1 EC611--Managerial Economics Optimization Techniques and New Management Tools Dr. Savvas C Savvides, European University Cyprus2 Models and Data Mode...

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Models and Data Model a framework based on simplifying assumptions Î it helps to organize our economic thinking based on a simplified picture of reality Î We focus on key elements

Data the economist’s link with the real world 1. time series 2. cross section Managerial Economics

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Real and Nominal Variables

Many economic variables are measured in money terms Nominal values measured in current prices

Real values adjusted for price changes compared with a base year measured in constant prices Managerial Economics

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Real & Nominal Values--Example 1960

1975

2003

Land Prices (Hilton Park Area, Nicosia)

£2,500

£27,000

£125,000

Price Index (2000=100) Real Land Price (in 2000 prices)

7.4

39.3

100.0

£33,783

£68,702

£125,000

£2,500

£5,084

£9,250

Real Land Price (in 1960 prices)

(2,500*100) / 7.4 = 33,783 Managerial Economics

(125,000*7.4) / 100 = 9,250

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Evidence in Economics Evidence collected and produced from empirical observation and testing may allow us to accumulate support for a theory, or to reject it, or indicate points for further research and investigation Scatter diagrams help us to test and validate economic theory with empirical reality Econometrics is a more sophisticated method that takes this task of empirically validating theory further using statistical techniques Managerial Economics

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Data & Scatter Diagrams Price

Year

Price

Quantity

1

6.0

100

2

5.5

105

3

6.0

90

7.0

X (7.0, 80)

X

4

6.5

85

5

6.0

87

6

7.0

80

7

6.5

88

X

X

6.0

X

X (6.0, 100) X

80

100

Quantity Managerial Economics

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Economic Models: An Example Examples: 1. Quantity of CDs demanded depend on (or is a function of): Î f (Prices, income, preferences) 2. Revenues are a function of Sales: Î f (Q)

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Expressing Economic Relationships

TR = 100Q - 10Q2

Equations:

e.g. if Q=1 Î TR = 100(1) – 10(31)2 = 90 if Q=3 Î TR = 100(3) – 10(3)2 = 210

Tables:

Q TR

0 0

1 90

2 3 4 5 6 160 210 240 250 240

TR 300

Graphs:

250 200 150 100 50 0 0

1

2

3

4

5

6

7 Q

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7

25

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Total, Average, & Marginal Cost

AC = TC/Q e.g. for Q=3 ÎAC = 180/3 =60

MC = ∆TC/∆Q For ∆Q from 3 to 4: ÎMC = (240-180)/(4-3) =60 / 1 = 60 Managerial Economics

Q 0 1 2 3 4 5

TC 20 140 160 180 240 480

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AC 140 80 60 60 96

MC 120 20 20 60 240 9

Total, Average, & Marginal Cost TC TC ($) 240 180 120 60 0 0

1

2

3

4

Q

AC, MC ($)

MC 120

AC 60

0 0

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1

2

3

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Q

10

Profit Maximization Profit = TR - TC Q 0 1 2 3 4 5 Managerial Economics

TR 0 90 160 210 240 250

TC Profit 20 -20 140 -50 160 0 180 30 240 0 480 -230

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Profit Maximization ($) 300

TC TR

240

180

120

60

0 60

Q 0

1

2

3

4

5

30 0 Profit

-30 -60

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Slope of a Line

Slope between A & B ∆P/∆Q = -5 / +5 = - 1

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Slope of a Line Price

Quantity Managerial Economics

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Slope of Non-Linear Relationships

Slope of TR at A is positive:

Total Revenue

Î Slope of tangency at pt. A

Slope of TR at B is negative Î Slope of tangency at pt. B

A

B

TR

Quantity Managerial Economics

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Concept of the Derivative (1) Optimization analysis can be conducted much more efficiently using differential calculus. This relies on the concept of the derivative, which resembles the concept of the margin. For example, if TR = Y and Q =X, Î the derivative of Y with respect to X is equal to the ∆Y w.r.t. X, as the ∆X approaches zero.

