Chapter 7 FORECASTING QUESTIONS & ANSWERS


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1 Chapter 7 FORECASTING QUESTIONS & ANSWERS Q7.1 Accurate company sales and profit forecasting requires careful consideration of firmspecific and broader influences. Discuss some of the microeconomic and macroeconomic factors a firm must consider in its own sales and profit forecasting. Q7.1 ANSWER The better a company can assess future demand, the better it can plan its resources. Every corporation is exposed to three types of factors influencing demand: company, competitive and macroeconomic factors. Microeconomic companyrelated factors include market share trends, changes in strategy and implementation, and changes in brand value. Microeconomic industryrelated factors include competitor advertising, competitor product offerings, market share. Macroeconomic factors that must be considered include income, economic growth, interest rates, and shocks. There are several methods used to assess and forecast demand. None yields demand numbers that are a 100% successful or guaranteed. However, using more than one imperfect method has proven helpful in improving forecast accuracy and confidence. Q7.2 Forecasting the success of new product introductions is notoriously difficult. Describe some of the macroeconomic and microeconomic factors that a firm might consider in forecasting sales for a new teeth whitening product. Q7.2 ANSWER To forecast market demand for any new product introduction, market size research must be combined with productspecific information. A useful approach would combine macroeconomic trend information with data on microeconomic and competitive performance. Customers will only buy a product if they perceive a need and are able to pay for the new good or service. Of course, ability to pay tends to be a strong determinant of demand for bigticket items, perceived need may be more important for smallticket items, like teeth whitening products. Advertising capability or brand name reputation is also apt to be important because consumers must be aware of new product or service offerings and perceive a given company's
2 offerings as having the best value. In practice, market size research is combined with market share research to forecast product and corporate demand. Econometric models are sometimes used for answering a wide variety of Awhat questions regarding the future. This stems from the fact that econometric models reflect the causal relation between Y (the forecast value) and a series of independent X variables. When a range of X values relating to various pessimistic to optimistic scenarios concerning future events is incorporated into a given econometric model, the resulting effects on Y become readily apparent. Thus, quantifiable answers to various Awhat questions can be obtained. Q7.3 Blue Chip Financial Forecasts gives the latest prevailing opinion about the future direction of the economy. Survey participants include 50 business economists from Deutsche Banc Alex Brown, Banc of America Securities, Fannie Mae, and other prominent corporations. Each prediction is published along with the average, or consensus forecast. Also published are averages of the 10 highest and 10 lowest forecasts; a median forecast; the number of forecasts raised, lowered, or left unchanged from a month ago; and a diffusion index that indicates shifts in sentiment that sometimes occur prior to changes in the consensus forecast. Explain how this approach helps limit the steamroller or bandwagon problems of the panel consensus method. Q7.3 ANSWER Although the panel consensus method often results in forecasts that embody the collective wisdom of consulted experts, it can be unfavorably affected by the forceful personality of one or a few key individuals. To mitigate such problems, the forecasting approach adopted by Blue Chip Financial Forecasts is similar to the Delphi method. In the Delphi method, members of a panel of experts individually receive a series of questions relating to the underlying forecasting problem. Responses are analyzed by an independent party, who then tries to elicit a consensus opinion by providing feedback to panel members in a manner that prevents direct identification of individual positions. Because the 50 business economists surveyed by Blue Chip Financial Forecasts are never collected at a single location, the steamroller or bandwagon problems of the panel consensus approach tend to be minimized. The force of personality is strongest in person, and surveys of 50 top economic forecasters are not apt to be as affected by group pressure as would be the case if Blue Chip Financial Forecasts were derived from regular group meetings. Q7.4 "Interest rates were expected to increase by 85% of all consumers in the May 2004 survey, more than ever before," said Richard Curtin, the Director of the University of Michigan=s Surveys of Consumers. "More consumers in the May 2004 survey cited the advantage of obtaining a mortgage in advance of any additional increases in interest rates than any other time in nearly ten said Curtin. Discuss this
