Part III. General Equilibrium
Intermediate Microeconomics (22014)
Part III. General Equilibrium
Instructor: Marc Teignier-Baqué
First Semester, 2011 Outline Part III. General Equilibrium Part III. General Equilibrium
Exchange
Production Welfare 1. Pure Exchange Economy (Varian, Ch 31) 1.1 Edgeworth Box 1.2 The Core 1.3 Competitive Equilibrium 1.4 Welfare Theorems 1.5 Walras' Law
2. Production (Varian, Ch 32)
3. Welfare (Varian, Ch 33) Topic 6. General Equilibrium Part III. General Equilibrium
Exchange I Up until now, partial equilibrium analysis: Production I markets for goods analyzed in isolation, ignoring eect of Welfare other prices on the market equilibrium;
I demand and supply functions of own price alone.
I In general, however, demand and supply in several markets interact to determine equilibrium prices of all goods.
I Substitutes and complements. I People's income aected by goods sold.
I In top 6, general equilibrium analysis: all markets clear simultaneously.
I Considerations of Pareto e ciency and also of welfare distribution and "social preferences." PURE EXCHANGE ECONOMY (Varian, Ch 31) Part III. General Equilibrium I Since very complex problem, simpli cations adopted:
Exchange I Only competitive markets studied, so that consumers and Edgeworth Box producers take prices as given. The Core Competitive I Situations with, at most, two goods and two consumers. equilibrium Welfare theorems I First, pure exchange economy: xed endowments, no Walras' Law description of resources conversion to consumables. Production I Afterwards, production introduced into the model. Welfare I Pure exchange economy:
I Two consumers, A and B, two goods, 1 and 2. I Endowments of goods 1 and 2: A A A B B B ω = ω1 ,ω2 , ω = ω1 ,ω2 .
I Given a price vector (p1,p2), consumers choose their favorite aordable allocation (as in topic 1): A A A A p1x1 + p2x2 ≤ p1ω1 + p2ω2 B B B B p1x1 + p2x2 ≤ p1ω1 + p2ω2
I Prices must be such that allocations chosen are feasible: A B A B A B A B x1 + x1 ≤ ω1 + ω1 , x2 + x2 ≤ ω2 + ω2 Edgeworth Box Part III. General Equilibrium I Edgeworth box is diagram showing all possible Exchange allocations of the available quantities of goods 1 and 2 Edgeworth Box The Core between the two consumers. Competitive equilibrium Welfare theorems Walras' Law I The dimensions of the box are the quantities available Production of the goods.
Welfare I The allocations depicted are the feasible allocations.
Height = A B 2 2
A B Width = 1 1 Endowment allocation Part III. General Equilibrium I The endowment allocation is the before-trade Exchange allocation: Edgeworth Box The Core Competitive equilibrium Welfare theorems B Walras' Law ω1 Production OB Welfare B A ω 2 ω 2 + Endowment B A ω 2 ω 2 allocation
OA A ω1 A B ω11+ ω Feasible reallocations Part III. General Equilibrium I Which reallocation will consumers choose?
Exchange I Feasible. Edgeworth Box The Core I Pareto-improving over the endowment allocation. Competitive equilibrium I An allocation is feasible if and only if Welfare theorems Walras' Law xA xB A B Production 1 + 1 ≤ ω1 + ω1 A B A B Welfare x2 + x2 ≤ ω2 + ω2
I All points in the box, including the boundary, represent feasible allocations of the combined endowments:
B x1 OB
ω A B 2 x2 + B ω 2 A x2 OA A x1 A B ω11+ ω Pareto-improving allocations Part III. General Equilibrium
Exchange I An allocation is Pareto-improving over the endowment Edgeworth Box The Core allocation if it improves the welfare of a consumer Competitive equilibrium without reducing the welfare of another. Welfare theorems Walras' Law Production I Preferences of consumers A and B: Welfare
Preferences of consumer B xA Preferences consumer A B 2 B 1 OB x1
B 2
A 2
OA A A x B 1 x1 2 Pareto-improving allocations Part III. General Equilibrium I An allocation is Pareto-improving over the endowment Exchange allocation if it improves the welfare of a consumer Edgeworth Box The Core without reducing the welfare of another. Competitive equilibrium Welfare theorems Walras' Law Production B Welfare 1 OB
A B 2 2
OA A 1 Set of Pareto‐improving allocations
I Since each consumer can refuse to trade, the only possible outcomes from exchange are Pareto-improving allocations. Contract curve Part III. General Equilibrium I An allocation is Pareto-optimal if the only way one Exchange Edgeworth Box consumer's welfare can be increased is to decrease the The Core Competitive welfare of the other consumer. equilibrium Welfare theorems Walras' Law The set of all Pareto-optimal allocations is called Production I
Welfare contract curve.
