Chu Thanh Duc – MDE 10
Chapter 2. Calculus and Optimization
2.1. Calculus
2.1.1 Functions of a Single Variable
Differentiable function: If the function is continuous and smooth, with no breaks and kinks.
Derivative f’(x): the slope or instantaneous rate of change in f(x):
dy
 f ' ( x) , the
dx
instantaneous amount by which y chances per unit change in x.
-
First Derivative tells us whether the value of f(x) is rising or falling as we increase x.
-
Second Derivative tells us what the curvature of the function is.
Differential (dy): Measure instantaneous amount by which f(x) change at the poin x
following from a “small” change in x. dy = f’(x)dx.
Theorem: Slope, Curvature and Differentials
For any twice continuous differentiable function, y = f(x), in the neighborhood of the point x,
and for all dx  0:
First Differentials:
dy  0  f’(x)  0  f is locally increasing
dy  0  f’(x)  0  f is locally decreasing
dy = 0  f’(x) = 0  f is locally constant
Second Differentials:
d2y  0  f’’(x)  0  f is locally concave
d2y  0  f’’(x)  0  f is locally convex
d2y = 0  f’’(x) = 0  f is locally linear.
2.1.2 Functions of Several Variables
We concern with real valued functions of several variables.
Partial Derivatives: Let y = f(x1, x2, …xn). The the partial derivative of f with respect to xi
is defined as:
f ( x)
f ( x1, x 2,..., xi  h,....xn)  f ( x1, x 2,..., xn)
 f i ( x)  lim
h 0
xi
h
Geoffrey A. Jehle – Advanced Microeconomic Theory
1
Chu Thanh Duc – MDE 10
-
Partial Derivative tells us that whether the function is rising or falling as we change
one variable alone, holding all others constant.
Total differential: Tells us value of the function is rising or falling as we change all or some
variables simultaneously.
n
dy   f i ( x)dxi
i 1
Using vector notation:
The Gradient Vector of all partial derivatives: Vector derivative with respect to
xi: f ( x)  ( f1 ( x), f 2 ( x)... f n ( x))1xn
 dx1
Vector changes in variables: dx   . 
dxn nx1
dy  f ( x).dx
Second – order partial derivative: f ij ( x) 
The Gradient Vector:
The Hessian Matrix:
 2 f ( x)
xixj
f j ( x)  ( f1 j ( x), f 2 j ( x),..., f nj ( x))
  2 f ( x) 
H ( x)  

 xixj  nxn
Theorem: Young’s Theorem
For any twice continuous and differentiable function f(x), we have:
 2 f ( x)  2 f ( x)

