CHAPTER 30EXCHANGE

Partial equilibrium analysis: The equilibrium conditions of ONE particular market, leaving other markets untreated.

General equilibrium analysis: The equilibrium conditions of ALL markets, allowing interactions between different markets.

30.1 The Edgeworth Box

Two consumers: A and B. Two goods: 1 and 2. Initial endowment: Allocation: Feasible allocation: total consumption does not

exceed total endowment for both goods.

1 2 1 2, , ,A A B B

1 2 1 2, , ,A A B Bx x x x

1 1 1 1A B A Bx x 2 2 2 2A B A Bx x

30.1 The Edgeworth Box

30.1 The Edgeworth Box

Each point in the Edgeworth box represents a feasible allocation.

From W to M:Person A trades units of good 1 for

units of good 2;Person B trades units of good 2 for

units of good 1.

1 1A Aw x 2 2

A Ax w

2 2B Bw x 1 1

B Bx w

30.2 Trade

30.2 Trade

Trade happens whenever both consumers are better off.

Starting from W, M is a possible outcome of the exchange economy because:Person A is strictly better off with than

with ;Person B is strictly better off with than

with .

1 2,A Ax x 1 2,A A

1 2,B Bx x 1 2,B B

30.3 Pareto Efficient Allocations

An allocation is Pareto efficient whenever:There is no way to make everyone strictly better

off;There is no way to make some strictly better off

without making someone else worse off;All of the gains from trade have been exhausted;There are no (further) mutually advantageous

trades to be made.

30.3 Pareto Efficient Allocations

30.3 Pareto Efficient Allocations

Pareto efficiency is given by the tangency of the indifference curves.

Contract curve: the locus of all Pareto efficient allocations.

Any allocation off the contract curve is Pareto inefficient.

30.4 Market Trade

Gross demand: Quantity demanded for a good by a particular consumer at the market price.

Excess demand: The difference between the gross demand and the initial endowment of a good by a particular consumer.

Disequilibrium: Excess demands by both consumers do not sum up to zero.

30.4 Market Trade

30.4 Market Trade

Competitive equilibrium: A relative price and an allocation , such that:The allocation matches the gross demands by both

consumers, given the relative price and initial endowments;

The allocation is feasible.

1 2 1 2, , ,A A B Bx x x x1 2p p

30.4 Market Trade

30.5 The Algebra of Equilibrium Consumer A’s demands: Consumer B’s demands: The equilibrium condition:

1 * * 1 * * 1 11 2 1 2( , ) ( , )A B A Bx p p x p p

2 * * 2 * * 2 21 2 1 2( , ) ( , )A B A Bx p p x p p

1 * * 2 * *1 2 1 2, , ,A Ax p p x p p

1 * * 2 * *1 2 1 2, , ,B Bx p p x p p

Re-arrangement:1 * * 1 1 * * 1

1 2 1 2[ ( , ) ] [ ( , ) ] 0A A B Bx p p x p p 2 * * 2 2 * * 2

1 2 1 2[ ( , ) ] [ ( , ) ] 0A A B Bx p p x p p

30.5 The Algebra of Equilibrium Net demand:

1 1 11 2 1 2( , ) ( , )A A Ae p p x p p

Aggregate excess demand:

Another expression:

1 1 11 2 1 2( , ) ( , )B B Be p p x p p

1 11 1 2 1 2 1 2( , ) ( , ) ( , )A Bz p p e p p e p p

2 22 1 2 1 2 1 2( , ) ( , ) ( , )A Bz p p e p p e p p

* *1 1 2, 0z p p

* *2 1 2, 0z p p

30.6 Walras’ Law Budget constraints:

1 2 1 21 1 2 2 1 2 1 2( , ) ( , )A A A Ap x p p p x p p p p

1 2 1 21 1 2 2 1 2 1 2( , ) ( , )B B B Bp x p p p x p p p p

Re-arrange the terms:1 2

1 1 2 2 1 2( , ) ( , ) 0A Ap e p p p e p p 1 2

1 1 2 2 1 2( , ) ( , ) 0B Bp e p p p e p p

Adding up:

1 1 1 2 2 2 1 2( , ) ( , ) 0p z p p p z p p

30.6 Walras’ Law Walras’ Law: The value of aggregate excess

demand is always zero. Applications of the Walras’ law:

implies ;Market clearing for one good implies that of the

other good;With k goods, we only need to find a set of prices

where k-1 of the markets are cleared.

* *1 1 2( , ) 0z p p * *

2 1 2( , ) 0z p p

30.7 Relative Prices

Walras’ law implies k-1 independent equations for k unknown prices.

Only k-1 independent prices. Numeraire prices: the price which can be

used to measure all other prices. If we choose p1 as the numeraire price, then it

is just like multiplying all prices by the constant t=1/p1.

EXAMPLE: An Algebraic Example of Equilibrium

The Cobb-Douglas utility function:1 2 1 2 1( , ) ( ) ( )a a

A A A A Au x x x x

11 2

1

( , , ) AA A

mx p p m a

p 2

1 22

( , , ) (1 ) AA A

mx p p m a

p

11 2

1

( , , ) BB B

mx p p m b

p 2

1 22

( , , ) (1 ) BB B

mx p p m b

p

The demand functions:

EXAMPLE: An Algebraic Example of Equilibrium Income from endowments:

1 21 2A A Am p p 1 2

1 2B B Bm p p Aggregate excess demand for good 1:

1 11 1

1 1

1 2 1 21 11 1

1 1

( ,1) A BA B

A A B BA B

m mz p a b

p p

p pa b

p p

EXAMPLE: An Algebraic Example of Equilibrium Equilibrium condition:

*1 1( ,1) 0z p

Equilibrium price:2 2

*1 1 1(1 ) (1 )

A B

A B

a bp

a b

30.8 The Existence of Equilibrium

The existence of a competitive equilibrium can be proved rigorously.

A formal proof is quite complicated and far beyond the scope of this course.

30.9 Equilibrium and Efficiency

Both indifference curves are tangent to the budget line at the equilibrium allocation.

The equilibrium allocation lies upon the contract curve.

The First Theorem of Welfare Economics: Any competitive equilibrium is Pareto efficient.

EXAMPLE: Monopoly in the Edgeworth Box A regular monopolist

EXAMPLE: Monopoly in the Edgeworth Box First degree price discrimination

30.11 Efficiency and Equilibrium

Reverse engineering: Starting from any Pareto efficient allocation; Use the common tangent line as the budget line; Use any allocation on the budget line as the initial

endowment.

The Second Theorem of Welfare Economics: For convex preferences, any Pareto efficient allocation is a competitive equilibrium for some set of prices and some initial endowments.

30.11 Efficiency and Equilibrium The Second Theorem of Welfare Economics

30.11 Efficiency and Equilibrium A Pareto efficient allocation that is not a competitive

equilibrium.