Managerial Economics- Optimization Techniques

8/14/2019 Managerial Economics- Optimization Techniques

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Managerial Economics (2) Role and Scope of Managerial Economics Mathematics Review Basic Concepts and Tools for Economic

Analysis


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Jul-06 S. Karna/PG06/IILM/Mgrl Eco/ME Intro/3 2

Optimal Decision: DMs Optimize

The optimal decision in managerialeconomics is one that brings the firmclosest to this goal.

Decision Makers Optimize

Practically in all managerial decisions the task ofthe manager is the same - each goal involves anoptimization problem.

The manager attempts either to maximize orminimize some objective function, frequentlysubject to some constraint(s).

And, for all goals that involve an optimizationproblem, the same general economic principlesapply!


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Basic Concepts:Maximizing the Value of a Firm

Value of a firm Price for which it can be sold Equal to net present value of expected

future profit

Risk premium Accounts for risk of not knowing future

profits The larger the rise, the higher the risk

premium, & the lower the firms value


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Basic Concepts:Maximizing the Value of a Firm

Value of a firm =

Maximizing firms profit in each time period

leads to maximizing value of firm Only if Cost & revenue conditions are

independent across time periods

If say for example in mining industry moreextraction in current time leads to higher cost

of extraction in future then higher profit now

will lead to lower profits in future.

1 2

21

...(1 ) (1 ) (1 ) (1 )

TtT

T ttr r r r

=

+ + + =+ + + +


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Economic OptimizationEconomic Optimization

Our first assumption Abstraction from Reality? A Simplification?

Economic agents (i.e., households, firms,

managers, etc.) have an objective that they

are trying to optimize. Individuals assumed to maximize utility.

For-profit firms maximize profits and minimize

costs.

Not-for-profit firms may maximize output giventhe budget or minimize cost given the output

Realistic?


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Economic Model of the Firm

Theory of the firm: Goal is to maximizefirm profits Use to represent profit

= Total Revenue Total Cost

= TR - TC or Simply R - C TR is determined by: sales and marketing

strategy, pricing, economy, etc.

TC is determined by: production methods,cost of capital, etc.


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Marginal Analysis

Marginal - change in the dependentvariable caused by a one-unit change inan independent variable

Marginal Revenue - change in total

revenue associated with a one-unitchange in output Marginal Cost - change in total cost

associated with a one-unit change in

output Marginal Profit - change in total profit

associated with a one-unit change inoutput



An Example with ProfitsExample:

Given Demand Function: P = 60 2Q

Derived Revenue Function: R = P*Q = 60Q-2Q2

Given Cost Function: C = 50Q - 12*Q2 + Q3

Derived Profit Function: = R-C = 10*Q + 10*Q2 Q3

Dem and and Marginal Revenue Lines

-10

0

10

20

30

40

50

60

70

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Quantity DemandedPrice Marginal Revenue



Output

TotalRevenue

MarginalRevenue

AverageRevenue

TotalCost

MarginalCost

AverageCost

TotalProfit

MarginalProfit

AverageProfit

0 0 0 0

1 58 58 58 39 39 39 19 19 19

2 112 54 56 60 21 30 52 33 26

3 162 50 54 69 9 23 93 41 31

4 208 46 52 72 3 18 136 43 34

5 250 42 50 75 3 15 175 39 35

6 288 38 48 84 9 14 204 29 34

7 322 34 46 105 21 15 217 13 31

8 352 30 44 144 39 18 208 -9 26

9 378 26 42 207 63 23 171 -37 19

10 400 22 40 300 93 30 100 -71 10

11 418 18 38 429 129 39 -11 -111 -1

12 432 14 36 600 171 50 -168 -157 -14

13 442 10 34 819 219 63 -377 -209 -29

14 448 6 32 1092 273 78 -644 -267 -46

15 450 2 30 1425 333 95 -975 -331 -65

16 448 -2 28 1824 399 114 -1376 -401 -86

17 442 -6 26 2295 471 135 -1853 -477 -109

Total, Marg. and Avg. Revenue, Cost & Profit at DifferentOutputs



Revenue Maximizing Output = 15; Maxm.Revenue= 450; Here MR=0 but Loss = 975

Maximization of Revenue

-100

0

100

200

300

400

500

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Output

Total,Mar

ginalandAverage

Revenue

Total Revenue Marginal Revenue Average Revenue



Cost Per Unit or AC is Lowest when MC=AC atoutput =6 but Profit = 204 (Below Max)

Minimization of Cost per uni

0

20

40

60

80

100

120

140

160

0 1 2 3 4 5 6 7 8Output

Total,

MarginalandAverage

Cost

Total Cost Marginal Cost Average Cos t



Maximum Total Profit 217 is at Output = 7

Total Profit =Total Rev enue - Total Cos

-300

-200

-100

0

100

200

300

400

500

600

700

0 1 2 3 4 5 6 7 8 9 10 11 12

OutputTotalReve

nue,

CostandProfit

Total Cost Total Revenue Total Profit



Maximum Profit =217 at output level of 7units at this Marg. Profit = MR - MC = 0

-50

0

50

100

150

200

250

0 1 2 3 4 5 6 7 8

OutputTotal,Margina

landAverageProfit

Total Profit Marginal Profit Average Profit



Profit Maximizing Output = 7 where MR= MC i.e. MC curve cuts MR from below

If MR>MC:increase output,

increase profit

If MR



Optimization Using Calculus

If y = f(x), the maxima or minima of yexists for that value of x (say at x=x*)where First derivative, dy/dx = 0

