\(\renewcommand{\epsilon}{\varepsilon}\) \(\renewcommand{\hat}{\widehat}\) \(\DeclareMathOperator*{\E}{\mathbb{E}}\)

Economic theory of abortion policies

[This blog post was originally a standalone webpage that I posted on 1/20/2020. I edited it a tiny bit for the blog.]

I recently read an interesting book by economist Phillip Levine. In Sex and Consequences, Levine studies the impact of abortion policies. Realistically, how are abortion rates, birth rates, and pregnancy rates likely to be affected by these government interventions?

The book addresses this question using both theoretical models and empirical evidence. The whole book is worth reading, but for this blog post I'm going to focus on the theoretical models. I'll provide some interactive visualizations and a little commentary.

The standard economic model of abortion

Levine describes the following simplified model of unwanted pregnancies, which he calls the "standard model of abortion." A woman chooses a value \(p\) between 0 or 1, representing the degree to which she ensures that she will not get pregnant that year. With probability \(p\), the woman successfully avoids pregnancy, while with probability \(1 - p\), the woman accidentally becomes pregnant. In the latter case, she chooses either to give birth or to get an abortion.

The tree of pregnancy possibilities in the standard model.

She makes her decisions by weighing the relevant costs. Here, "cost" should be understood to combine all relevant factors: money spent, time lost, pangs of conscience, physical discomfort, social ramifications, etc. I'll sometimes emphasize this point by putting "cost" in quotation marks.

To avoid pregnancy with probability \(p\), the woman incurs a cost \(C(p)\), representing all the negative aspects of refraining from intercourse or using contraception. We can expect \(C(p)\) to be increasing at an increasing rate. For simplicity, Levine sets \(C(p) = cp^2\) for some parameter \(c\) (see Figure ?.)

Cost of avoiding pregnancy coefficient (\(c\)):
The cost of avoiding pregnancy, as a function of the success probability \(p\). Use the slider to adjust the coefficient \(c\).

If she becomes pregnant and chooses abortion, she incurs a cost \(A\); if she chooses birth, she incurs a cost \(B\). The parameters \(A, B, c\) are specific to one individual woman, reflecting her particular circumstances, attitudes, environment, etc.

Admittedly, the model has limitations. For example, it doesn't properly account for miscarriages, pregnancy complications, multiple pregnancies in a single year, or pregnancy due to rape. Later, we'll see a more sophisticated model that addresses some (but not all) of these weaknesses. Hopefully, despite these weaknesses, the models still give some insights about people's aggregate behavior.

Anyway, we assume that the woman is rational, i.e., at each step, she acts to minimize her expected cost. So if she gets pregnant, she chooses whichever of birth or abortion is less costly, incurring the cost \(\min(A, B)\). For a given value of \(p\), her total expected cost is therefore \(C(p) + (1 - p) \cdot \min(A, B)\). Taking a derivative with respect to \(p\), we see that her total expected cost is minimized when \(C'(p) = \min(A, B)\), i.e., when the marginal cost of avoiding pregnancy, \(C'(p)\), exactly balances the cost of becoming pregnant, \(\min(A, B)\). (See Figure ?.)

Cost of abortion (\(A\)):
Cost of birth (\(B\)):
Cost of avoiding pregnancy coefficient (\(c\)):
The woman selects her probability \(p\) of avoiding pregnancy to optimize her total expected "cost." She balances the "cost" of avoiding pregnancy against the "cost" of pregnancy itself. Use the sliders to adjust the parameters.
Assuming the woman acts rationally, she will randomly move down the tree according to the depicted transition probabilities. Use the sliders to adjust the parameters.

Keep in mind that we're only considering costs to the woman herself, not costs to others ("externalities"). For example, the "cost" of abortion (\(A\)) only reflects the harm done to the baby insofar as that harm affects the mother (perhaps because she cares about it). As a pro-life advocate, I encourage everyone to adopt an ethical framework in which \(A = \infty\). That is, I think intentional abortion should be rejected no matter how costly the alternative is. But never mind my ethical views; we're trying to make realistic predictions here.

