Advanced Microeconomic Theory
Advanced Microeconomic Theory
Advanced Microeconomic Theory
- Theorems provice a compact and precise format for presenting the assumptions and
important conclusions of somtimes lengthy arguments, and so help to identify
immediately the scope and limitations of the result presented.
If A is not true, B must be not true. But that doesn’t mean that if B is not true, A must be
not true.
A true
B true
B not true
A not true
A is not true B must be not true B is not true is necessary for A is not true.
~ A ~ B .
- Sufficiency: “A is sufficient for B” means that A holds, B must hold. We can say “A
is true only if B is true”; or A implies B (A B).
B true
A true
A not true
B not true
B is not true A must be not true. B is not true is sufficient for A is not true
~B ~A.
A
B
- Contrapositive proof: Assume that B does not hold, then show that A cannot hold.
- A Subset: Set S is a subset of set T if every element of set S is also an element of set
T. Notation: S T.
- Complement Set: The complement of set S in an Universal set U is the set of all
elements in U which are not in S. Notation: Complemet set of S: cS.
- Index Set: The set of interger number starting with 1. S = (1, 2, 3, 4...n). Notation:
I {1,2,3...}
Consider the two sets: set S (s1, s2...) and set T (t1, t2,...)
The product of the two sets: SxT = {(s,t)| s S and t T} : Order pairs.
Any Collection or Set of Order pairs is said to constitute a Binary Relation (SxT) of the
sets S and T.
Meaningful Relationship: (sRt) : The Set of the order pairs that are constituted by elements
of the sets S and T are meaningful.
Completeness: A relation R on S is complete if, for all distinct elements x and y in S, xRy or
yRx ( the order pairs (x,y) or (y,x) are all meaningful.)
Reflexivity: A relation R on S is reflexive if, for all elements x in S, xRx (oder pairs (x,x) are
meaningful)
Transitivity: A relation R on S is Transitive if, for all three elements x, y and z in S, xRy and
yRz implies xRz
Odering: A binary relation which satisfies all properties of Completeness, Reflexivity and
Transitivity is called an Ordering.
A Function: Is a Relation that associate each element of one set with a single, unique element
of another set. f: D R: D is the Domain and R is calledthe Range.
The distance between the points: x1 ( x11 , x12 ) and x 2 ( x12 , x22 ) :
Open Ball: The open ball with the center x0 R n and radius e > 0 (a real number) is the
subset of poins in R n : Be ( x0 ) {x R n | d ( x0 , x) e} .
Closed Ball: the closed ball with center x0 R n and the radius e> 0 (a real number) is the
subset of points in R n : Be ( x0 ) {x R n | d ( x0 , x) e} .
Open Sets: S R n is an Open Set if for all x S, there exist some e > 0 such that the open
ball Be (x) S .
S is an open set. For every x S, choose some e x > 0 such that Bex ( x) S .
S Bex (x)
xS
Closed Sets: S R n is a closed set if and only if its complement cS is an Open set.
Proof:
Geoffrey A. Jehle – Advanced Microeconomic Theory 4
Chu Thanh Duc – MDE 10
If S R is closed cS is open. By the definition of an open set, for each x in cS,
we have: cS B (x) .
cS (ai , bi )
iI
S c (ai , bi )
iI
A lower bound: A real number l is called a lower bound for S if l x for all x S .
An Upper bound: A real number l is called an upper bound for S if l x for all
xS .
The greatest lower bound (g.l.b): the biggest number among those lower bounds for
S.
The least upper bound (l.u.b): the smallest number among those upper bounds for S.
1. Let S R is a bounded open set and let a be the g.l.b of S and b be the l.u.b of S.
Then a S and b S .
2. Let S is a bounded closed set in R. Let a is the g.l.b of S and b is the l.u.b of S.
Then a S and b S.
closed
and
bounded
1.3.1. Continuity
- In most economic application, we will either want to assume that the function we are dealing
with are continuous or we want to discover whether they are continuous when we are
unwilling to assume it.
f ( B ( xo) B ( f ( xo))
f(xo+ )
f(xo+ )
f(xo)
f(xo)+
f(xo+ ) f(xo)
xo xo+
Geoffrey A. Jehle – Advanced Microeconomic
xo xo+Theory 6
f(xo+ ) (f(xo), f(xo)+ )
Chu Thanh Duc – MDE 10
Continuity: Let D be a set, R be another set, and let f: D R. The function f is continuous
at the point xo D if, and only if, for all > 0, there exists a > 0 such that:
continuous function.
A function is continuous at a point xo if for all > 0, there exists > 0 such that any point
less than a distance away from xo is mapped by f into some point in the range which is less
than a distance away from f(xo).
Every point in B (xo) is mapped by f into some point no father than from f(xo).
Basically, a function is continuous if a “small movement” in domain does not cause a “big
jump” in the range.
It is not true that a continuous function always maps an open set in the domain set into an
open set in the range, or that closed set is mapped into closed sets. Ex: y = a. (the domain is
an open set but the range (image) set may not be an open set)
Proof:
Necessity: Let f is a continuous function and T be an open set in the range. We have to prove
that f 1 (T ) D is an open set.
Proof:
Let an image set T of f(x) is a closed set. We need to prove that f 1 (T ) D is a closed set
equivalent to f being a continuous function.
- All Existence Theorems specifies conditions if met, guarantee that something exists.
something
E.T.Cs
E.T.Cs do not hold
- Optimization theory
- The Weierstrass Theorem specifies sufficient conditions under which the existence of
a maximum and a minimum of a continuous function are assured.
The Weierstrass Theorem specifies sufficient conditions under which the existence of a
maximum and a minimum of a continuous function are assured.
f ( x * ) f ( x) f ( ~
x) for all x S .
Proof:
According to the theorem: The continuous image of a Compact set is a compact set (D is a
compact set, f is continuous the Range/image of D is a compact set. We have that S is
a compact set and the mapping is continuous the image of S is a compact set.
Obviously, the image set of S R Greatest Lower Bound = a (g.l.b) and Smallest
Upper Bound = b (s.u.b) of the set image of S there is x1 and x2 in S such that
f(a)=g.l.b and f(b)=s.l.b due to the property of a closed set S f (a) f ( x) f (b) .
.
Max
Min
x1 x2
some a R n , a 0 and 0 .
Geoffrey A. Jehle – Advanced Microeconomic Theory 9
Chu Thanh Duc – MDE 10
A Hyperplane H separates two sets S and T if: a.x for all x S and a.x for all
x T .
A hyperplance H in R n
Let S in R n be convex, closed and nonempty. Let y 0 R n and y 0 S . Then there exists
1. a.y 0
3. a.x
(1) and (3) tell that there is some hyperplane that separates the point y0 and the set S. (2)
means that the hyperplane will pass through a point on the boundary of S.
a = x’ – y0 and a.x’ = .
Lemma: Under the condition of the preceding theorem, there exists at least one point
x' S (boundary of set S) such that d(y0, x’) d(y0, x) for all x S and d(y0, x’)>0.
x2
S
xo
B*
A
yo
x1
Let xo be any poin in S and let be ||yo-xo||. Let B be a closed ball centered at yo with
*
radius . Let A be intersection of B and S. Because S and B are closed and nonempty
* *
Consider the function d(yo,x’) over domain A. This is a continuous function. Because A
– domain is a compact set According to Weierstrass theorem, there exists x’ in A such
that d(y0, x’) d(yo,x) for all x A . Because x’ is in S and yo is not in S, so d(yo.x’)>0.
So x’ is the closest point in A to yo . It is easy to realize that every point in S that is not in
the closed ball B* must be a distance from yo which is trictly greater than . So x’ is the
closest point in S set to yo. x’ must be on the boundary of S, ortherwise that there exists a
ball centered by x’ means there is other point which is closer to yo.
Prove the theorem: Separating a Point and a Convex set
Let a = x’ – yo, where x’ is the closest point in S to yo. a 0, due to x’ yo.
Let = a.x’
Due to x’ is the closest point in S to yo ||x’-yo|| ||xt – yo|| ||x’-yo||2 ||xt – yo||2
0 -2 ||x’-yo|| 2 +2(x’-yo)(x-yo)
0 (x’-yo).x’ - (x’-yo)x
Let S and T be two nonempty, disjoint, and convex sets in R n . Then there is an
a R n , a 0 and an R n such that a.x for all x S and a.x for all x T .
Theorem: The Brouwer Fixed Point Theorem
Let S R n be a compact and convex set. Let f: S S be a continuous mapping. Then
there exists at least one fixed point of f in S. That is, there exist at least one x* S such
that x* = f(x*).
S R n and f: S S , then f maps vectors in S back into other vector in the same set S.
y1 g 1 ( x1,...xn )
.
.
yn g n ( x1,...gn)
If one of these inequalities holds with equality, a or b will be a fixed point: a = f(a) or b=f(b).
Now we must consider the possibility of x = f(x) that g(a)<0 and g(b)>0.
