Transatlantic air travel in business class has an estimated elasticity of demand of 0.40 less than transatlantic air travel in economy class, with an estimated price elasticity of 0.62. Why do you think this is the case? For me, I feel that it is because business flights has a higher degree of necessity as compared to economy flights meant for leisure. As such, a change in airfare prices for business flights wouldn't impact much of the quantity demanded due to its higher indispensability.

I would say that it is simply because business class travelers care less about the price, given that they are already not buying the cheapest option.

Hey for the first set of elasticity, from going to G to H it says the % change in price is 7.81 but when I divide 10 by 125 I get 0.08 which is 8%, so am I doing it wrong or is the answer wrong?

You're right. It's supposed to be 8%. The answer is wrong.

What is the answer to critical question No.1

The change in price has a less impact on preferences of the people with higher incomes, who prefer the business class. The price is not the most important criterion for these people, that is why any given percentage change in price will cause a smaller percentage change in quantity demanded. The influence of the change in price is bigger for the people with less incomes, who prefer the economy class. This statistic shows us that the price has a remarkable significance for these people, because their purchasing power is less than people in business class, that is why the price should be a more important criterion for them. Additionally, despite the wage gap between the consumers the demand curve is inelastic for both classes in Transatlantic Air Travel.

to find the price elasticity of the entire demand curve, is it correct to find the elasticity for all points and find the average of those values?

No, there is no such thing as a price elasticity of the entire curve. The price elasticity is different for every point.

Are there answers to the problems?

The answers to the self-check questions are below the questions. The answers to the other questions aren't given.

could someone please give the answers to the problems?

Here's my answers to the problems: (1) E = |(1/5.5)/(-3/31.5)| = 1.91 Thus the demand curve is elastic here. (2) E1 = |(0.1/0.45)/(-1/4.5)| = 1 E2 = | [(1/36)/(17/72)]/(-1/8.5) | = 1 The demand curve at the two points are both unit elastic. In fact, the elasticity of price at *every point* of the curve equals to *1*. Since P always equals to 2/Q on this curve, ΔP = P(new) - P(old) = 2/Q(new) - 2/Q(old) = 2(Q(old) - Q(new))/Q(new)*Q(old) = 2ΔQ/[Q(new)*Q(old)]; similarly, P(average) = (Q(new) + Q(old)) / Q(new)*Q(old). Thus we get %ΔP = 2ΔQ/(Q(new) + Q(old)) and plug it into the formula E = %ΔQ/%ΔP, we get E = (ΔQ * (Q(new) + Q(old))) / (0.5 * (Q(new) + Q(old)) * 2ΔQ) = 1. (3) The method is the same as above. E1 = E2 = 1. (4) E = 0.41 Hope it helps! :)

Main content

Price elasticity of demand and price elasticity of supply

Google Classroom

How do quantities supplied and demanded react to changes in price?

Key points

Price elasticity measures the responsiveness of the quantity demanded or supplied of a good to a change in its price. It is computed as the percentage change in quantity demanded—or supplied—divided by the percentage change in price.
Elasticity can be described as elastic—or very responsive—unit elastic, or inelastic—not very responsive.
Elastic demand or supply curves indicate that the quantity demanded or supplied responds to price changes in a greater than proportional manner.
An inelastic demand or supply curve is one where a given percentage change in price will cause a smaller percentage change in quantity demanded or supplied.
Unitary elasticity means that a given percentage change in price leads to an equal percentage change in quantity demanded or supplied.

What is price elasticity?

Both demand and supply curves show the relationship between price and the number of units demanded or supplied. Price elasticity is the ratio between the percentage change in the quantity demanded,

Q_{d}

, or supplied,

Q_{s}

, and the corresponding percent change in price.

The price elasticity of demand is the percentage change in the quantity demanded of a good or service divided by the percentage change in the price. The price elasticity of supply is the percentage change in quantity supplied divided by the percentage change in price.

Elasticities can be usefully divided into five broad categories: perfectly elastic, elastic, perfectly inelastic, inelastic, and unitary. An elastic demand or elastic supply is one in which the elasticity is greater than one, indicating a high responsiveness to changes in price. An inelastic demand or inelastic supply is one in which elasticity is less than one, indicating low responsiveness to price changes. Unitary elasticities indicate proportional responsiveness of either demand or supply.

Perfectly elastic and perfectly inelastic refer to the two extremes of elasticity. Perfectly elastic means the response to price is complete and infinite: a change in price results in the quantity falling to zero. Perfectly inelastic means that there is no change in quantity at all when price changes.