Pareto‐optimal allocations are marked by . Convex indifference curves are tangent at . B 1 OB
A B 2 2
OA A 1 The contttract curve The Core Part III. General Equilibrium
Exchange I The core is the set of all Pareto-optimal allocations Edgeworth Box that are welfare-improving for both consumers relative The Core Competitive equilibrium to their own endowments. Welfare theorems Walras' Law Production
Welfare B 1 OB
A B 2 2
OA A 1 The Core: Pareto‐optimal trades not blocked by A or B.
I Rational trade should achieve a core allocation. Trade in competitive markets Part III. General Equilibrium I Speci c core alloation achieved depends upon the Exchange manner in which trade is conducted. Edgeworth Box The Core Competitive In perfectly competitive markets, each consumer is a equilibrium I Welfare theorems price-taker trying to maximize her own utility given Walras' Law Production (p1,p2) and her own endowment: Welfare A x2 Consumer A optimization A A A A px11 px 22 p 1 1 p 2 2
* A x2 A 2 A x1 OA x* A A 1 1
I Similarly for consumer B. Trade in competitive markets Part III. General Equilibrium I At equilibrium prices p1 and p2, both consumers Exchange maximize their own utility and both markets clear: Edgeworth Box The Core Competitive A B A B equilibrium x x Welfare theorems 1 + 1 = ω1 + ω1 Walras' Law xA xB A B Production 2 + 2 = ω2 + ω2
Welfare Budget constraint for consumer B
x*B B 1 1 OB
* A x *B x 2 2 A B 2 2 A OA * A x1 1 Equilibrium allocation Budget constraint for consumer A First fundamental theorem of welfare economics Part III. General Equilibrium
Exchange Theorem Edgeworth Box Given that consumers' preferences are well-behaved, trading The Core Competitive equilibrium in perfectly competitive markets implements a Pareto-optimal Welfare theorems Walras' Law allocation of the economy's endowment. Production
Welfare I Note: Indierence curves are tangent, which implies that the equilibrium allocation is Pareto optimal. Second fundamental theorem of welfare economics Part III. General Equilibrium Theorem Exchange Given that consumers' preferences are well-behaved, for any Edgeworth Box The Core Pareto-optimal allocation, there are prices and an allocation Competitive equilibrium of the total endowment that makes the Pareto-optimal Welfare theorems Walras' Law allocation implementable by trading in competitive markets. Production In other words, any Pareto-optimal allocation can be Welfare I achieved by trading in competitive markets provided that endowments are rst appropriately rearranged.
Pareto‐optimal allocation cannot be implemented by competitive trading from initial endowment but it can be implemented by competitive trading from alternative endowment .
OB
OA Walras' Law Part III. General Equilibrium
Exchange Edgeworth Box The Core Theorem Competitive equilibrium If consumer's preferences are well-behaved, so that for any Welfare theorems Walras' Law positive prices (p1,p2) consumers spend all their budget, the Production summed market value of excess demands is zero. This is Welfare Walras' Law.
