SD State University Optimization Techniques and Demand Theory Articles Analysis these the five topics that the teacher mentioned in the syllabus Optimiz

SD State University Optimization Techniques and Demand Theory Articles Analysis these the five topics that the teacher mentioned in the syllabus Optimization techniques, Demand Theory Regression Analysis, Application of Regression Models Production Theory, Cost Theory (including applications) Competition: Pure; Monopolistic, Monopoly , Forecasting Oligopoly and Game Theory, Pricing Topics the files are1. week 12. week 2 part 1 and 23. week 44. week 55. week 6 part 1 and 2 Optimization Techniques
SDSU – College of Business Administration
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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
➢ 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.
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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 )
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Typical graph of Benefits
TR
Q
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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: Δ(TR)/ΔQ.
This is shown graphically on the next 2
slides
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Marginal Analysis: Slopes and economic
concepts
D TR
TR
DQ
Q
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Marginal Analysis: Slopes and
Economic Concepts
TR
DTR / D Q
Q
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Marginal Analysis: Slopes and
Economic Concepts
As managers, we want to maximize profits,
π.
In this class, we will define profits to be
revenues – costs. Symbolically, π = B(Q)C(Q).
Managers would like to operate their
business at the point where profits (or net
benefits) are maximized.
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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
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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.
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Rules: Constants
C=a
dC / dQ = 0
Example:
C = 2
dC / dQ = 0
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Rules: Powers
C = aQn
dC / dQ = n a Q(n- 1)
Examples:
C = 2Q3
dC / dQ = (3)(2) Q(3 – 1) = 6Q2
C = 2Q
dC / dQ = (1)(2) Q(1 – 1) = 2
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Rules: Sums
C = f(Q) + g(Q)
dC / dQ = dC / dQ + dC / dQ
Example
C = 4 + 3Q
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dC / dQ = 3
13
A derivative is the same as a slope
30
B = 10Q – Q2
24
25
21
20
16
10
9
0
0
0
1
2
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4
5
14
Calculate the slope around the point (4,24)
If Q = 3.9
If Q = 4.1
B = 10(3.9) – (3.9)2 = 23.79
B = 10(4.1) – (4.1)2 = 24.19
Slope = (24.19 – 23.79) / (4.1 – 3.9)
= 0.4 / 0.2 = 2
dB/dQ = 10 – 2Q
At Q = 4 dB/dQ = 10 – 2(4) = 2
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B = 10Q – Q2
dB/dQ = 10 – 2Q
At Q = 1
At Q = 2
At Q = 3
At Q = 4
At Q = 5
dB/dQ = 10 – 2(1) = 8
dB/dQ = 10 – 2(2) = 6
dB/dQ = 10 – 2(3) = 4
dB/dQ = 10 – 2(4) = 2
dB/dQ = 10 – 2(5) = 0
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B = 10Q – Q2
30
Slope = 4
20
Slope = 2
Slope = 6
10
Slope = 8
0
0
1
2
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4
5
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Profit maximization example
Demand:
P = 1500 – 7Q
Q = 214.3 – 0.143P
Total Revenue:
TR = PQ = 1500Q – 7Q2
Total Cost:
TC = 225 + 38Q + Q2
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Total revenue, cost
100000
TR
80000
60000
40000
TC
20000
0
0
20
40
60
80
100
120
140
160
Output
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Profit maximization example
Demand:
P = 1500 – 7Q
Total Revenue:
TR = 1500Q – 7Q2
Total Cost:
TC = 225 + 38Q + Q2
Profit
p = [1500Q – 7Q2] – [225 + 38Q + Q2 ]
= -225 + 1462Q – 8Q2
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Marginal profit approach
p = -225 + 1462Q – 8Q2
Mp = 1462 – 16Q
At Q = 10
p= -225 + 1462(10) – 8(10)2 = $13,595
Mp = 1462 – 16(10) = $1,302
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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
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Marginal revenue, cost approach
Revenue:
TR = 1500Q – 7Q2
MR = 1500 – 14Q
Cost:
TC = 225 + 38Q + Q2
MC = 38 + 2Q
At Q = 10
MR = 1500 – 14(10) = $1,360
MC = 38 + 2(10) = $58
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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
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Bringing it all together!
Open the Excel Spreadsheet titled,
“Slide25-Problem-Week2”
Look at the sheet titled “Basics”
What is the first step to solving this
problem?
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Marginal Analysis Real World Example
Open the .pdf file,
“OptimizationApplication.pdf”
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Wrapping it all up!
Open the Excel Spreadsheet titled,
“Optimization-Inclass exercise-Set1.xls”
I will do these two problems on board.
Open the Excel Spreadsheet titled,
“Optimization-Inclass exercise-Set2.xls”
You can work with each other and I will
guide you.
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Appendix A: Product Rule
B = f(Q) g(Q)
dB/dQ = f(Q) · d[g(Q)]/dQ + d[f(Q)]/dQ · 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
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Appendix B: Extending to multiple
variables
Suppose revenue depends upon (1) price and
advertising or (2) sales of two products
“Partial derivatives” describe the effect of a small
change on revenue
Process:
➢ Derive marginal functions for each decision
variable.
