Review for Test 1

1
Review for Test 1

• Advanced Micro Theory
• ECO 6118

2
Assertion

• When you want to explain an observed behavior
  some aspects of the explanation are not
  observable
• Objectives of the agent
• Preferences of the agent
• You therefore make assertions about these
• These assertions, combined with observable facts
  about the reality of the agent (test conditions)
• make it possible to make theory predictions
• to compare to your behavioral observations

3
Scientific Pitfalls

• Ceteris Paribus
• useful in theory but does not hold in data
• auxiliary hypotheses
• Multiple Working Hypotheses and False Dichotomy
• Rejecting one hypothesis in favor of another
  neglects the possibility of non-included
  hypotheses
• builds support for one hypotheses falsely
• A causal explanation is confounded if there is
  more than one possible cause and the test does
  not control for that

4

• Step 1 Find x as the stationary point
• Step 2 Verify that this point is a maximum
• Step 3 Solve for xx(t)
• The explicit choice function
• Requires an inversion apply Implicit Function
  Theorem
• Step 4 Substitute x for x in FONC that is now
  an identity
• Step 5 Differentiate the identity
• Step 6 Solve for
• Step 7 Determine the sign by referring to SOSC
  plt0

5
Marginalism

• Comparative statics assumes changes are
  infinitesimally small
• The movement is from one optimum to another
  optimum
• No analysis of the path between the two optima
• FONC holds both before and after the change

6
Profit Maximization

• Max R(y) C(y), where yf(x)
• 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)
  Oligopoly, duopoly simultaneous optimization
  across multiple firms

7
Concavity

• For unconstrained maximization the objective
  function must be strictly concave
• The Hessian matrix must be negative definite
• where t is the direction of the change

8
Concavity for the n-dimensional case

• Principal minors of the determinant to the
  Hessian matrix of second-order partials of the
  objective function alternate in sign
• all diagonal elements fiilt0
• only the naturally ordered principal minors need
  to be checked
• the determinant to the Hessian is a principal
  minor and its sign depends on the value of n
• if n is even H gt 0
• if n is odd H lt 0

9
Maximize Profits

• SOSC requires strict concavity of production
  function
• needs to be asserted
• FONC defined over profits
• Production function does not need to have a
  stationary point

10
Refutable Hypotheses

• Comparative statics
• change in endogenous variable due to a change in
  an exogenous variable
• Follows from our assertion that the production
  technology is strictly concave
• Does not form a refutable hypothesis since no
  sign for f12 is asserted through strict concavity
• If f12gt0 (technological complements) then lt0

11
n-factors

• Identify the principal minors of the Hessian
• Substitute the explicit choice functions into the
  FONCs
• Totally differentiate the full system of
  equations wrt the parameter of interest
• Express in matrix notation
• Use Cramers rule to solve for the comparative
  statics
• Denominator is the determinant to the full
  Hessian it is a principal minor
• Numerator is the signed co-factor of a minor of
  the Hessian it is of a principal minor only if
  the rows and columns removed are numbered the
  same
• A signed co-factor is (-1)ij times the minor
• for a principal minor ij and therefore always
  even

12
More on cross-price effects

• n factor case
• Only if all factors are technological complements
• all off-diagonal elements in the Hessian are
  positive
• can we say that factors are economic complements,
  i.e.

13
All refutable hypotheses

• Factor demand functions are downward sloping
• Reciprocity in factor demands follows from
  Youngs theorem
• Supply function is upward sloping
• Reciprocity between factor demands and output
  supply

14
Comparative statics the general case

• f(x1, x2, a) is an arbitrary strictly concave
  objective function
• Example a species choice of attributes (x1,
  x2) for given environmental attributes (a)
• A neurological systems choice of responses (x1,
  x2) to given stimula (a)
• refutable only if either f1a0 or f2a0

15
Homogeneity

• Production function
• linearly homogeneous (H(1)) for Constant Returns
  to Scale
• homogeneous of degree lt1 for Decreasing Returns
  to Scale
• homogeneous of degree gt1 for Increasing Returns
  to Scale
• Factor demands and Output supply
• Homogeneous of degree 0 in (p,w) jointly due to
  stationarity in optimum (first order conditions)
• only relative prices matter

16
Euler Theorem

• if f(x) is homogeneous of degree r then
• for r1
• implies zero profits
• for r0
• Euler Theorem has implications for price
  elasticities

17
Restricted profit maximization

• Le Chatelier principle
• restricted optimization allows less adjustment
  than unrestricted optimization
• Proof based on the fundamental relationship
  between the restricted and unrestricted function

18
Duality in Profit Maximization

• Solution to maximization of direct (primal)
  objective function
• is the indirect (dual) objective function, called
  the Profit Function
• asserting profit maximization implies that the
  Profit Function is convex
• The Profit Function forms an envelope to the
  primal profit equations

19
Comparative statics

• By differentiating the Profit Function directly
  we can derive the factor demands and the output
  supply (Hotellings Lemma)
• Because of convexity we find the comparative
  statics results

20
General Comparative Statics using the Primal-Dual
objective maximization

• Find the optimum (the tangency point between the
  primal and the dual)
• it is a maximum since optimal profits are never
  less than non-optimal profits
• FONC
• the envelope theorem

21
Strict Concavity

• All diagonal elements in the Hessian determinant
  to this problem are negative by strict concavity
• Comparative statics wrt a (in the optimum treat a
  as exogenous)

22
Samuelsons Conjugate Pairs

• Forming refutable hypotheses
• if then

23
Properties of Profit Function

• Non-increasing in w, convex in w, continuous in w
• Non-decreasing in p, convex in p, continuous in p
• Linearly homogeneous in (p,w) jointly

24
Properties of Production Functions

• Continuous
• Differentiable
• Strictly increasing
• f(0)0
• Strictly quasi-concave
• isoquants are convex to origin
• strictly concave for unconstrained profit
  maximization

25

• Marginal Rate of Technical Substitution
• slope of the isoquant
• Elasticity of Substitution
• Constant Elasticity of Substitution production
  function
• Cobb-Douglas
• Leontief (fixed proportions)
• Homogeneity
• isoquants are parallel along a ray from the
  origin
• MRTS is a constant function of the input ratio
• Separability (weak and strong)
• Global and Local Returns to Scale