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Chapter 3: Answers to Questions and Problems 1. a. When P = $12, R = ($12)(1) = $12. When P = $10, R = ($10)(2) = $20. Thus, the price decrease results in an $8 increase in total revenue, so demand is elastic over this range of prices. b. When P = $4, R = ($4)(5) = $20. When P = $2, R = ($2)(6) = $12. Thus, the price decrease results in an $8 decrease total revenue, so demand is inelastic over this range of prices. c. Recall that total revenue is maximized at the point where demand is unitary elastic. We also know that marginal revenue is zero at this point. For a linear demand curve, marginal revenue lies halfway between the demand curve and the vertical axis. In this case, marginal revenue is a line starting at a price of $14 and intersecting the quantity axis at a value of Q = 3.5. Thus, marginal revenue is 0 at 3.5 units, which corresponds to a price of $7 as shown below. Price $14 $12 $10 $8 $6 $4 $2 Demand $0 0 1 2 3 MR 4 5 6 Quantity Figure 3-1 Managerial Economics and Business Strategy, 7e Page 1 2. a. At the given prices, quantity demanded is 700 units: Qxd 1000 2 154 .02 400 700 . Substituting the relevant information into Px 154 2 0.44 . Since this is less Qx 700 than one in absolute value, demand is inelastic at this price. If the firm charged a lower price, total revenue would decrease. b. At the given prices, quantity demanded is 300 units: Qxd 1000 2 354 .02 400 300 . Substituting the relevant information into the elasticity formula gives: EQx , Px 2 P 354 the elasticity formula gives: EQx , Px 2 x 2 2.36 . Since this is 300 Qx greater than one in absolute value, demand is elastic at this price. If the firm increased its price, total revenue would decrease. c. At the given prices, quantity demanded is 700 units: Qxd 1000 2 154 .02 400 700 . Substituting the relevant information into P 400 the elasticity formula gives: EQx , PZ .02 Z .02 0.011 . Since this 700 Qx number is positive, goods X and Z are substitutes. 3. a. The own price elasticity of demand is simply the coefficient of ln Px, which is – 0.5. Since this number is less than one in absolute value, demand is inelastic. b. The cross-price elasticity of demand is simply the coefficient of ln Py, which is – 2.5. Since this number is negative, goods X and Y are complements. c. The income elasticity of demand is simply the coefficient of ln M, which is 1. Since this number is positive, good X is a normal good. d. The advertising elasticity of demand is simply the coefficient of ln A, which is 2. Page 2 Michael R. Baye 4. % Qxd 2 . Solving, 5 we see that the quantity demanded of good X will decrease by 10 percent if the price of good X increases by 5 percent. % Qxd b. Use the cross-price elasticity of demand formula to write 6 . Solving, 10 we see that the demand for X will decrease by 60 percent if the price of good Y increases by 10 percent. % Qxd c. Use the formula for the advertising elasticity of demand to write 4. 2 Solving, we see that the demand for good X will decrease by 8 percent if advertising decreases by 2 percent. % Qxd d. Use the income elasticity of demand formula to write 3 . Solving, we 3 see that the demand of good X will decrease by 9 percent if income decreases by 3 percent. a. Use the own price elasticity of demand formula to write 5. 6. 50 5 . Solving, we see that the price %Py of good Y would have to decrease by 10 percent in order to increase the consumption of good X by 50 percent. Using the cross price elasticity formula, Using the change in revenue formula for two products, R $30,0001 2.5 $70,0001.1.01 $320 . Thus, a 1 percent increase in the price of good X would cause revenues from both goods to increase by $320. Managerial Economics and Business Strategy, 7e Page 3 7. Table 3-1 contains the answers to the regression output. SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 0.62 0.39 0.37 190.90 100.00 ANOVA degrees of freedom Regression Residual Total 2.00 97.00 99.00 Coefficients Intercept Price of X Income 187.15 -4.32 0.09 SS MS F 2,223,017.77 1,111,508.88 3,535,019.49 36,443.50 5,758,037.26 Standard Error t Stat 534.71 0.35 0.69 0.02 6.26 4.47 30.50 P-value 0.73 0.00 0.00 Significance F 0.00 Lower 95% -880.56 -5.69 0.05 Upper 95% 1,254.86 -2.96 0.14 Table 3-1 a. Qxd 187.15 4.32 Px .09 M . b. Only the coefficients for the Price of X and Income are statistically significant at the 5 percent level or better. c. The R-square is fairly low, indicating that the model explains only 39 percent of the total variation in demand for X. The adjusted R-square is only marginally lower (37 percent), suggesting that the R-square is not the result of an excessive number of estimated coefficients relative to the sample size. The F-statistic, however, suggests that the overall regression is statistically significant at better than the 5 percent level. 8. Page 4 The approximate 95 percent confidence interval for a is aˆ 2 aˆ 10 2 . Thus, you can be 95 percent confident that a is within the range of 8 and 12. The approximate 95 percent confidence interval for b is bˆ 2 bˆ 2.5 1 . Thus, you can be 95 percent confident that b is within the range of –3.5 and –1.5. Michael R. Baye 9. a. The t statistics are as follows: t aˆ 9369.45 1.36 0.848 ; t bˆ 2.429 ; and 11067.07 0.56 0.14 2.80 . 0.05 b. Since t aˆ 2 the coefficient estimate, â , is not statistically different from zero. t cˆ However, since t bˆ 2 and t cˆ 2 , the coefficient estimates b̂ and ĉ are statistically different from zero. c. The R-square and adjust R-square tell us the proportion of variation explained by the regression. The R-square tells us that 24 percent of the variability in the dependent variable is explained by price and income. The adjusted R-square confirms that fact and the R-square is not the result of estimating too many coefficients (i.e. few degrees of freedom). 