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General Equilibrium (without Production)
or
Exchange
(Chapter 31)
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core
General Equilibrium
- Events in one market have e↵ects 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 partial equilibrium analysis
- But its important to understand interdependence of markets– general equilibrium analysis Partial equilibrium analysis says that competitive markets yield e cient outcomes—is this still true in general equilibrium?
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
Endowment Economy
- (^) Consumers A and B; goods 1 and 2
- Endowments: !A^ = (! 1 A ,! 2 A ) and !B^ = (!B 1 , !B 2 )
- (^) Example: !A^ = (6, 4) and !B^ = (2, 2)
- (^) This means total endowment of good 1 is !A 1 + !B 1 = 6 + 2 = 8 and of good 2 is! 2 A + !B 2 = 4 + 2 = 6
Edgeworth Box
Edgeworth Box Feasible Reallocations
OA
OB
Z (^) 1 Z 1
A B
x
A
Z
Z
A
B
x
A
x
B
x
B
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core
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 and demand come from?
- (^) A and B can trade with each other
- (^) For everything to be balanced, the amount that A gives up has to equal amount that B receives (for each good, and vice versa)
- (^) 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
Utility maximization
- Preferences are given
- Given prices for each good, endowment bundle serves as income
- (^) Can write down budget constraint p 1 x 1 + p 2 x 2 p 1! 1 + p 2! 2
- (^) Solve utility maximization problem
- (^) Gives you optimal allocation, as a function of price ratio
- x 1 ⇤ ! 1 > 0 means person demands more of good 1
- (^) x 1 ⇤ ! 1 < 0 means person is willing to supply good 1
- (^) Key question: what prices will make it so that A demand exactly as much of each good as B supplies? General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core
Consumer A’s Preferences
Preferences of A
Adding Preferences to the Box
Z 2
A
Z 1
A
x
A
x
A
M
o
re
p
re
fe
rr
e
d
For consumer A.
OA
Putting Them Both Together
Putting both of them together Edgeworth’s Box Z (^) 2
A
Z 1
A
x
A
x
A
OA
Z (^) 2
B
Z 1
B
x
B
x
B
OB
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core
Pareto-improving allocations
- (^) Given a particular allocation, a Pareto-improving allocation improves the welfare of at least one consumer without reducing the welfare of another.
- How do we depict Pareto-improving allocations in the Edgeworth box?
Pareto-Improving Allocations
Putting both of them together Edgeworth’s Box Z (^) 2
A
Z 1
A
x
A
x
A
OA
Z (^) 2
B
Z 1
B
x
B
x
B
OB
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core
Pareto-Improving Allocations
Pareto-improving allocations Pareto-Improvements Z (^) 2
A
Z 1
A
x
A
x
A
OA
Z (^) 2
B
Z 1
B
x
B
x
B
OB
The set of Pareto-
improving allocations
The Set of Pareto-Optimal Allocations
Pareto-Optimal Allocations Pareto-Optimality Z (^) 2
A
Z 1
A
x
A
x
A
OA
Z (^) 2
B
Z 1
B
x
B
x
B
OB
All the allocations marked by
a are Pareto-optimal.
The contract curve
General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core
Pareto-optimal Allocations
- From the figures, we can see that an allocation at which the indi↵erence curves of the two consumers are tangent must be Pareto-optimal
- Tangency implies they have the same slope
- (^) What is the slope of an indi↵erence curve? The Marginal rate of substitution (MRS)!
- Condition for Pareto-optimality: MRS A = @uA(x 1 A ,x 2 A ) @x 1 A @uA(x 1 A ,x 2 A ) @x 2 A
@uB^ (x 1 B ,x 2 B ) @x 1 B @uB^ (x 1 B ,x 2 B ) @x 2 B
= MRS
B
- (^) We also require feasibility: x A 1 +^ x B 1 =^! A 1 +^! B 1 and^ x A 2 +^ x B 2 =^! A 2 +^! B 2
Example
Identifying Pareto-optimal allocations
- (^) Recall total endoments:! 1 A +! 1 B = 6 + 2 = 8 and !A 2 + !B 2 = 4 + 2 = 6
- (^) Let uA(x 1 A , x 2 A ) = ln(x 1 A ) + 2 ln(x 2 A ) and u B (x B 1 ,^ x B 2 ) = ln(x B 1 ) + 2 ln(x B 2 ).
- (^) MRS of consumer A: MRS A = 1 x 1 A 2 x 2 A
x 2 A 2 x A 1
- (^) Similarly, MRS B = 1 x 1 B 2 x 2 B
x 2 B 2 x 1 B
- So a Pareto-optimum is a feasible allocation for which x A 2 2 x 1 A
x B 2 2 x 1 B General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core
Example
Finding all Pareto-optimal allocations (deriving the contract curve)
- We can simplify tangency condition to: x A 2 x 1 A
x B 2 x 1 B
- (^) Recall endowment/feasibility requirement: x A 1 +^ x B 1 = 8^ and^ x A 2 +^ x B 2 = 6
- Re-write tangency condition, substituting x 1 B = 8 x 1 A and x B 2 = 6^ ^ x A 2 : x 2 A x 1 A
6 x 2 A 8 x 1 A or x A 2 =^
x A 1
- (^) This is the equation of the contract curve.
- (^) In this case, it’s just the diagonal of the rectangle
