Integration Effect on Gender Ratio

Quick Introduction

Currently, we often read in various media, and it is expressed by many people, that the gender ratio is not respected in many areas. For example, academia should set the right example. It’s common to hear that if the current gender ratio of graduates from a bachelor’s or master’s program is 50/50, the ratio among faculty should reflect the same. This leads to the introduction of so-called 'positive' bias. But how can a bias be positive? It just introduces more inequality—another form of inequality, but any kind of discrimination is still discrimination.

The goal here is not to debate whether a gender issue exists. That question is probably no longer open for debate. Instead, the focus is on discussing how long it will take for any decision to have a measurable effect and reach the desired goal.

Let’s assume a scenario where we aim for a ratio \(g\) (our goal). If at the bachelor’s entry level, we achieve \(g\) (with an integration period of 3 years), then 3 years later we should also achieve \(g\) at the master’s level. After an additional 2 years, we can expect \(g\) to be the ratio for PhD candidates. Assuming 4 years to complete the PhD and 5 extra years of postdoc, 14 years after introducing \(g\) at the bachelor’s level, we can expect candidates for professor positions to meet the target ratio. But that’s not entirely correct; the ratio will be much lower because people who obtained jobs earlier will still apply. However, in an ideal world where everyone gets a position (I dream of this!), only the new generation of faculty will have been hired under the \(g\) assumption. Assuming that everyone has a full academic career (let’s say 30 years after becoming a professor), and assuming a non-gender-biased hiring process, it would take 30 years to replace the older generations and fully achieve \(g\). So from the time we introduce \(g\), it will take 14+30=44 years before the faculty fully reaches this goal.

Let’s examine some biased scenarios to observe the effects. In the following analysis, we will assume that \(g\) was the gender ratio of bachelor’s graduates 14 years ago. Therefore, year 0 marks the start of professor positions. Moreover, this study assumes all positions are replaced, with no additional positions created.

At any time, we have: \[ g = i - \frac{xr}{l} + \frac{xh}{l} = i + \frac{x}{l}(h - r) \Leftrightarrow x = \frac{l(g - i)}{(h - r)} \] Where \(i\) is the initial rate, \(h\) is the hiring ratio, \(r\) is the retirement ratio, \(g\) is the expected rate, and \(l\) is the length of the career. All ratios are represented as proportions of the gender of interest.

Constant Scenario

Let's assume that only men retire until \(h\) is achieved. This is an unrealistic limit scenario: \[g = i + \frac{x h}{l} \Leftrightarrow x = \frac{l(g - i)}{h}\] Scenario: \(r = 0\%\), \(g = 50\%\), \(i = 30\%\), \(h = 100\%\), and \(l = 30\) years ⟹ 6 years
Scenario: \(r = 0\%\), \(g = 50\%\), \(i = 30\%\), \(h = 50\%\), and \(l = 30\) years ⟹ 12 years
Scenario: \(r = \%\), \(g = \%\), \(i = \%\), \(h = \%\), and \(l = \) years ⟹ (same units as \(l\)).

These are minimum durations assuming that only men retire, which is completely unrealistic. Now let's assume retirement is equal to the initial ratio, a typical scenario of regime change, where \(h\) is constant but suddenly changes from \(i = r\) to \(h\): \[x = \frac{l(g - i)}{(h - i)}\] Scenario: \(g = 50\%\), \(r = i = 30\%\), \(h = 100\%\), and \(l = 30\) years ⟹ 8.6 years
Scenario: \(g = 50\%\), \(r = i = 30\%\), \(h = 50\%\), and \(l = 30\) years ⟹ 30 years
Scenario: \(g = \%\), \(r = i = \%\), \(h = \%\), and \(l = \) years ⟹ (same units as \(l\)).

Evolutionary Scenario

Now we assume that in each step, the \(r\) is the previous ratio, forming a sequence. With \(u_0 = i\): \[u_{x+1} = u_x \left(1 - \frac{1}{l}\right) + \frac{h}{l}\] \[\Rightarrow u_x = h + (i - h)\left(1 - \frac{1}{l}\right)^x\] \[\Rightarrow \left(1 - \frac{1}{l}\right)^x = \frac{g - h}{i - h}\] \[\Leftrightarrow x = \frac{\ln(g - h) - \ln(i - h)}{\ln(1 - \frac{1}{l})}\] Scenario: \(g = 50\%\), \(i = 30\%\), \(h = 100\%\), and \(l = 30\) years ⟹ 10 years
Scenario: \(g = 50\%\), \(i = 30\%\), \(h = 50\%\), and \(l = 30\) years ⟹ Never
Let’s assume \(g = 49.5\%\), a 1% difference ⟹ 108 years
Scenario: \(g = \%\), \(i = \%\), \(h = \%\), and \(l = \) years ⟹ (same units as \(l\)).

Conclusion

Even in the most biased scenario, it takes 6 years to achieve the goal. In a scenario without hiring bias (but with a fully biased retirement scenario), we still need 12 years, assuming a 30% gender issue as the starting point. Any faculty achieving a faster transition either created new positions, had some professors leave before retirement, or biased the hiring process.