Suggested Solutions to Assignment 4 (Optional) - Trent University

ECON 3400H
Managerial Economics
Instructor: Sharif F. Khan
Department of Economics
Trent University
Fall 2012
Suggested Solutions to Assignment 4 (Optional)
Read each part of the question very carefully. Show all the steps of your calculations to
get full marks.
1. End-of-chapter “Problems” # 5 of Chapter 6 of the textbook – Salvatore (7e).
2. End-of-chapter “Problems” # 6 of Chapter 6 of the textbook – Salvatore (7e).
3. End-of-chapter “Problems” # 7 of Chapter 6 of the textbook – Salvatore (7e).
4. End-of-chapter “Problems” # 9 of Chapter 6 of the textbook – Salvatore (7e).
5. End-of-chapter “Problems” # 10 of Chapter 6 of the textbook – Salvatore (7e).
6. End-of-chapter “Problems” # 11 of Chapter 6 of the textbook – Salvatore (7e).
7. End-of-chapter “Problems” # 15 of Chapter 6 of the textbook – Salvatore (7e).

Table 2
Number of New-Housing Starts in the
United States in Each Year, 2005–2009
(in thousands of units)
Year
New-Housing Starts
2005 2,068.3
2006 1,800.9
2007 1,355.0
2008 905.5
2009 553.8
5. Since the problem asks to forecast the index of new housing starts for 1998 and compare
it to the index of 163 already known, we only forecast the index for 1998 without the
need to forecast the index for all previous years.
The only reason for forecasting all previous years would be to calculate the root-meansquare
error (RMSE) of the forecast obtained with different moving averages so as to be
able to choose the forecast of the moving average with the smallest RMSE. Since we
know here the actual value of the index for 1998, we can compare our results directly
with the actual index rather than using RMSE.
The three-year moving average index of new housing starts in the United States for 1998
is obtained by summing the indexes from 1995, 1996, and 1997 and dividing by 3. This
is (142 + 156 + 162)/3 = 460/3 = 153.3 or 153.
On the other hand, the five-year moving average is obtained by summing the indexes for
the five years from 1993 to 1997 and dividing by 5. This is (125 + 146 + 142 + 156
+ 162)/5 = 731/5 = 146.2 or 146.
Neither the three-year nor the five-year moving average provides a good forecast for
the actual index of 163 for 1998, but the three-year moving average is somewhat better.
In fact, the RMSE for the three-year moving average is 17, while the RMSE for the
five-year moving average is 24.
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6. (a) The exponential forecasts (F) of new housing starts (A) in the United States for
the years 1986 to 1998 with w = 0.3 and w = 0.7 are given in Table 3. They are obtained
by using Equation 6-12 in the text. To get the analysis started, we let F 1986 = 128
(the average of the actual index of housing starts between 1986 and 1997).
(b) From Table 3 we see that F 1998 = 145 with w = 0.3 and 159 with w = 0.7. Since A 1998 = 163,
the exponential forecast with w = 0.7 is better.
With w = 0.3, RMSE = 3 ,369 /12 = 17 and with w = 0.7, RMSE = 1 ,834 /12 = 12 .
Since the RMSE with w = 0.7 is smaller than the RMSE with w = 0.3, the former gives,
on the average, a better forecast.
Table 3
Exponential Forecasts of Index of New-Housing Starts in the United States for 1986–1997
(1)
Year
(2)
A
(3)
F with w = 0.3
(4)
A − F
(5)
(A −F) 2 (6) F
with w = 0.7
(7)
A −F
(8)
(A − F) 2
1986 116 128 −12 144 128 −12 144
1987 122 124 −2 4 120 2 4
1988 121 123 −2 4 121 0 0
1989 121 122 −1 1 121 0 0
1990 111 122 −11 121 121 −10 100
1991 97 119 −22 484 114 −17 281
1992 113 112 1 1 102 11 121
1993 125 112 13 169 110 15 225
1994 146 116 30 900 121 25 625
1995 142 125 17 289 139 3 9
1996 156 130 26 676 141 15 225
1997 162 138 24 576
3,369
1998 145 159
152 10 100
1,834
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7. The composite index is obtained by calculating the percentage change for each series
relative to the base month and then averaging these percentage changes. The percentage
change from the first to the second month is 10 for indicator A, 15 for indicator B, and
−10 for indicator C. Their simple average (since each indicator is given equal weight)
is 5 percent. Taking the first month as the base period with a composite index of 100,
we obtain the composite index of 105 for the second month.
