I now expand the basic model in several ways. First, I assume that there are two goods in the economy; one good is produced domestically and the other good is imported^{6}. Second, I allow for changes in product demand both because immigration may have changed the price of the domestically produced product (encouraging consumers to change their quantity demanded) and because immigrants themselves will consume the product. Finally, I explicitly introduce a supply curve of domestic capital. The resulting general equilibrium model has much in common with derivations of Marshall’s rules of derived demand. The technical details are summarized in the Mathematical Appendix.

Two goods are consumed in a large economy: good *q* is produced domestically and good *y* is imported^{7}. To fix ideas, I initially assume that the price of the imported good *y* is set in the global marketplace (or, alternatively, that it is produced at constant marginal cost). In this context, the price of *y* is the numeraire and set to unity. I will relax this assumption below and introduce an upward-sloping foreign export supply curve for *y*.

Each consumer *j* has the quasilinear utility function:

U\left(y,q\right)=y+{g}_{j}^{*}\phantom{\rule{0.12em}{0ex}}\frac{{q}^{\mathrm{\xi}}-1}{\mathrm{\xi}},

(8)

where the weight *g*^{*} reflects the consumer’s relative preference for the domestic good and may be different for different consumers. I assume that the utility function is quasiconcave, so that ξ< 1. Let *Z* be the consumer’s income. The budget constraint is given by:

Utility maximization implies that the product demand function for the domestic good is:

{q}_{j}={g}_{j}\phantom{\rule{0.12em}{0ex}}{p}^{-1/\left(1-\mathrm{\xi}\right)},

(10)

where *q*
_{
j
} is the amount of the good consumed by consumer *j;* and *g*
_{
j
} is the rescaled person-specific weight. The quasilinear functional form for the utility function implies that the consumer’s demand for the domestic product does not depend on his income. The assumption that there are no wealth effects will also be relaxed below.

Three types of persons consume good *q*: domestic workers, domestic capitalists, and consumers in other countries. Let *C*
_{
L
} be the number of domestic workers, *C*
_{
K
} be the number of domestic capitalists, and *C*
_{
X
} be the number of consumers in the “rest of the world”^{8}. I assume that all consumers have the same quasilinear utility function in (8), but that the weighting factor *g* may differ among the various types of consumers. The total quantity demanded by domestic consumers (*Q*
_{
D
}) and foreign consumers (*Q*
_{
X
}) is then given by:

{Q}_{D}=\left({g}_{L}\phantom{\rule{0.12em}{0ex}}{C}_{L}+{g}_{K}\phantom{\rule{0.12em}{0ex}}{C}_{K}\right){p}^{-1/\left(1-\mathrm{\xi}\right)},

(11a)

{Q}_{X}={g}_{X}{C}_{X}{p}^{-1/\left(1-\mathrm{\xi}\right)}.

(11b)

Balanced trade requires that expenditures on the imported good *y* equal the value of the exports of good *q*:

\mathit{wL}+\mathit{rK}-\left({g}_{L}{C}_{L}+{g}_{K}{C}_{K}\right){p}^{-\mathrm{\xi}/\left(1-\mathrm{\xi}\right)}={g}_{X}\phantom{\rule{0.12em}{0ex}}{C}_{X}{p}^{-\mathrm{\xi}/\left(1-\mathrm{\xi}\right)},

(12)

where (*wL* + *rK*) gives the total payment to domestic factors of production *L* and *K*. In a competitive market, the payment to each factor of production equals its value of marginal product. If the production function is linear homogeneous, Euler’s theorem implies that the expression in (12) can be rewritten as:

\mathit{wL}+\mathit{rK}=p\left({f}_{L}L+{f}_{K}K\right)=\mathit{pQ}=\left[{g}_{L}{C}_{L}+{g}_{K}\phantom{\rule{0.12em}{0ex}}{C}_{K}+{g}_{X}{C}_{X}\right]{p}^{-\mathrm{\xi}/\left(1-\mathrm{\xi}\right)}.

