denotes the m-dimensional vector of factor endowments, Q denotes the n-dimensional vector of outputs, p denotes the n-dimensional vector of goods prices, and $\Gamma \left(Q,v\right)\le 0$ denotes the production possibility set of the economy. Let $v^0$ be factor endowment before immigration and $v^e$ represent the number of factors flowing into and working in the economy. The total factors available to domestic firms are ${v}^t={v}^0+{v}^e$.

First, let us consider the case of a small country and the scenario where every good is tradable. Then, the vector of goods prices can be expressed as p ^w, which remains constant after international factor mobility. The profit maximization behavior of each firm can be interpreted as expenditure minimization behavior, and this implies a minimization of payments to factors. Thus, factor prices before migration, w ⁰, can be expressed as

$$ g\left({p}^w,{v}^0\right)=\underset{w}{\mathrm{Min}}\left\{wv:{p}_i^w\le {c}_i\left({w}^0\right),i= 1,.\dots, n\right\}={w}^0{v}^0, $$

where c _i (w) denotes the unit cost function of good i. Similarly, factor prices after immigration, w ¹, can be expressed as

$$ g\left({p}^w,{v}^t\right)={w}^1{v}^t. $$

Remember that w ⁰ is the vector of factor prices that minimize expenditure to employed factors $v^0$; w ¹ is the vector prices for factors $v^t$, not $v^0$. Thus, we can assert that

$$ {w}^0{v}^0<{w}^1{v}^0, $$

(A2.6)

under the assumption that ${w}^1\ne {w}^0$ and a strictly convex production possibility set. Now, let us define the indirect utility function of domestic residents as

$$ {V}^0\left(\,p,v,b\right)=\underset{C}{\mathrm{Max}}\left\{u(C):pC\le g\left(\,p,v\right)-b\right\}, $$

where C is the aggregate consumption bundle, u(C) is the social welfare level, and b is income transfers to immigrants. Before immigration, the factor endowment is $v^0$, and payment for those employed factors can be expressed as $v^0w^0$. Thus, the utility level of domestic residents before immigration yields $V^0(p^w,v^0,0)$. On the other hand, after immigration, the factor endowment is $v^t$, and payment for those employed factors can be expressed as $v^tw^t$. In this case, there exists transfer b, which equals $w^1v$. Thus, the utility level of domestic residents after immigration yields $V^0(p^w,v^t,b)$.

Equation A2.6 implies that the income of domestic workers will increase after immigration. Bearing in mind that the price vector remains constant and indirect utility is a strictly increasing function of income, we obtain

$$ {V}^0\left({p}^w,{v}^0,0\right)<{V}^0\left({p}^w,{v}^t,b\right), $$

(A2.7)

which implies that immigration enhances the welfare of domestic residents.