3.7: Problem Sets

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Problem Set 1: Use the Point Formula for Demand Elasticities.

Exercise $\PageIndex{1}$

Given the following:

$\textrm{The demand equation is:} Q_{1}= 150 -2P_{1} - 1.5 P_{2} + 0.1M$

$P_{1} = 20$

$P_{2} = 30$

$M = 100$

$\textrm{Calculate} \varepsilon _{11}, \varepsilon_{12}, \varepsilon_{1M}$

Is demand for Good 1 inelastic, unitary elastic, or elastic?

Is Good 2 a substitute or complement to good 1?

Is Good 1 inferior, a normal necessity, or a normal luxury?

Answer

$\varepsilon_{11} = -0.53$

$\varepsilon_{12} = -0.6$

$\varepsilon_{1M} = 0.13$

Inelastic

Complement

Normal necessity

Exercise $\PageIndex{2}$

Given the following:

$\textrm{The demand equation is:} Q_{1}= 75-2.5P_{1} - 0.5 P_{2} + 0.5M$

$P_{1} = 30$

$P_{2} = 40$

$M = 200$

$\textrm{Calculate} \varepsilon _{11}, \varepsilon_{12}, \varepsilon_{1M}$

Is demand for Good 1 inelastic, unitary elastic, or elastic?

Is Good 2 a substitute or complement to good 1?

Is Good 1 inferior, a normal necessity, or a normal luxury?

Answer

$\varepsilon_{11} = -0.94$

$\varepsilon_{12} = -0.25$

$\varepsilon_{1M} = 1.25$

Inelastic

Complement

Normal luxury

Exercise $\PageIndex{3}$

Given the following:

$\textrm{The demand equation is:} Q_{1}= 150-3P_{1} + 0.5 P_{2} - 0.5M$

$P_{1} = 40$

$P_{2} = 50$

$M = 300$

$\textrm{Calculate} \varepsilon _{11}, \varepsilon_{12}, \varepsilon_{1M}$

Is demand for Good 1 inelastic, unitary elastic, or elastic?

Is Good 2 a substitute or complement to good 1?

Is Good 1 inferior, a normal necessity, or a normal luxury?

Answer

$\varepsilon_{11} = -4.8$

$\varepsilon_{12} = 1$

$\varepsilon_{1M} = -1.2$

Elastic

Substitute

Inferior

Exercise $\PageIndex{4}$

Given the following:

$\textrm{The demand equation is:} Q_{1}= 100-2P_{1} + 1.5 P_{2} - 0.2M$

$P_{1} = 50$

$P_{2} = 40$

$M = 200$

$\textrm{Calculate} \varepsilon _{11}, \varepsilon_{12}, \varepsilon_{1M}$

Is demand for Good 1 inelastic, unitary elastic, or elastic?

Is Good 2 a substitute or complement to good 1?

Is Good 1 inferior, a normal necessity, or a normal luxury?

Answer

$\varepsilon_{11} = -5$

$\varepsilon_{12} = 3$

$\varepsilon_{1M} = -2$

Elastic

Substitute

Inferior

Exercise $\PageIndex{5}$

Given the following:

$\textrm{The demand equation is:} Q_{1}= 90-2.5P_{1} -0.5 P_{2} +0.5 M$

$P_{1} = 30$

$P_{2} = 40$

$M = 200$

$\textrm{Calculate} \varepsilon _{11}, \varepsilon_{12}, \varepsilon_{1M}$

Is demand for Good 1 inelastic, unitary elastic, or elastic?

Is Good 2 a substitute or complement to good 1?

Is Good 1 inferior, a normal necessity, or a normal luxury?

Answer

$\varepsilon_{11} = -0.79$

$\varepsilon_{12} = -0.21$

$\varepsilon_{1M} = 1.05$

Inelastic

Complement

Normal luxury

Exercise $\PageIndex{6}$

Given the following:

$\textrm{The demand equation is:} Q_{1}= 170-2.5P_{1} -1.5 P_{2} +0.1 M$

$P_{1} = 40$

$P_{2} = 30$

$M = 100$

$\textrm{Calculate} \varepsilon _{11}, \varepsilon_{12}, \varepsilon_{1M}$

Is demand for Good 1 inelastic, unitary elastic, or elastic?

Is Good 2 a substitute or complement to good 1?

Is Good 1 inferior, a normal necessity, or a normal luxury?

