Abstract
Ever since the publication of Harry Frankfurt’s “Equality as a Moral Ideal” (Ethics 98(1):21–43, 1987), the doctrine of sufficiency has attracted great attention among both ethical theorists and political philosophers. The doctrine of sufficiency (or sufficientarianism) consists of two main theses: the positive thesis states that it is morally important for people to have enough; and the negative thesis states that once everybody has enough, relative inequality has absolutely no moral importance. Many political philosophers have presented different versions of sufficientarianism that retain the general spirit of what Frankfurt had proposed in his seminal work. However, all of these different versions of sufficientarianism suffer from two critical problems: (a) they fail to give right answers to lifeboat situations, and (b) they fail to provide continuous ethical judgments. In this paper, I show a version of utilitarianism that solves these problems while retaining the major attractions of sufficientarianism. I call it “prospect utilitarianism.” In addition, I show that prospect utilitarianism can avoid standard objections to utilitarianism and has aspects that can appeal to both prioritarians and egalitarians as well.
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Notes
Throughout the paper, I will use the two expressions, “the doctrine of sufficiency” and “sufficientarianism”, interchangeably.
Actually, one may find that this is not entirely obvious under close examination. See Chung (2016), where I argue that Frankfurt's original doctrine of sufficiency is unable to solve the problems that Frankfurt had originally intended it to solve.
For instance, Frankfut writes, “Another response to scarcity is to distribute the available resources in such a way that as many people as possible have enough or, in other words, to maximize the incidence of sufficiency. This alternative is especially compelling when the amount of a scarce resource that constitutes enough coincides with the amount that is indispensable for avoiding some catastrophic harm—as in the example [lifeboat situation] just considered, where falling below the threshold of enough food or enough medicine means death” (Frankfurt 1987: 31).
In order to give right answers to lifeboat scenarios, the distributional principle must specify how we should distribute the available resources among those below the critical sufficiency threshold. Neither Crisp nor Huseby is clear on this issue: Crisp merely claims that “Absolute priority is to be given to benefits to those below the threshold at which compassion enters” (Crisp Roger 2003: 758), while Huseby merely claims that “strong priority should be given to those below the minimal sufficiency threshold.” (Huseby 2010: 184–185) Remember that in order to give the right answer to our six patient example, the distributional principle must give zero priority to the worst-off patient; this directly conflicts with giving either absolute or strong priority to the worst-off patient which Crisp’s and Huseby’s versions of sufficientarianism seem to respectively suggest.
See Gaertner (2009, A Primer in Social Choice Theory, p. 124) for different informational requirements of individual utility functions for different theoretical purposes.
See Rawls (1971/1999: Sections 5, 29).
Proof First, note that any distribution that wastes any positive amount of social wealth cannot be a solution to this problem, as we may simply distribute the wasted wealth to any of the two individuals and increase the value of the objective function. So, at a solution, the constraint \(x_{1} + x_{2} \le 10\) is binding (i.e. \(x_{1} + x_{2} = 10\)), and, by substituting \(10 - x_{1}\) for \(x_{2}\), the objective function becomes: \(\ln x_{1} + \ln \left( {10 - x_{1} } \right) = \ln x_{1} \left( {10 - x_{1} } \right)\). So, our constrained maximization problem becomes:
$$\begin{aligned} & \mathop {\hbox{max} }\limits_{{x_{1} \in {\mathbb{R}}}} \ln x_{1} \left( {10 - x_{1} } \right) \\ & {\text{subject to }}0 \le x_{1} \le 10 \\ \end{aligned}$$Since the natural log function is a strictly increasing function, whatever value of \({\text{x}}_{1}\) that maximizes x 1(10−x 1) will also maximize ln x 1(10−x 1). Since x 1(10−x 1) is a strictly concave function, the first-order condition is sufficient for it to obtain a maximum. By taking the derivative of x 1(10−x 1) with respect to x 1, we get:
$$\begin{aligned} - 2{\text{x}}_{1} + 10 = 0 \hfill \\ \Rightarrow {\text{x}}_{1} = 5 \hfill \\ \end{aligned}$$We can see that x 1 = 5 is within the range 0 ≤ x 1 ≤ 10. Therefore, it is a solution for individual 1. Since x 2 = 10–x 1 = 10–5 = 5, we have \(\left( {5, \,5} \right)\) as the unique solution for problem. □
A function \(F\) is concave if, for all \(\alpha \in \left[ {0,1} \right],\) \(F\left( {\alpha x + \left( {1 - \alpha } \right)y} \right) \ge \alpha F\left( x \right) + \left( {1 - \alpha } \right)F\left( y \right)\); convex if \(F\left( {\alpha x + \left( {1 - \alpha } \right)y} \right) \le \alpha F\left( x \right) + \left( {1 - \alpha } \right)F\left( y \right)\); and linear if \(F\left( {\alpha x + \left( {1 - \alpha } \right)y} \right) = \alpha F\left( x \right) + \left( {1 - \alpha } \right)F\left( y \right)\).
