2. Model

The theoretical setup features a static closed economy, with a perfectly competitive final goods sector and a monopolistically competitive intermediate goods sector, and four types of labor classified by skill and nativity. Intermediate good producers are heterogeneous in productivity, as in the closed economy in Melitz (Reference Melitz2003).

2.2 Final good

There is a stand-in firm that produces a consumption good using intermediate good $\omega \in \Omega$ , where $\omega$ is the name of the good and $\Omega$ is the endogenous set containing the names of all goods produced in the economy. The final good producer sells under perfect competition.

The production function follows a CES structure as in Dixit and Stiglitz (Reference Dixit and Stiglitz1977), with elasticity of substitution between intermediates $\sigma \gt 1$ ,

(1) \begin{equation} Q=\left [ \int _{\Omega _{}}q_{}(\omega )^{\frac {\sigma -1}{\sigma }}d\omega \right ]^{\frac {\sigma }{\sigma -1}}, \end{equation}

where $Q$ is the final consumption good and $q(\omega )$ is the quantity of intermediate good $\omega$ , for all $\omega \in \Omega$ . Let $P=\left [ \int _{\Omega }p_{}(\omega )^{1-\sigma }d\omega \right ]^{\frac {1}{1-\sigma }}$ , where $p(\omega )$ is the price of the intermediate good $\omega$ . The problem of the firm is:

(2) \begin{equation} \begin{aligned} \underset {\left \{ q(\omega ) \right \}}{\text {max}}\;\;\;\Pi &=PQ-\int _{\Omega }p_{}(\omega )q(\omega )d\omega \end{aligned} \end{equation}

subject to the technology in equation (1). The solution yields demand functions for intermediate goods $q(\omega )$ :

(3) \begin{equation} \begin{aligned} q_{}(\omega )=Q\left [ \frac {p_{}(\omega )}{P_{}} \right ]^{-\sigma }. \end{aligned} \end{equation}

2.3 Intermediate goods

Intermediate goods producers operate in a monopolistically competitive market and use labor as inputs in production. The production function is a CES aggregate of high-skilled and low-skilled labor. The skill demands are further nested into natives and immigrants within skill groups following Ottaviano and Peri (Reference Ottaviano and Peri2012). The production function is

(4) \begin{equation} q(\omega (\varphi ),A)=A\varphi \left[ \alpha S^{\frac {\eta -1}{\eta }}+(1-\alpha )U^{\frac {\eta -1}{\eta }} \right]^{\frac {\eta }{\eta -1}} , \end{equation}

where $A$ is aggregate productivity, common across all intermediate good producers, $\varphi$ is firm $\omega$ ’s idiosyncratic productivity level, $\eta$ is the elasticity of substitution between high- and low-skilled labor, and $S$ and $U$ are skill aggregates of native and immigrant labor defined as:

(5) \begin{align} S&=\left [\gamma _{s}S_{n}^{\frac {\rho _{s}-1}{\rho _{s}}}+(1-\gamma _{s})S_{i}^{\frac {\rho _{s}-1}{\rho _{s}}} \right ]^{\frac {\rho _{s}}{\rho _{s}-1}}, \text{ and} \end{align}

(6) \begin{align} U&=\left [\gamma _{u}U_{n}^{\frac {\rho _{u}-1}{\rho _{u}}}+(1-\gamma _{u})U_{i}^{\frac {\rho _{u}-1}{\rho _{u}}} \right ]^{\frac {\rho _{u}}{\rho _{u}-1}}. \end{align}

The parameters $\rho _{s}$ and $\rho _{u}$ represent the elasticity of substitution between natives and immigrants within the skill groups and $\gamma _{s}$ and $\gamma _{u}$ capture the average intensity of immigrant labor of either skill type in the production process. The reason for introducing the aggregate productivity parameter $A$ is to account for changes outside the model in our empirical application. While the model is static, we obtain our results by calibrating it to all years in the analysis, and allow $A$ to change to capture changes outside the scope of the model.