We know g(a) < 0, we always find out c > a such that g(x) < 0 for all x [a, c). Proving this
statement is rather easy. Beacase g(x) is a continuous function over [a, b]. g(x) is continuous
Similiarly, we can find c<b such that g(x) > 0 for all x (c, b] .
Let c* is the largest number to make the open interval [a, c*) such that g(x)<0 for all
x [a, c*) . Let x* is the least upper bound of [a,c*) x* = c.
Since x* = c, we know that a < x* < b and g(x) < 0 for all x [a, x*) .
We realize that g(x*) can not be negative. If g(x*)<0 -g(x*)>0. Because g(x) is continuous
function at x*, we take = -g(x*) > 0, there exists >0 such that g ( B ( x*)) B g ( x*) g ( x*) .
B g ( x*) g ( x*) (0,2 g ( x*)) , this contradics the assumption that g(x*)<0.
that g ( B ( x*)) Bg ( x*) g ( x*) . And Bg ( x*) g ( x*) (0,2 g ( x*)) . This vilolates the
condition g(x)<0 for all x [a, x*) .
So, g(x*) can not be both positive and negative g(x*) = 0 x* = f(x).
In the paragraph, the line f will cross the 450 curve at least one time. So according to
Brouwer’s Theorem, the fixed point is not unique.
Definition fo Real Valued Function: f: D R is a real valued function if D is any set and
R R.
This section we will restrict out attention to real valued functions whose domains are convex
sets.
Note: x 0 x means that at least one of components of vector x0 is greater than the same
ordered component of vector x.
The graph of a function is a related set which provices an easy and intuitive way of thinking
about the function.
Level Sets: L(y0) is a level set of the real valued function f: D R iff
L(y0) {x | x D, f(x) yo}, where y0 R .
L(x0) is a level sets relative to point x iff L(x0) = {x|x D , f(x) = f(x0)}.
3. S’(yo) = {x | x D , f(x) yo} is call strictly superior set for level yo.
4. I’(yo) = {x | x D , f(x) yo} is call strictly inferior set for level yo.
1. L(yo) S(yo)
2. L(yo) I(yo)
4. S’(yo) S(yo)
5. I’(yo) I(yo)
6. S’(yo) L(yo) = O
7. I’(yo) L(yo) = O
8. S’(yo) I’(yo) = O.
- The set of points beneath concave regions is a convex set. The set of points beneath
the non-concave region is not a convex set.
Theorem: Points On and Below the Graph of a Concave Function Always Form a
Convex Set
Prove: We have to show that f(x) is a concave function implies A is convex and A is
convex implies f(x) is concave function.
Take convex combination of the two point: (xt, yt) such that:
Consider two any points on the graph of f(x): (x1,f(x1)) and (x2,f(x2)).
Geometrically, these modification simply require the graph of the function to lie
everwhere strictly above the chord connecting any two points on the graph.
Quansiconcave functions:
Geometrically:
- When f(x) is an increasing function, it will be quansiconcave whenever the level set
relative to any convex combination of two points, L(xt) is always on or above the
lowest of the level sets L(x1) and L(x2).
x2
x1
xt
x2
x1
x2
x2
xt
x1
x1
f: D R is a quansiconcave function if and only if S(x) –( the superior set relative to point
x: {x| x D , f f(x)}) is a convex set for all x D .
Consider any two point x1 and x2 in set S(x). We need to prove that xt made by the convex
combination of the two vectors x1 and x2 is also in S(x) f(xt) f(x).
according to the definition of S(x), we have: f(x1) f(x) and f(x2) f(x).
f(xt) f(x).
We need to prove that f(xt) min[f(x),f(x2)] under the condition that S(x) is convex.
Consider any two points x1 and x2 in S(x). Without loss of generality, assume we have:
f(x1) f(x2).
xt = tx1 + (1-t)x2 (for all t [0,1] ) is also in S(x2) f(xt) f(x1) f(x2).
x2 x2
Increasing function
Decreasing function
S(x)
S(x)
L(x) L(x)
x1
x1
- A Strictly Concave Function forbid the convex combination of two points in the same
level set also lies in that level set
x2
x2
x2 Increasing Function
Not Strictly Quansiconcave xt
x1
xt
x2 L(x1)=L(x2)=L(xt)
x1
x1
x1
Strictly Quansiconcave Function
Convex Functions:
Convexity the region above the graph – set (x,y) or (D,R) is a convex set.
f: D R is a strictly convex function iff for all x1 and x2 in D, f(xt) < tf(x1) + (1-t)f(x2),
for all t (0,1).
Theorem. Points On and Above the Graph of a Convex Function Always Form a Convex
Set
Prove:
Take the convex combination of the two points, we have: (xt,yt), in which:
f: D R is a quansiconvex function iff I(x) – Inferior set is a convex set for all x D.
x2 x2
Increasing function
Decreasing function
Inferior set
Inferior set
x1 x1
Prove:
Sufficiency: We need to prove that f(x) is a quansiconvex I(x) Inferior set is a convex
set.
Consider the two points in I(x) set (x1,y1) and (x2,y2). Without loss of generality, we
assume that f(x1) f(x2) I(x1) I(x2) and f(x1)-f(x2) 0
I(x) is a convex set the convex combination of the two points is also in I(x).
Summaries:
5. f concave f quansiconcave.
6. f convex f quansiconvex.
f(x2)
f(x1)
f(xt)
X x
X1 Xt X2
f(x2)
f(x1)
X1 Xt X2
f(xt) f(x1)
f(x1) f(xt)
f(x2) f(x2)
x1 xt x2
x1 xt x2
Differentiable function: If the function is continuous and smooth, with no breaks and kinks.
dy
Derivative f’(x): the slope or instantaneous rate of change in f(x): 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.
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.
For any twice continuous differentiable function, y = f(x), in the neighborhood of the point x,
and for all dx 0:
First Differentials:
Second Differentials:
Partial Derivatives: Let y = f(x1, x2, …xn). The the partial derivative of f with respect to xi
is defined as:
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
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
2 f ( x)
Second – order partial derivative: f ij ( x)
xixj
2 f ( x)
H ( x)
xixj nxn
The Hessian Matrix:
2 f ( x) 2 f ( x)
xixj xjxi
The Young’s Theorem tells us that the Hessian matrix will be symmetric.
dx1
Let dx = d 2 y f11 ( x).(dx1) 2 d 2 y 0 f11 ( x) 0 .
0
f (tx ) f ( x)
If f(x) is homogeneous of degree 1, then for all t > 0.
xi xi
f (tx ) f ( x)
f (tx ) A.(tx1 ) (tx 2 ) A (tx1) 1 t.(tx 2 ) At 1x1 1 x2 t 1 .
x1 x1
f (tx ) f ( x)
1 the Cobb-Douglas function is homogeneous of degree 1.
x1 x1
n
f ( x)
k . f ( x) xi
i 1 xi
2. If f(x) is homogeneous of degree 1, then:
n
f ( x)
f ( x) xi
i 1 xi
Geoffrey A. Jehle – Advanced Microeconomic Theory 4
Chu Thanh Duc – MDE 10
Prove: Assume that f(x) is homogeneous of degree k, by definition: f(tx) = tkf(x)
f (tx ) n
f (tx )
xi
t i 1 txi
t k f ( x)
kt k 1 f ( x)
t
n
f (tx )
kt k 1 f ( x) xi
i 1 txi
n
f ( x)
k . f ( x) xi
Let t = 1, we have
i 1 xi
2.2. Optimization
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
Local Maximum: f(x) is 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
f(x’)
f(x*)
f(xt)
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*.
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).
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 .
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
Theorem: Second – Order Necessary Condition for Local Interior Optima of Real
Valued Fucntions
n n
d y dx H ( x*)dx f ij ( x*)dxidxj 0
2 T
i 1 j 1
n n
d y dx H ( x*)dx f ij ( x*)dxidxj 0
2 T
i 1 j 1
Proof:
f(x) reaches a critical point at x* , then g(t) reaches a critical at t = 0 g ' ' (0) 0 for
maximum target.
n
f ( x tdx )
For any t, g ' (t ) dxi
i 1 ( xi tdxi )
Geoffrey A. Jehle – Advanced Microeconomic Theory 8
Chu Thanh Duc – MDE 10
n n
2 f ( x tdx )
g ' ' (t ) dxidxj
i 1 j 1 ( xi tdxi ) ( xj tdxj )
n n
2 f ( x*)
At t = 0 g ' ' (0) dxidxj 0
i 1 j 1 xixj
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:
2. Local Minimum at x*: If f i ( x*) 0 and f(x) is strictly local convex at x*.
Let f(x) be twice differentiable function, and let Di (x) be ith-order principal minor
If the respective conditions hold for all x in the domain, the funtion is globally
strictly concave or globally strictly convex, respectively.
1. If f i ( x*) 0 and (1) i Di ( x*) 0 , then f(x) reaches a local maximum at x*.
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).
f ( x*)
At h=0 g ' (0) ( x'i xi *) f ( x*)dx (the product of two vectors)
xi
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.
