Topics in Microeconomic Theory General Equilibrium

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Topic 5 Part I
Topics in Microeconomic Theory General
Equilibrium
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General Equilibrium

• Until now we have covered a partial equilibrium
  model, where all prices other than the price of
  the good being studied are assumed to remain
  fixed
• We will now covered a general equilibrium model,
  where all prices are variable and equilibrium
  requires that all market clear

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General Equilibrium

• General equilibrium theory considers all
  interactions between markets, as well as the
  functioning of individual markets
• We will examine the special case of general
  equilibrium, the pure exchange economy, where all
  economic agents are consumers

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Pure Exchange Economy

• In a pure exchange economy, we have several
  consumers described by their preferences and the
  goods that they possess
• The agents trade the goods among themselves
  according to certain rules and attempt to make
  themselves better off

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Pure Exchange Economy

• Our goal is to describe what outcomes might arise
  through a process of voluntary exchange
• We will compare these results to the equilibria
  achieved under competitive market systems

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Pure Exchange Economy Assumptions

• No production sector
• Commodities exist but we do not ask how they came
  to be

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Pure Exchange Economy Assumptions

• Each consumer is endowed by nature with a certain
  amount of a finite number of consumable goods
• Suppose there are only 2 consumers, consumer 1
  and consumer 2
• Suppose there are only 2 goods, x1 and x2

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Pure Exchange Economy Assumptions

• Let e1 ? (e11, e21) denote the nonnegative
  endowment of the two goods owned by consumer 1
• Let e2 ? (e12, e22) denote the nonnegative
  endowment of the two goods owned by consumer 2
• The total amount of each good available in this
  society is e (e1 e2) (e11 e12, e21 e22)

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Pure Exchange Economy Assumptions

• Private ownership exists in this economy
• Redistribution of commodities from the initial
  endowment is possible only under voluntary
  exchange
• Barter economy

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Pure Exchange Economy Equilibrium

• We will search for the equilibrium in this barter
  economy what allocations may be reasonably
  result if consumers can freely exchange goods?

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Pure Exchange Economy Equilibrium

• We will represent this economy and analyze the
  equilibrium using the Edgeworth box as a tool

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Pure Exchange Economy Equilibrium

• Represent first separately consumers 1 and 2
• Consumer 1
• e21

x2
e1
U1(e1)
x1
e11
01
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Pure Exchange Economy Equilibrium

• Consumer 2
• e22

x2
e2
U2(e2)
x1
02
e12
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Pure Exchange Economy Equilibrium

• To represent both consumers in the Edgeworth box,
  rotate the representation of consumer 2 180
  degrees

02
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Pure Exchange Economy Equilibrium

• The representation of the initial endowments and
  preferences for the two consumers

e2 e21 e22
02
gains from trade
e (e1, e2)
U1(e1)
U2(e2)
01
e1 e11e12
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Pure Exchange Economy Equilibrium

• Every point in the box represent a feasible
  allocation among the two consumers

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Pure Exchange Economy Equilibrium

• Where the two consumers will end up?

e2 e21 e22
02
x0
e (e1, e2)
A
U1(e1)
U2(e2)
01
e1 e11e12
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Pure Exchange Economy Equilibrium

• Point e denotes the initial endowment
• Point A denotes an allocation where both
  consumers are better off w.r.t. e
• But there is an area between the two indifference
  curves that go through A where the both consumers
  are better off w.r.t. A

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Pure Exchange Economy Equilibrium

• The redistribution will end up when the
  indifference curves are tangent (at point x0) and
  gains from trade are exhausted
• Once x0 is achieved, no further gains from trade
  are possible, i.e., at x0 there is no feasible
  allocation that makes one of the consumers better
  off without making the other worse off (Pareto
  efficiency concept)
• So, x0 is a Pareto efficient allocation

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Pure Exchange Economy Equilibrium

• Define the contract curve or Pareto efficient
  allocation curve (curve CC) as the subset of
  feasible allocations where the consumers
  indifferent curves are tangent to each other

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Pure Exchange Economy Equilibrium
B
e2 e21 e22
02
C
e (e1, e2)
C
U1(e1)
U2(e2)
01
e1 e11e12
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Pure Exchange Economy Equilibrium

