# Partial derivatives are not total derivatives

When Borjas says, “immigration lowers the wage of competing workers,” he is only looking at the direct effect of immigrants on similarly skilled natives controlling for spillover effects. Immigrants affect natives in other indirect ways.

Please indulge my equation writing. How else am I to demonstrate my intellectual superiority make my point as concretely as possible?

$\120dpi w = \alpha + A * Comp + B * Lab + C * Educ + D * Exp + E * Cap$

Wages for a particular worker in a particular skill group are w. Those wages are affected by the number labor market competitors of that worker, Comp. These are the workers with exactly the same skills as the worker. But wages are also affected by other types of workers. For example, if the number of total workers, Lab, goes up it would probably increase wages for the worker because other workers on average make him more productive. Also, an increase in the number of workers with the same education but not the same experience, Educ, or workers with the same level of experience but different educations, Exp, may increase or decrease his wages. Finally, capital, Cap, makes the worker more productive.

(In general, there’s a ton more of these cross-labor group effects. Each education group, for example, may have a different impact on each of the other education groups. Its possible that high school workers and non-degreed workers are competitors but high school workers and college workers make each other more productive. Also, there may be skill-specific capital and so there may be additional cross-labor-capital group terms.)

Borjas’ skill group-based analysis is only estimating A. It is only estimating the direct effect of immigrants on natives in their own skill group. He ignores all the other effects of immigration and almost without exception these cross-effects are positive. The “area” studies, like Card’s Mariel boatlift study and the myriad cross-section studies, suffer similar problems. While all of them ignore spillover effects, some are estimating some terms in the equation but ignoring others. The difference between those studies (and perhaps the source of the variation in their estimates of the effect of immigration on wages) is in which subset of the terms they estimate ((It would be neat to see this hypothesis tested formally. Can we get the observed variation in estimates by adding and removing various fixed effects?)).

The hitch is that nobody cares about A. Analysts, policy makers and lay people care about the total effect of immigration on natives. To get the total effect, we need estimates of A, B, C, etc; estimates of the relevant cross-effects.