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Partial-Welfare

# Partial-Welfare - Partial Equilibrium and Welfare Analysis...

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Partial Equilibrium and Welfare Analysis Econ 210C UCSB May 16, 2011

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Partial Equilibrium Analysis Two Goods: the consumption good and the numeraire (e.g., expenditures on other goods). Endowments: ω i = ( 0, ω im ) , i = 1, 2, · · · , n . Preferences: u i ( x i , m i ) = φ i ( x i ) + m i , i = 1, 2, · · · , n . Production Technology: C j ( q j ) denotes the amount of the numeraire needed to produces q j units of the consumption good. Relative Shares: θ ij , i = 1, · · · , n , j = 1, · · · , m . A competitive equilibrium can be identified by the equilibrium price and quantities of the consumption good consumed by the consumers and produced by the firms. Econ 210C UCSB Paper
Assumptions: φ i is differentiable and concave and C j is differentiable and convex. Utility Maximization: max x i [ φ i ( x i ) + ω im + Profit Share - px i ] . By the Khun-Tucker Theorem, a solution satisfies φ 0 i ( x i ) - p 0 and [ φ 0 ( x i ) - p ] x i = 0. (1) Profit Maximization: max q j [ pq j - C j ( q j )] . By the Kuhn-Tucker Theorem, p - C 0 j ( q j ) 0 and [ p - C 0 j ( q j )] q j = 0. (2) Econ 210C UCSB Paper

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Competitive Equilibrium: ( x * , m * , q
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