General Equilibrium Microeconomics (Econ7790) Benson Tsz Kin Leung1 Fall 2021 1 Hong Kong Baptist University 1 Book Chapter • Chapter 13 of Nicholson Snyder. • Goal: • To analyze equilibrium of multiple (competitive) markets. • To derive welfare implications. . 2 Motivation • Ana’s demand curve for coffee depends on price of tea 3 Markets are Interlinked • Suppose an illness infects tea trees, causing supply curve of tea to shift to the left. • equilibrium price of tea will go up • How does this affect demand curve for coffee? • equilibrium price of coffee? • change in price of coffee shifts demand curve for tea • etc. etc. 4 General Equilibrium • Analyzes all markets at the same time • Assume every market is perfectly competitive • Goal: find equilibrium price and allocation for each market • equilibrium prices: quantity demanded in each market = quantity supplied in each market • equilibrium allocation: how much of each good each person/firm consumes/produces • Is this allocation desirable? 5 Exchange Economy with no production 6 Model of an Exchange Economy • endowment economy – no production • I people: i = 1, 2, ..., I • K goods: k = 1, 2, ..., K • individual i’s utility function u i • individual i’s endowment: how much of each good i owns initially think about market place with individuals trading with each other. prices? allocation? is the market efficient? 7 Model of an Exchange Economy for illustration • 2 people: Ann, Bob • 2 goods: x an y • endowment: exA , eyA for Ann, exB , eyB for Bob 8 Edgeworth Box 9 What is a good outcome? 10 Social Allocation • how much of each good each individual consumes: x A, y A , x B , y B • feasible: xA + xB ≤ exA + exB = ex yA + yB ≤ eyA + eyB = ey 11 Comparing Two Allocations • Allocation 1: x1A , y1A , x1B , y1B and allocation 2: x2A , y2A , x2B , y2B • Which allocation does Ann prefer? which does Bob prefer? 12 Pareto Criterion Allocation 1 is better than allocation 2 if everyone agrees that allocation 1 is better. Definition Allocation 1, x1A , y1A , x1B , x1B , is Pareto superior to allocation 2, x2A , y2A , x2B , x2B if no one is worse off under allocation 1 than under allocation 2: u A x1A , y1A ≥ u A x2A , y2A and u B x1B , y1B ≥ u B x2B , y2B and either Ann or Bob is strictly better off under allocation 1 than under allocation 2 (one of the inequality is strict). 13 Pareto Criterion Definition Allocation 1 is a Pareto improvement upon allocation 2 if allocation 1 is Pareto superior to allocation 2. 14 Pareto Criterion Only Ann and Bob. 15 Pareto Criterion Definition Allocation x∗A , y∗A , x∗B , x∗B is Pareto efficient if it is feasible and if there exists no other feasible allocation that is Pareto superior to x∗A , y∗A , x∗B , x∗B . 16 Pareto Criterion Definition Allocation star x∗A , y∗A , x∗B , y∗B is Pareto efficient if it is feasible and if there exists no other feasible allocation that is Pareto superior to x∗A , y∗A , x∗B , y∗B . 17 Social Allocation • how much of each good each individual consumes: x A, y A , x B , y B • feasible: xA + xB ≤ exA + exB = ex yA + yB ≤ eyA + eyB = ey • preference monotone: Pareto efficient allocation wastes no endowment =⇒ xB = ex − x A yB = ey − y A 18 Social Allocation xB = ex − x A yB = ey − y A 19 Pareto Criterion Is allocation Q Pareto superior to allocation E ? 20 Pareto Criterion Is allocation E Pareto efficient? 21 Pareto Criterion Allocation E is not Pareto efficient because allocation D is feasible and Pareto superior to E 22 Pareto Criterion We can find a Pareto improvement upon allocation E because Ann’s MRS at exA , eyA 6= Bob’s MRS at exB , eyB slope of Ann’s indifference curve at E 6= slope of Bob’s indifference curve at E 23 Pareto Efficient Allocations At a Pareto efficient allocation, Ann and Bob’s indifference curves have the same slope: A A B B Ann’s MRS at x , y = Bob’s MRS at x , y 24 Pareto Efficient Allocations 25 Pareto Efficient Allocation Can Be Unfair 26 The Set of All Pareto Efficient Allocations 27 Pareto Criterion • Often many Pareto efficient allocations exist • Does not say which Pareto efficient allocation is better • a Pareto efficient allocation may not be desirable — may be extremely unfair • The society may deem an extremely unfair Pareto efficient allocation worse than a fairer but Pareto inefficient allocation 28 Which Allocation will Happen? 