ECN 510 Regression Case SP 2013

ECN 510 Regression Case SP 2013

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Consulting Project (Applied Regression Analysis) Pricing and Production Decisions at PoolVac, Inc
Objectives of this Applied Consulting Case
Understand how to use and interpret the computerized regression output for an estimated general demand equation to advise management at Pool Vac on pricing and production decisions that are of interest to the firm, namely Pool Vac.

In order to accomplish these goals, you must Understand the theory of demand for a price-setting firm and related elasticity concepts.

Perform the following standard diagnostic checks of validity of sample regression:
o Determine whether the individual estimated parameters are statistically significant
o Evaluate how well the regression equation fits the data by Examining and interpreting the R statistic (also known as the coefficient of determination).
Determining whether the regression equation is statistically significant.Interpret the estimated slope parameters of estimated demand equation.
Derive elasticity point estimates from the estimated demand equation.
Apply the elasticity estimates to advise PoolVac on various pricing, cash flow, and production decisions.

Background Research and References
The following portions of the Thomas-Maurice text are important references that you will want to consult prior to and during the write-up of this case
Chapter 2 (mainly the theory of demand on pp. 32-45) and chapter 6 (Elasticity and Demand)
Chapter 4 (mainly the review of regression analysis on pp. 118-138).
Chapter 7, section 7.2 (particularly the discussion and formulas on pp. 250-251) and section 7.3
(particularly the case of the estimated demand function and demand elasticities for Checker Pizza illustrated on pp. 254-255 and pp. 258-259).

Setting the Scene
PoolVac, Inc. manufactures and sells a single product called theSting Ray, which is a patent-protected automatic cleaning device for swimming pools. PoolVacs Sting Ray accounts for 65 percent of total industry sales of automatic pool cleaners. Its closest competitor, Howard Industries, has captured 18 percent of the market.
Demand for Sting Rays is specified to be a linear function of its price (P), average income for households that have swimming pools in the U.S (MAVG) and the price of the competing pool cleaner sold by Howard Industries (PH). The general linear form of the demand function is
Qd

=

a + b P + c MAVG + d PH.

1

The attached Minitab worksheet presents the last 26 observations (monthly data) on the price charged for a Sting Ray (P), average income of households with pools (MAVG), and the price Howard industries charged for its pool cleaner (PH).
Your research department has run a regression analysis using the monthly data provided the following Minitab regression output for the estimated demand equation, obtained from a least squares multiple regression on the 26 observations (monthly data).
3/16/2010
Regression Analysis: Q versus P, MAVG, PH
The regression equation is
Q = 2729 – 10.8 P + 0.0214 MAVG + 3.17 PH
Predictor
Constant
P
MAVG
PH

Coef
2728.8
-10.758
0.021420
3.166

S = 73.0546

SE Coef
531.7
1.330
0.009452
1.344

R-Sq = 96.6%

T
5.13
-8.09
2.27
2.36

P
0.000
0.000
0.034
0.028

Analysis of Variance
Source
Regression
Residual Error
Total
Source
P
MAVG
PH

DF
1
1
1

DF
3
22
25

SS
3379846
117414
3497260

MS
1126615
5337

F
211.10

P
0.000

Seq SS
3327368
22878
29600

Consulting Report Guidelines
In your role as economic analyst for PoolVac, Inc., you must write a (typed) report addressing the following issues relating to the analysis of regression results and the applications of the estimated demand equations and demand elasticities. Your report (case) should be readable and understandable by a third party (such as PoolVac management) that does not have the project instructions or regression results.
Introduction
Start your consulting report with an introduction that includes a brief statement of objectives of the report, focusing on your role as consultant to Pool Vac.
Then move on to the two major sections of the report:

2

The Estimated Demand Equation and Diagnostic Checks Recommendations on Pricing, Cash Flows, and Production
The Estimated Demand Equation and Diagnostic Checks Start this section with a brief discussion of the general linear demand equation (recopied here from on p. 1) that will be estimated:
Qd

= a + b P + c MAVG + d PH.

o Define each of the variables (underneath the demand equation) in the context of this case, then briefly discuss the expected or predicted signs of each the three slope parameters based on demand theory and/or your economic intuition in this specific market of this case.

Describe the sample (data) used to estimate the general demand function. Then report the estimated general demand equation for PoolVacs Sting Ray obtained from the regression analysis (copied from the computer output):
Qd = 2729 – 10.8 P + 0.0214 MAVG + 3.17 PH

 Variable (Predictor) Coefficient Estimate Standard Error T-ratio P-value P -10.758 (10.8) 1.33 ….. MAVG ….. ….. 2.27 0.034 PH 3.166 (3.17) …. ….. 0.028

Important: before you address the numbered topics below, be sure you have completed the analysis in the bullets above the table (starting on the prior page).

1. Complete the typed table above using the regression output on the prior page. This table summarizing the major regression results of your study. (Give the table a title.)

