Managerial Economics Numerical's and Solutions

CHAPTER 3

DEMAND THEORY

1. A firm has estimated the following demand function for its product:

Q = 58 − 2P + 0.10I + 15A

where Q is quantity demanded per month in thousands, P is product price, I is an index of consumer income, and A is advertising expenditures per month in thousands. Assume that P = $10, I = 120, and A = 10. Use the point formulas to complete the elasticity calculations indicated below.

(i) Calculate quantity demanded.

(ii) Calculate the price elasticity of demand. Is demand elastic, inelastic, or unit

elastic?

(iii) Calculate the income elasticity of demand. Is the good normal or inferior? Is it a necessity or a luxury?

(iv) Calculate the advertising elasticity of demand.

Solution:

(i) Q = 58 − (2)(10) + (0.10)(120) + (15)(10) = 200

(ii) (−2)(10/200) = −0.10 so demand is inelastic

(iii) (0.10)(120/200) = 0.06 so the good is normal and a necessity

(iv) (15)(10/200) = 0.75

2. A firm has estimated the following demand function for its product:

Q = 100 − 5P + 5I + 15A

where Q is quantity demanded per month in thousands, P is product price, I is an index of consumer income, and A is advertising expenditures per month in thousands. Assume that P = $200, I =150, and A = 30. Use the point formulas to complete the elasticity calculations indicated below.

(i) Calculate quantity demanded.

(ii) Calculate the price elasticity for demand. Is demand elastic, inelastic, or unit

elastic?

(iii) Calculate the income elasticity of demand. Is the good normal or inferior? Is it a necessity or a luxury?

(iv) Calculate the advertising elasticity of demand.

Solution:

(i) Q = 100 − (5)(200) + (5)(150) + (15)(30) = 300

(ii) (−5)(200/300) = −3.33 so demand is elastic

(iii) (5)(150/300) = 2.50 so the good is normal and a luxury

(iv) (15)(30/300) = 1.50

3. A firm has kept track of the quantity demanded of its output during four time periods. Product price, consumer income, and advertising expenditures were also recorded for each time period. The information is provided in the table that follows. Use it to calculate the arc elasticity of demand with respect to price, income, and advertising.

Time Period          1       2        3          4

Quantity            120     80    100       80

Price                     20     30       30      30

Income              150    150    250    250

Advertising          50      50       50      30

Solution:

The price elasticity of demand (using time periods 1 and 2) is

[(120 − 80)/(20 − 30)][(20 + 30)/(120 + 80)] = −1

The income elasticity of demand (using time periods 2 and 3) is

[(80 − 100)/(150 − 250)][(150 + 250)/(80 + 100)] = 0.44

The advertising elasticity of demand (using time periods 3 and 4) is

[(100 − 80)/(50 − 30)][(50 + 30)/(100 + 80)] = 0.44

4. A firm has kept track of the quantity demanded of its output (Good X) during four time periods. The price of X and the prices of two other goods (Good Y and Good Z) were also recorded for each time period. The information is provided in the table that follows. Use it to calculate the own-price arc elasticity of demand and the two cross-price elasticities of demand. Determine whether Good Y and Good Z are complements or substitutes for Good X.

Time Period             1      2         3          4

Quantity of X       220    80     250     260

Price of X                15    25        15       25

Price of Y                10     10          5       10

Price of Z                20     20         20       30

Solution:

The own-price elasticity of demand (using time periods 1 and 2) is

[(220 − 80)/(15 − 25)][(15 + 25)/(220 + 80)] = −1.87

The cross-price elasticity of demand for Good X with respect to the price of Good Y (using time periods 1 and 3) is

[(220 − 250)/(10 − 5)][(10 + 5)/(220 + 250)] = −0.19

Good X and Good Y are complements.

The cross-price elasticity of demand for Good X with respect to the price of Good Y (using time periods 2 and 4) is

[(80 − 260)/(20 − 30)][(20 + 30)/(80 + 260)] = 2.65

Good X and Good Z are substitutes.

5. The price of a good increases from $9 to $11 and, as a result, the quantity of the good demanded declines from 120 to 80. Calculate the price elasticity of demand using the arc formula and determine whether demand is elastic, inelastic, or unit elastic.

Solution:

[(80 − 120)/(11 − 9)][(11 + 9)/(80 + 120)] = −2.00 so demand is elastic.

6. The demand function for a good is defined as Q = 20 − 0.5P. Calculate the price elasticity of demand using the point formula for P = 8 and determine whether demand is elastic, inelastic, or unit elastic.

Solution:

(−0.5)(8/16) = −0.25 so demand is inelastic.