3 Forecasting Q7.4 ANSWER statement and explain why consumer surveys are an imperfect guide to consumer expectations. Survey data can be highly useful in shortterm forecasting when carefully used to elicit consumer perceptions and attitudes. However, survey data are when they don't relate to actual market transactions and can be unreliable when consumers have incentives to misreport information. In the case of interest rate forecasting, consumers may have little tangible evidence upon which to base their expectations, and little expertise in interest rate forecasting. Moreover, even if consumers have an accurate fix on the future pattern of interest rates, they have incentives to complain about likely increases in the hope that by voicing this concern they might cause some moderation in tightening by the monetary authorities. Q7.5 Explain why revenue and profit data reported by shippers such as FedEx Corp. and United Parcel Service Inc. are apt to provide useful information about trends in the overall economy. Q7.5 ANSWER Revenue and profit data reported by shippers such as FedEx Corp. and United Parcel Service Inc. are apt to provide useful information about trends in the overall economy because the pace of goods shipped is a leading indicator of future sales. In a sense, FedEx and UPS find out about the sales revenues of major manufacturers before the stockholders of manufacturers whose goods are being shipped. Sales and profit numbers jump for shippers before sales and profit numbers tied to shipped goods reach the audited financial statements of manufacturers. For example, in mid2004, FedEx Corp. reported a 47% jump in fiscal fourthquarter profit and offered an increasingly optimistic outlook as the economic rebound continued to spread throughout its customer base. The Memphis, Tenn., company saw growing signs of what Frederick W. Smith, its chairman, president and chief executive, called a "solid, broadbased economic recovery" that includes industrial, durablegoods and retail shipments. FedEx results provided strong evidence that more businesses around the world were revving up their operations and replenishing inventories depleted during the economic slump, and that the economy's improvement was "strong and sustainable."
4 Chapter 7 Q7.6 AThe economy is on the verge of faster Federal Reserve Chairman Alan Greenspan testified. "We believe we are at a turning point. Our best judgment is that things will be improving after sluggish growth and a fitful recovery from What makes forecasting turning points difficult? What methods do economists use to forecast turning points in the overall economy? Q7.6 ANSWER All economic data have a strong trend elements, and turning points are, by definition, changes in trend. A basic shortcoming of trend projection is that the method is incapable of forecasting the magnitude or duration of divergences from trend and is not helpful for indicating fundamental changes in trend (i.e., turning points). Therefore, simple trend projection methods are incapable of forecasting the magnitude of cyclical fluctuations, seasonal variation, and irregular or random influences. To forecast the magnitude of such deviations from trend, managers often employ the barometric approach to forecasting. Q7.7 Would a linear regression model of the advertising/sales relation be appropriate for forecasting the advertising levels at which threshold or saturation effects become prevalent? Explain. Q7.7 ANSWER No, a linear model of the advertisingsales relation is not appropriate for estimating the advertising levels where or effects become prevalent. A nonlinear method of estimation is appropriate when advertising by a firm or an industry is subject to such influences. Quadratic, loglinear, or logistic models are often employed for this purpose. Q7.8 Perhaps the most famous early econometric forecasting firm was Wharton Economic Forecasting Associates (WEFA), founded by Nobel Prize winner Lawrence Klein. A spinoff of the Wharton School of the University of Pennsylvania, where Klein taught, WEFA was merged with Data Resources Inc. in 2001 to form Global Insight. Describe the data requirements that must be met if econometric analysis is to provide a useful forecasting tool. Q7.8 ANSWER
