Supplementary Notes for EC4101 c Indranil Chakraborty (Strictly not for further distribution) This note gives an overview of the mathematical details that advanced undergrad- uate stu dents would bene…t from. Thi s will not be covered in an examination and is being supplied only for the more mathematically inclined students. It is likely that most stude nts will hav e a hard time following the vector notati ons. In which case writ ing down the corresponding longhand expressions with just one or two constraints would help understand things clearly. In particular, the section on Lagrangian could be more of interest to you. It mig ht help if you set n = 2 and m = k = 1 in those section. Accordingly, you can ignore the notation Tin the superscripts of vectors, e.g., Tand T, tha t mea n transpose. If you are dea lin g wit h m= k = 1 then such constants are scalars rather than vectors. The section on grad ien t shoul d be skipped unless you are really into under stand ing things at a mu ch deeper lev el. None theless, I inclued ed them in here just in case some of you would like a quick summary of the related ideas and concepts. The mathematical concepts are best reviewed by Mathematics for Economistsby Carl P. Simon and Lawrence Blume (henceforth to be referred as “S&B”). For further clari…cations of the concepts that is hard to follow the best strategy is to Google it. Notations: 2 : “is in” 9: “there exists a/an” 8: “for all” 1 A quick review of standard de…nitio ns a nd results Set. Awell-de…nedcollection of elements is called a set. E.g., (i) set of integers (ii) set of positive numbers (iii) set of functions that take values in the interval [0,1]. Function. A functi on ffrom a set D to a set R is a relationship or mapping that associates each element ofD with a unique element in R. W e say that the functio n takes each element ofD to an element ofR and write f: D! R. This translation
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Supplementary Notes for EC4101c Indranil Chakraborty
(Strictly not for further distribution )
This note gives an overview of the mathematical details that advanced undergrad-
uate students would bene…t from. This will not be covered in an examination and is
being supplied only for the more mathematically inclined students. It is likely that most
students will have a hard time following the vector notations. In which case writing
down the corresponding longhand expressions with just one or two constraints would
help understand things clearly. In particular, the section on Lagrangian could be more
of interest to you. It might help if you set n = 2 and m = k = 1 in those section.Accordingly, you can ignore the notation T in the superscripts of vectors, e.g., T and
T , that mean transpose. If you are dealing with m = k = 1 then such constants are
scalars rather than vectors. The section on gradient should be skipped unless you are
really into understanding things at a much deeper level. Nonetheless, I inclueded them
in here just in case some of you would like a quick summary of the related ideas and
concepts.
The mathematical concepts are best reviewed by Mathematics for Economists by
Carl P. Simon and Lawrence Blume (henceforth to be referred as “S&B”). For furtherclari…cations of the concepts that is hard to follow the best strategy is to Google it.
Notations: 2 : “is in”
9: “there exists a/an”
8: “for all”
1 A quick review of standard de…nitions and results
Set . A well-de…ned collection of elements is called a set. E.g., (i) set of integers (ii) set
of positive numbers (iii) set of functions that take values in the interval [0,1].
Function . A function f from a set D to a set R is a relationship or mapping that
associates each element of D with a unique element in R. We say that the function
takes each element of D to an element of R and write f : D ! R. This translation
Closed set . We say that a set is closed if the limit of every convergent sequence in
the set is also in that set. E.g., the set R is closed, N is closed (because N does not
have a convergent sequence in it), [0; 1] is closed.
Theorem . (i) The union of a …nite number of closed sets is always closed. (ii) The
intersection of a …nite number of closed sets is always closed.
Open set . A set S is called open if its complement S c is closed. E.g., (0; 1).
Theorem . The union of a …nite number of open sets is always open. The intersection
of a …nite number of open sets is always open.
Example . The set [0; 1) is neither closed nor open.
