Week 1: Conceptual Framework of Microeconomic theory (Malivaud, September Chapter 6, 20151) / Mathemat 1 / 25. (Jehle and Reny, Chapter A1)

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1 Week 1: Conceptual Framework of Microeconomic theory (Malivaud, Chapter 1) / Mathematic Tools (Jehle and Reny, Chapter A1) Tsun-Feng Chiang *School of Economics, Henan University, Kaifeng, China September 6, 2015 Week 1: Conceptual Framework of Microeconomic theory (Malivaud, September Chapter 6, 20151) / Mathemat 1 / 25

2 1.1 Object of the Theory (Malivaud, Chapter 1) Definition of Economics By Malivaud, economics is the science which studies how scare resources are employed for the satisfaction the needs of men living in society: on the one hand, it is interested in the essential operations of production, distribution and consumption of goods, and on the other hand, in the institutions and activities whose object it is to facilitate these operations. The central role of microeconomics is price, which regulates the exchange of goods among agents. For the individual, the price reflects more or less exactly the social scarcity of the product which he buys and sells. That is why microeconomics is also called price theory. In this class, a model is a formal (and simple) represntation of the society. Our object is to find how the society attains the largest level of satisfaction, or efficiency. Week 1: Conceptual Framework of Microeconomic theory (Malivaud, September Chapter 6, 20151) / Mathemat 2 / 25

3 1.2 Elements of A Model: Goods, Agents, and Economy Goods (Commodities) The concept of goods is broad. It includes something that can be measured in an appropriate unit, such as car, pen, bag, and something abstract, like service. Labor time is also a kind of good used in production. Let h (h = 1, 2,, l) be the identity of each commodity, and for the hth commodity, there is an associated unit of quantity z h. A complex (bundle) of commodities can be representated by a vector z = [ z 1 z 2 z 3 z l ] We say, for example, that the price of the hth commodity is p h. We can also define a price vector for all goods. p = [ p 1 p 2 p 3 p l ] Week 1: Conceptual Framework of Microeconomic theory (Malivaud, September Chapter 6, 20151) / Mathemat 3 / 25

4 Goods (Commodities) Continued Therefore, the monetary value of a bundle z is p z = lh=1 p hz h Suppose p h z h = p 1 z 1. Let z h = 1, then z 1 = p h /p 1 (the relative price), which means how the quantity of good 1 must be given in exchange for one unit of h. Agents In most cases, the agents are divided into two categories: producers who transform certain goods into other goods, and consumers who use certain goods for their own needs. The former are also called firms, and the latter may represent either individuals or households. For each consumer i (i = 1, 2,, m), x i is the consumption bundle with l components; For each producer j (i = 1, 2,, n), y j is the production bundle with l components. Week 1: Conceptual Framework of Microeconomic theory (Malivaud, September Chapter 6, 20151) / Mathemat 4 / 25

5 A Prior (Endowment) The society has at its disposal certain quantities ω h of the different goods. These are the initial resources, also called endowmnet. In some models, endowment is given in the case for simplicity. Economy An economy is defined by a list of goods, a list of consumers, a list of producers, and initial rescoure ω = [ω 1, ω 2,, ω l ]. A state of economy is then defined when particular values are given for the m vectors x i and the n vectors y i. In optimum theory, we want to find how x i and y i are determined; in equilibrium theory, we want to see how price for x i and y i are determined. Before we start studying price theory, review mathematic tools for economics. Week 1: Conceptual Framework of Microeconomic theory (Malivaud, September Chapter 6, 20151) / Mathemat 5 / 25

6 A1.2 Elements of Set Theory A set is any collections of elements. For any set S, we write s S to indicate that s is a member of set S, and s / S to indicate that s is not in 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 x is a positive even integer greater than zero and less than 10}. The most 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 every element of S is also an element of T, e.g., S = {2, 4, 6, 8} and T = {1, 2, 4, 5, 6, 8, 10}. A set S is empty or is an empty set if it contains no elements at all. For example, if A = {x x 2 = 0 and x > 1}, then A is empty. We denote the empty set by the symbol and A =. Week 1: Conceptual Framework of Microeconomic theory (Malivaud, September Chapter 6, 20151) / Mathemat 6 / 25

7 Givens two sets A and B, new sets can be formed through set operations: A B, or A union B, is the set of all elements are either in A or in B (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 common to 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 A that are not in B: A B = {x x A and x / B} If it is clear that all sets under discussion are subsets of some universal set U, U A is often written as A c, and called the complement of A in U, e.g., U = {2, 4, 6, 8}, A = {4, 6}, then A c = {2, 8}. Week 1: Conceptual Framework of Microeconomic theory (Malivaud, September Chapter 6, 20151) / Mathemat 7 / 25

