Regression Analysis
In economics, regression analysis can be used to estimate demand functions and show a firm how to price to increase profits through the use of historical or cross-sectional data. Managers of a firm can utilize this information in order to quantify the impact of changes in price and income to the firm’s products or services and adjust their price to increase profits. Regression analysis helps us understand the relationship between a dependent variable and one or more independent variables. In order to better understand the estimation of the demand for a product or service let’s assume that a relationship exists between a dependent variable, Y, and an independent X and that relationship is expressed as Y = a + bX + e where “a” and “b” are unknown parameters and “e” is a random factor. This is also known as the Least Squares Regression Model. Because the parameters a and b are unknown we must try to find the values of these parameters to determine the relationship between X and Y. [Baye, 96-97]. Least Squares Regression Line The least squares regression line is the one that minimizes the squared deviation of the vertical distances between the data points and the line or the smallest sum of squared errors between the line and the data points. It is expressed as Y = ^a + ^bX Where ^a and ^b are estimates of parameters a and b, respectively, and results in a more accurate estimate between the line and the data. Case Study Example
SmallCable Co.
The company, SmallCable Co., is a relatively new internet service provider that has struggled to increase their profits in the Indianapolis area over the past year. The company’s CEO is wondering if his pricing is accurate for the Indianapolis area and has put together the quantity and average price (per client) of internet service over the past year as well as the average income (in thousands of dollars) of his customers. If SmallCable Co. were to increase their price by $10 dollars what would happen to their total revenue?
Month
Quantity
Price
Avg Member Income
1
20
57
71
2
35
85
76
3
38
76
75
4
56
59
74
5
53
54
74
6
61
52
75
7
64
49
74
8
59
51
76
9
54
50
75
10
67
48
75
11
84
46
76
12
68
49
75
With price being the independent variable and the quantity being the dependent variable, the results are plotted on the graph below
Using the least squares regression a line can be plotted to determine the smallest sum of squared errors below
Using the regression analysis package in Excel, the data from SmallCable Co. can be input to produce the output below.
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.926174
R Square
0.857799
Adjusted R Square
0.826198
Standard Error
7.118269
Observations
12
ANOVA
df
SS
MS
F
Significance F
Regression
2
2750.889
1375.444
27.14527
0.000154194
Residual
9
456.0278
50.66976
Total
11
3206.917
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Upper 95%
Lower 95.0%
Upper 95.0%
Intercept
-467.611
116.9319
-3.999
0.003115
-732.1290884
-203.092
-732.129
-203.092
Price
-1.05487
0.180103
-5.85704
0.000242
-1.462295077
-0.64745
-1.4623
-0.64745
Avg Income
7.794387
1.571674
4.959292
0.000781
4.239014793
11.34976
4.239015
11.34976
Review of the data The t-Statistic
The t-statistic is used to estimate the ratio of a parameter’s value to its standard error. If the t-statistic is greater than 2, then one can be 95% confident that the value is statistically significant. As you can see above, the t-statistic for price and average income is greater than 2 in absolute value, therefore price and income are significant factors in determining quantity demanded. The R-Square
The R-square explains overall fit of the estimated regression to the actual data. The closer the R-square is to 1, the better the fit. The R-square of .85 is fairly high, suggesting that the model explains 85 percent of the variation in the demand for internet service at SmallCable Co.. The F-Statistic
The F-statistic also provides a good measure of the overall fit of the regression line to the data. The significance of the F-statistic is extremely low so SmallCable Co. can be confident that this is an accurate model and does not include any random factors that may influence the data. Results
The estimated demand equation is Q = -467.61 -1.05P + 7.79M. This equation says that if SmallCable Co. were to increase their price by $10 then quantity demanded would decrease by 1.05 units. Likewise, if customers were to have an increase in income by $1,000 this would increase demand by 7.79 units.
