week 1: conceptual framework of microeconomic theory

Week 1: Conceptual Framework of Microeconomictheory (Malivaud, Chapter 1) / Mathematic Tools

(Jehle and Reny, Chapter A1)

Tsun-Feng Chiang

*School of Economics, Henan University, Kaifeng, China

September 6, 2015

Microeconomic Theory Week 1: Conceptual Framework of Microeconomic theory (Malivaud, Chapter 1) / Mathematic Tools (Jehle and Reny, Chapter A1)September 6, 2015 1 / 25

1.1 Object of the Theory (Malivaud, Chapter 1)

Definition of EconomicsBy Malivaud, economics is the science which studies how scareresources are employed for the satisfaction the needs of men living insociety: on the one hand, it is interested in the essential operations ofproduction, distribution and consumption of goods, and on the otherhand, in the institutions and activities whose object it is to facilitatethese operations.

The central role of microeconomics is price, which regulates theexchange of goods among agents. For the individual, the price reflectsmore or less exactly the social scarcity of the product which he buysand sells. That is why microeconomics is also called price theory.In this class, a model is a formal (and simple) represntation of thesociety. Our object is to find how the society attains the largest level ofsatisfaction, or efficiency.


1.2 Elements of A Model: Goods, Agents, andEconomy

Goods (Commodities)The concept of goods is broad. It includes something that can bemeasured in an appropriate unit, such as car, pen, bag, and somethingabstract, like service. Labor time is also a kind of good used inproduction. Let h (h = 1,2, · · · , l) be the identity of each commodity,and for the hth commodity, there is an associated unit of quantity zh. Acomplex (bundle) of commodities can be representated by a vector

z ′ =[z1 z2 z3 · · · zl

]We say, for example, that the price of the hth commodity is ph. We canalso define a price vector for all goods.

p′ =[p1 p2 p3 · · · pl

]Microeconomic Theory Week 1: Conceptual Framework of Microeconomic theory (Malivaud, Chapter 1) / Mathematic Tools (Jehle and Reny, Chapter A1)September 6, 2015 3 / 25

Goods (Commodities) ContinuedTherefore, the monetary value of a bundle z is

p′z =∑l

h=1 phzh

Suppose phzh = p1z1. Let zh = 1, then z1 = ph/p1 (the relative price),which means how the quantity of good 1 must be given in exchange forone unit of h.

AgentsIn most cases, the agents are divided into two categories: producerswho transform certain goods into other goods, and consumers whouse certain goods for their own needs. The former are also calledfirms, and the latter may represent either individuals or households.For each consumer i (i = 1,2, · · · ,m), xi is the consumption bundlewith l components; For each producer j (i = 1,2, · · · ,n), yj is theproduction bundle with l components.


A Prior (Endowment)The society has at its disposal certain quantities ωh of the differentgoods. These are the initial resources, also called endowmnet. Insome models, endowment is given in the case for simplicity.

EconomyAn economy is defined by a list of goods, a list of consumers, a list ofproducers, and initial rescoure ω′ = [ω1, ω2, · · · , ωl ]. A state ofeconomy is then defined when particular values are given for the mvectors xi and the n vectors yi .

In optimum theory, we want to find how xi and yi are determined; inequilibrium theory, we want to see how price for xi and yi aredetermined.

Before we start studying price theory, review mathematic tools foreconomics.


A1.2 Elements of Set Theory

A set is any collections of elements. For any set S, we write s ∈ S toindicate that s is a member of set S, and s /∈ S to indicate that s is notin the set S. Sets can be defined by enumeration of their elements,e.g., S = {2,4,6,8}, or by description of their elements, e.g., S = {x |xis a positive even integer greater than zero and less than 10}. Themost commonly used set in this course is the set R of all real number,denoted R = {x |−∞ < x <∞}.

A set S is a subset of another set T , where we write S ⊂ T , if everyelement of S is also an element of T , e.g., S = {2,4,6,8} andT = {1,2,4,5,6,8,10}.

A set S is empty or is an empty set if it contains no elements at all. Forexample, if A = {x |x2 = 0 and x > 1}, then A is empty. We denote theempty set by the symbol ∅ and A = ∅.


Givens two sets A and B, new sets can be formed through setoperations:

A ∪ B, or A union B, is the set of all elements are either in A or inB (or in both):

A ∪ B = {x |x ∈ A or x ∈ B}

A ∩ B, or A intersect B, is the set of all elements that are commonto both A and B,

A ∩ B = {x |x ∈ A and x ∈ B}

A− B, or A\B, is called A minus B, is the set of all elements of Athat are not in B:

A− B = {x |x ∈ A and x /∈ B}

If it is clear that all sets under discussion are subsets of someuniversal set U, U − A is often written as Ac , and called thecomplement of A in U, e.g., U = {2,4,6,8}, A = {4,6}, thenAc = {2,8}.


