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Expected social welfare maximization is where you try to obtain the set of parameters (such as climate change policies) that will maximize the expected value of a social welfare function.
A) Introduction to Social Welfare
There is a lot of uncertainty with respect to the issue of climate change. For example, the current (5th assessment report) subjective 95% confidence interval of Equilibrium Climate Sensitivity (ECS) according to the International Panel on Climate Change (IPCC) is 1.5 C to 4.5 C. Progress has been made recently in terms of constraining ECS, with the most recently empirical estimates suggesting that ECS is in the lower half of the IPCC’s confidence interval (Michaels and Knappenberger 2014). However, even with better estimates it seems that the uncertainty associated with ECS will remain relatively large. In addition, there are many other sources of uncertainty that are relevant to the issue of climate change such as uncertainty about the rate at which the climate reaches a new equilibrium, uncertainty about the rate of ocean uptake of CO2, uncertainty about the costs of mitigation policy, uncertainty about the cost-effectiveness of geoengineering and uncertainty about the economic impacts of climate change.
Despite all of this uncertainty, decisions about what to do about the issue of climate change are going to be made. As a result, it is desirable to have a method that can obtain a unique optimal policy response to climate change given all this uncertainty. Thus it is desirable to have a method that can rank policy responses in order to determine the best policy response given all the uncertainty. However, the scientific method alone does not rank policy responses since the scientific method doesn’t tell people what to do. In order to obtain a ranking of policy responses, at some point moral judgements need to be made. But in order to obtain agreement with others about the decision rule, one needs to limit moral judgements to well accepted moral principles in society. This post suggests that maximization of expected social welfare may be a reasonable method to rank policy options even if there is uncertainty. By appealing to egalitarianism, individualism, preference for model simplicity and empirical evidence, this post tries to justify reasonable social welfare functions for the basis of ranking policy responses.
So what is expected social welfare maximization? Expected social welfare maximization is where you try to obtain the set of parameters (such as climate change policies) that will maximize the expected value of a social welfare function. What is a social welfare function? A social welfare function (SWF) is a function that takes the utility of every individual in a society and outputs a real number, which represents the social welfare or wellbeing of society. What is the utility of an individual? The utility of an individual is a real number that represents the wellbeing of that individual. An individual’s utility may be thought of as a function of various things that influence the individual’s wellbeing in society, such as consumption, level of physical health, level of mental health, amount of leisure time, how much freedom they have, type of climate they live in, how many chocolate bars they eat on Wednesdays, etc. Such a function is often called the individual’s utility function and individuals in society tend to try to maximize their utility functions.
B) Additive Separability and Anonymity
Ultimately, it is desirable to obtain a unique SWF to be used for expected social welfare maximization. In order to do this, a reasonable first step may be to constrain the set of acceptable social welfare functions by applying some basic moral principles. Additive separability, in the context of SWFs, means that the social welfare of society is equal to the sum of the utilities of the individuals within a society. For example, if there are n individuals within society then society’s additively separable SWF is W = U1 + U2 + … + Un, where Ui is the utility of individual i. Additive separability may be justified on the basis of individualism and simplicity; a separable SWF treats people as individuals and additive separability is the simplest kind of separability.
The anonymity principle states that all individuals in society have the identical utility functions. Thus if the utility of individual i in society is a function of k parameters X1i, X2i, …, Xki then the additively separable SWF becomes W = Σi=1nU(X1i, …, Xki), where U is a utility function. The anonymity principle can be justified in two different ways. The first is that as information is limited, we cannot know the utility function of every single individual within society, thus the anonymity is a necessary simplifying assumption needed to be able to define a reasonable SWF. A second way to justify the anonymity principle is on the grounds of egalitarianism; under the anonymity principle, everyone is treated equally by the SWF.
C) Utility as a Function of What?
The wellbeing of an individual can depend on many things: from the number of grains of rice they eat per year to the number of freckles on their forehead. However, trying to make utility be a function of pretty much everything isn’t feasible, nor practical. It makes sense to restrict the utility function to be a function of only a few relevant parameters. Economic consumption is arguably the most relevant factor for individual utility in the context of climate change; mitigation policy, changes in average temperature, changes in precipitation patterns, the CO2 fertilization effect, sea level rise, and ocean acidification will all likely have an effect on economic output and thus consumption. Therefore, it makes sense to have consumption as one of the parameters in the utility function.
