# Approach #4: Structural models

The spatial correlations, natural experiments and skill cells approaches are called reduced-form approaches. In essence, economists decide on a linear relationship (like the best fit lines in Figures 8.1 and 8.3) to estimate and then turn to the data. These approaches impose very little structure on the equations that economists estimate. Over the past decade, there has been a renewed interest in using structural models to estimate the labor market impact of immigration. Researchers estimate or simulate a series of equations that build on key theoretical relationships. The set of equations is structural in that it specifies the structure of the labor market, or how different groups of workers interact with each other in the production process. These interactions are called cross effects. The benefit of this approach is that it is based in theory and can be used to simulate various policy scenarios, such as opening borders or restricting immigration based on a points-based system.

In order to estimate cross effects, researchers need to make assumptions about the production process in order to make the estimation tractable, or possible given the data available. The key to these models is to determine the appropriate level of labor market disaggregation (how broad or narrow the skill groups are) and utilize estimates of the elasticities of substitution to determine the wage effects of immigration. Thus, the elasticities of substitution are central to these models.

The intuition behind this approach is the following. The economist chooses a production function, which specifies how many types of labor there are and how those types interact with each other and with physical capital. The production function has elasticities of substitution between the various types of labor and capital built into it. The economist then assumes that workers are paid the value of their marginal product of labor and estimates or simulates a complete set of cross effects. Although the assumption that workers earn the value of their marginal product is standard, there is considerable disagreement about how many types of labor there are and what the various elasticities of substitution should be. Those assumptions have huge consequences for the estimated effects of immigration. It is often assumed that output is determined by a production function called a constant elasticity of substitution (CES) production function (see the appendix to Chapter 7). As the name suggests, at all output levels, there is a constant percentage change in factor proportions in response to changes in their relative marginal products. For one of the simplest CES functional forms with two types of labor, the production function is represented as follows:

where output, *Y,* is determined by a combination of technology (A), high-skilled labor (H) and low-skilled labor (*L*). Importantly for these studies, */и* is the share of income

**l**

paid to high-skilled labor and ~—— is the elasticity of substitution between high- and

low-skilled labor. Ifthe two types of labor are perfect substitutes, then cr= i. Ifthey are perfect complements, *a* approaches negative infinity (or -«>). Anywhere in between, they are imperfect substitutes (or complementary).^{4} In equation 8.2, there is only one (constant) elasticity of substitution because there are only two inputs to production. If there are more inputs to production, there can be multiple levels of CES functions, called nested CES functions. These cases require estimating multiple elasticities. Box 8.2 outlines the process of using structural estimation to calculate the wage effects of immigration based on the estimated elasticities.

In an important study using this approach, Borjas (2003) assumes that the functional form ofthe production process is a three-level nested CES production function. The three levels are the elasticity of substitution across experience groups within an education group; the elasticity of substitution across education groups; and the elasticity of substitution between capital and labor. Assuming that there are only three levels makes the model tractable, or computable. Once these elasticities are computed, the next step is to simulate how immigration affects the wage structure. Studies using these models usually simulate the impact of historical immigrant inflows. Simulation- based approaches compare the proportions of workers in a skill group to what they would be in the absence of immigration.

**Box 8.2 The structural approach: step-by-step**

Identification uses the model’s structure to impose a relationship between labor supply and wages. The magnitude depends on the specification ofthe production function and estimated parameters. The following is an intuitive four-step process for calculating wage effects using the structural approach.

Step i: After selecting the skill and experience levels to be assessed, use the skill cell approach to estimate the effects of immigration within-cells. For example, how does low-skilled immigration affect low-skilled native wages? Ask the same question for high-skilled immigrants and natives. Keep these estimates to use in the production function.

Step 2: Use regression analysis to estimate the elasticity of substitution between skill groups. Alternatively, use estimates from other studies.

Step 3: Using the production function and the estimates for the elasticities of substitution, derive wage equations for all types of workers.

Step 4: Determine estimates for the wage effects of immigration on specific types of workers. That is, as immigration of certain types of workers changes, how do wages for each type of worker respond? The overall effect of immigration on output can also be calculated.

A key feature of the structural approach is the extent of substitution (or complementarity) between different groups of workers. Estimates of elasticities of substitution vary widely, leading to a wide variety of wage effects. Using relatively high elasticities of substitution across education and experience groups, Borjas (2014) concludes that the magnitude of immigration that the United States experienced during 1990 to 2010 caused the average wage of high school dropouts to fall by 6.2 percent in the short run and 3.1 percent in the long run. For college graduates, the simulated wage effect is a 3.2 percent decline in the short run and o.i percent decline in the long run. Earlier studies by Borjas, Freeman and Katz (1992, 1996) also found substantial negative impacts for less-skilled workers (high school dropouts) from immigration but small effects on college and high school educated native-born workers.

