This is, essentially, the same argument that wearing a bicycle helmet will cause you to bike more recklessly. While this response may be true, wearing a bicycle helmet is weakly better for a rational rider (abstracting away from the cost of the helmet, unpleasantness of wearing it, etc.) based on a revealed preference argument. "Ride at the prudence I would ride without a helmet" is in the choice set of the helmet-wearer; if he chooses something else, that something else must be at least as good as riding carefully.

With the issue of PrEP, though, this analysis is incomplete because of externalities. If an individual engages in riskier sexual behavior, he puts others at risk as well. The HIV externality of PrEP could be negative only if the patient responded to PrEP by increasing his practice of risky sexual behaviors so much that he is

*more*likely to spread HIV to others. This is unlikely, given that PrEP seems to be very effective.

But of course, HIV is not the only sexually-transmitted infection (STI). PrEP does not protect against gonorrhea, chlamydia, or syphilis. Thus, an increase in unprotected sex, triggered by an increase in PrEP usage, could increase the prevalence of these STIs.

So, what can we say about the social optimality of PrEP usage?

Let's consider a simple model where a representative, atomistic man is choosing

**x**, the amount of unprotected sex to have. (To keep things simple, I'm not distinguishing between abstinence and "safe" sex, and I will model

**x**as continuous.) He has utility over unprotected sex

**u(x)**which is well-behaved, concave, and increasing over some region, but need not be increasing everywhere. He incurs cost

**c**every time he is diagnosed with HIV and

^{H}**c**every time he is diagnosed with some other STI. (This is to avoid having to deal with exponential distributions.) Assume that each unit of unprotected sex is associated with a probability

^{S}**p**of an HIV diagnosis and

^{H}**p**of a diagnosis of some other STI. Furthermore, suppose

^{S}**p**and

^{H}**p**depend on the share of individuals with HIV or other STIs,

^{S}**θ**and

^{H}**θ**respectively. Lastly, suppose that for each agent,

^{S}**p**is also increasing (in a differentiable way) on some parameter

^{H}**t**; I will interpret PrEP as a reduction in

_{i}**t**. Thus, his utility is:

_{i}**u(x) - x[p**

^{H}(θ^{H},t_{i})c^{H}+ p^{S}(θ^{S})c^{S}]What is the effect of a reduction in

**t**--- that is, a reduction in this person's risk of acquiring HIV--- on welfare? By the envelope theorem, it's just

_{i}**x*[d**

**p**

^{H}**/d**

**t**

_{i}**]**, where

**x***is the (privately) optimal choice of

**x**. This expression is unambiguously positive (assuming

**x***is positive). This is a special case of the bike-helmets-are-good result, in the case of a small change and well-behaved utility: whatever adjustments are made by the agent have no first order effect on utility and only the direct effect matters.

Now, suppose that

**t**is actually just

_{i }**t**--- that is, what happens when everyone's

**t**falls? The envelope theorem still holds, with the important modification that

**θ**and

^{H }**θ**now change when

^{S}**t**changes. The effect of a decrease in t is:

**x*[**

**c**

^{H}**(d**

**p**

^{H}**/dt + d**

**p**

^{H}**/d**

**θ**

^{H}*** d**

**θ**

^{H}**/dt) +**

**c**

^{S}**d**

**p**

^{S}**/d**

**θ**

^{S}*** d**

**θ**

^{S}**/dt ]**

The first term is positive, representing the reduced risk of HIV acquisition: both from the direct effect of PrEP as well as the herd immunity effect from reduced HIV share in the community (I am assuming, not deducing, the sign of this second term). The second term is negative.

**dθ**is how the prevalence of other STIs increases when HIV transmission becomes riskier: it is probably negative (so an decrease in

^{S}/dt**t**would increase other STIs), and

**d**

**p**

^{S}**/d**

**θ**is positive.

^{S}So, in the end, we have two offsetting forces: the benefits of the reduction in HIV transmission risk versus the cost of increasing other STIs. The sign is ambiguous, and the "helmets-are-good" result no longer holds unambiguously in the presence of these externalities.

What can we say about the magnitudes of these effects? My guess is that

**c**

^{H}**>>**

**c**, but

^{S}**dθS/dt**and

**d**

**p**

^{S}**/d**

**θ**could potentially be quite large. It is not clear which term wins the horse race.

^{S}
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