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General Equilibrium APEC 3001 Summer 2006 Readings: Chapter 16 1 Objectives • General Equilibrium – Exchange Economy – With Production • First & Second Welfare Theorems 2 General Equilibrium • Definition: – The study of how conditions in each market in a set of related markets affect equilibrium outcomes in other markets in that set. • Example of Exchange Economy – Two people: Mason & Spencer – Initial Endowments: • Mason: 75 pieces of candy & 50 pieces of gum. • Spencer: 25 pieces of candy & 100 pieces of gum. • Total: 100 pieces of candy & 150 pieces of gum. • Edgeworth Exchange Box: – A diagram used to analyze the general equilibrium of an exchange economy. 3 Graphical Example of Edgeworth Exchange Box 150 100 Spencer’s Gum 0 100 Spencer 0 75 25 Spencer’s Candy Mason’s Candy 0 Mason 100 0 50 150 Mason’s Gum 4 Question: Can Mason & Spencer do better? • To answer this question, we need to know something about Mason & Spencer’s preferences. • Assume: – – – – Complete Nonsatiable Transitive Convex • Implication: – Mason & Spencer have utility functions that produce indifference curves that • • • • • represent higher levels of satisfactions as we move away from the origin, are ubiquitous, are downward sloping, cannot cross, & are bowed toward the origin. 5 Edgeworth Exchange Box With Indifference Curves I2M > I1M > I0M 150 I2S > I1S > I0S 100 100 Spencer’s Gum 0 Spencer 0 I0S I1S 75 25 I2S Spencer’s Candy Mason’s Candy I2M I1M I0M 0 Mason 0 50 100 150 Mason’s Gum 6 How can Mason & Spencer do better? • Pareto Superior Allocation: – An allocation that at least one individual prefers and others like at least equally as well. • Pareto Optimal Allocation: – An allocation where it is impossible to make one person better off without making at least one other person worse off. • Consider the indifferences curves for Mason & Spencer that intersect the initial endowment. 7 Gains From Trade 150 100 Spencer’s Gum 0 100 Spencer 0 IES 75 25 Spencer’s Candy Mason’s Candy IEM 0 Mason 100 0 50 150 Mason’s Gum 8 Pareto Optimal Allocations 150 100 Spencer’s Gum 0 100 Spencer 0 IES 75 25 IPIS b Mason’s Candy Spencer’s Candy a IPIM IEM 0 Mason 100 0 50 150 Mason’s Gum 9 What are the Pareto Optimal allocations? • Contract Curve: – The set of all Pareto optimal allocations. 10 The Contract Curve 150 100 Spencer’s Gum 0 100 Spencer 0 Contract Curve IES 75 25 IPIS b Mason’s Candy Spencer’s Candy a IPIM IEM 0 Mason 100 0 50 150 Mason’s Gum 11 How can Mason & Spencer get to a Pareto Optimal allocation? • Suppose the price of candy is PC0 & the price of gum is PG0. • Implications: – Mason’s Income: M0M = PC075 + PG050 – Spencer’s Income: M0S = PC025 + PG0100 12 Income Constraint With Prices PC0 and PG0 for Candy and Gum 150 M0S/PG0 100 Spencer’s Gum 0 100 Spencer 0 75 25 Spencer’s Candy Slope = -PG0/PC0 Mason’s Candy 0 Mason 100 0 50 Mason’s Gum M0M/PG0 150 13 Mason’s and Spencer’s Optimal Consumption Given Prices PC0 and PG0 Spencer’s Gum 150 M0S/PG0 100 G0S 0 100 Spencer 0 I0S 75 25 C0S Spencer’s Candy Mason’s Candy C0M I0M Slope = -PG0/PC0 0 Mason 100 0 50 M G0 Mason’s Gum M0M /PG0 150 14 Is this a market equilibrium? • No! – C0M + C0S < 100 Excess supply of candy! – G0S + G0S > 150 Excess demand for gum! • So now what can we do? – Offer a higher price for gum or lower price for candy! – For example, PC1 < PC0 & PG1 > PG0. 