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Solution Manual for Managerial Economics in a Global

Economy 8th edition by Salvatore

Chapter 5

13. (a)
Multicollinearity arises when two or more explanatory variables are highly correlated in regression analysis. Multicollinearity leads to exaggerated standard errors and biased statistical tests.
14. (a)
Autocorrelation often arises in time series data when consecutive errors have the same sign or change sign frequently. This leads to exaggerated t statistics, and unreliable R2 and F statistics.
15. (a)
The domestic-currency price of a nation’s imports depend on the foreign-currency price of the nation’s imports and on the exchange rate for the nation’s currency.
(b) Multicollinearity can be easily detected when the estimated coefficients in the regression are statistically insignificant even though the coefficient of determination (R2) is very high.
(c) Multicollinearity could be overcome or reduced by increasing the sample size, utilizing a priori information, using a different functional form, or dropping one of the highly collinear variables. The latter method, however, can lead to mispecification of the regression model, which is an even more serious problem than multicollinearity.
(b) Autocorrelation can be detected by inspecting the plot of the errors or residuals against time or, more usually and precisely, by the Durbin-Watson test.
(c) Autocorrelation may be corrected by including a time trend or an important missing variable in the regression, using a nonlinear form, running the regression on first differences in the variables, or with more complex techniques.
(b) If the foreign-currency price of the nation’s imports increases and the nation’s currency depreciates, the domestic-currency price of the nation’s imports will increase for both reasons.
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CHAPTER 5 DEMAND ESTIMATION
PART TWO DEMAND ANALYSIS
1. (a)
ANSWERS TO PROBLEMS
In Figure 1, the observed price-quantity data points fall downward and to the right. Each point is the result of the interaction of demand and supply forces (, the result of the intersection of a demand and a supply curve of the commodity).
Since the observed data points fall downward and to the right, shifts in the supply curve are generally larger than shifts in the demand curve (for time-series data) or differences in supply curves are greater than differences in the demand curves across different individuals or areas.
By allowing the forces that cause the supply curve of the commodity to shift to operate un-hampered, while correcting for the forces that cause the demand curve to shift, we can derive the demand curve for the commodity (as, for example, the dashed demand curve in Figure 1).
This is accomplished by running a regression of the observed quantities of the commodity on the price of the commodity, consumers’ incomes, the price of related (, substitute and complementary) commodities, consumer tastes, and all the more specific forces that affect the demand for the particular commodity.
2. (a)
In such cases, correcting for the forces that cause the supply curve to shift will give or define, more or less, a single point on the demand curve rather than the demand curve itself.
On the other hand, correcting for the forces that cause the demand curve to shift would give or define, more or less, a single point on the supply curve of the commodity.
Thus, when the observed price-quantity data points are clustered or bunched together as in Figure 2, we can derive neither the demand curve nor the supply curve of the commodity by regression analysis. In such cases, the demand curve for a commodity may have to be derived by consumer surveys, consumer clinics, and market experiments. Such methods can also be used to collect additional data to resolve the identification problem.
See Figure 3.
(b) If the observed price-quantity data points are clustered or bunched together as in Figure 2, this means that both the demand curve and supply curve of the commodity do not shift very much (in time-series data), or different demand curves are very similar to one another and different supply curves are also very similar to one another (in cross-section data).
(b) See Figure 4.
(c) Because of changing and varying weather conditions, the supply curve of agricultural
commodities is likely to shift much more than the demand curve (since most foods are necessities). Thus, it may be easier to derive or identify the demand curve than the supply curve of agricultural commodities from the observed price-quantity data points.
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CHAPTER 5 DEMAND ESTIMATION
85
PART TWO DEMAND ANALYSIS
3. (a)
The same may be true in the markets for commodities, such as pocket calculators, where very rapid technological change shifted the supply curve of pocket calculators The same may be true in the market for commodities, such as calculators, where very rapid technological change shifted the supply curve of pocket calculators in a very short time during which demand conditions did not change much.
In the case of most industrial commodities, however, the demand curve is more likely to shift than the supply curve because of business cycles. Thus, it may be easier to derive or identify the supply curve than the demand curve from the observed price- quantity data points.
Since the price elasticity of demand for Florida Indian River oranges is , a 10 percent decrease in the price of these oranges would increase their quantity demanded by (-10%)() = percent.
