A bitter choice turned sweet: How acknowledging individuals’ concern at having a low relative income serves to align utilitarianism and egalitarianism

Abstract

When individuals’ utility is a convex combination of their income and their concern at having a low relative income (the weights attached to income and to the concern at having a low relative income sum up to one), the maximization of aggregate utility yields an equal income distribution. This alignment of utilitarianism and egalitarianism is obtained for any number of individuals, and for general utility functions that are convex combinations of a power function of income and the concern at having a low relative income. The alignment can also hold when the weights sum up to a number different than one.

Appendices

Appendix A: Proof of Proposition 1

With n individuals, there are n! possible orderings of incomes (permutations of the set {1,2,...,n}). Assuming a specific ordering ${x_{1}} \!\le \! {x_{2}} \!\le \! ... \!\le \!x_{n}$, we proceed to show that a corresponding result holds independently of the values of $\alpha _{1}^{~} ,...,\alpha _{n}$. By symmetry of the maximized function with respect to incomes and income weights, the result obtained below holds for other orderings of incomes.

Without loss of generality, and normalizing the sum of incomes if necessary, we assume that this sum is $\displaystyle \sum \limits _{i=1}^{n} {x_{i}} \,=\,1$. The set of admissible distributions $D\!\equiv \! \!\left \{ {x\in \mathbf {R}_{\ge 0}^{n} \text {: } x_{1}^{~} \le x_{2}^{~} \le ... \le x_{n} ,\displaystyle \sum \limits _{i=1}^{n} {x_{i}} =1} \right \}$ is convex (in fact, it is a convex polytope) and has a finite number of extremal points of the form

$$x^{k}=(\underbrace{0,0,...,0}_{n-k},\underbrace{1/k,1/k,...,1/k}_{k}),\,\, k\in \{ {1,2,...,n} \}, $$

where we have that x ⁿ = (1/n, 1/n,...,1/n) = x ^∗.

Over the closed set D, the function S W F is linear and therefore its maximum on D is attained in one of the extremal points x ^k. We have that

$$S{\kern-.5pt}W{}\!F(x^{k})=\underbrace{\frac{1}{k}\sum\limits_{i=n-k+1}^{n} {\alpha_{i}}}_{i{\kern-.5pt}n{\kern-.5pt}c{\kern-.5pt}o{\kern-.5pt}m{\kern-.5pt}e}\,\!-{\kern-.7pt}\!\underbrace{\frac{1}{n}\sum\limits_{i=1}^{n-k} {(1-\alpha_{i} )}}_{r{\kern-.5pt}e{\kern-.5pt}l{\kern-.5pt}a{\kern-.5pt}t{\kern-.5pt}i{\kern-.5pt}v{\kern-.5pt}e\,\,{\kern.5pt}d{\kern-.5pt}e{\kern-.9pt}p{\kern-.5pt}r{\kern-.5pt}i{\kern-.5pt}v{\kern-.5pt}a{\kern-.5pt}t{\kern-.5pt}i{\kern-.5pt}o{\kern-.5pt}n}, $$

namely, compared to x ⁿ, x ^k for k ≠ n represents taking away income from k individuals and distributing that income equally between the remainder n − k individuals.

A distribution x ⁿ constitutes a unique maximum if and only if S W F(x ⁿ) >S W F(x ^k) for any k ≠ n. To see this, note that

$$\frac{n-k}{n}=\left( {\frac{1}{k}-\frac{1}{n}} \right)k>\left( {\frac{1}{k}-\frac{1}{n}} \right)\sum\limits_{i=n-k+1}^{n} {\alpha_{i}} , $$

$$\frac{1}{n}\sum\limits_{i=n-k+1}^{n} {\alpha_{i}} >\frac{1}{k}\sum\limits_{i=n-k+1}^{n} {\alpha_{i}} -\frac{n-k}{n}, $$

$$\frac{1}{n}\sum\limits_{i=n-k+1}^{n} {\alpha_{i}} +\frac{1}{n}\sum\limits_{i=1}^{n-k} {\alpha_{i}} >\frac{1}{k}\sum\limits_{i=n-k+1}^{n} {\alpha_{i}} -\frac{n-k}{n}+\frac{1}{n}\sum\limits_{i=1}^{n-k} {\alpha_{i}} , $$

$$\frac{1}{n}\sum\limits_{i=1}^{n} {\alpha_{i}} >\frac{1}{k}\sum\limits_{i=n-k+1}^{n} {\alpha_{i}} +\frac{1}{n}\left[ {\sum\limits_{i=1}^{n-k} {\alpha_{i}} -(n-k)} \right]. $$