See Kahneman and Tversky (1979).
Proof Let \(\left( {u,\left( {\underline{x}_{i} } \right)_{i = 1}^{n} ,E,C} \right)\) be a distribution problem such that \(e_{i} <\, \underline{x}_{i}\) and \(e_{j} > \underline{x}_{j}\) and \(C > 0\); that is, individual \(i\)'s endowment is below his/her sufficiency threshold and individual \(j\)’s endowment is above his/her sufficiency threshold. Between the two individuals, we need to show that utilitarianism will start to distribute the remaining material resources to \(i\) first. Note that, between any two individuals, utilitarianism will always prioritize and give additional material resources to the individual who will generate a greater welfare gain from the additional material resources. As individual utility functions are strictly increasing in material resources, how much welfare each individual generates by having more additional material resources can be measured by the steepness of the slopes of the individuals’ utility functions. By our assumptions on our reference utility function \(u\) as well as our characterization of individual utility functions, the slope of any individual’s utility function below the individual’s sufficiency threshold will be steeper than the slope of any individual’s utility function above the individual’s sufficiency threshold. Since \(e_{i} < \underline{x}_{i}\) and \(e_{j} > \underline{x}_{j}\), the slope of \(u_{i}\) is greater than the slope of \(u_{j}\) at the initial endowment. Hence, utilitarianism will give individual \(i\) priority in distributing any additional amount of resources. \(\square\)
Proof Let \(\left( {u,\left( {\underline{x}_{i} } \right)_{i = 1}^{n} ,E,C} \right)\) be a distribution problem such that \(e_{i} < \underline{x}_{i}\) and \(e_{j} < \underline{x}_{j}\) and \(\left| {e_{i} - \underline{x}_{i} } \right| < \left| {e_{j} - \underline{x}_{j} } \right|\). In other words, the endowments of both individuals \(i\) and \(j\) are below his/her respective sufficiency threshold, but individual \(i\) is closer to his/her sufficiency threshold than individual j is to his/her sufficiency threshold at their respective endowments. We need to show that utilitarianism will prioritize individual \(i\) and start to distribute the remaining material resources to him/her rather than to individual j. Again, between any two individuals, utilitarianism will always prioritize and give additional material resources to the individual who will generate a greater welfare gain from the additional material resources. And, again, this implies that, between any two individuals, utilitarianism will start to distribute the additional material resources to the individual whose utility function has the steeper slope. Note that based on our assumptions, each individual’s utility function is strictly convex (i.e. its slope is increasing) below each individual’s sufficiency threshold. Hence, based on our characterization of individual utility functions along with the fact that \(\left| {e_{i} - \underline{x}_{i} } \right| < \left| {e_{j} - \underline{x}_{j} } \right|\), we have \(u^{\prime }_{j} \left( {e_{j} } \right) = u^{\prime } \left( {e_{j} - \underline{x}_{j} } \right) < u^{\prime } \left( {e_{i} - \underline{x}_{i} } \right) = u^{\prime }_{i} \left( {e_{i} } \right)\), implying \(u^{\prime }_{j} \left( {e_{j} } \right) < u^{\prime }_{i} \left( {e_{i} } \right)\). Hence, utilitarianism will give individual \(i\) priority in distributing any additional amount of resources. \(\square\)
See Roemer (2004: 273).