Other parameters may also be relevant. For example, the happiness of individuals in society may be directly affected by the climate; Tsutsui (2013) found that a temperature of 13.9 C maximizes happiness. In addition, how much individuals value the environment or how much individuals value their health may be relevant. While there are likely many relevant parameters, for simplicity, the rest of this post will treat utility as a function of only consumption. Thus, the SWF becomes W = Σi=1nU(Ci), where Ci is the consumption of individual i.
D) Simplicity and Empirical Evidence
What functional forms of U(C) may be reasonable? One individual may argue for a square root function while a second individual may argue for a logarithmic function. Who is correct? From here, you can take one of two positions. Either you know a priori everyone else’s preferences better than they do, or you don’t. If you don’t then perhaps your best option is to try to infer the average utility function of individuals in society by looking at empirical evidence. This reduces the problem of finding U(C) to a question of empiricism and is arguably a more democratic approach since you are looking at the average preferences of society. This post will assume that one does not know a priori everyone else’s preferences better than they do so will take the latter approach.
When trying to choose a model of U(C), it is important to try to balance model simplicity with the ability of the model to explain observations. A model that does not agree well with observations is arguably an unfit model of human preferences. On the other hand, simpler models often have more explanatory power; simpler models may make more falsifiable predictions or be easier to work with. Therefore, one should have preference for model simplicity. Trying to balance model simplicity with fit to observations may allow one to determine a unique best model of U(C) given the available empirical evidence.
E) Expected Utility Theory
One common and well known model of human behaviour is known as expected utility theory (EUT). Under EUT, an individual tries to maximize their utility, where the utility of an individual over a probability distribution of outcomes is equal to the expected value of the individual’s utility. The utility function that corresponds to EUT is a special type of utility function known as a von Neumann-Morgenstern (vNM) utility function. A vNM utility function is a utility function where U([pa;(1-p)b]) = pU(a) + (1-p)U(b), where a is one possible outcome, b is another possible outcome and [pa;(1-p)b] corresponds to a p probability of outcome a and a (1-p) probability of outcome b. A vNM utility function is unique up to positive affine transformation (i.e. multiplying a vNM utility function by a positive number or adding a constant will not change human behaviour). This means that a SWF that satisfies additive separability and anonymity, and uses a vNM utility function will be unique up to positive affine transformations. Positive affine transformations of the SWF do not change what maximizes the SWF, so using EUT to find a vNM utility function is sufficient for the purpose of social welfare maximization. Furthermore, nVM utility functions can be estimated empirically by looking at human behaviour under uncertainty. The rest of this post assumes that U(C) is a vNM utility function.
F) CRRA Utility Functions
The average utility function of individuals in society can be empirically estimated by looking at the decisions people make under uncertainty. Arguably, the two most common measures of risk aversion are the coefficient of absolute risk aversion and the coefficient of relative risk aversion (RRA). The coefficient of absolute risk aversion is defined as -(d2U/dC2)/(dU/dC) and the coefficient of RRA is defined as -(d2U/dC2)C/(dU/dC). A quick review of the literature suggests that the coefficient of absolute risk aversion is a decreasing function of consumption (Friend and Blume 1975). On the other hand, whether the coefficient of RRA increases, decreases or remains constant as a function of consumption is disputed and there is a fair amount of uncertainty about its magnitude (Outreville 2014). Given all the uncertainty and the desire to balance model simplicity with fit to empirical observations, it may be reasonable to assume that the coefficient of RRA is constant.