If . . .	It Is Called . . .
$\frac{% change in quantity}{% change in price} = \infty$ ‍	Perfectly elasti
$\frac{% change in quantity}{% change in price} > 1$ ‍	Elastic
$\frac{% change in quantity}{% change in price} = 1$ ‍	Unitary
$\frac{% change in quantity}{% change in price} < 1$ ‍	Inelastic
$\frac{% change in quantity}{% change in price} = 0$ ‍	Perfectly inelastic

Using the midpoint method to calculate elasticity

To calculate elasticity, instead of using simple percentage changes in quantity and price, economists sometimes use the average percent change in both quantity and price. This is called the Midpoint Method for Elasticity:

Midpoint method for elasticity = \frac{\frac{Q_{2} - Q_{1}}{(\frac{Q_{2} + Q_{1}}{2})}}{\frac{P_{2} - P_{1}}{(\frac{P_{2} + P_{1}}{2})}}

The advantage of the midpoint method is that we get the same elasticity between two price points whether there is a price increase or decrease. This is because the formula uses the same base for both cases. The midpoint method is referred to as the arc elasticity in some textbooks.

Using the point elasticity of demand to calculate elasticity

A drawback of the midpoint method is that as the two points get farther apart, the elasticity value loses its meaning. For this reason, some economists prefer to use the point elasticity method. In this method, you need to know what values represent the initial values and what values represent the new values.

Point elasticity = \frac{\frac{new Q - initial Q}{initial Q}}{\frac{initial P - new P}{initial P}}

Calculating price elasticity of demand

Let’s apply these formulas to a practice scenario. We'll calculate the elasticity between points

A

and

B

in the graph below.

First, apply the formula to calculate the elasticity as price decreases from $70 at point

B

to $60 at point

A

\begin{array}{ccc} % change in quantity & = & \frac{3, 000 - 2, 800}{(3, 000 + 2, 800) / 2} \times 100 \\ = & \frac{200}{2, 900} \times 100 \\ = & 6.9 \\ % change in price & = & \frac{60 - 70}{(60 + 70) / 2} \times 100 \\ = & \frac{- 10}{65} \times 100 \\ = & - 15.4 \\ Price elasticity of demand & = & \frac{6.9 %}{- 15.4 %} \\ = & 0.45 \end{array}

The elasticity of demand between point

A

and point

B

\frac{6.9 %}{- 15.4 %}

, or 0.45. Because this amount is smaller than one, we know that the demand is inelastic in this interval.

This means that, along the demand curve between point

B

and point

A

, if the price changes by 1%, the quantity demanded will change by 0.45%. A change in the price will result in a smaller percentage change in the quantity demanded. For example, a 10% increase in the price will result in only a 4.5% decrease in quantity demanded. A 10% decrease in the price will result in only a 4.5% increase in the quantity demanded.

Let's do another practice problem to cement this concept!

Calculate the price elasticity of demand for an increase in price from

G

H

in the graph above.

Step 1. Start with the formula for price elasticity of demand:

Step 2. Use the Midpoint Method formula to calculate percent change in quantity and price:

\begin{array}{ccc} % change in quantity & = & \frac{Q_{2} - Q_{1}}{(Q_{2} + Q_{1}) / 2} \times 100 \\ % change in quantity & = & \frac{1, 600 - 1, 800}{(1, 600 + 1, 800) / 2} \times 100 \\ = & \frac{- 200}{1, 700} \times 100 \\ = & - 11.76 \end{array}

\begin{array}{ccc} % change in price & = & \frac{P_{2} - P_{1}}{(P_{2} + P_{1}) / 2} \times 100 \\ % change in price & = & \frac{130 - 120}{(130 + 120) / 2} \times 100 \\ = & \frac{10}{125} \times 100 \\ = & 8.0 \end{array}

Step 4. Use the results of step three to fill in the formula for price elasticity of demand:

\begin{array}{ccc} Price elasticity of demand & = & \frac{% change in quantity}{% change in price} \\ = & \frac{- 11.76}{8} \\ = & 1.47 \end{array}

The elasticity of demand from

G

H

is 1.47.

Recall that the elasticity between point

A

point

B

was 0.45. So, demand was inelastic between points

A

and

B

and elastic between points

G

and

H

. This shows us that price elasticity of demand changes at different points along a straight-line demand curve.

Calculating the price elasticity of supply

Now let's try calculating the price elasticity of supply. We use the same formula as we did for price elasticity of demand:

\begin{array}{ccc} Price elasticity of supply & = & \frac{% change in quantity}{% change in price} \end{array}

Assume that an apartment rents for $650 per month and, at that price, 10,000 units are rented—you can see these number represented graphically below. When the price increases to $700 per month, 13,000 units are supplied into the market.

By what percentage does apartment supply increase? What is the price sensitivity?