5 Forecasting If the statistical analysis of economic relations, or econometrics, is to provide a fruitful tool for forecasting, a number of important conditions must be met. First, a sufficient number of sample observations must be available for analysis. For small populations and simple linear regression models, as few as 30 or 40 observations may be sufficient. Larger samples are needed for larger populations and when particularly difficult forecasting problems suggest the use of highly sophisticated econometric models, some of which entail many different structural relations (equations). Second, all relevant variables must be properly incorporated in the analysis. This involves data measurement and model specification issues that must be addressed. And third, there must be a high degree of stability over time between the dependent and independent variables under consideration. Q7.9 Cite some examples of forecasting problems that might be addressed using regression analysis of complex multipleequation systems of economic relations. Q7.9 ANSWER Econometric analysis of multipleequation systems of economic relations is a forecasting technique that is useful for reflecting the effects of important economic changes on related sectors, industries, or firms. It is most useful when indirect linkages between sectors are few in number and can be estimated with a great deal of precision. At the national level, for example, this type of econometric analysis has been used extensively to analyze changes in GDP, interest rates, energy, and water requirements. Similarly, firms might use a system method of analysis to measure the effects of changing energy, labor, or capital costs on demand conditions for related products. Q7.10 What are the main characteristics of accurate forecasts? Q7.10 ANSWER The main characteristics of accurate forecasts are a close correspondence, on average, between actual and forecast values and a high correlation between the actual and forecast series. When these two criteria are met, actual and forecast data will be closely related, and a desirable low level of average forecast error (root mean squared forecast error) will be apparent
6 Chapter 7 SELFTEST PROBLEMS & SOLUTIONS ST7.1 Gross Domestic Product (GDP) is a measure of overall activity in the economy. It is defined as the value at the final point of sale of all goods and services produced during a given period by both domestic and foreignowned enterprises. GDP data for the period shown in Figure 7.3 offer the basis to test the abilities of simple constant change and constant growth models to describe the trend in GDP over time. However, regression results generated over the entire period cannot be used to forecast GDP over any subpart of that period. To do so would be to overstate the forecast capability of the regression model because, by definition, the regression line minimizes the sum of squared deviations over the estimation period. To test forecast reliability, it is necessary to test the predictive capability of a given regression model over data that was not used to generate that very model. In the absence of GDP data for future periods, say , the reliability of alternative forecast techniques can be illustrated by arbitrarily dividing historical GDP data into two subsamples: a year test period, and a year forecast period. Regression models estimated over the test period can be used to actual GDP over the period. In other words, estimation results over the subperiod provide a forecast model that can be used to evaluate the predictive reliability of the constant growth model over the forecast period. A. Use the regression model approach to estimate the simple linear relation between the natural logarithm of GDP and time (T) over the subperiod, where ln GDP t = b 0 + b 1 T t + u t and ln GDP t is the natural logarithm of GDP in year t, and T is a time trend variable (where T 1950 = 1, T 1951 = 2, T 1952 = 3,..., and T 1999 = 50); and u is a residual term. This is called a constant growth model because it is based on the assumption of a constant percentage growth in economic activity per year. How well does the constant growth model fit actual GDP data over this period? B. Create a spreadsheet that shows constant growth model GDP forecasts over the period alongside actual figures. Then, subtract forecast values
7 Forecasting from actual figures to obtain annual estimates of forecast error, and squared forecast error, for each year over the period. Finally, compute the correlation coefficient between actual and forecast values over the period. Also compute the sample average (or root mean squared) forecast error. Based upon these findings, how well does the constant growth model generated over the period forecast actual GDP data over the period? ST7.1 SOLUTION A. The constant growth model estimated using the simple regression model technique illustrates the linear relation between the natural logarithm of GDP and time. A constant growth regression model estimated over the year period (tstatistic in parentheses), used to forecast GDP over the year period, is: ln GDP t = t, R 2 = 99.2% (188.66) (75.50) The R 2 = 99.2% and a highly significant t statistic for the time trend variable indicate that the constant growth model closely describes the change in GDP over the time frame. Nevertheless, even modest changes