Carl Shulman, one of the most impressive people I know, pointed me to a blog post he’d written a couple of weeks ago titled Turning log-consumption into a measure of short-run human welfare. Carl brought to my attention that a passage in my recent post titled how far are we from open borders?, used ambiguous language. Specifically, he pointed out that the passage:

These same estimates also suggest that much of the gain in production – and consumption – would be experienced by the world’s currently poorest people, leading to a significant reduction in, and perhaps an elimination of, world poverty. If we take utility to grow logarithmically with income, then this distributional aspect argues even more strongly in favor of the idea that open borders would increase global utility tremendously.

might suggest that I’m saying that taking utility as logarithmic points in the direction of the proportional gain in utility being higher than the proportional gain in world GDP. That was not my goal. Rather, my goal was to say that, if we take utility as the sum of logarithms of incomes, then for a given gain in world GDP, the gain in global utility resulting from that gain in world GDP would be higher if inequality was also reduced than if it wasn’t. Explicitly, having the poor’s income increase four-fold and the rich’s income stay the same, with overall GDP doubling, would give a higher utility gain than having everybody’s income double.

That’s the quick clarification. But Carl’s post raises a number of other points about the use of logarithms for considering utility, and I want to talk a bit more about some of the issues raised. The upshot, based on my reading, is that the considerations Carl raises point in favor of life-saving interventions (such as combating malaria) over interventions (such as open borders) that improve the quality of life of an existing population. But within the class of interventions that improve the quality of life of an existing population, the relative value of open borders to other interventions is not affected by the considerations Carl raises. Note also that the calculations in Carl’s original post explicitly adopt a short-run perspective, although he is elsewhere on the record stating that long-run considerations should dominate. Finally, population ethics is a fraught subject and there are a large number of issues that are somewhat related to this blog post that I do not get into, such as the question of how to value the potential existence of nonexistent people. See Nick Beckstead’s Ph.D. thesis for a detailed discussion of the far future and a summary of the philosophical literature on population ethics.

The rest of the post is fairly technical — following it properly requires a basic knowledge of calculus-level mathematics, though you can skip the quantitative statements and just consider the verbal statements.

I will consider six cases of progressively increasing complexity.

Case 1a: If you just have one person: taking logarithms is a monotone transformation that translates ratios into differences

Let’s begin with the case that we’re looking at just one person’s income. We want to understand, roughly, how the person’s “utility” grows with his or her income. We know that the greater the person’s income, the higher the person’s utility. In other words, utility is an increasing function of income. This in and of itself is good enough to tell us whether a given change in income leads to an increase or decrease in utility. What it doesn’t do is allow us to compare different changes in income with different starting and ending points. In other words, simply knowing that utility goes up with income says that income can be used as an ordinal scale for utility, but doesn’t allow us to answer questions such as: would increasing income from $10,000 to $11,000 matter more or increasing income from $100,000 to $101,000?

The assumption that utility grows logarithmically with income is an assumption that allows us to make cardinal comparisons between different changes in incomes. If we take utility to be logarithmic in income, then the increase in utility is the logarithm of the ratio of the final income by the initial income. This allows us to now meaningfully say that increasing income from $10,000 to $11,000 results in a bigger utility gain than increasing income from $100,000 to $101,000, because the ratio in the former case (1.1) exceeds that in the latter case (1.01). Note that we don’t need to take logarithms to answer the question of what gain is greater: we can just compare the ratios themselves.

The logarithm function is concave down, i.e., its second derivative is negative, so the average of the logarithms is less than the logarithm of the average. In other words, the gain in the logarithm for a given absolute gain in income is greater at lower income levels than at higher income levels. This can also be seen directly in terms of ratios as above: a $1,000 gain from $10,000 to $11,000 is larger as a proportional gain than a $1,000 gain the same absolute gain value) from $100,000 to $101,000.

There are two parameters to choose when setting up the logarithm-taking process, both of which are irrelevant for our purpose of comparing utility gains:

1. The base to which logarithms are taken. Changing the base of logarithm from one value to another corresponds to a scaling transformation.
2. The choice of “1” for income when taking logarithms, or equivalently, the choice of “0” for after taking logarithms, i.e., the income level whose logarithm we take to be zero. Changing this corresponds to a translation of the logarithmic scale, i.e., a change in the origin point.

