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Lecture on Optimization (Lecture 3-Week 3)
Todayâ€™s lecture materials can be found on blackboard within the folder titled, â€œLecture 3 (Optimization)Week 3â€ in â€œLecture Slidesâ€ under â€œCourse Documentsâ€.
Lecture Materials (This is what I would write on the left part of the white board during face-to-face
lecture):
1. Slides on optimization (Slides 1-17)
2. Break
3. Slides on optimization (Slides 18-24)
4. 1 in-class problem
(There are in-problems posted in the same folder (i.e., â€œLecture 3 (Optimization)-Week 3â€)
under the folder headed â€œIn-class problemsâ€ and the solution is also posted).
Todayâ€™s Lecture
We typically start each lecture with a theme. In broad sense, the term optimization refers to making the
best use out of a situation or resource. In our course, optimization refers to the process involved with
selecting the best decision from a set of alternatives. We can see that these two definitions are a lot
alike. They both refer to making the â€œbestâ€ decision and they both refer to selecting something or
making the best use out of a resource. These definitions guide the purpose of todayâ€™s lecture which is to
choose the â€œbestâ€ output from a set of alternatives. The word best is used in inverted quotes because
how we interpret what is the best depends upon our objective or what it is that we are trying to
achieve. We may be interested in selecting a strategic direction, minimizing cost, or maximizing profits.
These are all goals that involve the best output, as we mentioned during the first week of the course.
For the purposes of this course, we will focus on maximizing profit and maximizing revenue. We will also
focus on minimizing cost, but only within the context of maximizing profit. Regardless of what the
objective is, the manager would need to figure out how much output to produce in order to maximize
their earnings. The concept of the slope from algebra class from our high school classes will help us to
make this decision and gives way for our theme for todayâ€™s lecture. The theme in this lecture is that
slopes are useful to managers in helping them to make sound (i.e., well thought out) business decisions.
These ideas get introduced on slide 2 where find the purpose of optimization techniques and as
suggested above it provides us with a decision rule that enables us to choose from a set of alternatives.
A decision rule just guides us with how we are supposed to run our business given the economic
conditions at hand. An example would be Q=3 which is interpreted as: we need to produce 3 units-this is
the rule, to produce 3 units. The alternatives would be producing 4 units or 1 unit or 6 units etc. and Q
stands for Quantity of the control variable or as we see on slide 3, units of the control variable. You are
the manager and so you control or decide how much to produce. We use this variable to make
managerial type or business decisions. You can produce at any quantity you want, but the best thing you
can do for the operation, based upon economic theory, is to produce where your earnings objective is at
a maximum.
In slide 3, we also see some notation that will be useful throughout the semester. Before we dive into
this notation, we motivate this terminology to help us to remember these terms. When we produce
output, we hope to benefit from all the hard work and effort we put into making this output and this
benefit obtains from being able to sell your product in the market and earn income for your business.
Thus, we use the term benefit, denoted by B, as the sales or revenues earned from selling a certain
number of units denoted as Q and given in parentheses next to the B. The term B(Q), therefore, means
the benefits that I earn when I sell Q units. An example would look like the following: B(3) which means
the benefits that I earn when I produce and sell 3 units or B(6) which is interpreted as the benefits that I
earn when I produce and sell 6 units. We also assume that we are able to sell all the units we produce to
keep the analysis focused on production, output, and revenues or earnings (for instance, we do not then
consider inventory costs and how to incorporate these costs). C(Q) would then be, analogously, the
costs as they depend upon output produced. So, C(3) is the costs associated with producing 3 units and
C(6) is the costs associated with producing 6 units.
Slide 4 shows a picture of what the typical graph for benefits would look like for our class and during our
semester. Observe that it is an umbrella shape or upside down U-shape and can be thought of as â€“Q2
type of graph. Such a graph results from a linear demand curve as we also assume that total revenues
come from demand forecasts or from recognizing that if one is interested in revenues then one would
presumably analyze demand first.
Moreover and as motivated above, we would like to know, as managers, if we produce one more unit,
how much additional revenue can we earn? This is useful because assuming we have the resources by
producing one more unit if we can earn even 2 or 3 dollars, this is useful information. In slide 5, we
introduce the marginal benefit as the tool to help us here and observe that it is simply the slope of the
benefit function with respect to Q. It is interpreted as: if I produce one more unit of Q, how much more
revenue can I earn? Or equivalently, what is the benefit of producing one more unit of Q? Graphs of this
concept are given in slide 6 (in the form of the definition as rise/run) and 7 (in the form of the definition
as the tangent line slope at a single quantity value).
If benefit is B(Q) and cost is C(Q), then net benefit, i.e., the benefit after we adjust for cost or once we
take out cost is denoted as NB(Q)=B(Q)-C(Q). Once we subtract cost from benefit, we get net benefit
and since benefit is also revenue, net benefit is also profit. Managers would like to operate their
business where net benefit or profit is maximized. This information is provided in slide 8. In slide 9, we
see that this is where marginal profit is zero or where the slope of profit is a horizontal line. One point to
remember is that marginal profit is zero where total profit is maximized. Since NB(Q) = B(Q)-C(Q),
marginal NB(Q) = marginal B(Q) â€“ marginal C(Q) or MNB(Q) = MB(Q)-MC(Q). So, when MNB(Q)=0,
MB(Q)-MC(Q)=0 and so MB(Q)=MC(Q). The same above-mentioned concept works for revenue:
Marginal revenue is zero when total revenue is maximized. This concept is shown graphically in this slide
for profit.