in the intercept term and slope coefficient over the time frame can lead to large forecast errors. B. Each constant growth GDP forecast is derived using the constant growth model coefficients estimated in part A, along with values for each respective time trend variable over the period. Remember that T 2000 = 51, T 2001 = 52,..., and T 2004 = 55 and that the constant growth model provides predicted, or forecast, values for ln GDP t. To obtain forecast values for GDP t, simply take the exponent (antilog) of each predicted ln GDP t variable. The following spreadsheet shows actual and constant growth model GDP forecasts for the forecast period: Year GDP ln GDP Squared Forecast Forecast Error Forecast ln Forecast Error Time (GDP Forecast GDP GDP (GDP  Forecast Period GDP) GDP) $9, $9, $173.2 $29, , , ,
8 Chapter , , , ,049, , , , ,546, , , , ,499, Average $10, $11, $1,069.3 $1,662,463.7 Correlation 99.50% Mean squared error $1,289.4 The correlation coefficient between actual and constant growth model forecast GDP is r GDP, FGDP = 99.50%. The sample root mean squared forecast error is $1,298.4 billion (= $1, 662, 463.7), or 12.7% of average actual GDP over the period. Thus, despite the fact that the correlation between actual and constant growth forecast model values is relatively high, forecast error is also very high. Unusually modest economic growth at the start of the new millennium leads to large forecast errors when GDP data from more rapidly growing periods, like the period, are used to forecast economic growth. ST7.2 Multiple Regression. Branded Products, Inc., based in Oakland, California, is a leading producer and marketer of household laundry detergent and bleach products. About a year ago, Branded Products rolled out its new Super Detergent in 30 regional markets following its success in test markets. This isn't just a Ame product in a commodity market. Branded Products' detergent contains Branded 2 bleach, a successful laundry product in its own right. At the time of the introduction, management wondered whether the company could successfully crack this market dominated by Procter & Gamble and other big players. The following spreadsheet shows weekly demand data and regression model estimation results for Super Detergent in these 30 regional markets: Branded Products Demand Forecasting Problem Regional Market Demand in Cases, Q Price per Case, P Competitor Price, Px Advertising, Ad Household Income, I Estimated Demand, Q 1 1,290 $137 $94 $814 $53,123 1, , ,749 1, , ,881 1, , ,589 1, , ,799 1, , ,565 1, , ,959 1,
9 Forecasting 8 1, ,196 1, , ,163 1, , ,080 1, , ,263 1, , ,291 1, , ,343 1, , ,473 1, , ,501 1, , ,809 1, , ,471 1, , ,663 1, , ,839 1, , ,438 1, , ,348 1, , ,066 1, , ,166 1, , ,380 1, , ,656 1, , ,084 1, , ,000 52,249 1, , ,855 1, , ,546 1, , ,085 1,215 Average 1, ,788 1,286 Minimum 1, ,809 1,024 Maximum 1, ,000 55,839 1,478 Regression Statistics R Square 90.4% Standard Error Observations 30 Coefficients Standard Error t Stat Pvalue Intercept E06 Price, P E11 Competitor Price, P x E05 Advertising, Ad Household Income, I E
10 Chapter 7 A. Interpret the coefficient estimate for each respective independent variable. B. Characterize the overall explanatory power of this multiple regression model in light of R 2 and the following plot of actual and estimated demand per week. C. Use the regression model estimation results to forecast weekly demand in five new markets with the following characteristics: Regional Forecast Market Price per Case, P Competitor Price, Px Advertising, Ad Household Income, I A ,234 B ,845 C ,543 D ,560 E ,870 Average ,
11 Forecasting ST7.2 SOLUTION A. Coefficient estimates for the P, P x, Ad and I independent Xvariables are statistically significant at the 99% confidence level. Price of the product itself (P) has the predictably negative influence on the quantity demanded, whereas the effects of competitor price (P x ), advertising (AD) and household disposable income (I)are positive as expected. The chance of finding such large tstatistics is less than 1% if, in fact, there were no relation between each variable and quantity. B. The R 2 = 90.4% obtained by the model means that 90.4% of demand variation is explained by the underlying variation in all four independent variables. This is a relatively high level of explained variation and implies an attractive level of explanatory power. Moreover, as shown in the graph of actual and fitted (estimated) demand, the multiple regression model closely tracks weekbyweek changes in demand with no worrisome divergences between actual and estimated demand over time. This means that this regression model can be used to forecast demand in similar markets under similar conditions.. C. Notice that each prospective market displays characteristics similar to those of markets used to estimate the regression model described above. Thus, the regression model estimated previously can be used to forecast demand in each regional market. Forecast results are as follows: Regional Forecast Market Price per Case, P Competitor Price, Px Advertising, Ad Household Income, I Forecast Demand, Q A ,234 1,285 B ,845 1,298 C ,543 1,358 D ,560 1,223 E ,870 1,196 Average ,410 1,272 PROBLEMS & SOLUTIONS P7.1 Constant Growth Model. The U.S. Bureau of the Census publishes employment statistics and demand forecasts for various occupations