10. a. The own-price elasticity of demand is -1.36, so demand is elastic. b. The income elasticity of demandis-0.14, so X is an inferior good. 11. The result is not surprising. Given the available information, the own price elasticity 137 of demand for major cellular telephone manufacturer is EQ ,P 8.06 . Since 17 this number is greater than one in absolute value, demand is elastic. By the total revenue test, this means that a reduction in price will increase revenues. Managerial Economics and Business Strategy, 7e Page 5 12. The regression output is as follows: SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 0.97 0.94 0.94 0.00 49 ANOVA df Regression Residual Total Intercept LN Price LN Income 2 46 48 SS 0.00702 0.00044 0.00745 Coefficients Standard Error 1.29 0.41 -0.07 0.00 -0.03 0.09 MS 0.004 0.000 t Stat 3.12 -26.62 -0.33 F Significance F 370.38 0.0000 P-value 0.00 0.00 0.74 Lower 95% 0.46 -0.08 -0.22 Upper 95% 2.12 -0.07 0.16 Table 3-2 Thus, the demand for your batteries is given by ln Q 1.29 0.07 ln P 0.03ln M . Since this is a log-linear demand equation, the best estimate of the income elasticity of demand for your product is -.03: Your batteries are an inferior good. However, note the estimated income elasticity is very close to zero (implying that a 3 percent reduction in global incomes would increase the demand for your product by less than one tenth of one percent). More importantly, the estimated income elasticity is not statistically different from zero (the 95 percent confidence interval ranges from a low of -.22 to a high of .16, with a t-statistic that is well below 2 in absolute value). On balance, this means that a 3 percent decline in global incomes is unlikely to impact the sales of your product. Note that the R-square is reasonably high, suggesting the model explains 94 percent of the total variation in the demand for this product. Likewise, the F-test indicates that the regression fit is highly significant. 13. 14. Page 6 Based on this information, the own price elasticity of demand for Big G cereal is 3 EQ , P 1.5 . Thus, demand for Big G cereal is elastic (since this number is 2 greater than one in absolute value). Since Lucky Charms is one particular brand of cereal for which even more substitutes exist, you would expect the demand for Lucky Charms to be even more elastic than the demand for Big G cereal. Thus, since the demand for Lucky Charms is elastic, one would predict that the increase in price of Lucky Charms resulted in a reduction in revenues on sales of Lucky Charms. % Q d 1.75 . Solving, we see that coffee 4 purchases are expected to decrease by 7 percent. Use the income elasticity formula to write Michael R. Baye 15. To maximize revenue, Toyota should charge the price that makes demand unit elastic. Using the own price elasticity of demand formula, P EQ , P 1.25 1 . Solving this equation for P implies that the 100, 000 1.25P revenue maximizing price is P $40,000 . 16. Using the change in revenue formula for two products, R $6001 2.5 $400 0.2 .01 $9.8 million , so revenues will increase by $9.8 million. 17. The estimated demand function for residential heating fuel is d Q RHF 136.96 91.69 PRHF 43.88PNG 11.92 PE 0.05M , where PRHF is the price of residential heating fuel, PNG is the price of natural gas, PE is the price of electricity, and M is income. However, notice that coefficients of income and the price of electricity are not statistically different from zero. Among other things, this means that the proposal to increase the price of electricity by $5 is unlikely to have a statistically significant impact on the demand for residential heating fuel. Since the coefficient of PRHF is -91.69, a $2 increase in PRHF would lead to a 183.38 unit reduction in the consumption of residential heating fuel (since (-91.69)($2) = - 183.38 units). Since the coefficient of PNG is 43.88, a $1 reduction in PNG would lead to a 43.88 unit reduction in the consumption of residential heating fuel (since (43.88)(-$1) = -43.88). Thus, the proposal to increase the price of residential heating fuel by $2 would lead to the greatest expected reduction in the consumption of residential heating fuel. Managerial Economics and Business Strategy, 7e Page 7 18. The regression output is as follows: SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 0.97 0.94 0.94 0.06 41 ANOVA df Regression Residual Total Intercept ln (Price) SS 1 39 40 2.24 0.15 2.38 MS 2.24 0.00 F Significance F 599.26 0.00 Coefficients Standard Error t Stat P-value 4.29 0.12 37.17 0.00 -1.38 0.06 -24.48 0.00 Lower 95% Upper 95% 4.06 4.53 -1.50 -1.27 Table 3-3 Thus, the least squares regression line is ln Q 4.29 1.38 ln P . The own price elasticity of demand for broilers is –1.38. From the t-statistic, this is statistically different from zero (the t-statistic is well over 2 in absolute value). The R-square is relatively high, suggesting that the model explains 94 percent of the total variation in the demand for chicken. Given that your current revenues are $750,000 and the elasticity of demand is –1.38, we may use the following formula to determine how much you must change price to increase revenues by $50,000: Px Px P $50,000 $750,0001 1.38 x Px R Px Q x 1 EQx ,Px Px $50,000 0.175 . That is, to increase revenues by $50,000, Px $285,000 you must decrease your price by 17.5 percent. Solving yields Page 8 Michael R. Baye 19. The regression output (and corresponding demand equations) for each state are presented below: ILLINOIS SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 0.29 0.09 0.05 151.15 50 ANOVA degrees