(13)

where *f*
_{
i
} is the marginal product of factor *i*. It follows that aggregate market demand for the domestic good is given by:

Q=C\phantom{\rule{0.12em}{0ex}}{p}^{-1/\left(1-\mathrm{\xi}\right)},

(14)

where *C = g*
_{
L
}
*C*
_{
L
} *+ g*
_{
K
}
*C*
_{
K
} *+ g*
_{
x
}
*C*
_{
X
}
*,* the (weighted) number of consumers.

An important question arises: How does an immigration-induced increase in the size of the workforce affect the size of the consumer base for the domestic product?^{9} Let *C*(*L*) be the function that relates the number of consumers to the number of workers, and let ϕ = *d* log *C*/*d* log *L*. An important special case occurs when the elasticity ϕ = 1, so that the immigrant influx leads to a proportionately equal increase in the (weighted) number of consumers and the number of workers. I will refer to the assumption that ϕ = 1 as the case of *product market neutrality*. The “neutrality,” of course, refers to the fact that the immigration-induced supply shift leads to the same relative increase in the size of the consumer base and in the size of the workforce.

It is easy to allow for different product demand preferences between immigrants and natives by allowing for non-neutrality, i.e., by allowing for deviations from unity in the elasticity ϕ. For example, if immigrants prefer consuming the imported good, an immigrant influx that increases the size of the workforce by *x* percent may lead to a smaller percent increase in the number of “effective” consumers for the domestic good.

Equation (14) shows that an immigration-induced supply shift will have two distinct effects in the domestic labor market through product demand: First, the price of the domestic good might change, moving current consumers along the existing product demand curve; second, because immigrants are themselves “new” consumers, the market product demand curve will shift out and the magnitude of this shift will depend on ϕ^{10}.

It is analytically convenient to solve the model by using the inverse product demand function:

p={C}^{\mathrm{\eta}}\phantom{\rule{0.12em}{0ex}}{Q}^{-\mathrm{\eta}},

(15)

where η is the inverse price elasticity of demand, with η = 1 − ξ> 0, where the strict inequality follows from the assumed quasiconcavity of the utility function in (8).

The production technology for the domestic product is given by the CES production function:

Q={\left[\mathrm{\alpha}{K}^{\mathrm{\delta}}+\left(1-\mathrm{\alpha}\right){L}^{\mathrm{\delta}}\right]}^{1/\mathrm{\delta}},

(16)

where the elasticity of substitution between labor and capital is σ = 1/(1 − δ).

Finally, the supply of domestic capital is given by the inverse supply function:

r={K}^{\mathrm{\lambda}},

(17)

where λ ≥ 0, and is the inverse elasticity of supply of capital. The two special cases introduced in the previous section for the short run and the long run correspond to λ = ∞ and λ = 0, respectively.

In a competitive market, input prices equal the value of marginal product:

r=\mathrm{\alpha}\phantom{\rule{0.24em}{0ex}}{C}^{\mathrm{\eta}}\phantom{\rule{0.12em}{0ex}}{Q}^{1-\mathrm{\delta}-\mathrm{\eta}}\phantom{\rule{0.12em}{0ex}}{K}^{\mathrm{\delta}-1}.

(18a)

w=\left(1-\mathrm{\alpha}\right)\phantom{\rule{0.12em}{0ex}}{C}^{\mathrm{\eta}}\phantom{\rule{0.12em}{0ex}}{Q}^{1-\mathrm{\delta}-\mathrm{\eta}}\phantom{\rule{0.12em}{0ex}}{L}^{\mathrm{\delta}-1}.

