Review Advanced Micro Economic Theory

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Review Advanced Micro Economic Theory

• Professor Rutstrom
• Fall 2008

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Unifying themes

• Assertions
• Producers maximize profits and minimize costs
• Consumers behave according to the preference
  axioms and this can be modeled as utility
  maximization or expenditure minimization
• When there is uncertainty additional axioms apply
  to connect preferences over gambles to
  preferences over outcomes
• Test conditions
• Exogenous conditions that are manipulated in
  comparative statics exercises
• Predicted events
• Refutable hypotheses comparative statics
• With minimum specifications of the problem only
  assume optimization
• Test conditions must be observable and exogenous
• Unobservable parameters and variables must be
  stationary (preference parameters and technology
  parameters that are unobservable)

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Optimization problems

• Unconstrained profit maximization
• Constrained cost minimization
• Constrained utility maximization
• Constrained expenditure minimization
• Unconstrained maximization of utility over gambles

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Optimization problems

• Unconstrained profit maximization
• Technology defines the production function
• Test conditions output and input prices
• Constrained cost minimization
• Constrained utility maximization
• Constrained expenditure minimization
• Unconstrained maximization of utility over gambles

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Optimization problems

• Unconstrained profit maximization
• Technology defines the production function
• Test conditions output and input prices
• Constrained cost minimization
• Technology defines the production function
• Test conditions input prices and output quantity
• Constrained utility maximization
• Constrained expenditure minimization
• Unconstrained maximization of utility over gambles

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Optimization problems

• Unconstrained profit maximization
• Technology defines the production function
• Test conditions output and input prices
• Constrained cost minimization
• Technology defines the production function
• Test conditions input prices and output quantity
• Constrained utility maximization
• Preferences define the utility function
• Test conditions goods prices and income
• Constrained expenditure minimization
• Unconstrained maximization of utility over gambles

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Optimization problems

• Unconstrained profit maximization
• Technology defines the production function
• Test conditions output and input prices
• Constrained cost minimization
• Technology defines the production function
• Test conditions input prices and output quantity
• Constrained utility maximization
• Preferences define the utility function
• Test conditions goods prices and income
• Constrained expenditure minimization
• Preferences define the utility function
• Test conditions goods prices and utility level
• Unconstrained maximization of utility over gambles

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Optimization problems

• Unconstrained profit maximization
• Technology defines the production function
• Test conditions output and input prices
• Constrained cost minimization
• Technology defines the production function
• Test conditions input prices and output quantity
• Constrained utility maximization
• Preferences define the utility function
• Test conditions goods prices and income
• Constrained expenditure minimization
• Preferences define the utility function
• Test conditions goods prices and utility level
• Unconstrained maximization of utility over
  gambles
• Preferences define the expected utility function,
  the rank-dependent utility function, or the
  prospect value function
• Risk perception defines the probability
  transformations
• Risk attitudes, optimism/pessimism

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Optimization problems

• Unconstrained profit maximization
• Technology defines the production function
• Test conditions output and input prices
• Max p(x)f(x) - ?wx or Max p f(x) - ?wx
• p(p,w)
• Constrained cost minimization
• Constrained utility maximization
• Constrained expenditure minimization
• Unconstrained maximization of utility over gambles

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Optimization problems

• Unconstrained profit maximization
• Max p(x)f(x) - ?wx or Max p f(x) - ?wx
• p(p,w)
• Constrained cost minimization
• Technology defines the production function
• Test conditions input prices and output quantity
• Min ?wx s.t. yf(x)
• L ?wx ? (y -f(x))
• c(y,w)
• Constrained utility maximization
• Constrained expenditure minimization
• Unconstrained maximization of utility over gambles

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Optimization problems

• Unconstrained profit maximization
• Max p(x)f(x) - ?wx or Max p f(x) - ?wx
• p(p,w)
• Constrained cost minimization
• Min ?wx s.t. yf(x)
• L ?wx ? (y -f(x))
• c(y,w)
• Constrained utility maximization
• Preferences define the utility function
• Test conditions goods prices and income
• Max u(x) s.t. y ?px
• L u(x) ? (M - ?px )
• U(M, p)
• Constrained expenditure minimization
• Unconstrained maximization of utility over gambles

