Economics at Zero Marginal Cost

NP1: A research team interviewed 100 people and recorded their status regarding three conditions: I = Idle P = Poor W = Willing to work The results were as follows: Category Number of People Idle (I) 28 Poor (P) 33 Willing to work (W) 14 Idle and Poor (I ∩ P) 12 Idle and Willing to work (I ∩ W) 9 Poor and Willing to work (P ∩ W) 6 Idle, Poor, and Willing (I ∩ P ∩ W)    5 1. Check whether the data are consistent. 2. Find how many people are either idle, poor, or willing to work,  n(I ∪ P ∪ W). Solution : We use the inclusion-exclusion principle for 3 sets: n(I ∪ P ∪ W) = n(I) + n(P) + n(W) − n(I ∩ P) − n(I ∩ W) − n(P ∩ W) + n(I ∩ P ∩ W)  n(I ∪ P ∪ W) = 28 + 33 + 14 − 12 − 9 − 6 + 5 n(I ∪ P ∪ W) = 75 − 27 + 5=53  n(I ∪ P ∪ W) =  53  So, only 53 people are in I ∪ P ∪ W. That means: People not in any group, n(I'  ∩  P'  ∩  W')  = 100 − 53 = 47   ∵ The union count is ≤ 100 and all overlaps are consistent with the in...

NP1: Sets A and B are such that A has 25 members, B has 20 members and A ∪ B has 35 members. Draw a Venn diagram to represent the above situation. Find the number of elements in the intersection of sets A and B, i.e., A ∩ B  Solution: Given: n(A)=25 → number of elements in set A n(B)=20 → number of elements in set B n(A ∪ B) = 35 → number of elements in the union of A and B To find: • n(A ∩ B) ) → number of elements common to both A and B The formula for union of two sets is: n(A ∪ B) = n(A)+ n(B)− n(A ∩ B)  35 = 25 + 20 − n(A ∩ B) 35 = 45 − n(A ∩ B) n( A ∩ B) = 45 − 35 = 10 ∴ The number of elements in the intersection of sets A and B is 10 -------------------------------------------- NP2: In a class of 25 students of Economics and Politics 12 students have taken Economics. Out of these 8 have taken Economics but not Politics. Find the number of students who have taken Economics and Politics and those who have taken Politics but not Economics. Solution: Let, E →...

For Two Sets: Let: • n(A): Number of elements in Set A • n(B): Number of elements in Set B • n(A∪B): Number of elements in the union of A and B • n(A∩B): Number of elements common to both A and B Venn Regions: Region Set Expression Meaning Formula A 1 A ∩ B′  Only A n(A)  −  n( A ∩ B ) A 2 A ∩ B A and B n( A ∩ B ) A 3 B ∩ A′ Only B n(B)  −  n( A ∩ B ) A 4 A′ ∩ B′ Neither A nor B n( Ω)  −  n( A   ∪  B ) Union Formula: n(A ∪ B) = n(A) + n(B) − n(A ∩ B)  • If A and B are disjoint:  n(A ∩ B) = 0   ⇒ n(A ∪ B) = n(A) + n(B)  For Three Sets: Let the sets be A, B, and C. Venn Regions: Region Set Expression Meaning Formula A 1 A ∩ B′ ∩ C′ Only A n(A) − n(A ∩ B) − n(A ∩  C ) + n(A ∩ B ∩ C) A 2 B ∩ A′ ∩ C′ Only B n(B) − n(A ∩ B) − n(B ∩  C) +  n(A ∩ B ∩C...

What is a Set? A set is a group or collection of well-defined and distinct objects. These objects are called the elements or members of the set. Set theory was developed by Georg Cantor, a German mathematician. • A set is usually named using capital letters: A, B, C… • Elements are written in lowercase letters inside curly brackets: {a, b, c} If x is in set A, we write: x ∈ A If y is not in set A, we write: y ∉ A Example: Let A = {a, e, i, o, u} → the set of vowels in English. Ways to Represent a Set 1. Roster (Tabular or Listing) Method: This is a method of describing a set by listing each element of the set inside the symbol { }. In  listing the elements of the set, each distinct elements is listed once and the order of the elements does not matter. Example: Colors of a rainbow: A = {red, orange, yellow, green, blue, indigo, violet} 2. Set-Builder (Rule) Method: It is a method that lists the rules that determine whether an object is an element of the set rather than the actual el...

Mathematical Methods for Economics I Unit – I Basic Concepts : Sets and set operations; Relations and Functions; Types of Functions: quadratic, polynomial, power, exponential, logarithmic, convex, quasi-convex and concave functions; Graphs of functions of one real variable; Equations; Identities; Equilibrium condition; System of Simultaneous Linear Equations; The Straight line and its slope. Set Theory – Concepts and Definitions Number of Elements in a Set Numerical Problems I: Number of Elements in a Set (Two Sets) Numerical Problems II: Number of Elements in a Set (Three Sets) Relations and Functions Types of Functions Finding the Domain and Range of a Function Graphs of functions of one real variable Equations and Identities Equilibrium Conditions - Concept and Numerical Problems System of Simultaneous Linear Equations - Concept and Numerical Problems The Straight line and its slope Unit – II Differential Calculus: Limit and Continuity of a function; Differentiation: Meaning, Rules ...

Microeconomics II Unit II: Theory of Production Unit – II Theory of production: Isoquants and isocost lines; marginal rate of technical substitution; producer’s equilibrium; expansion path; elasticity of factor substitution; economies of scale; concept of producer’s surplus; output elasticity; concept of homogeneous production functions; concept and properties of Cobb-Douglas production function. Introduction to Production  Isoquants in Production Theory Iso-cost Lines and the Role of Input Prices Marginal Rate of Technical Substitution (MRTS) Producer’s Equilibrium Expansion path Elasticity of Substitution (σ) Economies of Scale Producer’s Surplus Output Elasticity Cobb-Douglas Production Function (1928) and Its Properties

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