Constraint set / Feasible set: the set (x1,x2) such that g(x1,x2) is satisfied.
L f ( x1, x 2) g ( x1, x 2)
x1 . 0
x1 x1
L f ( x1, x 2) g ( 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*).
- 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;
g 1 ( x1,...xn ) 0
g 2 ( x1,...xn ) 0
.
.
g m ( x1,...xn ) 0
L f ( x) m g j ( x)
j
xi xi j 1 xi
L g j ( x) 0
j
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
dx2 f ( x1, x 2)
1 : the slope of level set curve L(yo) through any point
dx1 along_ L ( yo ) f 2 ( x1, x 2)
(x1,x2).
dx 2 g ( x1, x 2)
1
dx1 along_ g ( o ) g 2 ( x1, x 2)
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
L(y)
x2*
L(y*)
x2*
g(x)=0
L(y)
L(y*)
g(x)=0
x1*
x1*
Minimization Problem
Maximization Problem
- 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.
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
f ' ( x*) 0
x * . f ' ( x*) 0 (Maximizatin Problem)
x* 0
f ' ( x*) 0
x * . f ' ( x*) 0 (Minimization Problem)
x* 0
f i ( x*) 0
xi * . f i ( x*) 0
x * 0
i
f i ( x*) 0
xi * . f i ( x*) 0
x * 0
i
Non – Linear Programming problem: there are no limitations on the forms of objective
function and constraint relations.
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:
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 non-
negative constraints (xi 0).
Lx 2 0 f 2 g 2 0 (3)
x 2.Lx 2 0 x 2( f 2 g 2 ) 0 ( 4)
Lz 0 0 (5) (maximization problem)
z.Lz 0 z 0 (6)
x1 0, x 2 0, z 0 (7 )
L 0 g ( x1, x 2) z 0
(8)
Lx 2 0 f 2 g 2 0 (3)
x 2.Lx 2 0 x 2( f 2 g 2 ) 0 ( 4)
(maximization problem)
g ( x1, x 2) 0 (5' )
z.Lz 0 g ( x1, x 2) 0 ( 6' )
x1 0, x 2 0, 0 (7 ' )
Condition (5’), (6’) and (7’) tell us that we are trying to minimize the Lagrangian in
subject to 0
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
z.Lz 0 g ( x1, x 2) 0 ( 6' )
x1 0, x 2 0, 0 (7 ' )
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
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
Maximization problem:
Denote: x=x(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:
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
j 1,..., m
a j a j
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)).
There are four fundamental building blocks in any model of consumer choice
A Feasible set
A preference relation
A behavioral Assumption
1. A consumption set
A cosumer set represents the set of all alternatives, or complete consumption plans, which
consumer is able to conceive of, whether some of them will be achievable in practice or not.
1. X 0(empty set )
2. X is closed
3. X is convex
5. 0 X
Feasible set: the set of alternatives which are achievable given the economic realities faced by
the consumer.
The feasible set B is a subset of the consumption set X which remains after we have
accounted for any constraints imposed on the consumer’s access to commodities by the
practical, institutional, or economic realities of the world.
3. A Preference Relation
It typically specifies the limits, if any, on the consumer’s ability to perceive in situations
involving choice, the form of consistency or inconsistency in the consumer’s choice, and
information about the consumer’s tastes for the different objects of choice.
4. Behavioral Assumption
This express the guiding principle the consumer uses to make final choices and so idetifies the
ultimate objectives in choice.
It is generally supposed that the consumer seeks to identify and select that available
alternative which is most prefered in the light of his personal tastes.
Assume that agents are motivated by self-interest is neither innocuous nor vacuous.
Nonetheless, the vast majority of economists are willing to do it despite many of the
criticisms and exceptions.
We assume self-interest as the principle guiding choice. Though this is the veiw of mankind
which many feel earns economics the title “the Dismal Science”.
The Preference Relation: specifies the capabilities and inclinations of the consuming agent
when faced with situations involving choice.
The “Law of Demand” was built upon some extremely strong assumption.
The axioms of choice are intended to give formal mathematical expression to three
fundamental aspects of consumer behavior and attitudes toward the objects of choice.
The consumer can make choice. He has ability to discriminate and the necessary knowledge
to evaluate alternatives.
Rational economic agents are ones who have the ability to make choices, whose internal
workings are at least minimally logical, and whose choices display a logical consistency.
Such a consumer can examine each alternative in the set and place it somewhere in a
hierarchy, or ranking.
The consumer’s preference enable him to construct such a complete and consistent ranking of
all alternatives in the consumption set by saying that the cosumer’s preferences can be
represented by a preference relation.
is called a preference relation if the symbol stands for the statement “is liked at least as
well as” and the relation "" satisfies Axioms 1, 2, and 3.
Is call a strict preference relation if is a preference relation, and is read “ is not at least as
good as”.
Using the two supplementary relations, for any pair x1 and x2, exactly one of three
mutually exclusive possibilities exist: either x1 x 2 or x 2 x1 , or x1 ~ x 2 .
Let x0 be any point in the consumption set, X. Relative to any such point, we can define the
following subsets of X:
Axiom 4: Continuity: For all x X , the “at least as good as” sets, (x) , and the “no better
than” sets, (x) , are closed and connected sets.
Continuity requires that there is no “gap” or discrete “jumps” in the indifference sets or,
equivalently, in the level curve of the utility function.
An open set S is called disconnected if there are two open, non-empty sets U and V
such that:
1. U V = 0
2. U V = S
A set S (not necessarily open) is called disconnected if there are two open sets U
and V such that
1. (U S) # 0 and (V S) # 0
2. (U S) (V S) = 0
3. (U S) (V S) = S
Note that the definition of disconnected set is easier for an open set S. In principle, however,
the idea is the same: If a set S can be seperated into two open, disjoint sets in such a way that
neither set is empty and both sets combined give the original set S, then S is called
disconnected.
To show that a set is disconnected is generally easier than showing connectedness: if you can
find a point that is not in the set S, then that point can often be used to 'disconnect' your set
into two new open sets with the above properties.
- The Axiom 4 guarantees that the indifference set ~ ( x) is closed, because ~ ( x) is the
intersection of (x) and (x) sets. The intersection of closed set is a closed set.
- Requiring the (x) and (x) sets are connected sets, we ensure that the indifference
sets, too, are connected sets, with no gaps or holes in them.
Axiom 5’: Local Non-Satiation: For all x 0 X , and for all 0 , there exists some
x B ( x 0 ) such that x x 0 .
- Within any vicinity of a given point x 0 , no matter how big or small that vicinity is,
there will be always exist at least one other point x which the consumer prefer to x 0
~(x)
X1
If x1 involves more at least one commodity and no less of any other commodity than x 0 does,
then x1 will be strictly preferred to x0.
X2
X1
- The slope (absolute value) of an indefference curve is sometimes called the Marginal
Rate of Substitution.
- Axiom 6 goes a bit further and requires that the MRS be constantly diminishing.
Summary:
- The Axioms of Completeness, Reflexivity and Transitivity formalize the notion that
the consumer is rational. Consumer can make comparisons among alternatives and his
choices are consistent.
- All other Axioms serve to characterize consumer’s tastes over the objects of choice.
- Through our basic assumptions on cosumer preferences are stated in terms of axioms on
the preference relation, preference relation can be represented by a nice, continuous real
valued function.
Geoffrey A. Jehle – Advanced Microeconomic Theory 6
Chu Thanh Duc – MDE10
- Any Preference relation which is complete, reflexive, transitive and continuous can
be represented by a continuous real valued utility function.
- This is only an existence theorem. Under the conditions stated, at least one continuous
real valued function representing the preference relation is guaranteed to exist.
- However, the theorem itself makes no statement on how many more function there
may be, nor does it indicate in any way what form any of them must take.
Proof:
Consider the vector e(1,1,…,1) X. The point t.e X. When preferences are monotonic,
for t1>t2, we have t1.e>>t2.e, so t1.e t 2.e by monotonicity.
u( x) {t | t R , x X , and t.e ~ x}
We need to prove that u(x) = t is unique for a given x and u(x) is a continuous function.
1. U(x) in order to be a function, the mapping satisfy two criteria: It must assign some
number in the range to every point in the domain, and that number must be unique.
x2
t1.e (t1>1)
e u(x) t.e=u(x).e
~(x)
t2.e (t2<1) 1 e
x1
x1
1
u(x)
If we begin at the origin and move out the ray through e, we must eventually encounte one
unique point t.e for some number t 0 such that t.e~x. That particular number will be image
of x under U, we denonte by u(x).
Since for all x X there exists a unique u( x) R , the mapping satisfies the requirements of
a function.
Consider two point in X and suppose that x1 x 2 , Then u(x1) will be the number such that
u(x1).e~x1 and u(x2) will be the number such that u(x2).e~x2. Thus u( x1 ).e u( x 2 ).e .
Recall that u(x1).e and u(x2).e are simply two points along the ray through e. Monotonicity
tells us that u( x1 ).e u( x 2 ).e only if u ( x1 ).e lies father out than u ( x 2 ).e , or only if
u ( x1 ) u ( x 2 )
We showed that a function is continuous if and only if the inverse image of every closed set in
the range is a closed set in its domain.