Closure of a set . The closure of a set S , denoted S , is simply the set obtained by
taking the union of all limit or accumulation points of S with S . Thus the closure of
an open set is a closed set. E.g., the closure of the set (0,1) is [0,1].
Theorem . The closure of a set is a closed set.
Interior of a set . An element x 2 S is said to be in the interior of the set if there
is an " > 0 such that for all y with the property jy xj < ", y 2 S (i.e., all elements
in a close neighborhood of x are also in S ). The set of all elements in the interior of aset is called the interior of the set. E.g., the set (0; 1) is the interior of the sets [0; 1],
[0; 1); (0; 1] and (0; 1).
Boundary of a set . The boundary of a set S , denoted @ S , consists of all elements in
the closure of the set that are not in the interior of the set. E.g., f0; 1g is the boundary
of the sets [0,1], (0,1], [0,1) and (0,1).
Important note . The notation @ is the same as that used in denoting partial deriv-
atives. It is generally clear what exactly @ denotes from the context in which it is
written.
Bounded set . A set S is called bounded if there is M 2 N such that jxj < M for all
x 2 S .
Convex set . A set is convex if for every two elements x and y in the set the element
x + (1 )y is also in the set for all 2 (0; 1). E.g., the set (0; 1] is convex.
Continuous functions . A function f is continuous in its domain D if for each x 2 D
and each " > 0 there is a > 0 such that whenever y 2 D satis…es jy xj < ;
jf (y) f (x)j < ". A function is said to be discontinuous at x if it is not continuous atx. E.g., the function f (x) = x2 is continuous on [0; 1) while the function f de…ned as
f (x) = x2 if x > 0
= x + 1 if x 0
de…ned on R is discontinuous at 0.
Theorem . A function f is continuous if and only if for every convergent sequence
fxn
g in D that converges to x in D,
ff (xn)
g converges to f (x).
Left and right derivatives . We de…ne the left derivative of a function f at x by
f 0
(x) limh!0
f (x + h) f (x)
h
and the right derivative by
f 0+(x) limh!0+
f (x + h) f (x)
h :
We say that the derivative f 0(x) of f exists at x if f 0
(x) = f 0+
(x). In that case the right
(i.e., the left) derivative is called the derivative of f at x. The function is also called
di¤erentiable at x :
Di¤erentiable functions . A function is called di¤erentiable if its derivative exists
everywhere in its domain, i.e., it is di¤erentiable everywhere in its domain.
Continuously di¤erentiable functions . A function f : D ! R is called continuously
di¤erentiable if its derivative f 0(x), as a function of x, is continuous everywhere in D.
The collection of all functions on D that are continuously di¤erentiable is denoted by
C 1(D) or simply C 1 if the domain is clear from the context.
Twice continuously di¤erentiable functions . A function f : D ! R is called twice
continuously di¤erentiable if its derivative f 0(x) is di¤erentiable everywhere on D and
that second derivative f 00(x) is continuous everywhere on D. The collection of all
functions on D that are twice continuously di¤erentiable is denoted by C 2.
Convex and concave functions . A di¤erential function f is called convex if f 00(x) 0
for all x 2 D and concave if f 00(x) 0 for all x 2 D. The function is called strictly
convex or strictly concave if the inequalities are strict. More generally, we call a functionf convex (resp., concave) if for all x; y 2 D and 2 (0; 1), f (x + (1 )y) (resp.,
) f (x) + (1 )f (y).
1.1 Functions of several variables (S&B ch - 10, 12, 13, 14)
The Real Line . It is simply the collection of all real numbers. It is denoted by R and
represented by a straight line that extends both in positive and negative directions.
Euclidean Space with Higher Dimensions . The direct product of n real lines gives
the n-dimensional Euclidean space Rn. When n = 2 this space is called the Euclidean
plane .
Vectors and scalars . An n-tuple (x1;:::;xn) 2 Rn is often called a vector (in the n
dimensional Euclidean space and often written as a column rather than a row) with the
interpretation that it has a sense of magnitude as well as a direction, rather than just
a location in the space. This interpretation comes from the area of mechanics and is
useful for doing multivariate calculus.