Gianmarco Ottaviano and Peri (2012) challenge the assumption that immigrants and natives are highly substitutable within education and experience groups. To do so, they introduce a fourth level of nesting to allow for complementarities (or imperfect substitution) between immigrants and natives within cells. They argue that given differences in language abilities and experiences in the source country, immigrants and native-born workers are unlikely to be perfect substitutes within education and experience classifications. Using Current Population Survey (CPS), decennial Census, and American Community (ACS) datasets between 1990 and 2006, they find evidence of complementarities, as seen by an increase in the wages of natives within a skill cell in response to an influx of immigrants. They uncover an elasticity of substitution of 20 between skilled immigrants and natives, indicating they are imperfect substitutes. Their estimated overall effect of immigration on low-skilled wages is positive, but relatively small (0.6 to 1.7 percent). Averaging the effects, they find that immigrants boost overall wages of native-born workers.

Meanwhile, Borjas, Jeffrey Grogger and Gordon Hanson (2012) report opposing results using the same data but employing alternative model specifications. They ascertain that skilled native-born workers and immigrants in the United States are more substitutable (that is, the elasticity of substitution is nearly infinite). As a result, skilled native-born workers experience negative wage effects from immigration. However, their analysis suggests that estimates of the elasticity of substitution between low-skilled native-born workers and immigrants are less reliable.

In fact, it may be important to treat high school graduates and high school dropouts as different education groups rather than pooling them. Card (2009), for example, suggests adding yet another level of nesting to the theoretical model to account for the fact that not all skill groups are equally substitutable. In particular, he argues that if high school dropouts are highly substitutable for high school graduates, U.S. wage effects may be much smaller (closer to zero) than if high school dropouts are not highly substitutable for graduates. Most U.S. natives have graduated from high school, but many immigrants have not. Pooling these education groups reduces the relative magnitude of the immigrant influx. Determining the elasticity of substitution between high school dropouts and high school graduates is difficult. Results are sensitive to the specifications used and to the assumptions made about the relative demand for the two groups. As a result, it is unclear what elasticity of substitution should be used for these types of workers in structural models.

Patterns of immigration to the United States may be country specific, so researchers also estimate elasticities in other countries. In the context of the United Kingdom, Marco Manacorda, Alan Manning and Jonathan Wadsworth (2012) analyze population data from the mid-igyos to the mid-2000s and find that native-born workers and immigrants are imperfect substitutes. They estimate the elasticity of substitution between the two groups to be around 7.8, much lower than for the United States. Overall, they find very little to no effect of increased immigration on the wages of UK- born workers.

The main strength of the structural approach is that it is strongly rooted in economic theory. However, this strong theoretical underpinning is also its biggest limitation. The results from structural studies tend to be sensitive to the form of the production function, which the economist must assume. As noted earlier, most researchers use the CES production function. Economists must also decide how many levels (or nests) to examine and which cross-group elasticities to consider and estimate. Thus, the results of these models depend critically on the assumptions made about the substitutability of different groups of workers. As noted earlier, there is considerable debate regarding the elasticities of substitution used to generate wage effects. These models present simulations of the impact of immigration—they are only as good as the assumptions that the economist makes. These models do not take into account how consumption, investment and other parts of the economy respond and like the third method, rely on defining skill cells. Most models assume a form of the production function that requires the long-run effect on average wages to be zero.’^{5}

In sum, structural models are sensitive to elasticity assumptions.’^{6} Borjas and his co-authors typically find high substitutability within cells, which results in large wage effects. Ottaviano and Peri (2012) and Manacorda, Manning and Wadsworth

(2012) disagree and believe there are complementarities across groups. The debate continues.

# Concluding remarks regarding wage effects

As seen through the four techniques presented earlier, there is a wide range of estimated impacts of immigration on the wages of native-born workers. Many wage effects are clustered around zero, but some are negative and a handful are positive.’^{7} Despite the different methodologies, some patterns emerge. Less-skilled native workers tend to be more negatively impacted by immigrants than their higher-skilled counterparts. Certain minority groups are especially disadvantaged; there is some evidence that the negative effect is more severe for low-educated black workers and Hispanic workers who have dropped out of high school and have limited English proficiency. Earlier cohorts of low-skilled immigrants, those without a high school degree, also see larger negative impacts compared to other low-skilled groups. Results in the long run are mixed. So which methodology is correct for assessing wage effects? The answer is “it depends” on the context, data and specific group under study. Empirical techniques are advancing within each approach, so there is hope that clearer patterns will emerge.