CRRA utility functions are defined, up to positive affine transformations, as U(C) = C1-η/(1- η) when η ≠ 1 and as U(C) = ln(C) when η = 1, where η is the coefficient of RRA. If η > 0 then the utility function is risk averse. Using a CRRA utility function in the SWF means that the SWF satisfies the Pareto principle (the property where if a policy can make 1 individual better off, without making any individuals worse off, then society is better off with that policy) and, if η > 0, satisfies the Pigou-Dalton principle (the property where if a policy can make a poor person richer by a small amount and a rich person poorer by an equally small amount then society is better off with that policy). η affects how the SWF deals with risk, intragenerational inequality and intergenerational inequality. η = 0 causes the SWF to be risk neutral and to not care about consumption inequality in society; this corresponds to the decision making under traditional cost-benefit analysis. η = ∞ causes the SWF to avoid all risk and avoid all inequality; this corresponds to the decision making under the strong precautionary principle. η between 0 and ∞ results in a SWF that has a moderate level of risk aversion and values both total consumption in society as well as consumption inequality. The value of η is a prior indeterminate. However, it can be estimated from empirical data. Below is a plot that shows CRRA utility functions for η = 0, 0.5, 1 and 2.
G) Time Separability
An individual may care about more than just their total consumption over their lifetime; they may care about the timing of that consumption. Humans tend to have a preference for present consumption over future consumption and tend to want to smooth consumption over time. The most common way to treat this timing issue is to treat the utility function as additively separable with respect to time. In this case, for an individual that lives for T periods, the individual’s lifetime utility function is Σt=1TwtU(Ct), where Ct is the individual’s consumption for period t and wt is the weight of period t. Often, the weights are represented by an exponentially decreasing function of time, which means that the individual’s lifetime utility function is Σt=1Te–ρtU(Ct), where ρ is a discount rate.
If one puts this into a SWF with constant RRA and assumes that the utility of a dead individual is zero then the SWF becomes W = Σt=1∞e–ρtΣi=1N(t)Cit1-η/(1- η) when η ≠ 1 and W = Σt=1∞e–ρtΣi=1N(t)ln(Cit) when η = 1, where η is coefficient of RRA, N(t) corresponds to the number of people alive in period t and Cit corresponds to the consumption of the ith individual alive in time period t. This specification reduces the problem of finding a SWF to the problem of finding 2 parameters, η and ρ, both of which may be estimated empirically.
H) Expected Social Welfare
Since maximizing expected utility is how individuals in society make decisions under uncertainty, it may make sense that maximizing the expected value of the SWF is how society should make decisions under uncertainty, as this is the natural extension of expected utility maximization to the SWF. However, before accepting expected social welfare maximization, it may make sense to look at some other common decision rules.
One common decision rule is traditional cost-benefit analysis. Under traditional cost-benefit analysis, one tries to choose the policy option that maximizes the net present value of benefits minus the net present value of costs. The theoretical justification for cost-benefit analysis is that if there is little uncertainty about the future and any policy under consideration will have relatively small impacts on the consumption of individuals in society then traditional cost-benefit analysis is approximately the same thing as expected social welfare maximization. However, in the case of climate change, the uncertainty is large and the impact of policies on consumption may be large, so traditional cost-benefit analysis is inadequate.
The strong precautionary principle is a decision rule that is frequently used in the context of climate change. Under the strong precautionary principle, if there is any risk that a policy may have a negative impact on society then that policy should not be taken. There are numerous problems with the strong precautionary principle. For one, the strong precautionary principle contradicts itself. For example, if one uses the strong precautionary principle to decide whether or not a new drug should be allowed on the market, one runs the risk of disallowing many perfectly good drugs, causing harm to society. Thus the strong precautionary principle has a risk of causing harm to society so should not be allowed under the strong precautionary principle. Secondly, the strong precautionary principle has many bizarre policy implications. For example, one cannot exclude the possibility that a giant flying spaghetti monster may appear and try to destroy New York; therefore, the US government should spend economic resources to ensure that it can fend off any attack by a giant flying spaghetti monster. Given these issues, the strong precautionary principle does not seem like a reasonable method to make decisions.
I) Empirical Estimates of η
It may be helpful to look at empirical evidence, to see what values of η may be reasonable. The literature on empirical estimates of η is vast and there are many different methods by which one can estimate it. If you wish to read a more extensive review of the literature of measures of η then I suggest Outreville (2014). However, this post will cover 4 different estimates, in order to keep things short.
J) Labour Market Behaviour
One way to estimate η is to look at labour market behaviour. One of the best studies that takes this approach is Chetty (2006), where Chetty invents a method to estimate η from uncertainty in labour market conditions and obtains a best estimate of 0.97. However, Chetty may be misestimating how willing people are to exchange consumption for leisure, so this estimate may be an underestimate. Even after taking this possibility into account, Chetty finds that a value of η greater than 2 is inconsistent with labour market behaviour.
