Will a government find it financially easier to neutralize a looming protest if more groups are involved?

Abstract

We study a policy response to an increase in post-merger social stress. If a merger of groups of people is viewed as a revision of their social space, then the merger alters people’s comparators and increases social stress: the social stress of a merged population is greater than the sum of the levels of social stress of the constituent populations when apart. We use social stress as a proxy measure for looming social protest. As a response to the post-merger increase in social stress, we consider a policy aimed at reversing the negative effect of the merger by bringing the social stress of the merged population back to the sum of the pre-merger levels of social stress of the constituent populations when apart. We present, in the form of an algorithm, a cost-effective policy response which is publicly financed and does not reduce the incomes of the members of the merged population. We then compare the financial cost of implementing such a policy when the merger involves more or fewer groups. We show that the cost may fall as the number of merging groups rises.

Appendix: The rationale and construction of the measure of social stress

Several recent insightful studies in social psychology (for example, Callan et al. 2011; Smith et al. 2012) document how sensing relative deprivation impacts negatively on personal wellbeing, but these studies do not provide a calibrating procedure; a sign is not a magnitude. For the purpose of constructing a measure, a natural starting point is the work of Runciman (1966), who argued that an individual has an unpleasant sense of being relatively deprived when he lacks a desired good and perceives that others with whom he naturally compares himself possess that good. Runciman (1966, p. 19) writes as follows: “The more people a man sees promoted when he is not promoted himself, the more people he may compare himself with in a situation where the comparison will make him feel deprived,” thus implying that the deprivation from not having, say, income y is an increasing function of the fraction of people in the individual’s reference group who have y. To aid intuition and for the sake of concreteness, we resort to income-based comparisons, namely an individual feels relatively deprived when others in his comparison group earn more than he does. An implicit assumption here is that the earnings of others are publicly known. Alternatively, we can think of consumption, which could be more publicly visible than income, although these two variables can reasonably be assumed to be strongly positively correlated.

As an illustration of the relationship between the fraction of people possessing income y and the deprivation of an individual lacking y, consider a population (reference group) of six individuals with incomes $\{1,2,6,6,6,8\}$. Imagine a furniture store that in three distinct compartments sells chairs, armchairs, and sofas. An income of 2 allows you to buy a chair. To be able to buy any armchair, you need an income that is a little bit higher than 2. To buy any sofa, you need an income that is a little bit higher than 6. Thus, when you go to the store and your income is 2, what are you “deprived of?” The answer is “of armchairs,” and “of sofas.” Mathematically, this deprivation can be represented by $P(Y > 2)(6 - 2) + P(Y > 6)(8 - 6),$ where P(Y > y _i ) stands for the fraction of those in the population whose income is higher than y _i , for $y_i=2,6$. The reason for this representation is that when you have an income of 2, you cannot afford anything in the compartment that sells armchairs, and you cannot afford anything in the compartment that sells sofas. Because not all those who are to your right in the ascendingly ordered income distribution can afford to buy a sofa, yet they can all afford to buy armchairs, a breakdown into the two (weighted) terms $P(Y > 2)(6 - 2)$ and $P(Y > 6)(8 - 6)$ is needed. This way, we get to the very essence of the measure of RD used in this paper: we take into account the fraction of the comparison group (population) who possess some good which you do not, and we weigh this fraction by the “excess value” of that good. Because income enables an individual to afford the consumption of certain goods, we refer to comparisons based on income.

Formally, let $y = (y_{1} , \ldots ,y_{m} )$ be the vector of incomes in a population of size n with relative incidences $p(y) = (p(y_{1} ), \ldots ,p(y_{m} )),$ where $m \le n$ is the number of distinct income levels in y. The RD of an individual earning y _i is defined as the weighted sum of the excesses of incomes higher than y _i such that each excess is weighted by its relative incidence, namely

$$ RD_{i} (y) \equiv \sum\limits_{{y_{j} > y_{i} }} {p(y_{j} )(y_{j} - y_{i} ).} $$

(8)

In the example given above with income distribution $\{1,2,6,6,6,8\}$, we have that the vector of incomes is $y=(1,2,6,8)$, and that the corresponding relative incidences are $p(y) = \left( {\frac{1}{6},\frac{1}{6},\frac{3}{6},\frac{1}{6}} \right).$ Therefore, the RD of the individual earning 2 is $\sum\nolimits_{{y_{j} > y_{i} }} {p(y_{j} )(y_{j} - y_{i} ) = p(6)(6 - 2) + p(8)(8 - 2) = \frac{3}{6} \cdot 4 + \frac{1}{6} \cdot 6 = 3.}$ By similar calculations, we have that the RD of the individual earning 1 is higher and is equal to $3\frac{5}{6},$ and that the RD of each of the individuals earning 6 is lower and is equal to $\frac{1}{3}.$