Abortion policies affect the "cost" of abortion

A new public policy governing abortion is modeled as a change in \(A\). For example, introducing public funding for abortions obviously decreases the cost of abortion. In the other direction, requiring abortion practitioners to offer women the opportunity to view a sonogram increases the psychological "cost" of abortion. Going further, if abortion is outlawed altogether, "...this does not mean that one cannot obtain an abortion, it just means that the cost of the procedure is very high." (Levine p. 41)

To better understand the effects of abortion policies, Figure ? shows the probability that a woman gets pregnant / gives birth / has an abortion in a given year, as a function of the cost of abortion.

Pregnancy outcome probabilities as a function of the "cost" of abortion (\(A\)), holding other parameters fixed. Use the sliders to adjust the fixed parameters.

Observe that if the cost of abortion is initially much lower than the cost of birth, then moderate anti-abortion laws decrease the chance of abortion without increasing the chance of an unwanted birth. The explanation for this paradox is that anti-abortion laws create a strong incentive to avoid unwanted pregnancy in the first place.

Of course, if the cost of abortion is raised above the cost of birth, then the model does predict some unwanted births. Still, overall, the chance of abortion potentially decreases much more than the chance of unwanted birth increases. I consider this good news.

Public policies can affect the "cost" of birth

The government can also set policies affecting the cost of birth (\(B\)), e.g., tax policies, family leave programs, welfare programs, etc. Figure ? again shows the probability that a woman gets pregnant / gives birth / has an abortion in a given year, but this time the horizontal axis is the cost of birth.

Pregnancy outcome probabilities as a function of the "cost" of birth (\(B\)), holding other parameters fixed. Use the sliders to adjust the fixed parameters.

Observe that for the purpose of reducing abortions, decreasing the cost of birth and increasing the cost of abortion are not equivalent. Suppose that initially, the cost of abortion is significantly less than the cost of birth. Slightly decreasing the cost of birth has no effect on the probability of abortion. There's only an effect if the cost of birth drops so low that births are substituted for abortions. In contrast, any little increase in the cost of abortion decreases the probability of abortion, because it decreases the probability of pregnancy occurring at all.

Public policies can affect the "cost" of avoiding pregnancy

Finally, government policies can also influence the cost of avoiding pregnancy coefficient (\(c\)). The obvious ways to do this involve contraceptives, but it could also be done in ways that my fellow Catholics can approve of, e.g., introducing subsidies for electronic fertility monitors, or working to reduce social pressure to engage in premarital sex. Figure ? shows the probability that a woman gets pregnant / gives birth / has an abortion in a given year, as a function of \(c\), the coefficient of the cost of avoiding pregnancy.

Pregnancy outcome probabilities as a function of the "cost" of avoiding pregnancy coefficient (\(c\)), holding other parameters fixed. The horizontal axis is on a logarithmic scale. Use the sliders to adjust the fixed parameters.

No surprises there: decreasing \(c\) decreases the probabilities of pregnancy, abortion, and birth.

Beyond the individual: Modeling a population

As I said before, the cost parameters \(A, B, c\) vary from one woman to the next. For example, some women want to have a child; for these women, \(B\) is negative and the standard model of abortion is essentially irrelevant. For simplicity, I'll assume that in the population of interest, \(A\), \(B\), and \(\log c\) are distributed according to three independent Gaussian distributions (the familiar "bell curve.") I set the variances to be pretty small, so you should think of a relatively narrow population of women, e.g., "pro-choice Protestant white college students in Nebraska" rather than "all women on Earth."