➢ Set each marginal function equal to zero.
➢ Solve for the decision variables.
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Revenue maximization
Revenue(x,y) = 200x + 100y – 10×2 – 20y2 + 20xy
Revenue / x = 200 – 20x + 20y = 0
Revenue / y = 100 – 40y + 20x = 0
200 – 20x + 20y = 0
100 + 20x – 40y = 0
300
– 20y = 0
300 – 20y = 0
y = 15
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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
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x
25
25
25
25
25
y
13
14
15
16
17
Revenue
$3,170
$3,230
$3,250
$3,230
$3,170
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Elasticity of Demand
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Downward Slope and the Demand Curve
Suppose that the price of some product increased by 10%.
What would happen to the quantity demanded for that
product?
By the law of demand, we know that prices and quantity
demanded are inversely related, and so an increase in price
would cause the demand to fall.
It would be useful to a manager to know by how much the
quantity would fall.
This primary tool used to determine such a change is
elasticity analysis.
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Overview of Elasticity of Demand
Elasticity of Demand
➢Price elasticity
➢What is it ?
➢How do we compute it?
➢Understanding factors that determine price
elasticity.
➢Relating price changes to changes in sales revenue
Pricing strategy
➢Using price elasticity to form pricing strategy
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Price Elasticity of Demand
Also known as “own price elasticity of
demand” or “price elasticity”
Measures the responsiveness of a good’s
sales to changes in its price
Equal to the percent change in quantity
demanded divided by the percent change in
price
E = |%DQ / %DP|, ceteris paribus
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Price Elasticity of Demand, cont.
Elastic -> |%ΔQ|>|%ΔP|->|E|>1
Unit Elastic -> |%ΔQ|=|%ΔP|=1
Inelastic -> |%ΔQ| 0 -> X and Y are substitutes
» Exy < 0 -> X and Y are complements
Cross-price elasticities play an important role in the pricing decisions
of firms that sell multiple products.
➢ 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.
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Other elasticities
Assume that firm’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:
ΔR = [Rx (1+EQx,Px) + RYEQY,Px ] x %ΔPx
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%?
Homework set 3, Problem #6 will help you learn more about how
firms can take advantage of this idea and boost their revenues!
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Concept Check
A positive income elasticity tells us that that
good is:
➢a. a normal good
➢b. a substitute good
➢c. an inferior good
➢d. an inelastic good
The cross-price elasticity of demand
between bread and crackers is 4.
➢What can we conclude about bread and
crackers ?
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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
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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
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TR and Price Elasticities
If you raise price, does TR rise?
Suppose demand is elastic, and raise price.
TR = P•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?
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Relating TR to price elasticity
d TR
dP
= P + Q
dQ
dQ
Q dP
MR = P 1 +
P dQ
MR = P [ 1 + 1 / Ep ]
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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 – bQ TR = PQ = aQ - bQ2 MR = a - 2bQ SDSU - College of Business Administration 30 Appendix A: Public Transportation example “How prices and other factors affect travel behavior” 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-don’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/
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Appendix B: Constant elasticity demand
functions
Q = kPb
dQ / dP = bkPb-1
Ep =
dQ P
dP Q
= bkPb-1
P
kPb
E=b
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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)
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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.
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ARC Elasticities
%DQ
Q2 – Q1
(Q1 + Q2)/2
P2 – P1
(P1 + P2)/2
%DP
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Arc elasticity
20
19
12
14
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Arc elasticity
%DQ = 2 / 13 = 15.38%
%DP = -1 / 19.5 = -5.13%
20
E = -3.0
19
12
14
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The Production
Process and Costs
McGraw-Hill/Irwin
Copyright © 2010 by the McGraw-Hill Companies, Inc. All rights reserved.
Overview
I. Production Analysis
–
–
–
–
Total Product, Marginal Product, Average Product.
Isoquants.
Isocosts.
Cost Minimization
II. Cost Analysis
– Total Cost, Variable Cost, Fixed Costs.
– Cubic Cost Function.
– Cost Relations.
III. Multi-Product Cost Functions
5-2
Production Analysis
▪ Production Function
– Q = F(K,L)
• Q is quantity of output produced.
• K is capital input.
• L is labor input.
• F is a functional form relating the inputs to output.
– The maximum amount of output that can be produced with K
units of capital and L units of labor.
▪ Short-Run vs. Long-Run Decisions
– Generally capital is fixed in the short run, although it is possible
for either factor to remain fixed.
▪ Fixed vs. Variable Inputs
– In the short run, we need to use more variable input to produce
more output, because increasing the fixed input is not possible.
5-3
Productivity Measures:
Total Product
▪ Total Product (TP): maximum output produced
with given amounts of inputs.
▪ Average Product of an Input: measure of output
produced per unit of input.
– Ave…
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