dY ∆Y = lim ∆X → 0 ∆ X dX Managerial Economics

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Concept of the Derivative (2) Let’s expand on the right hand side. Since Y depends on X, Î Y = f ( X ) Î ∆Y = ∆X ∆X = ∆X (tautology). Î Add & subtract X on RHS. ∆X = (X+∆X) – (X) ∆Y = f(X+∆X) – f(X) Î Divide both sides by ∆X Î ∆Y/ ∆X = [f(X+∆X) – f(X] / ∆X Substituting the RHS of the last expression in the derivative expression, we get Î dY/ dX = [f(X+∆X) – f(X] / ∆X Managerial Economics

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The Derivative – An Example If Y = X2 Î dY/ dX =

[(X+∆X)2 – X2 ] / ∆X

Î dY/ dX =

[ X2+ 2X * ∆X) + (∆X)2 - X2 ] / ∆X

Î dY/ dX =

[ (2X * ∆X) + (∆X)2 ] / ∆X

Î dY/ dX =

[ (2X * ∆X)/ ∆X ] + [(∆X)2 / ∆X]

Cancelling the ∆X terms Î dY/ dX =

(2X + ∆X)

This says that at the limit, i.e., as ∆X Î 0, the whole expression will approach 2X (since ∆X=0) Managerial Economics

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Rules of Differentiation Constant Function Rule: The derivative of a constant, Y = f(X) = a, is zero for all values of a (the constant). Y = f (X ) = a Y 10

dY =0 dX

Changes in X do not affect the value of Y. Horizontal lines have zero slope! Y = 10 X

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Rules of Differentiation Power Function Rule: The derivative of a power function, where a and b are constants, is defined as follows. Y = f (X ) = aX

b

dY = b ⋅ a X b −1 dX

Example: Y = 3X2 Derivative: Î dY/dX = 2 * 3X2-1 Î Managerial Economics

= 6X DR. SAVVAS C SAVVIDES

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Power Function --Example Equations:

TR = 100Q - 10Q2 Q TR

Tables:

0 0

1 90

2 3 4 5 6 160 210 240 250 240

TR 300

Graphs:

250

TR

200 150 100 50 0 0

1

2

3

4

5

6

MR

MR = dTR/dQ = 100 – 20Q Q

0

1

2

3

4

5

MR

100

80

60

40

20

0

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7 Q

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Rules of Differentiation Sum-and-Differences Rule: The derivative of the sum or difference of two functions U and V, is defined as follows. U = g( X )

V = h( X )

Y = U ±V

dY dU dV = ± dX dX dX Managerial Economics

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Rules of Differentiation Product Rule: The derivative of the product of two functions U and V, is defined as follows. U = g( X )

V = h( X )

Y = U ⋅V

dY dV dU =U +V dX dX dX Managerial Economics

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Rules of Differentiation Quotient Rule: The derivative of the ratio of two functions U and V, is defined as follows. U = g( X )

dY = dX Managerial Economics

V = h( X )

(

V dU

dX

) (

− U dV

V

U Y= V

dX

)

2

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Rules of Differentiation Chain Rule: The derivative of a function that is a function of X is defined as follows. Y = f (U )

U = g( X )

dY dY dU = ⋅ dX dU dX Managerial Economics

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Optimization With Calculus (1) Optimization often requires finding the max. or the min. of a function (e.g. maxTR, minTC, or maxΠ) Find X such that dY/dX = 0. This means that the curve of the function has zero slope Example: Given that TR = 100Q – 10Q2 Îd(TR) / dQ = 100 – 20Q ÎSetting dTR/dQ =0, we get 0 =100 – 20Q Î 20Q = 100 Î Q* = 5 Therefore, Total Revenues are maximized at Q* = 5 To find the optimum Price, we go to the demand equation from which the TR function derived: P = 100 – 10Q Î P* = 100 – 10 (5) = 50 Managerial Economics

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Optimization With Calculus (2) Equation:

TR = 100Q - 10Q2

TR 300 250

TR

200 150 100 50 0 0

1

2

3

4

MR = dTR/dQ = 100 – 20Q = 0

5

6

7 Q

MR

20Q = 100 Q=5 Managerial Economics

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Optimization With Calculus (2) To distinguish between a max and a min, we use the second derivative. Second derivative rules: If d2Y/dX2 > 0 (positive), then X is a minimum. If d2Y/dX2 < 0 (negative), then X is a maximum. In the example, we found d(TR) / dQ = 100 – 20Q

