Christianity’s fundamental promise is of eternal life, and the risk of refusing to accept God’s grace is generally accepted to be eternal damnation. While the truth of these statements is still subject to debate, there is little empirical evidence of the benefit of eternal life, and little research exploring the possible drawbacks of a decision to forego evil in exchange for the promise of eternal salvation. In a world of finite resources, decisions about how best to dispose of available resources while alive need to take into account the long-term and (if certain cosmological properties are shown to hold) potentially eternal consequences of the choice between good and evil. In this blog post, we will examine the costs and benefits of baptism and rejection of sin from an econometric standpoint. Of specific interest in this blog post is the relationship between the benefits of accepting God’s grace and the discount rate society applies to years of life not yet lived.

The immediate use of an analysis of the costs and benefits of accepting god’s grace is obvious, but from a wider perspective a clear understanding of the economic aspects of this theological decision may help us to understand the persistence of evil in a world where humans have free will, and to answer the eternal question: why does evil exist in a world shaped according to God’s will?

Methods

Standard cost-effectiveness analysis methods were applied to two simple decision problems. The first decision problem is the question of whether or not to baptize a child, on the assumption that baptism grants the child God’s grace, causing them to live a holy life but to lose the benefits that might accrue to an evil-doer. The analysis was then extended to consider a problem implicit in a great deal of modern rhetoric about the soul and sexuality, viz: if homosexuality is a choice, and that choice leads only to hell, is it cost-effective to choose to be homosexual? This question was answered in terms of numbers of partners foregone, and quality-adjusted life years gained from the sacrifice.

The basic decision problem: whether to baptize

The basic decision problem was addressed using standard measures of effectiveness. It was assumed that were a child to be baptized they would be eligible to enter heaven upon their death, and would thus be able to live forever. Were they not to be baptized, they are assumed to enter hell at death. Each year of life lived was assumed to grant the individual a full quality adjusted life year (QALY); each year in heaven (from now until the rapture, i.e. infinite years from now) was also assumed to grant 1 QALY; while entry into hell was considered to grant 0 QALYs. All QALYs were discounted using the standard formula, and the effect of the discounting rate on the benefits of each decision were calculated over three different life expectancies: 45 years (enlightenment-era), 70 years (biblical lifespan) and 80 years (the life expectancy granted by modern materialist living). Effectiveness was then assessed for a wide range of discount rates, varying from 0.5% to 5%. The difference in QALYs gained (the incremental effect) was then calculated for all these scenarios.

Cost-effectiveness calculation for the baptism problem

Having calculated the incremental effect of baptism, the cost was then calculated under the assumption that evil people make more money. This assumption is implicit in, for example, Mark 8:36, when Jesus asks

What good is it for a man to gain the whole world, yet forfeit his soul?

which suggests that doing good requires some form of material sacrifice. This is, of course, also obvious in the early doctrine of the Dominican and Franciscan orders, and much of pre-enlightenment religious debate was focused around this struggle between material goods and goodness.

This contrast was modeled by a variable \alpha, which represents the percentage of additional annual income an unbaptized sinner earns relative to a person living in grace. For example, if a sinner earns 10% more than a convert, then \alpha=0.1. Then, assuming a fixed average income for god-fearing individuals, we can calculate the lost income due to being good. This is the incremental cost of salvation. From this calculated incremental cost and the incremental benefit, we can estimate an incremental cost effectiveness ratio (ICER), and estimate whether the decision to baptize is cost-effective.

In keeping with standard practice as used by, for example, the National Institute for Health and Clinical Excellence, we set the basic income of one of the saved to be the mean income of the UK, and define baptism as “cost-effective” if its ICER falls below a threshold of three times the annual mean income of the UK. We also establish a formula for the cost-effectiveness of salvation, based on the relative difference in income between the good and the evil, the discount rate, and the human lifespan.

All income in future years was discounted in the same way as future QALYs.

The costs and benefits of voluntary homosexuality

Finally, we address a problem implicit in some forms of modern christian rhetoric, that of the wilful homosexual. Many religious theorists seem to think (either implicitly or openly) that homosexuality is a choice. If so, then the choice can be modeled in terms of an exchange of sexual partners for eternal damnation. In this analysis, we calculated the number of sexual partners a potentially homosexual male will forego over a 20 year sexual career commencing at age 15. We assumed that all life years before age 15 are irrelevant to the calculation (that is, we assumed that all individuals make a choice at age 15 as to whether to be good or evil), and that a person foregoing homosexuality will have 0 partners. Other assumptions are the same as those made above. The ICER for being good was then calculated as the cost in foregone sexual partners (discounted over a wide range of rates) divided by the QALYs gained through foregoing this lifestyle and gaining access to heaven.

