# Integration effect on gender ratio

## Quick intro

Currently, we can read in many media, and express my many people that gender ratio is not respected a little everywhere. As an example academia should give the right example. Currently, it's very common to read that if the current ratio of output of a bachelor or a master is 50/50 the ratio in a faculty should be the same. And use the fact that it's not to introduce all kinds of 'positive' bias. How a bias can be positive ? It just introduce more inequality. Sure, another kind of inequality, but any kind of discrimination is discrimination.

The goal here is not to debate if exist a gender issue or not. And this question is probably not to debate anymore. But the goal is to debate about the duration that any decision will take until to be simply measurable and to achieve a given goal.

Let assume some scenario, we are expecting a ratio $$g$$ (our goal). If at the entry of a Bachelor we get $$g$$ (we an integration period of 3 years), 3 years later we should get $$g$$ at the entry of Masters too 2 extra years we can expect $$g$$ as the ratio for PhD candidates. Assuming 4 years to complete the PhD and 5 extra years of postdoc. So 14 years after we introduce $$g$$ at the bachelor level, we can expect it from people that candidate for professor position. For sure it’s wrong, the ratio is much less because people that did get a job before still apply ! But let assume we have are in a perfect world where everyone gets a position. (I dream of it !) Then only the new generation in the faculty was hired under the $$g$$ hypothesis. Assuming that everyone has a complete academic life ( let say 30 years after getting professor position). Then assuming a non-gender-based hiring process we would trivially need 30 years to replace all the old generations and get $$g$$. So from the time we get $$g$$, it requires 14+30=44 years before faculty actually reach the goal.

Let study some bias scenarios to see the effects. For the following part, we will assume that $$g$$ is the ratio we get 14 years ago in bachelor. Therefore 0 is the start of a professor position. Furthermore, we assume in this study that all positions are renewed and no extra position created.

At any moment we have $g=i-{{xr} \over l} + {{xh} \over {l}}=i+{x \over l}(h-r) \Leftrightarrow x={l(g-i) \over (h-r)}$ With $$i$$ the initial rate, $$h$$ the hiring ratio $$r$$ the retirement ratio, $$g$$ the expected rate and $$l$$ the length of the career. All ratio as represented as proportion of the gender of interest.

## Constant scenario

Let assume that only man retires until $$h$$ is an achieved, this is an unrealistic limit scenario $g=i+{xh\over l} \Leftrightarrow x={l(g-i) \over h}$ 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 duration assuming that only man retires, this is completely unrealistic. This time let assume retirement equal to the initial ratio, typical scenario of change of regime, $$h$$ is a constant but suddenly change form $$i=r$$ to $$h$$ $x={l(g-i)\over(h-i)}$ 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 can assume that each step the $$r$$ is the previous ratio. Then it’s coming a sequence. With $$u_0=i$$ $u_{x+1}=u_{x}-{u_{x} \over l}+{h \over l}=u_{x} \left(1-{1\over l}\right)+{h\over l}$ $\Rightarrow u_{x}=h+(i-h)(\left(1-{1\over l}\right)^{x}$ $\Rightarrow (1-\frac{s}{l})^{x}=\frac{g-h}{i-h}$ $\Leftrightarrow x=\frac{(\ln(g-h)-\ln(i-h))}{\ln(1-\frac{s}{l})}$ 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 assume $$g$$=49.5% that is 1% difference. ⟹ 108 years
Scenario $$g$$=%, $$i$$=%, $$h$$=%   and $$l$$= years ⟹ (same units as $$l$$)

## Conclusion

Even in the most bias scenario we still 6 years to achieve the goal, and with a non-hiring bias scenario ( and still completely bias retiring scenario) we still need 12 years assuming 30% gender issue as starting points. Any faculty that achieve faster transition, either created new position, has some professors that left before retirement, or biased the hiring process.