Chapter 6 Exchange Economies Intermediate Microeconomics: A Tool-Building Approach Routledge, UK © 2016 Samiran Banerjee

Partial vs. general equilibrium Partial equilibrium analysis Looking at one market in isolation, is there a price so that this market is in equilibrium? Answer: See Chapter 1 General equilibrium analysis Looking at all markets, are there prices so that each market is in equilibrium simultaneously? Since markets are interrelated, it is not obvious if this is even possible!

Simplest general equilibrium model • Two consumers, a and b • Two goods, x1 and x2 • No production • Consumers exchange what they bring to the market Two-person pure exchange economy • Consumer a’s characteristic: ea = (ua, ωa) • Consumer b’s characteristic: eb = (ub, ωb) • The economy: e = (ea, eb) Utility function and endowment List of consumers’ characteristics

Pure exchange economy Consumer a • ωa = (5, 2) • ua = x1ax2a Consumer b • ωb = (7, 8) • ub = 2x1b + x2b a’s origin b’s origin

Pure exchange economy • Rotate b’s graph counterclockwise • Slide to the left until ωa and ωb coincide

Length and height of the Edgeworth box Pure exchange economy • Every point in the Edgeworth box is a feasible allocation • The point ω is the initial endowment of this economy • ω = (ωa, ωb) = ((5, 2), (7, 8)) • Aggregate endowment, Ω = (12, 10) Length and height of the Edgeworth box b’s origin Edgeworth box a’s origin

Individually rational allocations • An allocation (xa, xb) is individually rational means everyone is at least as well off as at their endowment: ua(xa) ≥ ua(ωa) and ub(xb) ≥ ub(ωb) • Draw a’s indifference curve through ω • Draw b’s indifference curve through ω • The lens-shaped area, IR, is the set of individually rational allocations

Pareto superior allocations • An allocation S is Pareto superior to (or a Pareto improvement over) R means everyone at S is at least as well off as at R and at least one person is better off. Both a and b are better off at S a is better off at S’, b is better off at S”

Pareto efficient allocations • An allocation E is Pareto efficient if no other allocation is Pareto superior. MRSa = MRSb at an interior Pareto efficient allocation • Fix b’s indifference curve arbitrarily • Maximize a’s utility given b’s indifference curve • Point E is Pareto efficient • Check zones I–IV for any Pareto superior allocations

Derive the Pareto set graphically • Fix one individual’s indifference curve, say, for a • Maximize b’s utility keeping a on this indifference curve • Pick a higher indifference curve for a • Maximize b’s utility again • Pick an even higher indifference curve for a • Maximize b’s utility again (a corner solution at T) • Join all these Pareto efficient points, PE • The set of Pareto efficient allocations is called the contract curve

Derive the Pareto set algebraically Interior Derive the Pareto set algebraically • Consumer a: ωa = (10, 20), ua = x1a (x2a)2 • Consumer b: ωb = (50, 10), ub = x1b (x2b)2 • Set MRSa = MRSb to obtain x2a/x1a = x2b/x1b • Use the supply constraints x1a + x1b = 60 and x2a + x2b = 30 to get rid of all b terms: x2a/x1a = (30 – x2a)/(60 – x1a) • Solve to obtain the contract curve: x2a = 0.5x1a

Walras equilibrium concept • Market mechanism • Walras prices: (p1, p2) • Walras allocation: (xa, xb) Given the Walras prices, ˆ a and b maximize their utilities ˆ 1. xa maximizes a’s utility given the budget constraint p1xa1 + p2xa2 ≤ p1ωa1 + p2ωa2 2. xb maximizes b’s utility given the budget constraint p1xb1 + p2xb2 ≤ p1ωb1 + p2ωb2 3. Demand equals supply for each good: xa1 + xb1 = ωa1 + ωb1 xa2 + xb2 = ωa2 + ωb2 ˆ ˆ ˆ The markets for goods 1 and 2 clear ˆ

Walras (dis)equilibrium – • Fix prices arbitrarily at (p1, p2) • Draw a’s budget line through ωa • Maximize a’s utility at A • S1a = supply of 1 by a • D2a = demand for 2 by a • Maximize b’s utility at B • S2b = supply of 2 by b • D1b = demand for 1 by b • S1a < D1b implies p1 increases • D2a < S2b implies p2 decreases p1/p2 increases!

Both markets clear at new prices! Walras equilibrium • Change old prices • Since new prices are (p1, p2), where p1/p2 > p1/p2, therefore new budget pivots around ω • Maximize a’s utility at E • S1a = supply of 1 by a • D2a = demand for 2 by a • Maximize b’s utility at E • S2b = supply of 2 by b • D1b = demand for 1 by b • S1a = D1b • D2a = S2b ˆ ˆ – Both markets clear at new prices!

Features of Walras equilibrium First Welfare Theorem: The Walras allocation is IR and PE (in the absence of externalities, asymmetric information, and public goods) • Walras prices: (p1, p2) • Walras allocation: E • E is individually rational: it is preferred to ω by both • E is Pareto efficient: MRSa = MRSb • (Walras’ Law corollary) If the market for good 1 clears, the market for good 2 clears automatically (and vice versa) • The Walras equilibrium depends on relative prices, p1/p2 ˆ ˆ

Deriving a Walras equilibrium Consumer a • ωa = (6, 4) • ua = x1ax2a • h1a = ma/(2p1) • h2a = ma/(2p2) Consumer b • ωb = (2, 8) • ub = (x1b)2 x2b • h1b = 2mb/(3p1) • h2b = mb/(3p2) Normalize one price, say, p2 = 1. Then ma = 6p1 + 4 and mb = 2p1 + 8 2. Select any one good for market clearing, say, good 1 3. Set total demand equal to total supply for good 1: h1a + h1b = 6 + 2 Solve to obtain p1 = 2 4. Substitute prices in demands to find the Walras allocation: (xa, xb) = ((4, 8), (4, 4)) ˆ ˆ ˆ

Two persons, three goods 1. Normalize one price, say, p3 = 1. 2. Select any two goods for market clearing, say, 1 and 2 3. Set total demand equal to total supply for goods 1 and 2, obtaining two equations in two unknowns, p1 and p2 4. Solve simultaneously to obtain p1 and p2 5. Substitute prices in demands to find the Walras allocation At any interior Pareto efficient allocation, we require MRS12a = MRS12b MRS13a = MRS13b MRS23a = MRS23b