Questions
1. What is the t-statistic used for?
a. Determining the standard deviation of the value of the parameter
b. Determining if the value of the parameter is statistically significant
c. Measuring the overall fit of the regression line to the data
d. Determining the slope of the line
2. Why would managers use regression analysis?
a.To understand the shift in supply
b.To determine own-price elasticity
c. To estimate a demand function
d.To understand consumer behavior
3. The lease squares regression line for the equation Y=a+bX + e, is
a. Y = ^a + ^bX
b. Y = ^a +^b + X
c. Y = ^b + ^aX
d.Y = ^a + ^bX + e
4. What R-squared below would be considered the best fit to its actual data?
a. .95
b. 3
c. .3
d. Unable to determine
5. The least squares regression line
a.Is used to plot the demand curve
b. Is used to determine the smallest sum of squared errors
c. Is used to plot the supply curve
d. Is used to determine the highest sum of squared errors
Answers: 1. B, 2. C, 3. A, 4. A, 5. B
References
Baye, Michael R. Managerial Economics and Business Strategy. (7th edition). New York. McGraw Hill, 2010
Summaries
1.“Auctions including English auction, First price sealed bid auction, second price sealed bid auction, Dutch auction, winners curse” by Kristin Westerfield
Kristin’s paper describes in great detail several types of auctions (English auction, First price sealed bid auction, second price sealed bid auction, Dutch auction) which are used in the real world to buy and sell products and services through bidding. The author offers real world examples for each type of auction and even mentions additional auction types, such as a Japanese auction. In her paper, Kristin breaks down both the advantages and disadvantages for each type of auction.
2. “Game theory -prisoners dilemma, dominant strategy and Nash equilibrium, coordination games Finitely repeated games, the end-of period problem, infinitely repeated games, sequential move games” – by Michael Okunade Ayodele
Michael’s paper outlines the basic premises of game theory including the prisoner’s dilemma and dominant strategy. The author provides good detail regarding Nash equilibrium however his paper loses focus on the overall topic of game theory and instead goes on to discuss pricing strategies for firms.
3. “Create and answer a case study similar to American Airlines Case (Chapters 5, 7-11, 13) - Cases Available under Assignments tab- that looks at pricing to drive out competitors and then raising prices in the future.” – Brad Nelson
The author’s paper discusses predatory pricing as a pricing strategy utilized to drive competition from the market. The author uses Wal-Mart’s pricing strategy as an example of predatory pricing as the firm attempts to put local sellers out of business by offering staple items such as milk and butter at below cost in order to steal the local customers.
4. “Government policies- Price ceilings, price floors, excise taxes, and the economic effects of prohibition (alcohol, drugs, kidneys, etc)”- Anthony Christofaro
The author’s paper discusses the history of prohibition and its economic effects. It appears that prohibition has played a major influence through history as had both positive and negative socioeconomic effects in the past and up through present day. After the amendment was repealed due to the US government needing cash (through taxation of alcohol) the government has repeatedly tried to seek new ways to obtain additional revenue from its people.
5. “Economies of scale, diseconomies of scale, constant returns to scale, Efficiency in production. Economies of scope and how to maximize profit when there’re two production plants, long-run equilibrium-Include Coverage from Chapters 2-6 of Science of Success”- Charles Hays
Charles paper provides the definition to the topics listed above and explains in more detail how each of these concepts work in an industry. Charles goes on to further discuss a book called, “Science of Success” by Charles Koch, which describes Market Based Management and how it breaks down into five different dimensions (vision, virtue and talents, knowledge processes, decision rights, and incentives) Each of these aspects contribute to the success of a company.
Notes
Create and answer a case study similar to Memo 14 from Time Warner (Most Relevant Chapter 11)- that uses regression analysis to estimate demand functions and then shows how to price to increase profits. John Slade
Memo 14
To: Pricing Manager, Midwest Region
From: Vice President, Marketing
Re: Pricing New Program Tiers
In response to the increated requests for a la carte pricing, we have decided to start a trial offering of smaller program tiers. As the first step of the trial, two small program packages will be offered to those using our basic package. The first is a sports package which includes NBA TV and the Soccer Channel. The second is a music package that includes MTV2 and GAC.