Given the above discussion, we can arbitrarily select several values for Q and use these values to obtain
values for B(Q) and C(Q) and use the algebraic formulas to compute the respective slopes and try to use
these definitions to pick the Q that makes the slopes equal, i.e., MB(Q)=MC(Q). This could take a long
time as there are a potentially large number of Q values to go through. Fortunately, there are rules from
calculus that can greatly simplify this process. These rules are given on slide 11 through 13 and are slides
to remember from this slide deck.
Slide 11 is all about fixed costs. The marginal value for a fixed cost is zero. We will not deal with fixed
quantities in computing the revenues and so this slide is focused on costs. So, as an example, if C=3,
then no matter how much I produce my cost is still 3. So, if Q=1, C=3. If Q=2, C=3. By producing one
more unit, i.e., boosting production incrementally (moving from 1 unit of production to 2 units of
production) leads to no change in cost (cost was \$3 initially when we produced 1 unit and it is still \$3
when we produce 2 units). This means the slope of cost is: (3-3)/(2-1) = 0/1=0. When costs are fixed the
marginal contribution to cost through an incremental increase in production is zero. The implication for
businesses is that fixed costs should not factor into production decisions. Even if we shut down
operation, we still have to pay them.
Slide 12 is for the power rule. By power rule, we just mean, how do we compute the slope when we
have exponents. Some students like using the formula in the first line of the slide. To use this formula,
we need to identify what is a an what is n and then where we see DC/DQ, we would plug these numbers
into the formula to the right of the DC/DQ. So, if C = 2Q3 , then a =2 and n=3 and so DC/DQ = 3*2*Q3-1 =
6*Q2. A derivative is a rate of change and so if the initial cost equation has highest power of 3 then by
performing the operation that represents a first order rate of change (this just means we subtract 1
from the exponent), we should end up with the highest power of 2. So, if C=2Q, we first notice there is
no exponent. If there is no exponent, we can always explicitly assume there is a 1 and so C=2Q=2Q1.
Thus, a=2 and n=1. So, DC/DQ = 1*2*Q1-1 = 2*Q0 = 2*1 = 2. Finally, DC/DQ=MC =Marginal cost and this is
a definition. We always want to remember that slope=derivative=marginal, this is an important relation
to remember in this course.
Finally, in slide 13, we see the addition rule and this just says that if we want to find the slope of some
formula which adds together two terms, we need to find the slope of each term individually using slide
11 (for a fixed cost) or slide 12 (for a term with an exponent). Then, we need to add these slopes
together if we are told to add them and subtract these slopes, one from another, if we are told to do so.
So, in our example, C(Q)=4+3Q and so we first find the slope of each term individually and then because
the these two terms have a plus sign between them we would add the slopes together. So, since 4 does
not depend upon Q, we use slide 11 to determine that the slope of 4 with respect to Q is zero. Since 3Q
does have Q next to it, we use slide 12 to compute its relative marginal contribution to cost. So, 3*Q =
3*Q1 and so a=3 and n=1 which means MC = 1*3*Q1-1 = 3*Q0 = 3*1 = 3. Thus, the slope of C(Q) = 0 + 3 =
3.
Slides 14-17 help us to understand the concept of derivative a little better. In slide 14, we set up an
example to help us to see graphically that a derivative (these are the formulas we used on slide 11
through 13) is the same as a slope. We are just using the formulas on slide 11 through 13 to compute
the slope at each value of Q. For instance, if B=10Q-Q2, then MB = 10-2Q and the values for B are given
in this slide. When Q=1, B=10*1-12 = 10-1=9.
The next slide, slide 15, shows using this formula for marginal benefits that the slope at Q=4 for the
above equation for B using the slope ratio by subtracting a small amount from 4 and adding the same
small amount and computing the rise to run ratio at 4 but around this change. In either case, we get a
value for slope of 2.
If we computed the slope at each of points Q=1 to 5 in the same manner, i.e., using the formula for MB
above, we can compute the slope values as given in slide 16. In slide 17, we show these computations
using the graph from which we see that Q=5 corresponds to where MB=0 and this is where our revenues
are the largest at \$25. We can also use equations to show this by setting MB=0 to get 10-2Q=0 and
solving for Q. MB=10-2Q=0 -> 10=2Q or 5=Q. Plugging 5 into the above equation for total benefits gives
us: B = 10*5-52 = 50-25 = 25.
Our in-class problem given in slide 18-24 is also given to you in the spreadsheet entitled â€œProfitMaximization Example-slide21-24.xlsxâ€ This is a good spreadsheet to look over and it is exactly what I
would write on the board if we were having a face to face class session. I encourage you to look over this
problem and to ask questions when and if any should come.
Optimization Techniques
1
Why should we study Marginal
Analysis?
Optimization techniques provide a decision rule that will allow us to
choose the best decision from a set of alternatives.
Optimal managerial decisions involve comparing benefits to a decision
against the associated costs of that decision.
Marginal Analysis is the tool that allows us to compare incremental
benefits of a decision against the incremental costs of that decision
As managers, tools provided by Marginal Analysis will allow us to use
optimization techniques to make sound business decisions
âž¢ If incremental benefit of staying open exceeds the incremental cost of staying open,
then it makes sense to continue to stay open, otherwise, reverse. Similar logic
applies to the decision to take on a new project.
2
Marginal Analysis: Introductory terms
The decisions of interest to managers often
involve revenues and costs, and so we will
generally refer to benefits as revenues and costs
as, well, costs.