Both choices are irrelevant for our main purpose: (2) is irrelevant because we are always looking at differences between points on the scale, so the location of the origin does not matter. (1) is irrelevant because we are comparing the differences with each other, not looking at their absolute magnitudes (in the same way as switching from meters to feet for length measurement will not change any of our fundamental analysis). (Technical note: We do need to impose the condition that the base of logarithms be greater than 1 for the analysis to hold, otherwise the scaling factor becomes negative and everything gets messed up).

A technical way of framing this is that we are treating the logarithm of income as an interval scale, i.e., a scale where it’s permissible to compare and take ratios of differences, but there is no natural zero, so it does not make sense to “double” a particular value of logarithm of income, nor does it make sense to add two values of logarithm of income. We can add, scalar-multiply, and take ratios between differences between logarithms of incomes, as these operations are invariant under the choice of origin. This is similar to how we treat temperature in practice: it does not make sense to add two temperatures and double a temperature, but we can perform the operations meaningfully on temperature differences.

However, once you delve deeper into physics, you discover that temperature actually does have an absolute zero and therefore can be measured on a ratio scale (the Kelvin scale being the standard choice in the case of temperature). If expressed in that scale, temperatures can legitimately be added and multiplied by scalars. Does there exist a similar natural choice of “absolute zero” for the logarithm of income? Not quite, but sort of. We now turn to some reasons for looking for such a zero.

Case 1b: Interpersonal utility comparisons

Let’s now consider the situation of comparing two people. We make the assumption not only that the utility functions of both are logarithmic in income, but also that the base of logarithms is the same. With these assumptions, we can compare an income change for one person to an income change for the other. If we also set an absolute zero for the log-income scale (i.e., a unit value for the income) for both the people (we could choose it to be the same for both) then we can also compare the income levels of the two people.

Case 1c: Considering the problem of zero income and non-existence

As income approaches zero, its logarithm approaches $latex -\infty$ (negative infinity). If we approximate death as switching to an income level of zero, then being dead corresponds to having a utility of negative infinity. This can pose problems when computing expected utilities in situations where there is a nonzero probability of death. Carl Shulman describes a standard way to get around the problem in his post. Explicitly, he suggests taking “subsistence income” as the absolute “1” for income, but with a twist: add a constant for the value of existing. Carl defines utility for a dead or non-existent person as 0, and utility for a living person as:

$latex s + \log(\text{income}) – \log(\text{subsistence income})$

where $latex s$ is the value of existence, and $latex \log(\text{income}) – \log(\text{subsistence income})$ is the additional value accrued from having income above the subsistence level. With this model, the effective “1” for income would be (subsistence income)/$latex b^s$ (where $latex b$ is the base of logarithms). People whose incomes are lower than that value have a negative value of existing. But in practice, we choose subsistence income and $latex s$ in a manner that nobody falls below subsistence income, let alone below (subsistence income)/$latex b^s$.

Once we have set up utility as a ratio scale as above, it makes sense to talk of the proportional change in utility. In particular, it makes sense to ask whether a given change in income causes utility to double, or more than double, or less. The answer to that would depend on the value of existence ($latex s$) and also on how far above subsistence income the person under consideration is. However, for reasonable choices, doubling income will lead to far less than a doubling of utility.

For instance, suppose we chose $latex b = 2$ as the base of logarithms, take subsistence income as $1/day, and take the existence value as $latex s = 5$. In this case, doubling income from $2/day to $4/day increases utility from 6 to 7, which is far less than doubling. Doubling income from $32/day to $64/day has an even smaller effect in terms of proportional utility gains: utility goes up from 10 to 11. (Technical aside: considering proportional gains in log-income is tantamount to taking differences of log-log-income.)

In fact, for a reasonably high choice of existence value, any change to the situation of an already existing person pales relative to a change that affects whether or not the person exists, such as birth and death. We’ll get back to this point once we consider the issue at the population level.

Case 2a: Getting multiple people into the picture, but abstracting away from the problem of people dying

We began by dealing with just one person who can earn different incomes, and then moved on to interpersonal utility comparisons, and also considered the possibility of death or non-existence, as well as . Let’s ignore the problem of death or non-existence right now, and consider a fixed population with more than one person.