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Optimization problems

• Unconstrained profit maximization
• Max p(x)f(x) - ?wx or Max p f(x) - ?wx
• p(p,w)
• Constrained cost minimization
• Min ?wx s.t. yf(x)
• L ?wx ? (y -f(x))
• c(y,w)
• Constrained utility maximization
• Preferences define the utility function
• Test conditions goods prices and income
• Max u(x) s.t. y ?px
• L u(x) ? (M - ?px )
• 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
• L ?px ? (u0 - u(x))
• M(u,p)

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Optimization problems

• Unconstrained profit maximization
• Max p(x)f(x) - ?wx or Max p f(x) - ?wx
• p(p,w), convex in p and w, H(1) in (p,w) jointly
• Constrained cost minimization
• Min ?wx s.t. yf(x)
• c(y,w) concave in w, H(1) in w
• Constrained utility maximization
• Max u(x) s.t. ?px M
• U(M, p) quasi-conves in (p,M) jointly
• Constrained expenditure minimization
• Min ?px s.t. u(x)u0
• M(p,u) concave in p, H(1) in p
• Unconstrained maximization of utility over
  gambles
• Max ?w(p)-w(r) u(x)

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• level function convex
• shadowprice of the constraint lagrange
  multiplier
• homothetic and homogeneous functions

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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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Duality

• Profit function
• p(p,w)
• Cost function
• c(w,y)
• Indirect utility function
• U(p,M)
• Expenditure function
• e(p,u)
• Envelope theorem and primal-dual optimization
• Comparative statics the easy way

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Primal-dual functions

• Unconstrained Profit maximization
• min F(w,p,x) F(w,p)
• F(w,p,x) pf(x)-wx
• Constrained Cost minimization
• max L(w,x,y) - F(w,y)
• L(w,x,y)Swx ?(y-f(x))
• Constrained Utility maximization
• min L(x,p,M)- F(p,M)
• L(x,p,M)u(x) ?(M- Spx)
• Constrained Expenditure minimization
• max L(p,x,u) - F(p,u)

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Envelope Theorem

• First-order condition of Primal-Dual optimization
• Fa Fa
• La Fa
• General Comparative Statics - Second-order
  condition of Primal-Dual optimization
• Faa - F aa lt 0 or Faa - F aa gt 0
• L aa - F aa lt 0 or L aa - F aa gt 0

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Concepts

• Le Chatelier principle
• Comparative statics when addition a just-binding
  constraint
• Samuelsons conjugate pairs theorem
• Homogeneity and Euler theorem
• Elasticities (price, output, income)
• Elasticity of substitution and marginal rate of
  substitution
• Interpreting Lagrange multipliers
• Separability (additive and multiplicative)
• Returns to scale in technology (global and local)

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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
• 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 simply the negative of the second-order
  partial of the dual objective function

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Consumer Theory

• Certainty or uncertainty
• ordinal or cardinal utility functions
• invariant to positive monotonic or only linear
  transformations
• transforming utility or transforming probabilities

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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
• Slutsky-Hicks decomposition
• Quasi-concave utility function convex
  indifference curves
• Ordinal preferences invariant to positive
  monotonic transformations

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Marshallian

• 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

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Hicksian

• 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
• Labor-leisure and opposite income effects
• Welfare analysis
• compensating and equivalent variations

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Revealed Preferences

• Weak axiom of revealed preferences
• x0 is revealed preferred to x1 if at p0 and M0
  they are both affordable and x0 is chosen
• If p0 x0 p0 x1 where x0 is chosen at p0 and x1
  is chosen at p1
• then x0 is revealed preferred to x1
• for a rational consumer it must then be the case
  that p1 x0 gt p1 x1

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Choice under uncertainty

• States, outcomes, and probabilities
• Gambles, prospects and lotteries
• Preference axioms
• Von Neumann-Morgenstern utility functions
• Expected utility property
• Strict concavity of the elementary utility
  function is consistent with risk averse choices
• Cardinality of elementary utility carries
  information
• Invariant only to LINEAR positive transformations
  of elementary utility functions
• Payoffs in monetary income terms elementary
  utility function is an indirect utility function

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VNM utility functions

• Invariant only to LINEAR positive transformations
• All valid utility functions do not have the
  expected utility property
• Utility numbers have more content than ordinality
• Risk attitudes and concavity