We have

$$\sum\limits_{i=1}^{n-k} {\alpha_{i}} -(n-k)=\sum\limits_{i=1}^{n-k} {(\alpha_{i} -\text{1)}} =-\sum\limits_{i=1}^{n-k} {(1-\alpha_{i} \text{)}} . $$

Consequently, we get

$$ S{\kern-.5pt}W{}\!F(x^{n})=\frac{1}{n}\sum\limits_{i=1}^{n} {\alpha_{i}} >\frac{1}{k}\sum\limits_{i=n-k+1}^{n} {\alpha_{i}} -\frac{1}{n}\sum\limits_{i=1}^{n-k} {(1-\alpha_{i} )} =S{\kern-.5pt}W{}\!F(x^{k}) $$

(A1)

for any k ≠ n, which completes the proof that x ⁿ = x ^∗ is the maximum. □

Appendix B: A loosened assumption on the weights in the individuals’ preferences

In this Appendix we attend to a model of preferences in which there are two (positive) parameters in the individual’s utility function rather than one: γ _i > 0 weight to be placed on individual’s i own income, and β _i > 0 weight to be placed on low relative income, that is, the individual’s utility function is

$$ u_{i} (x)=\gamma_{i} x_{i} -\beta_{i} R\!D_{i} (x). $$

(B1)

When this extra degree of freedom is allowed, we obtain the result that, as we increase the β weights, loosely speaking, the optimal utilitarian solution moves toward egalitarianism or, more correctly, social welfare decreases as we move away from egalitarianism.

Taking first as an example two individuals, and using the same notation as in the beginning of Section 2, for the case of x _i < x _j , when income is transferred from individual j to individual i, without changing the hierarchy of the two income earners, the marginal increase in i’s utility is γ _i + β _i , whereas the marginal decrease in j’s utility is γ _j . Unlike in the case of two parameters $\alpha _{1}^{~} ,\alpha _{2}^{~}$ in which the marginal gain of individual i, which is equal to 1, is greater than the marginal loss of individual j, which is equal to α _j ∈ (0, 1), here the marginal gain of i will be greater than the marginal loss of j if γ _i + β _i > γ _j , which translates into the condition that the marginal disutility of low relative income of the poorer individual, β _i , has to be larger than the difference between the two individuals’ marginal utilities of own income, γ _j − γ _i . From analyzing in a similar manner the case of x _i > x _j , we get an analogous condition on the marginal increase in social welfare, namely β _j > γ _i −γ _j .

Summing up: when the weights in the utility function of an individual do not sum up to 1, then for equality of incomes to be the optimal utilitarian solution, that is, for $x_{1}^{\ast } \,=\,x_{2}^{\ast } \,=\,\frac {A}{2}$ to obtain, we need to have that $\beta _{1}^{~} \!\!>\!\!\gamma _{2}^{~} \!-{\kern -.5pt}\gamma _{1}^{~}$ and that $\beta _{2}^{~} \!>\!\gamma _{1}^{~}\! -\gamma _{2}^{~}$.

This extension to a setting of the weights not summing up to one can be carried further and be applied to the case of more than two individuals. Let the utility of the i-th, i ∈ {1,...,n}, individual in a population with income vector x = (x ₁,...,x _n ) be described by (B1). In the following proposition we state and prove a condition on the weights γ _i , β _i in the population which, if satisfied, yields that the egalitarian income distribution maximizes utilitarian social welfare.

Proposition B1

If

$$ \gamma_{j}^{~} +\beta_{j}^{~} >\gamma_{l}^{\phantom{\frac{1}{2}}} \!\!+\frac{n-2}{n}\beta_{l} $$

(B2)

for all j ≠l, then the egalitarian income distribution maximizes social welfare.

Proof 3

The problem of the utilitarian social planner under a budget constraint A > 0 is

$$\begin{array}{@{}rcl@{}} &&\max SW\!F(x)\\ &&\text{for} x\in \mathbf{R}_{\ge 0}^{n} \text{s.t.} \sum\limits_{i=1}^{n} {x_{i}} = A, \end{array} $$

(B3)

where $S{\kern -.5pt}W{}\!F(x)\!\,=\,\!\sum \limits _{i=1}^{n} {u_{i} (x)}$. With n individuals, there are n! possible orderings of incomes (permutations of the set {1,2,...,n}). In what follows, we divide the maximization problem (4) into n! sub-problems corresponding each to a given ordering of incomes, and we show that in each sub-problem the egalitarian income distribution x ^∗ = (A/n,A/n,...,A/n) is optimal.