A scalar , on the other hand, is a real number with the interpretation that it has asense of magnitude only.
(See section 10.2 of S&B and the exercises therein.)
Important concepts to go over : Addition and subtraction of vectors, scalar multipli-
cation of a vector.
Dot (or inner) product of vectors . The dot product of two n-vectors u and v is
written and de…ned as u v =Xn
i=1uivi. A standard result interprets the dot product
u v = kuk kvk cos , where is the angle between the two vectors (the concept of angle
between vectors is tricky when we are in more than three dimensions, so just think onlyof n = 2 or 3).
Norm or metric . There is an easy way to measure the distance between two points
x and y in R. Simply calculate the absolute value jx yj of the di¤erence. When
we deal with higher dimensional Euclidean spaces, like Rn, there is no straightforward
de…nition of “distance between two points.” In that case, we construct an appropriate
de…nition for the “distance” and call it the metric or the norm of the Euclidean space.
Conventionally, the distance (as de…ned) between the point 0 of the vector spaceand a point x under consideration is called the norm of the space and denoted by kxk.
The idea is that this same de…nition can be used to calculate the “distance” d(x; y)
between two points x and y as d(x; y) = kx yk.
Several notions of distance are used depending on the nature of the problem at hand.
E.g., (i) kxk jxj (the absolute value) is a norm on the vector space R, (ii) kxk =q Xn
i=1x2i is a norm or metric on the vector space Rn. An alternative metric on Rn is
also given by kxk =Xn
i=1jxij. Note that the two di¤erent norms in the example mean
the same thing when n = 1, i.e., the vector space is the real line R
.
Bounded set . A set S is called bounded with respect to a norm kk if there is a
M 2 R+ such that kxk M for all x 2 S . A set that is not bounded is called an
unbounded set .
Convergence of Sequences in Rn. (i) Sequences in Rn; (ii) Convergence of a sequence;
(iii) Limit of a sequence.
Results
i. A sequence can have at most one limit.ii. lim(xn + yn) = lim xn + lim yn.
iii. lim cxn = c lim xn.
iv. lim xn yn = lim xn lim yn.
v. A sequence in Rn converges if and only if every component converges is R.
Subsequence of a sequence in Rn. Any sub-collection of the elements in the same
order is called the subsequence of the original sequence.
Results
i. A sequence converges if and only if every subsequence converges.
ii. The limit of a convergent sequence is the same as the limit of any of its subse-
quences.
Limit points or accumulation points of a set . The limit of a convergent sequence
constructed from the elements of a set is called a limit or accumulation point of the set.
Example . Consider the function f : S ! R given by f (x) =p
25 x21 x2
2 where
S = fxjx21 + x2
2 25g. Let us …nd its level curves corresponding to c 2 R++ and
describe these level curves. The level curve/set corresponding to level c is simply theset of x 2 R such that f (x) = c. It is described by the level set fxjx2
1 + x22 = 25 c2g.
As you can tell, the level set is de…ned only as long as jcj 5. So for c 2 (0; 5) the
level set corresponding to c is simply a circle centered around the origin with a radiusp 25 c2. As c increases from 0 to 5 we get a system of concentric circles all centered
at the origin but with smaller and smaller radii. Note that when x1 = 0 and x2 = 0
the level is at its highest, viz. 5. In fact, the level set corresponding to c = 5 is just the
singleton f(0; 0)g.
We call a set such as above a level set. However, when the set can be geometrically
represented by a curve, it is called a level curve. They are also called the contours of
the graph of f (x) =p
25 x21 x2
2.
Continuous functions . A function f is continuous at x 2 D Rn (that is a subset
of a normed linear space) if for each " > 0 there is a > 0 such that whenever y 2 D
satis…es ky xk < ; jf (y) f (x)j < ". The function f is said to be continuous on D
if it is continuous for every x in D.
