to have Pareto efficiency while also have high levels of inequality. Consider the following scenario: there is a pie and three persons; the most equitable way would be to divide the pie into three equal portions. However, if the pie is divided in half and shared between two people, it is considered Pareto efficient meaning that the third person does not lose out (despite the fact that he does not receive a piece of the pie). When making judgments, it is critical to consider a variety of aspects, including social efficiency, overall welfare, and issues such as diminishing marginal value. Pareto efficiency and market failure In order to fully understand market failure, one must first comprehend market success, which is defined as the ability of a set of idealized competitive markets to achieve an equilibrium allocation of resources that is Pareto-optimal in terms of resource allocation. According to the definition of market failure, it is a circumstance in which the conclusion of the first fundamental theorem of welfare is erroneous; that is, when the allocations made through markets are not efficient. In a free market, market failure is defined as an inefficient allocation of resources. Due to the fact that it is feasible to improve, market failure implies Pareto inefficiency. For example, excessive consumption of depreciating items (drugs/tobacco) results in external costs to non-smokers, as well as premature death for smokers who do not quit. An increase in the price of cigarettes could motivate people to quit smoking while also raising funds for the treatment of smoking-related ailments. Approximate Pareto efficiency Given some ε > 0, an outcome is called ε-Pareto-efficient if no other outcome gives all agents at least the same utility, and one agent a utility at least (1 + ε) higher. This captures the notion that improvements smaller than (1 + ε) are negligible and should not be considered a breach of efficiency. Pareto-efficiency and welfare-maximization Suppose each agent i is assigned a positive weight ai. For every allocation x, define the welfare of x as the weighted sum of utilities of all agents in x: Let xa be an allocation that maximizes the welfare over all allocations: It is easy to show that the allocation xa is Pareto-efficient: since all weights are positive, any Pareto improvement would increase the sum, contradicting the definition of xa. Japanese neo-Walrasian economist Takashi Negishi proved that, under certain assumptions, the