12 Chapter 7 Employment (1,000) Occupation Bill collectors Computer engineers Physicians assistants Respiratory therapists Systems analysts 617 1,194 A. Using a spreadsheet or handheld calculator, calculate the tenyear growth rate forecast using the constant growth model with annual compounding, and the constant growth model with continuous compounding for each occupation. B. Compare your answers and discuss any differences. P7.1 SOLUTION A. Using the assumption of annual compounding, E t = E 0 (1 + g) t 420 = 311(1 + g) = (1 + g) 10 ln(1.35) = 10 Η ln(1 + g) 0.300/10 = ln(1 + g) e = 1 + g = g g = or 3.1% Using the continuous compounding assumption,
13 Forecasting E t = E 0 e gt 420 = 311e 10g 1.35 = e 10g ln(1.35) = 10g g = /10 = 0.03 or 3.00% are: Using the same methods, continuous growth model estimates for various occupations Employment (1,000) Occupation Continuous Growth Model Annual Compounding Continuous Compounding Bill collectors % 3.00% Computer engineers % 7.32% Physicians assistants % 3.95% Respiratory % 3.57% therapists Systems analysts 617 1, % 6.60% B. For example, if the number of jobs jumps to 420,000 from 311,000 over a tenyear period, then a 3.05% rate of job growth is indicated when annual compounding is assumed. With continuous compounding, a 3.00% rate of growth leads to a similar growth in jobs over a tenyear period. Of course, this small difference is due to the amount of Either method can be employed to measure the rate of growth, but managers must make growth comparisons using a consistent basis. P7.2 Growth Rate Estimation. Almost 2 million persons per year visit wondrous Glacier National Park. Due to the weather, monthly park attendance figures varied widely during a recent year:
14 Chapter 7 Month Visitors Percent change January 7,481 February 9, % March 13, % April 24, % May 89, % June 255, % July 540, % August 528, % September 286, % October 57, % November 12, % December 6, % Average 152, % A. Notice that park attendance is lower in December than in January, despite a 42.4% average rate of growth in monthly attendance. How is that possible? B. Suppose the data described in the table measured park attendance over a number of years rather than during a single year. Explain how the arithmetic average annual rate of growth gives a misleading picture of the growth in park attendance. P7.2 SOLUTION A. The arithmetic average presents a distorted view of the rate of growth over time because upside growth is theoretically unlimited, but declines are limited to no more than 100%. In this case, December park attendance is lower than January attendance despite a 42.4% average monthly gain in park attendance. Big attendance gains in May (269.0%), June (186.2%), and July (111.85%) simply overwhelm smaller positive increases or declines in other months when arithmetic averages are taken. B. This simple example documents the difficulty involved with measuring growth using arithmetic averages. When compound growth rates are considered, managers rely
15 Forecasting on the geometric average rather than the arithmetic average rate of return. The arithmetic average presents a distorted view of the rate of growth over time because upside growth is theoretically unlimited, but declines are limited to no more than 100%. Notice that when sales increase from $250,000 to $500,000 (a 100% gain), but then fall back to $250,000 (a 50% loss), the arithmetic average growth is 25% (= (100%  50%)/2) despite the fact that no net growth has occurred. Similarly, when attendance falls from January to December levels, this decline in attendance is not captured by the 42.4% arithmetic average rate of growth in monthly attendance. P7.3 Sales Trend Analysis. Environmental Designs, Inc., produces and installs energyefficient window systems in commercial buildings. During the past ten years, sales revenue has increased from $25 million to $65 million. A. Calculate the company's growth rate in sales using the constant growth model with annual compounding. B. Derive a fiveyear and a tenyear sales forecast. P7.3 SOLUTION A. S t = S 0 (1 + g) t $65,000,000 = $25,000,000(1 + g) = (1 + g) 10 ln(2.6) = 10 Η ln(1 + g) 0.956/10 = ln(1 + g) e (0.0956)  1 = g B. FiveYear Sales Forecast g = or 10.0% S t = S 0 (1 + g) t
