MaximumValue Functions and the Envelope Theorem

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Maximum-Value Functions and the Envelope Theorem

• From Chiang and Wainwright, Chap. 13

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Maximum value functions

• A maximum-value function is an objective function
  where the choice variables have been assigned
  their optimal values.
• These optimal values of the choice variables are,
  in turn, functions of the exogenous variables and
  parameters of the problem.
• Once the optimal values of the choice variables
  have been substituted into the original objective
  function, the function indirectly becomes a
  function of the parameters only (through the
  parameters' influence on the optimal values of
  the choice variables).
• Thus the maximum-value function is also referred
  to as the indirect objective function.

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The Envelope Theorem for Unconstrained
Optimization

• Significance of the indirect objective function
• For any optimization problem
• the direct objective function is maximized (or
  minimized) for a given set of parameters.
• The indirect objective function traces out all
  the maximum values of the objective function as
  these parameters vary.
• Hence the indirect objective function is an
  "envelope" of the set of optimized objective
  functions generated by varying the parameters of
  the model.

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Illustration
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Essence of the Envelope Theorem

• At the optimum, only the direct effect of F on
  the objective function matters.
• The envelope theorem says that only the direct
  effects of a change in an exogenous variable need
  be considered, even though the exogenous variable
  may also enter the maximum-value function
  indirectly as part of the solution to the
  endogenous choice variables.

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The Profit Function
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The Profit Function
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The Profit Function
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The Envelope Theorem for Constrained Optimization

• Given an objective function (U), two choice
  variables (x and y) and one parameter f and the
  following constraint g(x,y f)0

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Interpretation of the Lagrangian Multiplier

• In consumer choice problem,
• the Lagrange multiplier ? represents the change
  in the value of the Lagrange function when the
  consumer's budget c is changed.
• ? is the marginal utility of income (shadow price
  of c).
• Derive a more general interpretation of the
  Lagrange multiplier using the envelope theorem.
• See Chiang and Wainwright p.434

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Duality and the Envelope Theorem

• Expenditure function and indirect utility
  function exemplify the minimum- and maximum-value
  functions for dual problems.
• Expenditure function specifies the minimum
  expenditure required to obtain a fixed level of
  utility given the utility function and the prices
  of consumption goods.
• An indirect utility function specifies the
  maximum utility that can be obtained given
  prices, income, and the utility function.

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The primal problem
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The primal problem
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The dual problem
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The dual problem
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Duality
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Roys Identity

• Roy's identity states that the individual
  consumer's Marshallian demand function is equal
  to negative of the ratio of two partial
  derivatives of the maximum-value function.

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Shephards Lemma

• A similar approach used to derive the Hotellings
  Lemma, when applied to the expenditure function
  yields Shephard's lemma.

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Derivation
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EXAMPLE 1
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EXAMPLE 2 This the dual problem of cost
minimization given a fixed level of
utility. Suppose U is the target level of
utility
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