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
o Evaluate how well the regression equation fits the data by
Examining and interpreting the R statistic (also known as the coefficient
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
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
a + b P + c MAVG + d PH.
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).
Regression Analysis: Q versus P, MAVG, PH
The regression equation is
Q = 2729 - 10.8 P + 0.0214 MAVG + 3.17 PH
S = 73.0546
R-Sq = 96.6%
R-Sq(adj) = 96.2%
Analysis of Variance
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.
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:
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:
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):
2729 - 10.8 P + 0.0214 MAVG + 3.17 PH
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.)
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).
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.
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?
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
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.
Appendix: Partial Summary of Ch 7 -Demand Estimation and Forecasting
(based largely on sections 7.2 and 7.3 of text)
This chapter presented the basic techniques of estimating demand functions and forecasting future
sales and prices. Estimation of demand functions is most often accomplished using the technique
of regression analysis. When demand is specified to be linear in form, the coefficients on each
of the explanatory variables measure the rate of change in quantity demanded as that explanatory
variable changes, holding all other explanatory variables constant. In linear form, the empirical
demand specification is
where Q is the quantity demanded, P is the price of the good or service, M is consumer income,
and PR is the price of some related good R. The estimated demand elasticities are computed as
As in any regression analysis, the statistical significance of the parameter estimates can be
assessed by performing t-tests or examining p-values.
The method of estimating the parameters of an empirical demand function depends on whether
the price of the product is market-determined or manager-determined. Managers of pricetaking firms do not set the price of the product they sell; rather, prices are endogenous or \"marketdetermined\" by the intersection of demand and supply.
Managers of price-setting firms set the price of the product they sell by producing the
quantity associated with the chosen price on the downward-sloping demand curve facing
the firm. The demand curve for price-setting firms is estimated using the ordinary
least-squares (OLS) method of estimation.
Econometric models use an explicit structural model to explain the underlying economic
relations. Econometric forecasting can be employed to forecast future demand for price-setting
firms. The three steps for forecasting the future demand for a price-setting firm are
1. Estimate the firm\\\'s demand function.
2. Forecast the future values of the demand-shifting variables.
3. Calculate the location of future demand.
Footnote: When making forecasts, analysts must be careful to recognize that the further into the
future the forecast is made, the greater the uncertainty. Incorrect specification of the demand
equation can seriously undermine the quality of a forecast. An even greater problem for accurate
forecasting is posed by the occurrence of structural changes that cause turning points in the
variable being forecast. Forecasts often fail to predict turning points. While there is no
satisfactory way to account for unexpected structural changes, forecasters should note that the
further into the future you forecast, the more likely it is that a structural change will occur.
See Case Study in CH 7: Estimating the Demand Facing a Pizza Firm (starts on p.255) .
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