Faustian discount rates and the problem of heavenly utilities

Commonly used discount rates range from 3 to 5%, but these are potentially inconsistent with the discount rates preferred by evil-doers. In this study we did not model differential discount rates between evil-doers and the elect, but we did consider one special case: that in which everyone observes a discount rate equal to that observed by Dr. Faust. As is well known, Dr. Faust sold his soul to Mephistopheles in exchange for earthly power, and after 24 years his soul was taken into hell. Since he knew the time frame at the beginning of the deal, this implies that he was following a discount rate sufficient to rate all time more than 24 years in the future at 0 value. Under standard discounting practice such a rate does not exist, but we can approximate it by the rate necessary to value all time more than 24 years in the future at no more than 5% of current value. This discount rate, which we refer to as the Faustian Discount Rate, is approximately 12.5%. All scenarios were also tested under this discount rate.

A further problem is the problem of calculating utility weights for a year spent in heaven or hell. Given the lack of empirical data on utility of a year in heaven, and the paucity of first hand accounts, we assumed that a year in heaven was equivalent to a year without pain or suffering of any kind, i.e. one full QALY. According to the site What Christians Want to Know, Revelations 4:8 describes heaven as

a constant chant of holy angels that are continually proclaiming Holy, Holy, Holy over the throne of God.  The Mercy Seat in heaven where God sits is surrounded by magnificent angels full of glory and power that proclaim and bless the holy name of God without ceasing.  Some of these are described as beasts, full of eyes, with six wings and neither rest day or night in their proclaiming the holiness of God.

For those of us who don’t enjoy doom metal, this would probably have a utility value of less than one. In the interests of a conservative analysis, we assign heaven a utility of 1.

A similar problem applies to assigning utilities for hell. Many people claim to have been to hell and back, but their accounts of their time at a Celine Dion concert are not convincing and it is unlikely that accurate data on the state of hell exists. Popular conception of hell suggests a realm of eternal torture, but it is worth noting that in standard burden of disease studies even the most unpleasant and torturous diseases – such as end states of cancer, AIDS, and severe disability – are assigned positive utility weights, often quite a lot higher than 0. It is therefore reasonable to suppose that hell should be assigned a positive but small utility. However, again in the interests of conservative analysis, we assign a utility weight of 0 to a year spent in hell – that is, it is equivalent to death.

Results

Incremental benefit of salvation

The formula for the incremental benefit of salvation can be derived as

LY_{g}=\frac{\exp(-rl)}{r}

where here,

  • LY_{g} is the incremental benefit of being good, in QALYs
  • r is the discount rate
  • l is the human life expectancy

Figure 1 charts this incremental benefit over a wide range of discount rates for three different life expectancies.

Figure 1: Incremental benefit of salvation for three different life expectancies

It is clear that as the discount rate increases the incremental benefit of salvation decreases rapidly. At the Faustian Discount Rate, the incremental benefit of salvation is a mere 0.03 QALYs for a 45 year life expectancy, or 0.0004 for a human with an 80 year life expectancy. That is, even if Faustus had been offered and then rejected his bargain at birth, and expected to live to 45 years only, he would have seen the benefit to himself as being only about 0.03 years of life, due to his tendency to discount the value of years far in the future.

The cost-effectiveness of baptism

We now consider the cost-effectiveness of baptism. Let the income of one of the saved be given by c_{g}, and that of an evil-doer be c_{e}=(1+\alpha)c_{g}. Then the income foregone in order to enter heaven is given by the formula

C=\alpha c_{g}(\frac{1-\exp(-rl)}{r})

where all parameters are defined as before. Then the incremental cost effectiveness ratio (incremental cost divided by incremental benefit) is

ICER=\alpha c_{g}(\exp(rl)-1)

The ICER is plotted in figure 2 for two common life expectancies across a range of values of the discount rate, assuming a mean annual income of 26,000 pounds and that evil-doers earn 10% more income than the saved.

Figure 2: Incremental cost-effectiveness of salvation for two different life expectancies

At a Faustian Discount Rate, life expectancy of 70 years, and 26,000 pound mean income, the ICER for baptism is 16,202,218 pounds per QALY gained.

