Dr. Hisham Abdelbaki Managerial Economics 1 ECON 340 Review of Mathematical Concepts

Dr. Hisham Abdelbaki Managerial Economics 2 Types of DATA Time series Show the values of an economic variable over period of time National income in Bahrain during the period 1995 – 2002 A consumer’s consumption for pizza during a week Gross section Show the values of an economic variable for different groups at a point in time National income for Arab countries in 2002 Households’ consumption of pizza on last Saturday. BOTH

Dr. Hisham Abdelbaki Managerial Economics 3 Time series : Q d for Ashraf’s family for orange during a week Day Quantity demanded (Kg) Saturday 5 Sunday 7 Monday 3 Tuesday 9 Wednesday 2 Thursday 6 Friday 9

Dr. Hisham Abdelbaki Managerial Economics 4 Gross section: Q d for orange for households in Manama on Saturday HouseholdsQd HH12 HH23 HH32 HH44 HH56

Dr. Hisham Abdelbaki Managerial Economics 5 Both: Q d for HHs for oranges during a week HHs Qd (Kg) Sat.Sun.Mon.Tue.Wed.Thu.Fri. HH HH HH HH HH

Dr. Hisham Abdelbaki Managerial Economics 6 Relationship between Variables Two variables / More than Two Type of the relationship: 1.Direct (Positive) Examples: 2. Inverse (Negative) Examples: 3. Unrelated Examples:

Dr. Hisham Abdelbaki Managerial Economics 7 Methods of Representation of the Relationship: THREE Methods (models) can be used to represent the relationship between variables: 1- By using table 2- By using Graph 3- By using Equation

Dr. Hisham Abdelbaki Managerial Economics 8 Functional Forms Linear equation: the independent variable is raised to the first power. Quadratic equation: the independent variable is raised to the second power (i.e. squared). Cubic equation: the independent variable is raised to the third power.

Dr. Hisham Abdelbaki Managerial Economics 9 Continuous / Step Functional Relationship A function can be said to be continuous if it can be drawn on a graph without taking the pencil off the paper.

Dr. Hisham Abdelbaki Managerial Economics 10 5 Functions Used in The Textbook 1.Demand (linear) 2.Total Revenue (quadratic) 3.Production (cubic) 4.Cost (cubic) 5.Profit (cubic)

Dr. Hisham Abdelbaki Managerial Economics 11 Slope (Gradient) of a Curve Change in the value of the variable measured on the –y axis divided by the change in the value of the variable measured on the – x axis. = ∆Y / ∆ X (read “delta Y over delta X”) = vertical distance between two points / horizontal distance between two points. Example: find the slope of the straight line passing through: A (1,2) and C (4, 1) = slope = (1-2) / (4-1) = -1 / 3

Dr. Hisham Abdelbaki Managerial Economics 12 Slope of Straight and Curved Line The slope of a straight line is constant. The slope of a curved line is NOT constant. Its slope depends on where on the line we calculate it. We can calculate the slope of a curved line by drawing a line tangent to the curve at that point (the slope of the curve will equal the slope of the tangent at that point)

Dr. Hisham Abdelbaki Managerial Economics 13 What is the difference between: 1- Intersection and tangent 2- Movement and Shifting

Dr. Hisham Abdelbaki Managerial Economics 14 Using Calculus (Derivative) Calculus can be applied only if the function is continuous. Calculus is Slope – Finding Calculus can be used to find the slope of tangent to any point on a line. The slope of the graph of a function is called the derivative. An alternative notation for the derivative is dY / d X (read” dee Y by dee X”)

Dr. Hisham Abdelbaki Managerial Economics 15 Finding the Derivative of a Function Rules: 1.The derivative of a constant = ZERO 2.Power Functions: Y = X n = n X n-1 (bring down the power and subtract one from the power) Example: Y = 5 X 6 + X

Dr. Hisham Abdelbaki Managerial Economics Sums Rule: Y = V + U where V = g(X) and U = h(X) dY/ dX = the sum of the derivative of the individual terms. ( Differentiate each function separately and add) Example: if U = 3X 2 and V = 4X 3, so dY/ dX = 6X + 12 X 2

Dr. Hisham Abdelbaki Managerial Economics The Difference Rule: Y = V - U where V = g(X) and U = h(X) dY/ dX = the difference of the derivative of the individual terms. ( Differentiate each function separately and subtract) Example: if U = 3X 2 and V = 4X 3, so dY/ dX = 6X - 12 X 2

Dr. Hisham Abdelbaki Managerial Economics Product Rule: Y = UV, U = g(X) and V = h(X) dY / dX = the first term times the derivative of the second + the second term times the derivative of the first. (multiply each function by the derivative of the other and add) Example: Y = 5X 2 (7- X) dY / dX = 5X 2 (-1) + (7- X) 10X

Dr. Hisham Abdelbaki Managerial Economics Quotient Rule: Y = U / V dY/ dX = (the denominator times derivative of the numerator minus numerator times the derivative of the denominator) / (the denominator times itself) (bottom times derivative of the top, minus top times derivative of bottom, all over bottom squared) Example: Y = (5X - 9) / 10 X2 dY / dX =

Dr. Hisham Abdelbaki Managerial Economics 20 Partial Derivatives Is used to find the change in dependent variable with respect to a change in a particular independent variable. ∂f /∂x (read “partial dee f by dee x) Example: Q = P + 50 I + 3 Ps + 2 N Find the impact of a change in P on Q

Dr. Hisham Abdelbaki Managerial Economics 21 Finding the Max & Min Values of a Function Optimization: TWO steps to find the optimized (optimum ) point of a function: 1- find out the first derivative of the function. 2- put the result equals ZERO Example: find the optimum quantity produced (quantity would the firm produce).

Dr. Hisham Abdelbaki Managerial Economics 22 Ex. 1 Y = X X X dY / d X = 12 X X X X + 12 = 0 = (12X + 12)(X + 1) = 0 X = -1 Example 2: Y = X X X

Dr. Hisham Abdelbaki Managerial Economics 23 But we do NOT know if the solution is Max or Min value. To do so, we find the SECOND derivative (derivative of the derivative). If the second derivative is NEGATIVE, the value is Max. whereas, if it is POSITIVE, the value is Min