29 Which Allocation will Happen if Bob is a Slave? 30 Which Allocations Might Happen if Trade Has to be Voluntary? 31 Which Allocation will Happen if Ann Has All the Bargaining Power? if Ann can choose any offer as long as Bob will agree to trade 32 Institution Determines Allocation • institution determines who can do what and payoffs given each person’s action, it affects allocation • example • slavery • farm communes • private property right and competitive market 33 Walrasian Equilibrium (Competitive Equilibrium) • Suppose there is a perfectly competitive market for each good (institutional assumption) • market for good k is perfectly competitive if each individual can buy and sell any amount of good k at the going price pk • reasonable assumption if there are many small buyers and many small sellers in each market, each taking price as given • Rationality assumption for individual behavior: • each consumer chooses the most preferred bundle in the budget set • each firm (if there is any) chooses input combinations (and output) to maximize profits • equilibrium price of the market for good k “clears the market”: at this price, quantity demanded = quantity supplied 34 Exchange economy (no production) 35 Walrasian Equilibrium (Competitive Equilibrium) A Walrasian equilibrium consists of price for each market, px∗ , py∗ , and allocation, x ∗A , y ∗A , x ∗B , y ∗B , such that • each market clears x ∗A + x ∗B = exA + exB y ∗A + y ∗B = eyA + eyB ; • each invidual chooses optimally x ∗A , y ∗A = Ann’s Marshallion demand at px∗ , py∗ , px∗ exA + py∗ eyA x ∗B , y ∗B = Bob’s Marshallion demand at px∗ , py∗ , px∗ exB + py∗ eyB 36 Walrasian Equilibrium (Competitive Equilibrium) • each market clears x ∗A + y ∗A = exA + exB y ∗A + y ∗B = eyA + eyB ; • there exists no x̃ A , ỹ A such that px∗ x̃ A + py∗ ỹ A ≤ px∗ exA + py∗ eyA u A x̃ A , ỹ A > u A x ∗A , y ∗A • There exists no x̃ B , ỹ B such that px∗ x̃ B + py∗ ỹ B ≤ px∗ exB + py∗ eyB u A x̃ B , ỹ B > u A x ∗B , y ∗B 37 Walrasian Equilibrium (Competitive Equilibrium) 38 First Welfare Theorem Theorem If x ∗A , y ∗A , x ∗B , y ∗B is a Walrasian equilibrium allocation, then it is Pareto efficient. 39 First Welfare Theorem Theorem If x ∗A , y ∗A , x ∗B , y ∗B is a Walrasian equilibrium allocation, then it is Pareto efficient. Proof • Let px∗ , py∗ be the equilibrium price that generates this allocation. • Suppose to the contrary that x ∗A , y ∗A , x ∗B , y ∗B is not Pareto efficient. • Then we can find another allocation x̃ A , ỹ A , x̃ B , ỹ B that is Pareto superior to x ∗A , y ∗A , x ∗B , y ∗B . • At least one person is better off under x̃ A , ỹ A , x̃ B , ỹ B than under x ∗A , y ∗A , x ∗B , y ∗B , and no one is worse off. • Either Ann is better off, or Bob is better off. 40 First Welfare Theorem — Proof continued Proof Suppose Ann better off. Then • px∗ x̃ A + py∗ ỹ A > px∗ exA + py∗ eyA because x ∗A , y ∗A maximizes Ann’s utility given prices px∗ , py∗ and her resulting income of px∗ exA + py∗ eyA • Bob is not worse off. So px∗ x̃ B + py∗ ỹ B ≥ px∗ exB + py∗ eyB . If x̃ B , ỹ B costs less than his income, then a bundle exists that costs no more than Bob’s income and is preferred to x ∗B , y ∗B . 41 First Welfare Theorem — Proof continued Proof • px∗ x̃ A + py∗ ỹ A > px∗ exA + py∗ eyA px∗ x̃ B + py∗ ỹ B ≥ px∗ exB + py∗ eyB . • So px∗ x̃ A + py∗ ỹ A + px∗ x̃ B + py∗ ỹ B > px∗ exA + py∗ eyA + px∗ exB + py∗ eyB 42 First Welfare Theorem — Proof Continued Proof • x̃ A , ỹ A , x̃ B , ỹ B must be feasible by definition of Pareto superiority. x ∗A , y ∗A , x ∗B , y ∗B is also feasible by the definition of equilibrium. So x̃ A + x̃ B = exA + exB = x ∗A + x ∗B ỹ A + ỹ B = eyA + eyB = y ∗A + y ∗B So px∗ x̃ A + x̃ B + py∗ ỹ A + ỹ B = px∗ exA + exB + py∗ eyA + eyB px∗ x̃ A + py∗ ỹ A + px∗ x̃ B + py∗ ỹ B = px∗ exA + py∗ eyA + px∗ exB + py∗ eyB . 43 Competitive Equilibrium and First Welfare Theorem equilibrium allocation affected by initial endowment E 44 Competitive Equilibrium and First Welfare Theorem equilibrium allocation affected by initial endowment E 45 Competitive Equilibrium and First Welfare Theorem equilibrium allocation affected by initial endowment E 46 Competitive Equilibrium and First Welfare Theorem First Welfare Theorem does not say equilibrium allocation is necessarily “better” than any other allocation. 