EC611--(Ch 02) Optimization Techniques

EC611--Managerial Economics

Optimization Techniques and New Management Tools Dr. Savvas C Savvides, European University Cyprus

Models and Data

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

Data the economist’s link with the real world 1 . tim e series 2 . c r os o s s s e c t io io n Managerial Economics

DR. SAVVAS C SAVVIDES

1

Models and Data

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

Data the economist’s link with the real world 1 . tim e series 2 . c r os o s s s e c t io io n 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 P rice (in 2000 prices)

7.4

39.3

100.0

£33,783

£68,702

£125,000

£2,500

£5,084

£9,250

Real Land P rice (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

10 0

2

5.5

10 5

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. Quantit y of CDs dem anded depend on (or is a function of):

f (P rices, incom e, preferences) 2. R evenues are a fun ction of Sales:

f (Q)

Managerial Economics


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Expressing Economic Relationships

TR = 100Q - 10Q2

Equations: e.g. if Q=1



TR = 100(1) – 10(31)2 = 90

if Q=3



TR = 100(3) – 10(3)2 = 210

Tables:

Q TR

0 0

1 90

2 160

3 210

4 240

5 250

6 240

3

4

5

6

TR 300

Graphs:

250 200 150 100 50 0 0

1

2

7

Q



7

25



8

Total, Average, & Marginal Cost

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

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

Q 0 1 2 3 4 5

TC 20 140 160 180 240 480


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


1

2

3


4

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



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Slope of Non-Linear Relationships

Slope of TR at A is positive:

Total Revenue

Slope of tangency at pt. A

Slope of TR at B is negative 

A

Slope of tangency at pt. B

B

TR



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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,  the derivative of Y with respect to X is equal to the ∆Y w.r.t. X, as the ∆X approaches zero. dY dX Managerial Economics

=

lim

∆ X →

0

∆

Y

∆

X


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Concept of the Derivative (2) Let’s expand on the right hand side. Since Y depends on X,  Y = f ( X )  ∆Y = ∆X ∆X

= ∆X (tautology).  Add & subtract X on RHS. ∆X = (X+∆X) – (X) ∆Y

= f(X+∆X) – f(X)  Divide both sides by ∆X  ∆Y/ ∆X = [f(X+∆X) – f(X] / ∆X Substituting the RHS of the last expression in the derivative expression, we get [f(X+∆X) – f(X] / ∆X  dY/ dX = Managerial Economics


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The Derivative – An Example If Y = X2



dY/ dX =

[(X+∆X)2 – X2 ] / ∆X



dY/ dX =

[ X2+ 2X * ∆X) + (∆X)2 - X2 ] / ∆X



dY/ dX =

[ (2X * ∆X) + (∆X)2 ] / ∆X



dY/ dX =

[ (2X * ∆X)/ ∆X ] + [(∆X)2 / ∆X]

Cancelling the X terms

dY/ dX =

(2X + X)

This says that at the limit, i.e., as X 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

dY

=

dX Y 10

0

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

X Managerial Economics


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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 ) = a X

b

dY dX

= b ⋅ a X

b −1

Example: Y = 3X2 Derivative:

dY/dX = 2 * 3X2-1 = 6X



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Power Function --Example

Equations:

TR = 100Q - 10Q2 Q TR

Tables:

0 0

1 90

2 160

3 210

4 240

5 250

6 240

TR 300

Graphs:

250

TR

200 150 100 50 0 0

1

2

3

4

MR = dTR/dQ = 100 – 20Q

5

6

7

MR

Q

Q

0

1

2

3

4

5

M R

100

80

60

40

20

0



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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 dY dX


=

=

h( X ) dU dX

±

Y

= U ± V

dV dX


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Product Rule: The derivative of the product of two functions U and V, is defined as follows. U

=

g ( X )

dY dX Managerial Economics

V

=

=U

h( X )

dV dX

+ V

Y

= U ⋅ V

dU dX


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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 V dU

=

=

h( X )

Y =

U V

− U ( dV ) ) dX dX

V

2


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Chain Rule: The derivative of a function that is a function of X is defined as follows. Y

=

f (U ) dY dX


U

=

=

g ( X )

dY dU ⋅

dU dX


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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 max Π) Find X such that dY/dX = 0. This means that the curve of the function has zero slope Example: Given that TR = 100Q – 10Q2 d(TR)

/ dQ = 100 – 20Q

we get 0 =100 – 20Q

Setting dTR/dQ =0, 20Q = 100

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

P* = 100 – 10 (5) = 50



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Optimization With Calculus (2) Equation:

TR = 100Q - 10Q2

TR 30 0 25 0

TR

20 0 15 0 10 0 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 d2 Y/dX2 > 0 (positive), then X is a minimum. If d2 Y/dX2 < 0 (negative), then X is a maximum. In the example, we found d(TR) / dQ = 100 – 20Q d2(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 – 3Y 2 + 100Y

To

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

To

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



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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 / π /

X = 80 – 4X – Y = 0

Y = 100 – X – 6Y = 0

To solve these simultaneous equations, we multiply the 1 st by (-6) and the 2nd by (-1) to get: - 480 + 24X +6Y = 0 100 – X – 6Y = 0 - 380 + 23X

=0

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 – 3Y 2 + 100Y

π

= 80(16.52) – 2(16.52)2 – 16.52 * 13.92 – 3(13.92) 2 + 100(13.92)

π

= 1,356.52



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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 w hat processes or techniques “excellent” firms use and adopt & adapt

Total Quality M anagement :

the constant improvem ents in product quality and processes to deliver consistently superior service and value to customers

Reengineering:

seeks to completely reorganize the firm (processes, departm ents, entire firm ). Radi cally redesigning processes to achieve significant gains in speed, qual ity , service, profit abili ty

The Learning Organiz ation :

continuous learning both on the individual level as w ell as on the collective level. 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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EC611--(Ch 02) Optimization Techniques

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