The trial offering will be limited to out Region 1 and Region 2 markets only. We estimate that our relevant incremental costs for the sports package are $1.45 per subscriber and the incremental costs for the music package are $1.20 per subscriber.
I am attaching a preliminary survey conducted by our marketing team that indicates anticipated sales at various pricing points. I would appreciate your recommendations regarding the pricing of these new program tiers. Thanks.
Brief discussion on the possibility of tiered pricing strategies for the internet based off of a paper, "How Many Tiers? Pricing in the Internet Transit Market", written by Nick Feamster et al. at Georgia Tech. Interesting article however the paper reviewed is more robust as it discusses demand functions, elasticity of demand, etc.
October 9th
Interesting to note that tiered pricing for the internet is very favorable to the cable and mobile phone companies. Short article mentioning tiered pricing and wireless companies.
Time Warner can enhance it's profits by packaging products (in the case of Memo 14, multiple channels) and offer these as a package deal instead of a la carte and thus forcing customers into making a decision of which package they must take (even though there may be channels that they don't watch).
Chapter 11 includes several pricing strategies that firms can use to enhance profits however tiered-pricing is not specifically mentioned in the chapter.
Regression analysis
Regression analysis will be used to determine the demand for the product offerings so that an appropriate price can be decided in order to maximize the profits of the company. Analysis of current internet usage of each customer related to uploading and downloading of files should be used as a quantity of measure.
Nov 27th
Cable companies are looking for more ways to lower consumer surplus and increase their own profits when it comes to internet services. By utilizing a mix of tiered pricing, two part-pricing, and package bundling these companies extract as much profit as possible.
Two-part pricing allows a firm to extract all consumer surplus from consumers
In economics, regression analysis can be used to estimate demand functions and show a firm how to price to increase profits through the use of historical or cross-sectional data. Managers of a firm can utilize this information in order to quantify the impact of changes in price and income to the firm’s products or services and adjust their price to increase profits.
Regression analysis helps us understand the relationship between a dependent variable and one or more independent variables. In order to better understand the estimation of the demand for a product or service let’s assume that a relationship exists between a dependent variable, Y, and an independent X and that relationship is expressed as
Y = a + bX + e
where “a” and “b” are unknown parameters and “e” is a random factor. This is also known as the Least Squares Regression Model. Because the parameters a and b are unknown we must try to find the values of these parameters to determine the relationship between X and Y. [Baye, 96-97].
Least Squares Regression Line
The least squares regression line is the one that minimizes the squared deviation of the vertical distances between the data points and the line or the smallest sum of squared errors between the line and the data points. It is expressed as
Y = ^a + ^bX
Where ^a and ^b are estimates of parameters a and b, respectively, and results in a more accurate estimate between the line and the data.
Case Study Example
SmallCable Co.
The company, SmallCable Co., is a relatively new internet service provider that has struggled to increase their profits in the Indianapolis area over the past year. The company’s CEO is wondering if his pricing is accurate for the Indianapolis area and has put together the quantity and average price (per client) of internet service over the past year as well as the average income (in thousands of dollars) of his customers. If SmallCable Co. were to increase their price by $10 dollars what would happen to their total revenue?
With price being the independent variable and the quantity being the dependent variable, the results are plotted on the graph below
Using the least squares regression a line can be plotted to determine the smallest sum of squared errors below
Using the regression analysis package in Excel, the data from SmallCable Co. can be input to produce the output below.
The t-Statistic
The t-statistic is used to estimate the ratio of a parameter’s value to its standard error. If the t-statistic is greater than 2, then one can be 95% confident that the value is statistically significant. As you can see above, the t-statistic for price and average income is greater than 2 in absolute value, therefore price and income are significant factors in determining quantity demanded.
The R-Square
The R-square explains overall fit of the estimated regression to the actual data. The closer the R-square is to 1, the better the fit. The R-square of .85 is fairly high, suggesting that the model explains 85 percent of the variation in the demand for internet service at SmallCable Co..