Since the price elasticity of demand for Florida interior oranges is , a 10 percent decrease in the price of these oranges would increase their quantity demanded by (-10%)() = percent.
Since the price elasticity of demand for California oranges is , a 10 percent decrease in the price of these oranges would increase their quantity demanded by (-10%)() = percent.
Since the demand for all three types of oranges is price elastic, a decline in price will increase the total revenue of the sellers of all three types of oranges because the percentage increase in quantity sold exceeds the percentage decrease in their prices.
Specifically, for the sellers of Florida Indian River oranges, the 10 percent decrease in price would result in an increase in the quantity sold of percent, so that their total revenue would increase by =
For the sellers of Florida interior oranges, the 10 percent decrease in price would result in an increase in the quantity sold of percent, so that their total revenue would increase by =
For the sellers of California oranges, the 10 percent decrease in price would result in an increase in the quantity sold of percent, so that their total revenue would increase by =
Total profits are equal to the total revenue minus the total costs. We know from part (b) that the sellers’ total revenue increases, but their total costs will also increase (to grow, transport, and sell more oranges). Their profits will increase if the increase in their total revenue exceeds the increase in their total costs. Since we do not know by how much total costs increase by selling more oranges, we cannot answer this question more precisely.
(b)
(c)
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4. Since the cross-price elasticity of demand between Florida Indian River oranges and Florida interior oranges is (the entry in the first row and second column of Table 5-1) a 10 percent increase in the price of Florida interior oranges would lead to a (10%)() = percent increase in the demand for Florida Indian River oranges.
On the other hand, since the cross-price elasticity of demand between Florida interior oranges and Florida Indian River oranges is (the entry in the second row and first column of Table 5-1), a 10 percent increase in the price of Florida Indian River oranges would lead to a (10%)() = percent increase in the demand for Florida interior oranges.
5. (a)
Table 1 shows the calculations to find a and b for the data in Table 5-6 in the text on sales revenues of the firm (Y) and its expenditures on quality control (for simplicity label this Z rather than X2 here).
By then using the value of b found below and the values of Y and Z found in Table 1, we get the value of a of
a=Y-bX =(5)=
Thus, the equation of the regression line is Yt = +
With quality-control expenditures of $3 million as in the first observation year (, with Z1 = $3 million), Y1 = + $(3) = $ million. On the other hand, with Z10 = $8 million, Y10 = $ + (8) = $ million. Plotting these two points (3, ) and (8, ) and joining them by a straight line, we have the regression line plotted in Figure 5 on age 69.
n
Σ(Xt-X)(Yt-Y)
b= t=1 __ = 110 =
n 32 Σ(Xt – X)2
t=1
(b)
CHAPTER 5 DEMAND ESTIMATION
87
PART TWO DEMAND ANALYSIS
Table 1
Calculations to Estimate Regression Line of Sales on Quality Control
Year
Yt
Zt
Yt – Y
Zt- Z
(Zt- Z)(Yt-Y)
Σ(Zt- Z)2
1
44
3
-6
-2
12
4
2
40
4
-10
-1
10
1
3
42
3
-8
-2
16
4
4
46
3
-4
-2
8
4
5
48
4
-2
-1
2
1
6
52
5
2
0
0
0
7
54
6
4
1
4
1
8
58
7
8
2
16
4
9
56
7
6
2
12
4
10
60
8
10
3
30
9
n=10
ΣYt = 500 Y = 50
ΣZt = 50 Z=5
Σ(Yt – Y ) =0
Σ(Zt- Z)=0
Σ(Zt – Z )(Yt – Y ) = 110
Σ(Zt – Z)2 = 32
(c) If the firm’s expenditure on quality control was $2 million, its estimated sales revenue would be
Yˆ t = + (2) = $ million
On the other hand, if the firm’s expenditure on quality control was $9 million, its estimated sales revenue would be
Yˆ t = + (9) = $ million
These results are greatly biased because we have seen in the text that the sales revenues of the firm also depend in an important way on its advertising expenditures. By including only the firm’s quality control expenditures in estimating the regression line we obtain biased estimates for the a and b parameters and, therefore, biased values for the forecast of the firm’s sales revenues.
