General Equilibrium and Efficiency

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General Equilibrium and Efficiency
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• General Equilibrium Analysis is the study of the
  simultaneous determination of prices and
  quantities in all relevant markets.

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• General Equilibrium (GE) occurs when
• There is no excess demand or excess supply in
  any output or input markets
• Consumers are maximizing utility subject to
  budget constraints
• Producers are maximizing profit subject to the
  production function
• Input suppliers are optimizing

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• Economic Efficiency
• An efficient (or Pareto efficient) allocation of
  goods is an allocation in which no one can be
  made better off without making someone else worse
  off.

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• Conditions of Economic Efficiency
• Efficiency in exchange
• Efficiency in production
• Efficiency in the output market

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• Efficiency in Exchange
• Efficiency in exchange occurs when
• MRSA MRSB
• where
• MRSi marginal rate of substitution of good y
  for good x for consumer i, i A, B
• amount of good y that consumer i is willing to
  give up for one more unit of good x

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• Example 2 consumers Ann Bob 2 goods X Y
• Ann is willing to trade 4Y for 1X (MRSA 4)
• Bob is willing to trade 2Y for 1X (MRSB 2)
• Ann and Bob can benefit from trading, e.g.,
• If Ann trades 3Y for 1X she is better off since
  she is willing to pay 4Y.
• If Bob receives 3Y for 1X he is better off
  since he would accept 2Y for 1X.
• When MRSA gt MRSB there are gains from trade.
• Only when MRSA MRSB can no one be made better
  off without making someone else worse off, and
  the Pareto efficient allocation occurs.

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• Efficiency in Production
• An allocation of inputs is technically
  efficient if the output of 1 good cannot be
  increased without decreasing the output of
  another good.
• Efficiency in production (or efficiency in the
  use of inputs in production) occurs when
• MRTSx MRTSy
• MRTSj marginal rate of technical substitution
  of labor (L) for capital (K) for good j , j x,
  y
• amount by which K can be reduced when 1 more
  unit of L is used, so that output remains
  constant.

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• Example Suppose MRTSx gt MRTSy
• e.g., MRTSx 4 and MRTSy 3
• For good x, producers can give up 4 units of K
  for 1 more unit of L, without changing output.
• For good y, producers can give up 3 units of K
  for 1 more unit of L, without changing output.
• Efficiency can be improved by using more K to
  produce good y and more L to produce good x.
• When MRTSx MRTSy, production efficiency
  cannot be improved by changing the input mix.

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Efficiency in the Output Market Efficiency in
the output market occurs when MRT MRSA
MRSB where MRT marginal rate of
transformation of good y for good x amount of
good y that must be given up to produce one
additional unit of good x
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• Example 2 individuals Ann Bob 2 inputs L
  K and 2 goods X Y
• MRSA MRSB 3, i.e., Ann and Bob are willing to
  trade 3Y for 1X.
• MRT 2, i.e., the economy can give up 2 units of
  Y to produce 1 more unit of X.
• It benefits society to produce more Y and less X
  until
• MRT MRSA MRSB

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• MRT MCx/MCy
• where
• MCj marginal cost of good j
• additional cost of producing 1 more unit of j
• Example MCx1, MCy 2, MCx/MCy ½
• The economy can produce 1 more unit of X for 1
  or 1 more unit of Y for 2.
• For 1, economy can produce 1 unit of X or ½ unit
  of Y.
• The amount of Y that must be given up to produce
  1 more unit of X is ½. (i.e., MRT ½ )
• MRT MCx/MCy

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Economic Efficiency Summary Conditions of
Economic Efficiency Efficiency in exchange
MRSA MRSB Efficiency in production MRTS1
MRTS2 Efficiency in the output market MRT
MRSA MRSB Note For the entire economy, MRS
must be equal for all consumers, MRTS must be
equal for all firms, and MRT must be equal to MRS
for all consumers .
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First Welfare Theorem The First Welfare Theorem
(the Invisible Hand Theorem) A competitive
equilibrium is efficient.
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• A competitive equilibrium satisfies the 3
  conditions for efficiency.
• Efficiency in exchange MRSA MRSB
• holds because constrained utility maximization
  requires
• MRSA Px/Py for Ann
• MRSB Px/Py for Bob and thus,
• MRSA Px/Py MRSB
• where Px is the price of good x and Py is the
  price of good y.

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• Efficiency in production MRTSx MRTSy
• holds in perfect competition because cost
  minimization requires
• MRTSx w/r for good x
• MRTSy w/r for good y , and thus
• MRTSx w/r MRTSy
• where w wage, r rental rate on capital

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• Efficiency in the output market MRT MRSA
  MRSB
• holds because of the following.
• MRT MCx/MCy
• Px MCx for profit maximization of firm 1
  which produces good x
• Py MCy for profit maximization of firm 2
  which produces good y
• These imply that
• MRT Px/Py
• For utility maximization
• MRSA Px/Py MRSB
• Thus
• MRT MRSA MRSB