Depends on how you define "price".
Sale price. Home values don't really change--that is to say, clearly, the value of purchasing a home over a payment term doesn't change. A $1300/month 30-year term house is a $1300/month 30-year term house. If interest rates go up, the market will bear $1300/month for that house (plus the inflation, of course), so the price must come down. In the end, you pay the same.
in which case, the average house cost to the buyer is the same at 3% or 14%
Right.
People buy houses based on monthly payment, not transaction amount.
Right again.
The interest rate just changes the numbers on the amortization table.
WRONG!
This is a simplified explanation: it's functionally correct, but technically incorrect. Think like the "principle and interest" explanation: you don't really pay $SOMEDOLLARS on principle and $OTHERDOLLARS on interest; rather, you accrue interest every day, increasing your loan balance, and then pay down the balance, with the relative difference between payment dates being the imaginary "principle", and the interest accrued that month being the "interest". It describes the effective behavior of your payments, yes, but the whole process is completely divorced from reality.
Let's say you take a $320,000 loan at 3.5% for 30 years. Down payment made, sale price, none of that matters; the fact is, after all of it, you have to borrow $320,000, at 3.5%, for 30 years. Your monthly payment is $1,436.94.
The first month, your payment is $933.33 interest and $503.61 principle, by the strange definition above. You accrue $933.33 before your first payment, and then pay down your balance by the $1,400-ish. In fact, your first six months are $933.33, $931.86, $930.39, $928.91 $927.43, $925.95. Your principle payments are $503.61, $505.08, $506.55, $508.03, $509.51, and $511.
If, in that first month, you pay an extra $2,540.17, you skip the next five months's payments. Your balance moves down to where it would be when you made your sixth payment, and so you skip accruing that interest--$4,644.45. Instead of a $1,400 payment, you have to find a $4,000 payment, but that's the effect.
Let's change the interest rate to 14%.
Let's say you take a $120,000 loan at 14% for 30 years. Down payment made, sale price, none of that matters; the fact is, after all of it, you have to borrow $120,000, at 14%, for 30 years. Your monthly payment is $1,421.85.
That's almost exactly the same payment.
The first month, your payment is $1,400.00 interest and $21.85 principle, by the strange definition above. You accrue $1,400 before your first payment, and then pay down your balance by the $1,400-ish. In fact, your first six months are $1,400, $1,399.75, $1,399.49, $1,399.23, $1,398.96, $1,398.70. Your principle payments are $21.85, $22.10, $22.36, $22.62, $22.88, $23.15.
If, in that first month, you pay an extra $113.11, you skip the next five months's payments. Your balance moves down to where it would be when you made your sixth payment, and so you skip accruing that interest--$6,996.13. Instead of a $1,400 payment, you have to find a $1,550 payment, but that's the effect.
Do you see the difference here?
In your early payments, if you throw down an extra $20, you save yourself $1,400. In fact, two and a half years in, dropping an extra $30 would save you $1,391.62, if you had only paid the minimum until then. Obviously, if you're doubling principle payment, you'd have to pay $45 two and a half years in, and would only save about $1,380 for that single extra payment.
It means if you raise your first payment by $258, you skip nearly a year of interest ($15,382). If you pay your second payment plus $297, you save $15,343 and skip ahead a whole year. Even if you just throw an extra $50 on every payment, you save yourself 8 years and $120,000 off the loan.
A 15-year loan on this house at 14% is $1,598, with a $287k final totalcost; a 15-year loan at 3.5% is $2,288, with a $412k final cost. On a 30-year loan, the final costs are $512k and $517k--the monthly payments are off by about $5,000 over 30 years.
Can you squeeze an extra $20 out of your wallet? Can you squeeze out an extra $180? An extra $180 will cut the cost of your house down by $225k in a 14% interest market where house prices are suppressed. The big joke in Europe used to be that Americans are so stupid they'll take a 30-year mortgage to save $120/month, because they can't comprehend that they're paying $50,000 for the same fucking house; now the interest rates are low and we just pay $200,000 more for the same fucking house outright.
High interest rates. We need high interest rates as the market norm--a high Federal Reserve rate to the banks--before we can get 15-year mortgages to the poorest homeowners (the ones who aren't going to get foreclosed anyway due to severe lack of financial planning) and 10-year mortgages to savvy middle classers. You might imagine the banks will just raise the rates because people are willing to pay $1,800 instead of $1,300; but people are willing to pay $1,800 FOR TEN YEARS to avoid paying $1,300 FOR THIRTY YEARS. Even if everyone jumps on that train, it'd be like if banks started marketing 45 and 60 year mortgages now; who the hell is that dumb?
You're going to ask how we got here in the first place.
Roosevelt cultured 30-year mortgages when most people rented, and only rich people had mortgages. Mortgages were 5-10 years at the time, but basically only like 30% of the country owned homes. Roosevelt got those numbers up to some 70% or something, I can't remember the exact figures. Don't take this as numerical gospel; it's mechanism I'm trying to describe. The point is he got THE MIDDLE CLASS into homeownership by giving them 30-year mortgages; it was never in our public mind as middle-class citizens that a mortgage should be 10, at most 15, years.
Today, we're more wealthy, and can afford that; it hasn't been proven, but is certainly possible, that the banks could somehow convince us that home monthly payments should stagnate for 2 or 3 generations while they slowly lengthen the loan terms. Such a campaign would have to divert attention away from the ideal of not having a payment, in the same way that we'd have to divert your mind from the ideal of retirement: we'd have to make you not think about 20 years of having thousands of extra dollars just sitting in your pocket every month.
Ten-year mortgages giving everyone over 30 an extra $2000+ discretionary spending would expand existing consumer markets and open up new markets: every business that's not a bank would viciously attack any information campaign suggesting consumers should move away from 10-year mortgages to longer 30-year mortgages. It would start cutting into their profits, quickly; even the banks can't stand up to literally everyone else, and the oil companies aren't on their side.
That's assuming people don't just up and move, although they'd then sell their house with less loss into interest, buy the next house for dirt cheap in total, pay off the mortgage in a year or three, and overall come out rich anyway. You'd have to move every 3-5 years to keep yourself as poor as a modern one-house homeowner.