Theory of the Firm Cost Minimization

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Topic 2 Part IV
Theory of the Firm- Cost Minimization
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Properties of the Cost Function

• Shephards Lemma (derivative property)
• Let xi(w, y) be the firms conditional factor
• demand for input i. If f is quasiconcave (unique
• solution) and the cost function is differentiable
  in
• w at (w, y) and wi gt 0 for i 1, , n.
• Then,
• ?c(w, y)/?wi xi(w, y), i 1, , n

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Properties of the Cost Function

• Properties of the cost function that follow from
  the structure of the cost minimization problem
• Restrictions on factor demand functions derived
  from these properties and from Shephards lemma

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Properties of the Cost Function
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Properties of the Cost Function
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Properties of the Cost Function
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Properties of the Cost Function
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Properties of the Cost Function
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Definiteness of a Matrix, Concavity and Convexity

• A matrix H(x) is negative definite (n.d.) iff its
  n (leading) principal minors are alternate in
  sign so that the odd order ones are lt 0 and the
  even order ones are gt 0
• If this holds for all x in the domain, then f is
  strictly concave
• A matrix H(x) is positive definite (p.d.) iff its
  n (leading) principal minors are all gt 0
• If this holds for all x in the domain, then f is
  strictly convex

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Second-Order Condition for Unconstrained
Optimization

• Let the Hessian matrix be defined as
• f11(x) f12(x) f1n(x)
• f21(x) f22(x) f2n(x)
  H(x)
  . . .
• fn1(x) fn2(x) fnn(x)

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Second-Order Condition for Unconstrained
Optimization

• Define a (leading) principal minor Di(x) of the
  Hessian matrix H(x) as the determinant of the
  matrix resulting when the last (n i) rows and
  columns of the Hessian matrix H(x) are deleted,
  for i 1, , n
• D1(x) ? f11
• D2(x) ? f11 f12
• f21 f22

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Second-Order Condition for Unconstrained
Optimization

• f11 f1i
• Di(x) ? . .
• fi1 fii
• .
• .
• .
• f11 f1n
• Dn(x) ? . .
• fn1 fnn

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Properties of the Cost Function
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Hessian Matrix of Cost Function

• c11 c12 c1n ?x1/?w1 ?x1/?w2 ?x1/?wn
• c21 c22 c2n ?x2/?w1 ?x2/?w2 ?x2/?wn
• H . .
  .
• cn1 cn2 cnn ?xn/?w1 ?xn/?w2
  ?xn/?wn

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Properties of the Cost Function
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Properties of the Cost Function
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Properties of the Cost Function