Let's graph the rates of pregnancy, birth, and abortion in this population (i.e., number per 1000 women per year) as a function of the median cost of abortion. The graph is basically the same as the corresponding graph for one individual except the curves are smoothed out:

Median cost of abortion (\(A\)):
Median cost of birth (\(B\)):
Median cost of avoiding pregnancy coefficient (\(c\)):
Pregnancy outcome rates as a function of the median "cost" of abortion (\(A\)), holding other parameters fixed. Use the sliders to adjust the fixed parameters.

Once again, if the initial cost of abortion is low, then anti-abortion laws decrease the pregnancy rate. Speaking a little loosely, when abortion is available at a very low cost, most abortions are of pregnancies that would never have occurred at all if abortion were more costly.

Anti-abortion laws can be expected to reduce abortions

To summarize what we've seen so far, the model predicts that anti-abortion laws decrease abortion rates through two mechanisms working together: they create a strong incentive to avoid pregnancy, and they give pregnant women a good reason to choose to give birth.

Some pro-choice advocates disagree with this prediction, arguing that even outlawing abortion would have little to no effect on abortion rates. These pro-choice advocates generally agree that outlawing abortion would greatly increase the "cost" of abortion. (They often make that point themselves, as part of their arguments in favor of allowing abortion.) Presumably, these pro-choice advocates believe that there are very few women who choose abortion by a "small margin" — women who choose legal abortion over giving birth, but who would choose giving birth over illegal abortion.

I find that hard to believe, but here's the thing. Even if it were true, it still wouldn't stop anti-abortion laws from working! To maintain that anti-abortion laws don't work, the pro-choice advocate must also believe that among women who get abortions, there are very few women who would have effectively reduced their risk of pregnancy if abortion had been illegal. (In mathematical terms, these pro-choice advocates need to believe that the second derivative \(C''(p)\) of the cost-of-avoiding-pregnancy-function is very large, like if there's a "kink" in the graph of \(C(p)\) located right at the chosen value of \(p\).) I find this hard to believe as well.

I'd say the most devastating blow to the "anti-abortion laws don't work" hypothesis is the substantial body of empirical evidence pointing in the opposite direction. Reviewing this (admittedly subtle) empirical evidence is beyond the scope of this blog post — I'll let you read Levine's book.

The extended model: Uncertain costs

Levine also studies a more sophisticated model of unwanted pregnancies, which he calls the "extended model." The extended model is just like the standard model, with one twist: the costs \(A\) and \(B\) of abortion and birth are modeled as random variables whose values are only determined after the woman gets pregnant.

This model captures the fact that in real life, a woman gains knowledge about the costs of birth and abortion after she becomes pregnant. A doctor might tell her that the baby has an unfortunate medical condition. She might learn that a pregnancy resource center near her can provide her with free maternity clothes and diapers. The father might abandon the mother and baby. She might become more emotionally attached to the baby than she expected. She might be surprised to learn that abortion is covered by her health insurance plan. Each of these pieces of information affects the "cost" of giving birth and/or the "cost" of getting an abortion.

The extended model can also crudely account for the horrific reality of rape. Before getting pregnant, one of the reasons a woman can't know for sure what it would "cost" to give birth or get an abortion is that she can't know for sure how she would hypothetically get pregnant. Knowing that the pregnancy is the result of rape is likely to increase the psychological cost of birth and decrease the psychological cost of abortion.

For this blog post, I'll assume that her prior knowledge of \(A\) and \(B\) is given by two independent Gaussian distributions.This is a bit different than the distributions Levine considers. See his article Abortion as Insurance, coauthored with Douglas Staiger, for more details. Let's return to an individual woman, not a whole population.The graphs for a population of women in the extended model look basically the same as the graphs for an individual woman. Rationally, the woman selects the probability \(p\) of avoiding pregnancy so that the marginal cost of avoiding pregnancy, \(C'(p)\), exactly balances the expected cost of pregnancy, \(\mathbb{E}[\min(A, B)]\).