The same, of course, is the case when sales revenues are regressed only on the firm’s advertising expenditures (as was done in the text). The only reason for running these simple regressions is to make it easier for the student to understand how regression analysis is performed and how its results are interpreted.
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6. (a)
Figure 5
Yˆ t = – + () ()
R2 = R 2 = F =
The value of b1 = indicates that a $1 decline in the price of the commodity will lead to a unit increase in the quantity demanded of the commodity. On the other hand, the value of b2 = , indicates that a $100 increase in consumers’ income will increase the quantity demanded of the commodity by units.
CHAPTER 5 DEMAND ESTIMATION
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PART TWO DEMAND ANALYSIS
(b) Since the t statistic for both b1 and b2 exceed the critical t value of for 17 df, both slope coefficients are statistically significant at the 5 percent level.
(c) The unadjusted and adjusted coefficients of determination are R2 = and R 2 =
, respectively. This means that the variation in price and income explains percent of the variation in the quantity demanded of the commodity when no adjustment is made for degrees of freedom, and percent when such an adjustment is made.
(d) Since the value of the F statistic exceeds the critical F value of with k – 1 = 3 – 1 = 2 and n – k = 20 – 3 = 17 df, we accept the hypothesis that the regression explains a significant proportion of the variation in the quantity demanded of the commodity (Y) at the 5 percent level of significance.
7. The statistical significance of the estimated slope coefficients, R 2, and the F-statistic in the
demand equation estimated in double-log form are very similar to those obtained in Problem 6 in linear form. The only advantage that the double-log formulation might have over the linear formulation of the demand function is that the estimated slope coefficients in the former represent elasticities rather than marginal effects (as in the latter).
Specifically, b1 indicates that a one percent increase in the price of the commodity leads to a percent decline in the quantity demanded of the commodity. Thus, the demand function of the commodity is price inelastic. On the other hand, b2 gives an income elasticity of Thus, the commodity is a necessity.
8. (a)
(b)
(c)
Since the price elasticity of demand is negative for all public goods, they all satisfy the law of (negatively sloped) demand. The demand is price elastic (, the price elasticity of demand exceeds the value of one, when disregarding the sign) for elementary school aid, parks and recreational areas, and highway construction and maintenance.
Since the income elasticity of demand is positive for all the public goods listed, all of them are normal goods. Furthermore, since the income elasticities of demand are smaller than one, they are all necessities, except for elementary school aid and parks and recreational areas, which are luxuries.
If the price or cost of college and university education increased by 10 percent, this, by itself, would result in an decline in the quantity demanded of college and university education (from times 10%). On the other hand, if incomes rose by 10 percent, this, by itself, would result in a percent increase in the demand for college and university education (from times 10%).
Thus, the net effect of a 10 percent increase in the price or cost of college and university education and a simultaneous 10 percent increase in incomes would be in a percent increase in the demand for college and university education.
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9. (a)
The result of the first regression clearly points to a multicollinearity problem in the regression. Specifically, the t-statistic of both slope coefficients are statistically insignificant at the 5 percent level, while the adjusted coefficient of determination is
very high ( R 2 = ).
The multicollinearity in regression 1 is also evident from the fact that when each explanatory variable is used alone in a simple regression (regressions 2 and 3) each is statistically highly significant (, each has a very high t statistic).
Removing one of the explanatory variables in regression 1 (as done in regressions 2 and 3), however, leads to biased estimates since economic theory postulates that both GNP and price are important determinants of imports.
The fourth regression attempts to remove the multicollinearity between GNP and P by dividing both M and GNP (Y) by P, and running the regression on the transformed variables. Thus, regression 4 tests the hypothesis that real imports are a positive function of real GNP. By doing so, regression 4 overcomes the multicollinearity problem without resulting in a misspecified relationship (since P was implicitly considered).
All the estimated slope coefficients have the correct sign (, the sign postulated by demand theory). Specifically, a one unit increase in real per capita consumption expenditures during a given year (Xt) results in a increase in the real per capita consumption expenditures on bus transportation during the same year (Qt).
Similarly, a one unit increase in Pt results in a decrease in Qt, and a one unit increase in St results in a decrease in Qt. Thus, as expected, the stock of cars per capita is a substitute for bus transportation. Finally, there has been a decline of in the value of the constant term in the regression during the post-war period.