A A A A p1x1 + p2x2 = p1ω1 + p2ω2 B B B B p1x1 + p2x2 = p1ω1 + p2ω2
⇒
A B A B A B A B p1 x1 + x1 − ω1 − ω1 + p2 x2 + x2 − ω2 − ω2 = 0 Implications of Walras' Law Part III. General Equilibrium I One implication of Walras' Law for a two-commodity Exchange exchange economy is that if one market is in equilibrium Edgeworth Box The Core then the other market must also be in equilibrium. Competitive equilibrium Welfare theorems Walras' Law A B A B A B A B p1 x1 + x1 − ω1 − ω1 +p2 x2 + x2 − ω2 − ω2 = 0 Production
Welfare A B A B A B A B ⇒ If x1 + x1 = ω1 + ω1 , then x2 + x2 = ω2 + ω2 .
I Another implication of Walras' Law for a two-commodity exchange economy is that an excess supply in one market implies an excess demand in the other market.
A B A B A B A B p1 x1 + x1 − ω1 − ω1 +p2 x2 + x2 − ω2 − ω2 = 0
A B A B A B A B ⇒ If x1 + x1 < ω1 + ω1 , then x2 + x2 > ω2 + ω2 . Outline Part III. General Equilibrium Part III. General Equilibrium
Exchange
Production Robinson Crusoe Economy Competitive Equilibrium 1. Pure Exchange Economy (Varian, Ch 31) Welfare Theorems Welfare 2. Production (Varian, Ch 32) 2.1 Robinson Crusoe economy 2.2 Competitive equilibrium 2.3 Welfare theorems
3. Welfare (Varian, Ch 33) PRODUCTION (Varian, Ch 32) Part III. General Equilibrium I Add input and output markets, rms' technologies. Exchange I Robinson Crusoe's Economy: Production One agent: Robinson Crusoe. Robinson Crusoe I Economy Endowment: a xed quantity of time. Competitive I Equilibrium Decision: use time for labor (production of coconuts) or Welfare Theorems I Welfare leisure.
I Technology: coconuts are obtained from labor according to the production function C = f (L).
Coconuts
Production function
Feasible production pllans 0 24 Labor (hours) Robinson Crusoe's preferences Part III. General Equilibrium I Indierence curves in the leisure-coconut diagram: Exchange coconut is a good, leisure is a good:
Production Robinson Crusoe Coconuts Economy Competitive More preferred Equilibrium Welfare Theorems Welfare
0 24 Leisure (hours)
I Indierence curves in the labor-coconuts diagram: coconut is a good, labor is a bad.
Coconuts More preferred
0 24 Labor (hours) Robinson Crusoe's choice Part III. General Equilibrium Robinson chooses time allocation and, as a result, his Exchange I
Production consumption of coconuts: Robinson Crusoe Economy Competitive Equilibrium Welfare Theorems Coconuts Welfare
MRS = MPL Production function C* Outpu t Labor Leisure 0 L* 24 Labor (hours) 24 0 Leisure (hours) Competitive equilibrium in the Robinson economy Part III. General Equilibrium
Exchange
Production Robinson Crusoe I Robinson esquizofrenia: Economy Competitive Equilibrium I We rst consider Robinson as a pro t-maximizing Welfare Theorems rm, who takes prices as given and decides how much Welfare hours to hire and how much to produce.
I Then, we consider Robinson as a utility-maximizing consumer who gets the rm pro ts and decides his hours of work and his consumption of coconuts.
I Let p be the coconuts price and w the wage rate.
I Use coconuts as the numeraire good; i.e. price of a coconut = 1. Robinson as a rm Part III. General Equilibrium I Optimization problem of the rm: given w, choose labor Exchange demand and coconut supply to maximize pro ts: Production ∗ Robinson Crusoe maxπ = C − wL = f (L) − wL ⇒ MP (L ) = w Economy L Competitive Equilibrium Welfare Theorems ∗ ∗ ∗ I Labor demanded: L , output supplied: C = f (L ). Welfare
I Graphically, rm demands L such that production function tangent to isopro t line:
Coconuts
w = MPL Isoprofit line: ** C wL * C* Production function *
0 L* 24 Labor (hours) Robinson as consumer Part III. General Equilibrium I Optimization problem of the consumer: choose labor Exchange supply and coconut demand to maximize utility subject Production to the budget constraint: Robinson Crusoe Economy U C L L Competitive ∗ ∂ ( , )/∂ Equilibrium maxU (C,L) s.t. C = π + wL ⇒ = w Welfare Theorems C,L ∂U (C,L)/∂C Welfare ∗ ∗ I Labor supplies: L , coconuts demanded: C .
