## 1. Introduction

The empirical analysis of cross-city and cross-state evidence in the United States has consistently found small and often insignificant effects of immigration on the wages of native workers.^{1} However, two recent influential contributions by Borjas (2003) and Borjas and Katz (2007) have emphasized the importance of estimating the effects of immigration using national level data and have found a significant negative effect of immigration on the wages of natives with no high school diploma.^{2} These studies have argued that wages across local labor markets are subject to the equalizing pressure that arises from the spatial arbitrage of mobile workers. As a result, the wage effects of immigration are better detected at the national level since one can exploit variation in wages and immigrants across groups of workers with different skills (as captured by education and experience) over time.

The underlying logic is that while it may be relatively easy for a US worker to react to local immigration by changing their residence within the United States it is much harder for her to do so by relocating across the US border or by changing her own skill mix. Accordingly, the estimation of the substitutability among workers with different skills should play a key role in the analysis of the wage effects of immigration. Our aim is to contribute to this approach at the national level in two ways: through an improved estimation of the substitutability among workers with different characteristics and through the clarification of a crucial distinction between the partial and the total wage effects of immigration, a distinction not fully appreciated in the existing literature.

First, in terms of substitutability and in contrast to Borjas (2003) and Borjas and Katz (2007), we estimate the elasticity of substitution between immigrant and native workers within the same education and experience group without assuming ex ante that they are perfectly substitutable. Given that natives and immigrants of similar education and age have different skills, often work in different jobs and perform different productive tasks, their substitutability is an empirical question, the answer to which has important implications since the degree of imperfect substitutability affects the impact that immigrants have on the wage of natives with similar skills.

Some recent papers have also estimated the native–immigrant elasticity of substitution. Card (2007), using US city data for year 2000, Raphael and Smolensky (2008), using US data over 1970–2005, and D’Amuri et al. (2010), using German data, all find small but significant values for the inverse of the native–immigrant elasticity implying less than perfect substitutability between these groups of workers (with an elasticity between 20 and 30).^{3} While our estimates are in the same ballpark, a closely related work by Manacorda, Manning, and Wadsworth (this issue) using UK data finds an even smaller substitutability between natives and immigrants (with elasticity between 5 and 10). This may be due to their use of yearly net inflows (rather than the ten-year flows we use) implying that the elasticity of substitution is identified on very recent immigrants, who are likely to be the most different from natives. On the other hand, Borjas, Grogger, and Hanson (2008) show that one can get small and insignificant estimates for the inverse of the native–immigrant elasticity, and therefore little evidence of imperfect substitutability, in specifications that are highly saturated with fixed effects.^{4}

We also reconsider the substitutability between workers of different schooling and experience levels. We produce new estimates and compare them with those found in the existing literature. In particular, since the inflow of immigrants to the United States in recent decades has been much larger among workers with no high school degree than among high school graduates, we emphasize the importance of distinguishing the substitutability between workers with no high school degree and workers with a high school diploma from the substitutability between those two groups taken together and workers with at least some college education. This distinction has a long tradition since Katz and Murphy (1992) argued that in order to understand the impact of changes in labor supply and demand on the wages of workers with different education levels it is important to consider highly educated and less-educated workers as imperfectly substitutable.^{5} This has been motivated by the observation that the wage time series of workers with and without high school degrees move together much more than do the wages of high school dropouts and college educated workers.^{6} The substitutability across alternative experience groups has been similarly investigated.^{7}

Our second contribution concerns the distinction between partial and total wage effects. While the former refers to the direct impact of immigration on native wages within a skill group given fixed supplies in other skill groups, the latter accounts for the indirect impacts of immigration in all other skill groups. Accordingly, the total wage effects on natives across skill groups depend on the relative sizes of the different skill groups, the relative strength of own- and cross-skill impacts and the pattern of immigration across skill groups.

To clarify the distinction between partial and total wage effects, we introduce an aggregate production function that produces marginal productivity equations that can be used to compute both sorts of effects of immigration on the wage of natives in each skill group. Because we consider a rich set of skills, a large number of cross-skill effects need to be estimated. Doing this with minimal structure is impossible given available data. For example, the 32 education-by-experience groups proposed in Borjas (2003) and Borjas and Katz (2007) imply 992 cross-skill effects. But US Census data only consists of 192 skill-by-year observations on employment and wages. Adding structure, like the nested-CES labor composite we introduce in what follows, allows the plethora of cross-skill effects to be expressed in terms of a limited number of structural parameters that can, in turn, be estimated with available data. In other words, the aggregate production function provides a structural foundation to the wage regressions used to assess workers’ substitutability and provides parametric interpretations of the estimated coefficients. That said, economic interpretation of estimates from any reduced-form equation requires assumptions on the form of the cross-skill interactions. So, by explicitly introducing the aggregate production function we are able to get the required estimates and we can discuss the pros and cons of the underlying assumptions.