Answer

$\varepsilon_{11} = -2.86$

$\varepsilon_{12} = -1.29$

$\varepsilon_{1M} = 0.29$

Elastic

Complement

Normal necessity

Exercise $\PageIndex{7}$

Given the following:

$\textrm{The demand equation is:} Q_{1}= 100-3P_{1} -0.5 P_{2} +0.5 M$

$P_{1} = 40$

$P_{2} = 20$

$M = 300$

$\textrm{Calculate} \varepsilon _{11}, \varepsilon_{12}, \varepsilon_{1M}$

Is demand for Good 1 inelastic, unitary elastic, or elastic?

Is Good 2 a substitute or complement to good 1?

Is Good 1 inferior, a normal necessity, or a normal luxury?

Answer

Answers:

$\varepsilon_{11} = -1$

$\varepsilon_{12} = -0.08$

$\varepsilon_{1M} = 1.25$

Unitary elastic

Complement

Normal luxury

Exercise $\PageIndex{8}$

Given the following:

$\textrm{The demand equation is:} Q_{1}= 200-2P_{1} +0.5 P_{2} -0.1 M$

$P_{1} = 20$

$P_{2} = 10$

$M = 200$

$\textrm{Calculate} \varepsilon _{11}, \varepsilon_{12}, \varepsilon_{1M}$

Is demand for Good 1 inelastic, unitary elastic, or elastic?

Is Good 2 a substitute or complement to good 1?

Is Good 1 inferior, a normal necessity, or a normal luxury?

Answer

$\varepsilon_{11} = -0.28$

$\varepsilon_{12} = -0.03$

$\varepsilon_{1M} = -0.14$

Inelastic

Substitute

Inferior

Exercise $\PageIndex{9}$

Given the following:

$\textrm{The demand equation is:} Q_{1}= 200-2.5P_{1} +1.5 P_{2} -0.5 M$

$P_{1} = 10$

$P_{2} = 20$

$M = 100$

$\textrm{Calculate} \varepsilon _{11}, \varepsilon_{12}, \varepsilon_{1M}$

Is demand for Good 1 inelastic, unitary elastic, or elastic?

Is Good 2 a substitute or complement to good 1?

Is Good 1 inferior, a normal necessity, or a normal luxury?

Answer

$\varepsilon_{11} = -0.16$

$\varepsilon_{12} = -0.19$

$\varepsilon_{1M} = -0.32$

Inelastic

Substitute

Inferior

Exercise $\PageIndex{10}$

Given the following:

$\textrm{The demand equation is:} Q_{1}= 60-3P_{1} +1P_{2} +0.1 M$

$P_{1} = 20$

$P_{2} = 30$

$M = 300$

$\textrm{Calculate} \varepsilon _{11}, \varepsilon_{12}, \varepsilon_{1M}$

Is demand for Good 1 inelastic, unitary elastic, or elastic?

Is Good 2 a substitute or complement to good 1?

Is Good 1 inferior, a normal necessity, or a normal luxury?

Answer

$\varepsilon_{11} = -1$

$\varepsilon_{12} = -0.5$

$\varepsilon_{1M} = -0.5$

Unitary elastic

Substitute

Normal necessity

Problem Set 2: Use Elasticities to Determine Whether to Increase Price.

Exercise $\PageIndex{1}$

Given the following:

Your current quantity is 1000 units.

Your current price is 25 dollars.

Your average variable cost is 15 dollars per unit.

You face an own-price demand elasticity of -3.

What will happen to profit above variable cost if you raise your price by 2 dollars?

Answer

Your current profit above variable cost is 10000 dollars.

Your price will change by 8 percent.

Your new quantity is 760 units.

If you raise price, the profit above variable cost will be 9120 dollars.

Exercise $\PageIndex{2}$

Given the following:

Your current quantity is 1100 units.

Your current price is 40 dollars.

Your average variable cost is 36 dollars per unit.

You face an own-price demand elasticity of -4.

What will happen to profit above variable cost if you raise your price by 1 dollars?

Answer

Your current profit above variable cost is 4400 dollars.

Your price will change by 2.5 percent.

Your new quantity is 990 units.

If you raise price, the profit above variable cost will be 4950 dollars.

Exercise $\PageIndex{3}$

Given the following:

Your current quantity is 1120 units.

Your current price is 50 dollars.

Your average variable cost is 40 dollars per unit.

You face an own-price demand elasticity of -5.

What will happen to profit above variable cost if you raise your price by 2 dollars?

Answer

Your current profit above variable cost is 11200 dollars.

Your price will change by 4 percent.

Your new quantity is 896 units.

If you raise price, the profit above variable cost will be 10752 dollars.

Exercise $\PageIndex{4}$

Given the following:

Your current quantity is 1200 units.

Your current price is 20 dollars.

Your average variable cost is 16 dollars per unit.

You face an own-price demand elasticity of -4.

