Why don't elves have more babies?

Question

Is there a particular reason the elves die off so fast? After the first war against Sauron, I recall the elves being decimated, to the point that they're almost useless army-wise in the trilogy. But I'm guessing men suffered equal or greater losses as well.

Anyways, other races just seem much more capable of repopulating, while is seems like there are incredibly few (if any?) elven children. Considering the fact that elves are immortal, wouldn't their population be the fastest to grow? Also the seem to be perpetually 40 years old, so aren't they eternally fertile as well? Why don't they have more kids and build bigger societies?

Answer 1

Tolkien himself addresses this in an essay called "Laws and Customs Among the Eldar". There are essentially two problems:

1. Elvish libido diminishes over time. Contrary to popular belief (and outward appearance, from a human perspective), Elves do age; and as they get older, they get less interested in procreation.
2. Having children is exhausting. The Elves believe that their child-bearing process draws a heavier physical and spiritual toll on them, compared to what it does to humans. They're understandable reluctant to have a lot of kids, because what's the point in having a dozen kids if doing so might literally kill you?

From "Laws and Customs":

It might be thought that, since the Eldar do not (as Men deem) grow old in body, they may bring forth children at any time in the ages of their lives. But this is not so. For the Eldar do indeed grow older, even if slowly: the limit of their lives is the life of Arda, which though long beyond the reckoning of Men is not endless, and ages also. Moreover their body and spirit are not separated but coherent. As the weight of the years, with all their changes of desire and thought, gathers upon the spirit of the Eldar, so do the impulses and moods of their bodies change. [...]

Also the Eldar say that in the begetting [conceiving], and still more in the bearing of children, greater share and strength of their being, in mind and in body, goes forth than in the making of mortal children. For these reasons it came to pass that the Eldar brought forth few children; and also that their time of generation was in their youth or earlier life, unless strange and hard fates befell them.

^{_{History of Middle-earth X Morgoth's Ring Part 3: "The Later Quenta Silmarillion" Chapter 2: "The Second Phase" Laws and Customs Among the Eldar}}

That last point about the toll on their mind and body is worth expanding on, and fortunately Tolkien does this later in the essay:

[A]ll the Eldar, being aware of it in themselves, spoke of the passing of much strength, both of mind and of body, into their children, in bearing and begetting. Therefore they hold that the fëa [soul], though unbegotten, draws nourishment directly from the fëa of the mother while she bears and nourishes the hrondo [body; later changed to the word hroä], and mediately but equally from the father, whose fëa is bound in union with the mother's and supports it.

^{_{History of Middle-earth X Morgoth's Ring Part 3: "The Later Quenta Silmarillion" Chapter 2: "The Second Phase" Laws and Customs Among the Eldar}}

Answer 2

Elves only have a small number of children in a small amount of time.

Elves only procreate for a certain period of their lives, as Tolkien said (in quotes):

The bad news is that elves tend to lose interest in sex after they've had kids. "With the exercise of the power (of generation), the desire soon ceases, and the mind turns to other things…they have many other urges of body and of mind which their nature urges them to fulfil." They do look back happily on the sexually-active time in their lives, though, a period of one to several hundred years.
_{What Tolkien Officially Said About Elf Sex; quote inside from "Laws and Customs of the Eldar", one of Tolkien's essays.}

When they do have children, there aren't many:

Seven children was the usual upper limit in ancient times, but in future ages, partly due to varying levels of corruption in various elven societies, these norms were often exceeded rarely, even when corrupted do they ever succumb to deeds of Lust.
_{Elf children}

So it's once and done, while Elves are young, and then things slow down. For most of their lives, Elves do not have young (even relatively young) children.

Answer 3

To supplement Jason Baker's excellent answer I wanted to add something about the number of children also from the same essay in Morgoth's Ring. I would like to point out that there seems to be something in HDE's answer that must have been sourced from a wiki; however commentator Harry Johnston in that answer asked if anyone could translate a quote in that answer. I did so but he and I both thought it didn't look like Tolkien. Since it talks about number of children Elves tend to have I thought I would add something from the same essay that Jason cited:

The Eldar wedded for the most part in their youth and soon after their fiftieth year. They had few children but these were very dear to them. Their families, or houses, were held together by love and a deep feeling for kinship in mind and body; and the children needed little governing or teaching. There were seldom more than four children in any house and the number grew less as the ages past but even in the days of old, while the Eldar were still few and eager to increase their kind Fëanor was renowned as the father of seven sons, and the histories record none that surpassed him(4).

Footnotes:

(4): There were actually two drafts of this essay; Christopher shows the main one and then shows divergences. In this essay at footnote four he has this to say about version A:

'.. while the Eldar were still few, and eager to increase their kind, before the weight of years lay on them, there is no record of any number more than seven', with 'seldom' written later above 'no'.

Summary

In the earlier ages they were more eager to increase their kind, when they had fewer. As the ages went by this desire decreased - as Jason's answer points out; the other answer suggests that seven was the upper limit but usually it was no more than four and as the ages went by the number grew less. Fëanor was one of the ones who actually had seven and none surpassed him in this.