pareto efficiency

Exchange

M. Utku UnverMicro Theory

Boston College

M. Utku Unver Micro Theory (BC) Exchange 1 / 23

General Equilibrium

So far we have been analyzing the behavior of a single consumer. In thischapter, we will see how consumers interact in a market setting and howthat affects the prices.

This kind of analysis is called General Equilibrium Analysis.

Suppose there are two consumers with their respective endowments. Theymeet at a market and trade some of their goods. How do we determinethe relative prices of the goods, and the consumers choice bundles?


An Exchange Economy

Consumers: A and BGoods: 1 and 2Endowments: = (A,B) = ((A1 ,

A2 ), (

B1 ,

B2 ))

Total endowment of goods in the economy:

Good 1: A1 +B1

Good 2: A2 +B2

Demands: X = (xA, xB) = ((xA1 , xA2 ), (x

B1 , x

B2 ))

Demands should be feasible:

Good 1: xA1 + xB1 =

A1 +

B1

Good 2: xA2 + xB2 =

A2 +

B2


Edgeworth Box

For equilibrium analysis we use a useful tool called Edgeworth Box

is endowment: (AI ,AL) is Agent As (BD,BG ) is Agent Bs.X is consumption: (AH,AK ) is Agent As, (BC ,BF ) is Agent Bs.Agent A sells HI units of good 1 and buys KL units of good 2.M. Utku Unver Micro Theory (BC) Exchange 4 / 23

The following figure shows the preferences of two agents in an Edgeworthbox.


Pareto efficiency

If we can make some agents better off without making any agent worseoff, then an allocation is not Pareto efficient.Formally, an allocation is Pareto efficient if it is not possible that we canmake some agents better off without making any agent worse off.


Allocation Y is Pareto efficient. We cannot make either agent better offthan she is at Y , without making the other worse off.

Allocation X is not Pareto efficient. Moreover, allocation Y Paretodominates allocation X , since it gives more utility to agent B and givesthe same utility to agent A. That is, it makes agent B better off withoutmaking A worse off.M. Utku Unver Micro Theory (BC) Exchange 7 / 23

Example: Suppose UA = (xA1 )(xA2 ) and U

B = (xB1 )(xB2 )

2. The initialendowments are given by A = (1, 1) and B = (2, 1). Find the set ofPareto efficient allocations (also known as the contract curve).Solution:

1 We need to equalize the MRS of agent A and MRS of agent B forPareto efficiency.

2 For the allocation to be feasible we needxA1 + x

B1 =

A1 +

B1 = 1 + 2 = 3 and

xA2 + xB2 =

A2 +

B2 = 1 + 1 = 2.

Then xB1 = 3 xA1 and xB2 = 2 xA2


MRSA = MRSB

UA/xA1

UA/xA2= U

B/xB1UB/xB2

xA2

xA1= x

B2

2xB1

Use the feasibility condition we found above

to isolate one persons choice, for example As

xA2

xA1= 2 x

A2

2(3 xA1 )

6xA2 2xA1 xA2 = 2xA1 xA1 xA2 xA2 =

2xA16xA1


The plot of this above equation (the contract curve:)

For example, if xA1 = 1 then xA2 =

25 and, by feasibility, x

B1 = 2 and

xB2 =85 .


What if there are kinks at preferences?Example: A has a perfect-complement utility function, while B has somesmooth strictly convex preferences.


What if there are corner solutions?Example: If both of the agents have perfect-substitutes preferences withdifferent MRS, we will have the set of Pareto efficient allocations as theedges of the Edgeworth box as shown as lighter mirrored L-shaped curve.

Note that if both of the agents have the same MRS in this case, all thepoints in the Edgeworth box are Pareto efficient. Why?M. Utku Unver Micro Theory (BC) Exchange 12 / 23

Example: UA = (xA1 )(xA2 ) and U

B = xB1 + 2xB2 . The initial endowments

are given as A = (1, 1) and B = (1, 1). Find the set of Pareto efficientallocations.Solution:

MRSA = MRSB =

UA/xA1

UA/xA2= U

B/xB1UB/xB2

=

xA2

xA1= 1

2= xA2 = 12xA1


The plot of above equation (the contract curve:) Observe that there areparts of contract curve where MRSs are not equal, corner solutions.


Competitive Equilibrium

A price vector (p1, p2) and an allocation (XA, XB) = ((xA1 , xA2 ), (x

B1 , x

B2 ))

is a competitive equilibrium if

1 each person is choosing the most preferred bundle in his budget setand

2 there is neither excess demand nor excess supply for any good. (i.e.,markets clear)


In the above figure, the price ratio p1/p2 specified by slope of the budgetline(s) and the allocation specified by (xA, xB) is not a competitiveequilibrium. While each agent is maximizing their utilities, the markets donot clear. There is excess demand for good 2 and excess supply for good 1.


The tangency point in the above Edgeworth box figure is the competitiveequilibrium for that economy. The price ratio p1/p2 together with theallocation x = (xA, xB) is a competitive equilibrium for this economy.


Example: Suppose UA = (xA1 )(xA2 ) and U

B = (xB1 )(xB2 )

2. Theendowments are given by A = (1, 1) and B = (2, 1). Find thecompetitive equilibrium in this economy.


Solution:Step 1 : First we find the demand functions of the agents for both goods.Let p1 = 1 (numeraire good,) and p2 = p (unknown, we can onlydetermine one of the prices)The demand functions of agent A are as follows (Cobb-Douglaspreferences):

xA1 =12mA

p1= 12

p1+p2p1

= 12 (1 + p) where mA =the value of the endowment

of agent A.xA2 =

12mA

p2= 12

1+pp .

The demand functions of agent B are as follows (Cobb-Douglaspreferences):

xB1 =13mB

p1= 13 (2 + p) and x

B2 =

23mB

p2= 23

2+pp where m

B = the value ofthe endowment of agent B.


Step 2 : Clearing the markets.

xA1 + xB1 =

A1 +

B1 = 3 =

1

2(1 + p) +

1

3(2 + p) = 3 =

p =11

5.

Then we can find the equilibrium allocation using the demand functions:xA1 =

85 , x

A2 =

811 , x

B1 =

75 and x

B2 =

1411 .

Therefore, the competitive equilibrium is (1,115 ) =competitive price and((85 ,

811 ), (

75 ,

1411 )) =competitive allocation.


Example: Suppose UA = (xA1 )(xA2 )

2 and UB = min{xB1 , xB2 }. The initialallocations are given as A = (0, 2) and B = (2, 0). Find thecompetitive equilibrium in this economy.


Walras Law

(Walras Law)

Suppose there are k goods in the exchange economy. If (k-1) marketsclear, then the kth market clears as well.


Welfare Economics

Theorem (First Fundamental Theorem of Welfare Economics)

Any competitive equilibrium allocation is Pareto efficient.

Theorem (Second Fundamental Theorem of Welfare Economics)

Supoose that preferences are convex. Then any interior Pareto efficientallocation can be obtained as a competitive equilibrium allocation fromsome initial endowment.


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