Expected cost of abortion (\(A\)):
Expected cost of birth (\(B\)):
Cost of avoiding pregnancy coefficient (\(c\)):
The dashed horizontal line shows the expected cost of pregnancy, \(\mathbb{E}[\min(A, B)]\). It's always a bit below \(\mathbb{E}[A]\) and \(\mathbb{E}[B]\), because it would be a bit surprising for both \(A\) and \(B\) to exceed their expectations. Use the sliders to adjust the parameters.
The transition probabilities in the extended model. Use the sliders to adjust the parameters.

Anti-abortion laws can theoretically decrease births

Let's graph the probability of pregnancy / abortion / birth as a function of the expected cost of abortion, to understand the effects of abortion policies.

Pregnancy outcome probabilities as a function of the expected "cost" of abortion (\(A\)), holding other parameters fixed. Use the sliders to adjust the fixed parameters.

If you set the cost-of-avoiding-pregnancy-coefficient (\(c\)) slider to a small value and the expected-cost-of-birth (\(B\)) slider to a moderate value, you can observe something strange. (Click to set the sliders appropriately.) As the expected cost of abortion increases, the probability of birth increases at first and then decreases again. Meanwhile, the probability of abortion simply decreases the whole time. So theoretically, anti-abortion laws can decrease the chances of abortion and birth simultaneously.

It was already interesting in the standard model to see the chance of birth increase by less than one might expect. The paradox in the extended model is more extreme: the chance of birth can actually decrease!

The explanation for this paradox is, once again, the effect of anti-abortion laws on \(p\). When the expected cost of abortion is very high, the woman virtually guarantees that she will not get pregnant, so the chance of giving birth is near-zero. With a more moderate expected cost of abortion, the prospect of pregnancy doesn't look quite so daunting, so she accepts a moderate risk of getting pregnant. But then conditioned on getting pregnant, there's also a decent chance that the cost of birth turns out to be less than the cost of abortion, in which case she gives birth.

Decreasing the "cost" of birth can theoretically increase abortions

There's a dark side to this latest paradox. Since increasing the "cost" of abortion can theoretically decrease the chance of birth, by symmetry, decreasing the "cost" of birth can theoretically increase the chance of abortion. Indeed, you can see this troubling phenomenon on the following graph by setting the cost-of-avoiding-pregnancy-coefficient (\(c\)) slider to a small value and the expected-cost-of-abortion (\(A\)) slider to a moderate value. (Click to set the sliders appropriately.)

Pregnancy outcome probabilities as a function of the expected "cost" of birth (\(B\)), holding other parameters fixed. Use the sliders to adjust the fixed parameters.

I consider this bad news. Providing free diapers, guaranteeing paid maternity leave, increasing daycare access, etc. make it easier for pregnant women to reject abortion; I'd like to think that these efforts therefore reduce abortions. The extended model does allow for this possibility, and I bet it's what happens in practice. However, the extended model also allows for the possibility that overall, these efforts increase abortions, because they make women feel less of a need to avoid pregnancy, and then those same women sometimes end up choosing abortion after getting pregnant and gaining information.

Let me stress that there are independent reasons to strive to reduce the "cost" of giving birth, aside from any effects on abortion rates. At a basic level, the motivation for reducing abortions in the first place is to promote people's welfare (namely, people in their mothers' wombs). That same motivation urges us to ease pregnant women's burdens and thereby promote their welfare.

Conclusions

Simple economic models paint a surprisingly complex picture. The relationship between abortion laws and birth rates is not straightforward, nor is the relationship between the cost of birth and abortion rates.

Some predictions from these models, however, are refreshingly straightforward. For example, increasing the "cost" of abortion reduces abortions. On a philosophical level, I think outlawing abortion is consistent with principles of justice. After reading Levine's book, I'm more convinced than ever that outlawing abortion is also consistent with principles of prudence. I think we should work to make abortion both unthinkable and illegal.

Thanks to Phillip Levine, Alicia Torres Hoza, and Matt Morgan for helpful comments on drafts of this blog post.