2. Now, use the reported p-values in your table to evaluate the statistical significance of the three estimated slope coefficients (variables). Your discussion should address the following questions or issues:
Which variables (estimated slope coefficients) are significant at the .05 or 5 percent significance level? Are any variables (estimated slope coefficients) significant at .01 or 1 percent level? Be sure to explain precisely how you decided the estimated slope coefficients (variables) were statistically significant or not.
Which of the three variables (estimated slope coefficients) is the most significant and why? (See the note below.)

3

Note: all analyses in the section above are based on p-values. The p-value approach compares the reported p-values on the t-ratios to the chosen significance level. (So be sure to review the decision rule for the p-value approach.)

3. Evaluate the overall fit of the sample regression equation to the data. A complete discussion should address the following:
Report the coefficient of determination (R2), and interpret the value of R2 in the context of this specific regression equation (dependent variable).

R2 tells us the predictive power of the regression equation. It equals the ratio of explained variation in Q and the total variation in Q. in our case the value is 96.6%, which is….

State whether the overall sample regression equation is significant at the 5 percent significance level and explain how you decided.

4. Discuss the interpretations of the algebraic signs and the numerical values of each of the three slope coefficients (parameters). Your discussion should address the following questions or issues:
Are the algebraic signs of the three slope coefficients consistent with your prior expectations based on consumer demand theory? (Explain for each slope sign.)
Interpret the numerical values of each of the three estimated slope parameters in the context of this specific regression (case).
Recommendations on Pricing, Cash Flows, and Production
In your job as economic analyst at PoolVac, you must use your regression results to advise the manager at PoolVac make a series of pricing and production decisions.

5. The manager of PoolVac, Inc. believes Howard Industries is going to price its automatic pool at \$240, and average income in the U.S is expected to be \$60,000. Based on this information, solve for the estimated (simple) demand function and the inverse demand function. Show all work (all steps in your derivations).

6. Assume the profit-maximizing quantity of Sting Rays is Q = 1650. Based on the information given in problem 5 and the estimated inverse demand function obtained, what price (P) should PoolVac charge for the Sting Ray if it wants to sell 1650 units?
Show all steps in your calculations.

7. Regardless of your prior answers, assume the current price (P) of a Sting Ray is \$290.00 and the current quantity (Q) of Sting Rays is 1650. Continue to assume that same average or expected income (M) of \$60,000 and that the current price of Howard Industries automatic pool cleaner (PH) is \$240.
a. Using the above assumed values of the variables (P, Q, M, PH ), the estimated slope coefficients from the regression output, and the relevant point elasticity formulas, compute each of the following estimated point elasticities:

The point price elasticity of demand for Sting Rays (E). Show all work (steps).
The point income elasticity of demand for Sting Rays (EM). Show all steps.
The point cross-price elasticity of demand for Sting Rays (EXH). Show all steps.

4

b. Is the algebraic sign of the estimated own price elasticity as expected? Briefly explain. Are the signs of the income elasticity and cross price elasticity as expected?

The sign of own price elasticity is negative as expected. This shows that the good obeys the law of demand. As price rises, quantity demanded falls.

The sign of income elasticity is positive, showing that as income rises demand rises.

The sign of cross elasticity is positive showing that the goods produced by both firms are substitutes of each other.
Briefly explain…

Use the relevant estimated point elasticities derived immediately above (in number 7) as needed to answer the following questions and provide economic consultation to the manager of PoolVac. Note: Again, you must use the estimated point elasticities and the general elasticity formulas to do the calculations necessary to answer all of the
remaining questions and all calculations and answers involve percentage changes. Again, as just noted immediately above, you must use the estimated point elasticities and general elasticity formulas to do the calculations necessary to answer all of the remaining questions, and all calculations and answers involve percentage changes.

8. The manager of PoolVac is interested in generating more cash flows (revenues) but is uncertain whether a price hike strategy or a price cut strategy will achieve this goal.
a. Advise the manager on whether a price hike or price cut is the appropriate strategy to increase total revenue for PoolVac. Clearly explain the economic rationale behind your pricing recommendation.
b. Suppose the manager wants to increase units sold of Sting Rays by 5 percent. Recommend a pricing strategy to achieve this target: that is, solve for the required percentage price increase or decrease to achieve the objective. Clearly show all steps in your calculations and briefly summarize your recommendation.

9. Assume your research department has forecast that average household income of pool owners is expected to rise by 2 percent over the next year. To advise the manager on production planning, calculate the predicted percentage increase or decrease in quantity demanded of Sting Rays as a result of the expected 2 percent increase in income
next year. (Show all steps in your calculations). Briefly summarize your calculations.

10. Suppose you learn that Howard Industries is expected to raise the price of its pool cleaner (PH) by 3 percent next period. Holding other factors constant, calculate the predicted percentage increase or decrease in quantity demanded of Stingrays as a result of the expected 3 percent rise in the price of pool cleaner (PH) sold by Howard Industries. (Show all steps in your calculations). Briefly summarize your calculations. 