We can define the benefits as B(Q) and the costs
then as C(Q), where Q is the number of units of
some control or decision variable (a variable that
we control or use to make a managerial decision)
A typical graph of B(Q) is shown on the next
slide. (This is the shape of all the graphs we will
consider in this class )
3
Typical graph of Benefits
TR
Q
4
Marginal Analysis: Slopes and
Economic terms
We might be interested in the additional
benefit (i.e. revenue) of adding one more
unit of some managerial control variable, Q,
on revenues.
The tool here is the marginal benefit and it
refers to the slope of the total revenue
function with respect to Q: Î”(TR)/Î”Q.
This is shown graphically on the next 2
slides
5
Marginal Analysis: Slopes and economic
concepts
D TR
TR
DQ
Q
6
Marginal Analysis: Slopes and
Economic Concepts
TR
DTR / D Q
Q
7
Marginal Analysis: Slopes and
Economic Concepts
As managers, we want to maximize profits,
Ï€.
In this class, we will define profits to be
revenues â€“ costs. Symbolically, Ï€ = B(Q)C(Q).
Managers would like to operate their
business at the point where profits (or net
benefits) are maximized.
8
Total profit
80000
Total profit
60000
Mp = 0
40000
20000
0
Slope = marginal profit
-20000
0
20
40
60
80
100
120
140
160
Output
9
Marginal Analysis and Differential
Calculus
Given, the previous slides, we can plug in values
of Q to obtain values for total revenues and total
costs and hence profits and then we can develop a
table to see where profits are maximized.
This can be very time consuming, since we could
have a potentially large number of Q values to put
into the formulas to evaluate our revenue and cost
functions.
Fortunately, there are rules (i.e. formulas) which
help us determine the value of Q which enables us
to determine the maximum profits.
10
Rules: Constants
C=a
dC / dQ = 0
Example:
C = 2
dC / dQ = 0
11
Rules: Powers
C = aQn
dC / dQ = n a Q(n- 1)
Examples:
C = 2Q3
dC / dQ = (3)(2) Q(3 â€“ 1) = 6Q2
C = 2Q
dC / dQ = (1)(2) Q(1 â€“ 1) = 2
12
Rules: Sums
C = f(Q) + g(Q)
dC / dQ = dC / dQ + dC / dQ
Example
C = 4 + 3Q
dC / dQ = 3
13
A derivative is the same as a slope
30
B = 10Q â€“ Q2
24
25
21
20
16
10
9
0
0
0
1
2
3
4
5
14
Calculate the slope around the point (4,24)
If Q = 3.9
If Q = 4.1
B = 10(3.9) â€“ (3.9)2 = 23.79
B = 10(4.1) â€“ (4.1)2 = 24.19
Slope = (24.19 â€“ 23.79) / (4.1 â€“ 3.9)
= 0.4 / 0.2 = 2
dB/dQ = 10 â€“ 2Q
At Q = 4 dB/dQ = 10 â€“ 2(4) = 2
15
B = 10Q â€“ Q2
dB/dQ = 10 â€“ 2Q
At Q = 1
At Q = 2
At Q = 3
At Q = 4
At Q = 5
dB/dQ = 10 â€“ 2(1) = 8
dB/dQ = 10 â€“ 2(2) = 6
dB/dQ = 10 â€“ 2(3) = 4
dB/dQ = 10 â€“ 2(4) = 2
dB/dQ = 10 â€“ 2(5) = 0
16
B = 10Q â€“ Q2
30
Slope = 4
20
Slope = 2
Slope = 6
10
Slope = 8
0
0
1
2
3
4
5
17
Profit maximization example
Demand:
P = 1500 – 7Q
Q = 214.3 – 0.143P
Total Revenue:
TR = PQ = 1500Q – 7Q2
Total Cost:
TC = 225 + 38Q + Q2
18
Total revenue, cost
100000
TR
80000
60000
40000
TC
20000
0
0
20
40
60
80
100
120
140
160
Output
19
Profit maximization example
Demand:
P = 1500 – 7Q
Total Revenue:
TR = 1500Q – 7Q2
Total Cost:
TC = 225 + 38Q + Q2
Profit
p = [1500Q – 7Q2] â€“ [225 + 38Q + Q2 ]
= -225 + 1462Q – 8Q2
20
Marginal profit approach
p = -225 + 1462Q – 8Q2
Mp = 1462 – 16Q
At Q = 10
p= -225 + 1462(10) â€“ 8(10)2 = \$13,595
Mp = 1462 â€“ 16(10) = \$1,302
21
Marginal profit approach
Mp = 1462 – 16Q = 0
Q = 1462 / 16 = 91.375 =
91
TR = 1500(91) – 7(91)2 = \$78,533
TC = 225 + 38(91) + (91)2 = \$11,964
Profit = 78,533 – 11,964 = \$66,569
22
Marginal revenue, cost approach
Revenue:
TR = 1500Q – 7Q2
MR = 1500 – 14Q
Cost:
TC = 225 + 38Q + Q2
MC = 38 + 2Q
At Q = 10
MR = 1500 â€“ 14(10) = \$1,360
MC = 38 + 2(10) = \$58
23
Marginal revenue, cost approach
Total Revenue:
Total Cost:
TR = 1500Q – 7Q2
TC = 225 + 38Q + Q2
MR = 1500 – 14Q
MC = 38 + 2Q
MR = MC
1500 – 14Q = 38 + 2Q
1462 = 16Q
Q = 91.375 = 91
24
Bringing it all together!
â€œSlide25-Problem-Week2â€
Look at the sheet titled â€œBasicsâ€
What is the first step to solving this
problem?
25
Marginal Analysis Real World Example
Open the .pdf file,
â€œOptimizationApplication.pdfâ€
26
Wrapping it all up!
â€œOptimization-Inclass exercise-Set1.xlsâ€
I will do these two problems on board.