xixj
xjxi
The Young’s Theorem tells us that the Hessian matrix will be symmetric.
Quadratic form:
d 2 y  d T x.H ( x).dx
Geoffrey A. Jehle – Advanced Microeconomic Theory
2
Chu Thanh Duc – MDE 10
Theorem: Curvature in Several Variables
Let f: D  R be twice continuous and differentiable and let x  D. Then,
2
T
d2y  0  f is concave at x  d y  d x.H ( x).dx  0 for all dx.
2
T
d2y  0  f is convex at x  d y  d x.H ( x).dx  0 for all dx.
2
T
d2y < 0  f is strictly concave at x  d y  d x.H ( x).dx  0 for all dx  0
2
T
d2y > 0  f is strictly convex at x  d y  d x.H ( x).dx  0 for all dx  0 .
The relations hold globally if they hold for all x  D .
H(x) is positive semi-definite:
d 2 y  d T x.H ( x).dx  0
H(x) is negative semi-definite:
d 2 y  d T x.H ( x).dx  0
H(x) is positive definite:
d 2 y  d T x.H ( x).dx  0
H(x) is negative definite:
d 2 y  d T x.H ( x).dx  0
Theorem: Concavity, Convexity and Second-Order Own Partial Derivatives
Let f(x) be a twice contiounously differentiable function.
1. If f(x) is concave, then f ii ( x)  0 , i = 1,2…n
2. If f(x) is convex, then f ii ( x)  0 , i = 1,2…n
3. If f(x) is strictly concave, then f ii ( x)  0 , i = 1,2…n
4. If f(x) is strictly convex, then f ii ( x)  0 , i = 1,2…n
Prove: If f(x) is concave, then f ii ( x)  0 , i = 1,2…n
F(x) is concave  d 2 y  dx T .H ( x).dx  0 for all dx
Without loss of generality, consider a two independent variable function.
Geoffrey A. Jehle – Advanced Microeconomic Theory
3
Chu Thanh Duc – MDE 10
 f ( x)
d 2 y  (dx1, dx2)  11
 f 21 ( x)
f12 ( x)   dx1
.
f 22 ( x) dx2
dx1
Let dx =    d 2 y  f11 ( x).( dx1) 2  d 2 y  0  f11 ( x)  0 .
 0 
This prove is similar for f ii ( x)  0 .
2.1.3. Homogeneous Functions
Homogeneous Functions: A real valued function is call
Homogeneous of degree k iff: f(tx) = tkf(x) for all t > 0.
Homogeneous of degree 1 iff: f(tx) = tf(x) for all t > 0.
Honogeneous of degree 0 iff: f(tx) = f(x) for all t > 0.
Theorem: Partial Derivative of Homogeneous
If f(x) is homogeneous of degree k, then its partial derivatives are homogeneous of
degree k-1.
Corollary: Linear Homogeneous Functions.
If f(x) is homogeneous of degree 1, then
f (tx) f ( x)

for all t > 0.
xi
xi
Example: Consider the Cobb-Douglas function: f ( x)  A.x1 x2
f (tx)  A.(tx1 ) (tx2 )  
f (tx)
f ( x)
 A (tx1)  1 t.(tx2 )   At    1x1 1 x 2   t    1 .
x1
x1
f (tx) f ( x)

     1  the Cobb-Douglas function is homogeneous of degree 1.
x1
x1
Theorem: Euler’s Theorem
1. If f(x) is homogeneous of degree k, then:
n
k . f ( x)  
i 1
f ( x)
xi
xi
2. If f(x) is homogeneous of degree 1, then:
n
f ( x)  
i 1
Geoffrey A. Jehle – Advanced Microeconomic Theory
f ( x)
xi
xi
4
Chu Thanh Duc – MDE 10
Prove: Assume that f(x) is homogeneous of degree k, by definition: f(tx) = tkf(x)
Differentiate the left - hand side with respect to t, we have:
n
f (tx)
f (tx)