(18b)

Let *d* log *L* represent the immigration-induced percent change in the size of the workforce. By differentiating Equations (18a) and (18b), and allowing for the fact that the supply of capital is given by Equation (17), it can be shown that:^{11}

\frac{d\text{log}w}{d\text{log}L}=\frac{-\mathrm{\lambda}\phantom{\rule{0.12em}{0ex}}\left(1-\mathrm{\delta}-\mathrm{\eta}\right)\phantom{\rule{0.12em}{0ex}}{s}_{K}}{\left(1+\mathrm{\lambda}-\mathrm{\delta}\right)-\left(1-\mathrm{\delta}-\mathrm{\eta}\right)\phantom{\rule{0.12em}{0ex}}{s}_{K}}-\frac{\left(1+\mathrm{\lambda}-\mathrm{\delta}\right)\phantom{\rule{0.12em}{0ex}}\mathrm{\eta}\phantom{\rule{0.12em}{0ex}}\left(1-\mathrm{\varphi}\right)}{\left(1+\mathrm{\lambda}-\mathrm{\delta}\right)-\left(1-\mathrm{\delta}-\mathrm{\eta}\right){s}_{K}}.

(19)

Consider initially the special case of product market neutrality (i.e., ϕ = 1), so that immigration expands the size of the consumer pool by the same proportion as its expansion of the workforce. The wage elasticity then reduces to:

\begin{array}{l}{\left.\frac{d\text{log}w}{d\text{log}L}\right|}_{\mathrm{\varphi}=1}=\frac{-\mathrm{\lambda}\phantom{\rule{0.12em}{0ex}}\left(1-\mathrm{\delta}-\mathrm{\eta}\right)\phantom{\rule{0.12em}{0ex}}{s}_{K}}{\left(1+\mathrm{\lambda}-\mathrm{\delta}\right)-\left(1-\mathrm{\delta}-\mathrm{\eta}\right)\phantom{\rule{0.12em}{0ex}}{s}_{K}}.\end{array}

(19a)

In the long run, λ = 0 and the wage elasticity goes to zero. Note also that the denominator of Equation (19a) is unambiguously positive^{12}. As long as there is incomplete capital adjustment (λ> 0), therefore, the wage elasticity will be negative if (1 − δ − η) > 0. Define η^{*} to be the elasticity of product demand (i.e., η^{*} = 1/η). It is then easy to show that (1 − δ − η) > 0 implies that:

\mathrm{\eta}*>\mathrm{\sigma}.

(20)

In other words, even after allowing for a full response by *all* consumers in the product market, the wage effect of immigration will be negative if there is incomplete capital adjustment and if it is easier for consumers to substitute among the available goods than it is for producers to substitute between labor and capital. This latter condition, of course, has a familiar ring in labor economics—as it happens to be identical to the condition that validates Marshall’s second rule of derived demand: An increase in labor’s share of income leads to more elastic demand “only when the consumer can substitute more easily than the entrepreneur” (Hicks, 1932, p. 246).

It turns out, however, that the condition in Equation (20) arises independently in a political economy model of immigration. In particular, the restriction that η^{*}> σ is a second-order condition to the problem faced by a social planner trying to determine the optimal amount of immigration in the context of the current model. One important feature of the competitive market model presented in this section is that the wage-setting rule ignores the fact that an additional immigrant affects product demand, so that the marginal revenue product of an immigrant is not equal to his value of marginal product. Suppose a social planner internalizes this externality and wishes to admit the immigrant influx that maximizes gross *domestic* product net of any costs imposed by immigration^{13}. More precisely, the social planner wishes to maximize:

\mathrm{\Omega}=\mathit{pQ}-\mathit{Fh}={C}^{\mathrm{\eta}}\phantom{\rule{0.12em}{0ex}}{Q}^{1-\mathrm{\eta}}-\mathit{Fh},

(21)

where *F* gives the number of immigrants and *h* gives the (constant) cost of admitting an additional immigrant (perhaps in terms of providing social services, etc.). For simplicity, consider the case with product market neutrality. In the Mathematical Appendix, I show that the second-order conditions for this maximization problem are satisfied if:^{14}

\left(1-\mathrm{\eta}\right)>0,\text{and}\phantom{\rule{0.25em}{0ex}}\left(1-\mathrm{\delta}-\mathrm{\eta}\right)>0.

(22)

In short, as long as the size of the immigrant influx is optimal, the wage elasticity in Equation (19a) must be negative. In that case, the scale effect resulting from immigration—regardless of whether it occurs through an expansion of the capital stock or through an expansion in product demand—can never be sufficiently strong to lead to a wage increase.

