A food-products company has recently introduced a new line of fruit pies in six

Question

A food-products company has recently introduced a new line of fruit pies in six U.S. cities: Atlanta, Baltimore, Chicago, Denver, St. Louis, and Fort Lauderdale. Based on the pie%u2019s apparent success, the company is considering a nationwide launch. Before doing so, it has decided to use data collected during a two-year market test to guide it in setting prices and forecasting future demand.

For each of the six markets, the firm has collected eight quarters of data for a total of 48 observations. Each observation consists of data on quantity demanded (number of pies purchased per week), price per pie, competitors%u2019 average price per pie, income, and population. The company has also included a time-trend variable for each observation. A value of 1 denotes the first quarter observation, 2 the second quarter, and so on, up to 8 for the eighth and last quarter.

A company forecaster has run a regression on the data, obtaining

the results displayed in the table.

Coefficient Standard Error of Coefficent Mean of ValueVariable

Intercept -4,516.3 4,988.2 ------

Price (dollars) -3,690.6 702.8 7.50

Competitor's

price (dollars) 4226.5 851.0 6.50

Income ($000) 777.1 66.4 40

Population(000) .40 .31 2,300

Time (1 to 8) 356.1 92.3 -----

N = 48. R2=.93. Standard error of regression = 1,442

a. Which of the explanatory variables in the regression are statistically significant? Explain. How much of the total variation in pie sales does the regression model explain? -

b. Compute the price elasticity of demand for pies at the firm%u2019s mean price: ($7.50) and mean weekly sales quantity (20,000 pies). Next, compute the cross-price elasticity of demand. Comment on these estimates.

c. Other things equal, how much do we expect sales to grow (or fall over the next year?

d. How accurate is the regression equation in predicting sales next quarter? Two years from now? Why might these answers differ?

e. How confident are you about applying these test-market results to decisions concerning national pricing strategies for pies?

Explanation / Answer

1. Demand Estimationand Forecasting.AjaiKurian Mathew Harshavardhan RPremRanjanShivraj Singh Negi

2. DEFINITIONEstimation of various demand function of a firm(industry) or market through various processes.For practical purposes ,demand function for a firm or market has to be estimated from the empirical data.

3. .Broadly there are two types methods of Estimation:Simple Method of Estimation(5 steps)Statistical method of Estimation(Econometric analysis,7 Steps).

4. STEPS FOR DEMAND ESTIMATIONStatement of a theory or hypothesis.Model specification.Data collection.Estimation of parameters.Checking goodness of fit.Hypothesis testing.Forecasting.

5. MODEL SPECIFICATIONWhat variables to be included and what mathematical form to followed.Need to formulate many alternative models.Deterministic(certainity) and Statistical relationshipIt is assumed to begin with that the relationship is deterministic. With a simple demand curve the relationship would therefore be:Q =f (P)

6. DATA COLLECTIONThis stage can only be performed after the demand model has been specified, otherwise it is not known for which variables we have to collect data.Types of data:Time series data Cross sectional data Pooled data

7. Estimation of parametersCoefficient of the variables.Relates the effects of Independent variable upon the dependent variable.Regression analysis is used to calculate these values.

8. Methods of Estimating DemandConsumer survey Market ExperimentStatistical methods

9. Consumer Survey Seeking information through questionnaire , interviews etc.Asking information about their consumption behavior ie, buying habits , motives etc.

10. Consumer survey AdvantagesThey give uptodate information about the current market scenario .Much useful information can be obtained that would be difficult to uncover in other ways; for example, if onsumersare ignorant of the relative prices of different brands, it may be concluded that they are not sensitive to price changes.This can be exploited by the firms for their best possible interest.DisadvantagesValidity ReliabilitySample Bias

11. Market Experiment Here consumers are studied in an artificial environment .Laboratory experiments or consumer clinics are used to test consumer reactions to changes in variables in the demand function in a controlled environment.Need to be careful in such experiments as the knowledge of being in the artificial environment can affect the consumer behavior.

12. Market experiment Advantages Direct observation of the consumers takes place rather than something of a hypothetical theoretical model .DisadvantagesThere is less control in this case, and greater cost; furthermore, some customers who are lost at this stage may be difficult to recover.Experiments need to be long lasting in order to reveal proper result.

