After reading some about how affordable housing is actually implemented, it looks to me like rich people could exploit it to avoid paying property and inheritance taxes, and generally get around the means testing requirements.
Affordable housing is about renting or selling homes well below market price, so if there were a large pool of affordability-restricted properties there would be a lot of incentive for people to figure out how to get around the spirit of the restrictions. I'm going to talk about buying here, but renting has a lot of similarities.
A typical buyer restriction today (Somerville example) is something like:
- Annual income no more than $71,400 for a household of two (80% AMI).
- Non-retirement assets no more than $250k.
- Haven't owned a home within 3y ("first-time homebuyer").
- No students.
- Preference for people who currently live or work in the city.
- No legal minimum income, but mortgage lenders will apply ones in practice.
Buyers who meet these restrictions are entered into a lottery, and the winner gets a 2-bedroom 2.5 bathroom 1,500 square-foot unit for $177k instead of $1,049k. Property taxes are also very low, ~$200/y instead of ~9k/y. 
These restrictions apply at purchase time: you have to have a relatively low income and assets to qualify, but then there are no further restrictions. This makes sense, because otherwise we would be requiring poor people to stay poor, but it also allows a lot of potential ways for rich people to 'legally cheat':
Intentionally keep a low income for several years. Three years at $70k instead of $140k loses you $210k, but you'd save more than that in property taxes alone long-term.
Arrange for deferred or speculative compensation. Stock that vests in four years, stock options, start a startup.
Get training that gives you high earning potential, but don't start your high paying job until after you have the house. This training is effectively an asset, but it's very hard for the affordable housing administrators to price it, so it's ignored.
Learn through self-study or apprenticeship to get around the prohibition on students.
Postpone transfers to your children until after they have qualified for affordable housing, since the income and assets of relatives are not considered.
Buy land, take advantage of density bonuses, build a large 100% affordable fancy building, and sell the units to your just-out-of-school currently-low-earning children.
There are also longer-term issues around resale. You can sell to anyone you want, as long as they meet the buyer restrictions and pay no more than the legal maximum price. This means sellers are in a position where they can effectively give a very large untaxed gift. This could let parents transfer large amounts of wealth to their children, untaxed.  You could also have problems with corruption, where I buy your property for $200k, but then I sneak you an extra $100k so you sell it to me instead of someone else.
Since these are implemented by deed restriction, they could be hard to fix if they're getting exploited. It's also not necessarily obvious whether or how much abuse there is, since the whole problem is that based on the city's verification legitimately poor people and artificially poor people look the same. (And what do we mean by "artificially poor," and do we want to include children of bankers who decide to become artists or low-paid academics?)
It's possible that the amount of hassle for the potential savings is too low for it to be worth it for rich people to subvert. If 90% of the units are used as intended and only 10% are tax shelters, I'd consider it not great but probably still good. But I'm very nervous about building a system that sets up so many opportunities for people with good lawyers to get around the spirit of the rules.
 The property is assessed at a low value because the city sets maximum resale prices. Since that's below the value of the city's residential exemption you're taxed as if the property is worth just 10% of it's assessed value. I calculate $8,830/year in property taxes for the market rate unit (after the residential exemption) and just $190/year for the affordable unit.
 Stow MA's Deed Restriction Program (faq) is an example of a way of doing this that seems especially prone to exploitation.
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