â€œOptimization-Inclass exercise-Set2.xlsâ€
You can work with each other and I will
guide you.
27
Appendix A: Product Rule
B = f(Q) g(Q)
dB/dQ = f(Q) Â· d[g(Q)]/dQ + d[f(Q)]/dQ Â· g(Q)
Example:
B = 3Q2 ( 3 – Q )
let f(Q) = 3Q2
g(Q) = 3-Q
dB / dQ = 3Q2 (- 1) + (3 – Q) (6Q)
= – 3Q2 + 18Q – 6Q2
= – 9Q2 + 18Q
28
Appendix B: Extending to multiple
variables
Suppose revenue depends upon (1) price and
advertising or (2) sales of two products
â€œPartial derivativesâ€ describe the effect of a small
change on revenue
Process:
âž¢ Derive marginal functions for each decision
variable.
âž¢ Set each marginal function equal to zero.
âž¢ Solve for the decision variables.
29
Revenue maximization
Revenue(x,y) = 200x + 100y – 10×2 – 20y2 + 20xy
ï‚¶Revenue / ï‚¶x = 200 – 20x + 20y = 0
ï‚¶Revenue / ï‚¶y = 100 – 40y + 20x = 0
200 – 20x + 20y = 0
100 + 20x – 40y = 0
300
– 20y = 0
300 – 20y = 0
y = 15
200 – 20x + 20(15) = 0
x = 25
30
Revenue maximization
Revenue(x,y) = 200x + 100y – 10×2 – 20y2 + 20xy
x
23
24
25
26
27
y
15
15
15
15
15
Revenue
\$3,210
\$3,240
\$3,250
\$3,240
\$3,210
x
25
25
25
25
25
y
13
14
15
16
17
Revenue
\$3,170
\$3,230
\$3,250
\$3,230
\$3,170
31
To find the equation of
the slope of the deman
VA-Intercept:
HA-Intercept:
Slope =
Elast. =
Region to the right of po
where we are unitary el
Region to the left of poi
where we are unitary el
If demand is elastic, an
price will lead to a redu
If demand is inelastic, a
If demand is unitary ela
revenues are maximize
These last three statem
is known as the total re
To determine the relationship between price, elasticity, and total revenue, we can also use the equation of the demand curve.
To find the equation of the demand curve, use the vertical axis intercept and the horizontal axis intercept to find
the slope of the demand curve
VA-Intercept:
HA-Intercept:
Slope =
Price =40 and Quantity=0
Price=0 and Quantity=80
-2
Q = -2P +b
Elast. =
-2* P/Q
Region to the right of point (Q>40)
where we are unitary elastic is inelastic
when Price=40, Q=0 -> b=80
when Price=0, Q=80 -> b=80
Equation is:
Region to the left of point (Q Q=40, P=20
If demand is unitary elastic (|E|=1), total
revenues are maximized
TR is Max when Q=40 and P=20
Or, when P/Q =1/2
These last three statements taken together
is known as the total revenue test
Q
P
0
10
20
30
40
50
60
70
P/Q
40
35
3.5
30
1.5
25 0.833333
20
0.5
15
0.3
10 0.166667
5 0.071429
TR
MR
0
350
600
750
800
750
600
350
30
20
10
0
-10
-20
-30
E
-7
-3
-1.66667
-1
-0.6
-0.33333
-0.14286
MR>0, elastic
MR |%Î”Q|>|%Î”P|->|E|>1
Unit Elastic -> |%Î”Q|=|%Î”P|=1
Inelastic -> |%Î”Q| 0 -> X and Y are substitutes
Â» Exy < 0 -> X and Y are complements
Cross-price elasticities play an important role in the pricing decisions
of firms that sell multiple products.
âž¢ E.g. selling a hamburger with a soda. These products are
complements-when a consumer purchases a burger, he or she will
typically purchase a soda with it.
22
Other elasticities
Assume that firmâ€™s revenues are derived from sales of two products, X
and Y, then we can use the following formula to approximate changes
in revenues that obtain from changes in the price of product X:
Î”R = [Rx (1+EQx,Px) + RYEQY,Px ] x %Î”Px
Suppose that a restaurant earns \$4,000 per week in revenues from
hamburger sales (Product X) and \$2,000 per week from soda sales
(Product Y). Own price elasticity of demand for burgers is -1.5 and
cross price elasticity of demand between soda and burgers is -4.0.
What would happen to total revenues if it reduced the price of
hamburgers by 1%?
firms can take advantage of this idea and boost their revenues!
23
Concept Check
A positive income elasticity tells us that that
good is:
âž¢a. a normal good
âž¢b. a substitute good
âž¢c. an inferior good
âž¢d. an inelastic good
The cross-price elasticity of demand
between bread and crackers is 4.
crackers ?
24
Elasticities from demand functions
Q = 6748 – 3P + 2.72Y + 1.68Po
Price
P = \$15,000
Income
Y = \$17,500
Related Price Po = \$14,500
Q = 6748 – 3(15,000) + 2.72(17,500) + 1.68(14,500)
Q = 33,708
25
Elasticities from demand functions
EP = -3 ( 15,000 / 33,708) = -1.335
EY = 2.72 ( 17,500 / 33,708 ) = 1.412
EPo = 1.68 ( 14,500 / 33,708 ) = 0.723
26
TR and Price Elasticities
If you raise price, does TR rise?
Suppose demand is elastic, and raise price.
TR = Pâ€¢Q, so, %DTR = %DP+ %DQ
If elastic, P , but Q a lot
Hence TR FALLS !!!