To illustrate, consider an example in which there are a total of 5 persons: 1 person who is in a critical condition; 3 persons who are in serious but less critical conditions; and 1 person who is perfectly healthy. Suppose that everybody except for the 1 healthy person will die if not treated immediately. Suppose that there is a limited supply of some super drug such that: (a) giving all of the available supply to the person suffering a critical condition will be enough to save him/her; (b) dividing the available supply to 3 portions will be enough to save the 3 persons in less critical conditions; (c) giving any amount to the perfectly healthy person will improve his/her complexion. By Proposition 1, the 4 patients below the critical sufficiency threshold for survival have priority over the 1 healthy person, who is above the critical sufficiency threshold for survival. Hence, no amount of the super drug will be distributed to the single healthy person before everybody is above the critical sufficiency threshold for survival. By Proposition 2, among the 4 persons below the critical sufficiency threshold, 3 of them whose conditions are relatively less severe have priority over the worst-off patient. Hence, the available medicine will be distributed to these 3 people, which is sufficient for them to meet their critical sufficiency threshold for survival. The lifeboat scenario is solved.
Proof Let \(\left({u,\left({\underline{x}_{i}} \right)_{i = 1}^{n},E,C} \right)\) be a distribution problem. Let \(X\) be the set of feasible distributions. So, \(X = \{\varvec{x} = \left({x_{1}, \ldots,x_{n}} \right) \in {\mathbb{R}}^{n}\) \(x_{1} \ge 0, \ldots,x_{n} \ge 0, x_{1} + \ldots + x_{n} \le C\}\). Note that X is non-empty, e.g. \(\left({0, \ldots,0} \right) \in X\). Pick any \(\varvec{x} = \left({x_{1}, \ldots,x_{n}} \right) \in X\). Define \(U\left(\varvec{x} \right) \equiv \mathop \sum \nolimits_{i = 1}^{n} u_{i} \left({e_{i} + x_{i}} \right)\). Let \(R\left(\varvec{x} \right)\) be the set of distributions that are weakly ethically preferred to \(x\) by prospect utilitarianism; that is, \(R\left(\varvec{x} \right) = \left\{{\varvec{y} = \left({y_{1}, \ldots,y_{n}} \right) \in X|U\left(\varvec{y} \right) = \mathop \sum \nolimits_{i = 1}^{n} u_{i} \left({e_{i} + y_{i}} \right) \ge \mathop \sum \nolimits_{i = 1}^{n} u_{i} \left({e_{i} + x_{i}} \right) = U\left(\varvec{x} \right)} \right\}\). We want to show that \(R\left(\varvec{x} \right)\) is closed; that is, we want to show that, for any sequence \(\left\{{\varvec{z}^{\varvec{n}}} \right\}\) in \(R\left(\varvec{x} \right)\) that converges to \(z = \left( {z_{1} , \ldots , z_{n} } \right)\), \(\varvec{z}\in R\left(\varvec{x}\right)\). Pick any sequence \(\left\{ {\varvec{z}^{\varvec{n}} } \right\}\) in \(R\left(\varvec{x} \right)\) that converges to Z, and, suppose, for a proof by contradiction, \(\varvec{z} \notin R\left( \varvec{x} \right)\). Then, \(U\left ( {\varvec{z}^{\varvec{n}} } \right) \ge U\left( \varvec{x} \right), \forall n \in {\mathbb{N}}\), but \(U\left( \varvec{z} \right) < U\left( \varvec{x} \right)\). Combining the two inequalities we have: \(U\left( {\varvec{z}^{\varvec{n}} } \right) \ge U\left( \varvec{x} \right) > U\left( \varvec{z} \right)\) \(,\forall n \in {\mathbb{N}}\). Let \(\in = \left| {U\left( \varvec{x} \right) - U\left( \varvec{z} \right)} \right|\) inequality, we have, \(\forall n \in {\mathbb{N}}\):