In what follows we drop the name of the good $\omega$ and refer to prices as functions of the productivity parameters $\varphi$ and $A$ , since decisions change only when these parameters change. It is convenient to define the aggregate price index in terms of these variables as follows:

(7) \begin{align} P(A) = \left [\int _0^\infty p(\varphi ,A)^{1-\sigma }M(A)g(\varphi )d\varphi \right ]^{\frac {1}{1-\sigma }} \end{align}

where $p(\varphi ,A)$ solves the maximization problem of each intermediate good producer. These behave monopolistically in their output market, solving the following problem:

(8) \begin{align} \max _{p,q,l_{sn},l_{si},l_{un},l_{ui}} pq-\sum _{k=sn,si,un,ui} w_k(A)l_k \end{align}

subject to the technology described in equations (4) through (6). The solution to this problem is that price is a constant mark-up over marginal cost. The marginal cost of a type $\varphi$ firm is $w(A)/\varphi$ , where

(9) \begin{align} w(A)=&\frac {1}{A}\biggl \{\alpha ^{\eta }\left [\big(\gamma _{s}^{\rho _{s}}w_{sn}(A)^{1-\rho _{s}}+(1-\gamma _{s})^{\rho _{s}}w_{si}(A)^{1-\rho _{s}}\big)^{\frac {1}{1-\rho _{s}}} \right ]^{1-\eta } \notag \\ &+ (1-\alpha )^{\eta }\left [\big(\gamma _{u}^{\rho _{u}}w_{un}(A)^{1-\rho _{u}}+(1-\gamma _{u})^{\rho _{u}}w_{ui}(A)^{1-\rho _{u}}\big)^{\frac {1}{1-\rho _{u}}}\right ]^{1-\eta }\biggr \}^{\frac {1}{1-\eta }} \end{align}

The pricing rule that maximizes profits is to set a constant mark-up, that is,

(10) \begin{align} p(A,\varphi ) = \frac {\sigma }{\sigma -1}\frac {w(A)}{A\varphi } \end{align}

2.4 Free entry condition

There is an infinite pool of potential entrants. To enter, a potential firm incurs a fixed entry cost of $f_{e}$ units of the final good to draw a productivity parameter $\varphi$ from a cumulative distribution function (CDF) $G(\varphi )$ , with probability distribution function (PDF) $g(\varphi )$ . In equilibrium, firms enter the market until the expected profits from entering are equal to the cost of entry, that is,

(11) \begin{equation} \overline {\pi }[1-G(\varphi ^*(A))]=f_{e}P(A) \end{equation}

where, $\overline {\pi }$ is the expected profits before knowing the productivity parameter $\varphi$ , that is,

(12) \begin{align} \overline {\pi }=\int _0^\infty \pi (A,\varphi )g(\varphi )d\varphi \end{align}

where $\pi (A,\varphi )$ is the profit of a type $\varphi$ firm when the aggregate productivity is $A$ .

Equation (11) implies that there are no profits to rebate back to the consumer. This is convenient, given that if this were not the case, we would have to determine how profits are rebated back to the four different types of consumers. Note that this does not prevent firms from making a profit. Net of entry costs, some firms will make positive profits and others negative in a way that aggregate profits are zero.

2.5 Welfare

In equilibrium, consumption (and welfare) is equal to the real wage:

(13) \begin{equation} C_{jk}(A)=\frac {w_{jk}(A)}{P(A)},\;\;\forall \;\;j=s,u\:;\:k=n,i. \end{equation}

We define aggregate welfare as the sum of individual welfare levels:

(14) \begin{equation} W(A)=\frac {w_{sn}(A)S_{n}+w_{un}(A)U_{n}+w_{si}(A)S_{i}+w_{ui}(A)U_{i}}{P(A)(S_n+U_n+S_i+U_i)}. \end{equation}

Similarly, native welfare is

(15) \begin{align} W_n(A) = \frac { w_{sn}(A) S_{n}+w_{un}(A) U_{n}}{P(A)(S_n+U_n)}. \end{align}

3. Equilibrium characterization

Before moving on to the quantitative analysis, we describe the potential mechanisms by which immigration can affect natives’ welfare. In particular, we are interested in how welfare changes from a change in immigration under different parameter values. The most important parameters determining welfare effects are the elasticity of substitution between varieties ( $\sigma$ ), the elasticity of substitution between domestic and foreign workers of a given type ( $\rho _u$ and $\rho _s$ ), and the elasticity of substitution between skills ( $\eta$ ). Symmetry implies that changes in $\rho _u$ affect low-skilled workers in the same way that $\rho _s$ affects high-skilled workers, so we only illustrate how different values of $\rho _u$ matter.

