David Beckworth directed me to this Nick Rowe post:
Let’s start with land, and assume we are not living in the Netherlands. The supply of land is fixed and hence the price is 100% demand determined. If consumer preferences do not shift, then the affordability does not change at all. But the concept of “price” is tricky, as it may be a composite variable that involves both the size of a required down payment (a function of the price of land) and the annual cost of a loan to buy land.
Now assume the interest rate falls in half for reasons unrelated to the land industry (to avoid reasoning from a price change.) If the price of land were to double, then land would be less affordable for the reasons suggested by Nick. But if consumer preferences did not change, then this cannot be the equilibrium outcome. Instead, price will rise by less than 100%, and the added negative of higher down payments will exactly offset the added benefit of lower monthly loan payments (due to lower interest rates.) The price of land will rise, but the composite cost of owning land (down payment plus interest) doesn’t change.
Now let’s think about how lower interest rates affect the price of mobile homes. Assume these homes are manufactured in industries where the long run supply curve is perfectly elastic. (That’s actually a reasonable assumption.) In that case, cutting the interest rate in half has no long run effect on mobile home prices. But it reduces the total cost of owning a mobile home and thus demand shifts right, which in the long run means higher quantity sold at the same price per unit.
Now think about “housing” as a house/land hybrid. The land is fixed in supply, and the house can be produced in the long run at constant cost. Now when interest rates fall in half there is some net decline in the total cost of housing (down payment plus interest) and some net rise in house prices and house output. In Texas, the effect of lower interest rates mostly shows up as higher quantity. In California, the main impact of lower rates is to increase house (i.e. mostly land) prices.
PS. This is not an academic exercise. Lower interest rates are the new normal.
PPS. Yes, it matters why rates fall. In this example, assume the fall in rates is caused by some combination of more saving and fewer opportunities for business investment.