Review Advanced Micro Economic Theory

1
Review Advanced Micro Economic Theory

• Professor Rutstrom
• Fall 2007

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Optimization problems

• Constrained cost minimization
• Technology defines the production function
• Test conditions input prices and output quantity
• Min ?wx s.t. yf(x)
• c(y,w)
• Constrained utility maximization
• Constrained expenditure minimization

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Optimization problems

• Constrained cost minimization
• Min ?wx s.t. yf(x)
• c(y,w)
• Constrained utility maximization
• Preferences define the utility function
• Test conditions goods prices and income
• Max u(x) s.t. ?px M
• U(M, p)

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Optimization problems

• Constrained cost minimization
• Min ?wx s.t. yf(x)
• c(y,w)
• Constrained utility maximization
• Max u(x) s.t. ?px M
• U(M, p)
• Constrained expenditure minimization
• Preferences define the utility function
• Test conditions goods prices and utility level
• Min ?px s.t. u(x)u0
• M(u,p)

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Duality

• Cost function
• Properties
• Shephards lemma
• Indirect utility function
• Properties
• Roys identity
• Expenditure function
• Envelope theorem and primal-dual optimization
• Comparative statics the easy way

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Concepts

• Le Chatelier principle
• Comparative statics when addition a just-binding
  constraint
• fundamental relationships
• Samuelsons conjugate pairs theorem
• Homogeneity and Euler theorem
• Elasticities (price, output, income)
• Interpreting Lagrange multipliers

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Technology

• Strict concavity of technology is SOSC for profit
  maximization no constant returns to scale
  (unless adding constraints such as zero profits)
• Quasi-concavity of technology is SOSC for cost
  minimization
• basic CES is CRTS
• Level functions (isoquants) are convex to the
  origin

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• Cost minimization
• Min C(x), s.t. f(x)y0
• Market structure plays no role in cost
  minimization
• Average and Marginal costs
• Longrun Competitive Markets

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Relationship between cost minimization and profit
maximization

• Producers transform inputs into outputs
• Technology
• Purpose maximize profits
• Two steps to profit maximization
• Choose inputs to minimize cost subject to output
  level
• Choose output level to maximize profits
• Derived relationships
• Factor demands
• Output supply

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Consumers

• Choice is guided by preferences
• Utility function is an abstraction for modeling
  preferences
• Maximize utility with expenditure constraint
• Minimize expenditures with utility level
  constraint
• Quasi-concave utility function convex
  indifference curves
• Ordinal preferences invariant to positive
  monotonic transformations

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Demand functions

• x(w, p), ?x/ ?wlt0, H(0) in (w,p) jointly
• x(w, p) xs(w1, x2(w,p), p)
• xc(w, y), ?xc/ ?wlt0, H(0) in w
• x(w, p) xc(w, y(w,p))
• xLC(w), ?xLC/ ?wltgt0
• xLC(w, p) x(w, p(w))
• xM(p,M), ?xM/ ?pltgt0, H(0) in (p,M) jointly
• xU(p,u), ?xU/ ?plt0, H(0) in p
• xM(p,M) xU (p, U(p,M))
• xU (p,u) xM(p,M(p,u))
• xUL (p,u) xUS (p1, x2(p,u),u)
• xML (p,M) xMS (p1, x2(p,M),M)

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Comparative statics

• Predictions based on marginalism
• Comparative statics
• Requires only local properties of technology or
  preferences
• Refutable hypotheses without adding further
  assumptions than FONC and SOSC of profit max
• parameter enters only one first-order condition
  and not the constraint
• With specific functional forms we can derive more
  comparative statics and also make global
  predictions (not just comparative statics)

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Parameters

• Profit maximization p, w
• Cost minimization w, y
• Utility maximization p, M
• Expenditure minimization p, u

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Principal Minors and SOSC

• Unconstrained maximization
• Principal minors alternate in sign (-1)k
• H of a 2 variable problem is positive
• H of a 3 variable problem is negative
• Unconstrained minimization
• Principal minors are all positive in sign
• Constrained maximization
• Border preserving principal minors alternate in
  sign (-1)k where kgtr (with r1 the smallest
  principal minor to be evaluated is two fonc
  whereof one is the constraint (i.e. the border),
  k2)
• Hb of a 2 variable problem with one constraint
  is positive
• Hb of a 3 variable problem with one constraint
  is negative
• Constrained minimization
• Border preserving principal minors are all either
  positive or negative (negative for odd number of
  constraints positive for even)

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Duality indirect objective functions

• Using primal-dual optimization we can show that
  we can perform comparative statics directly from
  Profit function or Cost function
• Hessian matrix of second-order partials
• SOSC of primal-dual with respect to parameters is
  the lower right submatrix of this Hessian
• faa ?aa
• When parameters enter primal objective function
  linearly the submatrix of second-order partials
  is simple the negative of the second-order
  partial of the Profit function or the Cost
  function

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Cost functions

• Dual function based on optimal adjustments of
  factor demands
• Cost function is upward sloping and concave in w
• Shephards lemma first-order partial of cost
  function is conditional factor demand
• Comparative statics from second-order partials of
  parameters that enter only cost minimization
  objective function
• Conditional factor demands are homogeneous of
  degree 0 in w
• ? as Marginal Cost
• d?/dy is indeterminate since y is a parameter in
  the constraint
• Envelope theorem cost function is tangent to
  primal lagrangean

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Short and Long Run

• Restricted and Unrestricted Conditional Factor
  demands
• Le Chateliêr
• Long Run Competitive Equilibrium zero profits,
  PAC min
• Comparative statics of factor demands now allow
  for indirect effect through adjustments of goods
  prices and output quantity

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Consumer choice

• Utility maximization
• Constrained maximization
• Expenditure minimization
• Constrained minimization
• Axiomatic approach to preference sets
• Utility functions constructed to be consistent
  with preference axioms
• Invariance to positive monotonic transformations

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• Indirect utility function
• Roys identity
• Marshallian demand functions are homogeneous of
  degree 0 in prices and income
• ? is marginal utility of income
• Homothetic utility functions
• Income expansion path is linear, ? is constant
  for all relative prices
• implies reciprocity result in comparative statics
  of demand functions

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• Expenditure function
• Concave in p
• Hicksian demand functions as the first order
  partials of expenditure function
• Relationship between utility maximization and
  expenditure minimization
• Slutsky equation income and substitution
  effects

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• Welfare analysis
• Consumer surplus
• Compensating variation