For each native type, immigration has two distinct mechanisms that can change welfare: a market effect and a substitution effect. We present figures as aids to our description. All parameters are set to the values described in Section 4 except for our parameters of interest ( $\rho _u, \eta$ and $\sigma$ ). We also set $\theta = 3.5$ , which is larger than the one we obtained in Section 4. A small value of $\theta$ limits the feasible range for $\sigma$ , since we require $\sigma \lt 1+\theta$ .Footnote ²

Consider first the substitution channel. Low-skilled natives are highly substitutable with low-skilled immigrants, so low-skilled immigrants increase the competition faced by low-skilled natives, lowering their wages. The degree of substitutability is governed by $\rho _u$ : the larger this is, the stronger the channel. Figure 2 shows how a 10% increase in low-skilled immigrants affects the wages of low-skilled natives under different values of $\rho _u$ . As the substitutability between native low-skilled workers and foreign low-skilled workers increases, the gains from immigration drop for domestic low-skilled workers. This is reflected in the fact that the increase in real wages for low-skilled individuals given a 10% increase in low-skilled immigration is decreasing in $\rho _u$ (all three lines show this).

Figure 2.

Change in low-skilled native wages from a 10% increase in low-skilled immigrants under different parameter values: Substitutability.

Next, consider the complementarity between high-skilled and low-skilled workers. The strength of this complementarity is governed by $\eta$ . The smaller this parameter, the higher the complementarity. Figure 2 shows the effect of increasing $\eta$ by comparing the red line to the blue line. As we increase the substitutability between workers of different skills, the gains for low-skilled domestic workers increase. This is because, when high-skilled and low-skilled workers are highly substitutable, immigrants compete with both skilled and low-skilled workers, so that labor demand changes will be spread out between skilled and low-skilled. As a result, the drop in demand for low-skilled domestic workers is lower. If this parameter is relatively small, Figure 3 shows that domestic low-skilled workers can lose from an inflow of low-skilled immigrants. The gain in real wages is decreasing in $\eta$ , and negative when the complementary channel is strong. This is because when substitution across skill groups is low, low-skilled immigrant workers compete only with low-skilled domestic workers, and the labor demand change accrues to only the low-skilled group.

Lastly, consider the market-size channel. Moving from the red line to the yellow line in Figure 2 shows the effects of increasing $\sigma$ : the higher this parameter, the lower the gains for low-skilled domestic labor. The reason is that when $\sigma$ is large, goods are easily substitutable among each other. Thus, the gains from varieties drop since one can easily replace across goods. In other words, the degree of increasing returns is decreasing in $\sigma$ , so that as $\sigma \to \infty$ , this model resorts to a representative firm model, with no increasing returns to scale.

4. Calibration

The discussion in Section 3 shows that domestic workers can gain or lose from immigration, depending on parameter values. This section assumes parameter values consistent with the literature to pin down the actual effect of the changes in 2016 and 2022. By using independent estimates, we provide an unbiased estimate of how welfare is likely to have been affected. More precise results would involve a proper estimation of these parameters, and this is beyond our scope.

We calibrate the model to the U.S. economy for each one of the years between 2010 and 2024. Our strategy divides our parameters into two sets. The first includes parameters whose value we draw from existing research, and the second set targets data moments. Table 1 reports the calibrated parameters. For parameters that change by year, we report the means and denote this with a bar, that is, $\bar x$ denotes the mean of $x$ between 2010 and 2024. We report the complete set of parameters in Table A1 in Appendix A.

Table 1.

Calibrated parameters

Table 1. Long description

A table with 12 rows and 4 columns. The columns are labeled Parameter, Value, Target, and Source. The table is divided into two sections: Parameters drawn from the literature and Parameters targeting data moments. The first section includes parameters with values drawn from existing research, and the second section includes parameters calibrated to target data moments. Each row lists a parameter, its value, the target it aims to achieve, and the source of the value. Notable parameters include sigma with a value of 3, eta with a value of 2, and rho sub s with a value of 4.3. The table also includes parameters targeting data moments such as firm size distribution, skill premium, and real GDP per capita.

