A concentration index for differentiated products: the case of religious competition

Abstract

In the past, scholars have used a Herfindahl–Hirschman Index using denominational market shares to measure the competitiveness of religious markets. However, this approach ignores both the imperfect substitutability between denominations and the degree of competition within denominations. These two shortcomings make the current index a suspect measure of religious competition; it often falsely identifies which market micro-economists would generally consider the more competitive one. We develop a new religious competition index that incorporates intra-denominational competition and creates a ‘substitutability parameter’ to better specify the appropriate degree of inter-denominational competition. While the model developed in this article applies specifically to religious markets, our index of competition could be expanded to other economic markets where such a substitutability parameter is meaningful.

Notes

¹ Actually, the literature most often uses the pluralism measure, p = 1 − ∑(σ_i )², that is, one minus the traditional HHI. For p, a larger number signifies a more competitive or pluralistic market.

² The range of HHI is taken to be [0, 1].

³ Papers such as Nauenberg et al. (1997) have dealt with ways to compensate for the HHI's shortcomings in other contexts.

⁴ Scholars such as Isaacs and Laband (1998) have also applied religious market structure data to the economic theory of club behaviour with respect to public goods.

⁵ Stark and Iannaccone (1996) pointed out in the context of religious castes the essence on which we build here – that pluralism and competition in religious markets are not identical concepts. They noted that pluralism does not capture the true nature of competition if there is a lack of substitutability between religious castes. We contend, however, that the divergence between the existing measure of religious pluralism and true religious competition is much wider than their example of castes.

⁶ At the other extreme, we recognize that there may be instances for which the assumption that churches within a denomination do not compete with each other may be reasonable if every aspect of the churches’ product – including songs, location, readings, décor, instruments, etc. – is controlled by a central authority. In such as case, it may be more appropriate to view such churches as plants being operated by a single firm. However, such rigid restrictions do not appear to be present in any of the major Christian denominations.

⁷ A great deal of the recent religious participation literature has involved empirically testing the competitive and institutional theses. Breault (1989), Land et al. (1991), Blau et al. (1992) and Olson (1999) found that local (city or county) denominational pluralism has a negative relationship with religious participation rates. Blau et al. (1997) analysed data at the level of U.S. states and found similar results. On the other hand, using evidence drawn from pluralism at the national level, Iannaccone (1991) and Smith et al. (1998) found a positive relationship. Smith and Sawkins (2003) found a positive relationship at the regional level. Voas et al. (2002) pointed out an important flaw in these studies; however, they noted that the currently employed denominational ‘pluralism index appears to be [a] valid measure’ of competition and is ‘not the source of the problems’ (2002, p. 218). We contend that future empirical studies need to use an improved index of religious competition.

⁸ See Kwoka (1985), for example, for a straightforward derivation.

⁹ This is Herfindahl's (1950) construction of and rationale for a concentration index.

¹⁰ Geroski (1983) examines ‘effective output’ in a somewhat different context. This effective output comes from conjectural variations and the downward pressure that own and rivals’ output causes on own price in differentiated goods markets. Per Geroski, a firm's share of its effective output equals own output/effective output. The smaller a firm's share of effective output, the more the downward pressure on own price that is caused by a unit increase in own output and the less competitively the firm acts.

¹¹ A ‘denomination’ may be a subdenomination or wing of a denomination as required by the possible diversity within a denomination.

¹² We take σ_d and s_dc to have the range [0,1], not the range [0,100], as is sometimes used.

¹³ The substitutability between two churches in the same denomination, b_ii , is assumed to be 1. We could relax this assumption and let b_ii  < 1 if denomination i successfully suppresses competition between churches in the denomination so that the level of substitutability between two churches in different denominations is greater than the substitutability among churches within the same denomination. In this case, b_ii (for a church), has a minimum of the church's market share of the denomination, s_ic . A high degree of centralization within a denomination could cause b_ii  < 1.

¹⁴ Shifts in denomination market shares also affect B_d . For example, when market shares shift from a less substitutable denomination to a more substitutable denomination, B_d rises.

¹⁵ Concentration measures often take the form ∑ [m_i  · h(m_i )], where m_i is the market share of organization i and h(m_i ) is an increasing function of m_i . For H_R , the market share is (σ_i · s_ij ), while its increasing function is (σ_i · s_ij )/B_i .

¹⁶ H_R can also be compared to the Lerner index as derived from Cournot profit-maximizing firms. The standard result is that the market-wide Lerner index = (1/η)·∑(m _i )², where m_i is the market share for firm i and η is the market price elasticity of demand. H_R incorporates the between-submarket (i.e. between-denomination) substitutability portion of η, so it is more closely related to the Lerner index of market power than is the standard HHI.

¹⁷ As previous research has used (denomination) market share as the measure of domination (by a denomination), this assumption is not new.

¹⁸ We thank an anonymous referee for pointing out that if efficiency is more broadly defined to account for the benefit religious consumers receive from variety, the ecumenical movement may actually decrease consumer welfare per resource dollar, since it reduces variety.

¹⁹ The denomination pluralism index and H_R may be equal by chance, despite the inequality of B_d and w_d . This occurs when

²⁰ A more complete discussion and formal proof of this is available on the journal's website.

²¹ A more complete discussion and formal proof of this is available on the journal's website.

²² At the opposite extreme, over 85% of Jewish respondents said they were Jewish at age 16. Almost 9% of the remaining had ‘no religion’ at age 16 while the other 6% came from the other major denominations, with Catholicism having a hefty plurality.

²³ The substitutability parameter within a given one of the ‘other’ denominations is 1.

²⁴ A straightforward way to test this hypothesis would be to have two independent variables – one accounting for the number of denominations and the other accounting for the average number of churches within each denomination.