xi
t
i 1 txi
Differentiate the right – hand side with respect to t, we have;
t k f ( x)
 kt k 1 f ( x)
t
n
 kt k 1 f ( x)  
i 1
f (tx)
xi
txi
n
Let t = 1, we have
k . f ( x)  
i 1
f ( x)
xi
xi
2.2. Optimization
Consider a single variable y = f(x) that is differentiable.
Local maximum at a point x*: means that f(x*)  f(x) for all x in some neighborhood of x*.
A unique local maximum at a point x*: means that f(x*) > f(x) for all x in some
neighborhood of x*.
Global maximum at a point x*: means that f(x*)  f(x) for all x in the Domain D.
A unique global maximum at a point x*: means that f(x*) > f(x) for all x in the Domain D.
Theorem: Necessary Conditions for Local Interior Optima in the Single – Variable Case
Let f(x) be a differentiable function of one variable. Then f(x) reaches a local interior
1. Maximum at x*
2. Minimum at x*
 f ' ( x*)  0( FONC )  dy  f ' ( x*)dx  0
 f ' ' ( x*)  0( SONC )  d 2 y  f ' ' ( x*)  0
(f(x) is local concave)
 f ' ( x*)  0( FONC )  dy  f ' ( x*)dx  0
 f ' ' ( x*)  0( SONC )  d 2 y  f ' ' ( x*)dx  0
(f(x) is local convex )
2.2.1 Real Valued Functions of Several Variables
Local Maximum: f(x) is local maximum at the point x* if for all x  B (x*) , f(x*)  f(x).
Global Maximum: f(x) is global maximum at the point x*  D  R n if for all x  D ,
f(x*)  f(x) .
Geoffrey A. Jehle – Advanced Microeconomic Theory
5
Chu Thanh Duc – MDE 10
Unique Local Maximum: f(x) is unique local maximum at the point x* if for all x
 B (x*) , f(x*) > f(x).
Unique Global Maximum: f(x) is global maximum at the point x*  D  R n if for all x
 D , f(x*)>f(x) .
Theorem: Local – Global Theorem
1. Let f(x) is a concave function. Then f(x) reaches a local interior maximum at x*
 f(x) reaches a global interior maximum at x*.
2. Let f(x) is a convex function. Then f(x) reaches a local interior minimum at x*
 f(x) reaches a global interior maximum at x*.
Prove: Let f(x) is a concave function. Then f(x) reaches a local interior maximum at x* 
f(x) reaches a global interior maximum at x*.
Sufficient condition: we need to prove that f(x) reaches a local interior maximum at x* 
f(x) reaches a global interior maximum at x*.
F(x) reaches a local interior maximum at x*  there exists   0 such that f(x*)  f(x) for all
x in
B (x*) .
If f(x) doesn’t reach a global interior maximum at x*  there exist x’ in D
such that f(x’) > f(x*). We need to prove this is a contradiction.
Take value of the function of the convex combination of x* and x’, and base on the definition
of a concave function, we have: f ( xt)  tf ( x' )  (1  t ) f ( x*)  f ( x*)  t[ f ( x' )  f ( x*)] .
Because f(x’) > f(x*)  f(xt) > f(x*) for all t  (0,1) .
If we take   0 such that xt in
B (x*)
 f(xt) > f(x*) for all t  (0,1) .
This is contradiction of the assumption that f(x) is local interior maximum at x* for some
  0 such that f(x*)  f(x) for all x in
B (x*) .
f(x)
f(x’)
f(x*)
f(xt)
x
Geoffrey A. Jehle – Advanced Microeconomic Theory x*
xt
x’
f(x*)>f(xt): contradiction of that f(x) is convace
6
Chu Thanh Duc – MDE 10
Necessary condition: Assume f(x) is global interior maximum at x*  f(x) is local interior
maximum at x*.
F(x) always reaches a local interior maximum at x* if f(x) reaches a global interior
maximum at x*.
Theorem: Strict Cocavity/Convexity and the Uniqueness of Global Optima
1. Let f(x) be a strictly concave function. If x* maximizes f(x), then x* is the unique
global maximizer and f(x*) > f(x) for all x  D.
2. Let f(x) be a strictly convex function. If x* minimizes f(x), then x* is the unique
global minimizer and f(x*) < f(x) for all x  D.
Prove: (1): Let f(x) be a strictly concave function. If x* maximizes f(x), then x* is the
unique global maximizer and f(x*)>f(x) for all x  D.
Let x* maximize f(x). Assume that there was x’ that also maximizes f(x)  f(x*)=f(x’).
F(x) is strictly concave function  f(xt) > tf(x*) + (1-t)f(x’) = f(x’). This violates the
assumption that x* and x’ are global maximizer of f(x).
So assumption that there was x’ that also maximizes f(x) is impossible.
Theorem: First – Order Necessary Condition for Local Interior Optima or Real Valued
Functions
Let f(x) be a differetiable function. If f(x) reachs a local interior maximum or minimum at
x*, then x* solves the system of simultaneous equations,
 f ( x*)
 x1  0

 f ( x*)  0
 x 2

.
.