Note: For the parameters calibrated to target data moments from 2010 through 2024, that includes A, $\alpha$ , $\gamma _{s}$ , $\gamma _{u}$ , $S_{i}$ , $U_{n}$ , $U_{i}$ , Table reports the average of the parameter values over 15 years. The complete set of these parameters for each year is reported in Appendix A, Table A1.

Figure 3.

Change in low-skilled native wages from a 10% increase in low-skilled immigrants under different parameter values: Complementarity.

We set the elasticity of substitution between intermediate varieties $\sigma$ equal to 3, which lies close to the median estimates by Broda and Weinstein (Reference Broda and Weinstein2006). Following the standards in the literature, we assume that the distribution of productivity is Pareto. This implies that in equilibrium the firm size distribution is Pareto, which closely resembles the upper tail of the firm size distribution in the U.S. (Axtell, Reference Axtell2001). More specifically, we assume

(16) \begin{align} g(\varphi ) = \theta \varphi ^{-\theta -1}, \ \ \ \varphi \in [1,\infty ), \ \theta \gt 0 \end{align}

We calibrate $\theta$ so that our firm size distribution is as close as possible to that in the data. For data, we use the U.S. manufacturing sector for the year 2012 following Rubini (Reference Rubini2014) which gives us a value of $\theta$ of 2.36. We obtain this by minimizing the sum of the squared difference between the employee size distribution of manufacturing firms in the data and the model in 2012. The data on the distribution of firms across the employee size bins comes from the 2012 Census of Manufacturing. The Census provides the number of firms hiring less than $L_{i}$ workers for a given employment threshold $L_{i}$ , where the thresholds are 4, 9, 19, 49, 99, 249, and 499, and the employment bins are denoted as $L_{i-1}-L_{i}$ . In the model, we consider $L_{i}$ as the summation of the four types of workers for each employment bin. To compare the workers in the model to the workers in the data, we compute the maximum number of workers in each bin with respect to the first bin in the model, such that the share of firms in that bin is the same in the model and in the data. Figure 4 shows the model fit of the firm size distribution observed in the data.

Figure 4.

Firm size distribution fit between the model and the data.

Note: The blue line plots the distribution of manufacturing firms across the employment thresholds from the data. The red line plots the distribution generation in the model equilibrium for the year 2012.

Source: 2012 Census of Manufactures.

The fixed entry cost $f_{e}$ is normalized to 1. Changing this normalization does not affect our results. The elasticity of substitution between high-skilled and low-skilled labor $\eta$ has been extensively studied, with estimates ranging from 1.5 (Katz and Murphy, Reference Katz and Murphy1992; Goldin and Katz, Reference Goldin and Katz2007) to 2.5 (Card and Lemieux, Reference Card and Lemieux2001; Ottaviano and Peri, Reference Ottaviano and Peri2012). We take the mid-point of these estimates and set $\eta = 2$ .

The elasticity of substitution between natives and immigrants across skill groups ( $\rho _{s},\rho _{u}$ ) that governs the effect of immigrant influx on the wages is available from the existing literature studying the wage impacts of immigration in the U.S. (Borjas, Reference Borjas2003, Reference Borjas2005; Borjas et al., Reference Borjas, Freeman and Katz2007; Raphael and Smolensky, Reference Raphael and Smolensky2009; Card, Reference Card2009; Ottaviano and Peri, Reference Ottaviano and Peri2012). The elasticity parameter most widely used is around 20, implying imperfect substitution between natives and immigrants when the elasticity is not allowed to vary by skill. Allowing for the substitution parameter to vary by skill, the substitution is higher between natives and immigrants within the high-skilled group which is estimated at around 33, compared to 11.1 for the low-skilled group (Ottaviano and Peri, Reference Ottaviano and Peri2012). Recent studies that focus on firm-level data have estimated elasticities much lower. Mahajan et al. (Reference Mahajan, Morales, Shih, Chen and Brinatti2024) studies college-educated H-1B immigrants and natives and finds an elasticity of substitution of 4.3 between these groups. This number is close to 4.28 as estimated by Brinatti and Morales (Reference Brinatti and Morales2025) for the case of Germany. Thus, we use $\rho _s=4.3$ in our calibration. Following Ottaviano and Peri (Reference Ottaviano and Peri2012) and meta-analysis by Kantova et al. (Reference Kantova, Havránek and Havránková2025), we adopt a conservative value of $\rho _u=11$ . This value lies between the lower-bound estimates $(=5)$ and upper-bound estimates $(=20)$ , and reflects substantial but imperfect substitutability between natives and immigrants within low-skilled groups.