16 Chapter 7 = $65,000,000 ( ) 5 = $65,000,000 (1.611) = $104,715,000 TenYear Sales Forecast S t = S 0 (1 + g) t = $65,000,000 ( ) 10 = $65,000,000 (2.594) = $168,610,000 P7.4 Cost Forecasting. Dorothy Gale, a qualitycontrol supervisor for Wizard Products, Inc., is concerned about unit labor cost increases for the assembly of electrical snapaction switches. Costs have increased from $80 to $100 per unit over the previous three years. Gale thinks that importing switches from foreign suppliers at a cost of $ per unit may soon be desirable. A. Calculate the company's unit labor cost growth rate using the constant rate of change model with continuous compounding. B. Forecast when unit labor costs will equal the current cost of importing. P7.4 SOLUTION A. C t = C 0 e gt $100 = $80e 3g 1.25 = e 3g ln(1.25) = 3g
17 Forecasting g = 0.223/3 B. Import Cost = C 0 e gt = or 7.4% $ = $100e (0.074)t = e (0.074)t ln(1.159) = 0.074t t = 0.148/0.074 = 2 years P7.5 Unit Sales Forecasting. Boris Badenov has discovered that the change in Product A demand in any given week is inversely proportional to the change in sales of Product B in the previous week. That is, if sales of B rose by X% last week, sales of A can be expected to fall by X% this week. A. Write the equation for next week's sales of A, using the variables A = sales of Product A, B = sales of Product B, and t = time. Assume that there will be no shortages of either product. B. Last week, 100 units of A and 90 units of B were sold. Two weeks ago, 75 units of B were sold. What would you predict the sales of A to be this week? P7.5 SOLUTION A. A t = A t1 + ΔA t1 A t = A t1 B B t A t1 t
18 Chapter 7 B. A t = A t1 = 100 B B t A t1 t = 80. P7.6 Revenue Forecasting. Gil Grissom must generate a sales forecast to convince the loan officer at a local bank of the viability of Marina Del Rey, a trendy westcoast restaurant. Grissom assumes that nextperiod sales are a function of current income, advertising, and advertising by a competing restaurant. A. Write an equation for predicting sales if Grissom assumes that the percentage change in sales is twice as large as the percentage changes in income and advertising but that it is only onehalf as large as, and the opposite sign of, the percentage change in competitor advertising. Use the variables S = sales, Y = income, A = advertising, and CA = competitor advertising. B. During the current period, sales total $500,000, median income per capita in the local market is $71,400, advertising is $20,000, and competitor advertising is $66,000. Previous period levels were $70,000 (income), $25,000 (advertising), and $60,000 (competitor advertising). Forecast nextperiod sales. P7.6 SOLUTION A. S t+1 = S t + Y A S Y A t t 21 St+ 21 t1 t1 t  CA CA 0.5 t  1 St t1 = S t + 2 Y A S S S  2S Y A t t t t t t t1 t
19 Forecasting  CA CA t 0.5 St + t1 1 S 2 t = Yt At 1 CAt 2 St + 2 St  St Yt1 At1 2 CAt12.5S t B. S t+1 = 2($500,000)(1.02) + 2($500,000)(0.80) ($500,000)(1.10) ($500,000) = $1,020,000 + $800,000  $275,000  $1,250,000 = $295,000 P7.7 Cost Forecasting. Dr. Clint Cassidy is supervising physician at the Westbury HMO, a New York Citybased medical facility serving the poor and indigent. Cassidy is evaluating the cost effectiveness of a preventive maintenance program, and believes that monthly downtime on the packaging line caused by equipment breakdown is related to the hours spent each month on preventive maintenance. A. Write an equation to predict next month's downtime using the variables D = downtime, M = preventive maintenance, t = time, a 0 = constant term, and a 1 = regression slope coefficien. Assume that downtime in the forecast (next) month decreases by the same percentage as preventive maintenance increased during the month preceding the current one. B. If 40 hours were spent last month on preventive maintenance and this month's downtime was 500 hours, what should downtime be next month if preventive maintenance this month is 50 hours? Use the equation developed in part A. P7.7 SOLUTION A. D t+1 = a 0 + a 1 M
20 Chapter 7 = D t  ΔD = D t   Mt Mt1 D t t1 M B. D t+1 = = 375 hours of downtime P7.8 Sales Forecast Modeling. Toys Unlimited Ltd., must forecast sales for a popular adult computer game to avoid stockouts or excessive inventory charges during the upcoming Christmas season. In percentage terms, the company estimates that game sales fall at double the rate of price increases and that they grow at triple the rate of customer traffic increases. Furthermore, these effects seem to be independent. A. Write an equation for estimating the Christmas season sales, using the variables S = sales, P = price, T = traffic, and t = time. B. Forecast this season's sales if Toys Unlimited sold 10,000 games last season at $15 each, this season's price is anticipated to be $16.50, and customer traffic is expected to rise by 15% over previous levels. P7.8 SOLUTION A. S t+1 = S t + ΔS = S t  ΔS P + ΔS T = S t  2(P t+1 /P t  1)S t + 3(T t+1 /T t  1)S t = 2(P t+1 /P t )S t + 3(T t+1 /T t )S t B. S t+1 =2(16.5/15)10, (1.15)10,
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