The product of two sets S and T is the set of "ordered pairs" in theform (s, t), where the first element in the pair is a member of S andthe second is a member of T . The product of S and T is denoted

S × T = {(s, t)|s ∈ S, t ∈ T}

The product of two sets of real numbers is called Cartesian plane,

R× R = {(x1, x2)|x1 ∈ R, x2 ∈ R}

then any point in the set can be identified with a point in the plane.The set R× R is also called "two-dimensional Euclidean space"and is commonly denoted R2. More generally, n-dimensionalEuclidean space is defined as the set product,

Rn = R× R · · · × R = {(x1, x2, · · · , xn)|xi ∈ R, i = 1,2, · · · ,n}

We usually denote (x1, x2, · · · , xn), or vectors in R with boldfacetype, so that x = (x1, x2, · · · , xn). We use the notation x ≥ 0 toindicate each component xi is greater or equal to zero; thenotation x� 0 indicates every component xi is strictly positive.


Definition A1.1 Convex Sets in Rn

S ⊂ Rn is a convex set if for all x1 ∈ S and x2 ∈ S, we have

tx1 + (1− t)x2 ∈ S

for all t in the interval 0 ≤ t ≤ 1.

The definition says that a set is convex if any points in the set, allweighted averages of those two points are also points in the same set.The kind of weighted average used in the definition is called a convexcombination. For visual examples (see the next slide), first considerthe case of R, let a point z be a convex combination of point x1 and x2,or z = tx1 + (1− t)x2, then z is between x1 and x2. The basic ideascarry over to sets of points in two dimensions as well.


Figure A1.3: Convex Combination in R


Consider the two vectors in R2, denoted x1 = (x11 , x

12 ) and

x1 = (x21 , x

22 ). The convex combination of x1 and x2 will be

z = tx1 + (1− t)x2. The point z will lie in that same proportion t of thedistance between x1 and x2 along the chord connecting them.Figure A1.4: Some convex combinations in R2


The definition of a convex set refers not just to some convexcombinations of two points, but to all convex combinations of thosepoints, i.e. different values of t , x1 and x2. We therefore have a verysimple and intuitive rule defining convex sets: A set is convex iff wecan connect any two points in the set by a straight line that lies entirelywithin the set. We say the convex sets are all "nice behaved." Theyhave no holes, no breaks, and no awkward curvatures on theirboundaries.Figure A1.5: Convex and nonconvex sets in R2


Theorem A1.1 The Intersection of Convex Sets is ConvexLet S and T be convex set in Rn. Then S ∩ T is convex set.

Let any ordered pair (s, t) associate an element s ∈ S to an elementt ∈ T . Any collection of ordered pairs is said to constitute a binaryrelation, denoted as <, between sets S and T . For example,S = {1,3,5}, T = {1,5,10}, the statement that defined a < is bothnumbers are equal, then for this relation, it contains the element{(1,1), (5,5)}. When s ∈ S bears the specified relationship to t ∈ T ,we denote membership in the relation < by (s, t) ∈ <. For the lastcase, (1,1) ∈ < and (5,5) ∈ <.


Definition A1.2 CompletenessA relation < is completeness if, for all elements x and y in S, (x , y) ∈ <or (y , x) ∈ <.

Definition A1.3 TransitivityA relation < is transitive if, for any three elements x , y and z in S,(x , y) ∈ < and (y , z) ∈ < implies (x , z) ∈ <.

For example, there is a set S = {a,b, c,d ,e, f ,g}, the relation < is "atleast as great as", (a, c) ∈ < and (c,g) ∈ < implies that "a is at leastas great as g."

A function is a relation that associates each element of one set with asingle, unique element of another set. We say that the function f is amapping from one set D to another set R and write f : D → R. Wecallec the set D the domain and the set R the range of the mapping.


Figure A1.7: (b) Functions and (a) nonfunctions


The image of f is that set of points in the range into which some pointin the domain is mapped, i.e. I ≡ {y |y = f (x), for somex ∈ D}. Theinverse image of a set of points S ⊂ I is defined asf−1(S) ≡ {x |x ∈ D, f (x) ∈ S}.

Figure A1.8: Domain, Range, and image


If every point in the range is assign to at most a single point in thedomain, the function is called one-to-one. For example. f (x) = x + 1is a one-to-one function, and f (x) = x2 + 1 is not (why?). If for eachelement x ∈ D, there is an element y ∈ R such that x = f−1(y), we sayf is an onto function. For example, f (x) = 4x − 1 is an onto functionwhere f : R→ R; f (x) = x2 − 3 is not an onto function where f : R→ R(why?) If a function f is one-to-one and onto, then an inverse functionf−1 : R → D exists that is also one-to-one and onto.


