Consulting Project (Applied Regression Analysis)Pricing and Production Decisions at PoolVac, IncObjectives of this Applied Consulting CaseUnderstand how to use and interpret the computerized regression output for an estimatedgeneral demand equation to advise management at Pool Vac on pricing and productiondecisions that are of interest to the firm, namely Pool Vac.In order to accomplish these goals, you mustUnderstand 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 statisticallysignificanto Evaluate how well the regression equation fits the data byExamining and interpreting the R statistic (also known as the coefficientof 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, andproduction decisions.Background Research and ReferencesThe following portions of the Thomas-Maurice text are important references that you will want toconsult prior to and during the write-up of this caseChapter 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 CheckerPizza illustrated on pp. 254-255 and pp. 258-259).Setting the ScenePoolVac, Inc. manufactures and sells a single product called theSting Ray, which is apatent-protected automatic cleaning device for swimming pools. PoolVacs Sting Rayaccounts for 65 percent of total industry sales of automatic pool cleaners. Its closestcompetitor, Howard Industries, has captured 18 percent of the market.Demand for Sting Rays is specified to be a linear function of its price (P), averageincome for households that have swimming pools in the U.S (MAVG) and the price of thecompeting pool cleaner sold by Howard Industries (PH). The general linear form of thedemand function isQd=a + b P + c MAVG + d PH.1The attached Minitab worksheet presents the last 26 observations (monthly data) onthe 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 providedthe following Minitab regression output for the estimated demand equation, obtainedfrom a least squares multiple regression on the 26 observations (monthly data).3/16/2010Regression Analysis: Q versus P, MAVG, PHThe regression equation isQ = 2729 â€“ 10.8 P + 0.0214 MAVG + 3.17 PHPredictorConstantPMAVGPHCoef2728.8-10.7580.0214203.166S = 73.0546SE Coef531.71.3300.0094521.344R-Sq = 96.6%T5.13-8.092.272.36P0.0000.0000.0340.028R-Sq(adj) = 96.2%Analysis of VarianceSourceRegressionResidual ErrorTotalSourcePMAVGPHDF111DF32225SS33798461174143497260MS11266155337F211.10P0.000Seq SS33273682287829600Consulting Report GuidelinesIn your role as economic analyst for PoolVac, Inc., you must write a (typed) reportaddressing the following issues relating to the analysis of regression results and theapplications of the estimated demand equations and demand elasticities. Your report(case) should be readable and understandable by a third party (such as PoolVacmanagement) that does not have the project instructions or regression results.IntroductionStart your consulting report with an introduction that includes a brief statementof objectives of the report, focusing on your role as consultant to Pool Vac.Then move on to the two major sections of the report:2The Estimated Demand Equation and Diagnostic ChecksRecommendations on Pricing, Cash Flows, and ProductionThe Estimated Demand Equation and Diagnostic ChecksStart 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 thecontext of this case, then briefly discuss the expected or predicted signsof each the three slope parameters based on demand theory and/or youreconomic 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 Rayobtained from the regression analysis (copied from the computer output):QdVariable(Predictor)PMAVGPH=2729 â€“ 10.8 P + 0.0214 MAVG + 3.17 PHCoefficientEstimateStandardErrorT-ratioP-valueImportant: before you address the numbered topics below, be sure you havecompleted 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. Thistable 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 significanceof the three estimated slope coefficients (variables). Your discussion should address thefollowing questions or issues:Which variables (estimated slope coefficients) are significant at the .05 or 5percent significance level? Are any variables (estimated slope coefficients)significant at .01 or 1 percent level? Be sure to explain precisely how you decidedthe estimated slope coefficients (variables) were statistically significant or not.Which of the three variables (estimated slope coefficients) is the mostsignificant and why? (See the note below.)3Note: all analyses in the section above are based on p-values. The p-value approachcompares the reported p-values on the t-ratios to the chosen significance level. (So besure to review the decision rule for the p-value approach.)3. Evaluate the overall fit of the sample regression equation to the data. A completediscussion should address the following:Report the coefficient of determination (R2), and interpret the value of R2 in thecontext of this specific regression equation (dependent variable).State whether the overall sample regression equation is significant at the 5 percentsignificance level and explain how you decided.4. Discuss the interpretations of the algebraic signs and the numerical values of each ofthe three slope coefficients (parameters). Your