One-to-one and onto functions . A function f : D
! R is called a one-to-one
function if whenever for x; y 2 D satisfy x 6= y then f (x) 6= f (y). A function is called
onto if for every y 2 R there is a x 2 D with f (x) = y.
Inverse functions . The inverse function for a function f : D ! R is a function
f 1 : R ! D such that f 1(f (x)) = x for all x 2 D and f (f 1(y)) = y for all y 2 R.
Results
i. The inverse of a function exists if and only if it is one-to-one and onto.
ii. Every strictly increasing function has an inverse.
(First order) partial derivative . The derivative of a function with respect to exactly
one variable, and while treating the other variables as constant, is called a partial
derivative. Speci…cally we de…ne the partial derivative of f on Rnwith respect to xi as
stone will move in the direction of the tangent to the circle where its position would be
at time t. This tangent is a vector with a direction and magnitude. The direction will
depend on the position of the stone on the circle at time t and intuitively the magnitudeof the vector will determine the distance that the stone will be carried once I let go
of it. Again, intuitively one would expect that higher the speed at which I rotate the
stone at the end of the string the greater the distance it will go once it is released.
This is easily seen by considering the path of the stone over time t as it rotates as
described by the parametric form
x = cos t
y = sin t
The speed at which I am rotating the stone covers an angle t in time t. If I rotate it at
a faster speed it covers an angle 2t in time t. In the …rst case, the position of the stone
at time t = =2 is (0; 1). The tangent vector at that point is
( sin =2; cos =2) = (1; 0):
That gives us the idea of the force acting on the stone as the stone is released. In the
second case when I am rotating the stone at a higher speed the position of the stone is
described as
x = cos 2t
y = sin 2t
and stone reaches the same position (0; 1) in time =4. Now let us calculate the tangent
vector at that point. It is given by
(2sin2=4; 2cos2=4) = (2; 0):
Thus the force acting on the stone in this case is twice that in the earlier case but in
the same direction.
Chain rule . Consider the function f (x) as its value changes on a curve written
parametrically as (x1(t);:::;xn(t)). The derivative of the function f (x(t)) of t with
Example . A person holds a stock for company A and a stock for company B. The
person’s utility when the prices of the stocks are pA and pB is U ( pA; pB) = 20 pA+ 5 p2A+20 pB. Suppose at time t prices are given by pA(t) = t and pB(t) = 2t2. How does the
person’s utility change over time?
Directional derivatives and Gradients . The chain rule is very useful for computing
how a function changes in a given direction. For instance, suppose we are interested
in knowing how a function f :Rn ! R changes as we move in the direction of the vector
v starting at a point x. The curve that this linear motion de…nes is described by the
parametric function
x(t) x + vt:
Now using the chain rule we have that the derivative of f at x in the direction v is
given byd
dtf (x(t)) =
Xn
i=1
@
@xi
f (x)vi
The array of all partial derivatives written and interpreted as a vector in its own
right is called the gradient (vector) of f at x and written as
Of (x) =0B@
@
@x1 f (x
)...@
@xnf (x)
1CA :
Now we can write the above directional derivative in a more compact form as Of (x)v.
Observe that the magnitude of the directional derivative depends on the magnitude
of v. So in order to make all directional derivatives comparable for all directions often
we consider vectors v with kvk = 1.
Interpreting the Gradient Vector . The directional derivative takes the greatest value
in the direction in which f increases most rapidly. Thus, Of (x) v is the greatest
when v points in the direction where f increases most rapidly. Recall that Of (x) v = kOf (x)k kvk cos where is the angle between the two vectors Of (x) and v.
