Equilibrium Analysis in Economics

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Equilibrium Analysis in Economics

• Equilibrium
• Static Analysis
• Partial Market Equilibrium
• General Equilibrium

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Equilibrium

• Equilibrium is a constellation of selected
  interrelated variables so adjusted to one another
  that no inherent tendency to change prevails in
  the model which they constitute

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Equilibrium

• Selected
• Some variables are not selected to be in the
  model
• Equilibrium is relevant only to the selected
  variables and may no longer apply if different
  variables are included (excluded)

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Equilibrium

• Interrelated
• Since the variables are interrelated, all the
  variables must be in a state of rest if
  equilibrium is to be achieved
• Inherent
• The state of rest refers to the internal forces
  of the model external forces (exogenous
  variables) are assumed fixed

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Equilibrium

• Since equilibrium refers to a lack of change, we
  often refer to equilibrium analysis as static
  analysis or statics

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Partial Market Equilibrium

• Constructing the model
• An equilibrium condition, behavioral equations,
  and restrictions must be specified
• Qd Qs
• Qd a - bP (a, b gt 0)
• Qs -c dP (c, d gt 0)

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Partial Market Equilibrium

• Solving the model
• Equilibrium tells us Qd Qs so we can substitute
  into the equilibrium equation and solve
• a - bP -c dP
• a c bP dP
• a c P(b d)
• a c P (equilibrium price)
• b d

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Partial Market Equilibrium

• Note the solution is entirely in the form of
  parameters - this is typical
• P is positive (as required by economics)
• a, b, c, d gt 0 therefore
• a c gt 0 as well
• b d

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Partial Market Equilibrium

• Find the equilibrium quantity by substituting the
  equation for price into one of the equations for
  Q
• Q a - b a c
• b d
• Q a(b d) - b a c
• b d b d

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Partial Market Equilibrium

• Q ab ad - ba - bc
• b d
• Q ad - bc
• b d
• The equilibrium value of Q should be gt 0
• b d gt 0 since b, d gt 0
• We have added restriction of ad gt bc for Q gt 0

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Partial Market Equilibrium

• Suppose we have the following model which results
  in a quadratic
• Qd Qs
• Qd 4 - P2
• Qs 4p - 1

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Partial Market Equilibrium

• Setting up equation to solve gives us
• 4 - P2 4P - 1
• P2 4P - 5 0
• The left-hand expression is a quadratic function
  of the variable P
• Can use the quadratic formula to solve the
  equation

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Partial Market Equilibrium

• General form of a quadratic equation is
• ax2 bx c 0
• Using the quadratic formula, two roots can be
  obtained from a quadratic equation, x1 and x2
• x1 and x2 provide solutions
• x1, x2 -b and - (b2 - 4ac)1/2
• 2a

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Partial Market Equilibrium

• Our expression is P2 4P - 5 0
• P1, P2 -4 and - (42 - 4(1)(-5))1/2
• 2(1)
• P1, P2 -4 and - (16 20)1/2
• 2
• P1, P2 -4 and - 6
• 2

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Partial Market Equilibrium

• P1 -4 6
• 2 2
• P1 -2 3 1
• P2 -4 - 6
• 2 2
• P2 -2 -3 -5
• Only P1 is relevant since P gt 0

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Partial Market Equilibrium

• If P 1 then Q 4P - 1 3

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General Equilibrium Model

• Our analysis can extend to n commodities
• There will be an equilibrium condition for each
  of the n markets
• There will be behavioral equations for each of
  the n markets

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General Equilibrium Model

• Equilibrium conditions
• Qd1 Qs1
• Qd2 Qs2
• . .
• . .
• . .
• Qdn Qsn

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General Equilibrium Model

• Behavioral equations
• Qd1 a0 a1P1 a2P2 anPn
• Qs1 b0 b1P1 b2P2 bnPn
• Qd2 c0 c1P1 c2P2 cnPn
• Qs2 d0 d1P1 d2P2 dnPn
• Qdn ?0 ?1P1 ?2P2 ?nPn
• Qsn ?0 ?1P1 ?2P2 ?nPn

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General Equilibrium Model

• Such a system is very difficult to solve with the
  method of substitution
• Can use matrix algebra to solve a system of
  linear equations