What will happen to profit above variable cost if you raise your price by 1 dollars?

Answer

Your current profit above variable cost is 4800 dollars.

Your price will change by 5 percent.

Your new quantity is 960 units.

If you raise price, the profit above variable cost will be 4800 dollars.

Exercise $\PageIndex{5}$

Given the following:

Your current quantity is 1300 units.

Your current price is 50 dollars.

Your average variable cost is 40 dollars per unit.

You face an own-price demand elasticity of -1.5.

What will happen to profit above variable cost if you raise your price by 1 dollars?

Answer

Your current profit above variable cost is 13000 dollars.

Your price will change by 2 percent.

Your new quantity is 1261 units.

If you raise price, the profit above variable cost will be 13871 dollars.

Exercise $\PageIndex{6}$

Given the following:

Your current quantity is 1200 units.

Your current price is 20 dollars.

Your average variable cost is 16 dollars per unit.

You face an own-price demand elasticity of -0.75.

What will happen to profit above variable cost if you raise your price by 2 dollars?

Answer

Your current profit above variable cost is 4800 dollars.

Your price will change by 10 percent.

Your new quantity is 1110 units.

If you raise price, the profit above variable cost will be 6660 dollars.

Exercise $\PageIndex{7}$

Given the following:

Your current quantity is 1120 units.

Your current price is 50 dollars.

Your average variable cost is 46 dollars per unit.

You face an own-price demand elasticity of -1.

What will happen to profit above variable cost if you raise your price by 1 dollars?

Answer

Your current profit above variable cost is 4480 dollars.

Your price will change by 2 percent.

Your new quantity is 1097.6 units.

If you raise price, the profit above variable cost will be 5488 dollars.

Exercise $\PageIndex{8}$

Given the following:

Your current quantity is 1200 units.

Your current price is 25 dollars.

Your average variable cost is 21 dollars per unit.

You face an own-price demand elasticity of -5.

What will happen to profit above variable cost if you raise your price by 1 dollars?

Answer

Your current profit above variable cost is 4800 dollars.

Your price will change by 4 percent.

Your new quantity is 960 units.

If you raise price, the profit above variable cost will be 4800 dollars.

Given the following:

Your current quantity is 1000 units.

Your price is 25 dollars.

Your average variable cost (AVC) is 21 dollars per unit.

Your elasticity of demand with respect to advertising is 1.5.

Your current advertising expenditure is $5,000 and is your only fixed cost.

What will happen to profit if you increase advertising by 300 dollars?

Answer

Your current profit is -1000 dollars.

Your advertising will change by 6 percent.

Your new quantity is 1090 units.

Profit will go up to -940 dollars (after subtracting the increse in advertising).

Problem Set 4: Short-Run and Long-Run Elasticities.

Exercise $\PageIndex{1}$

Given the following:

The short run demand equation is:

$\textrm{The short run demand equation is:} Q_{1}^{(t)} = 100 - 2P_{1} + 0.5 Q_{1}^{(t-1)}$

$P_{1} = 35$

$Q_{1}^{(t-1)} =60$

What is the habit parameter?

What is the long-run demand equation?

What is the short-run own-price elasticity of demand?

What is the long-run own-price elasticity of demand?

When is demand more elastic, in the short- or long-run?

Answer

$\textrm{Habit parameter}: \gamma = 0.5$

$Q_{1}^{LR} = 200-4P_{1}$

$\varepsilon_{11}^{SR} = -1.17$

$\varepsilon_{11}^{LR} = -2.33$

$\textrm{In the long-run.}$

Exercise $\PageIndex{2}$

Given the following:

$\textrm{The short run demand equation is:} Q_{1}^{(t)} = 100 - 3P_{1} + 0.6 Q_{1}^{(t-1)}$

$P_{1} = 20$

$Q_{1}^{(t-1)} =100$

What is the habit parameter?

What is the long-run demand equation?

What is the short-run own-price elasticity of demand?

What is the long-run own-price elasticity of demand?

When is demand more elastic, in the short- or long-run?

Answer

$\textrm{Habit parameter}: \gamma = 0.6$

$Q_{1}^{LR} = 250-7.5P_{1}$

$\varepsilon_{11}^{SR} = -0.6$

$\varepsilon_{11}^{LR} = -1.5$

$\textrm{In the long-run.}$

Exercise $\PageIndex{3}$

Given the following:

$\textrm{The short run demand equation is:} Q_{1}^{(t)} = 150 - 4P_{1} + 0.75 Q_{1}^{(t-1)}$

$P_{1} = 20$

$Q_{1}^{(t-1)} =280$

Answer

$\textrm{Habit parameter}: \gamma = 0.75$

$Q_{1}^{LR} = 600-16P_{1}$

$\varepsilon_{11}^{SR} = -0.29$

$\varepsilon_{11}^{LR} = -1.14$

$\textrm{In the long-run.}$

Exercise $\PageIndex{4}$

Given the following:

$\textrm{The short run demand equation is:} Q_{1}^{(t)} = 250 - 4P_{1} + 0.8Q_{1}^{(t-1)}$

$P_{1} = 40$

$Q_{1}^{(t-1)} =450$

What is the habit parameter?

