The merger of populations, the incidence of marriages, and aggregate unhappiness

Abstract

Let a society’s unhappiness be measured by the aggregate of the levels of relative deprivation of its members. When two societies of equal size, F and M, merge, unhappiness in the merged society is shown to be higher than the sum of the levels of unhappiness in the constituent societies when apart; merger alone increases unhappiness. But when societies F and M merge and marriages are formed such that the number of households in the merged society is equal to the number of individuals in one of the constituent societies, unhappiness in the merged society is shown to be lower than the aggregate unhappiness in the two constituent societies when apart. This result obtains regardless of which individuals from one society form households with which individuals from the other, and even when the marriages have not (or not yet) led to income gains to the married couples from increased efficiency, scale economies, and the like. While there are various psychological reasons for people to become happier when they get married as opposed to staying single, the very formation of households reduces social distress even before any other happiness-generating factors kick in.

Appendix

The results derived in the body of the paper are not dented when alternative measures of relative deprivation are employed. We consider two such measures: the aggregate of the excesses of incomes, and the distance from the highest income. We attend to these two measures in turn.

In population F, let the relative deprivation of an individual with income f _i , i = 1,2,...,n, be defined as

$$ \textit{RD}^\prime \left( {f_i ,{\textbf{\emph{f}}}} \right)\equiv \sum\limits_{k=i+1}^n \left( \vphantom{\boldsymbol{f}}{f_k -f_i } \right). $$

Then, the total relative deprivation of population F, as the sum of the excesses of incomes, is

$$ \textit{TRD}'\left( \boldsymbol{f} \right)\equiv \sum\limits_{i=1}^{n-1} \textit{RD}'\left( {f_i ,\boldsymbol{f}} \right)=\sum\limits_{i=1}^{n-1} \sum\limits_{k=i+1}^n \left(\vphantom{\boldsymbol{f}} {f_k -f_i } \right). $$

A property that is analogous to Lemma 1 characterizes this measure:

$$ \textit{TRD}^\prime \left( \boldsymbol{f} \right)=\sum\limits_{i=1}^{n-1} \sum\limits_{k=i+1}^n \left(\vphantom{\boldsymbol{f}} {f_k -f_i } \right)=\frac{1}{2}\sum\limits_{i=1}^n \sum\limits_{k=1}^n \left|\vphantom{\boldsymbol{f}} {f_k -f_i } \right|. $$

When a second population, M, merges with population F, we derive the superadditivity result once again.

Claim A1

$\textit{TRD}^\prime \left( {\boldsymbol{f}\cup \boldsymbol{m}} \right)\ge \textit{TRD}^\prime \left( \boldsymbol{f} \right)+\textit{TRD}^\prime \left( \boldsymbol{m} \right).$

Proof

We have that

$$ \textit{TRD}^\prime \left( {\boldsymbol{f}\cup \boldsymbol{m}} \right)=\frac{1}{2}\left[ {\sum\limits_{i=1}^n {\sum\limits_{j=1}^n {\left| {f_j -f_i } \right|} } +\sum\limits_{k=1}^n {\sum\limits_{l=1}^n {\left| {m_l -m_k } \right|} } } \right]+\sum\limits_{i=1}^n {\sum\limits_{k=1}^n {\left| {f_i -m_k } \right|} } . $$

Noting that $\frac{1}{2}\!\sum\limits_{i=1}^n \!{\sum\limits_{j=1}^n\! {\left| {f_j -f_i } \right|} } =\textit{TRD}^\prime \!\left( \boldsymbol{f} \right)$, that $\frac{1}{2}\sum\limits_{k=1}^n {\sum\limits_{l=1}^n {\left| {m_l -m_k } \right|} } =\textit{TRD}^\prime \left( \boldsymbol{m} \right)$, and that $\sum\limits_{i=1}^n {\sum\limits_{k=1}^n {\left|\vphantom{\boldsymbol{f}} {f_i -m_k } \right|} } \ge 0$, completes the proof. □

For the merged population H, introduced in the same manner as in Case 2 in the body of the paper, a claim akin to Claim 2 is now stated and proved.