 f ( x*)
 xn  0


Proof:
Geoffrey A. Jehle – Advanced Microeconomic Theory
7
Chu Thanh Duc – MDE 10
We suppose that f(x) reaches a local interior extremum at x* and seek to show that
f ( x*)  0 .
Consider the function: g(t) = f(x* + t.dx).
Note that (x*+t.dx) is a vector, so g(t) = f(x* + t.dx) is a some value of f(x) different from
f(x*).
We assumed that f(x) reaches a local interior extremum at x*  g(t) ~ f(x) reaches a local
interior extremum at t = 0 because g(0) = f(x*)  g ' (0)  0

g (0)
f ( x * tdx) ( x * tdx)
f ( x * tdx)
f ( x*)

.

dx  
dxi  0
t
( xi  tdx)
t
( xi  tdx)
xi
 f ( x*)  0 (matrix)
2.2.2 Second – Order Conditions
We have a maximum if the function is locally concave at the point.
We have a minimum if the function is locally convex at the point.
Theorem: Second – Order Necessary Condition for Local Interior Optima of Real
Valued Fucntions
Let y=f(x) be twice differentiable.
1. If f(x) reaches a local interior maximum at x*, then
n
n
d y  dx H ( x*)dx   f ij ( x*)dxidxj  0
2
T
i 1 j 1
2. If f(x) reaches a local interior minimum at x*, then
n
n
d y  dx H ( x*)dx   f ij ( x*)dxidxj  0
2
T
i 1 j 1
Proof:
Consider: g(t) = f(x* + tdx)
f(x) reaches a critical point at x* , then g(t) reaches a critical at t = 0  g ' ' (0)  0 for
maximum target.
n
For any t, g ' (t )  
i 1
f ( x  tdx)
dxi
 ( xi  tdxi)
Geoffrey A. Jehle – Advanced Microeconomic Theory
8
Chu Thanh Duc – MDE 10
n
 2 f ( x  tdx)
dxidxj
j 1  ( xi  tdxi) ( xj  tdxj)
n
 g ' ' (t )  
i 1
n
 2 f ( x*)
dxidxj  0
j 1 xixj
n
At t = 0  g ' ' (0)  
i 1
Note:
-
The both above theorems are only Necessary conditions. They state that if f(x)
reaches optimal value, then f’(x*) =0 and f’’(x*)  ()0 .
-
But the main target of us is to state that: “If such and such obtains at x, then x
optimizes the function” – Sufficient conditions.
Sufficient Conditions:
1. Local Maximum at x*: If f i ( x*)  0 and f(x) is strictly local concave at x*
2. Local Minimum at x*: If f i ( x*)  0 and f(x) is strictly local convex at x*.
Theorem. Sufficient Conditions for Strict Concavity or Strict Convexity of a Real
Valued Function
Let f(x) be twice differentiable function, and let Di (x) be ith-order principal minor
of the Hessian matrix H(x).
1. If (-1)i Di (x) > 0, then f(x) is strictly concave at x.
2. If Di (x) > 0, then f(x) is strictly convex at x.
If the respective conditions hold for all x in the domain, the funtion is globally
strictly concave or globally strictly convex, respectively.
Theorem. Sufficient Conditions for Local Interior Optima of Real Valued
Functions
1. If f i ( x*)  0 and (1) i Di ( x*)  0 , then f(x) reaches a local maximum at x*.
2. If f i ( x*)  0 and Di ( x)*  0 , then f(x) reaches a local minimum at x*.
Geoffrey A. Jehle – Advanced Microeconomic Theory
9
Chu Thanh Duc – MDE 10
Theorem. Sufficient Condition for Unique Global Optima
Let f(x) is differentiable
1. If f(x) is global strictly concave and f i ( x*)  0 then x* is the unique global
maximizer of f(x).
2. If f(x) is global strictly convex and f i ( x*)  0 then x* is the unique global
minimizer of f(x).
Proof: If f(x) is global strictly concave and f i ( x*)  0  x* is the unique global
maximizer of f(x).
Consider the two points: x* and x’
f (tx'(1  t ) x*)  tf ( x' )  (1  t ) f ( x*)
F(x) is strictly concave  f (tx'(1  t ) x*)  f ( x*)
for all t in (0,1).
 f ( x' )  f ( x*)
t
 lim
t 0
f ( x * t ( x' x*))  f ( x*)
 f ( x' )  f ( x*) (1)
t
Consider the function g(h) = f(x* + h(x’-x*))
g ' ( h)  
f ( x *  h( x' x*)
( xi ' xi *)
 ( xi *  h( x'i  x*))
At h=0  g ' (0)  
 g ' (0)  lim
t 0
f ( x*)
( x'i  xi *)  f ( x*)dx (the product of two vectors)
xi
f ( x * (t  0)( x' x*))  f ( x*)
 f ( x*)dx
t
Because f ( x*)  0  lim
t 0
f ( x * (t  0)( x' x*))  f ( x*)
 0 (2)
t
Combining (1) and (2), we have: f(x’) – f(x*) < 0 or f(x*) > f(x’).
Because x’ is chosen arbitriarily, this means that f(x*) > f(x’) for all x in the domain.
 x* is unique maximizer since f is global strictly concave.
Geoffrey A. Jehle – Advanced Microeconomic Theory
10
Chu Thanh Duc – MDE 10
2.3. Constrained Optimization
2.3.1 Equality Constraints
Max f(x1, x2) subject to g(x1, x2) = 0
Objective function / maximand: f(x1, x2)
Constraint: g(x1, x2)
Constraint set / Feasible set: the set (x1,x2) such that g(x1,x2) is satisfied.
2.3.2 Lagrange’s Method
Max f(x1, x2) subject to g(x1, x2) = 0
Lagrange Function: L(x1 ,x2,  ) = f(x1, x2) +  g(x1,x2)
Now we maximize L function when it is an ordinary function of three varibles (x1,x2,  ).
g ( x1, x 2)
 L f ( x1, x 2)
 .
0
 x1 
x1
x1

