ECN215

• Nature of mathematical economics:

  • Approach to analyzing economic phenomena using mathematical symbols, problem statements, hypothesizing, and drawing inferences from mathematical theorems
  • Mathematical Economics is a means to an end, not an end in itself
• Difference between Mathematical Economics and Nonmathematical Economics:

  • In Mathematical Economics, assumptions and deductions are stated in mathematical symbols instead of words
  • Equations are used instead of sentences
• Difference between Mathematical Economics and Econometrics:

  • Econometrics deals with empirical observations using statistical methods
  • Mathematical Economics deals with theoretical aspects of economic analysis and little to no statistical problems
• Components of Economic Models:

  • An economic model is an abstraction from real-world phenomena
  • Helps understand interrelationships of economic situations or variables
  • Consists of equations, variables, assumptions, and conclusions
• Variables in Economic Models:

  • Variables can change in magnitude or numerical value
  • Common variables in economics include price, profit, revenue, cost, national income, consumption, investment, imports, and exports
• Endogenous/exogenous Variables:

  • Endogenous variables are determined within the model
  • Exogenous variables are assumed or determined outside the model
• Constant and Parameter:

  • Constant is a magnitude that does not change
  • Parameter is a constant that is a variable without a specific value, related to exogenous variables
• Equations and Identities:

  • Equations are algebraic statements where two expressions are equal
  • Types of equations include definitional, behavioral, and equilibrium conditions
• Functions and Relation:

  • A function is a mathematical relationship between variables where the values of one are determined by another
  • Functions and relations denote relationships between variables in mathematical models
• A relation may have multiple values of y for a given x, while a function has only one corresponding y value for each x value
• A function uniquely determines a y value for every x value
• A function must be a relation, but a relation may not be a function
• In a relation, it is not always possible to determine a unique value of y for a given x
• An ordered pair associates a y-value with an x-value, forming a relation between y and x
• A relation is a function, but a function may not be a relation
• Mapping (or Transformation) of a function denotes the action of one variable with another
• In a function, x is the argument of the function, and y is the value of the function
• The domain of a function is the set of all permissible values that x can take
• The range of a function is the set of all values that the y-variable can take
• Utility function is the core of consumer theory, focusing on deriving satisfaction or utility from consuming goods
• Production function shows the technological relationship between inputs and outputs of a firm
• Types of functions in economics include Polynomial, Single-valued, Multi-valued, Increasing, Decreasing, Non-increasing, Non-decreasing, Constant, Step, Continuous, Discontinuous, Discrete, Kinked, Logarithmic, Exponential, Homogeneous, Rational, and Non-algebraic functions
• Equilibrium in Economics is a state of balance between the laws of Demand and Supply, where the quantity of goods and services bought equals the quantity supplied
• Equilibrium analysis aims to determine equilibrium values for a given market
• Equilibrium is a state of balance where no inherent tendency to change prevails in the model's variables
• Equilibrium analysis involves static equilibrium, where variables are at rest, and dynamic equilibrium, where changes in equilibrium positions are considered
• Partial Market Equilibrium involves finding equilibrium values in isolated markets, considering only one commodity
• In a Partial Market Equilibrium model, the variables are the amount of the commodity demanded, the amount supplied, and the price of the commodity
• A Linear Model in Partial Market Equilibrium considers one commodity and gives a straight line when plotted on a graph
• Equilibrium solution for a market is when the quantity demanded (Qd) equals the quantity supplied (Qs)
• For the given Supply and Demand functions:

  • Qs = -15 + 2.5P
  • Qd = 25 - P
• To find the equilibrium solution, set Qs equal to Qd and solve for P
• The equilibrium Price (P) is either -5 or 1
• Substitute the equilibrium Price back into the Demand function to find the Equilibrium Quantity (Q)
• The Equilibrium Quantity (Q) is either -21 or 3
• Choose the positive values for Price and Quantity to make economic sense
• Therefore, the Equilibrium Price is 1, while the Equilibrium Quantity is 3 units
• Properties of a limit:

  • Lim b = b
  • Lim x = c
  • Lim x^n = c^n
  • Lim √x = √c
• Laws/Rules of limits operation:

  • Constant multiple rule: lim [b(f)] = b[lim f(x)]
  • Additive rule: lim [f(x)+g(x)] = lim f(x) + lim g(x)
  • Multiplicative rule: lim [f(x) * g(x)] = [lim f(x)] * [lim g(x)]
  • Division/quotient rule: lim f(x)/g(x) = lim f(x)/lim g(x)
  • Power/index rule: Lim [f(x)]^n = [lim f(x)]^n
  • Radical rule: Lim n√f(x) = n√lim f(x)
• Direct Substitution:

  • If lim = f(x), the limit can be evaluated by direct substitution
  • A function that has this property is called continuous at c