I Graphically, consumer chooses C and L such that the indierence curve is tangent to the budget constraint:
Coconuts
MRS = w Budget constraint: C *wL. C* *
0 L* 24 Labor (hours) Market equilibrium Part III. General Equilibrium
Exchange I In equilibrium, wage rate must be such that Production Robinson Crusoe Economy quantity labor demanded = quantity labor supplied Competitive Equilibrium (quantity output supplied = quantity output demanded) Welfare Theorems Welfare Coconuts
MRS = w = MPL
C* *
0 L* 24 Labor (hours) First Fundamental Theorem of Welfare Economics Part III. General Equilibrium Theorem Exchange If consumers' preferences are convex and there are no Production externalities in consumption or production, a competitive Robinson Crusoe Economy market equilibrium is Pareto e cient. Competitive Equilibrium Welfare Theorems I Pareto e ciency: MRS = MP: Welfare I Competitive equilibrium achieves Pareto e ciency: w is the common slope of the ispro t line and the budget constraint.
Coconuts MRS = MP
0 24 Labor (hours) Second Fundamental Theorem of Welfare Part III. General Equilibrium Economics
Exchange
Production Robinson Crusoe Economy Competitive Equilibrium Welfare Theorems Welfare Theorem If consumers' preferences are convex, rms' technologies are convex, and there are no externalities in consumption or production any Pareto e cient economic state can be achieved as a competitive market equilibrium. Non-convex technologies Part III. General Equilibrium
Exchange I The First Welfare Theorems still holds if rms have Production non-convex technologies since it does not rely upon Robinson Crusoe Economy Competitive rms' technologies being convex. Equilibrium Welfare Theorems Welfare I The Second Welfare Theorem does not hold if rms have non-convex technologies.
Coconuts Coconuts MRS = MPL If compet iti ve equilib ri um exists, the common slope MRS = MPL. This Pareto optimal is the relative wage rate w allocation cannot be that i mpl ement s the PtPareto implemented by a efficient plan by competitive equilibrium. decentralized pricing.
0 24 Labor (hours) 0 24 Labor (hours) Outline Part III. General Equilibrium Part III. General Equilibrium
Exchange
Production
Welfare
1. Pure Exchange Economy (Varian, Ch 31
2. Production (Varian, Ch 32)
3. Welfare (Varian, Ch 33) WELFARE (Varian, Ch 33) Part III. General Equilibrium
Exchange
Production
Welfare I Social choice: Dierent economic states will be preferred by dierent individuals. How can individual preferences be aggregated into a social preference over all possible economic states?
I Fairness: Some Pareto e cient allocations are unfair (for example, one consumer eats everything). Under what conditions, competitive markets guarantee that a fair allocation is achieved? Social welfare functions Part III. General Equilibrium
Exchange Let ui (x) be individual i's utility from overall allocation x. Production
Welfare I Utilitarian social welfare function: n W = ∑ ui (x) i=1
I Weighted-sum social welfare function:
n W = ∑ ai ui (x), ai > 0 i=1
I Minimax welfare function:
W = min {u1 (x),u2 (x),....,un (x)} Fair allocations Part III. General Equilibrium
Exchange
Production
Welfare I An allocation is fair if it is
I Pareto e cient I envy free (no agent prefers the allocation of other agents to their own).
I If every agent's endowment is identical, then trading in competitive markets results in a fair allocation (may not be true for non-competitive markets).