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Risk aversion

• Jensens inequality
• EU(g)ltU(EV(g))
• Certainty Equivalent
• U(CE)EU(g)
• Risk Premium
• EV(g)-CE
• Arrow-Pratt measure of absolute risk aversion -
  (U/U)

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Probability weighting

• Convex probability transformation logically
  equivalent to concave utility transformation
• Original Prospect theory violates dominance and
  continuity
• Cumulative Prospect theory and Rank Dependent
  Utility theory does not
• Pessimism and Optimism
• Loss Aversion

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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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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
• Level functions (isoquants) are convex to the
  origin
• Marginal rate of technical substitution and
  elasticity of substitution
• Local and global returns to scale
• Separability

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• Profit maximization
• Max R(y) C(y), where yf(xi)
• Multi-plant firm R(y)-c1(y1)-c2(y2)
• Multi-market firm R(y1)R(y2)-c(y1y2)
• Multi-good firm R(y1)R(y2)-c1(y1)-c2(y2
• 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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Profit function

• Maps profits to parameters of profit maximization
  objective
• Restricts factor demands and output supplies to
  adjust to parametric changes optimally
• Convex in p,w (optimal profits are always higher
  than sub-optimal profits)
• Linearly homogeneous in p,w
• Increasing in p and decreasing in w

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Duality Profit functions

• Hessian matrix is negative definite by SOSC for
  unconstrained maximization of primal-dual
  objective function principal minors alternate in
  sign
• SOSC of primal-dual with respect to parameters is
  the lower right submatrix of this Hessian
• Samuelsons conjugate pairs results the
  fundamental comparative statics relation
• Reciprocity is simply Youngs theorem applied to
  the indirect objective function

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Dual comparative statics

• First order partials of profit function is supply
  and negative of factor demands (Hotellings
  lemma)
• Second order partials therefore are comparative
  statics of these functions

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Comparative statics of profit maximization problem

• Factor demands are downward sloping
• Output supply is upward sloping
• Cross price effect on factor demands is
  indeterminate
• Output price effect on each factor demand is
  indeterminate
• Reciprocity XpYw
• Homogeneity
• Unconditional factor demands are homogeneous of
  degree 0 in all prices (factor and output)
  irrespective of homogeneity of production function

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Short and Long Run

• Le Chateliêr
• Restricted and unrestricted unconditional factor
  demands and output supplies
• Fixed and variable factors
• X1(w1, w2, p)X1s(w1, p, x2(w1, w2, p))
• Y(w1, w2, p)Ys(w1, p, x2(w1, w2, p))
• Effect on variable factor from relaxing fixed
  factor constraint is indeterminate

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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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Reading guidance Silberberg

• Ch 4, 6, 7, 8, 10, 13
• 11.1 and 11.2 until p 325
• 11.5 until p 355
• You may skip the following
• 4.7 Finite changes
• Ch 4 appendix
• 6.4 more than one constraint
• 10.7 p 299-310
• 13.4

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Silberberg contd

• Homogeneity and elasticities
• Ch 4.5
• Ch 8.9
• Ch 10.6
• Chatelier
• Ch 4.6
• Ch 7.4 p 169
• Ch 8.8 p 205

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• Samuelsons conjugate pairs
• Ch 7.4 p 158 and p 169
• Interpreting Lagrange multipliers
• Ch 7.4 p 166
• Ch 8.6 p 189
• Ch 10.2 p 266
• Refutable hypotheses
• Ch 6 p 120
• Ch 6 p 126
• Ch 6 p 143
• Ch 8 p 196
• Ch 10.4 o 278

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• Concavity and second order conditions
• General equation for concavity
• Eq 4-2 p 71
• Eq 6-16 p 121
• Eq 6-63 p 135
• Eq 6-65 p 136

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Known typos in Silberberg

• Page 137 k2. should be k1.
• Page 162, equation 7-20b

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Reading guide to Jehle

• Ch 1.2
• Example 1.1, 1.2, 1.3, 1.4, 1.5
• Theorem 1.6, 1.7, 1.8, 1.9
• Example 2.1
• Ch 2.4
• Ch 3.2
• Example 3.3, 3.5, 3.6
• Theorem 3.2, 3.3, 3.4, 3.7

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• Also read the posted articles on non-expected
  utility theories