For the third month, the change of each indicator from the base period is 20 percent, 10
percent, and 10 percent, respectively, with an average of 13.3 percent. Adding this value
to 100 (the base-period value of the composite index), we get the composite index of 113.33
for the third month. These composite indexes are shown in the second column of Table 4.
Since two indicators rise and one falls from the first to the second period, the diffusion index
for the second period is 66.7 percent. On the other hand, since all three indicators rise from
the second to the third month, the diffusion index for the third month is 100 percent. These
diffusion indexes are shown in the third column of Table 4.
Thus, according to the composite and diffusion indexes, the level of economic activity
should improve in future months.
Table 4
Month
Composite
Index
Diffusion
Index
1 100.00 —
2 105.00 66.70
3 113.33 100.00
8. (a) Time series A is a coincident indicator of the index series of cereal sales of the Tasty
Food Company. The reason for this is that series A moves in step or coincidentally
with the time series of the sales index. When the time series of the sales index rises,
series A rises, and when the sales index falls, series A also falls in the same month.
(b) Time series B is a leading indicator of the index of cereal sales. That is, series B
anticipates or leads changes in the direction of the sales index and can, therefore, be
used to forecast changes in the firm's sales index. The lead time is two months.
For example, the decline in the sales index between the third and fourth month is
anticipated and can be forecasted from the decline in the value of time series B from
the first to the second month. Similarly, the rise in the index of cereal sales in the fifth
month can be forecasted by the rise in the value of series B in the third month, and so on.
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(c) Time series C is a lagging indicator of the index of cereal sales. That is, series C follows
or lags the changes in the direction of the sales index. The lag time is one month.
For example, the increase in the sales index between the first and the second month
is followed by the increase in series C from the second to the third month. Similarly,
the decline in the sales index in the third month is followed by the decline in series
C in the fourth month, and so on.
9. (a) By substituting the given values of the independent or explanatory variables and t = 24
for 1972 in the estimated demand equation, we get the quantity demanded (sales) of
sweet potatoes in the United States in 1972 of
QD s = 7,609 − 1,606(4.10) + 59(208.78) + 947(3.19) + 479(2.41) − 271(24)
= 11,013.74 in thousands (11.01 millions) hundredweight.
(b) By substituting the given values of the independent or explanatory variables and t = 25
for 1973 in the estimated demand equation, we get the quantity demanded (sales) of
sweet potatoes in the United States in 1973 of
QD s = 7,609 − 1,606(4.00) + 59(210.90) + 947(3.55) + 479(2.40) − 271(25)
= 11,364.55 in thousands (11.36 millions) hundredweight
Of course, the accuracy of these results depends on the accuracy of the forecasted
values of the independent or explanatory variables. The greater the errors in
forecasting the latter, the greater will be the forecasting error in the former.
10. (a) By substituting the given values of the independent or explanatory variables and
D = 1 (a postwar year) for 1962 in the estimated demand equation, we get the per
capita personal consumption expenditures on shoes of
D t = 19.575 − 0.0923(20) + 0.0289(1,646) − 99.569(0.4) − 4.06(1)
= $21.41 at 1954 prices
(b) By substituting the given values of the independent or explanatory variables and
D = 1 (a postwar year) for 1972 in the estimated demand equation, we get the per
capita personal consumption expenditures on shoes of
D t = 19.575 − 0.0923(30) + 0.0289(2,236) − 99.568(0.6) − 4.06(1)
= $17.63 at 1954 prices
It seems that real per capita personal consumption expenditures declined over
time as people used automobiles more and walked less (a change in taste).
