Originally, for CA, (60's and 70's) the increase in property values led to a direct increase in property taxes, which were assessed at a percentage (e.b., 1%) of property value.
Through the 70's of course, this would mean huge 20% increases every year for property taxes, for people living in existing homes. (it was not adjusted). A second problem was with some assessments that were artificially low, being suddenly corrected in one year.
The extra money went to the city budget. Big money rush for the city, likely resulting in spending on improvements with fire, police, roads, libraries, and sewers, water, etc.
The population became upset at rampant increases and expanding city budgets, and Prop 13 was passed by a 65% approval vote of 70% the population in 1978. A main change of Prop 13 was to limit the annual property tax paid by an individual, on an existing property, to 2% per year. (NOTE, not to CPI, but to a named 2%). Other changes were to revert to market assessment in purchase year as the basis of pricing, and to limit overall property tax rates and require 2/3 vote to pass future changes in taxation rates (making it hard to change in future).
The average home price in 1975 was close to 50,000, for example. That home would have paid a maximum of 1% or $500/year in 1978, and would therefore pay a maximum of $1060 per year in taxes today... and be worth >$350,000, with neighbors in identical property paying $3800/yr.
So -- Today, nearly 40 years later --
Yes, the extra increase goes to the bottom line of the city, BUT, because Prop13 has been around so long, using 2% instead of the actual city expenses increase (approx 3.9%/yr), the average property tax paid by homeowners NET over the entire group, is actually quite low.
I calculated that the average property tax in California Alameda Country, across all households new and old was $692/yr, (I am still not sure how it gets to be so low, maybe because of defaults or zero taxed homes? This is total revenue / number of households) but that the average property tax when buying a home today is $3800...
So, today, the extra money obtained would simply go to shore up underfunded programs due to the very low average tax rate, and reduced transfers from State funds. Note that property tax is only 14% of the total budget that the Alameda County administers, they also get transfer funds from the state, and in-area revenues.. which may be driving up state taxes to cover it all.
Meanwhile, when a decrease in net property values (2008/9) reduces many homeowners' tax, the city again is short of money.
This was a very interesting updated look at property taxes for me, I had not revisited these numbers since 2007, when I recall that the average property tax paid was a $400 per property, yet new owners (me looking for a home at that time) would be paying $8k/yr...