We can estimate a general condition on society’s discount rate for baptism to be cost-effective, in terms of the additional income gained by being evil and the life expectancy. This formula is given by:

r \le \frac{1}{l}ln\Bigl (\frac{3+\alpha}{\alpha}\Bigr)

For a life expectancy of 70 years, assuming that the damned earn 10% more than the saved, the required discount rate for baptism to be cost-effective is 4.3% or less; if the damned earn 20% more this threshold drops to 3.5%. It is clear that damnation doesn’t have to be much more materially rewarding before it becomes attractive even under quite reasonable discount rates.

The costs and benefits of voluntary homosexuality

We now consider the situation of a callow 15 year old youth, considering embarking on a life of sodomite sin. What should he choose? Obviously, from the perspective of a simple youth, the costs need to be weighed up in terms of foregone lovers. Assuming an average of five sexual partners a year, a sexual career beginning at age 15 (which is set to time 0 in this analysis) and lasting 20 years, and the same conditions on discount rates, eternal damnation, etc. as described above, a simple formula for the number of partners this man would be foregoing by refusing to choose the love that dare not speak its name can be derived as

p=\frac{5}{r}(1-exp(-20r))

and from this the incremental cost effectiveness ratio (measured in partners foregone per QALY gained) as

ICER=5\Bigl(\frac{1-exp(-20r)}{1-exp((15-l)r)}\Bigr)

Note that this ICER is not dependent on the human lifespan. It is in fact almost linear in the discount rate (Figure 3). At the Faustian Discount Rate, the potential gay man is looking at a cost of 4.6 lovers foregone for every QALY gained. Note these values change for different annual average numbers of lovers.

Figure 3: Incremental cost-effectiveness of foregoing a life of sodomy

It might be possible to construct an experiment that assessed individuals’ discount rates using this formula: their answers to the question “how many years of life would you give up to win an additional 5 lovers” could be used to identify their value of r.

Conclusion

In Mark 8:36, Jesus asks the rhetorical question

What good is it for a man to gain the whole world, yet forfeit his soul?

Although usually presented as a question with no clear answer, it is actually quite easy to investigate this question empirically, and to draw conclusions about its implied cost-effectiveness analysis. The results presented here show that, in general, the good gained by forfeiting one’s soul is quite great, and the decision to forego baptism and live a life of evil (including wilful homosexuality) is generally the best decision one would expect a rational actor to make. At very low life expectancies and unrealistically low discount rates it is more beneficial to forego evil and embrace salvation, but at the discount rates usually used by economists, and assumed to reflect rational decisions made by ordinary individuals, salvation is not a profitable course of action.

These findings have interesting theological implications. First, we note that the Church is most likely to gain converts in a society which has a very low discount rate – but in general, the societies where the Church first took hold were societies with high rates of infant mortality and all-cause mortality, which were likely to put a low value on the later years of life – that is, to have high discount rates. But such societies are not naturally sympathetic to the message of eternal damnation, unless they can be convinced to forego rationality in moral decision making. This might explain the Church’s historical resistance to scientific endeavour, and willingness to foment superstitious practices.

These findings also explain christianity’s historical opposition to usury. It is naturally the case that buying something today and paying for it later – i.e. borrowing – is inconsistent with a very low discount rate, which tends to value future years of lost income almost as much as now. Furthermore, usurers operating in the open market will set interest rates well above 0.05%, and it is likely that the practice of usury plus the publishing of interest rates will encourage a society with higher discount rates (in fact, it is likely that this would be encouraged by the lending class). This directly undermines the church’s lesson of salvation, which depends on very low discount rates to work.

Finally, low discount rates are often associated with environmentalism – care for future generations, priority setting that considers costs in the distant future, etc. – but on the central issue of our time (global warming) many of the born again religious organizations that most fervently preach the message of salvation also vehemently oppose any message of custodianship and environmental care. These organizations would probably make better progress in convincing people to give up the joys of the here-and-now for an indeterminate heaven (that seems to involve a lot of noise pollution) if they could find a theoretically consistent approach to discount rates.

This post has shown a simple explanation for the problem of evil: most people operate with discount rates closer to Dr. Faust than to St. Christopher, and as a result they are unlikely to accept the distant benefits of heaven over the joys of the material world. Until the church can find a way to convince us that all our tomorrows are as important as today, the problem of evil will never be solved.