g ( x1, x 2)
 L f ( x1, x 2)

 .
0


x
2

x
2

x
2

 L
   g ( x 2, x 2)  0

Solve this stimultaneous equation, we have a critical point (x1*,x2*,  ), such that
(x1*,x2*) is the critical point of f(x1,x2) along the constraint g(x1,x2)=0.
Proof:
We need to prove that (x1* , x2*) is also the critical of f(x1,x2)  df = 0 at (x1*,x2*).
f ( x1*, x 2*)
f ( x1*, x 2*)
dx1 
dx 2  g ( x1*, x 2*)d
x1
x 2
g ( x1*, x 2*)
 g ( x1*, x 2*)

 
dx1 
dx 2  0
x1
x 2


dL 
For all dx1, dx2 and d  .
Because g(x1,x2) = 0 for all dx1, dx2 
Take d  = 0  dL 
g ( x1*, x 2*)
g ( x1*, x 2*)
dx1 
dx 2  0
x1
x 2
f ( x1*, x 2*)
f ( x1*, x 2*)
dx1 
dx 2  df  0 at (x1*,x2*).
x1
x 2
Geoffrey A. Jehle – Advanced Microeconomic Theory
11
Chu Thanh Duc – MDE 10
 df(x1*,x2*) = 0 for all dx1*, dx2*./.
-
The critical points derived from the First Order condition can not alone decise to be
maxima or minima. To distinguish between two requires knowledge of the
“curvature” of the objective and constraint relations at the critical point in question.
General problem;
Max f(x1,x2….xn) subject to
g 1 ( x1,...xn)  0
g 2 ( x1,...xn)  0
.
.
g m ( x1,...xn)  0
m
Lagrange function with (n+m) variables:
L ( x,  )  f ( x )    j g j ( x )
j 1
First – Order Condition:
 L f ( x) m
g j ( x)
 xi  xi    j xi
j 1


