Topic 17 Equilibrium Liquid-vapour equilibrium The equilibrium law.
General Equilibrium (without Production) or Exchange...
Transcript of General Equilibrium (without Production) or Exchange...
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core Competitive Equilibrium Welfare
General Equilibrium (without Production)or
Exchange(Chapter 31)
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core Competitive Equilibrium Welfare
General Equilibrium
• Events in one market have effects on other markets (spillovers)
• Demand for x depends upon prices of complements,substitutes; income
• Supply of x depends upon factor prices
• Previously, we’ve taken these as given– doing partialequilibrium analysis
• But its important to understand interdependence of markets–general equilibrium analysis
Partial equilibrium analysis says that competitive markets yieldefficient outcomes—is this still true in general equilibrium?
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core Competitive Equilibrium Welfare
General Equilibrium
Our approach:
• Simple environment—the entire economy• 2 kinds of goods• 2 people
• Focus on exchange• Abstract away from production of new goods• Give people endowments• Specify preferences• Allow them to trade
• Make predictions about behavior of utility-maximizers
• Evaluate welfare
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core Competitive Equilibrium Welfare
Endowment Economy
• Consumers A and B; goods 1 and 2
• Endowments: ωA = (ωA1 , ω
A2 ) and ωB = (ωB
1 , ωB2 )
• Example: ωA = (6, 4) and ωB = (2, 2)
• This means total endowment of good 1 isωA1 + ωB
1 = 6 + 2 = 8 and of good 2 is ωA2 + ωB
2 = 4 + 2 = 6
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core Competitive Equilibrium Welfare
Allocations
• Endowment represents where people start, but through trade,their allocations may change
• General allocation or consumption: xA = (xA1 , xA2 ) and
xB = (xB1 , xB2 )
• (xA, xB) is feasible if it uses at most the aggregateendowment:
xA1 + xB1 ≤ ωA1 + ωB
1 and xA2 + xB2 ≤ ωA2 + ωB
2
• Helpful graphical tool: Edgeworth Box
• Allows us to simply depict all feasible allocations
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core Competitive Equilibrium Welfare
Edgeworth Box
Edgeworth Box
A( , )6 4
B( , )2 2
OA
OB
6
8
4
6
2
2
Theendowmentallocation
The Endowment Allocation
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core Competitive Equilibrium Welfare
Edgeworth Box
Edgeworth Box
Feasible Reallocations
OA
OB
1 1A B
xA2
2
2
A
B
xA1
xB1
xB2
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core Competitive Equilibrium Welfare
How do we think about equilibrium?
• In partial equilibrium analysis:• Treat each good separately• Find p and q that equate supply and demand
• But this is general equilibrium analysis: where do supply anddemand come from?
• A and B can trade with each other
• For everything to be balanced, the amount that A gives uphas to equal amount that B receives (for each good, and viceversa)
• In other words Supply = Demand for each good
• This will determine prices for each good
• How do we find supply and demand curves?
• Go back to utility maximization problem
• Need to specify preferences to do this
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core Competitive Equilibrium Welfare
Utility maximization
• Preferences are given
• Given prices for each good, endowment bundle serves asincome
• Can write down budget constraint
p1x1 + p2x2 ≤ p1ω1 + p2ω2
• Solve utility maximization problem
• Gives you optimal allocation, as a function of price ratio
• x∗1 − ω1 > 0 means person demands more of good 1
• x∗1 − ω1 < 0 means person is willing to supply good 1
• Key question: what prices will make it so that A demandexactly as much of each good as B supplies?
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core Competitive Equilibrium Welfare
Consumer A’s Preferences
Preferences of A
Adding Preferences to the Box
2A
1A
xA2
xA1
More
preferred
For consumer A.
OA
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core Competitive Equilibrium Welfare
Consumer B’s Preferences
Preferences of B
Adding Preferences to the Box
xB2
xB1
More
preferred
For consumer B.
OB
2B
1B
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core Competitive Equilibrium Welfare
Consumer B’s Preferences
Preferences of B
Adding Preferences to the Box
2B
1B
xB1
xB2
More
preferred
For consumer B. OB
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core Competitive Equilibrium Welfare
Putting Them Both Together
Putting both of them together
Edgeworth’s Box
2A
1A
xA2
xA1
OA
2B
1B
xB1
xB2
OB
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core Competitive Equilibrium Welfare
Pareto-improving allocations
• Given a particular allocation, a Pareto-improving allocationimproves the welfare of at least one consumer withoutreducing the welfare of another.
• How do we depict Pareto-improving allocations in theEdgeworth box?
