*Inquirer*featured it in his column and sung it high praises. Bottom line, it presents a mathematical argument for not passing the controversial RH bill, at least at this time, because there is a chance that contraceptives can actually increase the abortion rate.

The paper presents a model for the number of abortions as a function of six determinants: how frequently a woman has intercourse, when she begins sexual activity, cumulative time she is pregnant (infertile time on top of the monthly infertile period), cumulative time she is breastfeeding (they assume a woman is infertile while she is breastfeeding), the contraceptive failure rate, and the risk compensation factor. Notwithstanding the other problems with the paper, this post will just take a look at their model and the conclusions they gather from that.

**The Model in Brief (no pun intended)**

In a nutshell, their contraception-to-abortion argument hinges on the risk compensation mechanism they set forth in their paper. They argue, using maths, that women (curiously, they make no reference to men in their paper) will be more sexually active if contraception is available (economists will easily recognise this as the moral hazard problem). Sugon, McNamara and Intengan begin by building a model of a woman's likely number of abortions (they couch the discussion in individual language, although I would presume they are referring to population averages) in equation (10):

*Na*= (1/

*s*)[10(

*ns - np - nb*)(1 -

*ce*)]

where

*Na*= number of abortions with contraception (this is actually the number of pregnancies, which the authors say may be aborted; later on this is just the number of abortions);

*s*= interval between sexual intercourse in weeks (the lower this number, the more frequently she is having sex);

*ns*= number of "womb years" (a proxy for fertile period depending on time being sexually active, so the earlier a woman starts having sex the higher this number);

*n*p

*=*number of womb years a woman is pregnant;

*n*

*b*= number of womb years a woman is breastfeeding (she is assumed to be infertile during this time, even though LAM doesn't apply to all breastfeeding mothers); and

*ce*= the success probability of contraception, with 0 <

*ce*< 1, so (1 -

*ce*) is the contraceptive failure rate.

The authors then lay out their model of risk compensation in section 5 of their paper (page 7). The crucial link between contraceptive use and frequency of sexual intercourse can be seen in equation (12) under hypothesis 3:

*s*=

*k*(1 -

*ce*)^

*m*

where

*k*is a constant and the other variables are as earlier defined. In describing Hypothesis 3 and equation (12), the authors say that "we propose that a woman's intercourse interval

*s*is proportional to the

*mth*power of the contraceptive failure rate 1 -

*ce*" (page 7). The boundary of equation (12) for

*s*is then analysed when

*ce*approaches zero, or (1 -

*ce*) approaches one. This is done to examine the frequency of sexual intercourse when contraceptives fail miserably such that it is as if one is not using contraceptives at all. They find in equation (14) that

lim (

*ce*-> 0)

*s*=

*k =*

*s0*

*where*

*s0*= interval between sexual intercourse in the absence of contraception, or when

*ce*= 0. Thus, equation (12) becomes equation (15):

*s*=

*s0*(1 -

*ce*)^

*m*

Plugging in equation (15) into (10) and doing some algebra, they derive an equation linking the number of abortions with contraceptive use in equation (18):

*Na*=

*Na0*/ (1 -

*ce*)^(

*m*- 1)

where

*Na*= number of abortions with contraception;

*Na0*= number of abortions without contraception; and

*ce*and

*m*are as before. One can see that

*Na*'(

*ce*) =

*Na0*(

*m*- 1) / (1 -

*ce*)^

*m*

so that the sign of

*Na*'(

*ce*) depends on the sign of (

*m*- 1): any value of

*m*> 1 and we get

*Na*'(

*ce*) > 0, so contraceptive use will increase abortions. In other words, the way contraceptives affect the abortion rate depends on the magnitude of

*m*.