A1.3 A Little Topology

A metric is a measure of distance. A metric space is a set with anotion of distance defined among the points within the set. Forexample, (i) the set R, together with (ii) an appropriate functionmeasuring distance, is a metric space. One such distance function, ormetric, is just the absolute-value function. For any two points x1 and x2

in R, the distance between them, denoted d(x1, x2), is given by

d(x1, x2) = |x1 − x2|

The Cartesian plane, R2, is also a metric space with the commonlyused distance function

d(x1,x2) =√

(x21 − x1

1 )2 + (x2

2 − x12 )

2

where x1 = (x11 , x

12 ) and x2 = (x2

1 , x22 ) are in R2. Similarly, for the

general case of Rn, the distance function is usually

d(x1,x2) =√

(x21 − x1

1 )2 + (x2

2 − x12 )

2 + · · ·+ (x2n − x1

n )2


Once we have a metric, we can make precise what it means for pointsto be "near" each other. Let a distance ε > 0, then

Definition A1.4 Open and Closed ε-Balls

1. The open ε-ball with center x0 and radius ε > 0 (a real number) isthe subsets of points in Rn:

Bε(x0) ≡ {x ∈ Rn|d(x0,x) < ε}

2. The close ε-ball with center x0 and radius ε > 0 is the subsets ofpoints in Rn:

B∗ε (x0) ≡ {x ∈ Rn|d(x0,x) ≤ ε}

In R, the open ball with center x0 and radius ε is just the open intervalBε(x0) = (x0 − ε, x0 + ε). The corresponding ball is the close intervalB∗ε (x0) = [x0 − ε, x0 + ε]. In R2 (see the next slide for the visualexamples), an open ball Bε(x0) is a disk consisting of the set of pointsinside, or on the interior of the circle of radius ε around the point x0.


The corresponding ball in the plane, B∗ε (x0), is the set of points insideand on the edge of the circle.

Figure A1.10: Balls in R and R2


Definition A1.5 Open Sets in Rn

S ⊂ Rn is an open set if, for all x ∈ S, there exists some ε > 0 suchthat Bε(x) ⊂ S.

If a set is open if around any point in it we can draw some open ball, nomatter how small its radius may have to be, so that all the points in thatball will lie entirely in the set.


From the definition of open sets, it is immediate to have the followingtheorem,

Theorem A1.2 On Open Sets in Rn

1 The empty set, ∅, is an open set.2 The entire space, Rn, is an open set.3 The union of open sets is an open set.4 The intersection of any finite number of open sets is an open set.

Proof: The first and the second are trivial. For the third one, let anyx ∈ ∪iSi , where ∪iSi is the union of open sets for all i ∈ {1,2,3, · · ·}.Then it must be that x ∈ Si ′ for some i ′ ∈ {1,2,3, · · ·}. Because Si ′ isopen, Bε(x) ⊂ Si ′ for some ε > 0. Consequently, Bε(x) ⊂ ∪iSi , whichshows that ∪iSi is an open set. For the fourth one, let x ∈ ∩iSi , whichimplies x ∈ Si for all i ∈ {1,2, · · · ,n}. Since every Si is open,Bε(x) ⊂ Si for some ε > 0. Therefore, Bε(x) ⊂ ∩iSi , which shows that∩iSi is an open set.


An open set set can be described as a collection of different open sets.Since an open ball is itself an open set, there is an interesting theorem

Theorem A1.3 Every Open Set is a Collection of Open BallsLet S be an open set. For every x ∈ S, choose some εx > 0 such thatBεx(x) ⊂ S. Then,

S = ∪x∈SBεx(x)

Figure A1.11: An open ball is an open set


We use open sets to define closed sets.

Definition A1.6 Closed Sets in Rn

S is a closed set if its complement, Sc , is an open set.

Loosely speaking, a set in Rn is open if it does not contain any of thepoints on its boundary, and is closed if it contains all of the points on itsboundary (a point x is called a boundary point of a set S in Rn ifevery ε-ball centered at x contains points in S as well as points not inS). For example, consider the two sets A = {x |x ∈ Rn,−∞ < x < a}and B = {x |x ∈ Rn,b < x <∞}. By Theorem A1.2,A ∪ B = {x |x ∈ Rn,−∞ < x < a or b < x <∞} is an open set. Thenthe complement of A ∪ B, [a,b], is a closed set where a and b is theboundary points. The set of all boundary points of a set S is denoted∂S. For last case, ∂(A ∪ B)c = {a,b}.Figure A1.12: A closed interval is a closed set


A point x ∈ S is called an interior point of S if there is some ε-ballcentered at x that is entirely contained within S. The set of all interiorpoints of a set S is called its interior and is denoted int S. Looking atthings this way, we can see that a set is open if it contains nothing butinterior points, or if S = int S. By contrast, a set is closed if it containsall its interior points plus all its boundary points, or if S = int S ∪ ∂S.

Similar to open sets, closed sets also have properties corresponding toTheorem A1.2,

Theorem A1.4 On Closed Sets in Rn

1 The empty set, ∅, is a closed set.2 The entire space, Rn, is a closed set.3 The union of any fininte collection of closed sets is a closed set.4 The intersection of closed sets is a closed set.


week 1: conceptual framework of microeconomic theory

Documents