The F-Statistic
The F-statistic also provides a good measure of the overall fit of the regression line to the data. The significance of the F-statistic is extremely low so SmallCable Co. can be confident that this is an accurate model and does not include any random factors that may influence the data.
Results
The estimated demand equation is Q = -467.61 -1.05P + 7.79M. This equation says that if SmallCable Co. were to increase their price by $10 then quantity demanded would decrease by 1.05 units. Likewise, if customers were to have an increase in income by $1,000 this would increase demand by 7.79 units.
Questions
1. What is the t-statistic used for?
a. Determining the standard deviation of the value of the parameter
b. Determining if the value of the parameter is statistically significant
c. Measuring the overall fit of the regression line to the data
d. Determining the slope of the line
2. Why would managers use regression analysis?
a.To understand the shift in supply
b.To determine own-price elasticity
c. To estimate a demand function
d.To understand consumer behavior
3. The lease squares regression line for the equation Y=a+bX + e, is
a. Y = ^a + ^bX
b. Y = ^a +^b + X
c. Y = ^b + ^aX
d.Y = ^a + ^bX + e
4. What R-squared below would be considered the best fit to its actual data?
a. .95
b. 3
c. .3
d. Unable to determine
5. The least squares regression line
a.Is used to plot the demand curve
b. Is used to determine the smallest sum of squared errors
c. Is used to plot the supply curve
d. Is used to determine the highest sum of squared errors
Answers: 1. B, 2. C, 3. A, 4. A, 5. B
References
Baye, Michael R. Managerial Economics and Business Strategy. (7th edition). New York. McGraw Hill, 2010
Summaries
1.“Auctions including English auction, First price sealed bid auction, second price sealed bid auction, Dutch auction, winners curse” by Kristin Westerfield
Kristin’s paper describes in great detail several types of auctions (English auction, First price sealed bid auction, second price sealed bid auction, Dutch auction) which are used in the real world to buy and sell products and services through bidding. The author offers real world examples for each type of auction and even mentions additional auction types, such as a Japanese auction. In her paper, Kristin breaks down both the advantages and disadvantages for each type of auction.
2. “Game theory -prisoners dilemma, dominant strategy and Nash equilibrium, coordination games Finitely repeated games, the end-of period problem, infinitely repeated games, sequential move games” – by Michael Okunade Ayodele
Michael’s paper outlines the basic premises of game theory including the prisoner’s dilemma and dominant strategy. The author provides good detail regarding Nash equilibrium however his paper loses focus on the overall topic of game theory and instead goes on to discuss pricing strategies for firms.
3. “Create and answer a case study similar to American Airlines Case (Chapters 5, 7-11, 13) - Cases Available under Assignments tab- that looks at pricing to drive out competitors and then raising prices in the future.” – Brad Nelson
The author’s paper discusses predatory pricing as a pricing strategy utilized to drive competition from the market. The author uses Wal-Mart’s pricing strategy as an example of predatory pricing as the firm attempts to put local sellers out of business by offering staple items such as milk and butter at below cost in order to steal the local customers.
4. “Government policies- Price ceilings, price floors, excise taxes, and the economic effects of prohibition (alcohol, drugs, kidneys, etc)”- Anthony Christofaro
The author’s paper discusses the history of prohibition and its economic effects. It appears that prohibition has played a major influence through history as had both positive and negative socioeconomic effects in the past and up through present day. After the amendment was repealed due to the US government needing cash (through taxation of alcohol) the government has repeatedly tried to seek new ways to obtain additional revenue from its people.
5. “Economies of scale, diseconomies of scale, constant returns to scale, Efficiency in production. Economies of scope and how to maximize profit when there’re two production plants, long-run equilibrium-Include Coverage from Chapters 2-6 of Science of Success”- Charles Hays
Charles paper provides the definition to the topics listed above and explains in more detail how each of these concepts work in an industry. Charles goes on to further discuss a book called, “Science of Success” by Charles Koch, which describes Market Based Management and how it breaks down into five different dimensions (vision, virtue and talents, knowledge processes, decision rights, and incentives) Each of these aspects contribute to the success of a company.