 L  g j ( x)  0
  j
Theorem. Lagrange’s Theorem
Let f(x) and gj(x), j=1,2…m, be twice continuously differentiable real valued
function over some domain D  R n . Let x* be an interior of D and suppose
that x* is an optimum of f subject to the constraint, gj. If m < n and if the
gradient vectors g j (x*) , j = 1,…m, are linearly independent, then there exist
m unique numbers  j , such that L function has an optimum in x at x* and
L( x*,  ) f ( x*) m
g j ( x*)

  j
i=1,…n
xi
xi
xi
j 1
Geoffrey A. Jehle – Advanced Microeconomic Theory
12
Chu Thanh Duc – MDE 10
2.3.3 Geometrical Interpretation
- Represent the objective function by its Level sets.
L(yo) = {(x1,x2)| f(x1,x2) = yo}
The level set curve L(yo):

f ( x1, x 2)
f ( x1, x 2)
dx1 
dx 2  0
x1
x 2
f ( x1, x 2)
dx 2
 1
: the slope of level set curve L(yo) through any point
dx1 along _ L ( yo )
f 2 ( x1, x 2)
(x1,x2).
-
The slope of the constraint:
g ( x1, x 2)
g ( x1, x 2)
dx1 
dx 2  0
x1
x 2

g ( x1, x 2)
dx 2
 1
dx1 along _ g ( o )
g 2 ( x1, x 2)
-
Recall the First – Order Condition of the Lagrange function:
g ( x1*, x 2*)
 f ( x1*, x 2*)
 

x1
x1

g ( x1*, x 2*)
 f ( x1*, x 2*)
 

x 2
x 2

 g ( x1*, x 2*)  0


Deviding the first of these by the second to eliminate  , we have:
 f1 ( x1*, x 2*) g1 ( x1*, x 2*)


 f 2 ( x1*, x 2*) g 2 ( x1*, x 2*)
 g ( x1*, x 2*)  0

Thus, at (x1*,x2*), the slope of the level set curve of y equals the slope of the constraint
curve. However, this point must be on the constraint curve to satisfy the condition:
g(x1*,x2*)=0
Geoffrey A. Jehle – Advanced Microeconomic Theory
13
Chu Thanh Duc – MDE 10
L(y)
x2*
L(y*)
x2*
g(x)=0
L(y)
L(y*)
g(x)=0
x1*
x1*
Maximization Problem
Minimization Problem
2.3.4 Second – Order Conditions
- If (x*,  ) satisfy the second – order condition for a maximum of the Lagrange function,
we know we have a local maximum of f subject to the constraints.
- All we really need to know that we have a maximum is that the second differential of the
objective function at the point which solves the first order conditions is decreasing along
the constraint.
Theorem: Sufficient Conditions for Optima with Equality constraints
Let the objective function f(x) and m constraints be given by gj(x)=0, j=1,…,n. Let L
be the Lagrange function. Let (x*,  ) solve the First Order Condition. Then:
1. x* maximizes f(x) subject to the constraint if the principa minors alternate in sign
beginning with positive D3 >0, D4<0…. when evaluated at (x*,  ).
2. x* minimizes f(x) subject to the constraint if the principa minors are all negative
D3<0, D4<0…. when valuated at (x*,  ).
Hessian Matrix
 L11