$$\begin{aligned} \varepsilon & = \left| {U\left( x \right) - U\left( z \right)} \right| \\ & \le \left| {U\left( {z^{n} } \right) - U\left( z \right)} \right| \\ & = \left| {\mathop \sum \limits_{i = 1}^{n} u_{i} \left( {e_{i} + z_{i}^{n} } \right) - \mathop \sum \limits_{i = 1}^{n} u_{i} \left( {e_{i} + z_{i} } \right)} \right| \\ & = \left| {u_{1} \left( {e_{1} + z_{1}^{n} } \right) + \cdots + u_{n} \left( {e_{n} + z_{n}^{n} } \right) - u_{1} \left( {e_{1} + z_{1} } \right) - \cdots - u_{n} \left( {e_{n} + z_{n} } \right)} \right| \\ & = \left| {u_{1} \left( {e_{1} + z_{1}^{n} } \right) - u_{1} \left( {e_{1} + z_{1} } \right) + \cdots + u_{n} \left( {e_{n} + z_{n}^{n} } \right) - u_{n} \left( {e_{n} + z_{n} } \right)} \right| \\ & \le \left| {u_{1} \left( {e_{1} + z_{1}^{n} } \right) - u_{1} \left( {e_{1} + z_{1} } \right)\left| { + \cdots + } \right|u_{n} \left( {e_{n} + z_{n}^{n} } \right) - u_{n} \left( {e_{n} + z_{n} } \right)} \right| \cdots (*) \\ \end{aligned}$$Since each u i is continuous, \(\forall n \in {\mathbb{N}}\), for each \(i \in N\), there exists a \(\delta_{i}\) such that, for all \(z_{i} \in \left({e_{i} + z_{i}^{n} - \delta_{i}, e_{i} + z_{i}^{n} + \delta_{i}} \right)\),\(\left| {u_{i} \left( {e_{i} + z_{i}^{n} } \right)} - u_{1} \left( {e_{i} + z_{i} } \right)\right| < \frac{ \in }{n}\). Since, \(\varvec{z}^{\varvec{n}} = \left( {z_{1}^{n} , \ldots ,z_{n}^{n} } \right)\) converges to \(\varvec{z} = \left( {z_{1} , \ldots ,z_{n} } \right)\), each \(z_{i}^{n}\) converges to z i . Since each \(z_{i}^{n}\) converges to z i , for each \(i \in N\), there exists a \(N_{i} \in {\mathbb{N}}\) such that whenever \(n \ge N_{i}\), \(\left| {z_{i} - z_{i}^{n} } \right| < \delta_{i}\). Let \(M = \hbox{max} \left\{ {N_{1} , \ldots ,N_{n} } \right\}\). Then, whenever \(n \ge M\), we have \(\forall i \in N\), \(\left| {z_{i}^{n} - z_{i} } \right| < \delta_{i}\), which implies:
$$\begin{array}{c}\left| {u_{1} \left( {e_{1} + z_{1}^{n} } \right) - u_{1} \left( {e_{1} + z_{1} } \right)| + \ldots + |u_{n} \left( {e_{n} + z_{n}^{n} } \right) - u_{n} \left( {e_{n} + z_{n} } \right)} \right| \\< \underbrace {{\frac{\varepsilon }{n} + \cdots + \frac{\varepsilon }{n}}}_{{n\,times}} = \varepsilon \end{array}$$which contradicts the inequality in \(\left( * \right)\). Therefore, \(\varvec{z} \in R\left( \varvec{x} \right)\), and, hence, \(R\left( \varvec{x} \right)\) is closed. \(\square\)
Proof Let \(\left( {u,\left( {\underline{x}_{i} } \right)_{i = 1}^{n} ,E,C} \right)\) be a distribution problem where \(\underline{x}_{i} < e_{i}\) for all \(i\). If \(\left| {e_{1} - \underline{x}_{1} } \right| = \cdots = \left| {e_{n} - \underline{x}_{n} } \right|\), then, based on our assumption on individual utility functions, nobody is worse-off than anybody else in terms of welfare, and, hence, the proposition is satisfied vacuously. So, suppose there exist \(i, j \in N\) such that \(\left| {e_{i} - \underline{x}_{i} } \right| \ne \left| {e_{j} - \underline{x}_{j} } \right|\). Without loss of generality, suppose \(\left| {e_{i} - \underline{x}_{i} } \right| < \left| {e_{j} - \underline{x}_{j} } \right|\). Then, \(u_{i} \left( {e_{i} } \right) = u\left( {e_{i} - \underline{x}_{i} } \right) < u\left( {e_{j} - \underline{x}_{j} } \right) = u_{j} \left( {e_{j} } \right)\), which imply that individual \(i\) is worse-off in terms of welfare than individual \(j\) at the initial endowment. We need to show that, between individuals \(i\) and \(j\), prospect utilitarianism will prioritize individual \(i\) and give the remaining material resources to him/her first. Since \(u_{i} \left( {e_{i} } \right) = u\left( {e_{i} - \underline{x}_{i} } \right)\) and \(u_{j} \left( {e_{j} } \right) = u\left( {e_{j} - \underline{x}_{j} } \right)\) and \(0 < e_{i} - \underline{x}_{i} < e_{j} - \underline{x}_{j}\), by strict concavity of \(u\), we have \(u^{\prime } \left( {e_{i} - \underline{x}_{i} } \right) > u^{\prime } \left( {e_{j} - \underline{x}_{j} } \right)\), which implies \(u^{\prime }_{i} \left( {e_{i} } \right) > u^{\prime }_{j} \left( {e_{j} } \right)\). That is, at their respective endowments, individual \(i\)’s welfare increases faster than that of individual \(j\). Hence, in order to maximize the total sum of individual welfare, prospect utilitarianism will distribute the remaining material resources to individual \(i\) first. \(\square\)