Specifically, we divide the set $X\!\!\!\,=\,\!\!\!\left \{ {x\!\in \!\mathbf {R}_{\ge 0}^{n} :\displaystyle \sum \limits _{i=1}^{n} {x_{i}} {\kern -.5pt} ={\kern -.5pt} A} \right \}$ into sets X _P ,$X \!\,=\,\underset {P}{\bigcup } X_{P}$, where the summation is over all possible orderings of incomes. For x ∈ X _P , the income of individual P(i) is the i-th lowest in the population, namely $x_{P(1)}^{~} \le x_{P(2)}^{~} \!\le \! ...\!\le \! x_{P(n)}^{~}$; and $X_{P} \,=\,\{x\in X:x_{P(k)}^{~} <x_{P(l)}^{~} \Rightarrow k<l\}$.³ Obviously, x ^∗∈X _P for every P. For the ordering P, the maximization problem is

$$ \max\limits_{x\in X_{P}} S{\kern-.5pt}W{}\!F(x). $$

(B4)

For x ∈ X _P the social welfare function can be rewritten as

$$S{\kern-.5pt}W{}\!F(x)= \sum\limits_{i=1}^{n} {a_{P(i)}^{~} x_{P(i)}^{~}} , $$

where

$$ a_{P(i)}^{~} =\gamma_{P(i)}^{~} +\frac{(n-i)\beta_{P(i)}} {n}-\frac{1}{n}\sum\limits_{j=1}^{i-1} {\beta_{P(j)}} . $$

(B5)

Consider the income distribution x ∈ X _P which is not egalitarian, namely x ≠ x ^∗, and a set of individuals whose income is the lowest in the population under distribution x. Obviously, individual P(1) belongs to this set. We introduce $i_{x} \,=\,\max \{i\!:\! x_{P(1)}^{~} \,=\,x_{\!P(i)}^{~} \}$. The individuals P(1),...,P(i _x ) have the same income, the lowest in the population. Given that x ≠ x ^∗, we have that P(i _x ) < n, and that $x_{P(i_{x} )}^{~} \!\!<\!\!x_{P(i_{x} +1)}^{~}$. We construct an income distribution x ^′ ∈ X _P such that x ^′ ≠x in the following way: for i ≤ i _x + 1, ${x}^{\prime }_{\!P(i)} \,=\,\frac {1}{i_{x} +1}\displaystyle \sum \limits _{j=1}^{i_{x} +1} {x_{P(j)}^{\!}} \,=\,\frac {i_{x} x_{P(i_{x})}^{~} +x_{P(i_{x} +1)}^{~}}{i_{x} +1}$, and for i > i _x + 1, ${x}^{\prime }_{\!P(i)} \,=\,x_{P(i)}^{~}$. We thus obtain that

$$\begin{array}{@{}rcl@{}} &&{} S{\kern-.5pt}W{}\!F({x}^{\prime})-S{\kern-.5pt}W{}\!F(x) \\ &&{} =\!\sum\limits_{j=1}^{i_{x}} {a_{\!P(j)}^{~}\! \left( {\frac{i_{x} x_{\!P(i_{x} )}^{~} \!{\kern.5pt}+x_{P(i_{x} +1)}^{~}} {i_{x} \,+\,1}\,-\,x_{\!P(i_{x} )}^{~}} \!\right)}\!+a_{\!P(i_{x} +1)}^{~}\! \left( \frac{i_{x} x_{\!P(i_{x} )}^{~} \!{\kern.5pt}+x_{\!P(i_{x} +1)}^{~}} {i_{x} \,+\,1} \,-\,x_{\!P(i_{x} +1)}^{~}\! \right) \\ &&{} =\frac{x_{P(i_{x} +1)}^{~} -x_{P(i_{x} )}^{~}} {i_{x} +1}\left( {\sum\limits_{j=1}^{i_{x}} {a_{P(j)}^{~}} -i_{x} a_{P(i_{x} +1)}^{~} } \right) . \end{array} $$

(B6)