The second and sufficient condition formaxima or minima is For maxima, 2nd derivative should be

negative i.e. d2y/dx2 < 0 at x= x* For minima, 2nd derivative should be

positive i.e. d2y/dx2 > 0 at x= x*



Example: Maximize R= 60Q-2Q2

Here dR/dQ = 60 4Q(dR/dQ is basically MR) For maxima or minima dR/dQ=0 or 60-4Q=0 i.e. Q*=60/4 = 15 2nd derivative, d2R/dQ2 = -4



Marginal Revenue, Cost and Profit

Marginal Revenue, Cost & Profit:

MC-MRProfitMarginal

dQC)-R(d

dQdProfitMarginal

Profit,

and

=

===

=

==

===

dQdC

dQdR

CR

QCAC

dQdCMC

PQ

RARand

dQ

dRMR



Optimizing: Maximizing Revenue

For maximum revenue,

negativebeshouldcurveMRofSlopei.e.

0dQ

d(MR)Or,

00(ii)

0curverevenueofSlopei.e.

0MRi.e.0(i)

2

2

dQ

ACd

)ACmeanshisout that t(Check MC=

AC.ofSlopeMCofSlopewhenpossibleisthis:out(Check >

f


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Optimizing: Maximizing Profit orMinimizing Loss

belowfromcurveMRtheintersects

curveMCepoint wherby thegivenisoutput

maximizingprofitthatmeansy thisGraphicall

curveMCofSlopecurveMRofSlopei.e.

0)(dQ

d(MR)0dQd(ii)

andMC,MRor0MC-MR

0dQdProfitMarginal(i)

2

2

marginal cost Activity should be increasedto reach highest

net benefit

If marginal cost > marginal benefit Activity should be decreasedto reach

highest net benefit

Optimal level of activity When no further increases in net benefit are

possible Occurs when MB = MC


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Constraints on Optimization

Resource Constraints LimitedAvailability

Output Quantity and Quality Constraints

Legal Constraints

Environment Constraints

C t i d O ti i ti E i


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Constrained Optimization: Equi-marginal Principle

The ratio MB/Prepresents theadditional benefit per additional Respent on the activity

Ratios of marginal benefits to prices

of various activities are used toallocate a fixed amount of fundamong activities


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Constrained Optimization: Equi-marginalPrinciple

To maximize or minimize an objectivefunction subject to a constraint Ratios of the marginal benefit to price

must be equal for all activities

Constraint must be met

A B Z

A B Z

MB MB MB...

P P P = = =


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Constrained Maximization: An Example

Suppose that a firms production function isby Q = KL2 and the costs are given by C = wL+ rK, where K = capital, L = labor and w andr their per-unit costs.

a) Suppose that w = Rs.40, r = Rs.10 and that

the desires to produce 2000 units of output.How much capital and labor should be used, ifthe firm wants to produce at minimum cost?

b)Suppose now that instead of having the

objective of producing 2000 units the firmdecides to produce with a total cost budget ofRs.1800. How much capital and labor should beused to maximize output?

S l i C i d M i i i


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Solution: Constrained MaximizationExample

a) Objective Function Minimize C = wL + rK

Subject to Q = KL2 = 2000

Since Q = KL2 MP

L= dQ/dL = 2KL and MP

K= dQ/dK = L2

Cost will be minimum when MP

L/w = MP

K/r MP

L/MP

K=w/r (1)

MPL/MPK= 2KL/ L2=2K/L & w/r = 40/10 =4 Putting into eq. (1), we get 2K/L = 4

i.e. K = 2L

S l ti C t i d M i i ti


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Solution: Constrained MaximizationExample (Contd.)

Therefore Q = KL2 = 2L L2 = 2L3

Since Q = 2000, therefore 2000 = 2L3

Or, L3 = 1000 => L =10 Now, K = 2L = 2 x 10 = 20

Optimal combination is 20 units of K and10 units of L.

(The least cost C = wL + rK = 40x10 +

10x20 = 600 i.e. Rs. 600)



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b) Objective Function:Maximize output Q = KL2

Subject to C = wL + rK = 1800

As in (a), the condition for maximizingoutput MPL/w = MPK/r yields K = 2L

Putting K=2L, w= 40 and r =10 in costconstraint,

wL + r 2L = 1800 40L + 10 x 2L =1800 L = 1800/60 = 30 And therefore K = 2L = 2 x 30 = 60



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Optimal combination is K=60 units andL= 30 units.

(Maximum output, Qmax

= KL2 = 60 x 302=

54000 units)


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Managerial Economics- Optimization Techniques