Religious participation tends to decrease in the market share of a given religion in an area. This is because people participate more in order to preserve their identities when there is a higher threat of its extinction, which is when one’s religion is a small minority.

But how does religious pluralism (i.e. the abundance of religions in a region) affect religious participation? There are two main theories: participation either increases because of higher religious competition or it decreases because of a lack of credibility.

For instance if there is higher competition, churches may work harder to get their people to come to church as there is a higher chance of losing them to another denomination. Also, there is a larger set of religions for people to choose from, so more people may find a suitable denomination. On the other hand, lack of credibility could arise because religious pluralism might suggest that there is no absolute truth, as there are so many religions. So people may be less motivated to worship.

People have found evidence for both theories. In this post I write about a paper by Cohen-Zada and Elder (2014) (newer version from author’s website), which attempts to resolve this debate. Their idea is to connect religious market shares and religious pluralism together to religious participation in a model of school choice. They show that the specification used to establish results that support the second theory (that pluralism reduces participation) is only valid in a restrictive case. Consequently, researchers should use a more general specification, whose results actually support the first theory, that religious pluralism increases religious participation.

The model is a relatively simple model of households who belong to a denomination (or are secular). They have utility from consumption, the quality of education of their children and the probability that their child has the same religious identity as them.

Religious identity is affected by the home and school environments. Obviously, the home environment instills the parent’s religious values. The external environment, however, is more complex. If the child goes to a denominational school, then the school environment will also instill the parent’s religion’s values (in the model, a parent will never send their kid to a school with a denomination different from their own). If a child goes to public school, then they will be exposed to each religion’s values. And the exposure to a particular religion will be proportional to its “market share” (i.e. share in the population).

So what does this setup entail? For secular households, the results are straightforward, their kids go to public school, and there is an income threshold above which they send their kids to (higher quality) secular private school.

For religious people, the very same results hold, except that above the income threshold they send their kids to a denominational private school. The first important result that is derived is that the fraction of households in a given denomination who send their kids to religious schools is decreasing in the denomination’s share in the population. In other words, smaller groups are more likely to send their kids to religious school. This is because for these groups, sending their kids to public school carries a higher risk of losing their values.

The second important result is that the share of all people in the population who attend a school belonging to a particular denomination is an increasing but concave function of the denomination’s share in the population. In other words, more people will attend a larger denomination’s schools, at a decreasing rate. I.e. if you increase a denomination’s size by 10 percentage points, then the fraction of people attending that particular denomination’s schools will increase, but by fewer than 10 percentage points. This is because smaller groups have higher religious school attendance.

Now, with the assumption that the share of a group that attends religious schools is a (decreasing) linear function of that group’s share in the population, the authors show their most interesting result. With this assumption, overall religious enrollment (a proxy for religious participation) is shown to be a quadratic function of each denomination’s share.

In other words if *Q* is religious enrollment, *r_j* is the share of group *j* (excluding secular/non-religious groups) and the *a*‘s are constants, then

This is quite interesting because when studying the effect of religious pluralism on religious participation, what many researchers did was estimating an equation of the form

That is, they regressed religious participation on the % of people who belong to a denomination (first term) and on the Herfindahl index of religions to measure religious pluralism (second term), which is lower if there are more smaller religions than if there are only few large denominations.

This specification is only implied by the model of Cohen-Zada and Elder, however, if the constants *a_0* and *a_1* are the same for all religious groups. This is a strong assumption that at the very least needs to be tested after estimating the more general form (the first equation above).

Moreover, if a researcher regresses religious participation only on religious pluralism (as measured by the Herfindahl index) as follows

then they also implicitly assume that the sum of the *r_j* terms is 1. Now, since these *r_j*‘s exclude secular groups (as mentioned above), assuming that the share of all denominations adds up to 1 is equivalent to saying that there are no secular people in the population. So this specification as well, is based on a clearly wrong assumption.

Therefore, what Cohen-Zada and Elder showed is that if one wants to evaluate the effects of religious pluralism on religious participation then one must control for each denomination’s market share and market share squared *separately*. If this is not the case, then some quite likely wrong assumptions were made and the econometric specification used is wrong.

Finally, the authors take their model to the data. It is confirmed using US county-level data (with a sample size of roughly 3,000) that the fraction of households in a given denomination who send their kids to denominational schools is decreasing in their denomination’s share. I.e. smaller denominations send their kids to religious school more often. This holds for Catholics, Evangelical Protestants and Mainline Protestants. The effects are stronger in elementary than in secondary school.

The results are not the most robust, i.e. some variables slip in and out of significance depending on the exact denomination or amount of controls used. But they’re still believable, especially as the predictions are grounded in sound economic theory.

It is also confirmed that the share of people who attend a given denomination’s schools is an increasing concave function of the denomination’s market share. This holds for Catholics and Evangelicals, but not for Mainline Protestants. Mainline school enrollments tend not to be influenced much by market share, and tend to be uniformly low regardless of share.

This concave function prediction is also supported by individual-level data. It is verified for all grades (elementary, eighth grade and high school). But the effect is weakest in high school.

The point about wrong econometric specifications is tested last. The authors find with the county-level data that indeed the unrestricted specification (the first equation above) supports the theory that religious pluralism increases religious participation. Using the second specification above the same result is obtained. However, if a researcher uses the third specification, they will (wrongly) predict the opposite effect.

Using individual data the authors show that even the second specification above can predict a negative relationship between pluralism and participation. On the other hand, the unrestricted (first) specification stays consistent and robustly supports a positive relationship.

This paper therefore provides a nice theoretical basis for the specifications used in regressions evaluating the effects of religious pluralism on religious participation. The authors point out what errors other researchers have made, and show that the correct econometric model (the unrestricted one) supports a positive relationship between pluralism and participation.

This is a great example of how economic theory can guide applied econometric work.