.... living longer does increase the population, but it's linear not exponential.
We're all leaky bottles filled with the water of life. The birth rate is adding more bottles depending on how many there are (pink ones between 1/4 & 3/4 empty if you like). Increased life expectancy is adding bigger bottles - the total amount of bottles increases because there's a delay in removing the empties that only hits after the delay in getting into the zone of bottle adding water level. The additional rate of change in bottle quantity will level out as the quantity of smaller bottles tends towards zero, but it can't decrease.
If you decreased the bottle size you'd also create an increase bottle adding rate as the newest small bottles got in to "the zone" at the same time as the older, bigger bottles. As that advanced "zoning" would continue there'd be no corresponding decrease in population as the little bottles will still be replaced according to the birth rate operating when they reach empty.