Claim A2

$\textit{TRD}^\prime \left( \boldsymbol{h} \right)\le \textit{TRD}^\prime \left( \boldsymbol{f} \right)+\textit{TRD}^\prime \left( \boldsymbol{m} \right).$

Proof

Since TRD ^′ differs from TRD only by a scaling factor (the size of the population), TRD ^′ too is convex and of homogeneity of degree 1. Therefore, the proof of Claim A2 tracks the same steps as those undertaken in proving Claim 2. □

We next consider measuring relative deprivation as the distance from the highest income.

In population F, let the relative deprivation of an individual with income f _i , i = 1,2,...,n, be defined as

$$ \emph{RD}^{\prime\prime} \left( {f_i ,{\textbf{\emph{f}}}} \right)\equiv f^\ast -f_i , $$

where f* = max{f ₁, ...,f _n }. The total relative deprivation of population F, measured as the aggregate of the distances from the highest income, is

$$ \textit{TRD}^{\prime\prime} \left( f \right)\equiv \mathop \sum\limits_{i=1}^n \left( {f^\ast -f_i } \right). $$

When a second population, M, the total relative deprivation of which is measured in the same way as that of population F, merges with population F, we obtain the superadditivity result once again. The reasoning is straightforward. Unless the two populations have each the same highest income, a merger results in exposure of members of one of the populations to a higher post-merger highest income, while members of the other population continue to be exposed to the same highest income as prior to the merger. Thus, the aggregated distances from the highest income in the merged population will not be less than the sum of the corresponding distances in the constituent populations when apart. For the sake of completeness, we state this result in the following claim.

Claim A3

$\textit{TRD}^{\prime\prime} \left( {\boldsymbol{f}\cup \boldsymbol{m}} \right)\ge \textit{TRD}^{\prime\prime} \left( \boldsymbol{f} \right)+\textit{TRD}^{\prime\prime} \left( \boldsymbol{m} \right).$

Proof

The proof is contained in the preceding discussion.□

For the merged population H, introduced in the same manner as in Case 2 in the body of the paper, a claim akin to Claim 2 is now stated and proved.

Claim A4

$\textit{TRD}^{\prime\prime} \left( \boldsymbol{h} \right)\le \textit{TRD}^{\prime\prime} \left( \boldsymbol{f} \right)+\textit{TRD}^{\prime\prime} \left( \boldsymbol{m} \right).$

Proof

For the highest income in population H, h ^*, we have

$$ \begin{array}{rll} h^\ast&=&\max \left\{{f_1 +m_1^\prime,...,f_n +m_n^\prime }\right\}\le \max \left\{ \vphantom{m_1^\prime}{f_1,...,f_n } \right\}\\&&+\max \left\{{m_1^\prime,...,m_n^\prime}\right\}=f^\ast +m^\ast , \end{array} $$

where $m_1^\prime,...,m_n^\prime$ are the elements of the permutated $\boldsymbol{m}^\prime$ vector introduced in Case 2 of Section 2, and where $m^\ast =\max \left\{ {m_1^\prime,...,m_n^\prime } \right\}$, as the vectors $\boldsymbol{m}^\prime$ and $\boldsymbol{m}$ differ only in the order of their elements. Then,

$$ \begin{array}{rll} \textit{TRD}^{\prime\prime} \left(\boldsymbol{h}\right)&=&\mathop \sum\limits_{i=1}^n \left({h^\ast -h_i }\right)\mathop \sum\limits_{i=1}^n \left[{h^\ast -\left({f_i +m_i^\prime} \right)}\right]\le \mathop \sum\limits_{i=1}^n \left[ {\left( {f^\ast +m^\ast } \right)-\left( {f_i +m_i^\prime}\right)} \right] \\ &=&\mathop \sum\limits_{i=1}^n \left[ {\left( {f^\ast -f_i } \right)+\left( {m^\ast -m_i^\prime } \right)} \right]=\mathop \sum\limits_{i=1}^n \left( {f^\ast -f_i } \right)+\mathop \sum\limits_{i=1}^n \left( {m^\ast -m_i^\prime } \right)\\&=&\textit{TRD}^{\prime\prime} \left( \boldsymbol{f} \right)+\textit{TRD}^{\prime\prime} \left( {\boldsymbol{m}^\prime } \right). \end{array} $$

Noting that, obviously, TRD ^″($\boldsymbol{m}^\prime)$ = TRD ^″ ($\boldsymbol{m}$) completes the proof. □