of freedom Regression Residual Total 2 47 49 SS MS F Significance F 100540.93 1073835.15 1174376.08 50270.47 22847.56 2.20 0.12 t Stat P-value Lower 95% Coefficients Standard Error Intercept Price Income -42.65 2.62 14.32 496.56 13.99 6.83 -0.09 0.19 2.10 0.93 0.85 0.04 -1041.60 -25.53 0.58 Upper 95% 956.29 30.76 28.05 Table 3-4 The estimated demand equation is Q 42.65 2.62 P 14.32 M . While it appears that demand slopes upward, note that coefficient on price is not statistically different from zero. An increase in income by $1,000 increases demand by 14.32 units. Since the t-statistic associated with income is greater than 2 in absolute value, income is a significant factor in determining quantity demanded. The R-square is extremely low, suggesting that the model explains only 9 percent of the total variation in the demand for KBC microbrews. Factors other than price and income play an important role in determining quantity demanded. Managerial Economics and Business Strategy, 7e Page 9 INDIANA SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 0.87 0.76 0.75 3.94 50 ANOVA degrees of freedom Regression Residual Total 2 47 49 SS MS 2294.93 1147.46 729.15 15.51 3024.08 Coefficients Standard Error Intercept Price Income 97.53 -2.52 2.11 10.88 0.25 0.26 t Stat 8.96 -10.24 8.12 F 73.96 P-value 0.00 0.00 0.00 Significance F 0.00 Lower 95% 75.64 -3.01 1.59 Upper 95% 119.42 -2.02 2.63 Table 3-5 The estimated demand equation is Q 97.53 2.52 P 2.11M . This equation says that increasing price by $1 decreases quantity demanded by 2.52 units. Likewise, increasing income by $1,000 increases demand by 2.11 units. Since the t-statistics for each of the variables is greater than 2 in absolute value, price and income are significant factors in determining quantity demanded. The R-square is reasonably high, suggesting that the model explains 76 percent of the total variation in the demand for KBC microbrews. Page 10 Michael R. Baye MICHIGAN SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 0.63 0.40 0.37 10.59 50 ANOVA degrees of freedom Regression Residual Total 2 47 49 SS MS F Significance F 3474.75 5266.23 8740.98 1737.38 112.05 15.51 0.00 t Stat P-value Lower 95% Upper 95% 11.23 -3.28 4.09 0.0000 0.0020 0.0002 149.75 -1.65 0.72 215.12 -0.40 2.11 Coefficients Standard Error Intercept Price Income 182.44 -1.02 1.41 16.25 0.31 0.35 Table 3-6 The estimated demand equation is Q 182.44 1.02 P 1.41M . This equation says that increasing price by $1 decreases quantity demanded by 1.02 units. Likewise, increasing income by $1,000 increases demand by 1.41 units. Since the t-statistics associated with each of the variables is greater than 2 in absolute value, price and income are significant factors in determining quantity demanded. The R-square is relatively low, suggesting that the model explains about 40 percent of the total variation in the demand for KBC microbrews. The F-statistic is zero, suggesting that the overall fit of the regression to the data is highly significant. Managerial Economics and Business Strategy, 7e Page 11 MINNESOTA SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 0.64 0.41 0.39 16.43 50 ANOVA degrees of freedom Regression Residual Total 2 47 49 SS MS F Significance F 8994.34 12680.48 21674.82 4497.17 269.80 16.67 0.00 t Stat P-value Lower 95% Upper 95% 1.00 -0.05 5.68 0.32 0.96 0.00 -82.23 -5.19 2.20 245.62 4.94 4.62 Coefficients Standard Error Intercept Price Income 81.70 -0.12 3.41 81.49 2.52 0.60 Table 3-7 The estimated demand equation is Q 81.70 0.12 P 3.41M . This equation says that increasing price by $1 decreases quantity demanded by 0.12 units. Likewise, a $1,000 increase in consumer income increases demand by 3.41 units. Since the tstatistic associated with income is greater than 2 in absolute value, it is a significant factor in determining quantity demanded; however, price is not a statistically significant determinant of quantity demanded. The R-square is relatively low, suggesting that the model explains 41 percent of the total variation in the demand for KBC microbrews. Page 12 Michael R. Baye MISSOURI SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 0.88 0.78 0.77 15.56 50 ANOVA degrees of freedom Regression Residual Total 2 47 49 SS MS F Significance F 39634.90 11385.02 51019.92 19817.45 242.23 81.81 0.00 t Stat P-value Lower 95% Upper 95% 5.13 -1.36 12.73 0.00 0.18 0.00 75.57 -1.96 6.27 173.05 0.38 8.63 Coefficients Standard Error Intercept Price Income 124.31 -0.79 7.45 24.23 0.58 0.59 Table 3-8 The estimated demand equation is Q 124.31 0.79 P 7.45M . This equation says that increasing price by $1 decreases quantity demanded by 0.79 units. Likewise, a $1,000 increase in income increases demand by 7.45 units. The t-statistic associated with price is not greater than 2 in absolute value; suggesting that price does not statistically impact the quantity demanded. However, the estimated income coefficient is statistically different from zero. The R-square is reasonably high, suggesting that the model explains 78 percent of the total variation in the demand for KBC microbrews. Managerial Economics and Business Strategy, 7e Page 13 OHIO SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 0.99 0.98 0.98 10.63 50 ANOVA degrees of freedom Regression Residual Total 2 47 49 SS 323988.26 5306.24 329294.50 Coefficients Standard Error Intercept Price Income 111.06 -2.48 7.03 23.04 0.79 0.13 MS F 161994.13 1434.86 112.90 Significance F 0.00 t Stat P-value Lower 95% Upper 95% 4.82 -3.12 52.96 0.0000 0.0031 0.0000 64.71 -4.07 6.76 157.41 -0.88 7.30 Table 3-9 The estimated demand equation is Q 111.06 2.48P 7.03M . This equation says that increasing price by $1 decreases quantity demanded by 2.48 units. Likewise, increasing