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.

Figure 3-1

2.

a. At the given prices, quantity demanded is 700 units: . Substituting the relevant information into the elasticity formula gives: . Since this is less 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: . Substituting the relevant information into the elasticity formula gives: . Since this is 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: . Substituting the relevant information into the elasticity formula gives: . Since this 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.

4. a. Use the own price elasticity of demand formula to write . Solving, we see that the quantity demanded of good X will decrease by 10 percent if the price of good X increases by 5 percent. b. Use the cross-price elasticity of demand formula to write . Solving, we see that the demand for X will decrease by 60 percent if the price of good Y increases by 10 percent. c. Use the formula for the advertising elasticity of demand to write . Solving, we see that the demand for good X will decrease by 8 percent if advertising decreases by 2 percent. d. Use the income elasticity of demand formula to write . Solving, we see that the quantity demanded of good X will decrease by 9 percent if income decreases by 3 percent. 5. Using the cross price elasticity formula, . Solving, we see that the price of good Y would have to decrease by 10 percent in order to increase the consumption of good X by 50 percent.

6. Using the change in revenue formula for two products, . Thus, a 1 percent increase in the price of good X would cause revenues from both goods to increase by $320.

7. Table 3-1 contains the answers to the regression output.

Table 3-1

a. . 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. The approximate 95 percent