One of my favorite childhood books is Ender's Game. Throughout the book, having a third (and beyond) child is greatly discouraged and frowned upon. Even going as far as the world sanctioning Poland for its beliefs. I don't believe they mention a reason as to why this is the case and you would think with a war against bugs/aliens, they would need all the help they can get from the human population. Why is it that this government/movement discourages having more than two kids and in some cases allows for a third kid?
In Teacher's Pest, Ender's father proposes that the population controls are designed to ensure that the Hegemony is never such a comfortable yoke that it will endure past the end of the Formic Wars.
"I can tell them right now that population controls are about as stupid a move as they could make."
"Depending on the goal," said John Paul.
She thought about that for the moment. "You mean they might not be trying to make the Hegemony last?"
"If you intend to break up the Hegemony, you make sure as many nations and tribes as possible detest it and regard it as an oppressor."
Well, aren't I the stupid one, thought Theresa. In all these years, neither Father nor I has ever questioned the motive of the population laws. "Do you really think there's anybody in the Hegemony who's subtle enough to think of something like that?"
"It doesn't take a lot. A few key players. Why do they make such a divisive program the absolute linchpin of the war program? The population laws don't help the economy. We have plenty of raw materials, and we'd actually accomplish more, faster, if we had a steadily growing world population. On every account it's counterproductive.
In Ender's Game itself, we never see any actual signs of overpopulation or hear characters complaining about it. One description of the planet seems to suggest that the population has actually ebbed:
They sat in the back seat of the car together, driving along country roads to come at the airport from the back. "Back when the population was growing," said Graff, "They kept this area in woods and farms. Watershed land. The rainfall here starts a lot of rivers flowing, a lot of underground water moving around. The Earth is deep, and right to the heart it's alive, Ender. We people only live on the top, like the bugs that live on the scum of the still water near the shore."
and when the population ban is mentioned, it is in a political context:
That night Demosthenes published a scathing denunciation of the population limitation laws. People should be allowed to have as many children as they like, and the surplus population should be sent to other worlds, to spread mankind so far across the galaxy that no disaster, no invasion could ever threaten the human race with annihilation. "The most noble title any child can have," Demosthenes wrote, "is Third."
and after the war,
"I'm the new minister of Colonization.... We'll repeal the population limitation laws -"
"Which everybody hates -"
"And all those thirds and fourths and fifths will get on starships and head out for worlds known and unknown."
As an aside to an excellent current answer:
you would think with a war against bugs/aliens, they would need all the help they can get from the human population. Why is it that this government/movement discourages having more than two kids and in some cases allows for a third kid?
Remember that the "war" consists of already-built ships, long time ago launched against the Bugger worlds; and very very few talented soldiers in Solar system.
You simply don't need endless cannon fodder for this war, so population controls don't directly hurt in the sense you implied.
Mind you, they DID hurt in an ironic way - Graff was sadly "amused" that when they found their "savior" of the moment (Ender's future father, John Paul Wiezkorec/wiggin) in the "Polish Boy", he was so set against the Hegemony because his family AND country were punished for "non-compliance".
So, do population controls help?
One reason could be a subversive theory by John Paul Wiezkorec discussed in detail in another answer - which, in all fairness, is not necessarily a true fact in-universe (it's just a guess - of course it's a guess from an empathic savant who is rarely wrong and who topped the scores in Battle School admissions tests)
Another plausible reason (my own guess) is that it may be a policy established as a sop to a special interest group.
One such group benefitting from population controls would be the Hegemon (who represents USA), because in the multipolar world of Ender's Game, strategically, USA benefits from lower enforced birth rates by restricting both China's, and Muslim world's, and especially Russia's grouth - and given current birth rates and per-person productivity in USA, doesn't really get hurt by population controls that much... especially since they are easy-ish to violate in USA, as also discussed in "Polish Boy".
Another special interest group who would want to push this would be Malthusian-obsessed progressives (the whole official party line, as cited in other answers, about "overpopulation" and "resources", is basically Malthusianism. While Orson Scott Card is a Democrat on US political spectrum, he's severely on the "centrist" (aka "Moynihan") non-progressive wing of that party, and as a practicing member of LDS and a member of a very large family, clearly has major opposition to Malthusians. So it's not surprising that he worked in that world-view as a "kinda bad thing, as it was used to hurt Ender" angle, thus letting him cast Malthusians as a weakly-positioned antagonist.
Earth was getting overpopulated, so birth control laws were instituted similar to China's current law known as the One Child policy, but in Ender's world it is a Two Child policy. From the wiki:
Third was the term given to children who were the third born in their family. This term was considered an insult since the birth control laws in the Ender's Game Universe denote that any family cannot have more than two children.