income by $1,000 increases demand by 7.03 units. Since the t-statistics associated with each of the variables is greater than 2 in absolute value, price and income are significant factors in determining quantity demanded. The R-square is very high, suggesting that the model explains 98 percent of the total variation in the demand for KBC microbrews. Page 14 Michael R. Baye WISCONSIN SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 0.999 0.998 0.998 4.79 50 ANOVA degrees of freedom Regression Residual Total 2 47 49 SS Coefficients Standard Error Intercept Price Income 107.60 -1.94 10.01 MS F 614277.37 307138.68 13369.30 1079.75 22.97 615357.12 7.97 0.25 0.06 t Stat 13.49 -7.59 163.48 Significance F 0.00 P-value Lower 95% Upper 95% 0.00 0.00 0.00 91.56 -2.45 9.88 123.65 -1.42 10.13 Table 3-10 The estimated demand equation is Q 107.60 1.94 P 10.01M . This equation says that increasing price by $1 decreases quantity demanded by 1.94 units. Likewise, increasing income by $1,000 increases demand by 10.01 units. Since the t-statistics associated with price and income are greater than 2 in absolute value, price and income are both significant factors in determining quantity demanded. The R-square is very high, suggesting that the model explains 99.8 percent of the total variation in the demand for KBC microbrews. Managerial Economics and Business Strategy, 7e Page 15 20. Table 3-11 contains the output from the linear regression model. That model indicates that R2 = .55, or that 55 percent of the variability in the quantity demanded is explained by price and advertising. In contrast, in Table 3-12 the R2 for the log-linear model is .40, indicating that only 40 percent of the variability in the natural log of quantity is explained by variation in the natural log of price and the natural log of advertising. Therefore, the linear regression model appears to do a better job explaining variation in the dependent variable. This conclusion is further supported by comparing the adjusted R2s and the F-statistics in the two models. In the linear regression model the adjusted R2 is greater than in the log-linear model: .54 compared to .39, respectively. The F-statistic in the linear regression model is 58.61, which is larger than the F-statistic of 32.52 in the log-linear regression model. Taken together these three measures suggest that the linear regression model fits the data better than the log-linear model. Each of the three variables in the linear regression model is statistically significant; in absolute value the t-statistics are greater than two. In contrast, only two of the three variables are statistically significant in the log-linear model; the intercept is not statistically significant since the t-statistic is less than two in absolute value. At P = $3.10 and A = $100, milk consumption is 2.029 million d gallons per week Qmilk 6.52 1.613.10 .005100 2.029 . SUMMARY OUTPUT LINEAR REGRESSION MODEL Regression Statistics Multiple R 0.74 R Square 0.55 Adjusted R Square 0.54 Standard Error 1.06 Observations 100.00 ANOVA df Regression Residual Total Intercept Price Advertising 2.00 97.00 99.00 SS 132.51 109.66 242.17 MS 66.26 1.13 F Significance F 58.61 2.05E-17 Coefficients Standard Error t Stat P-value 6.52 0.82 7.92 0.00 -1.61 0.15 -10.66 0.00 0.005 0.0016 2.96 0.00 Lower 95% Upper 95% 4.89 8.15 -1.92 -1.31 0.00 0.01 Table 3-11 Page 16 Michael R. Baye SUMMARY OUTPUT LOG-LINEAR REGRESSION MODEL Regression Statistics Multiple R 0.63 R Square 0.40 Adjusted R Square 0.39 Standard Error 0.59 Observations 100.00 ANOVA df Regression Residual Total SS 2.00 97.00 99.00 MS 22.40 11.20 33.41 0.34 55.81 F Significance F 32.52 1.55E-11 Coefficients Standard Error t Stat P-value -1.99 2.24 -0.89 0.38 -2.17 0.28 -7.86 0.00 0.91 0.37 2.46 0.02 Intercept ln(Price) ln(Advertising) Lower 95% Upper 95% -6.44 2.46 -2.72 -1.62 0.18 1.65 Table 3-12 21. Given the estimated demand function and the monthly subscriptions prices, demand is d 172,000 subscribers Qsat 152.5 0.950 1.0530 1.1030 . Thus, revenues are $8.6 million, which are not sufficient to cover costs. Revenues are maximized when Psat 1 : Solving yields Psat $120.56 . Thus, the demand is unit elastic .9 217 . 9 P sat maximum revenue News Corp. can earn is $13,080,277.76 TR P Q 120.56 217 .9 120.56 1000 . News Corp. cannot cover its costs in the current environment. 22. The manager of Pacific Cellular estimated that the short-term price elasticity of demand was inelastic. In the market for cellular service, contracts prevent many customers from immediately responding to price increases. Therefore, it is not surprising to observe inelastic in the short-term. However, as contracts expire and customers have more time to search for alternatives, quantity demanded is likely to drop off much more. Given a year or two, the demand for cellular service is much more elastic. The price increase has caused Pacific to lose more customers than they initially estimated. 23. The owner is confusing the demand for gasoline for the entire U.S. with demand for the gasoline for individual gasoline stations. There are not a great number of substitutes for gasoline, but in large towns there are usually a very high number of substitutes for gasoline from an individual station. In order to make an informed decision, the owner needs to know the own price elasticity of demand for gasoline from his stations. Since gas prices are posted on big billboards, and gas stations in cities are generally close together, demand for gas from a small group of individual stations tends to be fairly elastic. Managerial Economics and Business Strategy, 7e Page 17 Chapter 4: Answers to Questions and Problems 1. a. The market rate of substitution is Px 10 0.25 . 