dr. hisham abdelbaki managerial economics 1 econ 340 review of mathematical concepts

Dr. Hisham Abdelbaki Managerial Economics

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ECON 340

Review of Mathematical Concepts


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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


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Time series : Qd for Ashraf’s family for orange during a week

DayQuantity demanded (Kg)

Saturday5Sunday7Monday3Tuesday9

Wednesday 2Thursday6

Friday9


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Gross section: Qd for orange for households in Manama on

Saturday

HouseholdsQd

HH12

HH23

HH32

HH44

HH56


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Both: Qd for HHs for oranges during a week

HHs

Qd (Kg)

Sat.Sun.Mon.Tue.Wed.Thu.Fri.

HH12321233

HH23322324

HH32131223

HH44432435

HH55645456


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Relationship between Variables

• Two variables / More than Two• Type of the relationship:1. Direct (Positive)Examples:2. Inverse (Negative)Examples:3. UnrelatedExamples:


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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


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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.


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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.


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5 Functions Used in The Textbook

1. Demand (linear)

2. Total Revenue (quadratic)

3. Production (cubic)

4. Cost (cubic)

5. Profit (cubic)


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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


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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)


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What is the difference between:

1- Intersection and tangent

2- Movement and Shifting


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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”)


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Finding the Derivative of a Function

• Rules:

1. The derivative of a constant = ZERO

2. Power Functions:

Y = Xn = n Xn-1

(bring down the power and subtract one from the power)

Example: Y = 5 X6 + X3 + 10


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3. 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 = 3X2 and V = 4X3,

so dY/ dX = 6X + 12 X2


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4- 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 = 3X2 and V = 4X3,

so dY/ dX = 6X - 12 X2


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5. 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 = 5X2 (7- X)

dY / dX = 5X2 (-1) + (7- X) 10X


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6. Quotient Rule:

Y = U / VdY/ 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 =


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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 = - 100 P + 50 I + 3 Ps + 2 N

Find the impact of a change in P on Q


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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).


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Ex. 1 Y = 10 + 4 X3 + 12 X2 + 12 X

dY / d X = 12 X2 + 24 X + 12

12 X2 + 24 X + 12 = 0

= (12X + 12)(X + 1) = 0

X = -1

Example 2: Y = 14 + 3 X3 + 12 X2 + 12 X


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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.

-+

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dr. hisham abdelbaki managerial economics 1 econ 340 review of mathematical concepts

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