Next, from (3) applied to j ∈ {1,...,i} and to l =i+1, we obtain by summation that

$$\sum\limits_{j=1}^{i} {(\gamma_{P(j)}^{~} +\beta_{P(j)}^{~} )} >i\gamma_{P(i+1)}^{~} +\frac{i(n-2)\beta_{P(i+1)}} {n}, $$

which implies

$$ \sum\limits_{j=1}^{i} {(\gamma_{P(j)}^{~} +\beta_{P(j)} )} >i\gamma_{P(i+1)}^{~} +\frac{i(n-i-1)\beta_{P(i+1)}} {n}. $$

(B7)

Recalling (B5), because

$$\sum\limits_{j=1}^{i} {a_{P(j)}^{~}} =\sum\limits_{j=1}^{i} {\gamma_{P(j)}^{~}} +\frac{n-i}{n}\sum\limits_{j=1}^{i} {\beta_{P(j)}} , $$

and because

$$ia_{P(i+1)}^{~} =i\gamma_{P(i+1)}^{~} +\frac{i(n-i-1)\beta_{P(i+1)}} {n}-\frac{i}{n}\sum\limits_{j=1}^{i} {\beta_{P(j)}} , $$

we get from (B7) that

$$ \sum\limits_{j=1}^{i} {a_{P(j)}^{~}} >ia_{P(i+1)}^{~} $$

(B8)

for all i∈ {1,...,n−1}.

Thus, by joining (B6) and (B8), we have that S W F (x ^′)−S W F (x) > 0. Consequently, distribution x such that x ≠ x ^∗ cannot be a solution to (B4). Because the set X _P is compact, S W F(x) attains its maximum over this set. Therefore, distribution x ^∗ is the solution to (B4) for any ordering P, and x ^∗ is the unique maximum of the social welfare function over the set X. □

Appendix C: Proof of Proposition 2

Akin to the proof of Proposition 1, and again without loss of generality, we assume a specific ordering of the incomes, $x_{1}^{~} \!\le \! x_{2}^{~} \!\le ... \!\le x_{n}$, and show that the result holds independently of the values of α ₁,...,α _n . The symmetry of the maximized function with respect to incomes and income weights implies that the result obtained below holds for other orderings of incomes. For the assumed ordering, the maximization problem of the utilitarian social planner is

$$\begin{array}{@{}rcl@{}} &&\max\limits_{x\in {\Omega}} \left\{ {\sum\limits_{i=1}^{n} {\left[ {\alpha_{i} x_{i}^{\gamma} -\frac{1-\alpha_{i}} {n}\sum\limits_{j=i+1}^{n} {\left( {x_{j} -x_{i}} \right)}} \right]}} \right\} \\ &&\text{s.t. } {\Omega} \text{=}\left\{ {\sum\limits_{i=1}^{n} {x_{i}} =A;\, x_{1}^{~} \ge 0;\, x_{i} \ge x_{i-1}^{~} \text{ for } i=2,...,n} \right\} . \end{array} $$

We denote $F(x)\,=\,-\!\displaystyle \sum \limits _{i=1}^{n} {\left [ {\alpha _{i} x_{i}^{\gamma } \!-\frac {1-\alpha _{i}} {n}\displaystyle \sum \limits _{j=i+1}^{n} {\left ({x_{j} \,-\,x_{i}} \right )}} \right ]}$, $h(x)\,=\,\!\displaystyle \sum \limits _{i=1}^{n} {x_{i} \,-\,A}$, and $g_{i} (x)=x_{i-1}^{~} -x_{i}$ for all i∈ {2,...,n}, $g_{1}^{~} (x)=-x_{1}^{~}$. Transforming the maximization problem to a minimization problem,

$$\begin{array}{@{}rcl@{}} &&\min F(x) \\ &&\text{s.t. } h(x)=0,\, g_{i} (x)\le 0\,\text{ for } \,i\!\!\in \!\{1,...,n\}, \end{array} $$

we obtain the Lagrangian

$$\begin{array}{@{}rcl@{}} L&=&F(x)+\sum\limits_{i=1}^{n} {\mu_{i} g_{i} (x)+\lambda h(x)} \\ &=&-\sum\limits_{i=1}^{n} {\left[ {\alpha_{i} x_{i}^{\gamma} -\frac{1-\alpha_{i}} {n}\sum\limits_{j=i+1}^{n} {\left( {x_{j} -x_{i}} \right)}} \right]} +\sum\limits_{i=1}^{n} {\mu_{i} g_{i} (x)+\lambda h(x)} . \\ \end{array} $$