40 Py b. See Figure 4-1. c. Increasing income to $800 (by $400) expands the budget set, as shown in Figure 4-1. Since the slope is unchanged, so is the market rate of substitution. Budget Set Y 25 20 15 Increase in income 10 5 0 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 X Figure 4-1 2. a. Since the slope of the line through point A is 20 1 and the price of good X 20 is $5, it follows that Py 5 . b. If the consumer spends all her income on good X she can purchase 20 units. Since these units cost $5 each, her income must be $100. c. At point A, the consumer spends ($5)(10) = $50 on good Y, which means that the remaining $100 - $50 = $50 is being spent on good X. Since good X costs $5 per unit, point A corresponds to 10 units of good X. d. The price of good Y decreased to $2.50. The consumer achieves a higher level of satisfaction at point B. Managerial Economics and Business Strategy, 7e Page 1 3. a. The consumer’s budget line is $250 $5 X $10Y . Rearranging terms and solving for Y results in Y 25 0.5 X . b. See in Figure 4-2. c. When the price of X increases to $10, the budget line becomes $250 $10 X $10Y , which is equivalent to Y 25 X (after rearranging and simplifying terms). This is shown in Figure 4-2. The market rate of substitution P P 5 1 10 changes from x to x 1 . 10 2 Py Py 10 Budget Set Y 30 25 20 15 10 5 0 0 5 10 15 20 25 30 35 40 45 50 X Figure 4-2 4. This is not always the case. For instance, if the consumer was initially consuming more of the inferior good than a gift certificate would purchase, then less of the inferior good will be consumed when given a gift certificate. 5. A half-price sale cuts the price of each and every unit in half. In contrast, a buy-one, get-one-free deal does not change the relative price of any units between 0 and 1 unit. Furthermore, it makes the price of units purchased between 1 and 2 units purchased zero. Page 2 Michael R. Baye 6. a. Px $50 , Py $100 and M = $300. b. M 300 3 units. Py 100 f. g. M 300 6 units. Px 50 1 unit (since the $50 gift certificate will purchase exactly one unit of good X). M $50 350 7 units. Px 50 D , B, C, A. Normal. a. b. c. d. Consumption of good X will decrease and consumption of good Y will increase. Consumption of good X will decrease and consumption of good Y will increase. Nothing will happen to the consumption of either good. Consumption of good X will increase and consumption of good Y will decrease. c. d. e. 7. 8. All properties hold except Property 4-3 (“Diminishing Marginal Rate of Substitution”) and Property 4-2 (“More is Better”). Managerial Economics and Business Strategy, 7e Page 3 9. a. The initial budget set is depicted in Figure 4-3. Y 125 Figure 4-3 250 X b. Doubling all income and price leaves the budget set unchanged. The increase in income is sufficient to offset the price increases. The market rate of substitution is unchanged. c. The consumer’s income is $500, the price of X is $2 per unit and the price of Y is $4 per unit. 10. 11. Page 4 a. The workers opportunity set in a given 24-hour period is E 320 5L . b. Since the worker is always willing to trade $12 dollars of income for one hour of leisure, the worker’s indifference curve does not exhibit diminishing marginal rate of substitution; the worker always trades between the two goods at the same rate. These preferences do not exhibit a diminishing marginal rate of substitution since consumers are always willing to substitute the same amount of store-brand sugar for an additional pound of producer-brand sugar. When store-brand sugar is $1 per pound and producer-brand sugar is $2 per pound, the consumer will purchase 10 pounds of store-label sugar and no producer-brand sugar. After the change, the consumer will purchase no store-label sugar and 10 pounds of producer-brand sugar. Michael R. Baye 12. See Figure 4-6. When there is no food stamp program, the market rate of substitution is –0.5. The Food Stamp program leaves the market rate of substitution unchanged, and a consumer can purchase $170 of food without spending her income. A dollarfor-dollar exchange of food stamps for money further expands a consumer’s opportunity set, potentially making her better off. Budget Constraint with and without Food Stamps Other 80 Goods Budget line when food stamps are sold on black market for $170 70 60 50 Budget line with $170 in food stamps 40 30 20 Initial budget line 10 0 0 10 20 30 40 50 60 70 80 90 100 110 120 130 Food Figure 4-6 13. See Figure 4-7. The offer expands the consumer’s budget set and allows her to purchase more tires. Budget Set with and without Buy 3, Get 4th Free Offer Income Spent on Other 600 Goods Budget line with "Buy 3, get the 4th Free Offfer" 500 400 300 200 Initial budget line 100 0 0 1 2 3 4 5 6 7 8 9 10 11 Tires Figure 4-7 Managerial Economics and Business Strategy, 7e Page 5 14. See Figure 4-8. The initial market rate of substitution is –0.5. Since, after the price P decrease, the MRS 1 0.625 EM (where PEM is the price of electronic media PT and PT the price of travel) equilibrium has not been achieved. To reach equilibrium, the business should increase its use of electronic media and decrease travel. Budget Set Quantity of Travel 7 6 5 New budget line 4 3 2 Initial budget line 1 0 0 1 2 3 4 5 6 7 8 9 10 Quantity of Electronic Media Figure 4-8 Page 6 Michael R. Baye 15. The impacts on the consumer’s budget sets are illustrated in Figure 4-9. As is shown in the diagram, if the consumer has a strong preference for other goods (so that the preferred quantity of other goods is greater than 7 units), the cash is preferred even though it is taxed. Otherwise, the non-taxable, employer-sponsored health insurance program allows an employee to achieve a higher