7. The demand function for Good X is defined as QX = 20 − 0.5PX + 1.2PY, where PY is the price of Good Y. Calculate the price elasticity of demand using the point formula for PX = 12 and PY = 10. Determine whether demand is elastic, inelastic, or unit elastic with respect to its own price and whether Good Y is a substitute or a complement with respect to Good X.

Solution:

(−0.5)(12/26) = −0.23 so demand is inelastic with respect to its own price.

(1.2)(10/26) = 0.26 so the two goods are substitutes.

8. The demand function for a good is defined as Q = 20 − 1.5P + 0.2I, where I is a measure of consumer income. Calculate the price elasticity of demand using the point formula for P = 16 and I = 110. Determine whether demand is elastic, inelastic, or unit elastic with respect to its own price and whether the good is normal or inferior and whether it is a luxury or a necessity.

Solution:

(−1.5)(16/18) = −1.33 so demand is elastic with respect to its own price.

(0.2)(110/18) = 1.22 so the good is normal and is a luxury.

9. A firm has estimated the following demand function for its product:

Q = 8 − 2P + 0.10I + A

where Q is quantity demanded per month in thousands, P is product price, I is an index of consumer income, and A is advertising expenditures per month in thousands. Assume that P = $10, I = 120, and A = 10. Use the point formulas to complete the elasticity calculations indicated below.

(i) Calculate quantity demanded.

(ii) Calculate the price elasticity of demand. Is demand elastic, inelastic, or unit

elastic?

(iii) Calculate the income elasticity of demand. Is the good normal or inferior? Is it a necessity or a luxury?

(iv) Calculate the advertising elasticity of demand.

Solution:

(i) Q = 8 − (2)(10) + (0.10)(120) + (1)(10) = 10

(ii) (−2)(10/10) = −2.0 so demand is elastic

(iii) (0.10)(120/10) = 1.2 so the good is normal and a luxury

(iv) (1)(10/10) = 1.0

10. A firm has estimated the following demand function for its product:

Q = 400 − 5P + 5I + 10A

where Q is quantity demanded per month in thousands, P is product price, I is an index of consumer income, and A is advertising expenditures per month in thousands. Assume that P = $200, I = 100, and A = 20. Use the point formulas to complete the elasticity calculations indicated below.

(i) Calculate quantity demanded.

(ii) Calculate the price elasticity of demand. Is demand elastic, inelastic, or unit

elastic?

(iii) Calculate the income elasticity of demand. Is the good normal or inferior? Is it a necessity or a luxury?

(iv) Calculate the advertising elasticity of demand.

Solution:

(i) Q = 400 − (5)(200) + (5)(100) + (10)(20) = 100

(ii) (−5)(200/100) = −10.0 so demand is elastic

(iii) (5)(110/100) = 5.0 so the good is normal and a luxury

(iv) (10)(20/100) = 2.0

11. The price of a good increases from $8 to $10, and as a result the quantity of the good demanded declines from 120 to 80. Calculate the price elasticity of demand using the arc formula and determine whether demand is elastic, inelastic, or unit elastic.

Solution:

[(80 − 120)/(10 − 8)][(10 + 8)/(80 + 120)] = −1.80 so demand is elastic

12. The demand function for a good is defined as Q = 20 – 0.5P. Calculate the price elasticity of demand using the point formula for P = 30 and determine whether demand is elastic, inelastic, or unit elastic.

Solution:

(−0.5)(30/5) = −3.0 so demand is elastic

13. The demand function for Good X is defined as QX = 75 − 2PX − 1.5PY, where PY is the price of Good Y. Calculate the price elasticity of demand using the point formula for PX = 20 and PY = 10. Determine whether demand is elastic, inelastic, or unit elastic with respect to its own price and whether Good Y is a substitute or a complement with respect to Good X.

Solution :

(−2)(20/20) = −2.0 so demand is elastic with respect to its own price.

(−1.5)(10/20) = −0.75 so the two goods are complements.

14. The demand function for a good is defined as Q = 45 − 2.5P − 0.2I, where I is a measure of consumer income. Calculate the price elasticity of demand using the point formula for P = 6 and I = 100. Determine whether demand is elastic, inelastic, or unit elastic with respect to its own price and whether the good is normal or inferior and whether it is a luxury or a necessity.

Solution:

(−2.5)(6/10) = −1.5 so demand is elastic with respect to its own price.

(−0.2)(100/10) = −2.0 so the good is inferior.

15. The demand function for a good is defined as Q = 50 − P. Calculate the price elasticity of demand using the point formula for P = 25 and determine whether the demand is elastic, inelastic, or unit elastic.

Solution:

(−1)(25/25) = −1.0 so demand is unit elastic.