K) Happiness Surveys
Arguably, one of the best estimates for η has been performed by Layard et al. (2008). Layard et al. take advantage of the fact that happiness is correlated with utility and use data on self reported happiness from over 200,000 individuals to estimate η. Initially, Layard et al. assume that happiness is a linear function of utility and estimate η as 1.26 (with 95% confidence interval of [1.15,1.37]). However, happiness is not necessarily a linear function of utility; more generally, it can be any positive monotonic transformation of utility. To check for this possibility, Layard et al. relax their linearity assumption slightly and find a better estimate of 1.24 (with 95% confidence interval of [1.14,1.35]). In both cases, Layard et al. control for various other explanatory factors such as leisure time. Overall, the Layard et al. result gives a surprisingly robust and well constrained estimate of η.
L) Ramsey Equation
If individuals in society have a lifetime utility function of Σt=1Te–ρt Cit1-η/(1- η) when η ≠ 1 and Σt=1Te–ρtln(Ct) when η = 1 then this has predictions about the interest rate of society. In such a society, r = ρ + ηg, where r is the real riskless after-tax interest rate, ρ is the discount rate, η is the coefficient of RRA, and g is the growth rate of real GDP per capita (see Creedy and Guest 2008 for a derivation). This equation is known as the Ramsey equation.
Using the Ramsey equation, it is possible to estimate both ρ and η by using data on r and g. However, in the short run, it is possible for a country’s interest rates to diverge from what is expected by the Ramsey equation, especially due to the policies of central banks (which tend to roughly follow a rule known as the Taylor rule). In the long run, the Ramsey equation should be roughly satisfied, so one may be able to estimate η by comparing r and g between countries. Anthoff et al. (2009) use data from 27 OECD countries over a 36 year period and obtain a best estimate of η of 1.18. This is similar to the estimate of Layard et al., although the uncertainty of the Anthoff et al. estimate is much higher.
If people do not value the future more than the present, then ρ cannot be less than zero. As a result, r/g provides an upper bound on the value of η. According to the World Bank, the average real riskless interest rate of the USA from 1995-2014 was 3.88%. In addition, the tax on capital gains in the USA is approximately 19.1% (http://taxfoundation.org/article/capital-gains-rate-country-2011-oecd). This suggests that the after-tax average real riskless interest rate for this period was 3.14%. By comparison, the average g for the USA over this period was 1.475%. This suggests that η greater than 2.14 is inconsistent with empirical evidence.
M) Statistical Value of Life
How people make decisions when it comes to the probability of death can also be used to estimate η. If the average individual in society is indifferent about taking a risk that has a (1 – α) chance of increasing their consumption by ΔC and an α chance of killing them, where α and ΔC/C are both very small, then ΔC/α is known as the statistical value of life (SVL). If the utility of death corresponds to a consumption level of zero then this suggests that U(C) = (1 – α)U(C + ΔC)
How people make decisions when it comes to the probability of death can also be used to estimate η. If the average individual in society is indifferent about taking a risk that has a (1 – α) chance of increasing their consumption by ΔC and an α chance of killing them, where α and ΔC/C are both very small, then ΔC/α is known as the statistical value of life (SVL). If the utility of death corresponds to a consumption level of zero then this suggests that U(C) = (1 – α)U(C + ΔC)
=> C1-η/(1 – η) = (1 – α)(C + ΔC)1-η/(1 – η) => C1-η = (1 – α)C1-η(1 + ΔC/C)1-η ≈ (1 – α)C1-η(1 + (1 – η)ΔC/C)
=> 1 ≈ 1 – α + (1 – η)ΔC/C => η = 1 – αC/ΔC = 1 – C/SVL
For the USA, the SVL is approximately 200 times annual per capita income (Cline 1992) and the life expectancy for the USA in 1992 was 72 years. If the average person is middle aged, then this suggests that they would have on average 36 years of remaining life. This means that total remaining consumption for the individual would be roughly 36 times annual per capita income. Putting this information into the above equation gives η = 0.82. Of course this is a very rough estimate with numerous problems, but it does illustrate how statistical value of life estimates can be used to estimate η. Overall, given the range of estimates given in this post and in the empirical literature, a value of η outside of the range [0.5,2.0] seems inconsistent with empirical observations (upper bound is from Chetty, lower bound is subjective).
