# Should Google hire more women?

So the NYT came out with an editorial detailing how Silicon Valley firms should become more diverse.

It seems wrong to blame firms for their hiring decisions in this case, however. Firms are rational players and they won’t (and neither should they) hire more diverse employees as long as the applicant pool itself doesn’t change.

Let me illustrate this with a very simple model. Suppose a firm’s output Y is a function of the average ability of its workers, A-bar, and of its workers’ diversity, D. Additionally, assume simply that output is increasing in both of these arguments,

$Y \equiv Y(\bar{A}, D)$

with

$Y'_{\bar{A}} > 0, Y'_D > 0.$

These are some very benign assumptions right here. Furthermore, suppose that mean ability is a function of diversity,

$\bar{A} \equiv \bar{A}(D)$

without any restrictions on the direction of this relationship. Under these set of assumptions, suppose a firm chooses its level of diversity in order to maximize output,

$D^* = \max_D Y(\bar{A}(D), D).$

After maximization, this optimization problem implies the following relationship at the optimum

$\frac{\partial \bar{A}}{\partial D} = - \frac{Y'_D}{Y'_{\bar{A}}} < 0.$

The inequality follows from our reasonable assumptions that the marginal products of both mean ability and diversity are positive.

So with this simple model – under a very loose set of assumptions -, we can show that the mean ability of workers at a firm is decreasing in diversity. Therefore, if a firm wishes to increase its level of diversity, it will have to sacrifice some level of mean ability, i.e. it will have to hire some workers with lower abilities.

Another way to see this is via a more concrete example. Suppose there are 10 workers applying for a job, and the firm wants to hire 5 of them. 80% of them are majority workers (e.g. men, White, Asian), 20% of them are minority workers (e.g. women, Black, Latino). Within both groups the ability distribution follows a bell curve-like distribution.

So the 8 majority workers have abilities of: A, B, B, C, D, E, E, F. Here, ability is measured from A to F, A being the best.

The 2 minority workers have abilities of A and F.

If the firm hires by ability, they will hire the two A ability workers (one majority, one minority), and three more majority workers (the ones with abilities B, B, and C). So altogether they will hire 4 majorities and 1 minority, exactly the 80-20 split we see in the applicant pool.

If the firm hires by diversity (making ability a secondary concern), and instead decides to hire both minority workers, then they will hire the minority with ability F and ditch the majority with ability C. Mean ability clearly declines.

These results hold as long as the ability distribution within the minority group is the same as (or worse than) the distribution in the majority group. In these cases, if only 20% of the applicants are minorities, then hiring more than 20% minorities will hurt mean ability.

Of course, a minority group with a superior ability distribution (say Asians) can be hired in higher shares. But minority groups with the same or worse distributions (which likely include women, Blacks and Latinos) cannot be hired in a higher fraction than their number in the applicant pool without hurting mean abilities.

In other words, firms like Google cannot hire more than 20% women if only 20% of computer science graduates are women without decreasing the mean ability of their hires. And if the ability distribution of women within the computer science field is actually worse than that of men (for whatever reasons), firms would actually need to hire even less than 20% women in order to maximize mean ability.

Now, I’m not saying this situation is desirable or unchangable or anything. It is just a fact. It shows that if one wants change, that probably shouldn’t start with Google and co. We should get more women to study the fields Silicon Valley needs, and we should ensure that female graduates of these fields are as competitive (in terms of ability) as males. If we can do that, Silicon Valley will hire more “minorities”.

I’m also not saying that we should necessarily incentivize women to enter into fields like computer science. This is just an option if we want more gender equality and whatnot in Silicon Valley. But there are theories implying that women in countries with more gender equality will self-select more into traditionally female fields for evolutionary reasons (if anybody knows of peer-reviewed empirical research on this, do let me know). So the lack of women in computer science may not be a big problem.

Finally, I’m not saying that there is no discrimination in the tech industry. It’s just that its role/magnitude may be much smaller than what certain people (would like us to) think.