47 Second Welfare Theorem Theorem Every Pareto efficient allocation can be the equilibrium allocation under some endowment exA , eyA , exB , eyB . 48 Second Welfare Theorem Theorem Every Pareto efficient allocation can be the equilibrium allocation under some endowment exA , eyA , exB , eyB . 49 Second Welfare Theorem Theorem Every Pareto efficient allocation can be the equilibrium allocation under some endowment exA , eyA , exB , eyB . 50 Second Welfare Theorem Theorem Every Pareto efficient allocation can be the equilibrium allocation under some endowment exA , eyA , exB , eyB . 51 First and second welfare theorem • Competitive equilibrium yields “good” outcome • All “good” outcomes can be implemented by a competitive equilibrium by changing the distribution of endowments 52 Example • Suppose x is amount of rice and y is the amount of vegetables • exA , eyA = (1, 0), exB , eyB = (0, 1) • Ann and Bob’s preference can be both represented by √ √ u (x, y ) = α x + y with α > 1 53 Example √ √ u (x, y ) = α x + y • feasibility and no wasted resources • allocation xB = exA + exB − x A = 1 − x A yB = eyA + eyB − y A = 1 − y A x A , y A , 1 − x A , 1 − y A on the edgeworth box: Ann’s MRS Bob’s MRS = ∂u ∂x ∂u ∂y = ∂u ∂x ∂u ∂y r yA xA r yB |(x A ,y A ) = α |(x B ,y B ) = α xB s =α 1 − yA 1 − xA 54 Example Pareto efficient allocation: Ann’s MRS at x A , y A = Bob’s MRS at x B , y B r α so yA =α xA s 1 − yA 1 − xA yA 1 − yA = xA 1 − xA y A − x Ay A = x A − x Ay A So xA = yA xB = 1 − xA = 1 − yA = yB 55 Example So every allocation x A , y A , x B , y B = ((t, t) , (1 − t, 1 − t)) where t ∈ [0, 1] is Pareto efficient 56 Example —- Competitive Equilibrium • Need to find px∗ , py∗ such that market clears when both Ann and Bob optimizes. • can normalize py∗ = 1 because general inflation does not change demand functions 57 Example —- Competitive Equilibrium • Given px∗ , py∗ = (px∗ , 1), Ann will choose x ∗A , y ∗A that maximizes her utility given her budget constraint px∗ x A + y A ≤ px∗ exA + eyA • FOC MRS = r α y ∗A = px∗ x ∗A So y ∗A = px∗ = px∗ py∗ px∗ α 2 x ∗A 58 Example —- Competitive Equilibrium • Substitute into budget line: px∗ exA + eyA = px∗ x ∗A + y ∗A = px∗ x ∗A + So x ∗A = px∗ α 2 x ∗A px∗ exA + eyA ∗ 2 px∗ + pαx 59 Example —- Competitive Equilibrium • Given px∗ , py∗ = (px∗ , 1), Ann’s optimal consumption bundle is px∗ exA + eyA ∗ 2 px∗ + pαx ∗ 2 px = x ∗A α x ∗A = y ∗A • Do the same for Bob: px∗ exB + eyB ∗ 2 px∗ + pαx ∗ 2 px = x ∗B α x ∗B = y ∗B 60 Example —- Competitive Equilibrium • Market for x has to clear: x ∗A + x ∗B = exA + exB exA + exB =x ∗A +x ∗B px∗ exA + exB + eyA + eyB = ∗ 2 px∗ + pαx • exA + exB = 1; eyA + eyB = 1 then px∗ α 2 =1 So px∗ = α • Plug it back into Ann and Bob’s demand function to find equilibrium allocation. 61 Example —- Competitive Equilibrium • The more people like rice, the more expensive it is • Write exA + exB = ex ; eyA + eyB = ey px∗ α 2 ex = ey r ey px∗ = α ex • More rare rice is, the more expensive it is 62 With production - one firm one consumer 63 Competitive equilibrium • One firm that produces y with production function y = f (x). • Consumer with utility function u(x, y ). • Consumer with endowment ex , ey and receives firm’s profit. 64 Competitive equilibrium • Firm maximize profit π = py∗ f (x) − px∗ x. (Solution x f ) • Consumer maximizes utility function u(x, y ) with price px∗ , py∗ and I . (Solution x c , y c ) • Income I = px∗ ex + py∗ ey + π. • Market clears: x c = ex − x f and y c = ex + f (x f ) 65 Example —- Competitive Equilibrium √ • f (x) = β x √ • u (x, y ) = xy • ex = 1, ey = 0 66 Example —- Competitive Equilibrium • Normalize py∗ = 1 • Firm: • FOC: py∗ f 0 (x f ) − px∗ = 0 • √β f = px∗ 2 x • xf = β2 4(px∗ )2 , yf = β2 2px∗ , π= β2 4px∗ 67 Example —- Competitive Equilibrium • Consumer: 2 β px∗ + 4p ∗ I ∗ x x = ∗ = ∗ 2px 2px ∗ I p β2 y∗ = ∗ = x + ∗ 2py 2 8px 68 Example —- Competitive Equilibrium • Market clears: y f = y ∗ + ey β px∗ β = + ∗ ∗ 2px 2 8px 3β = px∗ 4px∗ √ 3 ∗ px = β 2 • price of x increases in β. 69 Example —- Competitive Equilibrium • Check: xf = ∗ β2 1 = 4(px∗ )2 3 ex − x = 1 − β2 4px∗ 2px∗ px∗ + = 1 3 70