We have that

$$\frac{\partial L}{\partial x_{1}^{~}} =-\alpha_{1}^{~} \gamma x_{1}^{\gamma -1} -(n-1)\frac{1-\alpha_{1}} {n}-\mu_{1}^{~} +\mu_{2}^{~} +\lambda , $$

that

$$\frac{\partial L}{\partial x_{n}} =-\alpha_{n} \gamma x_{n}^{\gamma -1} +\sum\limits_{j=1}^{n-1} {\frac{1-\alpha_{j}} {n}} -\mu_{n} +\lambda , $$

and that

$$\frac{\partial L}{\partial x_{l}^{~}} =-\alpha_{l}^{~} \gamma x_{l}^{\gamma -1} +\sum\limits_{j=1}^{l-1} {\frac{1-\alpha_{j}} {n}} -(n-l)\frac{1-\alpha_{l}^{~}} {n}-\mu_{l}^{~} +\mu_{l+1}^{~} +\lambda $$

for l ∈ {2,...,n−1}.

Because the maximized function F is convex,⁴ and the feasible set D is described by affine constraints, the following set of the Karush-Kuhn-Tucker conditions is both sufficient and necessary for x to be a unique minimum:

$$\begin{array}{@{}rcl@{}} &&\frac{\partial L}{\partial x_{l}^{~}} (x)=0,\, l\in \left\{ {1,...,n} \right\} \!, \\ &&\mu_{l}^{~} g_{l}^{~} (x)=0,\, \mu_{l}^{~} \ge 0,\text{} g_{l}^{~} (x)\le 0,\, l\in \{1,...,n\} , \\ &&h(x)=0 . \end{array} $$

We now show that point x ^∗=(A/n,A/n,...,A/n) satisfies the preceding conditions and that, therefore, it constitutes the global minimum of this problem.

We note that because $x_{1}^{\ast } =A/n>0$, we set $\mu _{1}^{~} =0$, and hence the considered problem reduces to a set of linear equations in $\mu _{2}^{~} ,...,\mu _{n} ,\lambda$. We denote

$$C_{l} =\left\{ \begin{array}{ll} -\alpha_{1}^{~} \gamma A^{\gamma -1}n^{1-\gamma} -(n-1)\frac{1-\alpha_{1}^{~}}{n} & \qquad l=1, \\ -\alpha_{n} \gamma A^{\gamma -1}n^{1-\gamma} +\displaystyle\sum\limits_{j=1}^{n-1} \frac{1-\alpha_{j}}{n} & \qquad l=n,\\ -\alpha_{l}^{~} \gamma A^{\gamma -1}n^{1-\gamma} +\displaystyle\sum\limits_{j=1}^{l-1} \frac{1-\alpha_{j}} {n}-(n-l)\frac{1-\alpha_{l}^{~}}{n} & \qquad \text{otherwise.} \end{array} \right. $$

Thus, we obtain

$$\left\{\begin{array}{ll} C_{1} =-\mu_{2}^{~} -\lambda &\qquad l=1, \\ C_{n} =\mu_{n} -\lambda &\qquad l=n, \\ C_{l} =\mu_{l}^{~} -\mu_{l+1}^{~} -\lambda &\qquad \text{otherwise,}\\ \end{array}\right. $$

which can be written as

$$C=\mathrm{M}v, $$

where $C=(C_{l} )_{l=1}^{n}, v=\left ({\lambda ,\mu _{2}^{~},...,\mu _{n}} \right )$, and

$$\mathrm{M}=\left[ \begin{array}{ccccccc} {-1} & {-1} & 0 & {...} & {...} & {...} & 0 \\ {-1} & 1 & {-1} & 0 & {...} & {...} & 0 \\ {-1} & 0 & 1 & {-1} & 0 & {...} & 0 \\ {...} & {...} & {...} & {...} & {...} & {...} & {...} \\ {-1} & 0 & {...} & {...} & 0 & 1 & {-1} \\ {-1} & 0 & {...} & {...} & {...} & 0 & 1 \\ \end{array}\right]. $$

It can easily be verified that

$$\mathrm{M}^{-1}=\left[ \begin{array}{ccccccc} {-\frac{1}{n}} & {-\frac{1}{n}} & {-\frac{1}{n}} & {-\frac{1}{n}} & {...} & {-\frac{1}{n}} & {-\frac{1}{n}} \\ [12pt] {-\frac{n-1}{n}} & {\frac{1}{n}} & {\frac{1}{n}} & {\frac{1}{n}} & {...} & {\frac{1}{n}} & {\frac{1}{n}} \\ [12pt] {-\frac{n-2}{n}} & {-\frac{n-2}{n}} & {\frac{2}{n}} & {\frac{2}{n}} & {...} & {\frac{2}{n}} & {\frac{2}{n}}\\ {...} & {...} & {...} & {...} & {...} & {...} & {...} \\ {-\frac{1}{n}} & {-\frac{1}{n}} & {-\frac{1}{n}} & {-\frac{1}{n}} & {...} & {-\frac{1}{n}} & {\frac{n-1}{n}} \\ \end{array}\right]. $$

