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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
SaturdayHouseholdsQd
HH12
HH23
HH32
HH44
HH56
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Both: Qd for HHs for oranges during a week
HHsQd (Kg)
Sat.Sun.Mon.Tue.Wed.Thu.Fri.
HH12321233HH23322324HH32131223HH44432435HH55645456
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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
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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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)
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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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
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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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 NFind 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 ZEROExample: 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