N) Finite Statistical Value of Life?
The equation derived in M suggests that if η ≥ 1 then the SVL for a CRRA utility function cannot be finite. Yet in everyday society it is observed that people frequently do risky activities that put their lives at risk, from smoking to sky diving. To show the absurdity of an infinite SVL, take the simple activity of eating a potato chip. Potato chips are unhealthy and can increase your chance of dying. Even if the increase in the probability of dying from eating a potato chip is very small (say one in one sextillion), it is still finite, and the pleasure gained from eating a potato chip is finite. Thus an expected utility maximizing individual would never eat that potato chip if they assigned an infinite value to their own life. Given that people eat potato chips in society, the SVL is likely finite.
Is it possible to have η ≥ 1 and a finite SVL? It is possible provided that the constantness of η breaks down at low levels of consumption and the limit of η as consumption approaches zero is less than 1. One reason why this may occur is that people become increasingly desperate in extreme poverty and, in extreme poverty, the individual’s remaining life expectancy is strongly dependant on consumption. A person that is starving to death may only have a month or two of remaining life. However, if they take a risk then they may be able to afford enough food, which may give them many years of additional life. In such a scenario, individuals may even become risk loving (η < 0).
One may be able to observe a constant η ≥ 1 for most consumption levels and a finite SVL provided that the constantness of η breaks down in cases of extreme poverty. In particular, if the reason for this break down is due to starvation then there should be a discontinuity in human behaviour around the subsistence level of real GDP per capita. Caballero (2010) has found empirical evidence consistent with this. Caballero performs experiments with impoverished Colombians to try to determine their behaviour under risk. Caballero uses a subsistence level of real GDP per capita of 148,000 Colombian Pesos per month to try to see if there is a discontinuity in risk aversion around the subsistence level. Caballero finds that individuals just above the subsistence level are far more risk averse than people just below the subsistence level.
Given that the vast majority of the world has a level of income above the subsistence level, the usage of a SWF with constant η may be reasonable even if η ≥ 1. However, such a CRRA utility function would break down for dead people and for people in extreme poverty. A better utility function to be used for expected social welfare maximization might be a piecewise continuous function that is a CRRA utility function above the subsistence level of consumption and something else (such as a polynomial) below the subsistence level.
O) Discount Rate:
In addition to a reasonable value of η, a reasonable value of ρ is needed. As explained in section L, utility maximizing behaviour suggests that r = ρ + ηg. As a quick example, if one uses r = 3.14% and g = 1.475% (values from section L) and η = 1 then this suggests that one should use ρ = 1.665%. More generally, if η is in the interval [0.5,2.0] then the Ramsey equation suggests that ρ is in the interval [0.2%,2.4%]. By comparison, Nordhaus and Sztorc (2013) use η = 1.45 and ρ = 1.5% for DICE and Anthoff et al. (2009) use η = 1.47 and ρ = 1.07% for FUND. One possible explanation ρ is that ρ is due to the probability of a consumer dying in a given year. As a comparison, the average global life expectancy is 71 years for 2013; the inverse of this is 1.41% per year.
P) Which Moral Judgements Should be Made?
Ultimately, determining the SWF depends on moral judgements. In particular, one likely needs to choose η and ρ. I leave it to the reader to decide, based upon the information in this post, what η and ρ should be, although I would advise against being risk averse about the values of η and ρ as human risk aversion itself depends on η. However, η outside of [0.5,2.0] is inconsistent with empirical observations and the Ramsey equation suggests that ρ lies in the interval [0.2%,2.4%]. Given the uncertainty of these two moral parameters, what researchers could do is perform expected social welfare maximization for different values of η and ρ and leave it up to policy makers to choose the appropriate moral judgements.
References [link]
This essay is from a longer essay on this topic [Expected Social Welfare Maximization-2]
JC note: As with all guest posts, please keep your comments relevant and civil.
Filed under: Policy
