The Envelope Theorem by lindash


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									Andrew McLennan                                                          March 2, 1999

                        Economics 5113
            Introduction to Mathematical Economics
                          Winter 1999
                                     Lecture 13
                   The Envelope Theorem
  I. Introduction
       A. The envelope theorem is a general principle describing how the value of an
           optimization problem changes as the parameters of the problem change.
              1. In principle it is only a minor reduction in computational complexity.
              2. In practice it often clari es one's view of a problem, and it is particu-
                 larly important in allowing one to intuit the answer to certain important
       B. We will begin with a general statement of the principle for unconstrained max-
       C. The idea will then be applied to topics in consumer theory that are often
           described by the term duality."
       D. We then go on to consider the general result for constrained problems, formally
           substantiating the intuition that the Lagrangean multipliers are shadow prices
           associated with changing the allowed value of the constraint function.

 II. The General Principle
       A. Let f : IRn  IRk ! IR be a C 1 function.
            1. We think of IRk as a space of parameters, while IRn is the space of choice
      2. Thus we will consider the problem

                                            max f z; 

          where 2 IRk .
B. Let x : U ! IRn be a C 1 function de ned on an open set U IRk with
   the property that for each 2 U , x   solves the problem above, so that
   f x  ;   f z;  for all z 2 IRn.
       1. Assume that f is a C 2. Focusing on a particular point 0 2 U , the matrix
                 2     @2 f 2                              @2 f                          3
                     partialz1 x  0 ; 0       :::   partialz1 @zn x  0 ;       
                 6               .                                   .                    7
                 6               .                                   .                    7
                 4               .                                   .                    5
                       @ 2f
                                 x  0 ; 0    :::      @ 2f
                                                                   x  0 ; 0 
                   partialzn @z1                         partialzn

        is negative semi-de nite.
           a. It is nonsingular if and only if it is negative de nite. In this case
              the implicit function theorem implies that,
     2. De ne V : IRm ! IR by V   = f x  ; .
     3. Typically V is called the value function since it expresses the value of
        the problem as a function of the given parameter.
C. The envelope theorem is the assertion that for any 2 U and any i = 1; : : : ; k,
                                 @V   = @f  ; x  :
                                 @ i      @ i
      1. The proof is very simple: applying the chain rule yields
                  @V   = @f x  ;  + X @f x  ;   @xj  ;
                  @ i      @ i             j =1 @zj          @ i
         and the rst order conditions for optimization imply that
                               @f x  ;  = 0 j = 1; : : : ; k:
      D. To develop some appreciation for this principle, suppose you are a management
         consultant who has been asked to predict how a small increase in the cost of
         one of a rm's inputs will impact on the rm's pro ts.
            1. Initially it might seem that this is a very complicated problem, since, in
               response to the change in its input price, the rm may choose to change
               its input bundle. One might think that the only way to approach this
               problem is to look at the rm's production function, determine the new
               optimal bundle of inputs, and compute pro ts in the new situation.
            2. When you know the envelope theorem, your problem is very simple. You
               just determine how much of the input the rm is currently using, then
               multiply that amount by the price change.
            3. If the change in price is large, then the envelope theorem is not directly
               applicable, in the sense that the derivative of the pro t function is not
               a reliable guide to the total change in pro t. Still, the underlying in-
               tuition is useful, in that the idea is that at worst the rm could simply
               continue with its current practices, so that the price change multiplied by
               the current input usage is an upper bound on the reduction in pro ts.
III. Duality in the Theory of the Consumer
      A. We now treat some derived functions that emerge from natural problems in
          the theory of the consumer, and which involve interesting applications of the
          envelope theorem.
              1. These functions arise naturally in welfare economics.
              2. They have also found important applications in the econometric theory,
                 since they are useful foundations for the types of statistical models that
                 are applied there.
       B. Let u : IR2 ! IR be a consumer's utility function.

