Chapter 3Optimization Techniques and New

Management Tools

Presented By

Raheel Anjum

Optimization Techniques and New Management Tools      

• Economic relationship can be expressed in– Equation– Tables– Graphs

• Simple relationship– Tables or graph

• Complex relationship– Equations– Y = a + bX - cX2

– Quadratic eq:

– If c +ve concave upward– If c –ve Concave downward

• Suppose that the relationship between the total revenue (TR) of a firm and the quantity (Q), given by

• TR= 100Q-10Q2

Q 0 1 2 3 4 5 6TR 0 90 160 210 240 250 240

0

50

100

150

200

250

300

0 1 2 3 4 5 6 7

Q

TR

Total, Average, and Marginal Cost

• Average cost= TC/Q• Q=1, AC=TC/1= $140/1=

$140

• Marginal Cost= TC/Q• TC increases from $ 20 to $

140 when the firm produces the first unit of output. Thus MC=(140-20/1) =120 and so forth

• AC & MC first falls then rises

Q TC AC MC0 20 - -1 140 140 1202 160 80 203 180 60 204 240 60 605 480 96 240

• AC curve is U shaped•MC are plotted half way between successive levels of output.

•MC = 120 from Q (0 – 1)•MC= 20 from Q (1 – 2)

•MC is also U shaped• MC reaches smallest point at smaller level• Intercepts the AC curve through the lowest point.• This is always the case.

0

60

120

180

240

0 1 2 3 4Q

TC ($)

0

60

120

0 1 2 3 4 Q

AC, MC ($)AC

MC

Derivation of AC & MC from TC•AC curve is derived from TC by the slope of a ray from origin to the corresponding point.•Slope of OH provides H’.•H’, B’, C’, K’ and D’ provide AC.• AC falls up to point K’ and then rises.•MC is given by the slope of tangent on TC at that point.•As MC= TC/Q•H*= (180 – 140 /0.5)= 80•H*, B*, C*, K’, D* provide MC curve.•TC declines to point B (Inflection Point) then rises.•MC declines till point B then rises•TC or MC at point K are equal to slope of the ray from origin.

0

60

120

180

240

0 1 2 3 4Q

TC ($)

0

60

120

0 1 2 3 4 Q

AC, MC ($)AC

MC

H’

H* *

HB

K’

K

B*

** C*

D* *B’

C’ D’

Relationship B/W AC & MC

• As long MC is below AC, the AC falls.• When MC is above AC, the AC rises.• AC neither falling nor rising, AC is minimum.• MC intersects AC from below (AC=MC)

Profit Maximization By TR & TC App:

• TP=TR-TC• At Q=0, π=0-20=-20• Firm incurs losses• At Q=2, TR=TC, π=0• Break even point.• B/w Q=2 & Q=4, TR>TC• Shows profit for Firm, At

Q=3, π=30

Q TR TC Profit0 0 20 -201 90 140 -502 160 160 03 210 180 304 240 240 05 250 480 -230

OPTIMIZATION BY MARGINAL ANALYSIS •Most important concepts in managerial economics & optimisation analysis.•Firm Max: π when (MC=MR).

•MC given by the slope of TC, falls till B* (Point of inflection) rises again.

•MR is given by the slope of TR, the MR decreases, at E, the TR is highest and MR=0……. –ve.

•As long slope of MR > MC firm expand output.

•Firm adds to revenue, than cost. As Q=1 and Q=3 (Profit goes up)

•At Q=3, slope of MR=MC orTR=TC & vertical distance is profit.

•Max: profit at C*, Q=3 (MC=MR)•Slope of TR=TC or MR=MC at Q=1

•Firm Profit Max: Conditions…..•MC=MR•MC must intersect MR from below.

Concept of derivatives

The concept of derivation

•It is closely related to the concept of margin.•When output increases from 2 to 3 units, total revenue from $160 to $ 210. Thus, MR= TR/ Q = $ 210-$ 160/3-2 =$ 50.•The slope of chord BC on the total-revenue.

•When Q assumes values smaller than unity, as small as we want and even approaching zero in the limit, MR is given by the slope of shorter chords, and it approaches the slope of the TR curve at a point in the limit.