It is easy to measure the size of the scale effect triggered by immigration by considering the simple case of a Cobb-Douglas economy in the short run. The wage elasticity in (19a) then collapses to:

{\left.\frac{d\text{log}w}{d\text{log}L}\right|}_{\begin{array}{l}\mathrm{\varphi}=1\\ \mathrm{\delta}=0\\ \mathrm{\lambda}=\infty \end{array}}=-\left(1-\mathrm{\eta}\right){s}_{K}.

(23)

By contrasting this elasticity with the analogous effect in the one-good model presented in Equation (7), it is easy to see that the scale effect of immigration equals η*s*
_{
K
}. In the absence of the scale effect, the wage elasticity would equal −0.3. If the inverse elasticity of product demand is 0.5 (implying a product demand elasticity of 2.0), the wage elasticity would fall to −0.15. In other words, the short-run adverse effect of immigration on the wage can be greatly alleviated through increased product demand—as long as the product demand elasticity is sizable.

It is important to emphasize that the wage effect will not disappear in the long run if the product market neutrality assumption does not hold. Consider, for example, the case where immigration does not expand the size of the consumer base as rapidly as it expands the size of the workforce (i.e., ϕ< 1). The second term in (19) is then negative and does not vanish as λ goes to zero. In other words, there is a permanent wage reduction because there are “too many” workers and “too few” consumers. This result has interesting implications for the study of immigration when immigrants send a large fraction of their earnings to the sending country in the form of remittances. The negative effect of remittances on wages in the receiving country is permanent; it does not disappear even after capital has fully adjusted to the immigrant influx. Note, however, that it is also possible for immigration to generate permanent wage *gains* if ϕ> 1 and the immigrants are “conspicuous consumers” of the domestic product^{15}.

The wage consequences of even slight deviations from product market neutrality can be sizable. As an illustration, consider the long run effect in a Cobb-Douglas economy. The first term in Equation (19) vanishes and the wage elasticity reduces to:

{\left.\frac{d\text{log}w}{d\text{log}L}\right|}_{\begin{array}{l}\delta =0\\ \lambda =0\end{array}}=\frac{-\mathrm{\eta}\left(1-\mathrm{\varphi}\right)}{1-\left(1-\mathrm{\eta}\right)\phantom{\rule{0.12em}{0ex}}{s}_{K}}.

(24)

Suppose that ϕ = 0.8, so that an immigration-induced doubling of the workforce increases the size of the consumer pool by 80 percent. Suppose again that the inverse elasticity of product demand η is 0.5. Equation (24) then predicts that the *long-run* wage elasticity of immigration will equal −0.12.

### Immigration and prices

The wage elasticity in Equation (19) gives the wage impact of immigration in terms of the price of the imported product (i.e., the numeraire). It is also of interest to determine the impact of immigration relative to the price of the domestically produced good. After all, immigration has domestic product price effects both because the wage drops and because immigrants themselves shift the product demand curve outwards^{16}. By differentiating Equation (15) with respect to the immigration-induced supply shift, it can be shown that the effect of immigration on the domestic price is:

\frac{d\text{log}p}{d\text{log}L}=\frac{\mathrm{\lambda}\mathrm{\eta}{s}_{K}}{\left(1+\mathrm{\lambda}-\mathrm{\delta}\right)-\left(1-\mathrm{\delta}-\mathrm{\eta}\right){s}_{K}}-\frac{\mathrm{\eta}\left(1-\mathrm{\varphi}\right)\left[\mathrm{\lambda}+\left(1-\mathrm{\delta}\right){s}_{L}\right]}{\left(1+\mathrm{\lambda}-\mathrm{\delta}\right)-\left(1-\mathrm{\delta}-\mathrm{\eta}\right){s}_{K}}.

