Welfare effects of illegal immigration

Abstract

This paper analyzes the welfare effect of illegal immigration on the host country within a dynamic general equilibrium framework and shows that it is positive for two reasons. First, immigrants are paid less than their marginal product, and second, after an increase in immigration, domestic households find it optimal to increase their holdings of capital. It is also shown that dynamic inefficiency may arise, despite the fact that the model is of the Ramsey type. Nevertheless, the introduction of a minimum wage, which leads to job competition between domestic unskilled workers and immigrants reverses all of the above results.

Appendices

Appendix 1

1.1 The wage rate paid to immigrants

In this Appendix, I offer two alternative justifications for the relative wage rate paid to immigrants, namely, the equation

$$w_{m}=\beta w,\text{ \ \ }\beta \in \lbrack 0,1].$$

(6.1)

• (a) First, suppose that there is a cost associated with the employment of an undocumented immigrant (a similar approach is followed in Epstein and Heizler 2007). If caught employing an illegal immigrant, which occurs with probability p, an employer must pay a fine γ to the government. Then, maximizing again the representative competitive firm’s profit (Eq. 2.5) with respect to all inputs K, L and M

  $$\Pi =F(K,N)-rK-wL-w_{m}M-p\gamma\! M,$$

  taking all prices as given, yields, besides F _K  = r and F _L  = w, F _M  = w _m  + pγ. As F _L  = F _M , it follows that w − w _m  = pγ > 0; that is, as assumed in the main text, illegal immigrants are paid a wage (w _m ) that is lower than the wage rate (w) paid to domestic labor. In particular, as w _m  = βw and w − w _m  = pγ, it follows that

  $$\beta =\frac{w-p\gamma }{w}.$$

  (6.2)

  Hence, β is determined in equilibrium. Moreover, if either the fine or the probability of being caught is zero, that is, if the expected fine is equal to zero, then the immigrants will be paid as much as the natives. With this formulation, firms will not have any profit, and, thus, there will be no dividend. Hence, we have to examine if the results that are derived in the main text change. Instead of the firms, the revenue from the “exploitation” of the immigrants goes to the government. Indeed, the government raises total revenue equal to

  $$\Pi =p\gamma\! M.$$

  If we assume that this revenue is distributed to domestic households in a lump-sum manner, then

  $$\pi \equiv \frac{\Pi }{N}=p\gamma (1-\alpha ).$$

  (6.3)

  Substituting Eq. 6.3 in the private budget constraint (2.2), we get

  $$\dot{k}=f(k)-\alpha c-(1-\alpha )[w(k)-p\gamma ]-nk.$$

  (6.4)

  The equilibrium path is then determined by Eqs. 2.6 and 6.4. If we make the additional assumption that the fine γ imposed by the government is a multiple of the current wage rate, for example γ = θw, then Eq. 6.4 becomes

  $$\dot{k}=f(k)-\alpha c-(1-\alpha )(1-p\theta )w(k)-nk.$$

  (6.5)

  In this case, the analysis is identical to the one in the main text, as upon setting 1 − pθ ≡ β, Eqs. 6.5 and 2.8 become identical. Nevertheless, even if one does not accept the simplification γ = θw, all the results still hold. The only difference is that the term w(k) − pγ replaces now the term βw(k).

• (b) Suppose instead that the wage rate of the immigrant workers is determined via a bargaining process. Specifically, suppose that, each period, a representative firm bargains with a representative immigrant for the wage rate w _m . All negotiations are instantaneous and conducted according to a Nash bargaining rule, in which immigrants receive a share β and firms a share 1 − β in the surplus from a match. If the firm employs an immigrant worker, it gains Π = F(K,N + M) − rK − wL − w _m . Instead of the immigrant, the firm can employ a domestic worker, in which case it gains Π ^′ = F(K,N + M) − rK − wL − w. Hence, the gain to a firm employing an illegal immigrant is w − w _m . On the other hand, an immigrant can work for the firm and receive w _m or return to her country and receive w _f . Thus, the gain to an illegal immigrant from accepting to work for a firm in the host country is w _m  − w _f . The wage rate w _m derived from the Nash bargaining solution is such that it maximizes $(w-w_{m})^{1-\beta}(w_{m}-w_{\!f})^{\beta }$, $\beta \in \lbrack 0,1]$. Performing the suggested differentiation leads to w _m  = βw + (1 − β)w _f . This equation coincides with Eq. 6.1 if w _f  = 0. This can be justified with the additional assumption that the cost for the immigrant of returning to her country is high relative to the benefit. Thus, her outside option is not w _f but zero.

Appendix 2

1.1 Immigrants’ saving behavior

In this Appendix, I show that making the alternative assumption, as in Hazari and Sgro (2003) and Moy and Yip (2006), according to which immigrants do not save, instead of sending all their savings abroad, does not change our results. Let us start by specifying the resource constraint in aggregate terms for the economy:

$$\dot{K}+C+C_{M}+S_{M}=F(K,N),$$

(6.6)

where C _M and S _M denote, respectively, the consumption and savings of the immigrants. Consider the following two alternatives: (a) Immigrants do not save, and hence, their consumption is equal to their income, that is,

$$S_{M}=0\quad \text{and}\quad C_{M}=\beta Mw,$$

(6.7)

or (b) immigrants save but they channel all their savings abroad, that is,

$$S_{M}>0\quad \text{and}\quad C_{M}+S_{M}=\beta Mw.$$

(6.8)

Substituting either of the two Eqs. 6.7 and 6.8 in Eq. 6.6 leads to

$$\dot{K}+C=F(K,N)-\beta Mw,$$

or if we divide both sides by N

$$\dot{k}+ac=f(k)-b(1-\alpha )w-nk,$$

which is the resource constraint (2.8) considered in the main text. Therefore, both alternatives lead to the same resource constraint. Of course, Eq. 2.6, which determines the consumption rule of the natives, also does not change if either assumption is made; thus, the equilibrium remains the same under either assumption. The reason is simply that this is a growth model with full employment. As long as capital accumulation is not affected, whether part of immigrants’ income is spent within the country or abroad does not alter the equilibrium.