Any t 0 will be the image under U of the point x=t.e (t.e~x), or more compactly u(te)=t for
all t 0 .
Let T be any closed set in the range R of U. We need to prove that U 1 (T ) is a closed set in
the domain X.
By the theorem, any closed set in R will be the intersection of some collections of unitions
U 1 (T ) U 1 [0, ti ] U 1 [ti ,)
iI
We have: U [0, t i ] (t i .e) and
1
U 1 [t i ,] (t i .e)
So U 1 (T ) (t i .e) (t i .e) .
iI
U
1
(T ) (t i .e) (t i .e) is a closed set.
iI
The theorem 3.2.1 is very important. It frees us to choose the form in which we would like to
represent preferences.
If all we require of the preference ordering is that it order the bundles in the consumption set,
and if all we require of a utility function reperesenting that preference relation is that it reflect
that ordering of bundles by the ordering of numbers which it assigns to them, then any other
function which assigns bundles numbers in the same order as U does will also represent that
preference relation and itself be just as good a utility function as U.
If we have some function U which we believe represents some set of preferences, it frees us to
transform U into other, perhaps more covenient or easily manipulated forms, so long as the
transformation we choose is order-preserving.
Let "" be a continuous preference ordering and suppose that u(x) is a utility function
which represents it. If Z(x) is any strictly increasing function of a single variable, then the
composite function Z(u(x)) called a positive monotonic transform of u(x), is also a utility
function representing "" .
Let "" be complete, reflexive, transitive, and continuous, and let u(x) represent "" . Then:
For any preference ordering satisfying the conditions of Theorem 3.2.1(complete, reflexive,
transitive, continuous and monotonic), there will exists a strictly increasing and twice
continuously differentiable utility function if and only if the indifference sets are twice
continuously differentiable.
We assume that the individual consumer is atomistic, with no significant weight on the
markets in which it transacts.
Under the assumption on consumer preferences, the utility function u(x) is real valued and
continuous. The budget set B is a non-empty, closed, bounded and thus compact subset of Rn.
By the Weierstrass Theorem, we are assured that a maximum of u(x) over B exists. Moreover,
since B is convex and the objective function is strictly quansiconcave, the maximizer of u(x)
over B is unique.
The Weierstrass Theorem specifies sufficient conditions under which the existence of a
maximum and a minimum of a continuous function are assured.
f ( x * ) f ( x) f ( ~
x) for all x S .
Since preference are monotonic, the solution x* will satisfy the budget constrain with
equality, lying on, rather than inside, the boundary of the budget set.
x2
x1
The solution vector x* depends on the parameters to the problem. Since it will be unique for
any particular values of p and y, we can view the solution as a function from the set of prices
and income: xi * xi ( p, y)
L u ( x*)
x x pi 0
i i
L u ( x*)
xi * . xi * . p i 0
xi xi
L
y p . x* 0
L
[ y p.x*] 0
0, xi * 0
u j ( x*) pj
We can derive that:
: the ratio of marginal utilities must equal the ratio of
u k ( x*) pk
prices.
For simplicity, we will sometimes just assume that the consumer’s problem admits of an
interior solution. So the solution resulted from the following system of equations:
L u ( x*)
x x pi 0
i i
L y p.x* 0
The direct Utility Function: The ordinary utility function, u(x), is defined over the
consumption set X and represents the consumer’s preferences directly.
the solution x* =x*(p,y) The Indirect Utility Function: u(x*) = u*(p,y) = v(p,y)
The indirect Utility Function represent the relation between prices, income and the
highest level of utility achieved.
This function is clearly well-defined since, when preferences are monotonic and
strictly convex, a unique solution x(p,y) to the consumer’s problem exists.
Let preferences be monotonic and differentiable, and let p>>0 and y>0. Then v(p,y) has
these properties:
2. Increasing in y
3. Non-increasing in p
4. Quansiconvex in p
Geoffrey A. Jehle – Advanced Microeconomic Theory 1
Chu Thanh Duc – MDE10
Proof:
Equiproportionate changes in all p and y leave the consumer’s budget set unchanged.
y p1 x1 p2 x2 ty tp1 x1 tp 2 x2 (t 0)
The set of feasible choices, and so the maximal level of utility the consumer can achieve,
must therefore also remain the same.
Changing all p an y by proportion t>0 must leave the maximal utilitly unchanged.
v( p, y ) L
x* 0
pi pi ( x *, )
as the proof.
v( p, y ) L
0
y y ( x *, )
3. Quansiconvex in p
The Lagrangian multiplier ( ) will measure the sensitivity of the objective function u(x) to
changes in the constraint constant (y). (See the Exercise 2.29).
Thus the value of the Lagrangian multiplier at the solution measure the marginal utility of
income.
Let B1, B2, and Bt be the budget sets available when the consumer has income y and faces
prices p1, p2, and p t tp 1 (1 t ) p 2 , then:
B1 {x | p 1 .x y}
B 2 {x | p 2 .x y}
B t {x | p t .x y}.
So we will show that every choice the consumer can possibly make when she faces budget B t
is a choice which could have been made when she faced either budget B 1 or budget B 2 . It
would be the case that every level of utility she can achieve faving B t is a level she could
have achieved either when facing B 1 or when facing B 2 . Then the maximum level of utility
Geoffrey A. Jehle – Advanced Microeconomic Theory 2
Chu Thanh Duc – MDE10
that she can achieve over B t could be no larger than at least one of the maximum level of
utility that she can achieve over B 1 or the one she can achieve over B 2 .
The indirect utility function tells us the maximal level of utility the consumer can
achieve facing different prices and incomes.
The demand functions give us the utility maximizing choices of each commodity he
will make facing different prices and incomes.
To get the indirect utility function, we simply substitute the demand functions into the
direct utility function.
To get the indirect utility function, we simply substitute the demand functions into the
direct utility function.
There is a question that how to derive the direct utility function from the indirect
utility function ? This theorem will answer this question.
Theorem: Let v(p,y) be any indirect utility function satisfying the conditions of Theorem
4.1.1. Then,
v( p, y) / pi
x i ( p, y )
v( p, y) / y
Proof:
Let x* and solve the Kuhn – Tucker conditions. The solution ( p, y) gives us the
v( p, y ) L
( p , y ) x i ( p , y )
pi pi ( x *, )
v( p, y ) L
( p, y )
y y ( x *, )
v( p, y) / pi
x i ( p, y )
v( p, y) / y
4.1.2 The Expenditure Function
What is the minimum level of money expenditure, or outlay, which the consumer
must make facing a given set of prices in order to achieve a given level of utility?
X2
X1
If preferences are monotonic and strictly convex, the solution will be unique.
The lowest expenditure necessary to achieve utility u at prices p will be equal to cost
of the bundle x h ( p, u) : e( p, u) p.x h ( p, u) .
If we fix the level of utility the consumer is permitted to achieve at some arbitrary
level u, how will his purchases of each good behave as we change the prices he faces?
h
x1 ( p, u) : x1h ( p10 , p20 , u) , x2h ( p10 , p20 , u) .
Let preferences be monotonic and let p>>0. Let u>u(0) and let e(p,u) be defined as in
4.1.3. Then e(p,u) is:
1. Increasing in u
2. Non-decreasing in p
3. Homogeneous of degree 1 in p
4. Concave in p
5. Also, the price partial derivatives of e(p,u) are the Hicksian demand functions
e( p, u )
xih ( p, u )
pi
Proof:
L u ( x h )
pi 0
xi xi
L u ( x h )
xih xih [ pi ]0
xi xi
L
u u( x h ) 0
L
[u u ( x h )] 0
u ( x h )
From the Kuhn-Tucker conditions we must therefore have: p j 0 for at least one
x j
j.
u ( x h ) pj
By the monotonicity of preference, we have: 0 0.
x j u ( x h )
x j
e( p, u ) L
By the Envelop theorem: 0
u u
( x , )
h
e(p,u) is increasing in u.
e( p, u ) L
By the Envelop theorem we have: xih ( p, u ) 0
pi pi ( x h , )
e(p,u) is in-decreasing in p.
p 1 .x1 p1 .x t tp 1 .x1 tp 1 .x t
p 2 .x 2 p 2 .xt (1 t ) p 2 .x 2 (1 t ) p 2 .x t
t.e( p1 , u ) (1 t ).e( p 2 , u ) e( p t , u )
Let v(p,y) and e(p,u) be the indirect utility function and expenditure function for some
consumer. Then the following relations between the two obtain for all prices p, incomes y,
and utility level u:
e( p, v( p, y )) y
v( p, e( p, u )) u
This theorem points us to an easy way to derive either one directly from knowledge of
the other, thus requiring us to solve only one optimization problem and giving us the
choice of which one we care to solve.
