Editing Pareto efficiency (section)

== Pareto-efficiency and welfare-maximization ==
{{See also|Pareto-efficient envy-free division}}
Suppose each agent ''i'' is assigned a positive weight ''a<sub>i</sub>''. For every allocation ''x'', define the ''welfare'' of ''x'' as the weighted sum of utilities of all agents in ''x'':

: <math>W_a(x) := \sum_{i=1}^n a_i u_i(x).</math>

Let ''x<sub>a</sub>'' be an allocation that maximizes the welfare over all allocations:

: <math>x_a \in \arg\max_x W_a(x).</math>

It is easy to show that the allocation ''x<sub>a</sub>'' is Pareto-efficient: since all weights are positive, any Pareto improvement would increase the sum, contradicting the definition of ''x<sub>a</sub>''.

Japanese neo-[[Léon Walras#General equilibrium theory|Walrasian]] economist [[Takashi Negishi]] proved<ref>{{cite journal |last=Negishi |first=Takashi |date=1960 |title=Welfare Economics and Existence of an Equilibrium for a Competitive Economy |journal=Metroeconomica |volume=12 |issue=2–3 |pages=92–97 |doi=10.1111/j.1467-999X.1960.tb00275.x }}</ref> that, under certain assumptions, the opposite is also true: for ''every'' Pareto-efficient allocation ''x'', there exists a positive vector ''a'' such that ''x'' maximizes ''W<sub>a</sub>''. A shorter proof is provided by [[Hal Varian]].<ref>{{cite journal |doi=10.1016/0047-2727(76)90018-9 |title=Two problems in the theory of fairness |journal=Journal of Public Economics |volume=5 |issue=3–4 |pages=249–260 |year=1976 |last1=Varian |first1=Hal R. |hdl=1721.1/64180 |hdl-access=free }}</ref>