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Since property taxes are based on the assessed value of a property at the time of acquisition, a current market value that is higher than the previously assessed Proposition 13 adjusted base year value will increase the property taxes. Conversely, if the current market value is lower than the previously assessed Proposition 13 adjusted base year value, then the property taxes on that property will decrease.
On November 3, 2020, California voters approved Proposition 19, The Home Protection for Seniors, Severely Disabled, Families, and Victims of Wildfire or Natural Disasters Act.
There is also Prop 13 in CA, which assesses the property at the value of when it was acquired, and the assessment cannot increase more than 1% per year regardless of what the fair market value may be. Which goes to what you were saying - if you buy a new home you likely also have new mortgage principle and interest payments and a re-assessment of the property for the new owner, and for the seller a new property tax assessment on the new house. Which goes to what I was saying - no incentive to move unless you have to; especially for people in their homes a long time. Better to stay in it, remodel it, put a chair lift or elevator if you need to, or just rent it out or bequeath it to your family.
Hence, lack of inventory and that situation is likely to continue. It’s still very much a sellers market, but buyers are drying up.
This will probably lead to greater out-migration from places like New York and California before you see any major correction in housing prices.