We expand the vector y to include incomes with their possible respective repetitions, that is, we include each y _i as many times as its incidence dictates, and we assume that the incomes are ascendingly ordered, that is, $y = (y_{1} , \ldots ,y_{n} )$ such that $y_{1} \le y_{2} \le \ldots \le y_{n} .$ In this case, the relative incidence of each y _i , p(y _i ), is $\frac{1}{n}$, and, (8) becomes exactly as given in (1):

$$ RD_{i} (y) \equiv \left\{{\begin{array}{*{20}l} {\frac{1}{n}\sum\limits_{j = i + 1}^{n} {(y_{j} - y_{i})}} \hfill &\quad {\text{for}\,\,i = 1, \ldots,n - 1,} \hfill \\ 0 \hfill &\quad {\text{for}\,\,i = n.} \hfill \\ \end{array}} \right. $$

Looking at incomes in a large population, we can model the distribution of incomes as a random variable Y over the domain $[0,\infty )$ with a cumulative distribution function F. We can then express the RD of an individual earning y _i as

$$ RD_{i} (y) = [1 - F(y_{i} )]E(Y - y_{i} |Y > y_{i} ). $$

(9)

The formula in (9) is quite revealing because it casts RD in a richer light than the ordinal measure of rank, which have been studied intensively in sociology and beyond. The formula informs us that when the income of individual A is, say, 10, and that of individual B is, say, 16, the RD of individual A is higher than when the income of individual B is 15, even though, in both cases, the rank of individual A in the income hierarchy is second. The formula also informs us that more RD is sensed by an individual whose income is 10 when the income of another is 14 (RD is 2) than when the income of each of four others is 11 $\left(RD\,\hbox{is}\,\frac{4}{5}\right),$ even though the excess income in both cases is 4. This property aligns nicely with intuition: it is more painful (more stress is experienced) when the income of half of the population in question is 40 percent higher, than when the income of $\frac{4}{5}$ of the population is 10 percent higher. In addition, the formula in (9) reveals that even though RD is sensed by looking to the right of the income distribution, it is impacted by events taking place on the left of the income distribution. For example, an exit from the population of a low-income individual increases the RD of higher-income individuals (other than the richest) because the weight that the latter attach to the difference between the incomes of individuals “richer” than themselves and their own income rises. The often cited example from a three tenors concert organized for Wembley Stadium in which Pavarotti reputedly did not care how much he was paid so long as it was one pound more than Domingo was paid does not invalidate the logic behind our measure because, in light of the measure, Pavarotti’s payment request can be interpreted as being aimed at ensuring that no RD will be experienced when he looks to the right in the pay distribution.

Similar reasoning can explain the demand for positional goods (Hirsch 1976). The standard explanation is that this demand arises from the unique value of positional goods in elevating the social standing of their owners (“These goods [are] sought after because they compare favorably with others in their class.” Frank 1985, p. 7). The distaste for relative deprivation offers another explanation: by acquiring a positional good, an individual shields himself from being leapfrogged by others which, if that were to happen, would expose him to RD. Seen this way, a positional good is a form of insurance against experiencing RD.

There can, of course, be other, quite intuitive ways of gauging RD, and in some contexts and for some applications, a measure simpler than (1) can be adequate. Suppose that an individual’s income is I, and the average income of the individual’s reference group is R. We can then define RD as a function of I and R, namely

$$ RD(I,R) = \left\{{\begin{array}{*{20}l} {R - I} \hfill &\quad {{\text{if}}\,\,I < R} \hfill \\ 0 \hfill &\quad {{\text{if}}\,\,I \ge R.} \hfill \\ \end{array}} \right. $$

(10)

This representation captures the intuitive requirements

$$ \frac{\partial RD(I,R)}{\partial I} < 0,\quad \frac{\partial RD(I,R)}{\partial R} > 0\quad {\text{for}}\,\,R > I, $$

namely that, holding other things the same, for a relatively deprived individual (that is, for an individual whose income is lower than the average income of the individual’s reference group), RD decreases with his own income, and increases with the average income of his reference group. Examples of the use of (10) are in Fan and Stark (2007), Stark and Fan (2011), and Stark and Jakubek (2013). However, the advantage of using (1) is that it is based on an axiomatic foundation which is, essentially, a translation of Runciman’s (1966) work, let alone that it is nice in economics to draw on a foundation laid out in social psychology.