Geoffrey A. Jehle – Advanced Microeconomic Theory 7
Chu Thanh Duc – MDE10
v( p, e( p, u)) u . Invert the indirect utility function in its income variable, we have:
e( p, u) v 1 ( p : u)
e( p, u ) is strictly increasing in u.
v( p, u) e 1 ( p : y)
Example: The CES direct utility function gives the indirect utility function:
v( p, y ) ( p1r p 2r ) 1 / r . y
For an income level equal to e(p,u), we must have: v( p, e( p, u)) ( p1r p2r ) 1 / r .e( p, u)
For any prices p, income y, and utility level u, the following identical relations hold
between the consumer’s Hicksian and Marshallian demand functions:
The Hicksian demands are: xih ( p, u) ( p1r p2r )1/ r )1 pir 1u
r 1 / r
The indirect utility function is: v( p, y ) ( p1 p 2 )
r
y
pir 1 y
We have: xi ( p, y) xih ( p, v( p, y ) ( p1r p 2r ) (1 / r )1 pir 1 ( p1r p 2r ) 1 / r y
p1r p 2r
Relative prices and real income are two such real measures
Relative price: By the relative price of some good, we mean the number of units of
some other good which must be foregone in order to acquire 1 unit of the good in
question.
pi $ / unit i unit j
p j $ / unit j unit i measure the units of good j forgone per unit of good i acquired.
Consumer’s real income: We mean the total number of units of some commodity
which could be acquired if the consumer spent his entire money income on that
commodity.
y $
Real income interm of good j: units of j
p j $ / unit of j
Proof:
Equiproportionate changes in all p and y leave the consumer’s budget set unchanged.
y p1 x1 p2 x2 ty tp1 x1 tp 2 x2 (t 0)
Application:
1
With t 0
pn
p1 p y
We have: xi ( p, y ) xi (tp , ty ) xi ( ,, n1 ,1, )
pn pn pn
Demand for each of the n goods depends only on (n-1) relative prices and the consumer’s
real income.
Ordinarily, a consumer will buy more of a good when its price declines, and less when its
price increases. However, these cases are not necessarily true.
Substitution effect:
Since all goods are taken to be desirable by the consumer, even if the consumer’s total
command over goods were unchanged, we would expect him to substitute more of the
good which has become relatively cheapter for less of the goods which are now
relatively more expensive.
Income effect:
When the price of any one good declines, the consumer’s total command over all
resources is effectively increased, allowing him to change his purchases of all
goods in any way he sees fit. The effect on quantity demanded of this generalized
increase in purchasing power is called the income effect.
The income effect is defined as the residual out of the total effect which is left
after the substitution effect.
Total effect:
Let x(p,y) be the consumer’s Marshallian demand system. Let u* be the level of utility the
consumer achieves at prices p and income y. Then,
x j ( p, y ) x hj ( p, u*) x j ( p, y )
x i ( p, y )
p p y
i i
TE SE IE
Proof:
x hj ( p, u*) x j ( p, e( p, u*))
By the assumption , u* is the level of utility the consumer achieves facing prices p and
having income y (see the theorem) u*=v(p,y)
the minimum expenditure at prices p and utility u* will therefore be the same as the
minimum expenditure at price p and utility v(p,y)
e( p, u*)
We have: xi ( p, y) (3)
pi
x j ( p, y ) x hj ( p, u*) x j ( p, y )
x i ( p, y )
p p y
i i
TE SE IE
xih ( p, u )
0
pi
Proof:
e( p, u )
xih ( p, u )
pi
2 e( p , u ) xih
pi
2
pi
Inferior goods: consumption decreases as real income increases, holding relative prices
constant.
Let preferences be complete, transitive, reflexive, monotonic, and strictly convex. If a good
is a normal good, then a decrease in price will cause an increase in quantity demanded. If a
decrease in price causes a decrease in quantity demanded, then the good must be an
inferior good.
xih ( p, u ) x j ( p, u )
h
p j pi
Proof:
xih ( p, u ) 2 e( p, u )
p j pip j
2 e( p , u ) 2 e( p , u )
By the Young’s Theorem:
pi p j p j pi
xih ( p, u ) x j ( p, u )
h
p j pi
x h ( p, u )
Let x h ( p, u) be the consumer’s system of Hicksian demands, and let i
p j i , j 1n
represent the entire matrix of Hicksian substitution terms. Then this matrix is negative
semi-definite.
Proof:
xih ( p, u ) 2 e( p, u )
p j pip j
The expenditure functions is concave in prices. From the theorem 2.1.3, the matrix of second-
order partials (the Hessian) of a concave function is negative semi-definite
Let x(p,y) be the consumer’s Marshallian demand system. Define the Slutsky matrix as the
n n matrix of price and income responses given by:
xi ( p, y ) x ( p, y )
x j ( p, y ) i
p j y
i , j 1,n
Then the theory of the preference-maximizing, atomistic consumer requires that the Slutsky
matrix be negative semi – definite.
If preferences are monotonic, at least one good will be bought in a positive amount.
xi ( p, y ) y
Income elasticity: i
y x i ( p, y )
xi ( p, y ) p j
Price elasticity: ij
p j x i ( p, y )
n
p i x i ( p, y )
And Income Share: si
y
so that si 0 and s
i 1
i 1.
Let x(p,y) be the consumer’s Marshallian demand system. Let i , ij , and si be income
elasticity, cross-price elasticity and income share. Then the following relations must hold
between income shares, price, and income elasticities of demand:
Cournot Aggregation: s
i 1
i ij s j
Proof:
Prove (1)
y p.x( p, y)
n
xi ( p, y) n
p x x ( p, y) y n
1 pi i i i si i
i 1 y i 1 y y xi i 1
Prove (2)
n
xi ( p, y )
0 pi xj
i 1 p j
n
xi ( p, y )
x j pi
i 1 p j
xj pj n
pi xi ( p, y ) n
p x x ( p, y ) p j n
pj i i i si ij
y i 1 y p j i 1 y p j xi i 1
n
si ij s j
i 1
There is a question: Starting with an expenditure or an indirect utility function, can we “work
backwards” to discover the underlying direct utility function that would have generated it?
v( ~
p ) is called normalized indirect utility function, and depends on normaliz
price alone.
With direct utility function u(x), the normalized indirect utility function is defined
as:
v( ~
p ) max u( x) s.t. ~
p.x 1
x
Let v( p, y) be any indirect utility function, and form the normalized indirect utility
function v( ~
p ) . Then the implied direct utility function is given by the following minimum
Proof:
Theorem 4.3.5 (Hotelling, Wold) Duality and the System of Inverse Demands
Let u(x) be the consumer’s direct utility function. Then the inverse demand function for
good i is given by:
~ u ( x) / xi
pi ( x) n
x
j 1
j (u ( x) / x j )
Proof:
u( x) min
~
v( ~
p ) s.t. ~
p.x 1
p
L v( ~
p) [1 ~
p.x]
u ( x) L
Applying the Envelope Theorem: * ( x). ~
p j * ( x) (1)
x j x j ~
p*( x ), *( x )
n
u ( x) n
j 1
xj
x j
* ( x ) ~
j 1
p j * ( x).x j * ( x) (2)
~ u ( x) / x j
p j * ( x) n
x (u( x) / x )
j 1
j j
4.4 Uncertainty
Certainty: the consumer knows the prices of all commodities and knows that any feasible
consumption bundles can be obtained with certainty.
Many eoconomic decisions contain some element of uncertainty: future income, future
prices…
4.4.1 Preferences
Beforem, the consumer was assumed to have a preference ordering over all
consumption bundles x in the consumption set X. Implicit in our statement that
“bundle xi is preferred to bundle xj” was the assumption the individual chooes between
xi with certainty and xj with certainty.
Let’s first define an outcome as a result of some uncertain situation. For example,
outcomes of betting are win and loss.
A {a1 , a2 ,..., an } : the set of all mutually exclusive ultimate outcomes that an
individual could endup with.
Gambles: G [ p1oa1p2 oa2 pn an ] denotes the entire gamble involving a1 with
Where:
The space g(A) is the set of all possible gamble which can be constructed from the outcome
n
set A by varying the probabilities 0 pi 1 of each ai A while ensuring that i 1
pi 1 .
Since each a i is the special gamble in g(A) where pi 1 and p j 0 , i j , the set of all
The problem of choice under uncertainty can be veiwed as a choice between alternative
gambles in g(A).
We can then define a binary relation on g(A). Where the symbol stands for the
statement “is atleast as well as”.
We again suppose that these preferences obey certain rules which we’ll lay down in the form
of axiom, called the “axioms of choice under uncertainty”.
Axiom G1: (Completeness) For any distinct gambles G1 and G2 in g(A), either G1 G2 or
G2 G1 .
Axiom G3: (Transitivity) For any three gambles G1, G2, and G3 in g(A), if G1 G2 and
G2 G3 , then G1 G3 .
One important consequence of this Axiom is that there must exist a best and a worst outcome
in A. Note that best and worst outcomes need not be unique.
Axiom 3, there therefore must exist a best and a worst outcome in A. A best outcome a B A
satistfies a B a j for all a j A . A worst outcome aW A statisfies a w a j for all a j A .