• Note that not all Pareto efficient allocations
  are candidates for being an equilibrium
  allocation in the barter economy
• Trade occurs somewhere in the contract curve
  between the two initial indifference curves
  (which pass through e)
• For example, at point B, consumer 1 receives all
  gains from trade and consumer 2 is not worse off
  w.r.t. to his/her initial situation

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Pure Exchange Economy Equilibrium

• Clearly, there are many barter equilibria and we
  will identify the conditions under which an
  allocation is a barter equilibrium in a two-good,
  two-consumer barter economy
• The assignment of goods should not exceed the
  amount available
• The allocation should be Pareto-efficient

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Pure Exchange Economy Equilibrium

• No equilibrium allocation can make any consumer
  worse off than he/she would be consuming his/her
  initial endowment

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Pareto Efficiency Calculus Conditions

• A Pareto efficient allocation is one for which
  there is no way to make all agents better off
• In other words, a Pareto efficient allocation is
  one for which each agent is as well off as
  possible, given the utility of the other agent

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Pareto Efficiency Calculus Conditions

• We are looking for a point in the Edgeworth box
  that represents an allocation of x1 and x2
  between the two agents that is Pareto efficient
• Start with the point C ((x11, x21), (x12, x22))

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Pareto Efficiency Calculus Conditions

• One way to find a Pareto equilibrium within the
  Edgeworth box is to hold the utility of one of
  the consumers constant, for example the utility
  of consumer 2 U2 at ?U2, and then maximize the
  utility of consumer 1 subject to the constraint
  that consumer 2 gets ?U2
• Graphically, it is represented by a movement to a
  higher and higher indifference curve for consumer
  1 along the surface of ?U2

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Pure Exchange Economy Equilibrium
C
C
e2 e21 e22
02
?U2
x22
x22
U1
U1
01
x11
x11
e1 e11e12
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Pareto Efficiency Calculus Conditions

• The highest utility agent 1 can achieve without
  reducing agent 2s utility below?U2 is U1,
  yielding an allocation
• C ((x11, x21), (e1 - x11, e2 - x21))

allocation for agent 1
allocation for agent 2
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Pareto Efficiency Calculus Conditions

• Then, the calculus conditions describing an
  interior Pareto efficient allocation is
• max U1(x11, x21)
• x11,x21, x12, x22
• s.t. U2(x12, x22) ?U2
• x11 x12 e1
• x21 x22 e2

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Pareto Efficiency Calculus Conditions

• The maximization problem asks to find the
  allocation ((x11,x21), (x12, x22)) that makes
  agent 1s utility as large as possible, given a
  fixed level for agent 2s utility, and given that
  the total amount of each good is equal to the
  amount available (sum of endowments for each
  good)

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Pareto Efficiency Calculus Conditions

• The Lagrangian is then
• L U1(x11, x21) ? (?U2 - U2(x12, x22)
• ?1 (e1 - x11 - x12)
• ?2 (e2 - x21 - x22),
• where, ? is the Lagrangian multiplier on the
  utility
• constraint, and ?i (i 1, 2) are the Lagrangian
• multipliers on the resource constraints

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Pareto Efficiency Calculus Conditions

• By FOC,
• ?U1/?x11/ ?U1/?x21 ?1/?2
• i.e., MRS1 ?1/?2
• and
• ?U2/?x12/ ?U2/?x22 ?1/?2
• i.e., MRS2 ?1/?2

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Pareto Efficiency Calculus Conditions

• Then, MRS1 MRS2
• Therefore, at an (interior) Pareto-efficient
  allocation, the MRS between the two goods must be
  the same for both agents
• Otherwise, there will be some trade that would
  make each consumer better off

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Pareto Efficiency Calculus Conditions

• Therefore, given Ui (i 1,2), goods x1 and x2,
  and initial endowments e1 and e2, we can find the
  set of Pareto efficient allocations by using the
  following conditions
• (1) MU11/MU21 MRS1 MRS2 MU12/MU22
• (2) x11 x12 e1
• x21 x22 e2