              1. We will always assume that it is C 1 and strictly increasing, in the sense
                   that the partials @u and @u are everywhere positive.
                                      @x       @y
                2. In what follows we will also assume that the derived functions we discuss
                   are C 1, but you should be warned that the question of what assumptions
                   on u justify this is quite complicated.
                3. The derived functions of interest are as follows:
             vpx; py ; I  = maxpxx+pyyI ux; y | the indirect utility function
             xpx ; py ; I ; ypx ; py ; I  = argmaxpxx+pyyI ux; y | the Marshallian de-
             mand functions
             epx ; py ; v = minux;yv pxx + py y | the expenditure function
             hpx ; py ; v; kpx; py ; v = argminux;yv pxx + py y | the Hicksian compen-
             sated demand functions
                4. We now have the following consequences of the envelope theorem:
Proposition 1:
             @v                                      @v
       a   @px px ; py ; I  = xpx ; py ; I   @I px ; py ; I ;
             @v                                      @v
       b   @py px ; py ; I  = y px ; py ; I   @I px ; py ; I .
Proof: To achieve the framework of the envelope theorem unconstrained optimization
we use the constraint to eliminate one variable, so that

                                   vpx; py ; I  = max ux; I , pxx :
                                                                p         y
Apaplying the envelope theorem yields
                                 @v = , x @u and @v = , 1 @u
                                 @px   py @y     @I    py @y
which yields a when x = xpx ; py ; I . The proof of b is similar.

Proposition 2:
        a @px px ; py ; v = hpx; py ; v;
       a @py px ; py ; v = kpx; py ; v.
Proof: In this case the constraint ux; y  v does not allow one to solve for y explicitly.
We use the implicit function theorem to de ne a function f such that ux; f x; v = v.
                           epx; py ; v = min pxx + py f x; v
and the envelope theorem yields a when x = hpx; py ; v. The envelope theorem also
implies b once we realize that kpx; py ; v = f hpx ; py ; v; v.

 IV. The Envelope Theorem for Constrained Optimization
       A. Suppose now that we are given

                              f : IRn  IRk ! IR and g : IRn  IRk ! IRm :

               1. We are studying the problem of choosing z to maximize f z;  subject
                  to the constraint gz = 0.
               2. Let L : IRn  IRk  IRm be the Lagrangean function for this problem:

                                          Lz; ;  := f z;  +   gz; :
        B. Suppose that U IRk is open, and that x : U ! IRn and  : U ! IRm
           satisfy the rst order conditions for optimization:
                             @ L x  ; ;    = 0 for i = 1; : : : ; n, and
                               @ L x  ; ;    = 0 for j = 1; : : : ; m.
               1. We have not, in fact, assumed that x   solves the maximization prob-
                  lem, but of course in almost all applications this will be the case. De ne
                  the value function to be the function V : U ! IR de ned by

                                                V   = f x  ; :
Theorem: Under the assumptions described above,
                  @V   = @f x  ;  + X    @gj x  ; :
                  @ h      @ h                  j @ h
                                           j =1

Proof: We have the following computation in which, after a certain point, we do not write
the points at which the various functions are evaluated, sicne they may be inferred:
         @V = @ Lx  ; ;   
        @ h @ h
                 n                       m
                X @f @x    i + @f + X @j gj +   X @gj @xi + @gj 
              = @z  @                                 j
                i=1 i       h @ h j =1 @ h                  i=1 @zi @ h @
                 n          m
                Xh @f X
  @gj i @x X @         m                    m
              =          +      j  @z  @   i +        j g + @f + X   @gj
                                                            j @
                i=1 @zi j =1           i      h j =1 @ h           h j =1 j @
                 n               m
                         @x X @ @ L                  m
              = @ L  @ i + @ j @ + @@f +   @gj
                X                                      X
                i=1 zi      h j =1 h j             h j =1 j @
                 @f + X   @gj :
                   h j =1 j @

        C. With this result, the facts about the indirect utility function and the expendi-
           ture function follow directly.
              1. For the optimization problem de ning the indirect utility function

                                      vpx ; py ; I  := maxxx x+pyy=I ux; y
                 we have f x; y; px ; py ; I  = ux; y and gx; y; px ; py ; I  = I , pxx , py y.
                 From the Theorem we get
                               @v p ; p ; I  =  p ; p ; I x p ; p ; I  and
                              @px x y                 x y            x y

                                         @v p ; p ; I  =  p ; p ; I :
                                         @I x y                  x y

                 Assuming that the partials of u are everywhere positive, the Lagrangean
                 conditions imply that  6= 0, after which the equation @px = x @v@v
                 follows from division.
2. For the optimization problem de ning the expenditure function

                        epx ; py ; v = uminv pxx + py y

  we have f x; y; px ; py ; v = pxx + py y and gx:y:px; py ; v = v , ux; y.
  The equation @px = hpx; py ; v now follows from the theorem simply
  because px does not a ect g.


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