Class+12+Partial+Equilbrium+II+_Pareto+Efficiency+and+First+Welfare+Theorem+bw

Class+12+Partial+Equilbrium+II+_Pareto+Efficiency+and+First+Welfare+Theorem+bw

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Partial Equilibrium II: Pareto Efficiency, Surplus, Welfare Theorem
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The Partial Equilibrium Model
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A Quasilinear Economy One Consumer: Consumes limes (q) and money (m). (Price of money is always 1.) One Firm: Produces limes to earn profits. The “Primitives”: Y Consumer’s income (endowment) U=m + v(q) Consumer’s preferences c(q) Firm’s costs of producing limes Primitives: Define what is possible (feasibility constraints) and what is desirable Partial Equilibrium Model (preferences). Assumptions: v(q) increasing and strictly concave, v’(q) is continuous , v’(0) >0 , lim q Æ v’(q)=0 m negative amounts are possible c(q) increasing and strictly convex, c’(q) is continuous, c’(0) < v’ (o), lim q Æ v’(q)= c(0) = 0 (fixed costs are sunk)
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Allocation An allocation x = [q s ,q d , ,m] determines: q d Limes for consumer q s Limes made by firm Money for firm (profit) m Money left for consumer (to spend) Allocations: what we care about. Preferences Consumer and firm have preferences over allocations. Firm: The more money (profit) the better. Consumer: Whatever maximizes U=m + v(q) Feasible Allocation An allocation x= [q s d , ,m] is feasible if: (i) q s q d Allocations: What is good? What is feasible? All limes are produced (ii) Y – c(q s ) +m Money (endowment) is used to produce the limes, c(q s ), to pay profit, , and to consume m.
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Market
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0 1 2 0 2 4 6 8 () 5 2 D Pq q = ( ) 2 S Pq q = p* q * Example: Economy given by Y=10 U(m,q) = m + 5 q c(q)= q 2 Competitive Allocation Price quantity Demand and Supply Firm’s supply: P S (q)=c’(q) Consumer’s demand: P D (q)=v’(q) Competitive Allocations Competitive price and quantity: p*= 2.32, q*= 1.16 Competitive Allocation: m* = Y - p*q* = 10 – 2.32(1.16) = 7.31 * = p*q* - c (q*)= 2.32(1.16) - (1.16) 2 = 1.35 The competitive allocation is [q * , q * , * ,m*]= [1.16, 1.16, 1.35 ,7.31]
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Competitive Allocation An allocation x * = [q * , q * , * ,m*] is competitive if there is some price p* s.t. (i) Q D (p*) = Q S (p*) = q* “Demand =Supply” (ii) m* = Y - p*q* (iii) * = p*q* - c (q*) Competitive Allocations: General
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Observe I: Transfers Competitive Allocation An allocation x * = [q * , q * , * ,m*] is competitive if there is some price p* s.t. (i) Q D (p*) = Q S (p*) = q* “Demand =Supply” (ii) m* = Y - p*q* (iii) * = p*q* - c (q*) Transfers: Money moved around. Competitive Allocations: General The competitive allocation involves a transfer: Transfer from the consumer T D = Y – m* = p*q* Transfer to the firm T S = * + c(q) = p*q* The transfer from the consumer is equal to the transfer received by the firm.
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Observe II: MB=MC Competitive Allocation An allocation x * = [q * , q * , * ,m*] is competitive if there is some price p* s.t.
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Class+12+Partial+Equilbrium+II+_Pareto+Efficiency+and+First+Welfare+Theorem+bw

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