Notes
Create and answer a case study similar to Memo 14 from Time Warner (Most Relevant Chapter 11)- that uses regression analysis to estimate demand functions and then shows how to price to increase profits.
John Slade
Memo 14
To: Pricing Manager, Midwest Region
From: Vice President, Marketing
Re: Pricing New Program Tiers
In response to the increated requests for a la carte pricing, we have decided to start a trial offering of smaller program tiers. As the first step of the trial, two small program packages will be offered to those using our basic package. The first is a sports package which includes NBA TV and the Soccer Channel. The second is a music package that includes MTV2 and GAC.
The trial offering will be limited to out Region 1 and Region 2 markets only. We estimate that our relevant incremental costs for the sports package are $1.45 per subscriber and the incremental costs for the music package are $1.20 per subscriber.
I am attaching a preliminary survey conducted by our marketing team that indicates anticipated sales at various pricing points. I would appreciate your recommendations regarding the pricing of these new program tiers. Thanks.
October 2nd
From The Economist
"Will the internet burst into tiers?" Sep 5th 2011, 15:40 by G.F. | SEATTLE
http://www.economist.com/blogs/babbage/2011/09/internet-traffic
Brief discussion on the possibility of tiered pricing strategies for the internet based off of a paper, "How Many Tiers? Pricing in the Internet Transit Market", written by Nick Feamster et al. at Georgia Tech. Interesting article however the paper reviewed is more robust as it discusses demand functions, elasticity of demand, etc.
October 9th
Interesting to note that tiered pricing for the internet is very favorable to the cable and mobile phone companies. Short article mentioning tiered pricing and wireless companies.
From iSuppli
"Tiered Pricing Becomes Business Model of Choice for Wireless Operators" September 30th 2011 by Jagdish Rebello, PhD
http://www.isuppli.com/Mobile-and-Wireless-Communications/MarketWatch/pages/Tiered-Pricing-Becomes-Business-Model-of-Choice-for-Wireless-Operators.aspx
Will need to dig further into specifics for the regression analysis.
Article from CNET
"What AT&T's tiered pricing means for you (FAQ)" by Marguerite Reardon | June 4, 2010 4:00 AM PDT
http://news.cnet.com/8301-30686_3-20006780-266.html
October 17th
From Bloomberg BusinessWeek
"Time Warner Cable Expands Internet Usage Pricing" March 31, 2009 By Tom Lowry
http://www.businessweek.com/technology/content/mar2009/tc20090331_726397.htm
"Time Warner's Net-Metering Precedent" June 4, 2008, by Catherine Holahan
http://www.businessweek.com/technology/content/jun2008/tc2008063_767960.htm
"Time Warner's Pricing Paradox", January 18, 2008, by Catherine Holahan
http://www.businessweek.com/technology/content/jan2008/tc20080118_598544.htm
October 29th
Chapter 11 in Baye, Two-part Pricing
Time Warner can enhance it's profits by packaging products (in the case of Memo 14, multiple channels) and offer these as a package deal instead of a la carte and thus forcing customers into making a decision of which package they must take (even though there may be channels that they don't watch).
Chapter 11 includes several pricing strategies that firms can use to enhance profits however tiered-pricing is not specifically mentioned in the chapter.
Regression analysis
Regression analysis will be used to determine the demand for the product offerings so that an appropriate price can be decided in order to maximize the profits of the company. Analysis of current internet usage of each customer related to uploading and downloading of files should be used as a quantity of measure.
Nov 19th
http://wirelessforamerica.org/
Nov 27th
Cable companies are looking for more ways to lower consumer surplus and increase their own profits when it comes to internet services. By utilizing a mix of tiered pricing, two part-pricing, and package bundling these companies extract as much profit as possible.
Two-part pricing allows a firm to extract all consumer surplus from consumers