13. Statistical methodsThese are various quantitative methods to find the exact relationship between the dependent variable and the independent variable(s).The most common method is regression Analysis :Simple (bivariate) Regression: Y = a + bXMultiple Regression: Y = a +bX1 + c X2 +dX3 +..

14. Limitations of Statistical methodsThey require a lot of data in order to be performed.They necessitate a large amount of computation.

15. Linear Regression %u2013 OLS Method Applicable when our model employs a linear relationship between X and Y.Find a line %u0176 = a + bX which minimizes sum of square errors %u03A3(Yi%u2013%u0176i)2.Find a and b by partial differntiation.

16. Goodness of FitRegression %u2013 type of relationshipCorrelation %u2013 strength of relationshipAn alternative to visual inspectionMeasures:Correlation coefficient (r)Coefficient of Determination (R2)

17. Correlation CoefficientMeasures the degree of linear correlationSmall correlation may imply weak linear, but strong non-linear relationship.Hence, visual inspection is also important.Does not talk about causationCausation may be reversed, circular, endogenic or third-partyHence, correlation cannot tell you how good a model is.

18. Correlation CoefficientIt can be calculated as follows:r varies from 0 to 1.A high value of r implies that the points are very closely scattered around the regression line.

19. Coefficient of Determination (R2)The proportion of the total variation in the dependent variable that is explained by the relationship with the independent variable.

21. Coefficient of Determination (R2)TD: Total DeviationED: Explained DeviationUD: Unexplained DeviationTD = ED + UD%u03A3TD2 = %u03A3ED2 + %u03A3UD2

22. Coefficient of Determination (R2)R2 also varies from 0 to 1.Low R2 values imply that:The model is not a good fit. Perhaps a power regression model is needed?We are missing important variables. Look at Multivariate regression?R2is preferred to Correlation Coefficient (r)

23. Power RegressionMathematical form: Y=aXbCannot directly use the OLS method. However by ignoring error terms and taking logarithm we get a linear model.log(Y) = log(a) + b*log(X)

24. Significance Testingt-test: Test of significance of a particular variable.t-stat = estimated coefficient/standard errorRule of thumb for a 95% confidence interval: >2Implies that the independent variable truly impacts the dependent variableSpecially useful in Multivariate regressionF-test: Checks if variation in X explains a significant amount of the variation in Y.

25. The Pizza DillemnaEstimate the demand for Pizza by college students.Select variables for the model that you believe are:Relevant, and for whichData can be found

26. The Pizza DillemnaAverage number of pizza slices consumed per month by students (Y)Average Selling Price of a Pizza slice (X1)Annual Course Fee %u2013 proxy for student income (X2)Average price of a soft drink %u2013 complementary product (X3)Location of the campus %u2013 proxy for availability of substitutes (X4) (1 for city campus, 0 for outskirts)

27. The Pizza DillemnaY = a + b1X1 + b2X2 + b3X3 + b4X4Results of linear regression based on actual dataY = 26.67 %u2013 0.088 X1 + 0.138 X2 - 0.076 X3 - 0.0544 X4 (0.018) (0.087) (0.020) (0.884)R2 = 0.717 Adjusted R2 = 0.67 F= 15.8Std Error of the Y-estimate = 1.64(The standard errors of the coefficients are listed in parenthesis)

28. The Pizza DillemnaValues of Elasticity:Price Elasticity -0.807Income Elasticity 0.177Cross-price Elasticity -0.767T-test: b2 and b4 are not significant.R2 = 0.717

29. Demand ForecastingEstimation or prediction of future demand for goods and services. Nearer it is to its true value, higher is the accuracy. Active and Passive forecasts. Short term, long term and medium term. Capacity utilization, Capacity expansion and Trade Cycles. Different forecasts needed for different conditions, markets, industries. Approaches to Forecasting: Judgmental, Experimental, Relational/Causal, Time Series Approaches.