15 Mason’s and Spencer’s Optimal Consumption Given Equilibrium Prices PC1 and PG1 Spencer’s Gum 150 M0S/PG0 M0S/PG1 100 G S G S 0 1 0 100 Spencer 0 I0S 75 25 I1S C0S Mason’s Candy C1S M C1 Spencer’s Candy C0M Slope = -PG1/PC1 0 Mason 0 50 M G1 G0 M M0M /PG1 Mason’s Gum I1M I0M Slope = -PG0/PC0 M0M /PG0 100 150 16 Is this a market equilibrium? • Yes! – C0M + C0S = 100 There is no excess demand or supply of candy! – G0M + G0S = 150 There is no excess demand or supply of gum! • What is true at this point? – MRSM = MRSS – We are on the contract curve, so we are at a Pareto Optimal allocation! • First Welfare Theorem: – Equilibrium in competitive markets is Pareto Optimal. • Second Welfare Theorem: – Any Pareto optimal allocation can be sustained as a competitive equilibrium. 17 General Equilibrium with Production • Production Possibility Frontier: – The set of all possible output combinations that can be produced with a given endowment of factor inputs. 18 Edgeworth Box for Candy and Gum Production Firm G’s Labor G2 > G1 > G0 C2 LE KE 0 Firm G (Gum) 0 C1 C0 Firm G’s Capital Firm C’s Capital G0 G1 0 G2 KE LE Firm C 0 (Candy) C 2 > C1 > C 0 Firm C’s Labor 19 Efficient Production of Candy and Gum Production Firm G’s Labor LE KE 0 C2 C1 Firm G (Gum) 0 More candy with same amount of gum! Firm C’s Capital Firm G’s Capital More gum with same amount of candy! 0 G2 G1 KE LE Firm C 0 (Candy) Firm C’s Labor 20 Contract Curve for Candy and Gum Production G2 > G1 > G0 Firm G’s Labor LE C2 KE MRTSC = 0 0 C1 MRTSG Firm G (Gum) C0 Firm G’s Capital Firm C’s Capital G0 G1 0 Firm C 0 (Candy) C 2 > C1 > C 0 G2 Firm C’s Labor KE LE 21 Competitive Cost Minimizing Production • • • • MRTSC = MPLC/MPKC = w/r MRTSG = MPLG/MPKG = w/r So, MRTSG = w/r = MRTSG Competitive production will result in Pareto Efficient production! 22 Graphical Example of Production Possibility Frontier Candy Slope = C/G C2 C1 C0 G0 G1 G2 Gum 23 Production Possibility Frontier • Marginal Rate of Transformation: – The rate at which one output can be exchanged for another at a point along the production possibility frontier: |C/G|. 24 Note that TCG = wLG + rKG and TCC = wLC + rKC TCG = wLG + rKG and TCC = wLC + rKC Also, LG = LE – LC and KG = KE – KC LG = – LC and KG = –KC Therefore, TCG = -wLC - rKC = -TCC TCG/ (GC) = -TCC/ (CG) MCG/C = -MCC/G |C/G| = MCG/MCC The Marginal Rate of Transformation is the ratio of Marginal Cost! 25 Profit Maximization in Competitive Industry • MCC = PC • MCG = PG • Implications: – MRT = MCG/MCC = PG/PC 26 Utility Maximization with Competitive Markets • MRSM = PG/PC • MRSS = PG/PC • Implications: – MRT = MRSM = MRSS 27 Competitive Equilibrium with Production Candy |Slope| = PG/PC GS Spencer IS CM CS IM Mason GM Gum 28 Summary • For a general equilibrium with production to be Pareto Efficient, three types of conditions must hold: – Firms must equate their marginal rates of technical substitution. – Consumers must equate the marginal rates of substitution. – Consumers’ marginal rates of substitution must equal the marginal rate of transformation. Competitive Markets Yield This Outcome! 29 Adding Production Does Not Change The Implications of The First and Second Welfare Theorems! • Competitive markets result in the Pareto efficient production and distribution of goods and services! • Any Pareto efficient production and distribution of goods can be supported by a competitive market. 30 So, is there anything that can mess up these welfare theorems? • Yes! • Government Intervention – Taxes – Subsidies • Market Failure – Externality: Either a benefit or a cost of an action that accrues to someone other than the people directly involved in the action. – Public Goods: (1) nondiminishability and (2) nonexcludability of consumption. • Noncompetitive Behavior – Monopoly – Oligopoly 31 What You Should Know • General Equilibrium Conditions – Exchange Economy – With Production • Pareto Optimal Allocations • First & Second Welfare Theorems & Caveats 32