The goal is to consider two different income configurations for this population, and compare them to find out which one is better. Now, at the individual level, the knowledge that utility is increasing in income was enough to say which of two income levels is greater, and the logarithmic assumption was necessary only to answer the question of how differences compared. However, in order to effectively aggregate the individual data, we do need to use a cardinal scale. In this case, since utility is assumed to be logarithmic in income, the “total utility” is the sum of all log-income values. We can then compare these totals across different configurations. Note that this case relies, albeit indirectly, on our being able to execute interpersonal utility gain comparisons (the case 1c above), and that reliance is reflected in our choice of using the same base for logarithms for all members of the population.

Now, although we are taking the sum, we are still using only the interval scale properties, and in particular, the location of the zero does not matter. This is because we are adding the same number of terms (corresponding to the members of the population) in all configurations. If we shift the location of the “zero” then that affects our “sum of log-incomes” for all configurations by an equal amount. Perhaps a better way to think than the sum is the average, for which the conclusion is clearer.

If open borders were to double every individual’s income, it would increase the average value of log-income by $latex \log 2$ and it would increase total utility by $latex \log 2$ times the population size. If, however, open borders doubled world GDP with its effect concentrated on people with low incomes, it would increase the average value of log-income by more than $latex \log 2$ (this follows from the remarks made in the discussion of Case 1a about the logarithm function being concave down).

The above is the situation that I consider by default and that is the context in which the quoted passage from my earlier post was written.

Case 2b: Comparing different populations

The remarks above continue to apply to the case of comparing improvements for different populations, including populations of different sizes, with the following catch: unless we fix an absolute zero, we can only compare changes in one population with changes in the other population. We cannot compare the absolute level of one population against the absolute level of the other, except in the following cases:

• If both populations have the same size, we can compare absolute utility levels for the populations with one another assuming they have the same absolute zero, but we do not need to specify this absolute zero.
• If the populations have different sizes, we need to specify absolute zeros for both populations in order to compare their absolute utility levels.

Case 2c: How the absolute zero allows us to compare absolute levels for different populations and introduce the possibility of death

If we embrace the model used by Carl described in Case 1c, we can tackle both the situation of comparing different populations and dealing with the problem of a nonzero probability of death. In this context, it actually does make sense to ask questions such as:

• What is the ratio of the utility levels of two different populations?
• What is the ratio of the utility levels of two different configurations for the same population?
• What is the expected utility gain for a population in a scenario (where there is a nonzero probability of death or new births in the population)?

In this context, therefore, it makes sense to ask whether doubling world GDP would indeed double utility. The general answer would be that it falls far short of doing so, even if the gains are concentrated on the poorest people. This is for the reasons already discussed in Case 1c: for any individual, doubling income has a far smaller effect on utility than doubling. Note that the gains being concentrated on the poorest people still has more effect on utility than having the gains evenly distributed or concentrated on the richest people, but that “more” still isn’t sufficient to cause doubling.

The main reason why open borders gets to look a lot worse from the point of view of the ratios of utility levels than otherwise is simply that existence itself carries huge value, and open borders, by simply moving people, doesn’t add value commensurate with the value of existing. But this argument applies generally: for a reasonably high estimate of the value of existence, any measure that makes an existing population somewhat better (without increasing births or eliminating deaths) would look a lot worse than a measure that increased the size of the population. In fact, from the existence value perspective, there’s very little that’s more promising than pro-natalism and mortality reduction — which could range from combating malaria to ending aging. Thus, the comparative analysis of open borders with most other typically considered interventions, except interventions that directly and significantly affect mortality, still points strongly in favor of open borders.

With that said, it’s worth noting that it’s likely that, by reducing poverty, open borders would reduce child mortality and increase life expectancy worldwide, and therefore could also increase global utility more greatly through that channel than the GDP estimates indicate. Also, the effects on global population growth would need to be considered: it’s likely that open borders would lead to a short-term increase in population (as first-generation migrants resemble the fertility rates of their source countries but would have lower mortality rates), meaning a significant gain in utility, but within a few generations, migrants would assimilate to native fertility rates, which may well mean lower world population than in the status quo scenario. The effect of open borders on long-term demographic trends is an important topic but is outside the scope of the current post.