(c) We would expect the error for the 1972 forecast to be larger than for the 1962
forecast. The reason is that the further into the future the dependent variable
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is forecasted, the more the estimated regression equation is likely to lead to
error because it does not reflect the structural changes (i.e., changes in the values
of the estimated coefficients) that usually occur over time.
11. (a) The reduced-form equation for C t is obtained by substituting Equation 6-23 for GNP t
into Equation 6-21 for C t . That is:
Substituting Equation 6-23 for GNP t into Equation 6-21 for C t (and omitting u 1t
because its expected value is zero), we get
C t = a 1 + b 1 (C t + I t + G t )
(6-24')
C t = a 1 + b 1 C t + b 1 I t + b 1 G t (6-24")
By then substituting Equation 6-22 for I t into Equation 6-24" (and omitting u 2t ), we have
C t = a 1 + b 1 C t + b 1 (a 2 + b 2 FS) + b 1 G t
(6-25')
C t = a 1 + b 1 C t + b 1 a 2 + b 1 b 2 π t − 1 + b 1 G t (6-25")
Collecting the C t terms to the left in Equation 6-25" and isolating C t , we have
C t (1 − b 1 ) = a 1 + b 1 a 2 + b 1 b 2 π t − 1 + b 1 G t
(6-26')
Dividing both sides of Equation 6-26' by 1 − b 1 , we finally obtain
C t = a 1 + b 1 a 2 + b 1 b 2 π t − 1 + b 1 G t (6-27')
1 − b 1 1 − b 1 1 − b 1 1 − b 1
Equation 6-27' is the reduced-form equation for C t . It is expressed only in terms
of π t − 1 and G t (the exogenous variables of the model) and can be used in forecasting
C t + 1 by substituting the known value of π t and the predicted value of G t+1 into
Equation 6-27' and solving for C t + 1 .
(b) Equation 6-22 is already in reduced form because π t − 1 is exogenous (i.e., it is
known in period t). The lagged (and therefore known) value of a variable is
sometimes known as a predetermined variable.
12. Wait for June 2011 when the 2011/1 issue of the OECD Economic Outlook will be out and
compare the forecasts presented in Table 6-10 in the text with the actual results reported
for 2010 in June 2011.
13. Prepare a table similar to Table 6-10 in the text of the macroforecasts in the 2011/2 issue of
the OECD Economic Outlook.
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14. (a) Wait for June 2012 when the 2012/1 issue of the OECD Economic Outlook is out and
compare the forecast presented in the table that you prepared for Problem 13 with the actual
results reported in the 2012/1 issue of the OECD Economic Outlook.
(b) Compare the accuracy of the forecasts in your answer to Problem 12 with your answer to
part (a) of this problem. The accuracy of the forecasts for 2011 will be better than for those
for 2010 if economic conditions were more stable in 2011 than in 2010. It is easier to forecast
the level of economic activity when economic conditions are stable than when they are
less stable or unsettled.
15. (a) For ease of reference, the estimated regression and the meaning of the variables are
repeated below:
ln Q t = 1.2789 − 0.1647 ln P t + 0.5115 ln Y t + 0.1483 ln P' t − 0.0089t
(−2.14) (1.23) (0.55) (−3.36)
− 0.0961 D 1t − 0.1570 D 2t − 0.0097 D 3t
(−3.74) (−6.03) (−0.37)
R 2 = 0.80 D–W = 2.08
where
Q t = quantity (in pounds) of coffee consumed per capita (for population over 16) in quarter t
P t = relative price of coffee per pound in quarter t, at 1967 prices
Y t = per capita disposable personal income in quarter t, in thousands of 1967 dollars
P' t = relative price of tea per quarter pound in quarter t, at 1967 prices
t = time trend; t = 1 for first quarter of 1963 to t = 58 for second quarter of 1977
D 1t = dummy variable equal to 1 for first quarter (spring) and 0 otherwise
D 2t = dummy variable equal to 1 for second quarter (summer) and 0 otherwise
D 3t = dummy variable equal to 1 for third quarter (fall) and 0 otherwise
and the numbers in parentheses below the estimated coefficients are values of the t statistic.