Data on labor supply in the U.S. comes from the American Community Survey (ACS) for the years 2010 through 2024. Labor is classified by citizenship as immigrants or natives, and is classified into skill groups of high-skilled (more than schooling) and low-skilled (schooling and less) by years of education. The supply of high-skilled natives $S_{n}$ is normalized to 1 for every year, and the labor supply of the other three types of labor $S_{i},U_{n},U_{i}$ is calibrated to match their levels relative to high-skilled natives for each year.Footnote ³

The remaining parameters that are calibrated to match the U.S. economy are $\alpha$ , $\gamma _{s}$ , $\gamma _{u}$ , and $A$ . These parameters determine the technology in production and how it has changed over the years. The parameter governing the skill share in production $\alpha$ is calibrated to match the skill premium in data, that is, the wage ratio between all high-skilled and all low-skilled workers in the US for the years 2010–2024. The parameter governing the share of high-skilled natives in production $\gamma _{s}$ is calibrated to match the ratio of high-skilled immigrant wages to high-skilled native wages in the U.S. for the years 2010–2024 and $\gamma _{u}$ is calibrated to match the change in the wage ratio between immigrants and natives in the low-skilled group over the years. The common productivity parameter $A$ captures other factors that might affect changes in the technological mix in the economy over the years. It is calibrated to match the GDP per capita for the U.S. economy over each year between 2010 and 2024. We use data from the Bureau of Economic Analysis’ (BEA) gross domestic product data series.

6. A two-sector economy

This section expands our benchmark model by introducing a second sector that is more intensive in low-skilled workers. We do this to explore how sectoral heterogeneity affects our results.

There are two intermediate goods sectors operating independently in a perfectly competitive market. A representative final goods sector aggregates these sectoral composites via a Cobb-Douglas technology.

(18) \begin{equation} Q \;=\; Q_1^{\beta }\, Q_2^{1-\beta }, \qquad \beta \in (0,1). \end{equation}

Each sectoral composite $Q_m, m\in \{1,2\}$ is a CES aggregate of a continuum of intermediate varieties $\omega \in \Omega _m$ :

(19) \begin{equation} Q_m \;=\; \left [\int _{\Omega _m} q_m(\omega )^{\frac {\sigma _m-1}{\sigma _m}} \, d\omega \right ]^{\frac {\sigma _m}{\sigma _m-1}}, \qquad \sigma _m\gt 1. \end{equation}

Cost minimization by the final-good firm gives the demand for each sectoral aggregate as

(20) \begin{align} Q_1 \;=&\; Q \left ( \frac {\beta }{1-\beta } \right )^{1-\beta } \left (\frac {P_{2}}{P_{1}}\right )^{1-\beta }, \end{align}

(21) \begin{align} Q_2 \;=&\; Q \left ( \frac {\beta }{1-\beta } \right )^{-\beta } \left (\frac {P_{2}}{P_{1}}\right )^{-\beta } \end{align}

where the sectoral price indices are $P_m = \left [\int _{\Omega _m} p_m(\omega )^{1-\sigma _m}\, d\omega \right ]^{\frac {1}{1-\sigma _m}}$ , for $ m=1,2$ . This implies constant expenditure shares for each sector, $E_m \;=\; \lambda _m \, P Q$ , where $\lambda _1=\beta ,\ \lambda _2=1-\beta$ , and the economy-wide price index $P$ is given as,

(22) \begin{align} P &\;=\; \left (\frac {P_1}{\beta }\right )^\beta \left (\frac {P_2}{1-\beta }\right )^{1-\beta } \end{align}

Similar to the one-sector setup, the intermediate varieties are produced using CES technology in each sector, where a firm with draw $\phi _{m}$ and sectoral productivity $A_m$ produces output according to,