Since cos takes the highest value of 1 when = 0, this means that Of (x) v takes
the greatest value when v points in the same direction as Of (x). In other words, the
Hessians . The Hessian of a function f of n variables is the matrix of all its second
order partial derivatives. It is written as
f xx(x) = D2f x(x) =
0BBBB@
@ 2
@x21
f (x) @ 2
@x2@x1f (x) @ 2
@xn@x1f (x)
@ 2
@x1@x2f (x) @ 2
@x22
f (x) @ 2
@xn@x2f (x)
... ...
. . . ...
@ 2
@x1@xnf (x) @ 2
@x2@xnf (x) @ 2
@x2nf (x)
1CCCCA
Young’s Theorem . This result essentially tells us that in the kind of situations that
we will encounter the order in which derivatives are taken in mixed partial derivatives
is irrelevant, i.e.,
@ 2
@xi@x jf (x) = @
2
@x j@xi
f (x):
Taylor’s Theorem (For a function of many variables). Let f : X ! R, where X is
an open subset of Rn be a C 2 function. Then for x 2 X and h 6=0 with x + th 2 X
for all t 2 [0; 1] we have
f (x + h) = f (x) + f x(x + h)h
for some
2(0; 1), and
f (x + h) = f (x) + f x(x)h + 1
2!hT f xx(x + h)h
for some 2 (0; 1) where f xx is the Jacobian of f .2
1.3 Quadratic forms
Quadratic form . A function of many variables of the form Q(x) =Xn
i=1aijxix j is
called a quadratic form. Incidentally a quadratic form can be written as
Q(x) = xT Ax
where A is a symmetric matrix.
2 Note that the Taylor’s theorem for functions of many variables can be stated to approximate afunction with the k -th and lower order derivatives. However, in that case the formula must be writtenin the longhand.
the output it should produce and the price it should charge to maximize its pro…t.
Without even looking at the details of the functions we can write down the necessary
condition (viz. MC = MR) that the …rm needs to solve in order to make its price-output decision. The rule is obtained by solving the seller’s pro…t maximization problem
involving a single variable.
Consider the problem of the same monopolist if it sells its product to two di¤erent
markets that are disjoint with no arbitrage the inverse demand for the second market
being P (q ) = 100 2q . The problem becomes slightly more complex in that it now
involves two variables q 1 and q 2 representing the outputs in the two markets.
Now suppose that the government stipulates that the …rm must produce 20 units
of the output in total. How should the …rm set its price-quantity decisions for the
two markets? The constrained optimization problem involved in this case is much more
complex than the ones described above. So how does one solve such problems in general,
and use them in economic analysis is what we are going to talk about in the next several
lectures. Before we consider the more complex area of constrained optimization, let us
refresh our memory of unconstrained optimization, in a rather formal way. In the
process, we will try to develop a di¤erent approach that will be helpful for solving and
analyzing the constrained optimization problems.
3.1. Unconstrained Static Optimization
Consider the problem maxx2S f (x) of maximizing a function on an open subset S
of Rn.
Theorem 3.1.1. (First order necessary condition). If f : S ! R is C 1 on an open
set S Rn and f has a local maximum at x 2 S , then f x(x) = 0.
Theorem 3.1.2. (Su¢cient condition). If f : S ! R is C 2 (S is an open subset
of Rn) and at x satis…es the conditions (i) f x(x) = 0 and (ii) the Hessian f xx(x) is
negative semi-de…nite in an open neighborhood of x then f has a local maximum at
x.
We will use only the strict version of this theorem:
Theorem 3.1.2a. (Su¢cient condition). If f : S ! R is C 2 (S is an open subset
of Rn) and at x satis…es the conditions (i) f x(x) = 0 and (ii) the Hessian f xx(x) is
Theorem 4.1.2. (Theorem 18.2 of S&B: necessary conditions with multiple equal-
ity constraint). Let f , h1;:::;hm be C 1 functions on an open subset of Rn (i.e., functions
of n variables). Suppose that x
is a solution of the problem
maxx
f (x)
s.t. h(x) = 0
Suppose further that the Jacobian hx(x) has rank m (as large as it can be)4 and de…ne
the Lagrangian function
L(x; ) f (x) T h(x):
Then, there is a 2 Rm such that Lx;(x; ) = 0.