Suppose demand is inelastic, and we decide
to raise price. What happens to TR?
27
Relating TR to price elasticity
d TR
dP
= P + Q
dQ
dQ
Q dP
MR = P 1 +
P dQ
MR = P [ 1 + 1 / Ep ]
28
Relating TR to price elasticity
MR = P [ 1 + 1 / Ep ]
E = -1
MR = 0
Implication: Revenues are maximized when E=-1.
|E| > 1
MR > 0
|E| < 1 MR < 0 SDSU - College of Business Administration 29 Relating TR to price elasticity P = a â€“ bQ TR = PQ = aQ - bQ2 MR = a - 2bQ SDSU - College of Business Administration 30 Appendix A: Public Transportation example â€œHow prices and other factors affect travel behaviorâ€ http://www.vtpi.org/elasticities.pdf Why would you want to use public transportation? If using Metrolink, must have good reason (for e.g. not many available substitutes-donâ€™t have alternative modes of transportation)->inelastic demand
North County Transit District (NCTD) had been losing
ridership in 2010, and so it gambled on reducing fares to
increase ridership.
http://www.signonsandiego.com/news/2010/sep/15/nctdgamble-lower-fares-build-ridership/
31
Appendix B: Constant elasticity demand
functions
Q = kPb
dQ / dP = bkPb-1
Ep =
dQ P
dP Q
= bkPb-1
P
kPb
E=b
32
Constant elasticity demand functions
Q = P -2.5 Y 1.7 A 0.4
Ep = -2.5
EY = 1.7
EA = 0.4
Log-linear transformation
Log Q = -2.5(Log P) + 1.7(Log Y) + 0.4(Log A)
33
Appendix C: Computing elasticity
without a demand function.
It turns out that if we are just given a range
of prices and quantities, we can do a
reasonable job computing elasticity using a
formula called Arc elasticity.
This can be useful for a manager who wants
to introduce a new product, but does not
have much data on sales or quantity
demanded.
34
ARC Elasticities
%DQ
Q2 – Q1
(Q1 + Q2)/2
P2 – P1
(P1 + P2)/2
%DP
35
Arc elasticity
20
19
12
14
36
Arc elasticity
%DQ = 2 / 13 = 15.38%
%DP = -1 / 19.5 = -5.13%
20
E = -3.0
19
12
14
37
Lecture on Theory of Elasticity of Demand (Lecture 8 (Elasticity of Demand)-Week 9&10)
Todayâ€™s lecture materials can be found on blackboard within the folder titled, â€œLecture 8 (Elasticity of
Demand)-Week 9&10â€ in â€œLecture Slidesâ€ under â€œCourse Documentsâ€.
Lecture Materials (This is what I would write on the left part of the white board during face-to-face
lecture):
1. Slides on Elasticity of Demand (Slides 1-26)
2. Break
3. Slides on Elasticity of Demand (Slides 27-30)
(covering the relationship between total revenue, elasticity of demand, and price)
Todayâ€™s lecture
All of the materials needed to follow along in this lecture are given to you under â€œLecture Materialsâ€.
With each lecture, we begin with a theme and the theme in todayâ€™s lecture is â€œby how muchâ€ the
concept of elasticity represents a sensitivity measure that allows you as the manager to build directly on
your toolkit from microeconomic theory of demand and in particular supply and demand. Therefore,
tools from supply and demand help us to determine trends in important economic variables like price
and income. When we use words like trends what we are saying is that we are able to qualitatively
define relationships between economic variables. For instance, prices go up and quantity demanded
comes down. Or, we know that for a normal good, an increase in income leads to an increase in quantity
demanded. In this lecture on elasticity, we take these ideas one step further and say okay we know that
a price of some good goes up, this means that the number of units of this good demanded by customers
goes down. This is by virtue of the Law of Demand. It would be useful for you as the manager to know
â€œby how muchâ€. So, in particular, if we know that prices for some good went up by 10%, it would be
useful for managers to know by how much (i.e., by how many units) did quantity demanded come
down? These questions are motivated on slide 2. This is the first theme for our lecture today, elasticity
tells us â€œby how muchâ€; it is a sensitivity measure. Please remember to check â€œclick to add notesâ€ for
In slide 3, we present an over of this lecture presentation. This lecture covers the price elasticity of
demand which is our first and basic measure of the quantitative effect of changes in price on the change
in quantity demanded. We begin by defining what is meant by this term and how we compute it. The
factors that determine price elasticity of demand are given on slides 16-20 and these slides will be for
your term project only and they will not be covered on your final exam project. So, in regards to the
term project, this presentation is helpful for question #7 and in specific question #7a and b which ask us
to describe the four factors (these are given in slide 16-20) that impact the elasticity of demand for our
firmâ€™s primary product and in part b we need to explain how these effects vary with the customer
segment defined in question 2c. Finally, we will wrap up our discussion in slides 27-30 and using the
The content of our lecture begins in slide 4 with what we call the â€œown price elasticity of demandâ€. It is
also called â€œprice elasticityâ€. This is measuring the effect on the number of units sold (we can also think
of this in terms of sales or revenue as this is defined as price x number of units sold) when there is a
change in price. It is also called â€œown price elasticity of demandâ€ because it measures the
responsiveness of the good for which we writing down the demand equation to changes in its own price.
This means that the effect of the price of a substitute or complementary good are not being measured,
but, rather, the effect of the price of the good for which we have written down the demand equation.
So, if we wrote down the demand equation for pizza, then this measure would assess the
responsiveness to changes in the price of pizza. Finally, elasticity is a unit less change (i.e., it is just a
number like 3 or 4 or 1.4) as it is defined in this slide as a percentage change. Again, it has the latin
terms ceteris paribus meaning all else equal.
