Edgeworth Box Analysis: Two Consumers

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Edgeworth Box AnalysisTwo Consumers

• Lecture 25
• November 19 2002
• Slides adapted from slide set for Microeconomics
  by Pindyck and Rubinfeld, Prentice Hall, 1998.

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The Edgeworth Box Analysis

• Francis Edgeworth developed this method of
  analysis in the last portion of the 19th century.
• Provides a powerful way of graphically studying
  exchange and the role of markets.
• Understanding the Edgeworth Box is critical to
  understanding exchange and markets.

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To Form and Edgeworth Box

• Rotate one of the graphs onto the other one until
  it forms a box.

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Here the axes for Jane have been rotated
Jane
y1
Bill
x1
5
y1
x1
6
The Edgeworth Box
x2
Jane
y2
Total Fixed Supply of y
A
y1
Bill
x1
Total Fixed Supply of x
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Consider two consumers and two products
Bill Jane
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The Edgeworth Box
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The Edgeworth Box
10
The Edgeworth Box
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The Edgeworth Box
12
The Edgeworth Box
13
The Edgeworth Box
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The Edgeworth Box
x2
III1 II1 I1
Jane
y2
Trading area?
A
C
I2 II12 III12
y1
Bill
x1
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The Edgeworth Box
x2
III1 II1 I1
Jane
y2
A
What about A here?
C
I2 II12 III12
y1
Bill
x1
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The Edgeworth Box
x2
III1 II1 I1
Jane
y2
B
PARETO OPTIMAL
A
C
I2 II12 III12
y1
Bill
x1
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Pareto Optimal

• When no change can make one better off without
  making the other worse off.

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The Edgeworth Box
x2
IV2 III2 II2 I2
Jane
y2
B
E
E
Contract line
A
E
C
y1
I1 II1 III1 IV1
Bill
x1
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Contract Line

• Is the locus of Pareto optimal points

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The Edgeworth Box
x2
IV2 III2 II2 I2
Jane
y2
B
E
E
A
E
C
y1
I1 II1 III1 IV1
Bill
x1
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The Edgeworth Box
x2
x2 x2
III2 II2 I2
Jane
y2
B
E
E
Pareto improving--from A or B to E or E
y2 y2
y1 y1
A
E
C
y1
I1 II1 III1
Bill
x1
x1 x1
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The Edgeworth Box
x2
III2 II2 I2
Jane
y2
B
E
E
A
E
C
y1
I1 II1 III1
Bill
x1
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Understanding the Picture

• Any point in the Edgeworth box indicates a
  particular distribution of the two goods among
  the two individuals, e.g., Bill and Jane.
• Each individual has an indifference curve going
  through that point.
• If the distribution is Pareto optimal, those two
  indifference curves are tangent at that point.

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Prices that are consistent with the Pareto
optimal point

• At that tangency of the two indifference curves,
  the slope of the tangency line--the straight line
  drawn through the point of tangency--represents
  the relative prices for the two goods. Hence,
  there are relative prices that will be consistent
  with the Pareto optimum.

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The Edgeworth Box
x2
III2 II2 I2
Jane
y2
B
E
E
A
Price or budget line
E
C
y1
I1 II1 III1
Bill
x1
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Tangent line is really a budget line for both
individuals

• If one extends the tangent line to each axis, we
  now have a budget line.
• For example, the budget line for Jane is
  IJane Pxxjane PyyJane where I is the income
  Jane could get from selling the X and Y she holds
  at the Pareto optimum point.

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Budget Line for Jane
Ijane/Py
IJane Pxxjane PyyJane
E
Price or budget line
C
y
Jane
Ijane/Px
x
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Marginal Rate of Substitution

• MRSxy the number of units of y one is willing
  to give up per unit of x and stay on same
  indifference curve.
• Slope of indifference curve gives the marginal
  rate of substitution.

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Marginal Rate of Substitution
Slope of indifference curve gives the marginal
rate of substitution
Ijane/Py
D
Price or budget line
y
Jane
Ijane/Px
x
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MARGINAL RATE OF SUBSTITUTION
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At point D

• Slope of indifference curve equals the slope of
  the budget line or
• MRSxy -Px/ Py

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NOW RETURN TO EDGEWORTH BOX ANALYSIS

• At point E, the indifference curve for Jane is
  just tangent to the indifference curve for Bill,
  and the price line is the tangency line at E. In
  other words, Slope(Indifference curve for
  Jane) Slope(Indifference curve for Bill)
  Slope of price line

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The prices (ratio of prices) can produce the
optimum
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The Edgeworth Box
x2
III2 II2 I2
Jane
y2
B
E
E
A
Price or budget line
E
C
y1
I1 II1 III1
Bill
x1