Axiom G4: (Continuity) For any gamble G g (A) , there exists some probability z,
0 z 1 , such that G ~ [ z o a B (1 z ) o aW ] .
Indifference probability
For any gamble G there is some other gamble, involving only the best and the worst outcomes
in A, which the agent ranks indifferent to G.
Axiom G5. (Monotonicity) For any two best-worst gambles, G1 [ poa B (1 p)oaW ] and
Axiom G5 states that given the choice between any two best – worst gambles with different
probabilities attached to the same best outcome, an individual will never prefer the gamble
with the lower probability of the best outcome.
Together, Axiom G4 and G5 rule out some kinds of behavior which, at first glance, might
appear quite reasonable. Example, let A = {“death”, $10, $1000}; aW death , a B $1000
and a B $10 aW . Consider the intermediate gamble G3 = $10. According to axiom G4,
Axiom G6. (Substitutability) For any outcome ai A and any gamble G j g (A) , if
ai ~ G j , then
Axiom G6 states that if the individual is indifferent between and outcome promised with
certainty and some gamble, then he mus also be indifferent between two otherwise identical
gambles offering each of these with the same probability.
Then
Whether we can represent those preferences with a continuous real valued function? Say Yes!
Axiom G1, G2, G3 and some kind of continuity assumption should be sufficient to ensure the
existence of a simple numerical function representing .
Instead of asking whether there is a certain kind of function, with a certain specific
mathematical property, representing .
Let G [ p1oa1 pi oai pn oan ] be any gamble in g(A), and suppose that the function
n
U (G ) p i U (ai ) , we say that the utility function U possesses the extra, expected
i 1
utility property.
A Utility Function possesses the Expected Utility Property if and only if the Utility number it
assigns to any gamble can be expressed as the Expected Value of the Utility numbers it
assigns to the Ultimate outcomes in that gamble.
Theorem 4.4.1 Existence of a Von Neumann – Morgenstern (VNM) Utility Function over
Gambles
Let preferences over gambles, , satisfy Axiom G1 through G7. Then there exists a
function U : g ( A) R such that, for all G1 and G 2 in g(A), G1 G2 if and only if,
U (G1 ) U (G2 ) , and where, moreover, for any gamble G [ p1oa1 pi oai pn oan ] ,
n
U (G ) p i U (ai ) .
i 1
Proof:
Geoffrey A. Jehle – Advanced Microeconomic Theory 19
Chu Thanh Duc – MDE10
Let G be any gamble in g(A), where: G [ p1oa1 pi oai pn oan ] (P.1)
By the Axiom G1, G2, and G3, there is a complete ordering over g(A), and we can therefore
identify a best outcome a B and worst outcome a w in A.
It is easy to prove that these indifference probabilities are unique (using Axiom G5).
According to Axiom G6, we can substitute from the right-hand side of (P.2) of each a i in
(P.1):
n n
G ~ pi z i oa B 1 pi z i oaW (P.4) (remind that p 1 ).
i 1
i
i 1
Take note that this mapping is indeed a function, since the indifference probabilities from
which it is constructed always exist and are unique.
Applying the mapping in (P.5) and (P.6) to the gamble G1, we obtain:
n n
U (G1 ) qiU (ai ) qi zi
i 1 i 1
3. Let U (ai ) z i
Example 4.4.1
Under this mapping, the utility of the best outcome must be (identically) 1, and that of
the worst outcome must be (identically) 0. The utility assigned to intermediate outcomes
will depend on the individual’s attitude toward taking risks.
G1 G2
We can rank any of the infinite number of gambles that could be constructed from the three
outcomes in A./.
The VNM mapping does its assignment of numbers to gambles in two distinct stages:
1. First, all gambles in g(A) that offer one outcome with certainty are assigned
utility numbers that reflect th agent’s ordering of those alternatives with
certainty.
2. Then, all other gambles in g(A) are assigned utility numbers via the expected
utility calculation.
The VNM utility numbers assigned to ultimate outcomes must not only properly reflect the
agent’s ranking of those particular outcomes relative to each other, they must also be capable
of properly reflectig the agent’s ranking of gambles comprised of them through the (special)
expected utility calculation.
It should not, therefore, be terribly suprising that we are less free to transform VNM utility
functions if the ranking of every gamble is to be preserved.
Theorem 4.4.2 VNM Utility Functions are Unique Up to Positive Affine Tranformations
Let satisfy Axiom G1 through G7, and suppose that the VNM utility function U(G)
represents . Then the VNM utility function, V(G), represents those same preferences if,
and only if, V (G) U (G) , for some arbitrary scalar and some scalar >0.
Proof:
By the proof of Theorem 4.4.1 that if the VNM utility function U(G) represents , it
possesses the expected utility property and so, for any G G(A) , we can write:
n n
U (G) piU (ai ) pi z i where z i satisfies: ai ~ [ zi oa B (1 zi )oaW ] .
i 1 i 1
Suppose that V(G) is another VNM utility function which represents , we must have the
following:
i 1 i 1
For any outcome set A and VNM utility function V, the numbers V (a B ) and V (aW ) are
Theorem 4.4.2 tells us that VNM utility functions are not completely unique, nor are
they entirely ordinal. We can still find an infinite number of them that will rank
gambles in precisely the same order and also possess the expected utility property.
However, unlike ordinary utility functions, here we must limit the posivite
transformation of VNM utility function under the form of V (G) U (G) with
, 0 .
Yet the less than complete ordinality of the VNM utility function must not tempt us
into attaching undue significance to the absolute level of a gamble’s utility, or to the
differene in utility between one gamble and another. With what little weve required of
the agent’s binary comparisons between gambles in the underlying preference
ordering, we still cannot use VNM utility functions for interpersonal comparisons of
well – being, nor can we measure the “intensity” with which one gamble is perferred
to another.
The VNM utility function we created reflected some desire to avoid risk.
We shall assume that the VNM utility function is both increasing and differentiable over the
appropriate domain of wealth concerned.
n
The expected value of G: E[G ] pi wi .
i 1
Now suppose that the agent is given a choice between accepting the gamble G on the one
hand, or receiving with certainty the expected value of G on the other. We can evaluate these
two alternative as follows:
When someone would rather receive the expected value of a gamble with certainty than face
the risk inherent in the gamble itself, we say they are risk averse.
If these relationships hold for all gambles G g (A) , these definitions apply globally.
We will be asked to show that the agent is risk averse, risk neutral, or risk
loving over some subset of gambles if, and only if, their VNM utility function
is strictly concave, linear, or strictly convex, respectively, over the appropriate
domain of wealth.
U(w2)
U(E[G])
U(G)
U(w1)
P
w1 CE E(G) w2
We can see that, strict concavity of the VNM utility function, U(E[G])>U(G),
so the individual is risk averse.
A risk – averse person will “pay” some positive amount of wealth in order to
avoid the gamble’s inherent risk. This willingness to pay to avoid risk is
measured by the Risk Premium.
The Certainty Equivalent of any gamble G is an amount of wealth, CE, offered with
certainty, such that U (G) U (CE ) . The Risk Premium is an amount of wealth, P, such
that U (G) U ( E[G] P) . Clearly, the two are related, and P E[G] CE .
Example 4.4.2
Suppose that U (w) log( w) U is strictly concave in wealth, the individual is risk averse.
Let G offers 50-50 odds of winning or losing some amount of wealth, h, so that:
Risk aversion and concavity of the VNM utility function in wealth are equivalent.
The sign of the second derivative U’’(w) does tell us whether the individual is risk
averse, risk loving, or risk neutral, its size is entirely arbitrary.
The size of U’’(w) depends on the positive affine transformations of U(w) and the
units in which the outcome is measured.
Arrow (1965) and Pratt (1964) have proposed a measure a risk aversion which is
based on the second derivative, but which avoids these non-uniqueness problems.
U ' ' ( w)
Ra ( w)
U ' ( w)
Ra (w) is positive, negative, or zero as the agent is Risk Averse, Risk Loving, or Risk Neutral
repectively.
Any positive affine transformation of utility will leave the measure unchanged.
Ra (w) is only a local measure of risk aversion, so it need not be the same at every level
wealth.
The firms buys its inputs on factor markets at prices determined on those markets, and
these expenditures are the firm’s costs. The firm sells its output on product markets
and earns revenue from these sales.
The way inputs are combined to produce outputs is partly a matter of choice and partly
a matter of what is technologically possible. There is therefore a close relationship
between the prices at which the firm can acquire inputs, the amounts of these inputs it
decides to acquire, the technological possibilities in combining inputs to produce
outputs, and the cost at which the firm is able to produce outputs.
Similarly, the prices at which the firm is able to sell its output, the quantity of output it
decides to sell, and any other stratergies it may use in marketing its products affect its
revenues.
Circumstances and decisions affecting its costs and/or its revenues obviously affect the
difference between the two, firm profits.
Profit maximization is not the only conceivable motive of firm behavior. Sales, market
share, or even prestige maximization are possibilities.The majority of economists
continue to embrace the hypothesis of profit maximization. Why?