8 The product of two sets S and T is the set of "ordered pairs" in the form (s, t), where the first element in the pair is a member of S and the 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 = {(x 1, x 2 ) x 1 R, x 2 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 R 2. More generally, n-dimensional Euclidean space is defined as the set product, R n = R R R = {(x 1, x 2,, x n ) x i R, i = 1, 2,, n} We usually denote (x 1, x 2,, x n ), or vectors in R with boldface type, so that x = (x 1, x 2,, x n ). We use the notation x 0 to indicate each component x i is greater or equal to zero; the notation x 0 indicates every component x i is strictly positive. Week 1: Conceptual Framework of Microeconomic theory (Malivaud, September Chapter 6, 20151) / Mathemat 8 / 25

9 Definition A1.1 Convex Sets in R n S R n is a convex set if for all x 1 S and x 2 S, we have for all t in the interval 0 t 1. tx 1 + (1 t)x 2 S The definition says that a set is convex if any points in the set, all weighted averages of those two points are also points in the same set. The kind of weighted average used in the definition is called a convex combination. For visual examples (see the next slide), first consider the case of R, let a point z be a convex combination of point x 1 and x 2, or z = tx 1 + (1 t)x 2, then z is between x 1 and x 2. The basic ideas carry over to sets of points in two dimensions as well. Week 1: Conceptual Framework of Microeconomic theory (Malivaud, September Chapter 6, 20151) / Mathemat 9 / 25

10 Figure A1.3: Convex Combination in R Week 1: Conceptual Framework of Microeconomic theory September (Malivaud, Chapter 6, ) / Mathemat 10 / 25

11 Consider the two vectors in R 2, denoted x 1 = (x 1 1, x 1 2 ) and x 1 = (x 2 1, x 2 2 ). The convex combination of x1 and x 2 will be z = tx 1 + (1 t)x 2. The point z will lie in that same proportion t of the distance between x 1 and x 2 along the chord connecting them. Figure A1.4: Some convex combinations in R 2 Week 1: Conceptual Framework of Microeconomic theory September (Malivaud, Chapter 6, ) / Mathemat 11 / 25

12 The definition of a convex set refers not just to some convex combinations of two points, but to all convex combinations of those points, i.e. different values of t, x 1 and x 2. We therefore have a very simple and intuitive rule defining convex sets: A set is convex iff we can connect any two points in the set by a straight line that lies entirely within the set. We say the convex sets are all "nice behaved." They have no holes, no breaks, and no awkward curvatures on their boundaries. Figure A1.5: Convex and nonconvex sets in R 2 Week 1: Conceptual Framework of Microeconomic theory September (Malivaud, Chapter 6, ) / Mathemat 12 / 25

13 Theorem A1.1 The Intersection of Convex Sets is Convex Let S and T be convex set in R n. Then S T is convex set. Let any ordered pair (s, t) associate an element s S to an element t T. Any collection of ordered pairs is said to constitute a binary relation, denoted as R, between sets S and T. For example, S = {1, 3, 5}, T = {1, 5, 10}, the statement that defined a R is both numbers 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 R by (s, t) R. For the last case, (1, 1) R and (5, 5) R. Week 1: Conceptual Framework of Microeconomic theory September (Malivaud, Chapter 6, ) / Mathemat 13 / 25

14 Definition A1.2 Completeness A relation R is completeness if, for all elements x and y in S, (x, y) R or (y, x) R. Definition A1.3 Transitivity A relation R is transitive if, for any three elements x, y and z in S, (x, y) R and (y, z) R implies (x, z) R. For example, there is a set S = {a, b, c, d, e, f, g}, the relation R is "at least as great as", (a, c) R and (c, g) R implies that "a is at least as great as g." A function is a relation that associates each element of one set with a single, unique element of another set. We say that the function f is a mapping from one set D to another set R and write f : D R. We callec the set D the domain and the set R the range of the mapping. Week 1: Conceptual Framework of Microeconomic theory September (Malivaud, Chapter 6, ) / Mathemat 14 / 25

15 Figure A1.7: (b) Functions and (a) nonfunctions Week 1: Conceptual Framework of Microeconomic theory September (Malivaud, Chapter 6, ) / Mathemat 15 / 25

16 The image of f is that set of points in the range into which some point in the domain is mapped, i.e. I {y y = f (x), for somex D}. The inverse image of a set of points S I is defined as f 1 (S) {x x D, f (x) S}. Figure A1.8: Domain, Range, and image Week 1: Conceptual Framework of Microeconomic theory September (Malivaud, Chapter 6, ) / Mathemat 16 / 25

17 If every point in the range is assign to at most a single point in the domain, the function is called one-to-one. For example. f (x) = x + 1 is a one-to-one function, and f (x) = x is not (why?). If for each element x D, there is an element y R such that x = f 1 (y), we say f is an onto function. For example, f (x) = 4x 1 is an onto function where f : R R; f (x) = x 2 3 is not an onto function where f : R R (why?) If a function f is one-to-one and onto, then an inverse function f 1 : R D exists that is also one-to-one and onto. Week 1: Conceptual Framework of Microeconomic theory September (Malivaud, Chapter 6, ) / Mathemat 17 / 25