Since the above equation is expressed in terms of logarithms, the first step in the solution
is to change all the predicted values of the independent or explanatory variables (except the
dummy variables and t) into their natural logs. These are shown in Table 5.
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Table 5
Quarter ln P Ln Y ln P'
1977.3 0.62 1.27 0.10
1977.4 0.55 1.28 0.08
1978.1 0.47 1.29 0.07
1978.2 0.38 1.30 0.05
By substituting the given values of the independent or explanatory variables,
t = 59, D 3 = 1, and D 1 = D 2 = 0, for the third quarter of 1977 in the estimated
demand equation, we get the third-quarter 1977 forecast of the demand for coffee of
ln Q 59 = 1.2789 − 0.1647(0.62) + 0.5115(1.27) + 0.1483(0.10) − 0.0089(59)
− 0.0961(0) − 0.1570(0) − 0.0097(1)
ln Q 59 = 1.3064. The antilog of 1.3064 is 3.69 lb.
This is the forecasted number of pounds of coffee consumed per capita for the
population over age 16 in the third quarter of 1977.
(b) By substituting the given values of the independent or explanatory variables, t = 60
and D 1 = D 2 = D 3 = 0, for the fourth quarter of 1977 in the estimated demand
equation, we get the fourth-quarter 1977 forecast of the demand for coffee of
ln Q 60 = 1.2789 − 0.1647(0.55) + 0.5115(1.28) + 0.1483(0.08) − 0.0089(60)
− 0.0961(0) − 0.1570(0) − 0.0097(0)
ln Q 60 = 1.3209. The antilog of 1.3209 is 3.75 lb.
This is the forecasted number of pounds of coffee consumed per capita for the population
over age 16 in the fourth quarter of 1977.
(c) By substituting the given values of the independent or explanatory variables, t = 61,
D 1 = 1, and D 2 = D 3 = 0, for the first quarter of 1978 in the estimated demand
equation, we get the first-quarter 1978 forecast of the demand for coffee of
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ln Q 61 = 1.2789 − 0.1647(0.47) + 0.5115(1.29) + 0.1483(0.07) − 0.0089(61)
− 0.0961(1) − 0.1570(0) − 0.0097(0)
ln Q 61 = 1.2327. The antilog of 1.2312 is 3.43 lb.
This is the forecasted number of pounds of coffee consumed per capita for the
population over age 16 in the first quarter of 1978.
(d) By substituting the given values of the independent or explanatory variables,
t = 62, D 1 = D 3 = 0, and D 2 = 1, for the second quarter of 1978 in the estimated
demand equation, we get the second-quarter 1978 forecast of the demand for coffee
ln Q 62 = 1.2789 − 0.1647(0.38) + 0.5115(1.30) + 0.1483(0.05) − 0.0089(62)
− 0.0961(0) − 0.1570(1) − 0.0097(0)
ln Q 62 = 1.1799. The antilog of 1.1799 is 3.25 lb.
This is the forecasted number of pounds of coffee consumed per capita for the
population over age 16 in the second quarter of 1978.
(e) Since the estimated regression equation seems to fit the data reasonably well (it
explains 80 percent of the variation in Q t ) and we are using it to forecast Q t for
the next four quarters only, we can be reasonably confident in our forecast.
A sudden and unforeseen change in tastes (due, for example, to the finding that coffee
causes heart ailments) could cause the forecasted error to become very large. In that
case, the regression equation would have to be reestimated to reflect this structural
change before we could use it to generate forecasts in which we could have a great
deal of confidence.
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