(23) \begin{equation} q_m(\phi _{m};\,A_m) \;=\; A_m \,\phi _{m} \,\left [ {\alpha _m} S_{m}^{\tfrac {\eta -1}{\eta }} + (1-\alpha _m) U_{m}^{\tfrac {\eta -1}{\eta }} \right ]^{\tfrac {\eta }{\eta -1}}, \end{equation}

where $S$ is the high-skill aggregator and $U$ is the low-skill aggregator. Each of these skill aggregates nests natives and immigrants:

(24) \begin{align} S_{m} &\;=\; \Big [\gamma _{s,m} S_{n,m}^{\tfrac {\rho _s-1}{\rho _s}} + (1-\gamma _{s,m}) S_{i,m}^{\tfrac {\rho _s-1}{\rho _s}} \Big ]^{\tfrac {\rho _s}{\rho _s-1}} \end{align}

(25) \begin{align} U_{m} &\;=\; \Big [\gamma _{u,m} U_{n,m}^{\tfrac {\rho _u-1}{\rho _u}} + (1-\gamma _{u,m}) U_{i,m}^{\tfrac {\rho _u-1}{\rho _u}} \Big ]^{\tfrac {\rho _u}{\rho _u-1}} \end{align}

6.1 Gaining intuition: a sector with only low-skilled labor

This section examines two changes to the baseline model. First, it investigates how introducing a low-skill intensive sector affects the results, emphasizing the role of its relative size. Second, it considers how changing the labor composition of natives and immigrants in the low-skill intensive sector affects the results.

To explore the effects of changing the immigrant composition in sector 2, we set $\gamma _{u,1}=\gamma _{u,2}$ as a baseline for comparability, and then double the immigrant share parameter in sector 2 so that $1-\gamma _{u,2}=2((1-\gamma _{u,1}))$ . We calibrate $\alpha _{1}$ to match the skill premium in the US, and $\gamma _{s,1}$ to match the wage ratio between high-skilled immigrants and high-skilled natives in the US. We assume $A_{1}=A_{2}$ for simplicity, and calibrate the common productivity parameter to match US GDP per capita. Table A2 in Appendix A shows all parameter values.

Figure 13.

Skill complementarity and welfare impacts of counterfactual growth in low-skilled immigrants: a two-sector economy.

Note: Parameters targeting data moments vary by year. Variables are normalized to 1 in 2016.

Figure 13 plots our results. The top panel plots the welfare change for high-skilled natives, and the bottom panel for low-skilled natives. The black dashed lines show our estimates of the increase in consumption. The black solid line shows the counterfactual path consumption would have followed had immigration not slowed down.

Introducing the second sector amplifies the gains from immigration (or the losses from the lack of it). High-skilled natives gain about 0.6% of consumption-equivalent units, and low-skilled natives 0.07% (compared to 0.5% and 0.05% in the baseline, respectively).

Figure 14.

Relative output of sector 1 to sector 2.

Note: Parameters targeting data moments vary by year. Variables are normalized to 1 in 2016.

The reason why the gains are larger is that this adds a margin of adjustment to the economy: the relative amount of good 2. In fact, we show in Figure 14 that the ratio $Y_1/Y_2$ changes. In particular, resources flow towards the good $Y_2$ that is relatively more intensive in low-skilled immigrants.Footnote ⁷

The red lines in Figure 13 replicate these results for a larger immigrant share in sector 2. This amplifies all effects even further. Doubling the immigrant share parameter in sector 2 triples the gains for high-skilled natives (by 1.55%). Gains for the low-skilled natives increase by 1.05%. This amplification happens because, by decreasing $\gamma _{u,2}$ , we increase the marginal productivity of low-skilled immigrants. Thus, the same increase in employment produces a larger increase in the production possibility set.

These findings are consistent with Brinatti and Morales (Reference Brinatti and Morales2025), who finds that when immigrant intensity differs across firms, and the elasticity of substitution between natives and immigrants is larger than the demand elasticity, immigrants make the production more efficient by relocating natives to native-intensive firms. This amplifies the welfare gains from immigration.

We explore changes to the size of sector 2 in Figure 15. Our benchmark sets $\beta = 0.8$ , so that sector 2 accounts for 20% of GDP. Making this sector larger amplifies the gains for both native types. This is because when $\beta$ is smaller, the marginal productivity of low-skilled immigrants is larger. Thus, as in the case with the share parameter, under a smaller value of $\beta$ , the expansion of the production set is larger given the same increase in low-skilled immigrants.