indifference curve. Budget Line with Employer Sponsored Health Insurance Other Goods 9 Budget line with (taxable) cash equivalent health insurance benefit 8 7 6 5 Budget line with health insurance benefit 4 3 2 Initial budget line 1 0 0 1 2 3 4 5 6 7 8 Quantity of Health Insurance Figure 4-9 16. Under the existing plan, a worker that does not “goof off” produces 3 copiers per hour and thus is paid $9 each hour. Under the new plan, each worker would be paid a flat wage of $8 per hour. While it might appear on the surface that the company would save $1 per hour in labor costs by switching plans, the flat wage would be a lousy idea. Under the current plan, workers get paid the $9 only if they work hard during the hour and produce 3 machines that pass inspection. Under the new plan, workers would get paid $8 an hour regardless of how many units they produce. Since your firm has no supervisors to monitor the workers, you should not favor the plan. Managerial Economics and Business Strategy, 7e Page 7 17. As shown in Figure 4-10, the budget line when more than 10 dozen bagels are purchased annually under the frequent buyer program is always greater than the budget line when the firm sells each dozen bagels at a 3 percent discount. However, the budget line for consumers who purchase fewer than 10 dozen bagels per year is greater under the 3 percent per dozen discount. Comparison of Budget Lines Under Different Offers Income Spent on Other Goods 160 140 120 100 Budget line under the frequent buyer program 80 60 Budget line with 3 percent discount 40 20 0 0 5 10 15 20 25 30 Quantity of Bagels (dozens) Figure 4-10 18. Page 8 Yes. Since pizza is an inferior good, if the consumer is given $30 in cash she will definitely spend it entirely on CDs – just as she would if given a $30 gift certificate at a local music store. Michael R. Baye 19. Figure 4-11 illustrates a consumer’s budget line when a firm offers a “quantity discount.” A consumer will never purchase exactly 8 bottles of wine, since at this kink in the opportunity set the consumer would always be better off by buying more or less wine. Budget Line with Quantity Discount Quantity of Other Goods 110 100 90 80 70 60 50 40 30 20 10 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Quantity of Wine Figure 4-11 Managerial Economics and Business Strategy, 7e Page 9 20. Figure 4-12 contains profit as a function of output. Output when managers are compensated based solely on output is 25 units and profits are zero. In contrast, when managers’ compensation is based solely on profits, output is 12.5 units and profits are $156.25. When managers’ compensation is based on a combination of output and profit, output ranges between 12.5 and 25 units and profit will be between zero and $156.25. The exact combination of output and profit depends on how these variables are weighted. Profit ($) 180 160 140 120 100 80 60 40 20 0 0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 Output (Q) Figure 4-12 Page 10 Michael R. Baye 21. Figures 4-13a and 4-13b, respectively, illustrate Albert’s and Sid’s opportunity sets. Since there are 24 hours per day, at the new wage rate of $18 per hour Albert will supply 12 hours of labor per day (24-12), and Sid will supply 8 hours of labor per day (24-16). This seemingly contradictory result is explained by decomposing the wage change into the substitution effect and income effect. The diminishing marginal rate of substitution between income and leisure implies that the substitution effect will increase the amount of leisure consumed by each worker (decrease the amount of labor supplied). Since after the wage change Albert is observed consuming less leisure (supplying more labor), the income effect dominates the substitution effect. In contrast, the substitution effect dominates the income effect for Sid; since Sid is observed consuming more leisure (supplying less labor) after the wage change. Income Albert’s Opportunity Set 480 432 12 14 24 Leisure Figure 4-13a Income Sid’s Opportunity Set 480 432 14 16 24 Leisure Figure 4-13b Managerial Economics and Business Strategy, 7e Page 11 22. Gift cards are not merely a fad. Retailers experience significant benefits from gift cards since they minimize product returns; independent of whether the good is normal or inferior. Gift cards can also benefit consumers. A gift card does not impact the amount purchased for one good (say the good on the Y axis), but shifts out the budget constraint for the other good (the good on the X axis) by the face value of the gift card. The expanded budget constrain permits the consumer to reach a higher indifference curve; resulting in greater utility. 23. AOG Flat-Rate Plan Old Plan AOG A 1,499 43,200 1,499 43,200 Under the Old Plan, consumers consumed 1,499 of online monthly minutes for $14.99. The budget line under the Flat-Rate Plan, however, is significantly different. Consumers can choose to now spend all their income on all other goods (AOG), represented by point A on the AOG axis or consume the same about AOG and any amount of online minutes up to the maximum number of minutes in a month. Optimizing consumers will choose the corner solution represented by the same number of units of AOG as the Old Plan and 43,200 online monthly minutes. Thus, UK consumers are necessarily better off (assuming no busy signals). AOL UK, however, gains no additional revenues and presumably must increase it network capacity. Therefore, AOL UK may earn lower profit (ignoring other factors). Page 12 Michael R. Baye Chapter 5: Answers to Questions and Problems 1. a. When K = 16 and L = 16, Q 16 16 16 . Thus, APL = Q/L = 16/16 = 0.75 0.25 1. When K = 16 and L = 81, Q 16 81 8 3 24 . Thus, APL = 0.75 0.25 24/81 = 8/27. 