ð¸ð‘ =
%Î”ð‘„
Î”ð‘„ ð‘ƒ
=
ð‘¥
%Î”ð‘ƒ
Î”ð‘ƒ ð‘„
Slope
Elasticity and slope are different. (See solution to study guide question 1 on page 48 for a good problem
that highlights this difference). A slope is a per unit change and elasticity is a percentage change. In slide
5, we define different degrees of sensitivity for the elasticity of demand. We define the elasticity of
demand to be elastic if |ð¸ð‘ | > 1, we use absolute value for interpretation but not computation. The
reason for this is because we know from the law of demand that an increase in price leads to a reduction
in quantity demanded and that this is the reason for the negative sign so we use the absolute value to
help us to focus on the interpretation through the magnitude. If we are elastic, then we are more price
sensitive. If |ð¸ð‘ | = 1 , we are unit elastic and this rarely happens in practice in business, but when
marginal revenues are zero and total revenues are maximized (as we saw on midterm #1), then we are
unit elastic, but only at this single points. Finally, if |ð¸ð‘ | < 1, then we are inelastic and in this case we are less price sensitive. Goods that do not conform to laws of demand are called Veblan goods and these correspond to certain luxury goods. An example of an elastic demand is given on slide 6. The elasticity of pepsi is -2. So, immediately, we see the negative sign and this is a result of the law of demand so we keep the sign for the purposes of calculation but use absolute value for its interpretation in the following sense. |âˆ’2| = 2>1 and so since
the absolute value of the elasticity is greater than one, we have an elastic demand, this means we are
more price sensitive. What does this mean? If we were to increase the price of pepsi by 10%, then the
number of units sold of pepsi would be 20%. If we were to increase the price of pepsi by 1%, then the
number of units sold of pepsi would be 2%. This ratio is always 2 and so â€œmore price sensitiveâ€ means
that a percentage change in price would lead to a greater than exactly proportional percentage change
in quantity. If we are inelastic, the reverse would hold. A percentage change in price would lead to a
smaller than exactly proportional percentage change in quantity. Please try the question on slide 7 on
your own and the answer is d. This is an exactly proportional effect, meaning a percentage change in
price leads to an exactly proportional change in the percentage change in quantity. This is a unit elastic
demand curve.
A more complete list of definitions are given in slide 8. Examples of perfectly inelastic demand curves
correspond to products that are absolutely essential, perhaps even for living like medical equipment.
Certain types of equipment are needed for people to continue to live and so we are perfectly in sensitive
to changes in price. Perfectly elastic demand curves on the other side correspond to perfect substitutes
like for instance fruit. If you go to vons to get bananas or to ralphs and they do not have the same price
people should go to the cheaper price because a banana is banana it doesnâ€™t matter if Vons or Ralphs is
selling it. It may be different if its organic, but this is potentially a different product.
Sample elasticities are given in slide 9. Products that are considered more as necessities meaning we
need them for our every day life (e.g., eggs butter or chicken) would be more inelastic. Anything that is a
necessity would be more inelastic. Perishable goods like fruit tend to be more elastic. Ground beef is
close, but not exactly, unit elastic. We interpret that alcohol is a necessity from here also. Moving onâ€¦.
An explanation is needed here as we know that alcohol is not a necessity as is milk, butter or eggs. So,
instead of thinking about alcohol having an elasticity as below 1 as implying that it is a necessity, we
instead would think about it in terms of data and the pure mathematical definition for this good. What
an elasticity of demand that is less than one tells us is that the data on alcohol consumption indicate
that the consumption of alcohol is not very sensitive to price. This means that whether prices go up or
down does not seem to affect the purchase of alcohol. The reasoning behind why we tend to observe
this phenomena is a subject of research and this research is beyond the scope of the course content.
We now begin to discuss how to calculate elasticity. As we can infer from the equation representing the
definition of elasticity above, we need the slope and a price-quantity pair to compute elasticity. This
means that elasticity is calculated for each price-quantity pair given by the demand curve and so as we
move up or down the demand curve we calculate a different elasticity. This point is given in the first
bullet on slide 10. The second two bullet points we mentioned above when we defined elasticity and so I
So, as is shown on slide 11, we would calculate elasticity by taking the product of the slope (which
comes from the demand equation) and the price-quantity pair. Either price or quantity must be given,
because if we have one we can use the demand equation to compute the other. Since the slope
represents a relative change of quantity with regard to price (holding all other factors constant) is it
simply interpreted as a per unit change in quantity relative to per unit change in price. If we have a
demand equation, then we would write this using the delta (the triangle looking symbol, we have been
using thus far) on slide 12. If we have are given a demand function (which includes all factors that you
think impact the sales for your product), then we represent the same quantity using a sea shell looking
symbol denoting a partial change. This symbol allows us to acknowledge that there are factors other
than price that impact our demand and we are holding them constant and this a partial change because
a total change would require changing all inputs and then seeing the affect on demand. It may not be as
informative from the standpoint of making a business decision because we do not know where the
source of the change in sales when we change three different inputs (as an example) at one time. So, we
isolate the impact of each underlying factor influencing quantity demanded using a sea shell looking
symbol given in slide 12 through the term on the extreme right most side of the statement of equalities.