30. Demand ForecastingRequirements for Demand Forecasting. Elements related to Consumers.Elements concerning the Suppliers.Elements concerning the Markets or Industry. Other Exogenous Elements like taxation, government policies, international economic climate, population, income etc. Estimating general conditions, estimating the total market demand and then calculating the firm%u2019s market share. Multiple methods of forecasting, used depending upon suitability, accuracy and other factors. Subjective methods used when appropriate data is not available.

31. Demand ForecastingSubjective methods depend on intuition based on experience, intelligence, and judgment. Expert%u2019s opinion survey, consumer%u2019s interview method and historical analogy method. Survey MethodsUsing questionnaires with either complete enumeration or sample survey method. Using consumers, suppliers, employees or experts (Delphi method). Problems of survey methods. Less reliable and accurate due to subjectivity, but give quick estimates and are cost saving.

32. Demand ForecastingHistorical Analogy Method.Forecasting for new product or new market/area. Difficulties in finding similar conditions. Test Marketing involves launching in a test area which can be regarded as true sample of total market. Difficulties of cost, time, variation of markets and imitation by competitors.

33. Demand ForecastingSystematic forces may show some variation in time series of sales data of a product. Basic parameters like population, technology. Business cycles, seasonal variations and then random events. Main focus is to find out the type of variation and then use it for long term forecasting. Use judgment to extrapolate the trend line obtained from sales data. OLS method to prepare a smooth curve is a better option. We may obtain a linear trend, quadratic trend, logarithmic trend or exponential trend each of which gives us a different information about the behavior of demand.

34. Demand ForecastingLinear: Y = a0 + a1(t)Quadratic: Y = a0 + a1(t) + a2(t)2Logarithmic :Log Y = b0 + b1 log (t)Exponential :Log Y = c0 + c1 (t)Choice of the equation is based on multiple correlation coefficient (R) of OLS. Averaging is used to remove any large scale fluctuations.

35. Demand ForecastingThe sales curve eventually is an S shaped %u2018product life cycle curve%u2019. Price elasticities vary in different stages. Highest in later stages as substitutes are available. All these stages give exponential shape to the curve. Trend method assumes little variations in business conditions. Knowledge of curve helps in planning marketing and planning for the product.

36. Demand ForecastingLeading Indicators or Barometric method. Time as a explanatory variable may not always show a liner relation, so we use another commodity as an indicator for sales. Regression method : Identify the demand factors for commodity and expected shape of the demand function. Use regression to fit the time series data. Higher the R2 the better is explanation. Inadequacy of data, multi-collinearity, auto-correlation, heteroscedasticity and lack of direct estimates of future values of explanatory variables.

37. Need for ForecastingLong Range Strategic PlanningCorporate Objectives: Profit, market share, ROCE,strategic acquisitions, international expansion, etc.Annual BudgetingOperating Plans: Annual sales, revenues, profitsAnnual Sales PlansRegional and product specific targetsResource Needs PlanningHRM, Production, Financing, Marketing, etc

38. Factors affecting Method SelectionCost-benefit for developing forecasting modelComplexity of behavioral relationships to be forecastedThe accuracy of forecasts requiredThe lead time required for making decisions dependent on results of the model

39. Box Jenkins MethodAlso known as ARIMA(%u2018Auto-Regressive Integrated Moving Average%u2019) models, this is an empirically driven method of systematically identifying, estimating, analyzing and forecasting time series.Used only for short term predictions . Suitable only for demand with stationary time series sales data,i.e the one that does not reveal the long term trend.The models are designated by the level of autoregression,integration and moving averages(P,d,q) where P is the order of regression,d is the order of integration and q is the order of moving average.

40. Box Jenkins MethodThere are 3 components of the ARIMA process:AR(Autoregressive) process.MA(Moving Average) process.Integration process.

41. Box Jenkins MethodAR process: Of order %u2018p%u2019, generates current observations as a weighted average of the past observations over p periods, together with a random disturbance in the current period. Yt=%u03BC+a1Yt-1+a2Yt-2+%u2026.+apYt-p+et

42. Box Jenkins MethodMA process: Order q, each observation of Yt is generated by the weighted average of random disturbances over the past q periods. Yt= %u03BC +et-c1et-1-c2et-2+%u2026.-cqet-qIntegrated Process: Ensures that the time series used in the analysis is stationary. The previous 2 equations are combined to form:Yt=a1Yt-1+a2Yt-2+...+apYt-p+%u03BC+et-c1et-1-c2et-2+%u2026-cqet-q

43. Input-output modelAn input-output model uses a matrix representation of a nation's (or a region's) economy to predict the effect of changes in one industry on others and by consumers, government, and foreign suppliers on the economy.One who wishes to do work with input-output systems must deal skillfully with industry classification, data estimation, and inverting very large, ill-conditioned matrices.Wassily Leontief, won the Nobel Memorial Prize in Economic Sciences for his development of this model in 1973.