18 A1.3 A Little Topology A metric is a measure of distance. A metric space is a set with a notion of distance defined among the points within the set. For example, (i) the set R, together with (ii) an appropriate function measuring distance, is a metric space. One such distance function, or metric, is just the absolute-value function. For any two points x 1 and x 2 in R, the distance between them, denoted d(x 1, x 2 ), is given by d(x 1, x 2 ) = x 1 x 2 The Cartesian plane, R 2, is also a metric space with the commonly used distance function d(x 1, x 2 ) = (x1 2 x 1 1)2 + (x2 2 x 2 1)2 where x 1 = (x1 1, x 2 1) and x2 = (x1 2, x 2 2) are in R2. Similarly, for the general case of R n, the distance function is usually d(x 1, x 2 ) = (x1 2 x 1 1)2 + (x2 2 x 2 1)2 + + (xn 2 xn 1 ) 2 Week 1: Conceptual Framework of Microeconomic theory September (Malivaud, Chapter 6, ) / Mathemat 18 / 25

19 Once we have a metric, we can make precise what it means for points to be "near" each other. Let a distance ɛ > 0, then Definition A1.4 Open and Closed ɛ-balls 1. The open ɛ-ball with center x 0 and radius ɛ > 0 (a real number) is the subsets of points in R n : B ɛ (x 0 ) {x R n d(x 0, x) < ɛ} 2. The close ɛ-ball with center x 0 and radius ɛ > 0 is the subsets of points in R n : B ɛ (x 0 ) {x R n d(x 0, x) ɛ} In R, the open ball with center x 0 and radius ɛ is just the open interval B ɛ (x 0 ) = (x 0 ɛ, x 0 + ɛ). The corresponding ball is the close interval B ɛ (x 0 ) = [x 0 ɛ, x 0 + ɛ]. In R 2 (see the next slide for the visual examples), an open ball B ɛ (x 0 ) is a disk consisting of the set of points inside, or on the interior of the circle of radius ɛ around the point x 0. Week 1: Conceptual Framework of Microeconomic theory September (Malivaud, Chapter 6, ) / Mathemat 19 / 25

20 The corresponding ball in the plane, B ɛ (x 0 ), is the set of points inside and on the edge of the circle. Figure A1.10: Balls in R and R 2 Week 1: Conceptual Framework of Microeconomic theory September (Malivaud, Chapter 6, ) / Mathemat 20 / 25

21 Definition A1.5 Open Sets in R n S R n is an open set if, for all x S, there exists some ɛ > 0 such that B ɛ (x) S. If a set is open if around any point in it we can draw some open ball, no matter how small its radius may have to be, so that all the points in that ball will lie entirely in the set. Week 1: Conceptual Framework of Microeconomic theory September (Malivaud, Chapter 6, ) / Mathemat 21 / 25

22 From the definition of open sets, it is immediate to have the following theorem, Theorem A1.2 On Open Sets in R n 1 The empty set,, is an open set. 2 The entire space, R n, 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 any x i S i, where i S i is the union of open sets for all i {1, 2, 3, }. Then it must be that x S i for some i {1, 2, 3, }. Because S i is open, B ɛ (x) S i for some ɛ > 0. Consequently, B ɛ (x) i S i, which shows that i S i is an open set. For the fourth one, let x i S i, which implies x S i for all i {1, 2,, n}. Since every S i is open, B ɛ (x) S i for some ɛ > 0. Therefore, B ɛ (x) i S i, which shows that i S i is an open set. Week 1: Conceptual Framework of Microeconomic theory September (Malivaud, Chapter 6, ) / Mathemat 22 / 25

23 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 Balls Let S be an open set. For every x S, choose some ɛ x > 0 such that B ɛx (x) S. Then, S = x S B ɛx (x) Figure A1.11: An open ball is an open set Week 1: Conceptual Framework of Microeconomic theory September (Malivaud, Chapter 6, ) / Mathemat 23 / 25

24 We use open sets to define closed sets. Definition A1.6 Closed Sets in R n S is a closed set if its complement, S c, is an open set. Loosely speaking, a set in R n is open if it does not contain any of the points on its boundary, and is closed if it contains all of the points on its boundary (a point x is called a boundary point of a set S in R n if every ɛ-ball centered at x contains points in S as well as points not in S). For example, consider the two sets A = {x x R n, < x < a} and B = {x x R n, b < x < }. By Theorem A1.2, A B = {x x R n, < x < a or b < x < } is an open set. Then the complement of A B, [a, b], is a closed set where a and b is the boundary 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 Week 1: Conceptual Framework of Microeconomic theory September (Malivaud, Chapter 6, ) / Mathemat 24 / 25

25 A point x S is called an interior point of S if there is some ɛ-ball centered at x that is entirely contained within S. The set of all interior points of a set S is called its interior and is denoted int S. Looking at things this way, we can see that a set is open if it contains nothing but interior points, or if S = int S. By contrast, a set is closed if it contains all its interior points plus all its boundary points, or if S = int S S. Similar to open sets, closed sets also have properties corresponding to Theorem A1.2, Theorem A1.4 On Closed Sets in R n 1 The empty set,, is a closed set. 2 The entire space, R n, 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 September (Malivaud, Chapter 6, ) / Mathemat 25 / 25