Question. Can you tell why the statements involve open subsets rather than
closed?
Theorem 4.1.3. (Su¢cient condition for a strict local maximum) Let f , h1;:::;hm
be C 2 functions on an open set in Rn. If there exist vectors x and such that
Lx;(x; ) = 0 and the Hessian Lxx(x; ) is negative de…nite then f has a strict
local maximum at x.
Example. Consider again the example of the …rm that must maximize its pro…t
subject to an expenditure of $200. Let us solve the same problem now using theLagrangian method. First of all, observe that the NDCQ is satis…ed. We have L(x; ) =
100 (x1 10)2 (x2 15)2 (20x1 + 5x2 200). Suppose that a maximum exists
at x, then the necessary conditions for maximization can be written as
Lx1(x; ) = 2(x1 10) 20 = 0
Lx2(x; ) = 2(x2 15) 5 = 0
L(x; ) = 20x1 + 5x2 200 = 0
Solving these equations we have x1 =
110
17 ; x2 =
240
17 , and =
6
17 .Now we check if the su¢ciency condition for maximization is satis…ed at this solu-
tion. The relevant Hessian is given by
Lxx(x; ) =
2 00 2
:
4 This condition is the non-degenerate constraint quali…cation (NDCQ) for this result.
Since the Hessian is negative de…nite at x1 = 11017 ; x2 = 240
17 , and = 617 , therefore, the
aforementioned outputs maximize the seller’s pro…t under the given constraint.
Observe that outputs cannot possibly be negative. So we should have really imposedthe constraints x1 0 and x2 0, as well. In this case, we were just lucky that
that did not cause any problem, but there can be situations where not imposing these
constraints explicitly may end up giving negative outputs which would be meaningless
in this context. Later, we will see such a situation.
3.2 Optimization with inequality constraint
Theorem 4.2.1. (Theorem 18.3 of S&B: optimization under a single inequality con-
straint). Suppose that f and g are C 1 functions on R2 and that (x1; x2) maximizes f
on the constraint set g(x1; x2) 0. If g(x1; x2) = 0, suppose that
@g
@x1(x1; x2) 6= 0 or
@g
@x2(x1; x2) 6= 0:
Then there is a multiplier such that:
(i) @L
@x1(x1; x2; ) = 0
(ii) @L
@x2
(x1; x2; ) = 0
(iii) g(x1; x2) = 0
(iv) 0
(v) g(x1; x2) 0
where the Lagrangian L(x1; x2; ) = f (x1; x2) g(x1; x2).
Theorem 4.2.2. (Theorem 18.4 of S&B: optimization under multiple inequality
constraints). Suppose that f and g1;:::;gk are C 1 functions on Rn and that x maximizes
f on the constraint set g(x) = (g1(x);:::;gk(x))
T
0. Suppose also that the …rst k0constraints are binding and that the last k k0 constraints are not binding, and that
the rank of the Jacobian of the binding constraints0B@
where ~L(x; ) = f (x) T g(x) is the Kuhn-Tucker Lagrangian of the problem.
Theorem 4.4.3. (Kuhn-Tucker su¢ciency condition) Let f be a concave C 1 func-
tion and gi, i = 1;:::;k be convex C 1 functions on an open convex subset S of Rn. If
x 2 S satis…es the Kuhn-Tucker conditions
@ ~L
@x(x; ) 0
@ ~L
@(x; ) 0
xi
@ ~L
@xi
(x; ) = 0, i = 1;::;n
j@ ~L
@ j (x
;
) = 0, j = 1;::;k
then f has a global maximum at x subject to the constraints
g(x) 0
x 0:
Example. Consider the problem of a price discriminating monopolist with a cost
function C (q ) = q 2 who sells to two disjoint markets (without arbitrage) that have
inverse demands P 1(q 1) = 100 q 1 and P 2(q 2) = 150 q 2. Suppose that environmental
regulations prohibit the monopoly …rm from producing more than 25 units of outputin total. Calculate how much this pro…t maximizing monopolist will be selling in each
market.
