When the subject of curing aging is brought up, someone usually responds along the lines of “what about overpopulation?”. It’s hard to define what constitutes overpopulation, since it depends not just on the number of people but also on our fluctuating ability to convert the sun’s energy into stuff we want, and our consumption of said stuff. But let’s assume that with our current consumption, rate of technological progress, population and its growth rate, limits to the Earth’s sustainable output either have already been reached or will be in say the next 100 years.
If this assumption is wrong then the original question is irrelevant anyway. But if it’s right, then it seems intuitively obvious that eliminating the majority of deaths would greatly exacerbate the problem. Obvious, but only because our intuition is next to useless in dealing with anything exponential (i.e. where some amount is multiplied by a constant value for every unit of time). Let’s take a look at two examples of population growth. We’ll simplify by rounding some numbers, and looking only at the number of women while assuming an equal number of mates. In A, every woman has one daughter after a certain period of time (say 20 years), and the daughters reproduce after the same period. Nobody dies. B is closer to our current situation: each woman has two daughters after 20 years, and when her great-grandchildren are born (60 years later) she dies.
Quite obviously, B results in a vastly greater population because the birth rate is an exponential increase (in this case doubling), while the deaths are a linear product of the current population. And that’s the essence of what I’m trying to describe here: linear effects are usually negligible in comparison with exponential effects.
So how important is the mortality rate in comparison with birth rate? In the graph below I plot three curves based on similar simplified math1. In each case I start with a population of 1000 women, and project 500 years into the future. The red line is our current situation (it’s not quite smooth because we’re rounding everything to steps of 20 years, and nobody dies for the first 80 years): at 20 each woman has the current global average of 2.5 kids, and at 80 she dies. The green line represents the complete absence of death but with the birth rate reduced to 2.38, which results in roughly the same population growth. The blue line is the “extreme” scenario of nobody dying, but each woman having only two kids in her lifetime2.
What does this mean? That the effect on population of eliminating ALL deaths (not just aging) can be abolished by one woman out of every eight having one or two kids instead of two or three. And if everyone stuck with two kids there would be 20-fold fewer people at the end of the period. So the answer to our question is that mortality rate is not at all important relative to birth rate.
Thus far everything is simple math (with a few unassailable assumptions), and the conclusion is not a matter of opinion. It is of course hypothetical that population could feasibly be controlled by a reduction of births, but it seems quite likely3. In fact, I think it’s reasonable to assume that any country that had the capacity to eliminate aging and disease would have very low birth rates. As mentioned, the global average birth rate is 2.5. The EU average is 1.6, the US 2. Japan, Hong Kong, Taiwan and South Korea are all below 1.4. The majority of African countries lie between 4 and 7, which brings up the average. Empirically, birth rates have a very strong negative correlation with child mortality and overall level of education. An example of this effect can be seen in South Korea, which in 1970 had a birth rate of 4.53. Concurrent with its explosive economic growth, this rate dropped to its present value of 1.2.
It thus seems reasonable to assume that any society advanced enough to eliminate aging and disease would already have low birth rates, quite possibly well below 2 children per woman (the blue line above)4. Other societal changes, both planned and unplanned, are likely to affect the population dynamics of a society where death is voluntary. One possibility is that when we gain the right to live indefinitely, we lose the right to reproduce without limits. A simple version would be a default of one child per family, à la present-day China. Another version might be that you can live as long as you want, but when you have children you start your own aging process and expire in say 100 years. This latter idea sounds kind of crazy, until you consider that it’s the situation we have now (plus an option to delay the process). But as we saw earlier, this solution would a greater effect psychologically than on the actual population unless the number of children is also limited. On the other hand, we might see unexpectedly strong effects from simple psychological adaptations to indefinite lifespan. In the absence of an expiration date there would be less reason to have children in your twenties. Everyone, but women especially, would have the choice to postpone parenthood in order to establish a career, travel the world or simply feel fully prepared. A big deal for the individual, certainly, but for the population? Well, if we delay the green line’s age of childbirth from 20 to 40 in the graph above, its final value becomes ~8.4% of the red scenario…
I won’t pretend that I can predict exactly how societal changes will influence population dynamics. Any number of possible (and seemingly impossible) scenarios might play out, with or without aging and disease. But the one scenario that we don’t have to worry about is that eliminating aging will lead to “everything the same, but with more people around”. Dying, it turns out, is not a viable solution to overpopulation. On the bright side this means that you shouldn’t feel compelled to die for the sake of the environment, although it also means that any viable solution will require you to think about how you live.
 At this point the math is officially oversimplified. The example is true if every woman has 2.38 kids, but it doesn’t work if that’s the average value of some distribution and kids have the same birth rates as their parents: because the growth is exponential, going above the average has a greater effect than going below. But that just means that the real value is even closer to 2.5, so the point stands.
 For example, while China’s one-child certainly hasn’t worked perfectly it does seem to have had a significant effect. China’s birth rate is around 1.6, down from ~6 in 1950. By comparison, India’s is 2.5 (also down from ~6). As a result, although China’s population was ~50% in 1950, the UN estimates that India will be the most populous country by 2028.
 In addition to the fact that delaying aging would make for a population that is effectively younger, in terms of dependents vs. contributors.