CHAPTER FOUR
ESTIMATING DEMAND
ANSWERS TO EVEN-NUMBERED PROBLEMS
2.a.
Coca Cola’s management is likely to conclude that consumers will
prefer New Coke to Coke Classic. (However, as part b shows, they
may be wrong.)
b. Yes, these rankings are consistent with the information in part a.
Consumers prefer Pepsi to Coke Classic by 58 to 42 (types A and C)
and New Coke to Pepsi by 58 to 42 (B and C). However, a blind taste
test between Classic and New Coke would have Classic preferred 84
to 16 (A and B)!
c. It would be a big mistake to replace Classic by New Coke. The
obvious strategy is to retain Classic but also offer and promote New
Coke. New Coke will attract type C consumers away from Pepsi. As
the text indicated, blind taste tests do not tell the whole story about
consumer buying behavior; brand-name allegiance and loyalty is also
important.
4.a.
False. A high R2 indicates that the equation closely tracks the past
data, but this is only one part of performance. A complete evaluation
would address these questions: i) Does the equation make economic
sense? ii) Are the signs and magnitudes of the coefficients
reasonable? iii) How well does it forecast, short-term and long-term?
b. Partly True. More data is better as long as the time-series relationship
is stable. However, such behavior often changes over time. If two
time periods (say, one decade versus another) show very different
behavior, one should estimate separate time-series for each.
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c. False. Throwing in everything but the kitchen sink is bad on
theoretical grounds and empirical grounds. Including irrelevant or
insignificant variables will lower the adjusted R2 (a better measure of
performance) and will typically worsen the equation’s forecasting
accuracy.
d. Partly True. But there are exceptions. i) Even forecasts that
accurately track the past can produce implausible long-term
predictions. See, for example, the prediction in problem 8 of Chapter
5 for the mile record in the year 2050. ii) No matter how good the
past fit, an equation will generate poor forecasts if it relies on
explanatory variables that are themselves difficult to predict.
6. a. Yes, the equation makes economic sense. Growth in tire sales is
fueled by growth in miles driven and growth in new car sales.
b. The equation performs well in explaining the past data (R 2 = .83).
The coefficients of the two explanatory variables are highly
significant, and the Durbin-Watson statistic indicates no serial
correlation.
c. The t-statistics for the respective coefficients are: (1.41 - 1)/.19 =
2.15 and (1.12 - 1)/.41 = .29. The first coefficient is significantly
different than one; the second is not. If the second coefficient is
taken to be one, this means that tire sales are proportional to new
auto sales.
d. The forecast is: .45 + (1.41)(-2) + (1.12)(-13) = -16.93. An actual
drop of 18% would not be surprising; it’s well within the margin of
forecast error.
8. a. The t-statistics for each of the explanatory variables are:
Price
-5.11
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Comp. Price
Income
Population
Time
4.97
11.70
1.29
3.85
Using a cutoff of 1.68 (41 degrees of Freedom), we see that all the
explanatory variables are statistically significant except population.
The regression model explains 93% of the total variation.
b. Price elasticity is (dQ/dP)(P/Q) = (-3590.6)(7.5/20,000) = -1.35.
Cross-price elasticity is
(dQ/dPc)(Pc/Q) = (4,226.5)(6.5/20,000) = 1.37
c. According to the regression, pie sales should increase by
approximately (4)(356) = 1424 pies next year. (Remember one
year equals 4 quarters.)
d. You might be fairly confident in predicting sales for the next quarter
given that 93% of the variation is explained by the regression but
only if accurate information about the explanatory variables can be
obtained. Of course, you control your own price. However,
competitors’ prices and other variables are not in your control. As
for two years from now, predictions as to the values of the
explanatory variables become even more difficult. Furthermore, the
demand relationship itself is subject to change as tastes change over
time. This makes prediction two years from now much more
uncertain.
e. Your confidence would depend on how well these test markets
represent the national market.
Discussion Question.
One way of comparing the qualitative and quantitative approaches is to put
them to a side-by-side forecasting test. Ask each approach to make forecasts
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of the average weekly revenues for a series of upcoming films. Then keep a
careful track record of which forecasts are closer to the actual, realized
figures.
Another powerful test is to combine the approaches in estimating movie
demand. While human decision makers might be skilled at identifying
important qualitative factors, they are less competent in estimating the
magnitude of these factors’ impacts on demand. This is where statistical
techniques excel. Fortunately, almost all the likely qualitative factors can be
introduced in the regression analysis as measured variables or dummy
variables. For instance, the director (or the script) could be rated on a one to
four star scale, and this rating could be introduced as a explanatory variable
in the regression equation. If the director makes a large difference in the
quality of the film and its box office revenues, then the estimated coefficient
for this variable should be positive and highly significant.
ANSWERS TO SPREADSHEET PROBLEMS
(You may download any and all spreadsheets from the John Wiley,
Samuelson and Marks Website.)
S1. a. To estimate price elasticity, we compare 2001 and 2004, two years
in which the level of income was the same:
EP = [(1.90 – 2.00)/2.00]/ [(22 – 20)/20] = -5%/10% = -.5.
b. Comparing 2001 and 2003 when prices were constant, we find:
EY = [(1.94 – 2.00)/2.00]/ [(97 – 100)/100] = -3%/-3% = 1.0.
c. dQ/Q = EP(dP/P) + EY(dY/Y)
= (-.5)[(24 – 22)/22] + (1.0)[(105 – 100)/100]
= -4.5 + 5 = .5 percent.
The forecast calls for a very slight increase in sales, whereas actual
sales were unchanged.
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d. The OLS regression produces the equation: Q = 1 - .05P + .02Y
with an R2 of 1.00! Surprisingly, this equation provides a perfect fit
of the five years of observations. Obviously, this degree of
accuracy is more than a bit unrealistic.
S2. a. The OLS estimated equation is: W = 18.25 - .41t, where the
t-statistic associated with the “year” coefficient is -1.616. This
t-value fails significance. We cannot reject the null hypothesis of a
zero “year” coefficient. Thus, there is insufficient evidence to
establish a downward trend in the water table.
b. Average rainfall in the last 5 years was 46 inches compared to 38.4
inches in the first 5 years. Thus, accounting for variation in rainfall
is crucial. Adding rainfall (R) as an explanatory variable would
likely reveal a more pronounced yearly drop in the water table.
c. The multiple regression equation is:
W = 9.69 - .51t + .216R.
This equation (with an R2 of .82) provides a much better fit to the
data than the simpler part-a regression (R2 of .246). The t-values for
the coefficients are –3.81 and 4.75 respectively, and both are highly
significant. Notice that the equation identifies a stronger negative
time trend.
S3. a. The estimated OLS equation is: Q = 332.5 – 506.6P. The equation
is statistically significant (R2 = .941). If price is cut by $.10,
quantity increases by 50.66 units.
b. A careful plot of the points shows a slight degree of curvature.
c. The Log-Log equation is:
Log(Q) = 2.975 – 2.02Log(P).
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This provides a better fit of the data (R2 = .992) than the linear
equation. The Log-Log equation implies the demand equation:
Q = 19.6P-2.02.
(The antilog of 2.975 is 19.6.) The price elasticity of demand is –2.02.
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