6.2 Construction vs. Non-construction

To explore quantitatively the welfare effects of low-skilled immigration in a context where one firm is intensive in their use of low-skilled workers, we map the low-skilled intensive sector with construction in the data. The choice of the construction sector is grounded in the observed pattern of immigrant employment in the U.S., as shown in Table 4. First, the construction sector on average employs twice the number of low-skilled relative to high-skilled workers, whereas on average a sector in the U.S. employs more high-skilled workers as opposed to low-skilled workers. Second, considering the ratio of low-skilled immigrants to low-skilled natives, the construction sector employs more low-skilled immigrants than the average U.S. sector, for any given number of low-skilled natives.Footnote ⁸ This ratio is reversed when looking at the high-skilled immigrant employment pattern in the U.S., but the ratios are close to the rest of the economy. The construction sector employs fewer high-skilled immigrants than the average U.S. sector. These patterns suggest that the construction sector becomes an important channel through which low-skilled immigration changes can affect the economy. In fact, Howard et al. (Reference Howard, Wang and Zhang2024) finds that increased immigration enforcement in the form of deportations of low-skilled immigrants reduces new housing construction and drives up the price of existing houses. Thus, to capture how the drop in low-skilled immigrants can affect welfare in the presence of heterogeneous sectors, we focus on construction and non-construction sectors.

Table 4.

Construction vs non-construction intensities

Table 4. Long description

A table comparing construction and non-construction sectors based on skill ratios and immigrant to native ratios. The table has three rows and four columns. The columns are labeled Sector, Ratio of high-skilled to low-skilled, Ratio of immigrant to native in low-skilled, and Ratio of immigrant to native in high-skilled. The rows are labeled Construction, Non-construction, and All sectors. Row 1: Construction, 0.51, 0.37, 0.15. Row 2: Non-construction, 1.56, 0.26, 0.17. Row 3: All sectors, 1.45, 0.28, 0.17.

Note: The ratios correspond to the average of the time period 2010–2019. The construction sector is identified using the ACS industry classification, and the non-construction sector includes all other industry classifications. All sectors represent the national average for the U.S.

Figure 15.

Welfare gain between 2016 and 2024 with different values of $\beta$ .

Note: The welfare gains correspond to the scenario where $(1-\gamma _{u,2})$ is doubled and the model is initially calibrated for $\beta =0.8$ , with marginal changes in the counterfactual. The welfare gain is calculated as the difference between consumption with an increase in low-skilled immigrants and consumption without an increase.

This section maps the low-skilled sector (sector 2) with the construction sector in the data. Sector 1 maps to the rest of the economy. We calibrate the production function parameters in sector 1 to match the observed wage ratios for workers across all industries in the U.S. The corresponding parameters in sector 2 are calibrated to match the labor demands observed in the construction sector in the U.S. We set $\beta =0.95$ to match a share of the construction sector in the U.S. GDP of 5%. All other parameters are calibrated following the targets used in our benchmark model.

Figure 16.

Skill complementarity and welfare impacts of counterfactual growth in low-skilled immigrants: construction vs. non-construction.

Note: Parameters targeting data moments vary by year. Variables are normalized to 1 in 2016.

Figure 16 plots the welfare for high-skilled and low-skilled natives over the years in the equilibrium and the counterfactual with more low-skilled immigrants. We compare these welfare values with the one-sector equilibrium where $\beta$ is equal to 1.

Welfare changes remain practically the same as in our baseline, mostly because of the small size of the new sector. In fact, expanding this sector would increase the gains for low-skilled natives of further low-skilled immigration, although these changes are likely to be small. The robustness of our results to heterogeneous sectors ratifies our confidence in our baseline results.

Both high-skilled and low-skilled natives gain from an inflow of low-skilled immigrants. Thus, even when immigrant workers are disproportionately distributed across sectors, the inflow of low-skilled immigrants generates welfare gains for natives.

Figure 17.

Welfare gain between 2016 and 2024 with different values of $\beta$ .

It is important to mention that these results hardly change as we increase the size of the low-skilled sector. Figure 17 shows that increasing this share from 5% to 20% has very little effect on the welfare changes for all natives.Footnote ⁹