We'll start by using the Midpoint Method to calculate percentage change in price and quantity:

\begin{array}{ccc} % change in quantity & = & \frac{13, 000 - 10, 000}{(13, 000 + 10, 000) / 2} \times 100 \\ = & \frac{3, 000}{11, 500} \times 100 \\ = & 26.1 \\ % change in price & = & \frac{$ 700 - $ 650}{($ 700 + $ 650) / 2} \times 100 \\ = & \frac{50}{675} \times 100 \\ = & 7.4 \end{array}

Next, we take the results of our calculations and plug them into the formula for price elasticity of supply:

\begin{array}{ccc} Price elasticity of supply & = & \frac{% change in quantity}{% change in price} \\ = & \frac{26.1}{7.4} \\ = & 3.53 \end{array}

Again, as with the elasticity of demand, the elasticity of supply is not followed by any units. Elasticity is a ratio of one percentage change to another percentage change—nothing more. It is read as an absolute value. In this case, a 1% rise in price causes an increase in quantity supplied of 3.5%. The greater than one elasticity of supply means that the percentage change in quantity supplied will be greater than a one percent price change.

It is a common mistake to confuse the slope of either the supply or demand curve with its elasticity.

The slope is the rate of change in units along the curve, or the rise/run—change in

y

over the change in

x

. For example, in the graph above, at each point shown on the demand curve, price drops by $10 and the number of units demanded increases by 200. So the slope is

- 10 / 200

along the entire demand curve; it does not change.

The price elasticity, however, changes along the curve. Elasticity between point

A

and point

B

was 0.45 and increased to 1.47 between points point

G

and point

H

. Elasticity is the percentage change, which is a different calculation from the slope and has a different meaning.

When we are at the upper end of a demand curve, where price is high and the quantity demanded is low, a small change in the quantity demanded—even, say, one unit—is pretty big in percentage terms. A change in price of a dollar is going to be much less important in percentage terms than it would have been at the bottom of the demand curve. Likewise, at the bottom of the demand curve, that one unit change when the quantity demanded is high will be small as a percentage.

So, at one end of the demand curve, where we have a large percentage change in quantity demanded over a small percentage change in price, the elasticity value would be high, or demand would be relatively elastic. Even with the same change in the price and the same change in the quantity demanded, at the other end of the demand curve the quantity is much higher, and the price is much lower, so the percentage change in quantity demanded is smaller and the percentage change in price is much higher. That means at the bottom of the curve we have a small numerator over a large denominator, so the elasticity measure would be much lower, or inelastic.

As we move along the demand curve, the values for quantity and price go up or down, depending on which way we are moving, so the percentages for a $1 difference in price or a one unit difference in quantity will change as well, which means the ratios of those percentages will change.

Summary

Price elasticity measures the responsiveness of the quantity demanded or supplied of a good to a change in its price. It is computed as the percentage change in quantity demanded—or supplied—divided by the percentage change in price.
Elasticity can be described as elastic—or very responsive—unit elastic, or inelastic—not very responsive.
Elastic demand or supply curves indicate that the quantity demanded or supplied responds to price changes in a greater than proportional manner.
An inelastic demand or supply curve is one where a given percentage change in price will cause a smaller percentage change in quantity demanded or supplied.
Unitary elasticity means that a given percentage change in price leads to an equal percentage change in quantity demanded or supplied.

Self-check questions

Using the data shown in the table below about demand for smart phones, calculate the price elasticity of demand from point

B

to point

C

, point

D

to point

E

, and point

G

to point

H

. Classify the elasticity at each point as elastic, inelastic, or unit elastic.

Points	P	Q
A	60	3,000
B	70	2,800
C	80	2,600
D	90	2,400
E	100	2,200
F	110	2,000
G	120	1,800
H	130	1,600

From point

B

to point

C

, price rises from $70 to $80, and

Qd

decreases from 2,800 to 2,600. Using these numbers, we can fill in the formulas for percent change in quantity and price:

\begin{array}{ccc} % change in quantity & = & \frac{2600 - 2800}{(2600 + 2800) \div 2} \times 100 \\ = & \frac{- 200}{2700} \times 100 \\ = & - 7.41 \\ % change in price & = & \frac{80 - 70}{(80 + 70) \div 2} \times 100 \\ = & \frac{10}{75} \times 100 \\ = & 13.33 \end{array}

Now, we can use the results of our calculations above to fill in the formula for elasticity of demand:

\begin{array}{ccc} Elasticity of demand & = & \frac{- 7.41 %}{13.33 %} \\ = & 0.56 \end{array}

The demand curve is inelastic in this area; that is, its elasticity value is less than one.