discussion should address the followingquestions or issues:Are the algebraic signs of the three slope coefficients consistent with your priorexpectations based on consumer demand theory? (Explain for each slope sign.)Interpret the numerical values of each of the three estimated slope parameters inthe context of this specific regression (case).Recommendations on Pricing, Cash Flows, and ProductionIn your job as economic analyst at PoolVac, you must use your regression results toadvise 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 itsautomatic pool at $240, and average income in the U.S is expected to be $60,000. Basedon this information, solve for the estimated (simple) demand function and the inversedemand function. Show all work (all steps in your derivations).6. Assume the profit-maximizing quantity of Sting Rays is Q = 1650. Based on theinformation 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 thatsame average or expected income (M) of $60,000 and that the current price of HowardIndustries automatic pool cleaner (PH) is $240.a. Using the above assumed values of the variables (P, Q, M, PH ), the estimated slopecoefficients 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.4b. Is the algebraic sign of the estimated own price elasticity as expected? Brieflyexplain. Are the signs of the income elasticity and cross price elasticity as expected?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 themanager of PoolVac. Note: Again, you must use the estimated point elasticities and thegeneral elasticity formulas to do the calculations necessary to answer all of theremaining questions and all calculations and answers involve percentage changes.Again, as just noted immediately above, you must use the estimated point elasticitiesand general elasticity formulas to do the calculations necessary to answer all of theremaining questions, and all calculations and answers involve percentage changes.8. The manager of PoolVac is interested in generating more cash flows (revenues) but isuncertain 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 toincrease total revenue for PoolVac. Clearly explain the economic rationale behind yourpricing 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 requiredpercentage price increase or decrease to achieve the objective. Clearly show all stepsin your calculations and briefly summarize your recommendation.9. Assume your research department has forecast that average household income of poolowners is expected to rise by 2 percent over the next year. To advise the manager onproduction planning, calculate the predicted percentage increase or decrease inquantity demanded of Sting Rays as a result of the expected 2 percent increase in incomenext 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 poolcleaner (PH) by 3 percent next period. Holding other factors constant, calculate thepredicted percentage increase or decrease in quantity demanded of Stingrays as aresult of the expected 3 percent rise in the price of pool cleaner (PH) sold by HowardIndustries. (Show all steps in your calculations). Briefly summarize your calculations.5Appendix: 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 futuresales and prices. Estimation of demand functions is most often accomplished using the techniqueof regression analysis. When demand is specified to be linear in form, the coefficients on eachof the explanatory variables measure the rate of change in quantity demanded as that explanatoryvariable changes, holding all other explanatory variables constant. In linear form, the empiricaldemand specification isQ=a+bP+cM+dPRwhere 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 asAs in any regression analysis, the statistical significance of the parameter estimates can beassessed by performing t-tests or examining p-values.The method of estimating the parameters of an empirical demand function depends on whetherthe 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 â€œmarketdeterminedâ€ by the intersection of demand and supply.Managers of price-setting firms set the price of the product they sell by producing thequantity associated with the chosen price on the downward-sloping demand curve facingthe firm. The demand curve for price-setting firms is estimated using the ordinaryleast-squares (OLS) method of estimation.Econometric models use an explicit structural model to explain the underlying economicrelations. Econometric forecasting can be employed to forecast future demand for price-settingfirms. The three steps for forecasting the future demand for a price-setting firm are1. Estimate the firmâ€™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 thefuture the forecast is made, the greater the uncertainty. Incorrect specification of the demandequation can seriously undermine the quality of a forecast. An even greater problem for accurateforecasting is posed by the occurrence of structural changes that cause turning points in thevariable being forecast. Forecasts often fail to predict turning points. While there is nosatisfactory way to account for unexpected structural changes, forecasters should note that thefurther 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) .