Let me clarify this. Suppose there are two individuals: individual 1 and individual 2. Suppose individual 1’s critical sufficiency threshold is 10 while individuals 2’s critical sufficiency threshold is 100. Suppose individual 1 has 20 units of resources while individual 2 has 100 units of resources. Suppose there is an additional unit of resource to distribute. In such case, prospect utilitarianism will distribute the additional resource to individual 2 rather than individual 1 despite the fact that individual 2 currently has much more material resources than individual 1. This is because although individual 2 has much more material resources than individual 1, he/she is still the worse-off person in terms of their respective welfare levels. (Individual 1 is 10 units over his/her critical sufficiency threshold, while individual 2 is barely at his/her critical sufficiency threshold.) This is generally in accordance with Michael Otsuka’s claim that “egalitarian justice calls for the equalization of opportunity for welfare rather than the equalization of anything other than, or in addition to, that” (Otsuka 2003: 25) To illustrate his point, Otsuka gives an example of dividing an unowned large blanket between two individuals, one of whom is twice as large as the other. Otsuka argues that “it would be unfair for the smaller person to acquire half of the blanket rather than that lesser portion which would leave him as comfortable as the other.” (Otsuka 2003: 26) Here, the larger person would correspond to individual 2 whose critical sufficiency threshold requires 100 units of material resources, 10 times of that of individual 1.
Benbaji (2005: 312).
For instance, see Roger Crisp’s “Beverly Hills Case” (Crisp 2003: 755).
Proof Let \(\left( {u,\left( {\underline{x}_{i} } \right)_{i = 1}^{n} ,E,C} \right)\) be a distribution problem such that \(\underline{x}_{i} < e_{i}\) for all \(i\), and assume that it is not the case that \(\left| {e_{1} - \underline{x}_{1} } \right| = \ldots = \left| {e_{n} - \underline{x}_{n} } \right|\). Then, there exist \(i, j \in N\) such that \(\left| {e_{i} - \underline{x}_{i} } \right| \ne \left| {e_{j} - \underline{x}_{j} } \right|\). Without loss of generality, suppose \(\left| {e_{i} - \underline{x}_{i} } \right| < \left| {e_{j} - \underline{x}_{j} } \right|\). Define \(\Delta \equiv u_{j} \left( {e_{j} } \right) - u_{i} \left( {e_{i} } \right) = u\left( {e_{j} - \underline{x}_{j} } \right) - u\left( {e_{i} - \underline{x}_{i} } \right)\). That is, \(\Delta\) is the welfare difference between individuals \(i\) and \(j\) at endowment \(E\). Let \(E^{\prime } = \left( {e_{1}^{\prime } , \ldots ,e_{n}^{\prime } } \right)\) where \(e_{i}^{\prime } = e_{i} + K (K > 0)\) for all \(i\). That is, \(E^{\prime }\) is another endowment in which every individual has \(K\) more material resources than what he/she has in \(E\). Define \(\Delta^{\prime } \equiv u_{j} \left( {e_{j}^{\prime } } \right) - u_{i} \left( {e_{i}^{\prime } } \right) = u\left( {e_{j}^{\prime } - \underline{x}_{j} } \right) - u\left( {e_{i}^{\prime } - \underline{x}_{i} } \right) = u\left( {e_{j} + K - \underline{x}_{j} } \right) - u\left( {e_{i} + K - \underline{x}_{i} } \right)\). We need to show that \(\Delta > \Delta^{\prime }\). That is, we need to show that the welfare differences between individuals \(i\) and \(j\) becomes smaller when we increase both of their material resource levels by a constant \(K > 0\).