While the nested-CES approach imposes restrictions on the form of the cross-elasticities, it is still flexible enough to allow for the exploration of alternative nesting structures in terms of number of cells, order of nesting and skill grouping. In particular, we explore four different nesting models, which together span most of the structures used to estimate the substitutability among skill groups in the existing literature. Model A augments the structure proposed by Borjas (2003, Section VII) by allowing for imperfect substitutability between US- and foreign-born workers of equal education and experience. This model assumes the same substitutability between any pair of education groups and between any pair of experience groups with identical education. While the latter assumption is standard in the labor literature, the former is rather unusual as it is more common to divide workers into two broad education groups of workers, those with high education (some college education and more) and those with low education (high school education or less).^{8} This alternative partition is considered in model B. Models C and D cover plausible alternatives that are not much used by the existing literature. Model C considers the possibility that some experience groups may be closer substitutes than others by allowing for the elasticity across broad experience groups to differ from the elasticity across narrow experience groups. Finally, in model D the nesting order of education and experience in Borjas (2003, Section VII) is inverted with respect to model A.

We estimate the relevant elasticities of substitution for the four models using data from the Census in 1960, 1970, 1980, 1990, and 2000, and from the American Community Survey (ACS) 2006 downloaded from IPUMS (Ruggles et al. 2009). As this set of data generates only six time-series observations, in order to better estimate the elasticities of substitution between large aggregate groups we also use Current Population Survey (CPS) yearly data for the period 1962–2006 (downloaded from IPUMS-CPS, King et al. 2009). We then use the different nested-CES models to compute the effects of immigration on the wages of natives and previous immigrants in the period 1990–2006 based on the corresponding estimated elasticities.^{9}

While overall the elasticity estimates and, therefore, the computed wage effects are somewhat sensitive to model specification, some results are robust across specifications. First, we find a small but significant degree of imperfect substitutability between natives and immigrants within the same education and experience group. When we constrain the native–immigrant elasticity to be the same for all education groups, our preferred estimate is 20. It becomes much lower (around 12.5) for less educated workers once we remove that constraint. Using model A, such large but finite elasticities imply that the negative wage impact of immigration on less-educated natives is −1.1% to −2.0% over the period 1990–2006. This model would imply a wage loss of less educated natives of −3.1% when the elasticity of substitution between natives and immigrants is infinite, as in Borjas (2003) and Borjas and Katz (2007). Hence, allowing for imperfect substitutability reduces the impact of immigration on native wages by no less than a third. This imperfect substitutability also implies that, on average, immigrants already in the United States suffer much larger wage losses than natives as a consequence of inflows of new immigrants. Based on model A, their average wage losses due to immigration are calculated to be around 6.7% for the period 1990–2006.

Second, while model A is a useful tool to assess the effects of introducing imperfect native–immigrant substitutability in the framework proposed by Borjas (2003), the data suggest that model B should be preferred instead. The key evidence for this is gathered when the different models are estimated on CPS data. That sample is large enough to allow for the separate estimation of the elasticity of substitution between broad education groups and between narrow education groups. These elasticities are indeed estimated to be quite different from each other, with the first evaluated around 2 and the second evaluated above 10. Using these estimates in model B generates wage effects that are rather different from those obtained from model A. In particular, the effect of immigration on the wages of natives with low education is now a small positive effect (between 0.6% and 1.7%). This result is due to the balanced inflow of immigrants between the broad high-education and low-education groups together with the imperfect substitutability between natives and immigrants, especially those with low education levels.

Finally, there is not much support for model C as the elasticity across broad experience groups is not very different from the elasticity across narrow experience groups (both being estimated around 5). There is no reason to favor model D either, as this leads to similar parameter estimates as model A. Indeed, for given parameter estimates, both models C and D generate wage effects that are very similar to those of model A.

The rest of the paper is organized as follows. In Section 2 we introduce the aggregate production function and the alternative nested-CES models. We also derive the equations used to estimate workers’ substitutability as well as those needed to calculate the partial and total effects of immigration on wages. Section 3 presents the data and describes how we compute the relevant variables. Section 4 details the empirical estimation of the relevant elasticities of substitution among different groups of workers. Section 5 uses the estimated elasticities in the alternative models to compute the wage effects of immigration. Section 6 concludes.