44. Input-output model Consider 4 industries, Industry 1: X1=X11+X12+X13+X14+C1 Industry 2: X2=X21+X22+X23+X24+C2 Industry 3: X3=X31+X32+X33+X34+C3 Industry 4: X4=X41+X42+X43+X44+C4Xij= Output of the industry i which is purchased by industry j for the producion of its output.Ci = Demand of the customers for products for final use .

45. Input-output modelLet Xij=aijXj,i=1 to 4,j=1 to 4 or Xij/Xj=aij where aij is the output of ith industry required to produce unit output of jth industry. Thus X1=a11X1+a12X2+a13X3+a14X4+C1 X2=a21X1+a22X2+a23X3+a24X4+C2 X3=a31X1+a32X2+a33X3+a34X4+C3 X4=a41X1+a42X2+a43X3+a44X4+C4

46. Input-output modelI=Unit Matrix A=Technology Coefficient MatrixX=Output VectorC=Final Demand Vector

47. Input-output modelX=AX+C[I-A]X=CX=[I-A]-1C

48. Input-output modelIf we know/get a forecast for X, total output, we can easily find labor, capital & other requirements. This makes Input-Output method a powerful tool for planning.To find the component D(represented as C before),Demand, one may use the previously discussed methods or a simple projection method.

49. Input-output modelDit=Di0(1+ %u03C1i)t Dit-Level of Final Demand%u03C1i= Growth rate of final Demand Pt=P0(1+s)t Pt-Population at time t s = Rate of growth of Populationdit=di0(1+x)t dit = Per-capita consumption in time t x = rate of growth of per-capita consumption in time t.

50. Input-output modeleyi=(%u2206dit/dit)/(%u2206 y/y)%u2206 eyi=Income elasticity of Demand r=%u2206 y/y= Rate of growth of per capita income. Thus eyi=x/r;x= eyi *rThus dit=di0(1+eyi*r)tdit=Dit/Pt, di0=Di0/P0

51. Input-output modelWe get,Dit/Pt=Di0/P0*(1+eyi*r)tDit=Di0/P0*(1+eyi*r)t * P0*(1+s)ti.eDit=Di0*(1+eyi*r)t*(1+s)tComparing with the original eqn. for Demand,%u03C1i=[(1+eyi*r)(1+s)]-1

52. Input-output modelThis eqn. gives the growth rate of final demand for the ith commodity in terms of its income elasticity of demand, target rate of growth of per capita income and population growth.If these parameters are known exogenously then %u03C1i can be computed and final demand Dit can be predicted.

53. Input-output modelAdvantages:It gives sector wise breakdown of demand forecasts for commodities.Helps the firm to formulate its marketing policies in a better way by taking into account various market segment strengths for its products.

54. Input-output modelDisadvantages:Input-output tables are not available every year. Sometimes there may be large gap between the years for which input-output coefficients are available and the years for which the forecasts are needed. Larger the timegap,less stable will be the coefficients, thus reducing the forecasting accuracy.Also changes in the production technology,tastes,preferences etc during the period makes the forecast less valid.

55. Controlling the ForecastControl of forecasting is the process of comparison,evaluation,interpretation and auditing the performances of the firm against objectives and standards forecasted.We measure the inaccuracy in forecasting in terms of Percentage Forecasting Inaccuracy(PFI).

56. Controlling the ForecastPFI1=(|Yt-Yt%u2019|*100)/YtPFI2=( *100)/PFI1 stands for one period forecast and PFI2 stands for multi-period forecasts, t for time, k for length of time.Based on these ratios we fix some acceptable limits for them which depends on the commodity type, market nature, forecasting method.

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