So, in slide 13, if our demand equation is Q = 60 â€“ 10P and we want to find the price elasticity of demand
when price is 5 and quantity is 10; this means that P=5 and Q=10. We would begin with our definition of
elasticity which we repeat for convenience below:
%Î”ð‘„
Î”ð‘„
ð‘ƒ
ð¸ð‘ = %Î”ð‘ƒ = Î”ð‘ƒ ð‘¥ ð‘„
The first step would be to find the slope which is given in the demand equation. The slope is a relative
change meaning it tells a change in one variable relative to a change in another. So, here, since Q is in
the numerator and P is in the denominator, the slope tells us the change in Q relative to a change in P.
If I change price by one unit (i.e., one dollar or our measure of currency), how does quantity change
(i.e., the units demanded or sold or purchased). In a linear demand equation, all exponents are equal to
one and so the slope is just a constant equal to the number coming before the variable in the
denominator of our above expression, in this case P. Our slope is -10. To see this using more information
than just the demand equation and its slope. We can assume that P=\$1 and compute Q. When P = \$1, Q
= 60 â€“ 10*1=50. Then because we only want to know the relative change in Q due to the change in P
(this is the defn of slope), we only need to make P=\$2, i.e., we only need to increment it by \$1 to keep
the numbers simple in identifying the slope. So, when P=\$2, Q = 60 â€“ 10*2=40. So, when we increase P
from \$1 to \$2, Q reduces from 50 units to 40 units; a \$1 change in price leads to a 10 unit change in
quantity. The relative change in quantity due to a \$1 change in price is 10 units. If you try other
numbers, you will still get 10, this is what it means to have a constant slope or rate or change. So, the
elasticity is going to be -10 *( P/Q). Here either P or Q has to be given. So, we are told P is 5, from the
demand equation Q = 60 â€“ 10*5 = 60-50=10, and so Q is 10 units. Even though we are told this, even if
we were not we could calculate it from the demand equation as shown in the prior sentence. So, our
elasticity is -10*(5/10)= -5. In absolute value this number is greater than 1 and so we are elastic at this
point on demand curve. We can calculate elasticities at each point of the demand curve in an analogous
manner and the numbers we would obtain are given in slide 14. Notice that as we move down the
demand curve (this means reducing price and increasing quantity) we move from being elastic to unit
elastic (only at a single point) to being inelastic. This will help us with addressing the second theme of
todayâ€™s lecture.
At higher values for price and necessarily (by law of demand) lower quantities, the ratio of P/Q is
relatively large and so with a constant slope this means the product of slope and P/Q would be a
relatively large number and this means its elastic. When prices are lower and necessarily quantities are
higher (by law of demand), the ratio of P/Q is relatively small and so with a constant slope this means
the product of slope and P/Q would be relatively a small number and this means its inelastic. Hopefully
between this explanation with words and the one given in the table on slide 14 this important
relationship between price, quantity, and elasticity is clearer. For managers, this relationship is
important for boosting sales which is where we will conclude this lecture shortly.
On slide 15, we add additional factors impacting the demand for our product to show how to compute
elasticity with a demand function. Its very similar as we illustrate below. So, as we have more variables
now to work through, we first compute Q using the demand equation and in this case what is given for
income and price. So, income=0 and price=5, plugging these into the demand function, we have: Q =
150-10*5+5*0 = 150-50+0=100. So, Q = 100, P=5, and the slope corresponding to the change in Q
relative to a change in P is again -10 (this is the coefficient before the P). Thus, the elasticity is: -10*
(5/100) =-10*(1/20)=-0.5. In absolute value, this is less than one and so we are inelastic. A one percent
change in price leads to a reduction in quantity demanded by half a percent.
Slide 16-20 are for your term project only and to help you, specifically, with question 7a and 7b. After
listing the four main factors affecting elasticity in slide 16, we begin with substitutes in slide 17 for which
we see that the better and more numerous the availability of substitutes the more elastic the demand
is. So, if we cannot really find substitutes for something like certain medical drugs or pharmaceutical
drugs needed to live, then we have an inelastic demand. Also, its hard to find a substitute for
transportation and so transportation has an inelastic demand; it is a broadly defined good. Analogously,
it is easy to find a substitute for a very specifically defined good like a Pontiac Catalina since it is easy to
find another product which delivers the same or similar value as a GM Pontiac Catalina. Whereas it is
not so easy to find another product which delivers the same or similar value as transportation since we
need some form of transportation to move from place to place and get to work for instance.
On slide 18, we see that when something costs more (as a proportion of our budget), the more elastic
the demand is, if you spend 15% of your income on something, you will be more sensitive to changes in
price. If you spend 1% of your income on something (like salt), you wont be as price sensitive. In slide 19,
we see that if you have more time to adjust to changes in price, then you are more elastic and as in the
longer term you have more time to seek out substitutes you tend to be more elastic in the longer term
versus the shorter term. So, gasoline is a good example here as when we first move to a city we may not
know it well enough and it may be hard to find a substitute but as we get acquainted with the city we
can find more suitable substitutes and this happens over time. The is the subject of the concept check in
slide 20 for which the answer is choice b. Demand tends to be more elastic in the longer term because
consumers have more time to seek out substitutes. Finally, the more durable or long lasting a product is,
the more elastic it is. So, cars are durable because used ones in many cases can be good substitutes for a
new one.
In slide 21 and 22, we discuss other elasticities. This includes elasticities related to income in slide 21
and related goods known as cross-price elasticities in slide 22. These are calculated the same way as
price elasticity, but now we do not use the absolute value because the sign plays an important role in
interpreting the result. For income elasticity, a positive sign means a normal good and a negative sign
means an inferior good and for cross-price elasticity, a positive sign means a substitute and a negative
sign means a complement. Other than this, these measures are computed in exactly the same way as
price elasticity of demand.
In slide 23, we show how cross-price elasticities play a role in pricing decisions for a firm with two goods.
