## Tamilnadu Samacheer Kalvi 11th Economics Notes Chapter 12 Mathematical Methods for Economics Notes

→ Economic analysis is a systematic approach to (a) determine the optimum use of scarce resources and (b)choose available alternatives and select the best alternative to achieve a particular objective.

→ A function is a mathematical relationship in which the values of a dependent variable are determined by the values of one or more independent variables.

→ A statement of relationship between two quantities is called an equation, e.g., Y = 100 – 10X.

→ Slope or Gradient of the line represents the ratio of the changes in vertical and horizontal lines.

→ Demand Function: Q_{d} = f (P_{x}) where Q_{d} stands for Quantity demand of a commodity and P_{x}

is the price of that commodity.

→ Supply Function: Q_{s} = f (P_{x} ) where ‘Q_{s} ’ stands for Quantity supplied of a commodity and P_{x} is the price of that commodity.

→ The point of intersection of demand line and supply line is known as equilibrium

→ ‘Matrix’ is a singular while ‘matrices’ is a plural form. Matrix is a rectangular array of numbers systematically arranged in rows and columns within brackets.

→ In a matrix, if the number of rows and columns are equal, it is called a square matrix.

→ For every square matrix, there exists a determinant. This determinant is an arrangement of same elements of the corresponding matrix into rows and columns by enclosing vertical lines.

→ Cramer’s rule provides the solution of a system of linear equations with ‘n’ variables and ‘n’ equations. It helps to arrive at a unique solution of a system of linear equations with as many equations as unknowns, \(x=\frac{\Delta x}{\Delta}, y=\frac{\Delta y}{\Delta}, z=\frac{\Delta z}{\Delta}\)

→ If the determinant Δ = 0, then solution does not exist.

→ The fundamental operation of calculus is differentiation. Derivative is used to express the rate of change in any function.

→ (Any non-zero real number)° = 1

→ Marginal concept is concerned with variations of Y (on the margin of X), that is; it is the variation corresponding in Y to a very small variation in X. (X is the independent variable and Y is the dependent variable)

→ Marginal product of a factor of production refers to addition to total product due to the use of an additional unit of a factor.

MP = d(TP)/dQ = ΔTP/ΔQ

→ Marginal cost is an addition to the total cost caused by producing one more unit of output. In symbols: MC = \(\frac{d(T C)}{d Q}\)or MC = \(\frac{T C}{Q}\) Where, ΔTC represents a change in total cost and ΔQ represents a small change in output or quantity.

→ Marginal Revenue is the revenue earned by selling an additional unit of the product. In other words, Marginal Revenue is an addition made to the total revenue by selling one more unit of the good.

\(M R=\frac{d(T R)}{d Q} \text { or } M R=\frac{\Delta T R}{\Delta Q}\)

Where ΔTR stands for change in the total revenue, and AQ stands for change in output.

→ Elasticity of Demand is the ratio of the proportionate change in quantity demanded to the proportionate change in price. In mathematical terms,

\(\mathrm{e}_{d}=\left(\frac{p}{x}\right)\left(\frac{d x}{d p}\right)\)

In demand function Q = a – bP. So, e = (dQ/dP)(P/Q)

a. J is used to denote the process of integration. In fact, this symbol is an elongated ‘S’ denoting sum.

b. The differential symbol ‘dx’ is written by the side of the function to be integrated.

c. ∫f(x) dx = F(x) + C, C is the integral constant ∫f(x)dx means, integration of f(x) with respect to x.

→ Consumer’s Surplus theory was developed by the Alfred Marshall. The demand function P(x) reveals the relationship between the quantities that the people would buy at given price. It can be expressed as P =f(x). Mathematically, the consumer’s surplus (CS) can be defined

as \(\mathrm{CS}=\left[\int_{0}^{x_{0}} p(x) d x\right]-\mathrm{x}_{0} \mathrm{p}_{0}\)

→ MS Excel 2007 version supports up to 1 million rows and 16 thousand columns per work sheet.