(25)

Suppose there is product market neutrality. The second term of (25) then vanishes, and immigration has no price effects in the long run (λ = 0). However, Equation (25) shows that immigration must *increase* prices as long as the product demand curve is downward sloping (η > 0) and capital has not fully adjusted. The inflationary effect of immigration is attenuated (and potentially reversed) if ϕ< 1 and product demand does not rise proportionately with the size of the immigrant influx.

The prediction that domestic prices rise at the same time that wages fall seems counterintuitive. However, it is easy to understand the economic factors underlying this result by noting that the derivative in (25) can also be expressed as:

\frac{d\text{log}p}{d\text{log}L}=\mathrm{\eta}\phantom{\rule{0.12em}{0ex}}{s}_{K}\left(1-\frac{d\text{log}K}{d\text{log}L}\right)-\mathrm{\eta}\left(1-\mathrm{\varphi}\right).

(26)

As long as there is product market neutrality, the price of the domestic good must rise whenever capital adjusts by less than the immigration-induced percent shift in supply. The intuition is clear: In the absence of full capital adjustment, the immigration-induced increase in domestic product demand cannot be easily met by the existing mix of inputs, raising the price of the domestic product^{17}.

An important question, of course, is: what happens to the *real* wage defined in terms of the price of the domestic product (or *w*/*p*)? By combining results from Equations (19) and (25), it is easy to show that:

\phantom{\rule{0.12em}{0ex}}\dot{w}=\frac{d\text{log}\left(w/p\right)}{d\text{log}L}=\frac{-\mathrm{\lambda}\phantom{\rule{0.12em}{0ex}}\left(1-\mathrm{\delta}\right)\phantom{\rule{0.12em}{0ex}}{s}_{K}}{\left(1+\mathrm{\lambda}-\mathrm{\delta}\right)-\left(1-\mathrm{\delta}-\mathrm{\eta}\right){s}_{K}}-\frac{\mathrm{\eta}\phantom{\rule{0.12em}{0ex}}\left(1-\mathrm{\varphi}\right)\phantom{\rule{0.12em}{0ex}}\left(1-\mathrm{\delta}\right)\phantom{\rule{0.12em}{0ex}}{s}_{K}}{\left(1+\mathrm{\lambda}-\mathrm{\delta}\right)-\left(1-\mathrm{\delta}-\mathrm{\eta}\right){s}_{K}}.

(27)

Note that if the product market neutrality assumption holds, the second term in (27) vanishes and immigration *must* reduce the real wage as long as capital does not fully adjust. This result does not depend on the relative magnitudes of the elasticities of substitution and product demand. The negative impact of immigration on the real wage is not surprising. After all, immigration reduces the nominal wage and increases the domestic price simultaneously. To simplify the discussion, I will refer to the elasticity in (27) as the *real* wage elasticity of immigration.

In order to get a sense of the magnitude of the real wage elasticity, it is instructive to refer back to the simplest example: a Cobb-Douglas economy in the short run. Equation (27) collapses to:

{\left.\frac{d\text{log}\left(w/p\right)}{d\text{log}L}\right|}_{\begin{array}{l}\mathrm{\delta}=0\\ \mathrm{\lambda}=\infty \end{array}}=-{s}_{K}.

(28)

The short-run real wage elasticity is identical to that implied by the simplest one-good Cobb-Douglas model in Equation (7). Even after the model accounts for the fact that immigrants increase the size of the consumption base proportionately and that immigration-induced price changes move the pre-existing consumers along their product demand curve, the short-run real wage elasticity is still −0.3.

The theory of factor demand clarifies an important misunderstanding: the often-heard argument that the outward shift in product demand induced by immigration will somehow return the economy to its pre-immigration equilibrium does not have any theoretical support. Instead, the theory reveals that immigration has an adverse effect on the real wage^{18}. Put differently, the number of domestically produced widgets that the typical worker in the receiving country can potentially buy will decline as the result of the immigrant influx—even after one accounts for the fact that immigrants themselves will increase the demand for widgets. And, under some conditions, the decline in the number of widgets that can be purchased is exactly the same as the decline found in the simplest factor demand model that ignores the role of immigrants in the widget product market.