Starting with the name of ALLAH, the most
beneficent and the most merciful
Institute of Business and Management Sciences
Optimization Techniques
By: Maria Saeed Khattak
Shehla afridi
Uzma izhar
Huma
Ch 2: Optimisation Techniques
Optimization: What consumer
chooses
3
Managerial Economics in a Global Economy
© Dominick Salvatore; ed. 2007;2011, 2012/13, Sami Fethi, EMU, All Right Reserved.
Ch 2: Optimisation Techniques
Optimization: What the consumer
chooses


Goal is to understand how a consumer make
choices.
We have the two pieces necessary for the analysis:
1: the consumer budget constraint. (how much
he can afford to spend)
2: the consumer preferences (what he want to
spend it on
4
Managerial Economics in a Global Economy
© Dominick Salvatore; ed. 2007;2011, 2012/13, Sami Fethi, EMU, All Right Reserved.
Ch 2: Optimisation Techniques
Optimizaton


Maximization and minimization of
some functions subject to cost and
output (Q) is optimization.
Best allocation of resources as
resources are limited, consumer will
purchase optimal combination of
goods.
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Managerial Economics in a Global Economy
© Dominick Salvatore; ed. 2007;2011, 2012/13, Sami Fethi, EMU, All Right Reserved.
Ch 2: Optimisation Techniques
Optimization Techniques and New Management Tools

The first step in presenting optimisation
techniques is to examine ways to express
economic relationships. Economic relationship
can be expressed in the form of equation, tables,
or graphs. When the relationship is simple, a
table and/ or graph may be sufficient. However,
if the relationship is complex, expressing the
relationship in equational form may be
necessary.
6
Managerial Economics in a Global Economy
© Dominick Salvatore; ed. 2007;2011, 2012/13, Sami Fethi, EMU, All Right Reserved.
Ch 2: Optimisation Techniques
Optimization Techniques and New Management Tools

Expressing an economic relationship in equational
form is also useful because it allows us to use the
powerful techniques of differential calculus in
determining the optimal solution of the problem.

More importantly, in many cases calculus can be
used to solve such problems more easily and with
greater insight into the economic principles underlying
the solution. This is the most efficient way for the firm
or other organization to achieve its objectives or reach
its goal.
7
Managerial Economics in a Global Economy
© Dominick Salvatore; ed. 2007;2011, 2012/13, Sami Fethi, EMU, All Right Reserved.
Ch 2: Optimisation Techniques
Example 1

Suppose that the relationship between the total
revenue (TR) of a firm and the quantity (Q) of the
good and services that firm sells over a given
period of time, say, one year, is given by
TR= 100Q-10Q2
(Recall: TR= The price per unit of commodity times
the quantity sold; TR=f(Q), total revenue is a
function of units sold; or TR= P x Q).
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Managerial Economics in a Global Economy
© Dominick Salvatore; ed. 2007;2011, 2012/13, Sami Fethi, EMU, All Right Reserved.
Ch 2: Optimisation Techniques
Example 1

By substituting into equation 1 various
hypothetical values for the quantity sold, we
generate the total revenue schedule of the firm,
shown in Table 1. Plotting the TR schedule of
table 1, we get the TR curve as in graph 1. In
this graph, note that the TR curve rises up to
Q=5 and declines thereafter.
9
Managerial Economics in a Global Economy
© Dominick Salvatore; ed. 2007;2011, 2012/13, Sami Fethi, EMU, All Right Reserved.
Ch 2: Optimisation Techniques
Example 1
TR = 100Q - 10Q2
Equation1:
Table1:
Q
TR
0
0
1
90
2
3
4
5
6
160 210 240 250 240
TR
300
Graph1:
250
200
150
100
50
0
0
1
2
3
4
5
6
7
Q
10
Managerial Economics in a Global Economy
© Dominick Salvatore; ed. 2007;2011, 2012/13, Sami Fethi, EMU, All Right Reserved.
Ch 2: Optimisation Techniques
Example 2

Suppose that we have a specific relationship
between units sold and total revenue is precisely
stated by the function: TR= $ 1.50 x Q. The
relevant data are given in Table 2 and price is
constant at $ 1.50 regardless of the quantity sold.
This framework can be illustrated in graph 2.
11
Managerial Economics in a Global Economy
© Dominick Salvatore; ed. 2007;2011, 2012/13, Sami Fethi, EMU, All Right Reserved.
Ch 2: Optimisation Techniques
Example 2
Revenue per tim e
period
Graph of the relationship betw een total revenue and
units sold
9
7.5
6
4.5
3
1.5
0
1
2
3
4
5
6
Unit Sold
TR
Price
1
1.5
1.5
2
3
3
4.5
4
6
5
7.5
6
9
Unit sold for time period
Graph2:
Table2:
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Managerial Economics in a Global Economy
© Dominick Salvatore; ed. 2007;2011, 2012/13, Sami Fethi, EMU, All Right Reserved.
Ch 2: Optimisation Techniques
Total, average and marginal cost
13
Managerial Economics in a Global Economy
© Dominick Salvatore; ed. 2007;2011, 2012/13, Sami Fethi, EMU, All Right Reserved.
Ch 2: Optimisation Techniques
Total, Average, and Marginal Cost