 .
L
H   n11
 g1
 .
 1
 gn
. L1n
.
.
. Lnn
. g1m
.
.
. g nm
. g1m 

. . . 
g 1n . g nm 

0 0 0 
0 0 0 

0 0 0  ( n m) x ( nm)
g11
Geoffrey A. Jehle – Advanced Microeconomic Theory
14
Chu Thanh Duc – MDE 10
2.3.5. Inequality Constraints:
Consider the simplest problem: max f(x) subject to x  0 ( D  R ).
Let x* be the maximum of the problem. There exists three cases:
Case 1: x*=0 and f ’(x*)<0.
Case2: x* = 0 and f ’(x*) = 0
Case 3: x* >0 and f ’(x*) = 0
Making a convenient set of conditions to summarize all three posibilities:
Conditions: x* must to satisfy all these conditions:
 f ' ( x*)  0

 x * . f ' ( x*)  0 (Maximizatin Problem)
 x*  0

To solve this inequation system, we should concentrate on (2) condition: x*f’(x*)=0 to
have a set of solutions x, then use the conditions (1) and (3) to determine which x* is.
Similarly, we can make conditions for Minimization Problem:
 f ' ( x*)  0

 x * . f ' ( x*)  0 (Minimization Problem)
 x*  0

Theorem: Necessary Conditions for Optima of Real Valued Funtions Subject to
Non-negativity Constraints (xi  0)
Let f(x) be continuously differentiable.
1. If x* maximizes f(x) subject to x  0, then vector x*(x1*,x2*…xn*) satisfies:
 f i ( x*)  0

 xi * . f i ( x*)  0
x *  0
 i
2. If x* minimizes f(x) subject to x  0, then vector x*(x1*,x2*…xn*) satisfies:
Geoffrey A. Jehle – Advanced Microeconomic Theory
15
Chu Thanh Duc – MDE 10
 f i ( x*)  0

 xi * . f i ( x*)  0
x *  0
 i
2.3.6 Kuhn – Tucker Conditions
Non – Linear Programming problem: there are no limitations on the forms of objective
function and constraint relations.
Max f(x1,x2) subject to g(x1,x2)  0 and x1  0 , x2  0 .
There is an above theorem that tells us that maximum of a function subject to equality
constrains concides with the maximum of its corresponding Lagrangian with no
constraints. There is also an above theorem that show us how to characterize the
maximum of a function with non – negativity constraints only.
To solve the non – linear programming problem, we will convert the problem to one with
equality constraints and non-negativity constraints and apply what we know.
The trick:
Because g  0  there is z such that g – z = 0 and z  0 (z must be positive due to g  0.)
Now, our problem is converted to:
Max f(x1,x2) subject to g(x1,x2) – z = 0 and x1  0, x2  0, z  0.
Theorem tell us that the maximum over x of f subject to equality constraints concides with
the unconstrainted maximum over x of the associated Lagrangian.
Then theorem tell us how to solve the problem of finding optima of f(x) subject to nonnegative constraints (xi  0).
The problem is converted to;
Max L(x1, x2, z,  ) = f(x1,x2) +  [g(x1,x2)-z] subject to x1  0, x2  0 and z  0.
The first – order condition on x1, x2, z,  :
Geoffrey A. Jehle – Advanced Microeconomic Theory
16
Chu Thanh Duc – MDE 10
(1)
 Lx1  0  f1  g1  0
 x1.L  0  x1( f  g )  0
( 2)
x1
1
1

 Lx 2  0  f 2   g 2  0
(3)

 x 2.Lx 2  0  x 2( f 2  g 2 )  0 ( 4)

(5) (maximization problem)
 Lz  0     0
 z.Lz  0   z  0
( 6)

(7 )
 x1  0, x 2  0, z  0
 L  0  g ( x1, x 2)  z  0
(8)
 
We don’t impose any sign restriction on 
By (8) condition, we can eliminate z by substituting z by g(x1,x2):
 Lx1  0  f1  g1  0
 x1.L  0  x1( f  g )  0
x1
1
1

