In a sense you are both right, you just don't set the threshold at the same place.
The bank will look at how much you earn and how much you owe and say "We can pre-approve you for $X at Y%". They consider that you can pay that rate and they are most likely right.
Now if you do take $X, you have effectively leveraged your entire earnings, which for most people is a pretty bad idea because stuff happens and you can end up with a mortgage you can't pay because turns out you also want a car.
That being said (at least in Canada), your broker won't really let you take the full bank offers, or at least they don't expect you to do it.
In the US, most lenders will do between 36 and 43% of your earnings pre-tax. Assuming you pay another 25 to 30 in taxes, this should be half of your disposable income, leaving 50% or cars food Etc.
For a $5,000 mortgage, that's a $5,000 a month buffer
Um. Look at your numbers again. Assuming your 36-43% is correct (I haven't shopped a mortgage in almost 15 years), that puts the combined total at 61-73%. Also, that combined tax rate feels really low for anyone with the income to effectively bid in this current market. Add up SS and Medicare taxes, federal income taxes, state income taxes, and sales taxes and 30% feels like a floor rather than a ceiling.
You are not going to get an 800,000 loan with a repayment of $4,000/mo on a 120K income. For one, that's in jumbo loan territory with more onerous requirements for DTI above and beyond 20% down payment.
You haven't factored in property taxes, home insurance, or anything. Putting these numbers into a calculator, your monthly loan payment is going to be closer to $5,400 a month.
You're taking home $6,800 a month.
You're not living on $1,400 a month in the Bay Area.
The bank will look at how much you earn and how much you owe and say "We can pre-approve you for $X at Y%". They consider that you can pay that rate and they are most likely right.
Now if you do take $X, you have effectively leveraged your entire earnings, which for most people is a pretty bad idea because stuff happens and you can end up with a mortgage you can't pay because turns out you also want a car.
That being said (at least in Canada), your broker won't really let you take the full bank offers, or at least they don't expect you to do it.