Economics at Zero Marginal Cost

 

Integration by Parts  The formula for integration by parts is: The integral of 'v' with respect to 'u' is equal to 'uv' less the integral of 'u' with respect to 'v'. Guidelines for Choosing u and dv When deciding which part of the integrand should be u and which should be dv, follow the LIATE rule. This is a priority list: L – Logarithmic functions I – Inverse trigonometric functions A – Algebraic functions (polynomials, powers of xxx) T – Trigonometric functions E – Exponential functions How to Apply the LIATE Rule • Look at the product of functions in your integral. • The function that appears first in the LIATE list becomes u. •   The remaining part of the integrand becomes dv. Example Find: Here, we have: •  x³   → Algebraic function (A in LIATE) • log x → Logarithmic function (L in LIATE) Since L comes before A in the LIATE list, we choose: u = log x   and dv =  x³  dx We then proceed with the integration by parts formula.

Producer’s Surplus (PS) refers to the difference between the price a producer actually receives for a good or service and the minimum amount they would be willing to accept to produce that good or service. In essence: Producer’s Surplus = Actual Revenue – Minimum Acceptable Revenue It is a measure of producer welfare or economic benefit, representing the extra earnings producers receive above their marginal cost of production. Graphical Representation Producer’s surplus is visually represented in a supply and demand diagram: • It is the area above the supply curve (which reflects marginal cost) • And below the equilibrium price line • From zero to the quantity sold Illustration: • The market equilibrium price (P)* is determined by the intersection of the demand and supply curves. • The area between the supply curve and the horizontal line at Pe  (up to the equilibrium quantity Qe) is the producer surplus. As the market price increases, the producer surplus increases, because: • M...

Consumer’s Surplus refers to the difference between the maximum price a consumer is willing to pay for a good or service and the actual price they pay in the market. In essence:  Consumer’s Surplus = Total Willingness to Pay − Actual Expenditure  It is a measure of consumer welfare or economic benefit, representing the extra satisfaction (monetary value) consumers receive over and above what they actually pay. Graphical Representation Consumer’s surplus is visually represented in a demand and supply diagram: • It is the area below the demand curve (which reflects willingness to pay) • And above the equilibrium price line • From zero to the quantity purchased Illustration: • The market equilibrium price (Pe​) is determined by the intersection of the demand and supply curves. • The area between the demand curve and the horizontal line at Pe​ (up to equilibrium quantity Qe​) is the consumer surplus. Effect of Price Changes: • As the market price decreases, consumer surplus increa...

Equation An equation is a statement that shows two quantities are equal. It uses the equal sign === to indicate equality between the left-hand side (LHS) and right-hand side (RHS). Form : LHS = RHS   Example : 5 x = 15  This means 5 × x is exactly equal to 15.  An equation is true only for certain values of the variable(s).  In the example above, it is true only when x = 3. Identity An identity is a special type of equation that is true for all possible values of the variable(s). It expresses a universal relationship between quantities. Form: LHS ≡ RHS (The symbol ≡ means “identically equal to.”) Example: No matter what a and b are, this equality always holds. Key Difference between Equation and Identity: ○ Equation : True for some values. ○ Identity : True for all values. Types of Equations (Common in Economics & Mathematics) 1. Definitional Equation It states a definition in mathematical form. Both sides have exactly the same meaning — one is just a dif...

Q. What Are Systems of Simultaneous Linear Equations? A system of simultaneous linear equations is a collection of two or more linear equations involving multiple variables, where the equations are solved together because the variables are interdependent. Each equation represents a relationship between variables, and the solution is a set of values that satisfies all equations simultaneously. Its uses in Economics In economics, many variables are interrelated. For instance: • Consumption depends on income, • Income depends on investment, • Investment may depend on interest rates. These relationships are modelled using simultaneous equations to understand how changes in one variable affect others within an economic system. ----------------------------------------------------------------------------- NP1: Solve the given system of  simultaneous equations: 2x + 3y = 12  x + y = 5 Solution: From (2): x = 5 − y  Substitute into (1): 2(5 − y) + 3y = 12 ⇒10 − 2y + 3y = 12 ⇒ y = ...

A system can be said to be in equilibrium when the various important variables in it show no change, and  when there are no pressures or forces working which will cause any change in the values of important  variables.  By consumer’s equilibrium, we mean that regarding the allocation of money expenditures among various  goods, the consumer has reached the state where he has no tendency to re-allocate his money expenditure.  A firm is said to be in equilibrium when it has no tendency to change its level of output, that is, when it has no  tendency either to increase or to contract its level of production.  Equilibrium in economic activities may never be realized in actual practice. But the importance of the  equilibrium analysis lies in the fact that if other things remain the same, the economy would tend towards the  equilibrium values. What happens is that before the final equilibrium is reached changes occur in the  determining factors...

Introduction: A function defines a relationship between an independent variable (input) and a dependent variable (output). Different types of functions model different types of relationships in real-world phenomena, such as cost curves, production functions, and growth models. Types of Functions: 1. Constant Functions A constant function is a function whose output value remains the same, regardless of the input value. It can be expressed as: f(x) = c  Where c is a constant real number. Example: f(x)=7  This means that for any value of x, f(x) will always equal 7. Graphical Representation: In the Cartesian coordinate system, a constant function is represented as a horizontal straight line parallel to the x-axis. 2. Polynomial Functions A polynomial function is an algebraic expression involving a sum of powers of the variable x, each multiplied by a coefficient. The general form is: Linear Function: • Form: f(x)=a + bx • Graph: Straight line with slope b and y-intercept a Us...

Introduction In mathematics, the concept of relation and function helps us understand how two sets of objects are connected through specific rules or correspondences. These are fundamental ideas in algebra and calculus, widely used in computer science, economics, and data analysis. Relations A relation R from a non-empty set A to a non-empty set B is a subset of the Cartesian product A × B. That means a relation is any collection of ordered pairs (a, b), where a ∈ A and b ∈ B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B. The set of all first elements in a relation R, is called the domain of the relation R, and the set of all second elements called images, is called the range of R. Mathematically: If R ⊆ A×B, then R is a relation from A to B. Example : Let: • Set A = {1,4,5}  • Set B = {2,3}  Let R= {(1,2), (4,3), (5,3)} • Domain of R = Set of all first elements = {1,4,5}  • Range of R = ...

What is the Difference between Indefinite and Definite Integrals? Indefinite Integrals:  If the function F(x) is an anti-derivative of f(x), then the expression F(x) + C, where C is an arbitrary constant, is called the indefinite integral of 𝑓(𝑥) with respect to x and is denoted by  ∫ 𝑓(𝑥) 𝑑𝑥 , i.e.,  ∫𝑓(𝑥)𝑑𝑥 = F(x) + C . The function  f(x)  is called the integrand, ʃ the integral sign, x is called the variable of integration, and C is the constant of integration. Definite Integrals:  A definite integral is simply an indefinite integral, but with numbers written to the upper and lower right of the integral sign. A definite integral is usually a number. We define the indefinite integral of the function 𝑓(𝑥) with respect to 𝑥 from 𝑎 𝑡𝑜 𝑏 to be: Where, 𝐹(𝑥) is the anti-derivative of 𝑓(𝑥). We call 𝑎 𝑎𝑛𝑑 𝑏 the lower and upper limits of integration, respectively. The function being integrated, 𝑓(𝑥), is called the integrand. Note that i...