From an empirical point of view, the assumption that firms profit maximize leads to
predictions of firm behavior which are time and again borne out by the evidence. From
a theoretical point of view, there is first the virtue of simplicity and consistency with
the hypothesis of self – interested, utility maximization on the part of consumers.
There are identifiable market forces which coerce the firm toward profit maximization
even if its owners or managers are not themselves innately inclined in that direction.
For suppose that some firm did not conduct its activities to maximize profits. Then if
the fault lies with the managers, and if at least a working majority of the firm’s owners
are no-satiated consumers, those owners have a clear interest in ridding themselves of
that management and replacing it with a profit – maximizing one. If the fault lies with
the owners, then there is an obvious incentive for any non – satiated entrepreneur
outside the firm to acquire it and change its ways.
5.2 Production
The state of techonoloty determines and restricts what is possible in combining inputs to
produce output, and there are several ways what we can represent this constraint. The most
general way is to conceive of the firm as posseing a production possibility set Y.
1. 0 Y
2. Y Rm {0}
3. Rm Y
The first axiom is called the possibility of inaction. It’s possible for the firm to
acquire No inputs and produce no output. One immediate implication of this axiom is
that firm profits in the long run need never be negative.
The axiom 3 is called free disposal. It has little practical significance, it is included
for mathematical competeness. It says, in effect, that the firm can always use
unlimited amounts of inputs to produce no output.
Axiom 4 is also for mathematical purposes. It ensures that the production possibility
set contains its boundary so that there will always be an efficient frontier, giving a
well-defined maximum amount of output that can be obtained from a given level of
inputs.
V ( y) {x | x Rn , y R , ( y, x) Y }
The input requirement set is defined as all combinations of inputs which produce an
output level of at least y units.
t.x1 (1 t ).x 2 V ( y)
The fist axiom is both a continuity requirement and an implication of the “no free
production” axiom.
Monotonicity says that adding more of any input can never reduce the amount of
output produced and it is implied by the axiom of free disposal.
Convexity: If the production process is time divisible, any convex combination of two
processes in V(y) can be viewed as a hybrid production run where one process is run
some fraction of the relevant time period and the other process run the remaining
fraction of the period.
x2
V(y)
x V (x)
x1
Input Regularity ensures that the lower boundary of V(y) is solid, unbroken, and
contained in V(y).
Under monotonicity, all points northeast of that boundary must produce an output of y
or more and the boundary must not be positively sloped.
The isoquant is the efficient frontier of the input requirement set and is where we would
always expect a firm producing y units of output to choose to operate whenever inputs are
costly.
V ( y) {x | f ( x) y}
Q( y) {x | f ( x) y}
When V(y) is input regular, the production function is continuos and f(0)=0. If
y=f(x) and y>0, then xi 0 for at least one input i.
f ( x)
is called Marginal Product of factor i
xi
f ( x)
If V(y) is monotonic 0.
xi
The marginal Rate of Technical Substitution (MRTS): is the slope of the Isoquant.
f ( x) f ( x)
0 dx1 dx 2
x1 x 2
dx1 f ( x) x1
(1)
dx 2 f ( x) x 2
dx j f ( x) xi
MRTS ij (1)
dxi along Q ( y )
f ( x) x j
In general, the MRTS between any two factors depends on the amounts of all factors
employed.
Let N = {1,…,n} index the set of all factors, and suppose that these factors can be
partitioned into s mutually exclusive and exhaustive subsets, N1 ,, N S .
The production function is called weakly separable if the MRTS between two factors within
the same group is independent of factor usage in other groups:
f i ( x) / f j ( x)
0 for all i, j N s and k N s .
x k
The production function is called strongly separable if the MRTS between two factors from
different groups is independent of all factors outside those two groups:
f i ( x) / f j ( x)
0 for all i N s , j N k and k N s N t , ( s t ).
x k
For a production function f(x), the elasticity of substitution between factors i and j at the
point x is defined as
d log( x j / xi ) d ( x j / xi ) f i ( x) / f j ( x)
ij
d log( f i ( x) / f j ( x)) x j / xi d ( f i ( x) / f j ( x))
Between two factors x i and x j , holding all other factors and the level of output
The closer is to zero, the more strictly convex the isoquants and the more
“difficult” substitution between factors.
The larger is, the flatter the isoquants and the “easier” substitution between factors
Geoffrey A. Jehle – Advanced Microeconomic Theory 5
Chu Thanh Duc – MDE10
x2 x2 x2
0 0
Q(y)
Q(y)
Q(y)
x1 x1 x1
Example 5.2.1:
1 1
log( x2 / x21 ) log( x2 ) log( x1 ) d log( x1 / x2 ) dx1 dx2
x1 x2
1
f1 ( x1 x 2 )1 / 1 . x1 1 ( x1 x 2 )1/ 1 x1 1
f 2 ( x1 x 2 )1 / 1 x 2 1
x1 1
log( f1 / f 2 ) log 1 ( 1)log( x1 ) log( x 2 )
x2
1 1
d log( f1 / f 2 ) ( 1) dx1 dx 2
x1 x2
1
12
1
We can see that the degree of substitution between factors always be the same. This is
therefore on the one hand a somwhat restrictive characterization of the CES
technology.
1/
n n
y i xi , where i 1
i 1 i 1
1
In the CES form, ij for all i j
1
When 0 , ij 1 , and this CES form reduces to the linear homogeneous Cobb-
n
Douglas form: y xi
ai
i 1
Let f(x) be a production function and suppose that it is homogeneous of degree 1. Then f(x)
is a concave function of x.
Proof:
With 0 t 1
f ( x1 ) y1 0 f (tx 1 ) ty 1 0
f ( x 2 ) y 2 0 f ((1 t ) x 2 ) (1 t ) y 2 0
tx 1 1 (1 t ) x 2 1
f 1 1 f (tx ) 1 and f
1
2
f ((1 t ) x 2 ) 1
(1 t ) y (1 t ) y
2
ty ty
tx 1 (1 t ) x 2 tx 1 (1 t ) x 2
f 1 f
2
1 and are in V(1).
ty (1 t ) y ty 1 (1 t ) y 2
tx 1 (1 t ) x 2
V(y) is convex set f z. 1 (1 z ) 1 where 0 z 1 .
ty (1 t ) y 2
ty 1 (1 t ) y 2
Let z 1 and (1 z ) 1
ty (1 t ) y 2 ty (1 t ) y 2
tx 1 (1 t ) x 2 tx 1 (1 t ) x 2
f 1 1 f 1 1
ty (1 t ) y
2
ty 1 (1 t ) y 2 ty (1 t ) y
2
tx 1 (1 t ) x 2 1
We have: f 1 1
2
f (tx 1 (1 t ) x 2 )
ty (1 t ) y ty (1 t ) y
2
f (tx 1 (1 t ) x 2 ) ty 1 (1 t ) y 2 tf ( x1 ) (1 t ) f ( x 2 )
We are interested in how output responds as the amounts of different factors are varies.
Returns to variable proportions: In the short run, at least one factor is fixed to the firm. So
output can be varied only by changing the amounts of some factors. As amounts of the
variable factors are changed, the proportion in which the fixed and variable factors are used is
changed.
Returns to Scale: How output responds when all factors are varies in the same proportion.
1. Constant return to scale if, and only if, f(tx) = tf(x) for all t >0 and all x.
2. Increasing return to scale if, and only if, f(tx) > tf(x) for all t > 0 and all x.
3. Decreasing return to scale if, and only if, f(tx) < tf(x) for all t <0 and all x.
A production function has constant return to scale if and only if it is a (positive) linear
homogeneous function of degree 1.
Suppose that every technology falls into just one of these catergories since each requires that
output always respond to proportional changes in factor usage in the same qualitative way,
regardless of the current level of output or scale of the inputs.
fx n
d log f (tx ) i 1 i i
( x) lim i ( x)
t 1 d log t f ( x) i 1
Returns to scale are locally constant, increasing, or decreasing, as (x) is equal to, greater
than, or less than 1.
The elasticity of scale and the output eslasticity of factors are related as follows:
n
( x) i ( x)
i 1
Many technologies exhibit increasing, constant, and decreasing returns over only certain
ranges of output. It is therefore often useful to have a local measure of return to scale. It is
therefore often useful to have a local measure of return to scale.
The elasticity of scale or the (overall) elasticity of output tells us the instantaneous
percentage change in output that occurs with a 1 percent increase in all inputs.
5.3 Cost
The firm maybe is faced with upward –sloping supply curve for its factors, so the more it
hires, the higher the per-unit price it must pay. In the other case, the firm is astomistic or
perfectly competitive on its input markets.
