4. What is the “envelope theorem” for static optimization problems and how is it proved?
Preliminary point: The implicit function theorem is a result from mathematics that economists simply
use. For a proof, see H. Nikaido, Convex Structures and Economic Theory, Academic Press, 1968. The
theorem states that a solution for a static optimization problem exists if the Jacobean of the system of
equations that defines FOC is non-zero at the point where they are mutually satisfied. The Jacobean is a
matrix. Its rows contain the first order derivatives of each first order condition with respect to the
choice variables and Lagrange multipliers, evaluated at the optimal point. The Jacobean is a square
matrix because there are
first order conditions and each is a function of choice variables and
Lagrange multipliers. The Jacobean is the bordered Hessian (no border in an unconstrained problem).
Assuming sufficient SOC guarantees that the bordered Hessian has a non-zero determinant and so
ensures a non-zero Jacobean for the system of FOC at the point where these conditions are satisfied.
With the condition for the implicit function theorem satisfied, write the solution for the optimal choice
variables as functions of the parameters of the problem:
where is a
complete list of all the parameters that appear in the objective function and the constraints. A different
set of parameters, in general, alters the solution of some or all choice variables. For example, in the
unconstrained net revenue maximization problem referred to in the answer to question 1, includes all
the parameters that define the production function plus the price of output and the prices of
for a total of
inputs,
in the case of a CES production function (which has four parameters). With
constraints, Lagrange multipliers also vary with the parameters and so their optimal values are also
written as functions of the vector of parameters:
.
The objective function and constraints (if any) can be written as
and
for
.
This notation (parameters listed after a semi-colon) simply means that the parameters are “variable
constants” as in the equation for a straight line:
. For each set of parameters,
defines a different straight line. In this most general case, the list of parameters is common to the
objective function and the constraints, if any. However, in the three standard problems microeconomic
problems (unconstrained net revenue or profit maximization; constrained utility maximization; and
constrained cost or expenditure minimization) there is no such overlap, and the resulting simplicity is
central to the analysis of these problems. Such a separation can be indicated in a problem with two
constraint as follows:
subject to
and
. The vectors , , and are
mutually exclusive. In developing the envelope theorem, the uncommon general case will be assumed in
order to show how important the separation of parameters is.
The essential step in the proof of the envelope theorem is to substitute the optimal solution for the
choice variables back into the objective function:
The function
is called the indirect objective function and this is the function that usually estimated
in empirical work. The envelope theorem concerns the derivatives of the indirect objective function and
is therefore essential to the interpretation of empirical results. To obtain the theorem, define a new
“primal-dual” Lagrangian for a new optimization problem:
There are two ways to proceed. One is to replace
with
in which case the above Lagrangian is
identically zero. Its total derivative with respect to a parameter is therefore also zero and it is this fact
that allows
to be written in terms of partial derivatives of the objective function and
the constraints, evaluated at the optimal point.
n
r
n
r
r
d j
dL*
f dxi*
f
g j dxi*
g j



 j 
 j
  g j ()

0
d k i 1 xi d k  k j 1 i 1 xi d k j 1  k j 1
d k  k
n 
r
r
r
d j
f
g j  dxi*
f
g j

j

  
 j

g
(

)




0


j

xi  d k j 1
d k  k j 1  k  k
i 1  xi
j 1
All but the last three terms vanish at the optimal point where FOC are satisfied:
r
f
g j
 j
0
xi j 1 xi
g j ()  0
i  1,...n
j  1,...r
The envelope theorem follows. At the optimal point:
r

f
g j

  j
 k  k j 1  k
A second way to prove the theorem is to write down the FOC for an extreme point of the primal-dual
Lagrangian. This Lagrangian is no longer identically zero. Choice variables and parameters are treated
as variables (no semi-colon). FOC for an extreme point of
are:
There are choice variables, constraints, and parameters each of which, in the general case, can
occur in the objective function and every constraint. The key to obtaining the envelope theorem is to
note that, if the first two sets of FOC are satisfied, then
. This means that if all the FOC for the
primal-dual problem are satisfied, then in the last conditions, the partials of the objective function and
the constraints must be evaluated at the optimal point:
These are partial derivatives of the indirect or optimized objective function. The point of the envelope
theorem is that the changes in the optimal choice variables and Lagrange multipliers can be ignored,
treating them as if they were constants set equal to their optimal values.
5. What does the envelope theorem reveal about the importance of the location of parameters of
interest in a static optimization problem?
Special cases of the envelope theorem are now easily stated. Assume there is no overlap in the list of
parameters that appear in the objective function and each of the constraints and restrict attention to
the case of two constraints. Let
,
, and
. Then:
There are parameters in the objective function, parameters in the first constraint, and
parameters
in the second constraint. The most important implication of the special cases, which are common in
economics, is that when a parameter of interest occurs in the objective function only, its effect on the
indirect, optimized objective function can be analyzed without knowing how Lagrange multipliers vary.
Whenever a problem can be reformulated without changing its essential structure, so that parameters
of interest occur only in the objective function, it will always be an analytical advantage to do so. The
standard example is to replace “primal” utility maximization subject to a budget constraint with “dual”
expenditure minimization subject to a utility constraint if the focus of interest is on the effect of changes
in prices and income. If the focus of interest is on the effect of a shift in some parameter of the utility
function, the “primal” problem has an analytical advantage over the “dual” problem. Similarly, cost
minimization subject to an output constraint is the better formulation if the effect of input price changes
is the focus of interest; whereas, output maximization subject to a cost constraint is preferred if the
effect of changes in parameters of the production function is the focus of interest.
6. How are comparative static results obtained using envelopes? What symmetry properties hold
among comparative static results and what relevance does not have for empirical work?
Comparative static results are obtained by substituting the optimal choice variables and associated
Lagrange multipliers into the FOC from which they are obtained by the implicit function theorem. This
turns the FOC into identities. Differentiating them with respect to choice variables, Lagrange multipliers,
and parameters therefore leaves them as equations. Suppose there are variables, parameters, and
constraint. The associated matrix of second order cross partials of
is square of dimension
. The order of the FOC is immaterial.
Here the FOC in choice variables make up the first group; the envelope FOC make up the second group;
and the FOC in the single Lagrange multiplier (the constraint) comes last. Each FOC is differentiated first
with respect to choice variables, then parameters, and finally the Lagrange multiplier. Then the matrix
of second order cross partials appears as a bordered Hessian. Below, the dimensions of each matrix and
each vector on the left are given on the right.
 f x x  g x x
i j
 i j
 f  x  g  x
i j
 i j
gxj


f xi  j   g xi  j
f i j  g i j   i j
g j
g xi   dx   n  n n  m n  1   n  1 

g i   d   m  n m  m m  1  m  1  0

0   d   1  n 1  m 1  1   1  1 

Clearly, a non-trivial solution requires the matrix on the left to have a zero determinant from which all
comparative static results follow, using envelopes to identify each
so that each
can be
interpreted. If the indirect objective function is twice continuously differentiable in parameters, then a
solution for
is a solution for
so that an empirical estimate of
is likewise an empirical
estimate of
.