In fact, the current structure of mortgages backed by the government takes wealth into account indirectly. So-called "conforming" loans that government mortgage agencies have traditionally backed are limited by size. The problem is that this size is currently much too great. At this time, conforming loans in high-cost areas can be as high as $729,750. And remember, that's just the mortgage -- not the home price, which is likely at least 10% to 20% higher in most cases. Anyone purchasing a home at that price can afford a slightly higher interest rate or a bigger down payment to substitute for the government guarantee.
In fact, the conforming loan limit has grown much more quickly than home prices. In 1989, the median U.S. home price was $94,000, which had increased to $172,500 by the end of 2009, according to data from the National Association of Realtors. Over that period, the conforming limit increased from $187,600 to between $417,000 and $729,750 (depending on where you live) over that same period, according to Fannie Mae (.pdf). In other words, the limit went from 2x the median home price to between 2.4x and 4.2x.
So the first step is to bring the conforming loan limit back down. In this case, it should probably be somewhere around $350,000 -- but possibly lower if home prices fall further. But there's another important step needed.
Create a Conforming Loan Income Limit
There may be some situations where very wealthy people obtain mortgages that lie even below the conforming limit. Yet, these loans still obviously don't need a government guarantee. As a result, income should also be a consideration as the government considers which loans need its backing.
Really, banks shouldn't have any trouble worrying about losses from these mortgages -- they can just demand that they are full recourse. For example, if someone with an income of $500,000 per year defaults on his mortgage, then he shouldn't just be able to shrug and walk away. Since he can afford to pay for the debt he owes, he should have to hold up his end of the contract.
Really, I would argue that this principle should hold for all mortgage borrowers; the sanctity of a contract would suggest that debts should be paid as promised in general. But if there's any argument to allow non-recourse loans, it would be in the case where borrowers would not be able to afford to pay for the loss on a home in a full recourse scenario, since lenders' default risk would be too great. Those would likely be the cases under which the government would guarantee the loan anyway, since their income would be below the conforming income limit.
So a new housing finance system should have two characteristics. First, mortgages guaranteed by the government should have a much lower conforming limit, so that people who can afford a large mortgage aren't obtaining a taxpayer subsidy to buy a home that's much more expensive than average. Second, any household's income that exceeds a certain income threshold, say $250,000 per year, should be forbidden from obtaining a government-provided mortgage guarantee. If they default, then they should be on the hook for whatever debt they owe, rather than sticking taxpayers with the bill.