Numericals Solved - Demand Theory - Unit -3 2j356
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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.
Managerial Economics: Principles and Worldwide Applications, 7/e
Copyright (c) Oxford University Press, 2012
(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 Managerial Economics: Principles and Worldwide Applications, 7/e
Copyright (c) Oxford University Press, 2012
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.
Managerial Economics: Principles and Worldwide Applications, 7/e
Copyright (c) Oxford University Press, 2012
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.
Managerial Economics: Principles and Worldwide Applications, 7/e
Copyright (c) Oxford University Press, 2012
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
Managerial Economics: Principles and Worldwide Applications, 7/e
Copyright (c) Oxford University Press, 2012
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. Managerial Economics: Principles and Worldwide Applications, 7/e
Copyright (c) Oxford University Press, 2012
Managerial Economics: Principles and Worldwide Applications, 7/e
Copyright (c) Oxford University Press, 2012
(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 Managerial Economics: Principles and Worldwide Applications, 7/e
Copyright (c) Oxford University Press, 2012
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.
Managerial Economics: Principles and Worldwide Applications, 7/e
Copyright (c) Oxford University Press, 2012
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.
Managerial Economics: Principles and Worldwide Applications, 7/e
Copyright (c) Oxford University Press, 2012
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
Managerial Economics: Principles and Worldwide Applications, 7/e
Copyright (c) Oxford University Press, 2012
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. Managerial Economics: Principles and Worldwide Applications, 7/e
Copyright (c) Oxford University Press, 2012
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