The relationship between total, average, and
marginal concepts and measures is crucial in
optimisation analysis. The definitions of totals
and averages are too well known to warrant
restating, but it is perhaps appropriate to define
the term marginal.
 A marginal relationship is defined as the
change in the dependent variable of a function
associated with a unitary change in one of the
independent variables.
14
Managerial Economics in a Global Economy
© Dominick Salvatore; ed. 2007;2011, 2012/13, Sami Fethi, EMU, All Right Reserved.
Ch 2: Optimisation Techniques
Total, Average, and Marginal Cost
(Recall: Total cost: total fixed cost plus total variable costs;
Marginal cost: the change in total costs or in total variable
costs per unit change in output).
Table3:
AC = TC/Q
MC = TC/Q
Managerial Economics in a Global Economy
Q
0
1
2
3
4
5
TC AC MC
20 140 140 120
160 80 20
180 60 20
240 60 60
480 96 240
15
© Dominick Salvatore; ed. 2007;2011, 2012/13, Sami Fethi, EMU, All Right Reserved.
Ch 2: Optimisation Techniques
Total, Average, and Marginal Cost

The first two columns of
Table
3
present
a
hypothetical
total
cost
schedule of a firm, from
which the average and
marginal cost schedules are
derived in columns 3 and 4
of the same table. Note
that the total cost (TC) of
the firm is $ 20 when
output (Q) is zero and rises
as output increases (see
graph 3 to for the graphical
presentation
of
TC).
Average cost (AC) equals
total
cost
divided
by
output. That is AC=TC/Q.
Thus, at Q=1, AC=TC/1=
$140/1= $140. At Q=2,
AC=TC/Q =160/2= £80
and so on. Note that AC
first falls and then rises.
Table3:
Q
0
1
2
3
4
5
TC AC MC
20 140 140 120
160 80 20
180 60 20
240 60 60
480 96 240
16
Managerial Economics in a Global Economy
© Dominick Salvatore; ed. 2007;2011, 2012/13, Sami Fethi, EMU, All Right Reserved.
Ch 2: Optimisation Techniques
Total, Average, and Marginal Cost

Marginal cost (MC), on the other
hand, equals the change in total cost
per unit change in output. That is,
MC= TC/Q where the delta ()
refers to “a change”. Since output
increases by 1unit at a time in column
1 of table 3, the MC is obtained by
subtracting successive values of TC
shown in the second column of the
same table. For instance, TC increases
from $ 20 to $ 140 when the firm
produces the first unit of output. Thus
MC= $ 120 and so forth. Note that as
for the case of the AC and MC also
falls first and then rises (see graph 4
for the graphical presentation of both
AC and MC). Also, note that at Q=3.5
MC=AC; this is the lowest AC point.
At Q=2; that is the point of inflection
whereas the point shows MC at the
lowest point.
Managerial Economics in a Global Economy
Table3:
Q
0
1
2
3
4
5
TC AC MC
20 140 140 120
160 80 20
180 60 20
240 60 60
480 96 240
17
© Dominick Salvatore; ed. 2007;2011, 2012/13, Sami Fethi, EMU, All Right Reserved.
Ch 2: Optimisation Techniques
Total, Average, and Marginal Cost
T C ($ )
240
Graph3:
180
120
60
0
0
1
2
3
4
Q
MC
A C , M C ($ )
AC
120
60
Graph4:
0
0
1
2
3
4
Q
MC=AC
18
Managerial Economics in a Global Economy
© Dominick Salvatore; ed. 2007;2011, 2012/13, Sami Fethi, EMU, All Right Reserved.
Ch 2: Optimisation Techniques
Total,average and marginal
revenue:
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Managerial Economics in a Global Economy
© Dominick Salvatore; ed. 2007;2011, 2012/13, Sami Fethi, EMU, All Right Reserved.
Ch 2: Optimisation Techniques
Average Revenue:



Total revenue per unit of output.
When all output is sold at the same
price, average revenue will be the
same as price.
AV=TR/units of out put(Q)
average revenue curve (AR),is also the
market demand curve
20
Managerial Economics in a Global Economy
© Dominick Salvatore; ed. 2007;2011, 2012/13, Sami Fethi, EMU, All Right Reserved.
Ch 2: Optimisation Techniques
Marginal Revenue:


Marginal revenue (R') is the
additional revenue that will be
generated by increasing product
sales by 1 unit.
It indicates the extra revenue
received for selling each extra unit
21
Managerial Economics in a Global Economy
© Dominick Salvatore; ed. 2007;2011, 2012/13, Sami Fethi, EMU, All Right Reserved.
Ch 2: Optimisation Techniques
Total, Average, and Marginal revenue

In the total revenue function, marginal revenue is the
change in total revenue associated with a one-unit
change in units sold. Generally, we analyse an
objective function by changing the various independent
variables to see what effect these changes have on the
dependent variables. In other words, we examine the
marginal effect of changes in the independent variable.
The purpose of this analysis is to determine that set of
values for the independent or decision variables which
optimises the decision maker’s objective function.
22
Managerial Economics in a Global Economy
© Dominick Salvatore; ed. 2007;2011, 2012/13, Sami Fethi, EMU, All Right Reserved.
Ch 2: Optimisation Techniques
Total, Average, and Marginal revenue


Marginal revenue can be positive, zero or negative
When the average revenue (demand) curve is
elastic, marginal revenue is positive and total
revenue is increasing.

When the average revenue (demand) curve is
inelastic, marginal revenue is negative and total
revenue is decreasing.

When average revenue (demand) curve is unit
elastic, marginal revenue is zero and total revenue
is not changing.
23
Managerial Economics in a Global Economy
© Dominick Salvatore; ed. 2007;2011, 2012/13, Sami Fethi, EMU, All Right Reserved.
Ch 2: Optimisation Techniques
Profit Maximization

Table 4 indicates the relationship between TR, TC and
Profit. In the top panel of graph 5, the TR curve and the
TC curve are taken from the previous graphs. Total
Profit () is the difference between total revenue and
total cost. That is = TR-TC. The top panel of Table 4
and graph 5 shows that at Q=0, TR=0 but TC=$20.
Therefore, = 0-$20= -$20. This means that the firm
incurs a loss of $20 at zero output. At Q=1, TR=$90
and TC=$ 140. Therefore, = $90-$140= -$50. This is
the largest loss. At Q=2, TR=TC=160. Therefore, = 0
and this means that firm breaks even. Between Q=2
and Q=4, TR exceeds TC and the firm earns a profit.
The greatest profit is at Q=3 and equals $30.
24
Managerial Economics in a Global Economy
© Dominick Salvatore; ed. 2007;2011, 2012/13, Sami Fethi, EMU, All Right Reserved.
Ch 2: Optimisation Techniques
Profit Maximization
Q
0
1
2
3
4
5
TR
0
90
160
210
240
250
TC Profit
20
-20
140
-50
160
0
180
30
240
0
480 -230
Table 4:
Managerial Economics in a Global Economy
Table
4 indicates the relationship
between TR, TC and Profit. In the
top panel of graph 5, the TR curve
and the TC curve are taken from the
previous graphs. Total Profit () is
the difference between total revenue
and total cost. That is = TR-TC.
The top panel of Table 4 and graph 5
shows that at Q=0, TR=0 but
TC=$20. Therefore, = 0-$20= -$20.
This means that the firm incurs a
loss of $20 at zero output. At Q=1,
TR=$90 and TC=$ 140. Therefore,
= $90-$140= -$50. This is the
largest loss. At Q=2, TR=TC=160.
Therefore, = 0 and this means that
firm breaks even. Between Q=2 and
Q=4, TR exceeds TC and the firm
earns a profit. The greatest profit is
at Q=3 and equals $30.
25
© Dominick Salvatore; ed. 2007;2011, 2012/13, Sami Fethi, EMU, All Right Reserved.
Ch 2: Optimisation Techniques
Profit Maximization
Graph5:
($)
300
TC
240
TR
180
MC
120
60
MR
0
Q
0
1
2
3
4
5
60
30
0
-30
Profit
-60
vertical difference between TR and TC is profit
Managerial Economics in a Global Economy
26
© Dominick Salvatore; ed. 2007;2011, 2012/13, Sami Fethi, EMU, All Right Reserved.
Ch 2: Optimisation Techniques
Profit maximization:
Two conditions for profit maximization:
1: MC must be equal to MR (this is a necessary condition)
2: slope of MC should be greater than MR
(this is sufficient condition)
27
Managerial Economics in a Global Economy
© Dominick Salvatore; ed. 2007;2011, 2012/13, Sami Fethi, EMU, All Right Reserved.
Ch 2: Optimisation Techniques
The End
Thanks
28
Managerial Economics in a Global Economy
© Dominick Salvatore; ed. 2007;2011, 2012/13, Sami Fethi, EMU, All Right Reserved.