Comparative Statics and Duality of the Cost Function

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Comparative Statics and Duality of the Cost
Function

• Lecture VII

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Comparative Statics

• Comparative statics with respect to changes in
  input prices.
• The most common results of the comparative
  statics with respect to input prices involve
  intuition about derived demand functions.
• From the primal approach, we expect the demand
  functions for each input to be downward sloping
  with respect to input prices.

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• Starting from the cost function
• By Shephards lemma.
• In addition, we know that if ij then by the
  concavity of the cost function in input prices

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• In addition, we know that by Youngs theorem, the
  Hessian matrix for the cost function is
  symmetric
• The Hessian matrix for the cost function is also
  singular.

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• Eulers Theorem Eulers theorem is based on the
  definition of homogeneity
• Differentiating both sides with respect to t and
  applying the chain rule yields

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• Letting t1 then yields
• Coupling this result with the observation that if
  a function is homogeneous of degree r, then its
  derivative is homogeneous of degree r-1. We know
  that the input demand functions are homogeneous
  of degree zero in prices. Thus,

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• Multiplying this expression by xi(w,y) yields
• Given that we know that eiilt 0, this result
  imposes restrictions on the cross-price
  elasticities.

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• Briefly, let us prove the homogeneity of the
  marginal cost function. Starting with the
  marginal cost, differentiate with respect to each
  price

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• Comparative statics with respect to output
  levels.
• Following from the restrictions on the cross
  price elasticities above, the comparative statics
  with respect to output levels imply that not all
  inputs can be inferior or regressive.
• An inferior input is an input whose use declines
  as production increases while the use of a normal
  input increases as production increases.

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• Like the cross-price results above, we start with
  the sum of the differences of individual demand
  functions with respect to the level of output

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• In order to develop the effect of output on total
  cost, we start with the original Lagrangian from
  the primal problem
• Solving for the output using the first-order
  condition on the Lagrange multiplier and
  differentiating the solution then yields

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• From the first set of first-order conditions, we
  see

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• Which proves the definition of l consistent with
  the envelope theory. Therefore,
• Given this optimum l, we can then sum over the
  initial first-order conditions

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• Introducing the notion of an elasticity of scale

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• Chambers defines
• as the cost flexibility (the ratio between the
  marginal and average costs).

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• Remember the elasticity of scale in the
  production function
• which is the ratio between the marginal physical
  and average physical products.

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• What is developed here is not quite the same, but
  is actually the elasticity of size.
• It answers the question Do I build one large
  plant or several small ones?
• Defining yy/m
• is related to the homogeneity
  of the cost function in terms of scale. Taking

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• If n(y,w) gt 1 then e(y,x(w,y)) lt 1 there are no
  efficiencies to centralization diseconomies of
  scale.
• If n(y,w) lt 1 then e(y,x(w,y)) gt 1 there are
  efficiencies to centralization economies of
  scale.

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• Just like the MPP-APP comparison, we can envision
  the ratio be marginal cost and average cost. It
  is clear that
• Also, it is apparent that average cost equals
  marginal cost at the minimum of the average cost
  curve

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Duality between the Cost and Production Functions

• In our discussion of the primal we demonstrated
  how the production function placed restrictions
  on economic behavior.
• The question posed in duality is whether the
  optimizing behavior can be used to recover or
  reconstruct the properties of the production
  function.

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• Minkowskis Theorem A closed, convex set is the
  intersection of half-spaces that support it
• The half space H(m,k) is defined as

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• Thus, based on the definition of a cost function
  we have
• This definition actually recovers the original
  production set

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• If V(y)V(y), the original technology can be
  recovered.