What is the long-run demand equation?

What is the short-run own-price elasticity of demand?

What is the long-run own-price elasticity of demand?

When is demand more elastic, in the short- or long-run?

Answer

$\textrm{Habit parameter}: \gamma = 0.8$

$Q_{1}^{LR} = 1250-20P_{1}$

$\varepsilon_{11}^{SR} = -0.36$

$\varepsilon_{11}^{LR} = -1.78$

$\textrm{In the long-run.}$

Exercise $\PageIndex{5}$

Given the following:

$\textrm{The short run demand equation is:} Q_{1}^{(t)} = 120 - 2P_{1} + 0.6Q_{1}^{(t-1)}$

$P_{1} = 40$

$Q_{1}^{(t-1)} =100$

What is the habit parameter?

What is the long-run demand equation?

What is the short-run own-price elasticity of demand?

What is the long-run own-price elasticity of demand?

When is demand more elastic, in the short- or long-run?

Answer

$\textrm{Habit parameter}: \gamma = 0.6$

$Q_{1}^{LR} = 300-5P_{1}$

$\varepsilon_{11}^{SR} = -0.8$

$\varepsilon_{11}^{LR} = -2$

$\textrm{In the long-run.}$

Exercise $\PageIndex{6}$

Given the following:

$\textrm{The short run demand equation is:} Q_{1}^{(t)} = 150 - 3P_{1} + 0.8Q_{1}^{(t-1)}$

$P_{1} = 30$

$Q_{1}^{(t-1)} =300$

What is the habit parameter?

What is the long-run demand equation?

What is the short-run own-price elasticity of demand?

What is the long-run own-price elasticity of demand?

When is demand more elastic, in the short- or long-run?

Answer

$\textrm{Habit parameter}: \gamma = 0.8$

$Q_{1}^{LR} = 750-15P_{1}$

$\varepsilon_{11}^{SR} = -0.3$

$\varepsilon_{11}^{LR} = -1.5$

$\textrm{In the long-run.}$

Exercise $\PageIndex{7}$

Given the following:

$\textrm{The short run demand equation is:} Q_{1}^{(t)} = 200 - 3P_{1} + 0.75Q_{1}^{(t-1)}$

$P_{1} = 45$

$Q_{1}^{(t-1)} =260$

What is the habit parameter?

What is the long-run demand equation?

What is the short-run own-price elasticity of demand?

What is the long-run own-price elasticity of demand?

When is demand more elastic, in the short- or long-run?

Answer

$\textrm{Habit parameter}: \gamma = 0.75$

$Q_{1}^{LR} = 800-12P_{1}$

$\varepsilon_{11}^{SR} = -0.52$

$\varepsilon_{11}^{LR} = -2.08$

$\textrm{In the long-run.}$

Exercise $\PageIndex{8}$

Given the following:

$\textrm{The short run demand equation is:} Q_{1}^{(t)} = 80 - 2P_{1} + 0.6Q_{1}^{(t-1)}$

$P_{1} = 30$

$Q_{1}^{(t-1)} =50$

What is the habit parameter?

What is the long-run demand equation?

What is the short-run own-price elasticity of demand?

What is the long-run own-price elasticity of demand?

When is demand more elastic, in the short- or long-run?

Answer

$\textrm{Habit parameter}: \gamma = 0.6$

$Q_{1}^{LR} = 200-5P_{1}$

$\varepsilon_{11}^{SR} = -1.2$

$\varepsilon_{11}^{LR} = -3$

$\textrm{In the long-run.}$

Exercise $\PageIndex{9}$

Given the following:

$\textrm{The short run demand equation is:} Q_{1}^{(t)} = 120 - 3P_{1} + 0.5Q_{1}^{(t-1)}$

$P_{1} = 30$

$Q_{1}^{(t-1)} =60$

What is the habit parameter?

What is the long-run demand equation?

What is the short-run own-price elasticity of demand?

What is the long-run own-price elasticity of demand?

When is demand more elastic, in the short- or long-run?