3 4 b. The marginal product of labor is MPL 2 L . When L = 16, MPL 2 16 3 4 1/ 4 . When L = 81, MPL 2 81 3 4 2 / 27 . Thus, as the number of units of labor hired increases, the marginal product of labor decreases MPL 16 1/ 4 2 / 27 MPL 81 , holding the level of capital fixed. c. We must equate the value marginal product of labor equal to the wage and solve for L. Here, VMPL P MPL $100 2 L equal to the wage of $25 gives 200 L quantity of labor is L = 16. 3/ 4 Managerial Economics and Business Strategy, 7e 3/ 4 200 L 3/ 4 . Setting this 25 . Solving for L, the optimal Page 1 2. See Table 5-1. (1) (2) (3) Q (4) Marginal Product of Capital MP K (5) Average Product of Capital AP K (6) Average Product of Labor AP L (7) Value Marginal Product of Capital VMPK Capital Labor Output K L 0 1 2 3 4 5 6 7 8 9 10 11 20 20 20 20 20 20 20 20 20 20 20 20 0 50 150 300 400 450 475 475 450 400 300 150 -50 100 150 100 50 25 0 -25 -50 -100 -150 -50 75 100 100 90 79.17 67.86 56.25 44.44 30 13.64 -2.50 7.50 15 20 22.50 23.75 23.75 22.50 20 15 7.50 -100 200 300 200 100 50 0 -50 -100 -200 -300 Table 5-1 a. Labor is the fixed input while capital is the variable input. b. Fixed costs are 20($15) = $300. c. To produce 475 units in the least-cost manner requires 6 units of capital, which cost $75 each. Thus, variable costs are ($75)(6) = $450. d. Using the VMPK = r rule, K = 5 maximizes profits. e. The maximum profits are $2(450) $15(20) $75(5) $225 . f. There are increasing marginal returns when K is between 0 and 3. g. There are decreasing marginal returns when K is between 3 and 11. h. There are negative marginal returns when K is greater than 7. 3. The law of diminishing marginal returns is the decline in marginal productivity experienced when input usage increases, holding all other inputs constant. In contrast, the law of diminishing marginal rate of technical substitution is a property of a production function stating that as less of one input is used, increasing amounts of another input must be employed to produce the same level of output. 4. a. FC = 50. 2 3 b. VC 10 25 10 30 10 5 10 $8, 250 . c. C 10 50 2510 3010 510 $8,300 . $50 d. AFC 10 $5 . 10 VC 10 $8, 250 $825 . e. AVC 10 10 10 f. ATC 10 AFC 10 AVC 10 $830 . 2 3 g. MC 10 25 6010 1510 $2,125 . 2 Page 2 Michael R. Baye w , the firm is not using the cost minimizing combination of labor r and capital. To minimize costs, the firm should use more labor and less capital since MPL 50 MPK 75 the marginal product per dollar spent is greater for labor: . 6 12 w r 5. Since MRTS KL 6. See Table 5-2. (1) (2) (3) (4) (5) (7) (8) Average Fixed Cost (6) Average Variable Cost Quantity Fixed Cost Variable Cost Total Cost Average Total Cost Marginal Cost Q FC VC 0 100 200 300 400 500 600 10,000 10,000 10,000 10,000 10,000 10,000 10,000 0 10,000 15,000 30,000 50,000 90,000 140,000 TC AFC AVC ATC MC 10,000 20,000 25,000 40,000 60,000 100,000 150,000 -100 50 33.33 25 20 16.67 -100 75 100 125 180 233.33 -200 125 133.33 150 200 250 -100 50 150 200 400 500 Table 5-2 Managerial Economics and Business Strategy, 7e Page 3 7. a. For a quadratic multi-product cost function, economies of scope exist if f aQ1Q2 0 . In this case, f 75 and a 0.25 , so economies of scope exist since f is fixed cost, which is always nonnegative. b. Cost complementarities exist since a 0.25 0 . c. Since a 0.25 0 , the marginal cost of producing product 1 will increase if the division that produces product 2 is sold. 8. Fixed costs are associated with fixed inputs, and do not change when output changes. Variable costs are costs associated with variable inputs, and do change when output changes. Sunk costs are costs that are forever lost once they have been paid. 9. a. When K = 2 and L = 3, Q = 4 units. b. The cost-minimizing mix of K and L that produce Q = 4 is K = 2, L = 1. c. Since K and L are perfect complements in the production process, the costminimizing levels of K and L do not depend on the rental rates of K and L. Therefore, the cost-minimizing levels of K and L do not change with changes in the relative rental rates. 10. a. With K = 2 and L = 3, Q = 16. b. Since the MRTSKL is 2, that means a company can trade two units of capital for every one unit of labor. This production function does not exhibit diminishing marginal rate of technical substitution. The perfectly substitutability between capital and labor means that only input will be utilized. Since MPL MPK 4 2 , the company should hire all capital. w r 30 10 c. The company should hire only labor. 11. An investment tax credit would reduce the relative price of capital to labor. Other w things equal, this would increase , thereby making the isocost line more steep. This r means that the cost-minimizing input mix will now involve more capital and less labor, as firms substitute toward capital. Labor unions are likely to oppose the investment tax credit since the higher capital-to-labor ratio will translate into lost jobs. You might counter this argument by noting that, while some jobs will be lost due to substituting capital for labor, many workers will retain their jobs. Absent the plan, automakers have an incentive to substitute cheaper foreign labor for U.S. labor. The result of this substitution would be a movement of plants abroad, resulting in the complete loss of U.S. jobs. 12. Since MRTS KL Page 4 w , the firm was not using the cost minimizing combination of labor r and capital. To achieve the cost minimizing combination of inputs, the previous Michael R. Baye manager should have used fewer units of capital and more units of labor, since MPL 100 MPK 100 . w r 8 16 13. The profit-maximizing level of labor and output is achieved where VMPL w . Here, VMPL 2 $100 4 1/ 2 L 1 2 $400 L 1/ 2 and w $100 per day. Solving yields L = 16. The profit-maximizing level of output is Q 24 16 16 units. The firm’s fixed costs are $10,000, its variable costs are $100(16) = $1,600, and its total revenues are $200(16) = $3,200. Profits are $3,200 – $11,600 = – $8,400. The firm is suffering a loss, but the loss is lower than the $10,000 that would be lost if the firm shut down its operation. 