In view of the very high values of the t-statistics, all of the estimated coefficients are statistically significant at better than the 1 percent level. The regression explains over 99 percent of the variation in Qt.
Since all of the estimated slope coefficients are statistically significant at better than the 1 percent level, multicollinearity does not seem to be a problem. Finally, since the value of the D-W statistic falls within the level of dL and dU, the test for evidence of autocorrelation is indeterminate.
The 4% increase in the price of TV sets in Japan and the 5% depreciation of the dollar lead to a net increase of 9% in the dollar price of imported TV sets in the United States, from $300 to $327.
Since the price elasticity of demand for the TV sets in the United States is , the 9% increase in price will lead to a ()() = or reduction in the quantity demanded of imported TV sets in the United States.
(b)
10. (a)
(b)
(c)
11. (a)
(b)
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CHAPTER 5 DEMAND ESTIMATION

PART TWO DEMAND ANALYSIS
12. (a)
The 4% increase in the price of TV sets in Japan and the 5% appreciation of the dollar lead to a net decrease of 1% in the dollar price of imported TV sets in the , from $300 to $297.
13. (a)
By itself, the 10% increase in the price of household natural gas reduces the quantity demanded by 14% in the short run; the 10% increase in consumers’ income increase demand for natural gas by 12%, the 10% increase in the price of electricity increases the demand for natural gas by 8%, and the 10% increase in population increases the demand for natural gas by 10%. thus, the 10% increase in all variable increases the demand for house-hold natural gas for the california power company by 16% (–14% + 12% + 8% + 10%).
(c) Since consumers’ income increases by 3% in the United States and the income elasticity of demand for the imported TV sets is 2, the demand for the imported TV sets increases by 6% as a result of the increase in consumers’ income alone.
(d) Since the price increase leads to a decline in the quantity demanded, while the 3% increase in incomes leads to a 6% increase in demand, the net effect of both the price and income increases is a reduction in the demand for imported TV sets in the United States.
(b) Since the price elasticity of demand for the TV sets in the United States is , the 1% decrease in price will lead to a ()() = or increase in the quantity demanded of imported TV sets in the United States.
(c) Since consumers’ income increases by 3% in the United States and the income elasticity of demand for the imported TV sets is 2, the demand for the imported TV sets increases by 6% as a result of the increase in consumers’ income alone.
(d) Since the price decrease leads to a increase in the quantity demanded and the 3% increase in incomes leads to a 6% increase in demand, the net effect of both the price and income changes is a increase in the demand for imported TV sets in the United States.
(b) In the long run, the demand for household natural gas is expected to increase by 9% (-21% + 12% + 8% + 10%).
14. The California Power Company management should ask its statistical department to provide it with the t- values of the estimated parameters so as to be able to determine their statistical significance. It should also ask for the coefficient of determination to see what percentage of the variation in the demand for its household natural gas is in fact “explained” by the variation in the variables included in the analysis. Finally, the management should ask for the D-W statistics of the regression to determine if the assumption of regression analysis are satisfied (so that the significance of the estimated parameters are not overestimated).
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15. (a) We have chosen to include the price of commodity Z into the regression equation because economic theory postulates that the quantity demanded of a commodity per unit of time is a function of or depends not only on the price of the commodity, the consumers’ income, but also on the price of related commodities.
(b) From the t-test, we conclude that the estimated slope coefficient of Px and Y are statistically significant at the 5 percent level (the critical value from the table of the t distribution for n – k = 20 -4 = 16 df is ), and the regression explains a very high and significant proportion of the variation in Qx.
(c) Since the estimated slope coefficient of Pz is negative, commodity Z seems to be a complement of commodity X. That is, when Pz rises, less of both commodities X and Z are purchased. Thus, the relationship between X and Z seems to be one of complementarity.
However, since the estimated slope coefficient of Pz is not statistically significant, we cannot make much of the negative sign of Pz. Indeed, were we not to suspect multicollinearity between Px and Pz, we would have to conclude that commodities X and Z are unrelated.
There is however, strong collinearity between Pz and Px as indicated by the simple correlation coefficient, r = Therefore, we cannot be sure whether commodity Z is a complement, a substitute, or is unrelated to commodity X.