The cost function for the firm facing fixed factor prices w >>0 is defined as the minumum –
value function,
Suppose that x* solves this problem. Forming Kuhn – Tucker necssary conditions, x* must
satisfy:
f ( x*)
wi x 0 (1)
i
f ( x*)
xi * wi 0 (2)
x i
y f ( x*) 0 (3)
y f ( x*) 0 (4)
For all y>0, input regularity requires that xi * 0 for at least one factor
f ( x*)
(2): wi 0
xi
For every pair of factors which the firm chooses to employ in positive amounts, we have:
f ( x*) / xi w
i
f ( x*) / x j w j Technical substitution
The firm’s factor demand functions – Conditional Factor demand: x(w,y) is the amount
of each factor that the cost – minimizing firm will buy to produce output y when it faces
factor prices w.
Example 5.3.1
Assuming that y>0 and an interior solution, the firs – order Lagrangian conditions reduce to
the two conditions:
w x 1
1 1
w2 x 2
1/
y ( x1 x 2 )
c(w, y) w1 x1 (w, y) w2 x2 (w, y) y w1 /( 1) w2 /( 1)
( 1) /
Let V(y) be input regular and monotonic and suppose that w>>0 and y>0. Let c(w,y) be the
cost function as defined in Definition 5.3.1. Then c(w,y) is continuous and:
1. Increasing in y
2. Non – decreasing in w
3. Homogeneous of degree 1 in w
4. Concave in w
c( w, y )
xi ( w, y )
wi
Proof:
Let the cost function be twice differentiable and let x(w,y) be the vector of conditional
factor demands. Then:
xi ( w, y )
3. The own – substitution effect, 0 for all i.
wi
xi ( w, y ) x j ( w, y )
4. The cross – substitution effects are symmetric, so that: for all
w j wi
i and j.
Proof:
Each of these may be proved just as the corresponding properties of the consumer’s Hicksian
demands.
Theorem 5.3.4 Cost and Conditional Factor Demands When Production Is Homothetic
a) The cost function is multiplicatively separate in factor prices and output and can be
written c(w,y) = h(y)c(w,1), where h’(y)>0 and c(w,1) is the unit cost function, or the
cost of 1 unit of output.
b) The conditional factor demands are multiplicatively separable in factor prices and
output and can be written x(w, y) h( y) x(w,1) where h’(y)>0 and x(w,1) is the
conditional factor demand for 1 unit of output.
a) c(w, y) y.c(w,1)
b) x(w, y) y.x(w, y)
Proof:
Let the production function be F(x) and suppose it is homothetic. Then F(x) = f(g(x)), where
f’>0 and g(x) is homogeneous of degree one.
x
min w.x s.t. g 1 1
x
f (1)
x
min w.x s.t. f 1 ( y ) g 1 f 1 ( y )
x
f (1)
f 1 ( y )
min w.x s.t. g 1 x f 1 ( y )
x
f (1)
f 1 ( y )
min w.x s.t. f g 1 x y
x
f (1)
f 1 (1) f 1 (1) f 1 ( y )
min w. s.t. f g 1 x y
f 1 ( y ) x f 1 ( y ) f (1)
f 1 (1)
min w.z s.t. f g ( z ) y
f 1 ( y ) x
f 1 (1)
c( w, y )
f 1 ( y )
f 1 ( y )
So c( w, y) c( w,1) h( y)c( w,1)
f 1 (1)
The optimal cost of the variable factors, w.x(w, w , y; x ) , is called total variable cost. The
cost of the fixed factors, w.x , is called total fixed cost.
There is a duality between production and cost just as there is between utility and expenditure.
The principles are identical: If we begin with a technology and derive its cost function, we can
take that cost function and use it to generate a technology.
Any function will all the properties of a cost function implies some technology for which it is
the cost function.
This is the most significant developmets in modern theory and has had important implications
for applied work. Applied research need no longer begin their study of the firm with detailed
knowledge of the technology and with access to relatively obscure data. Instead, they can
concentrate on devising and estimating flexible functions of observable market prices and
output and be assured that they are carrying along all economically relevant aspects of the
underlying technology.
Let V(y) be input regular and monotonic. Let c(w, y) min w.x s.t. x V ( y) .
x
and let c * (w, y) min w.x s.t. x V * ( y) . Then for all y > 0:
2. c(w, y) c * (w, y)
E1.3
1. c(S T ) cS cT
c(S T ) {x | x S and T } {x | x S T }
cS cT {x | x S or x T } {x | x S and T } {x | x S T }
c(S T ) cS cT
2. c(S T ) cS cT
c(S T ) {x | x S T }
cS cT {x | x cS cT } {x | x S T }
c(S T ) cS cT
E 1.5
Let A and B be convex set. Show by counter-example that A B need not be a convex sex.
We have: [1,3] and [6,8] are convex sets. However, with t=0.5, the convex combination of 1
and 6: 1*0.5+6*0.5=3.5 is not in the set: [1,3] [6,8] [1,3] [6,8] is not a convex set.
E 1.6
S i {x | x S1 and S 2 and S n }
i 1
n
Let x1 and x2 in S i x1 and x2 S i . Because Si is a convex set tx1 (1 t ) x2 S i
i 1
n
S i is a convex set.
i 1
E 1.7
Graph each of the sets given below. If the set is convex, give a proof. If it is not convex, give
a counter-example
a. {( x, y) | y e x }
(0.5*0+0.5*1,0,5*1+0,5*e)=(0.5,0.5+0.5*2.718)=(0.5,1.86)
b. {( x, y) | y e x }
{( x, y) | y e x } {( x, y) | y e x }
c. {( x, y) | y 2 x x 2 , x 0, y 0}
x1 1 y1 1
x2 1.5 y 2 0.75
{( x, y) | y 2 x x 2 , x 0, y 0}
d. {( x, y) | xy 1, x 0, y 0} {( x, y) | y 1 / x, x 0, y 0}
The convex combination of (x1,y1) and (x2,y2) with t=0.5 is: (0.75,1.5).
E 1.9
Let A and B be two sets in the domain D, and suppose that B A . Prove that f ( B) f ( A)
for any mapping f : D R .
Consider the set A2 {x | x A and x A B} for all x in A2, there is an unique f(x)
the range f ( A2) f ( B) 0 .
E 1.10
Let A and B be two sets in the range R, and suppose that B A . Prove that
f 1 ( B) f 1 ( A) for any mapping f : D R .
E 1.13
Let f : D R be any mapping and let B be any set in the range R. Prove that
f 1 (cB) c( f 1 ( B)) .
f 1 (cB) c( f 1 ( B))
E 1.14
For any mapping f : D R , and any two sets A and B in the range of f, show that:
1. f 1 ( A B) f 1 ( A) f 1 ( B)
Let some y in A B y in A or B or A B .
f 1 ( y) in f 1 ( A) f 1 ( B)
f 1 ( A B) f 1 ( A) f 1 ( B)
2. f 1 ( A B) f 1 ( A) f 1 ( B)
f 1 ( y) in f 1 ( A) f 1 ( B)
f 1 ( A B) f 1 ( A) f 1 ( B)
Exercises in Chapter 2. Calculus and Optimization
E 2.11
A real valued function is called homothetic if it can be written in the form y g ( f ( x)) where
1 1
We have: g 1 ( y) f ( x) f (t ( y ) x) f ( .x ) f ( x) 1 due to f(x) is
f ( x) f ( x)
homogeneous of degree 1.
E 2.12
Let F(z) be a monotonic increasing function of the single variable z. Form the composite
function (or “transform”), F(f(x)). Show that x* is a local maximum (minimum) of f(x), if and
only if, x* is a local maximum (minimum) of F(f(x)).
F ( f ( x))
Because F(z) is a monotonic increasing function of single variable z 0 for all
f ( x)
f(x).
Consider:
E 2.13
Suppose that f(x) is a concave function and M is the set of all points in R n which give global
maxima of f. Prove that M is a convex set.
E 2.21
Find the local extreme values and classify the stationary points as maxima, minima, or
neither:
f1 2 2 x1 0 x1 1
f 2 2 x2 0 x 2 0
2 0
Hessian Matrix: H
0 2
f1 3x12 0 x1 0
f 2 2 x2 2 0 x2 1
6 x 0
The Hessian Matrix: H 1
0 2
0 0
At the critical point, the Hessian matrix is: H
0 2
D1=0, D2=0
f1 3x1 6 x 2 0
2
f 2 6 x1 3x 2 0
2
6 x 6
The Hessian matrix: H 1
6 6 x2
0 6
At the point (0,0), H D1=0; D2=-36 The point (0,0) is not neither
6 0
maxima or minima.
12 6
At the point (2,2), H D1=12>0; D2>0 the point (2,2) is minima.
6 12
E 2.22
L1 2 x1 x 2 0
L2 2 x 2 x1 0 the critical points: (1,1,-2) and (-1,-1,-2)
L x x 1 0
1 2
2 2 1
At the point (1,1,-2), H 2 2 1 , D3=0
1 1 0
2 2 1
At the point (-1,-1,-2), H 2 2 1 , D3=0
1 1 0
L1 x 2 2x1 0
L2 x1 2x 2 0
L x1 x 2 1 0
2 2
2 1 2 x1
The border Hessian matrix: H 1 2 2 x 2
2 x1 2 x2 0