58
Economics
Proof of EnvelopeTheoremfor constrainedoptimization problems(from Varian)
Consider a parameterized maximization problem of the form
max g(x1,x2,a)
M(a ):
s u ch that h(n1,n2,o) :0.
The Lagrangian for this problem is
E:
g ( rr,az,a) - l h(o1, rz,a),
and the first-order conditions are
Idxt -.ril:o
drr
(27.4)
3 -)3 :o
OfrZ
Ofiz
h(ay,x2,@ ): 0.
These conditions determine the optimal choice functions (a{a),rz(o)),
which in turn determine the ma;<imum nalue function
M (o) : s@L(a),
r2(a),a).
(27.5)
The envelope theorem gives us a formula for the derivative of the
value function with respect to a parameter in the maximization problem.
Specifically the formula is
d.M(a) _ ?E(x,a)
1
lx=x(o)
da
0a
0
g
(a
r,a
z,o)
_
|
_ ,i l h(xuxz,a) 1
lx=x(c).
l*=x(c) "
A"
Ao
As before, the interpretation of the partial derivatives needs special ca,re:
they are the derivatives of g and h with respect to a holding t1 ond x2
fiied, at their optirnal aalues.
The proof of the envelope theorem is a straightforwa,rd calculation. Differentiate the identity (27.5) to get
#:y^*.H**H,
and substitute from the first-order conditions (27.Q to find
d,M
E:
,f Ah dq
, 0h da2l , 0s
" La^ at - a- dol* a"'
(27.6)
Now observe that the optimal choice functions must identically satisfy
the constraint h(a1(a),r2(a),a) : 0. Difierentiatingthis identity with
respectto a, we have
A h d q ,0hd,a2,A h_^
a.rE - 6;rE
-r
6;
= v'
Substitute (27.7) into (27.6)to find
ry: -t* *P.
aa
which is the required result.
oa
da'
(27.7)