Now follow the same process using the data from point

D

to point

E

\begin{array}{ccc} % change in quantity & = & \frac{2200 - 2400}{(2200 + 2400) \div 2} \times 100 \\ = & \frac{- 200}{2300} \times 100 \\ = & - 8.7 \\ % change in price & = & \frac{100 - 90}{(100 + 90) \div 2} \times 100 \\ = & \frac{10}{95} \times 100 \\ = & 10.53 \\ Elasticity of demand & = & \frac{- 8.7 %}{10.53 %} \\ = & 0.83 \end{array}

The demand curve is inelastic in this area; that is, its elasticity value is less than one.

And from Point

G

to point

H

\begin{array}{ccc} % change in quantity & = & \frac{1600 - 1800}{1700} \times 100 \\ = & \frac{- 200}{1700} \times 100 \\ = & - 11.76 \\ % change in price & = & \frac{130 - 120}{125} \times 100 \\ = & \frac{10}{125} \times 100 \\ = & 8 % \\ Elasticity of demand & = & \frac{- 11.76 %}{8 %} \\ = & - 1.47 \end{array}

The demand curve is elastic in this interval.

Using the data shown in in the table below about supply of alarm clocks, calculate the price elasticity of supply from: point

J

to point

K

, point

L

to point

M

, and point

N

to point

P

. Classify the elasticity at each point as elastic, inelastic, or unit elastic.

Point	Price	Quantity Supplied
J	$8	50
K	$9	70
L	$10	80
M	$11	88
N	$12	95
P	$13	100

From point

J

to point

K

, price rises from $8 to $9, and quantity rises from 50 to 70. Using these numbers, we can fill in the formulas for percent change in quantity and price:

\begin{array}{ccc} % change in quantity & = & \frac{70 - 50}{(70 + 50) \div 2} \times 100 \\ = & \frac{20}{60} \times 100 \\ = & 33.33 \\ % change in price & = & \frac{$ 9 - $ 8}{($ 9 + $ 8) \div 2} \times 100 \\ = & \frac{1}{8.5} \times 100 \\ = & 11.76 \end{array}

Now, we can use the results of our calculations above to fill in the formula for elasticity of supply:

\begin{array}{ccc} Elasticity of supply & = & \frac{33.33 %}{11.76 %} \\ = & 2.83 \end{array}

The supply curve is elastic in this area; that is, its elasticity value is greater than one.

Now follow the same process using the data from point

L

to point

M

; the price rises from $10 to $11, while the

Qs

rises from 80 to 88.

\begin{array}{ccc} % change in quantity & = & \frac{88 - 80}{(88 + 80) \div 2} \times 100 \\ = & \frac{8}{84} \times 100 \\ = & 9.52 \\ % change in price & = & \frac{$ 11 - $ 10}{($ 11 + $ 10) \div 2} \times 100 \\ = & \frac{1}{10.5} \times 100 \\ = & 9.52 \\ Elasticity of supply & = & \frac{9.52 %}{9.52 %} \\ = & 1.0 \end{array}

The supply curve has unitary elasticity in this area.

And finally, from point

N

to point

P

, the price rises from $12 to $13, and

Qs

rises from 95 to 100.

\begin{array}{ccc} % change in quantity & = & \frac{100 - 95}{(100 + 95) \div 2} \times 100 \\ = & \frac{5}{97.5} \times 100 \\ = & 5.13 \\ % change in price & = & \frac{$ 13 - $ 12}{($ 13 + $ 12) \div 2} \times 100 \\ = & \frac{1}{12.5} \times 100 \\ = & 8.0 \\ Elasticity of supply & = & \frac{5.13 %}{8.0 %} \\ = & 0.64 \end{array}

The supply curve is inelastic in this region of the supply curve.

Review Questions

What is the formula for calculating elasticity?
What is the price elasticity of demand? Can you explain it in your own words?
What is the price elasticity of supply? Can you explain it in your own words?

Critical-thinking questions

Transatlantic air travel in business class has an estimated elasticity of demand of 0.40 less than transatlantic air travel in economy class, which has an estimated price elasticity of 0.62. Why do you think this is the case?
What is the relationship between price elasticity and position on the demand curve? For example, as you move up the demand curve to higher prices and lower quantities, what happens to the measured elasticity? How would you explain that?

Problems

The equation for a demand curve is $P = 48 - 3 Q$ ‍. What is the elasticity in moving from a quantity of 5 to a quantity of 6?
The equation for a demand curve is $P = 2 / Q$ ‍. What is the elasticity of demand as price falls from 5 to 4? What is the elasticity of demand as the price falls from 9 to 8? Would you expect these answers to be the same?
The equation for a supply curve is $4 P = