Chap # 5 :

Optimization Techniques

Tahir IslamAssistant Professor in Economics

Kardan Institute of Higher Education, Kabul

Introduction of Optimization

• Objective of the firm is to maximize the profit or minimize the cost of production

• In this chapter we present the optimization technique or method for maximizing the profit or minimizing the cost of an organization

• first step in presenting optimization techniques is to examine ways and methods to express economic relationship

Introduction of Optimization (cont….)

• We examine the relationship between marginal, total ,average concepts and measures such as revenue, product, cost and profit

• Finally optimization technique is useful in managerial making decision

Method of expressing economic relationship

• Equation, graph and tables are used for expressing economic relationship

• We use graph and table for simple relationship and equation is used for a complex relationship

• Showing relationship through equation is very useful in economic as it allows us to use powerful differential technique in order to determine the optimal solution of the problem

• Suppose we have total revenue equation• TR=100Q-10Q

2

Method of expressing economic relationship (cont..)

• Substituting values for quantity sold, we generate the total –revenue schedule of the firm

Q 100Q – 10Q2 TR

0 100(0)- 10 (0) $ 0

1 100(1)-10 (1) 90

2 100(2)-10 (2) 160

3 100(3)-10 (3) 210

4 100(4)-10 (4) 240

5 100(5)-10 (5) 250

6 100(6)- 10 (6) 240

2

2

2

2

2

2

Relationship between total, marginal and average

• Relationship between total, marginal and average concepts and measures is crucial in managerial economic

• Total cost is equal to total fixed cost plus total variable cost or average cost multiply by total number of units produced

• TC=TFC+TVC or TC= AC.Q• Marginal cost is the change in total cost resulting from

one unit change in output• Average cost shows per unit cost of production or total

cost divide by number of units produced

Relationship between total, marginal and average (cont..)

Q TC TR Profit AC MC

0 $ 20 0 -20 - -

1 140 90 -50 140 120

2 160 160 0 80 20

3 180 210 30 60 20

4 240 240 0 60 60

5 480 250 -230 96 240

Relationship between total, marginal and average (cont..)

• Geometrically MC is the slop of TC and AC is a line straight to the origin from a particular point on TC curve

output

TCTC

Q1 Q2 Q4Q3

TFC

Optimization analysis

• By optimization analysis we mean a process through which a firm determines the output level at which it maximizes total profits

• Two approaches are used for optimization• Total revenue and total cost approach• Marginal revenue and Marginal cost approach

Optimization analysis (cont…)

1. Profit maximization by Total revenue and total cost approach

• According to this approach total profit is maximum at that level of output where the difference between TR and TC is maximum

• Π =TR-TC ,when output is zero ,TR=0, but TC=$20, so total loss is $20

• When output is equal to one, TR=$90, and TC= 140 ,so total loss is equal to $50

• And at Q2, TR=TC=$160, therefore profit is equal to zero

Optimization analysis (cont…)

• When profit is equal to zero, it means that firm reached at break even point

• The same is true at Q =4, at which TR=TC=240 dollars• Between Q=2 and Q=4,TR exceeds TC, and firm earns a

profit• Profit is maximum at Q=3,because difference is +ive as

well as greatest• Profit is equal to $30

Optimization through marginal revenue and marginal cost approach

• According to TC and TR approach profit is maximum at that level of output where the difference between these term is maximum

• Marginal revenue and marginal cost concepts are very useful in managerial economics in general and in optimization technique in particular

• According to marginal analysis, profit is maximum at a level of output where MR is equal to MC

Optimization through marginal revenue and marginal cost approach

• Marginal cost is the change in total cost resulting from one unit change in output and is the slope of total cost curve

• Where as marginal revenue is the change in total revenue resulting from one unit change in sale, and is the slope of total revenue curve

Optimization through marginal revenue and marginal cost approach

• If we look at the picture at point “C* ” both curves intersect one another

• So this point is our equilibrium point and the corresponding level of output is equilibrium level of output

• According to marginal analysis ,as long as MR exceeds the MC, it pays the firm to expand the output and sales

• So in this case total profit of a firm would increase, in figure this is true between Q=1 and Q=3

Optimization through marginal revenue and marginal cost approach (cont…..)

• Between Q=3 and Q=4 MR is smaller than MC so each unit of output produced/sale would add less to total revenue than to its total revenue

• Total profit would be less

• At Q=3,

• MR=MC , and vertical distance between TR and TC is maximum, so profit is maximum at this level of output

Optimization through marginal revenue and marginal cost approach (cont…..)

• Thus according to marginal analysis, as long as marginal benefit of an activity is greater than marginal cost, It pays for an organization to increase the activity (output)

• The total net benefit is maximum when the MR or marginal benefit equals the MC

Optimization with calculus

• In this section we want to examine the process of optimization with calculus

• First we examine how we determine the point at which a function is maximum or minimum

• Then we will distinguish between a maximum and minimum

Optimization with calculus (Cont…)

• Determining a Maximum or Minimum by Calculus

• Optimization requires finding the maximum or minimum value of a function

• For example a firm wants to maximize its total revenue, minimize the cost of producing a given output ,or more likely to maximize its total profit

• For a function to be at its maximum or minimum, the derivative of the function must be zero

Optimization with calculus (cont…)

• Geometrically this corresponds to the point where the curve has zero slope

• TR=100Q-10Q

• Taking derivative of the function

• d(TR)/dQ =100-20Q

• Setting d(TR)/dQ =0 we get Q=5

2

• As we know that the derivative (slope) of a function (curve) is zero at both maximum and a minimum point

• So in order to distinguish between a maximum and minimum point, we use the second derivative

• We have a general function i.e. Y= f (X)

• The second derivative is written as d Y/dX

Optimization with calculus (cont…)

2 2

Optimization with calculus (cont…)

• Geometrically derivative mean slope of the function

• Where as second derivative refers to the change in the slope of the function

• Value of second derivative is important to determine whether we have a maximum or minimum

Optimization with calculus (cont…)

• If the value of second derivative is positive ,the function is minimum

• If the value of second derivative is negative, the function is maximum

Optimization with calculus (cont…)

• Conclusion:

• we will take derivative of a function • Equate the function to zero• Then we will take the 2nd derivative• We will look at the value• If value is +ive function is minimum• If value is –ive function is maximum