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Externality-Public Good

# Externality-Public Good - Externalities and Public Goods...

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Externalities and Public Goods Econ 210C UCSB May 23, 2011

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Externalities An externality is present if the well-being of a consumer or the production possibilities of a producer is directly affected by actions of another agent (a consumer or a producer). Example (Coase, 1960): A farmer and a rancher had adjoining properties. There was no fence. The rancher’s cattle could wander over to the farmer’s property and eat some the his wheat. The more cattle there were, the more wheat was eaten. Profit Functions: Π f ( w , c ) for the farmer and Π r ( w , c ) for the rancher, where w denotes a generic quantity of wheat and c the number of cattle. Assume Π f ( w , c ) c < 0 and Π r ( w , c ) w > 0. (1) Econ 210C UCSB Paper
Econ 210C UCSB Paper

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Econ 210C UCSB Paper
Non-Cooperative Solution: Let ( c * , w * ) be a Nash equilibrium such that w * > 0 and c * > 0. Then, ( c * , w * ) satisfies Π f ( w , c ) w = 0 and Π r ( w , c ) c = 0. (2) Pareto Optimal Solution: A PO solution can be found by solving max [ λ f Π f ( w , c ) + λ r Π r ( w , c )] for some choice λ = ( λ f , λ r ) ∈ < 2 + . At the interior solution, λ f Π f ( w , c ) w + λ r Π r ( w , c ) w = 0 λ f Π f ( w , c ) c + λ r Π r ( w , c ) c = 0. (3) By (1), (2), and (3), Nash equilibrium is not Pareto optimal. Econ 210C UCSB Paper

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Econ 210C UCSB Paper
A Simple Model with Externality E = { X i , u i , ω i } 2 i = 1 where X i = < 2 + and ω i = ( ω ix , ω im ) , u i ( x i , m i , h ) = φ i ( x i , h ) + m i h ∈ < + is a level of an activity consumer 1 may choose. There is no market for externality (i.e. for activity h ). Given h ∈ < + and market prices p x for the consumption good, consumer i ’s utility maximization problem is max φ i ( x i , h ) + m i s . t . p x x i + m i = p x ω ix + ω im where v i ( · ) is consumer i ’s indirect utility function. Since u i ( x , m i , h ) is quasi-linear, the consumer’s indirect utility function is also quasi-linnear v i ( p , w i , h ) = ψ i ( p , h ) + w i . Econ 210C UCSB Paper

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Consumer 1’s Optimal Activity Level: max h 0 ψ 1 ( p , h ) . At the optimal level, ψ 0 1 ( p , h ) 0 and h ψ 0 1 ( p , h ) = 0. (4) Pareto Optimal Activity Level: Due to quasi-linearity of consumers’ utility functions, the Pareto optimal activity level is determined by max h 0 [ φ 1 ( p , h ) + ψ 2 ( p , h )] .
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