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core Competitive Equilibrium Welfare
Pareto-Improving Allocations
Putting both of them together
Edgeworth’s Box
2A
1A
xA2
xA1
OA
2B
1B
xB1
xB2
OB
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core Competitive Equilibrium Welfare
Pareto-Improving Allocations
Pareto-improving allocations
Pareto-Improvements
2A
1A
xA2
xA1
OA
2B
1B
xB1
xB2
OB
The set of Pareto-improving allocations
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core Competitive Equilibrium Welfare
Pareto-optimal allocations
• An allocation is Pareto-optimal if it is feasible and there isother feasible allocation that is a Pareto-improvement over it.
• In other words, there is no was to make anyone better offwithout making someone worse off.
• The set of all Pareto-optimal allocations is called the contractcurve.
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core Competitive Equilibrium Welfare
A Pareto-Optimal Allocation
A Pareto-Optimal Allocation
Pareto-Optimality
A is strictly better offbut B is strictly worse
off
B is strictly betteroff but A is strictlyworse off
Both Aand B are
worseoff
Both A andB are worseoff
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core Competitive Equilibrium Welfare
The Set of Pareto-Optimal Allocations
Pareto-Optimal Allocations
Pareto-Optimality
2A
1A
xA2
xA1
OA
2B
1B
xB1
xB2
OB
All the allocations marked bya are Pareto-optimal.
The contract curve
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core Competitive Equilibrium Welfare
Pareto-optimal Allocations
• From the figures, we can see that an allocation at which theindifference curves of the two consumers are tangent must bePareto-optimal
• Tangency implies they have the same slope
• What is the slope of an indifference curve? The Marginal rateof substitution (MRS)!
• Condition for Pareto-optimality:
MRSA =
∂uA(xA1 ,xA2 )
∂xA1∂uA(xA1 ,x
A2 )
∂xA2
=
∂uB(xB1 ,xB2 )
∂xB1∂uB(xB1 ,xB2 )
∂xB2
= MRSB
• We also require feasibility:
xA1 + xB1 = ωA1 + ωB
1 and xA2 + xB2 = ωA2 + ωB
2
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core Competitive Equilibrium Welfare
ExampleIdentifying Pareto-optimal allocations• Recall total endoments: ωA
1 + ωB1 = 6 + 2 = 8 and
ωA2 + ωB
2 = 4 + 2 = 6• Let uA(xA1 , x
A2 ) = ln(xA1 ) + 2 ln(xA2 ) and
uB(xB1 , xB2 ) = ln(xB1 ) + 2 ln(xB2 ).
• MRS of consumer A:
MRSA =
1xA12xA2
=xA2
2xA1
• Similarly,
MRSB =
1xB12xB2
=xB2
2xB1
• So a Pareto-optimum is a feasible allocation for which
xA22xA1
=xB2
2xB1
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core Competitive Equilibrium Welfare
Clicker Vote
Which of these allocations is Pareto Optimal?
A) (xA1 , xA2 , x
B1 , x
B2 ) = (4, 2, 6, 3)
B) (xA1 , xA2 , x
B1 , x
B2 ) = (4, 1, 4, 5)
C) (xA1 , xA2 , x
B1 , x
B2 ) = (4, 4, 5, 1)
D) (xA1 , xA2 , x
B1 , x
B2 ) = (4, 3, 4, 3)
E) (xA1 , xA2 , x
B1 , x
B2 ) = (6, 4.5, 2, 1.5)
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core Competitive Equilibrium Welfare
Example
Identifying Pareto-optimal allocations
• Recall that a Pareto-optimum is a feasible allocation for which
xA22xA1
=xB2
2xB1
• In other words, A and B need to have the same x1 : x2 ratio,which makes sense, because they have identical preferences
• Clicker option A meets the tangency condition, but is notfeasible.
• B is feasible, but does not meet the tangency condition
• C satisfies neither condition.
• However, both D and E satisfy both conditions, so they bothare Pareto-optimal.
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core Competitive Equilibrium Welfare
ExampleFinding all Pareto-optimal allocations (deriving the contract curve)
• We can simplify tangency condition to:
xA2xA1
=xB2xB1
• Recall endowment/feasibility requirement:
xA1 + xB1 = 8 and xA2 + xB2 = 6
• Re-write tangency condition, substituting xB1 = 8− xA1 andxB2 = 6− xA2 :
xA2xA1
=6− xA28− xA1
or
xA2 =3
4xA1
• This is the equation of the contract curve.• In this case, it’s just the diagonal of the rectangle
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core Competitive Equilibrium Welfare
Core Allocations
How can we narrow down our prediction about the outcomeresulting from trade?
• We’ve derived the contract curve: xA2 = 34x
A1
• Note that clicker options D and E are both on this curve.