For a proof by contradiction, suppose Δ ≤ Δ′. Then,
$$u\left( {e_{i} - \underline{x}_{j} } \right) - u\left( {e_{i} - \underline{x}_{i} } \right) \le u\left( {e_{j} + K - \underline{x}_{j} } \right) - u\left( {e_{i} + K - \underline{x}_{i} } \right) \cdots (*).$$By strict concavity of u (above 0), we have:
$$\begin{aligned} &u^{\prime } \left( {e_{i} - \underline{x}_{i} } \right) > \frac{{u\left( {e_{j} - \underline{x}_{j} } \right) - u\left( {e_{i} - \underline{x}_{i} } \right)}}{{u\left( {e_{j} - \underline{x}_{j} } \right) - u\left( {e_{i} - \underline{x}_{i} } \right)}} > u^{\prime } \left( {e_{j} - \underline{x}_{j} } \right) \hfill \\ &> u^{\prime } \left( {e_{i} + K - \underline{x}_{i} } \right) > \frac{{u\left( {e_{j} + K - \underline{x}_{j} } \right) - u\left( {e_{i} + K - \underline{x}_{i} } \right)}}{{u\left( {e_{j} + K - \underline{x}_{j} } \right) - u\left( {e_{i} + K - \underline{x}_{i} } \right)}} > u^{\prime } \left( {e_{j} + K - \underline{x}_{j} } \right) \hfill \\ \end{aligned}$$which implies
$$\begin{aligned}& \frac{{u\left( {e_{j} - \underline{x}_{j} } \right) - u\left( {e_{i} - \underline{x}_{i} } \right)}}{{u\left( {e_{j} - \underline{x}_{j} } \right) - u\left( {e_{i} - \underline{x}_{i} } \right)}} > \frac{{u\left( {e_{j} + K - \underline{x}_{j} } \right) - u\left( {e_{i} + K - \underline{x}_{i} } \right)}}{{u\left( {e_{j} + K - \underline{x}_{j} } \right) - u\left( {e_{i} + K - \underline{x}_{i} } \right)}} \hfill \\ &\quad \Rightarrow \frac{{u\left( {e_{j} - \underline{x}_{j} } \right) - u\left( {e_{i} - \underline{x}_{i} } \right)}}{{u\left( {e_{j} - \underline{x}_{j} } \right) - u\left( {e_{i} - \underline{x}_{i} } \right)}} > \frac{{u\left( {e_{j} + K - \underline{x}_{j} } \right) - u\left( {e_{i} + K - \underline{x}_{i} } \right)}}{{u\left( {e_{j} - \underline{x}_{j} } \right) - u\left( {e_{i} - \underline{x}_{i} } \right)}} \hfill \\ &\quad \Rightarrow u\left( {e_{j} - \underline{x}_{j} } \right) - u\left( {e_{i} - \underline{x}_{i} } \right) > u\left( {e_{j} + K - \underline{x}_{j} } \right) - u\left( {e_{i} + K - \underline{x}_{i} } \right) \hfill \\ \end{aligned}$$contradicting ( * ). Hence, \(\Delta > \Delta^{\prime }\), as desired. □
To illustrate with a simple example, consider the natural logarithmic utility function \(u\left( x \right) = \ln x\), which is strictly concave. Assuming that people share the same utility function, the welfare difference of somebody who has 5 units of resources and somebody who has 1 unit of resource is: \(u\left( 5 \right) - u\left( 1 \right) = \ln 5 - \ln 1 \approx 1.609\). The welfare difference of somebody who has 10005 units of resources and somebody who has 10001 units of resources is: \(u\left( {10005} \right) - u\left( {10001} \right) = \ln 10005 - \ln 10001 \approx 0.001\). In both cases, the difference in resource level was 4 units; however, the difference in welfare levels that 4 units of difference in resource levels generates starts to become smaller, and, in the end, negligible as people have more and more resources.
For instance, John Roemer has expressed his dissatisfaction with characterization (6). Namely, For all \(x < 0 < y\), \(u^{\prime } \left( x \right) > u^{\prime } \left( y \right)\). This entails that the derivative of u as x becomes arbitrarily large and negative is greater than the right-hand derivative of u at zero. The practical implication of this is that an additional unit of resource will always benefit a person suffering extreme penury more than a person who has barely met sustainability, which may seem implausible.
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Acknowledgments
I would like to thank (in alphabetical order) Jacob Barrett, Jerry Gaus, Adam Gjesdal, Brian Kogelmann, John Roemer, Jack Sanders, Alex Schaefer, Stephen Stich, and John Thrasher for helpful comments on earlier drafts of this paper.
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Chung, H. Prospect utilitarianism: A better alternative to sufficientarianism. Philos Stud 174, 1911–1933 (2017). https://doi.org/10.1007/s11098-016-0775-3
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DOI: https://doi.org/10.1007/s11098-016-0775-3