Then,

$$\lambda =-\frac{1}{n}\sum\limits_{l=1}^{n} {C_{l}} , $$

and for i∈{1,2,...,n−1}

$$\mu_{i+1} =-\frac{n-i}{n}\sum\limits_{l=1}^{i} {C_{l}} +\frac{i}{n}\sum\limits_{l=i+1}^{n} {C_{l}} \equiv f_{i+1} (\alpha_{1} ,...,\alpha_{n} ). $$

Obviously the functions f _{i + 1} are linear and, moreover, for i∈{1,2,...,n − 1}, we have that f _i+1(1,...,1) = 0. We calculate for j = l

$$\frac{\partial C_{l}} {\partial \alpha_{l}^{~}} =-\gamma A^{\gamma -1}n^{1-\gamma} +\frac{n-l}{n}, $$

for j <l

$$\frac{\partial C_{l}} {\partial \alpha_{j}} =-\frac{1}{n}, $$

and for j > l

$$\frac{\partial C_{l}} {\partial \alpha_{j}} =0. $$

Therefore, for j > i we obtain

$$\frac{\partial f_{i+1}} {\partial \alpha_{j}} =\frac{i}{n}\sum\limits_{l=j}^{n} {\frac{\partial C_{l}} {\partial \alpha _{j}} } =\frac{i}{n}\left( {\frac{\partial C_{j}} {\partial \alpha_{j}} +\sum\limits_{l=j+1}^{n} {\frac{\partial C_{l}} {\partial \alpha_{j}} }} \right)=-i\gamma A^{\gamma -1}n^{-\gamma} <0, $$

whereas for j ≤i, we have

$$\begin{array}{@{}rcl@{}} &&\frac{\partial f_{i+1}} {\partial \alpha_{j}} =-\frac{n-i}{n}\left( {\frac{\partial C_{j}} {\partial \alpha_{j}} +\sum\limits_{l=j+1}^{i} {\frac{\partial C_{l}} {\partial \alpha_{j}} }} \right)+\frac{i}{n}\sum\limits_{l=i+1}^{n} {\frac{\partial C_{l}} {\partial \alpha_{j}} } \\ &&=-\frac{n-i}{n}\left( {-\gamma A^{\gamma -1}n^{1-\gamma} +\frac{n-j}{n}-\frac{i-j}{n}} \right)-\frac{i(n-i)}{n^{2}} \\ &&=\frac{n-i}{n}\left( {\gamma A^{\gamma -1}n^{1-\gamma} -1} \right). \end{array} $$

From the assumption that A≥n/e we get that $A\ge n\gamma ^{\frac {1}{1-\gamma } }$, where this last inequality is due to the fact that $\lim \limits _{\gamma \to 1} \gamma ^{\frac {1}{1-\gamma } }\,=\,1/e$,⁵ and that $\gamma ^{\frac {1}{1-\gamma } }$ as a function of γ is non- decreasing.⁶ Thus, we have that $\left ({\gamma A^{\gamma -1}n^{1-\gamma } \,-\,1} \right )\!\le \left [ {\gamma \! \left ({n\gamma ^{\frac {1}{1-\gamma } }} \right )^{\gamma -1}n^{1-\gamma } \,-\,1} \right ]$ =0. Therefore, $\frac {\partial f_{i+1}} {\partial \alpha _{j}} \!\le \! 0$ for any i ∈ {1,2,...,n−1} and j = {1,...,n}. Thus, because f _i+1(1,...,1) = 0, for any set of weights $(\alpha _{1}^{~} ,...,\alpha _{n} )\!\!\in \!\! (0, 1)^{n}$ we obtain that f _i+1 $(\alpha _{1}^{~} ,...,\alpha _{n} )\ge 0$ for i ∈ {1,2,...,n−1}.

In sum, we obtained a set of non-negative multipliers $\mu _{2}^{~} ,...,\mu _{n}$ for inequality constraints $g_{2}^{~} ,...,g_{n}$ which are satisfied by x ^∗ with equality. This completes the proof that x ^∗ is the global maximum of the considered problem. □