**The**

*s*(*ce*) Function and the RH BillThe value of

*m*is crucial for the paper because this is used to argue that the country should hold off on passing the RH bill. Since contraceptives could reduce, increase, or have no impact on abortions depending on the value of

*m*, they argue that the government should hold off on the RH bill pending an estimation of

*m*. After all, if

*m*happens to be empirically greater than unity then the RH bill will increase abortions, rather than decrease them as the bill's supporters argue. Other than the obvious line of critique against this argument (risk compensation also applies to seat belts and health insurance), the problem is that there is no explanation for the functional relationship between sexual frequency and contraceptive effectiveness.

Their argument against the RH bill hinges on their arbitrary definition of the risk compensation mechanism as seen in equation (12), which introduces

*m*and leads to equation (18). It is arbitrary because it is neither derived from other equations, the outcome of an optimisation exercise, nor taken from previous literature. While the other parts of the model were derived from other equations, they just assumed a functional form for

*s*(

*ce*) and went from there. There is also no reason, theoretical or empirical, to say that the risk compensation mechanism should follow the power law.

That said, their incorporation of the risk compensation mechanism is a valid exercise: it is reasonable to say that intervals between sexual intercourse could be inversely proportional to the effectiveness of contraception, especially those who do not wish to have children at the moment. Looking at equation (12), one can see that the risk compensation mechanism, through the

*s*(

*ce*) function, needs to satisfy two conditions:

*Condition 1:*

*s*'(

*ce*) < 0

Condition 2: lim (

*ce*-> 0)

*s*(

*ce*) =

*k =*

*s0*

Taken together, the two conditions state that (1) contraceptive effectiveness is inversely related to the intercourse interval and (2) if contraceptives are totally ineffective

*s*(

*ce*) should revert to the no-contraception case of

*s0*. This would be a more general formulation of their Hypothesis 3.

Using their exact same model, but being just as arbitrary as the authors in defining

*s*(

*ce*), one can define the risk compensation mechanism as

*s**(

*ce*) =

*k*/ (1 +

*ce*)

which one can easily verify satisfies the two conditions above. This then becomes

*s**(

*ce*) =

*s0*/ (1 +

*ce*)

after the limits operation. The equation implies that, on average, women (men are not part of the authors' model) will be twice as sexually active with 100% effective contraception as they are without them. Plugging this into equation (10) gives us the following function for the number of abortions:

*Na** = (1/

*s0*)[10(

*ns - np - nb*)(1 -

*ce*^2)] =

*Na0*(1 -

*ce*^2)

So how does increasing contraceptive effectiveness affect the number of abortions relative to the no-contraceptives situation? One can easily see that

*Na**'(

*ce*) = - 2

*Na0*

*ce*< 0

so promoting contraceptive use will

*unambiguously*reduce the number of abortions. In other words, using their same model while slightly changing the risk compensation function, but keeping the same properties, one can arrive at a totally opposite conclusion: approve the RH bill now because it will reduce abortions. In fact, if contraceptives were 100% effective there would be zero abortions!

**So What's the Point?**

The Sugon, McNamara and Intengan paper essentially sets out a thought experiment with the aim of bolstering the argument that contraceptives can increase, rather than decrease, the number of abortions. However, what we show here is that the driver of their thesis-- their formulation of the risk compensation mechanism-- is an arbitrary assertion meant to arrive at their desired conclusion. There was no attempt to buttress this formulation through a risk minimisation (or sexual utility maximisation) operation, or to generalise the functional form of the mechanism. What we show here is that, using essentially their same model and being as arbitrary as the authors, one can do a similar thought experiment and arrive at a totally different conclusion and policy prescription. Upon closer examination, the authors' conclusions are really just assertions dressed in the language of mathematics. Take out those assertions and the paper falls apart.

+++

[

*Postscript:*

*As a critique, it would have been easy to just point out their unfounded assumptions, especially in equation (12), and do away with the paper, but it would be unfair to the authors. I really admire their effort to further their side through the use of hard logic-- this is really much better than the fear mongering and name-calling we have been hearing from other anti-RH campaigners. A bit more of this, and a lot less of Sotto's unoriginal dramatics, would be much appreciated.*]