 Lx 2  0  f 2  g 2  0

 x 2.Lx 2  0  x 2( f 2  g 2 )  0
 g ( x1, x 2)  0

 z.Lz  0  g ( x1, x 2)  0

 x1  0, x 2  0,   0
(1)
( 2)
(3)
( 4)
(maximization problem)
(5' )
( 6' )
(7 ' )
Condition (5’), (6’) and (7’) tell us that we are trying to minimize the Lagrangian in 
subject to   0
 Maximization Problem: Necessary Condition: A Maximum of Lagrangian in the
variables xi and a minimum of Lgrangian in the multiplier  .
Saddle Point (x1*,x2*,  )
(1)
 Lx1  0  f1  g1  0
 x1.L  0  x1( f  g )  0
( 2)
x1
1
1

 Lx 2  0  f 2  g 2  0
(3)

 x 2.Lx 2  0  x 2( f 2  g 2 )  0 ( 4) (Minimization Problem)
 L  0  g ( x1, x 2)  0
(5' )
 z
( 6' )
 z.Lz  0  g ( x1, x 2)  0

(7 ' )
 x1  0, x 2  0,   0
Theorem Kuhn – Tucker: Necessary Conditions for Optima of Real Valued
Functions Subject to Inequality and Non – negativity Constraints
Geoffrey A. Jehle – Advanced Microeconomic Theory
17
Chu Thanh Duc – MDE 10
Let f(x) be continuously differentiable problem
1. Consider the maximation problem,
Max f(x) subject to gj (x)  0, j= 1,2,…m, and x  0 (1)
m
with associated Lagrangian: L = f(x) +
 g
j 1
j
j
( x)
(2)
If x* solves (1) and if the gradient vector for all binding constraints at x* are linearly
independent, then there exist m numbers  j *  0 , such that ( x*, *) is a saddle point of
the Lagrangian satisfying the Kuhn – Tucker conditions:
 Li ( x*, *)  0 and xi * .Li ( x*, *)  0, i  1,...n

 Li ( x*, *)  0 and  j * .Li ( x*, *)  0 j  1,...m
2. Consider the minimization problem:
Min f(x) subject to gj (x)  0, j= 1,2,…m, and x  0 (3)
m
with associated Lagrangian: L = f(x) +
 g
j 1
j
j
( x)
(4)
If x* solves (3) and if the gradient vector for all binding constraints at x* are linearly
independent, then there exist m numbers  j *  0 , such that ( x*, *) is a saddle point of
the Lagrangian satisfying the Kuhn – Tucker conditions:
 Li ( x*, *)  0 and xi * .Li ( x*, *)  0, i  1,...n

 Li ( x*, *)  0 and  j * .Li ( x*, *)  0 j  1,...m
Proof: Find Luenberger (1973).
2.4. Value Functions
Maximization problem:
Max f(x,a) subject to g(x,a) = 0 and x  0.
In which: x is a vector of choice variable; a is a vector of parameters.
The solution of this problem will depend on the vector of parameters a.
Denote: x=x(a)
The maximization problem:
Geoffrey A. Jehle – Advanced Microeconomic Theory
18
Chu Thanh Duc – MDE 10
Max M(x(a),a) subject to g(x(a),a).
Theorem of the Maximum: If f(x) and g(x) are continuous in parameters, and if the
domain is a compact set, then M(a) and x(a) are continuous functions of the parameters
a.
Envelope Theorem:
The problem Max f(x,a) subject to g(x,a) = 0 and x  0.
If f(x,a) and g(x,a) are continuously differentiable in a. Let x(a) > 0 solve the problem and
assume that it is continuously differentiable in a. Let L(x, a,  ) be the problem’s
associated Lagrangian function and let (x(a),  (a)) solve the Kuhn – Tucker conditions.
Finally, bet M(a) be the problem’s associate maximum value function. Then the Envelope
Theorem states that:
M (a) L

a j
a j
j  1,..., m
x ( a ), ( a )
Where the right hand side denotes the partial derivative of the Lagrangian function with
respect to the parameter aj evaluated at the point (x(a),  (a)).
Geoffrey A. Jehle – Advanced Microeconomic Theory
19