• However, Pareto Optimal, but given her endowment,(ωB = (2, 2), Person B would never agree to trade toallocation E = (6,4.5,2,1.5).
• We need to restrict attention to Pareto-optimal allocationsthat are Pareto-improvements over the initial endowment.
• These allocations are called core allocations, or the core—theset of all PO allocations that are welfare improving for bothconsumers relative to their own endowments
• An allocation will be in the core if it is feasible and it is notblocked by any consumer
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core Competitive Equilibrium Welfare
Core Allocations
Core Allocations
The Core
2A
1A
xA2
xA1
OA
2B
1B
xB1
xB2
OB
Pareto-optimal trades blockedby B
Pareto-optimal trades blockedby A
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core Competitive Equilibrium Welfare
The Core
Core Allocations
The Core
2A
1A
xA2
xA1
OA
2B
1B
xB1
xB2
OB
Pareto-optimal trades not blockedby A or B are the core.
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core Competitive Equilibrium Welfare
Competitive General EquilibriumA competitive general equilibrium is defined by prices (p1, p2) andallocations (xA, xb) such that
1. Utility Maximization: Given (p1, p2), the allocation(xA,XB) solves each consumer’s utility maximizationproblem. That is, xA is the solution to
max(xA1 ,x
A2 )UA(xA1 , x
A2 )
such thatp1x
A1 + p2x
A2 = p1ω
A1 + p2ω
A2 ,
and similarly for B.
2. Market Clearing: the total demand of good 1 is equal to thetotal endowment (supply) of good 1 and similarly for good 2:
xA1 + xB1 ≤ ωA1 + ωB
1 and xA2 + xB2 ≤ ωA2 + ωB
2
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core Competitive Equilibrium Welfare
Trade in Competitive MarketsA’s utility maximization:
A’s Utility Maximization
Trade in Competitive Markets
2A
1A
xA2
xA1
OA
For consumer A.
p x p x p pA A A A1 1 2 2 1 1 2 2
x A2*
x A1*
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core Competitive Equilibrium Welfare
Trade in Competitive MarketsB’s utility maximization
B’s Utility Maximization
Trade in Competitive Markets
2B
1B
xB2
xB1
For consumer B.
OB x B1*
x B2*
p x p x p pB B B B1 1 2 2 1 1 2 2
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core Competitive Equilibrium Welfare
Trade in Competitive MarketsA’s utility maximization:
A’s Utility Maximization
Trade in Competitive Markets
2A
1A
xA2
xA1
OA
2B
1B
xB1
xB2
OB
Budget constraint for consumer A
x A2*
x A1*
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core Competitive Equilibrium Welfare
Trade in Competitive MarketsB’s utility maximization
B’s Utility Maximization
Trade in Competitive Markets
2A
1A
xA2
xA1
OA
2B
1B
xB1
xB2
OB
Budget constraint for consumer B
x A2*
x A1*
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core Competitive Equilibrium Welfare
Trade in Competitive MarketsMarket Clearing:
Market Clearing
Trade in Competitive Markets
2A
1A
xA2
xA1
OA
2B
1B
xB1
xB2
OB
x A2*
x A1*
x B1*
x B2*
Sox xA B A B
1 1 1 1* *
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core Competitive Equilibrium Welfare
What can we say about welfare?
Two important results:
1. First Welfare Theorem: Given that consumers’ preferences arewell-behaved, trading in perfectly competitive marketsimplements a Pareto-optimal allocation of the economy’sendowment. That is, CE ⇒ PO.
2. Second Welfare Theorem: any Pareto-optimal allocation (i.e.any point on the contract curve) can be achieved by tradingin competitive markets provided that endowments are firstappropriately arranged among the consumers. That is,PO ⇒ CE .
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core Competitive Equilibrium Welfare
What can we say about welfare?First Welfare Theorem: CE must be on contract curve
First Welfare Theorem
First Fundamental Theorem
2A
1A
xA2
xA1
OA
2B
1B
xB1
xB2
OB
A*2x B*
2x
A*1x
B*1x
Implemented by competitive
trading from the endowment .
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core Competitive Equilibrium Welfare
What can we say about welfare?Second Welfare Theorem: any PO allocation is a CE . . .
Second Welfare Theorem
Second Fundamental TheoremxA
2
xA1
OA
xB1
xB2
OB
But this allocation is implemented
by competitive trading from .
A1
B2
B1
A2
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core Competitive Equilibrium Welfare
What can we say about welfare?Second Welfare Theorem: any PO allocation is a CE . . . from anappropriate starting point!
Second Welfare Theorem
Second Fundamental TheoremxA
2
xA1
OA
xB1
xB2
OB
But this allocation is implemented
by competitive trading from .
A1
B2
B1
A2