Kuhn-Tucker necessary conditions for the problem are given by
instance, in the principal-agent model of designing the optimal contract the principal
would be interested to know how the reserve utility of the agent (which is often a
function of the agent’s outside options) would impact the principal’s pro…t. The impactof the di¤erent constraints faced in decision making on our chosen objective is also
interesting, among other things, for a better understanding and appreciation of the
trade-o¤s we face in economic contexts. In what we have seen so far, the Lagrangian
multiplier that accompanied a constraint in the optimization problem was simply a
tool that facilitated our computation. In this section we will see that the Lagrangian
multiplier is also loaded with information about the tradeo¤s made in the optimization
process. We will also see results that tell us how to compute the relationship between
the parameters of the model and the optimal value of the objective function.
First consider the following two theorems that tell us how to interpret the Lagrangian
multipliers.
Theorem 5.1. (Theorem 19.2 of S&B: Interpretation of the multiplier: equality
constraints). Let f; h1;::;hm be C 1 functions on Rn. Let a = (a1;:::;am) be an m-tuple
of exogenous parameters, x(a) be the solution of the maximization problem
maxx
f (x)
h(x) = a
with the corresponding multipliers being (a). Suppose that x(a) and (a) are
di¤erentiable function of a and that the relevant NDCQ holds at a. Then the multiplier
corresponding to the j -th constraint satis…es:
j(a) = @
@a jf (x(a)):
Thus the multiplier is the rate at which the maximum value of the objective functionchanges when the constraint is relaxed. It can be seen as the price we will be willing to
pay to relax the constraint a little bit. It is, therefore, called the shadow price of the
corresponding constraint. Increasing the level of ai may have a positive or a negative
e¤ect on the objective. To see the intuition behind this observation consider the case of
a manufacturer who is constrained to use a …xed number of workers (due to government
regulations or worker union demands). It is possible that the number of workers the
…rm is allowed to employ is too few and increasing the number of workers will help the
…rm to organize things better (e.g. perform division of labor, etc.) In that case, the…rm’s pro…t can be expected to increase if it is allowed to add one more employee. On
the other hand, consider the situation where the …rm has many more employees than it
needs. Imagine the extreme scenario, where the …rm is so over-sta¤ed that some of the
employees cannot be put into any productive activity. If the …rm is constrained to add
one more employee then that will not increase its output or revenue. This simply adds
to the …rm’s cost. In other words, adding the employee in this case is actually going to
decrease the …rm’s pro…t. It is due to this reason that the shadow price corresponding
to an equality constraint can be negative or positive.
Next consider the case of the inequality constraints:
Theorem 5.2. (Theorem 19.3 of S&B: Interpretation of the multiplier: inequality
constraints). Let f ; g1;::;gm be C 1 functions on Rn. Let a = (a1;:::;am) be an m-tuple
of exogenous parameters, x(a) be the solution of the maximization problem
maxx
f (x)
s:t: g(x) awith the corresponding multipliers being (a). Suppose that x(a) and (a) are
di¤erentiable function of a and that the relevant NDCQ holds. Then the multiplier
corresponding to the j -th constraint satis…es:
j(a) = @
@a jf (x(a)):
Example. Consider a monopoly seller who uses l units of labor and r units of
a certain raw material to produce lr units of it product. The seller faces an inversedemand of P (q ) = 100q in the market. Let us calculate the optimal choice of the seller
if it is allowed to use at most 10 units of its raw material by government regulations
when the prices for labor and the raw material are given by $2 and $1 per unit. In this