Catholic doctrine permits the use of the rhythm method, avoiding intercourse during the woman's fertile period, but regards all other forms of contraception as sinful. Critics argue that adhering to that rule results in rapid population growth in Catholic countries, which they view as a major cause of poverty. In evaluating that argument it is important to recognize that how useful a form of contraception is depends on what you are using it for. Contraception intended for family planning, to hold down the number of children to the number a married couple want to produce, does not need to be as reliable as contraception intended to permit an unmarried woman to have regular intercourse with no significant risk of pregnancy. 

As best I could tell by a little online research, there are about four days during a woman's cycle when intercourse has about one chance in four of leading to pregnancy, with a much lower chance on a few more days. Imagine that a married couple is having intercourse twice a week with no attempt to avoid the wife's fertile period. That should, on average, produce a pregnancy about every four months, hence reproduction at almost the biological maximum. 

Suppose instead that they are Catholics trying to hold down the number of children they produce by avoiding intercourse during the wife's fertile period. They do not do a perfect job of calculating the fertile period and keeping track of it, so one month a year they end up having intercourse during it. The result is one pregnancy about every four years. A woman cannot get pregnant when she is already pregnant and fertility is substantially reduced while she is nursing an infant, which reduces it to about one pregnancy every five years. About 15-20% of pregnancies end in miscarriages so about one child every six years. 

Fertility starts to drop in the early thirties, declines faster in the late thirties. Since this is a back of the envelope calculation I will assume that a woman marries at twenty and becomes infertile at forty. One child every six years for twenty years produces, on average, three and a third children. I am considering the situation in a poor society, so about a third of children will die before they reach reproductive age. We are now down to each couple producing just over two adult children, hence a population growing very slowly—well below one percent a year.

I have left out a variety of complications. Some births produce twins, pushing the number up a little. Some husbands or wives are infertile and some women never marry or marry late, pushing it down a good deal. But the bottom line seems to be that, while other forms of contraception make it easier to control the timing of births, the rhythm method is probably adequate to give married couples who want children a reasonably effective way of controlling how many they have.

That leads me to suspect that neither side of the controversy over contraception is being entirely honest about its objectives. The Catholic church defends its position on doctrinal grounds but it can be interpreted, perhaps more plausibly, as social engineering. Limiting contraception to unreliable methods — rhythm, which the church approves of, and interruptus, which it disapproves of but has no way of preventing — makes casual sex riskier without imposing large burdens on marital sex, making the former less attractive as a substitute for the latter. Critics of the church's position claim that their concern is with overpopulation and poverty but support contraceptive technologies that enable, arguably have created, the modern pattern of sex largely outside of long-term relationships.

This leads to another question: Why have birth rates in at least some poor Catholic countries been much higher than my calculations suggest? One possible answer is that most using the rhythm method are doing it incompetently, either through careless calculation or inadequate willpower. Another, and I think more plausible, answer, is that most couples in such societies chose to have large families.

That fits with my view of a similar issue in a different context. Back when contraception and abortion in the U.S. faced significant legal barriers the most prominent argument for legalizing them was to prevent "unwanted children." The implicit assumption was that most births to unmarried women were unintended, would not have occurred if the women had access to adequate contraception or, if that failed, legal abortion. As someone put it, "mistakes cause people."

If that assumption was correct, legalized abortion and the widespread availability of contraception should have led to a sharp drop in the non-marital birthrate. What actually happened was the precise opposite. In 1965, when Griswold v. Connecticut established a constitutional right to access to contraception (for married couples, but a case a few years later extended it to the unmarried), the rate of births to unmarried women in the U.S. was below 8%. It is currently about 40%.

The obvious conclusion is that births to unmarried mothers, for the most part, are not and were not unwanted. That explains why they did not fall. A possible explanation of why they instead rose can be found in an old article by Akerlof, Yellen and Katz or, in a less elaborate mathematical form, in Chapter 13 of my Law's Order (search for "Akerlof").

On general principles, of course, I believe contraception should be legal. On the question of whether improved contraception has had, on net, good or bad effects I am agnostic; I can see legitimate arguments in both directions. My point is not to support either side of that question but only to point out reasons to suspect that neither side of the controversy over contraception is being entirely honest about its motives.