Answer

$\textrm{Habit parameter}: \gamma = 0.5$

$Q_{1}^{LR} = 240-6P_{1}$

$\varepsilon_{11}^{SR} = -1.5$

$\varepsilon_{11}^{LR} = -3$

$\textrm{In the long-run.}$

Exercise $\PageIndex{10}$

Given the following:

$\textrm{The short run demand equation is:} Q_{1}^{(t)} = 100 - 2P_{1} + 0.5Q_{1}^{(t-1)}$

$P_{1} = 40$

$Q_{1}^{(t-1)} =40$

What is the habit parameter?

What is the long-run demand equation?

What is the short-run own-price elasticity of demand?

What is the long-run own-price elasticity of demand?

When is demand more elastic, in the short- or long-run?

Answer

$\textrm{Habit parameter}: \gamma = 0.5$

$Q_{1}^{LR} = 200-4P_{1}$

$\varepsilon_{11}^{SR} = -2$

$\varepsilon_{11}^{LR} = -4$

$\textrm{In the long-run.}$

Problem Set 5: Multiple Choice.

Exercise $\PageIndex{1}$

When is demand unitary elastic?

a) When the income elasticity is 1.

b) When the own-price elasticity of demand is -1.

c) When demand is totally unresponsive (inverse demand is a vertical line).

d) All of the above.

Answer: b

Exercise $\PageIndex{2}$

If price goes up by 4 percent and quantity demanded falls by 2 percent then

a) This is a substitute good.

b) This good has an elastic demand.

c) This good is an inferior good.

d) This good has an inelastic demand.

Answer: d

Exercise $\PageIndex{3}$

Suppose that the own-price elasticity of demand is -1. Which statement is true?

a) Profits are maximized.

b) Revenue is maximized.

c) Consumer surplus is maximized.

d) All of the above.

Answer: b

Exercise $\PageIndex{4}$

Suppose the own-price elasticity of demand is -0.75. Which statement is true?

a) An increase in price will decrease revenue.

b) An increase in price will increase revenue.

c) The good in question is a substitute in consumption.

d) The good in question is an inferior good.

Answer: b

Exercise $\PageIndex{5}$

Which of the following elasticity numbers is consistent with a normal good?

a) An income elasticity of demand that is -0.5.

b) An income elasticity of demand that is 0.5.

c) An own-price elasticity of 0.28.

d) Both a and c.

Answer: b

Exercise $\PageIndex{6}$

If an increase in income causes an increase in demand, then we know that

a) The good in question is a normal luxury good.

b) The good in question is a normal necessity good.

c) The good in question is a normal good but we can’t tell whether it is classified as a necessity or a luxury without further information.

d) The good in question violates the law of demand.

Answer: c

Exercise $\PageIndex{7}$

Which demand elasticity number tells you that two products are substitutes in consumption?

a) A cross-price elasticity of -0.5.

b) A cross-price elasticity of 0.5.

c) An income elasticity of 1.5.

d) An own-price elasticity of -0.75.

Answer: b

Exercise $\PageIndex{8}$

One problem with using the arc elasticity formula to compute an own-price elasticity is:

a) It is very hard to compute because you need lots of data points.

b) Actually, there are no problems with using the arc formula to compute elasticities.

c) One must assume that all other factors that affect demand, other than own price, remain the same.

d) If demand has not shifted, the arc formula will often return a positive value for the own-price elasticity.

Answer: c

Exercise $\PageIndex{9}$

Given that a firm has some control over the price it charges (it is not a price taker) and it faces a positive marginal cost, which case would be consistent with profit maximization?

a) It sets its price to make demand inelastic.

b) It sets its price to make demand elastic.

c) It sets its price to make demand unitary elastic.

d) It sets its price so that both demand and supply are in the inelastic range.

Answer: b

Exercise $\PageIndex{10}$

As described in class, the supply schedule for van Gogh paintings:

a) Is perfectly inelastic.

b) Is downward sloping.

c) Maybe quite inelastic but is probably not perfectly inelastic.

d) Is a horizontal line.

e) Choices a and d only.

Answer: c

Exercise $\PageIndex{11}$

Other things equal, in the long run

a) Demand is less elastic than in the short run.

b) Supply is less elastic than in the short run.

c) Demand is more elastic than in the short run.

d) Both demand and supply are less elastic than in the short run.

Answer: c

Exercise $\PageIndex{12}$

Which best describes the concept of elasticity?

a) The responsiveness of demand or supply to own-price or some other shift variable.

b) The idea that demand schedules always slope downwards.

c) The idea that consumer surplus is the area under the demand schedule.

d) The idea that supply schedules always have non-negative slopes.

Answer: a

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