Chap # 5 :
Optimization Techniques
Tahir Islam
Assistant 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
2
• TR=100Q-10Q
Method of expressing economic relationship
(cont..)
• Substituting values for quantity sold, we generate the
total –revenue schedule of the firm
Q
0
100Q – 10Q2
TR
2
100(0)- 10 (0)
$0
2
1
100(1)-10 (1)
2
100(2)-10 (2)
90
2
160
2
3
100(3)-10 (3)
210
2
4
100(4)-10 (4)
240
2
5
100(5)-10 (5)
250
6
100(6)- 10 (6)
240
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
0
$ 20
0
1
140
90
2
160
3
Profit
AC
MC
-
-
-50
140
120
160
0
80
20
180
210
30
60
20
4
240
240
0
60
60
5
480
250
-230
96
240
-20
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
TC
TC
TFC
Q1
Q2 Q3
Q4
output
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
2
• TR=100Q-10Q
• Taking derivative of the function
• d(TR)/dQ =100-20Q
• Setting d(TR)/dQ =0 we get Q=5
Optimization with calculus (cont…)
• 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)
2
2
• The second derivative is written as d Y/dX
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