So, lets say these goods are X and Y, where X =hamburgers and Y=sodas. The equation in bullet point
number 2 gives us a formula that allows us to use elasticities and cross-price elasticities to determine
revenues that can result from price changes. So, if ð‘…ð‘¥ =4,000 (this is the revenues from hamburger sales)
and ð‘…ð‘¦ =2,000 (this is revenue from soda sales) and EQx,Px = ð¸ð‘ = -1.5 and EQY,Px = EQdx,Py (using our
notation) = -4 (the negative sign means these numbers are complements which tend to be consumed
together. When someone gets a hamburger he/she tends to get soda with it. The interpretation of this
number is analogous to price elasticity of demand defined above). So, if we reduce the price of
hamburgers by 1%, then using the equation given in bullet point 2, we can find the effect on revenues as
shown below:
Î”R = [Rx (1+EQx,Px) + RYEQY,Px ] x %Î”Px
= [4,000(1+-1.5)+2,000(-4)]x-0.01
= [4,000(-0.5)-8,000]x-0.01
=[-2,000-8,000]x-0.01
=20+80
=100
So, when the business reduces the price of hamburgers and because when someone gets a hamburger
they tend to get a soda with it and the business earns more revenue from the soda sales (owing to cross
price elasticity) than the hamburger sales.
Slide 24 contains a concept check. If you have questions, please let me know, the answers are a and
substitutes.
Slide 25 is an example problem, please try this and let me know if you have questions. The answer is on
slide 26.
Finally, we conclude our lecture with our second and final theme. How do we use all these relationships
to boost our firm level revenue?
To help us address this theme, we turn to our spreadsheet entitled, â€œElast., MR, and TR.xlsxâ€ to help us
to visualize the graphs and equations needed to understand this theme as it relates to business.
We begin in slide 27 by addressing the question: when I increase price, what happens to total revenues?
So, via equations, in bullet point 2, we see how the percentage change in Q and percentage change in P
jointly affect the percentage change in revenue. This equation comes from the top of the sheet titled
â€œEquationsâ€ in the spreadsheet mentioned above. It is derived using the same percentage change
definition as elasticity except for revenue and after applying the chain rule in algebraic form and doing
more algebra to simplify the equation we end up with a relationship between total revenue, price, and
quantity. Our second equation which is underneath it is the definition of elasticity. Putting these two
equations together will allow us to answer our question posed at the beginning of this paragraph.
Two examples illustrating the relationship between price, elasticity, and revenue are given in the rest of
the sheet on the tab titled â€œEquationsâ€. So, in the first equation beginning underneath the box with the
two equations, we see that when we are elastic (E=-1.5) and we boost price (%Î”ð‘ƒ =6%), then %Î”Q =
Ex%Î”ð‘ƒ = -1.5 x 6% = -9% (from equation (2)). Then, percentage change in revenue will be the sum of the
percentage change in Q and P (this is from equation (1)) and this is -9%+6% = -3%. So, %Î”TR =-3%.
When we are elastic and we increase price we reduce total revenues. Think about traveling for leisure or
fun, when we are more sensitive to changes in price, if the airline company boosts prices, since we donâ€™t
need to travel, we will decide to postpone travel or to not travel and this reduces the revenues of airline
companies. This is shown the two graphs underneath of demand (with the ranges of elasticity) and
revenues with the big red arrows. When we are in the elastic region, we only look at the region on the
left where it says elastic. In this region, when we boost price (by law of demand this means reduce
quantity). The first red arrow and only red arrow in the demand equation plot is moving towards the top
of the price axis and towards the origin on the quantity axis in the elastic region. This means that
quantity is reducing and so we move to the origin of the revenue graph and this moves us away from the
top of the revenue graph and we reduce revenue. The reason we place these economic concepts on two
different graphs is because while the horizontal axis is quantity for both and this is measured in units,
the vertical axis for demand is price and this is measured in dollars per unit while the vertical axis for
revenues is dollars in sales and this measured in dollars. The units for the vertical axis variable is
different for demand and revenue and so they go on two different graphs.
When we are inelastic in the example below this so our elasticity , E=-0.5 and we boost price again
(%Î”ð‘ƒ=6%), then %Î”Q=-0.5×6=-3%. Change in revenues, %Î”TR=-3%+6%=3%. This means when we are
inelastic, and we boost price we boost revenues. If you have to travel to Dallas tomorrow morning for
work, even if the airline boost price since you have to go, you still go and the company still earns
revenue. We can see this graphically in the graph below the above described example. Now, we see the
red arrow on the demand plot on the left side since we are inelastic in this region, when we boost price
and reduce quantity we move in the same direction as in the example above except starting closer to
the right side of the graph rather than the left. In the revenue graph underneath we see that now we
start on the right side of the horizontal axis and reduce quantity and this moves us back towards the top
of the revenue graph thereby boosting revenues. When we are inelastic, we boost price then we
increase our revenues.
Finally, when we are unit elastic E=-1 and at this point and only at this point we have maximal revenues.
The marginal revenue values in these ranges is given in the tab called â€œGraphsâ€ which shows that when
we are elastic marginal revenues are positive when we are unit elastic marginal revenues are zero and
when we are inelastic marginal revenues are negative.
In slide 28, we use equations to show this relationship between marginal revenue, price, and elasticity
again using the chain rule from calculus. So, in slide 29, we see that when E=-1, MR=0 and when E0.
Finally in slide 30, we see that Demand and MR are, for linear equations, related via the slope and
intercept. The intercept is the same for demand and MR and the slope of MR is double the slope of
demand by a factor of 2. This is because we used a linear demand equation.