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Dr. Hisham Abdelbaki Managerial Economics
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ECON 340
Review of Mathematical Concepts
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Dr. Hisham Abdelbaki Managerial Economics
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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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Dr. Hisham Abdelbaki Managerial Economics
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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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Dr. Hisham Abdelbaki Managerial Economics
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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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Dr. Hisham Abdelbaki Managerial Economics
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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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Dr. Hisham Abdelbaki Managerial Economics
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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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Dr. Hisham Abdelbaki Managerial Economics
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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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Dr. Hisham Abdelbaki Managerial Economics
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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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Dr. Hisham Abdelbaki Managerial Economics
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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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Dr. Hisham Abdelbaki Managerial Economics
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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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Dr. Hisham Abdelbaki Managerial Economics
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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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Dr. Hisham Abdelbaki Managerial Economics
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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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Dr. Hisham Abdelbaki Managerial Economics
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What is the difference between:
1- Intersection and tangent
2- Movement and Shifting
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Dr. Hisham Abdelbaki Managerial Economics
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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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Dr. Hisham Abdelbaki Managerial Economics
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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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Dr. Hisham Abdelbaki Managerial Economics
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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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Dr. Hisham Abdelbaki Managerial Economics
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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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Dr. Hisham Abdelbaki Managerial Economics
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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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Dr. Hisham Abdelbaki Managerial Economics
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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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Dr. Hisham Abdelbaki Managerial Economics
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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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Dr. Hisham Abdelbaki Managerial Economics
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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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Dr. Hisham Abdelbaki Managerial Economics
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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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