12 12 14. The higher wage rate in Europe induces Airbus to employ a more capital intensive input mix than Boeing. Since Airbus optimally uses fewer workers than Boeing, and profit-maximization entails input usage in the range of diminishing marginal product, it follows that the lower quantity of labor used by Airbus translates into a higher marginal product of labor at Airbus than at Boeing. 15. Table 5-3 provides some useful information for making your decision. According to the VMPL = w rule, you should hire five units of labor and produce 90 units of output to maximize profits. Your fixed costs are ($10)(5) = $50, your variable costs are ($50)(5) =$250, and your revenues are ($5)(90) = $450. Thus, your maximum profits are $450 - $300 = $150. (1) (2) (3) Q (4) Marginal Product of Labor MP L (5) Average Product of Labor AP L (6) Average Product of Capital AP K (7) Value Marginal Product of Labor VMPL Labor Capital Output L K 0 1 2 3 4 5 6 7 8 9 10 11 5 5 5 5 5 5 5 5 5 5 5 5 0 10 30 60 80 90 95 95 90 80 60 30 -10 20 30 20 10 5 0 -5 -10 -20 -30 -10 15 20 20 18 15.8 13.6 11.3 8.9 6 2.7 -2 6 12 16 18 19 19 18 16 12 6 -50 100 150 100 50 25 0 -25 -50 -100 -150 Table 5-3 16. The $1,200 per month that could be earned by renting out the excess rental space. Managerial Economics and Business Strategy, 7e Page 5 17. Had she not spent the $6,000 on advertising but instead collected the $65,000 refund, her total loss would have been limited to her sunk costs of $10,000. Her decision to spend $6,000 on advertising in an attempt to fetch an extra $5,000 was clearly foolish. However, the $6,000 is a sunk cost and therefore irrelevant in deciding whether to accept the $66,000 offer. She should accept the $66,000 offer because doing so makes her $1,000 better off than obtaining the $65,000 refund. 18. Facility “L” produces 6 million kilowatt hours of electricity at the lowest average total cost, so this is the optimal facility for South-Florida. Facility “M” produces 2 million kilowatt hours of electricity at the lowest average total cost, so this is the optimal facility for the Panhandle. There are economies of scale up to about 3 million kilowatts per hour, and diseconomies of scale thereafter. Therefore, facility “M” will be operating in the range of economies of scale while facility “L” will be operating in the range of diseconomies of scale. 19. To maximize profits the firm should continue adding workers so long as the value marginal product of labor exceeds the wage. The value marginal product of labor is defined as the marginal product of labor times the price of output. Here, output sells for $50 per panel, so the value marginal product of the third worker is $50(290) = $14,500. Table 5-4 summarizes the VMPL for each choice of labor. Since the wage is $7,000, the profit maximizing number of workers is 4. Machines 5 5 5 5 5 5 5 Workers Output MPL VMPL Wage 0 0 – – – 1 600 600 $30,000 $7,000 2 1,000 400 $20,000 $7,000 3 1,290 290 $14,500 $7,000 4 1,480 190 $9,500 $7,000 5 1,600 120 $6,000 $7,000 6 1,680 80 $4,000 $7,000 Table 5-4 20. Page 6 The rental rate of capital is ¥475,000, computed as r MPK P .5 950,000 475,00 . Therefore, the marginal product of labor is MPL 0.5 0.0014 cars per hour, which is found by solving . Costs are 1,330 475,000 minimized when the marginal rate of technical substitution is 0.0028. Michael R. Baye 21. Given the tightly woven marine engine and shipbuilding divisions, economies of scope and cost complementarities are likely to exist. Eliminating the unprofitable marine engine division may actually raise the shipbuilding division’s costs and cause that division to become unprofitable. For this argument to withstand criticism, you must show the CEO that the quadratic multi-product cost function exhibits cost complementarities and economies of scope, which occurs when a 0 and f aQ1Q2 0 , respectively, and compare profitability under the different scenarios. 22. Taking into account both implicit and explicit costs, the total fixed cost from operating the kiosk is $6,000; the $2,000 in rent plus the $4,000 in forgone earnings. Total variable costs are $1.23 per gallon. The cost function is C Q 6,000 1.23Q . dC Q $1.23 ; the wholesale price. The average The marginal cost is MC Q dQ C Q 1.23Q $1.23 . The average fixed cost is variable cost is AVC Q Q Q $6000 AFC Q . The entrepreneur will earn a profit when revenues exceed costs, Q which occurs when 2Q 6,000 1.23Q . Solving for Q implies the entrepreneur earns a profit when she sells Q > 8571.43 gallons, or 8572 gallons. The average fixed cost $6000 of selling Q = 8572 is AFC 8572 $0.70 . 8572 23. Assuming that the optimal mix of unskilled and semi-skilled labor were being utilized at the time the legislation passed, in the short run, a higher minimum wage paid to unskilled labor implies that to minimize costs the retailer should increase its use of semi-skilled worker and decrease its use or unskilled workers. In the longer run, the retailer may want to consider substituting capital for labor (invest in some machines to automate a portion of your boxing needs). Obviously, additional information would be required to conduct a net present value analysis for these long-run investments, but it is probably worth getting this information and running some numbers. Managerial Economics and Business Strategy, 7e Page 7