Îd2(TR)/dQ2 = - 20 (negative) Therefore, we know that the TR function is at a maximum (“top of the hill”) at Q = 5 Managerial Economics

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Multivariate Optimization Multivariate functions: TR = f (Sales, Advertising, prices, …) TC = f ( wages, interest, raw materials, …) Demand = f (price, income, P of substitutes, …)

To optimize a function that has more than one independent variables, we use the partial derivative. Managerial Economics

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Multivariate Optimization (2) The Partial Derivative: The partial derivative (indicated by ∂) is used in order to isolate the marginal effect of each one of the independent variables. The same rules of differentiation apply, except that when we differentiate the dependent variable w.r.t. one variable, we hold all other variables constant. Managerial Economics

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Partial Derivative--Example Suppose that Profits (π) are a function of the sales of products X and Y as follows: π = f (X, Y) = 80X – 2X2 – XY – 3Y2 + 100Y ÎTo find the partial derivative of Π w.r.t X, we hold Y constant (i.e. ∆Y =0) to get: ∂π / ∂X = 80 – 4X – Y ÎTo find the partial derivative of Π w.r.t Y, we hold X constant (i.e. ∆X =0) to get: ∂π / ∂Y = 100 – X – 6Y Managerial Economics

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Max or Min Multivariate Functions Example (cont) To max or min a multivariate function, we set each partial derivative equal to zero and solve the resulting simultaneous equations: ∂ π / ∂ X = 80 – 4X – Y = 0 ∂ π / ∂ Y = 100 – X – 6Y = 0 To solve these simultaneous equations, we multiply the 1st by (-6) and the 2nd by (-1) to get: - 480 + 24X +6Y = 0 100 – X – 6Y = 0 - 380 + 23X

=0

Î Therefore, X = 380 / 23 = 16.52 Managerial Economics

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Max or Min Multivariate Functions Example (cont) Substituting X = 16.52 into the first equation, we find the value of Y: 80 – 4 (16.52) – Y = 0 80 – 66.08 – Y = 0 Y = 13.92 Thus, the firm maximize Profits when it sells 13.92 unit of Y and 16.52 units of X. Thus: π = 80X – 2X2 – XY – 3Y2 + 100Y π = 80(16.52) – 2(16.52)2 – 16.52 * 13.92 – 3(13.92)2 + 100(13.92) π = 1,356.52 Managerial Economics

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Constrained Optimization So far, we dealt with unconstrained optimization However, in most real life situations, firms are faced with a series of constraints (budget, capacity, lack of raw materials, etc). In these cases, we need to optimize (max or min) the objective function (profits, revenues, costs, market share, etc) subject to the constraints faced by the firm. We have two methods to solve constrained optimization problems: 1. Substitution Method (used for simple functions) 2. Lagrangian Method (used for complex functions) Managerial Economics

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New Management Tools Benchmarking: finding out what processes or

techniques “excellent” firms use and adopt & adapt

Total Quality Management: the constant

improvements in product quality and processes to deliver consistently superior service and value to customers

Reengineering: seeks to completely reorganize the

firm (processes, departments, entire firm). Radically redesigning processes to achieve significant gains in speed, quality, service, profitability

The Learning Organization: continuous learning

both on the individual level as well as on the collective level. Î It is based on five ingredients: a new mental model achieve personal mastery – develop system thinking – develop shared vision – strive for team learning Managerial Economics

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Other Management Tools Broad Banding: eliminating multiple layers of salary levels, and

increasing labor flexibility

Direct Business Model: dealing directly with the consumer,

eliminating distributors and saving on time and costs (e.g, Dell )

Networking: the formation of strategic alliances to increase the synergies and capitalize on individual competences

Pricing Power: being able to increase prices faster than costs thus

increasing profits

Small-World Model: large firms may gain efficiency by simulating the

operation of small firms by breaking up the process in smaller scale and linking the units or individuals through organizational systems

Virtual Integration: the blurring of traditional boundaries between manufacturer and its suppliers and manufacturer and customer Î

supply chain management

Virtual Management: the simulation of the production process and consumer behavior using computer models Managerial Economics

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