•At B, as the change in quantity approaches zero, the change in total revenue or marginal revenue approaches the slope of the TR curve at point B.•MR= TR/ Q = $ 60- the slope of tangent BK to the TR curve at point B.•Similarly MR at Pt C,D,E are 40, 20, 0.

Slope of BD= 40Slope of BC= 50Slope at B = 60

• In general, if TR=Y and Q=X, • Derivative of Y with respect to X is given by

the change in Y with respect to X, as the change in X approaches zero.

• The derivative of Y with respect to X is equal to the limit of the ratio Y/X as X approaches zero.

0limX

dY Y

dX X

• Note: Smaller the change in X, closer will be the value of derivative at a point on the curve.

• Example: Change in X from 2-4 the average dY/dX =40 (Slope of chord BD).

• While, Change in X from 2-3 the average change =50 (slope of chord BC) i-e closer to the slope of curve at point B i-e (60).

Differentiation

• Differentiation:

The process of determining derivatives of a function,

(Finding change in Y for a change in X, when change in X approaches Zero).

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

0dY

dX

Example

• Y=2 • dY/dX=0

• The slope of the line Y is zero.

1 2 3 40

0.5

1

1.5

2

2.5Y

Y

Power Function Rule

• The derivative of a power function, where “a” and “b” are constants, is defined as:

Exponent “b” multiplied by coefficient “a” times the variable “X” raised to the power b-1.

( ) bY f X a X

1bdYb a X

dX

Example• Y=2x1

• dY/dX=1.2X1-1

• dY/dX=2

1 20

0.51

1.52

2.53

3.54

4.5

Y=2x

Sum-and-Differences Rule• The derivative of the sum or difference of

two functions U and V, is defined as:

( )U g X ( )V h X

Y U V

dY dU dV

dX dX dX

Example

• U=2x and V=x2

• Y=U+V=2x+ x2

• dY/dX=1.2.x1-1+2.1.x2-1

(By power rule)

• dY/dX=2+2x

U=2x and V=x2

Product Rule

• The derivative of the product of two functions U and V, is defined as;

First function multiplied by derivative of the second function plus second function multiplied by derivative of

the first function.

( )U g X ( )V h X

Y U V

dY dV dUU V

dX dX dX

Example• Y=2 x2 (3-2 x)

• Let U=2 x2 and V=3-2 x

• dY/dX=2x2(dV/dX)+(3-2x)(dU/dX)

• dY/dX=2 x2(0-1*2x1-1)+ (3-2 x) (2*2x2-1)

• dY/dX=2 x2(-2)+ (3-2 x) (4x)

• dY/dX= - 4x2+ 12x - 8 x2

• dY/dX= 12x-12 x2

• Y=2 x2 (3-2 x)• Let U=2 x2 and V=3-2 x

Quotient Rule• The derivative of the ratio of two functions

U and V, is defined as

Denominator multiplied by derivative of the numerator, minus the numerator times derivative of denominator, all divided by the

denominator squared.

( )U g X ( )V h XU

YV

2

dU dVV UdY dX dXdX V

Example• Y=3-2x / 2x2

• Let V=2 x2 and U=3-2 x

• dY/dX= (2 x2(dU/dX)- (3-2 x) (dV/dX)) / V2

• dY/dX= 2 x2(-2)- (3-2 x) (4x) / (2 x2)2

• dY/dX= -4x2-12x+ 8x2 /4x4

• dY/dX= 4x2 -12x / (4x4)

• dY/dX= 4x (x-3) / 4x(x3)

• = x-3 / x3

Chain Rule

• The derivative of a function that is a function of X is defined as,– If Y=f(U) and U=g(X) then the derivative of Y

with respect equals, derivative of Y with respect to U, multiplied by the derivative of U with respect to X.

( )Y f U ( )U g X

dY dY dU

dX dU dX

Example

• Y=U3+10 and U=2X2

• then• dY/dU=3U2 and dU/dX=4X

• dY/dX=dY/dU.dU/dX=(3U2) 4X

• dY/dX=3(2X2)2(4X)=48X5