Positive externalities

In economics, an externality is the cost or benefit that affects a party who did not choose to incur that cost or benefit. Economists often urge governments to adopt policies that "internalize" an externality, so that costs and benefits will affect mainly parties who choose to incur them.

For example, manufacturing activities that cause air pollution impose health and clean-up costs on the whole society, whereas the neighbors of an individual who chooses to fire-proof his home may benefit from a reduced risk of a fire spreading to their own houses. If external costs exist, such as pollution, the producer may choose to produce more of the product than would be produced if the producer were required to pay all associated environmental costs. Because responsibility or consequence for self-directed action lies partly outside the self, an element of externalization is involved. If there are external benefits, such as in public safety, less of the good may be produced than would be the case if the producer were to receive payment for the external benefits to others. For the purpose of these statements, overall cost and benefit to society is defined as the sum of the imputed monetary value of benefits and costs to all parties involved. Thus, unregulated markets in goods or services with significant externalities generate prices that do not reflect the full social cost or benefit of their transactions; such markets are therefore inefficient.

Two British economists are credited with having initiated the formal study of externalities, or "spillover effects": Henry Sidgwick (1838-1900) is credited with first articulating, and Arthur C. Pigou (1877-1959) is credited with formalizing the concept of externalities.

Suppose that there are ${\displaystyle K}$ different possible allocations and ${\displaystyle N}$ different agents, where ${\displaystyle K,N<\infty }$ and ${\displaystyle N\geq 2}$. Suppose that each agent has a type ${\displaystyle \theta _{i}\sim F_{i}[0,1]}$ and that each agent gets payoff ${\displaystyle v_{i}(\theta _{i},k)+t_{i}}$, where ${\displaystyle t_{i}}$ is the transfer paid by the ${\displaystyle i}$-th agent. A map ${\displaystyle f=(\kappa ,t_{1},...,t_{N})}$ is a social choice function if

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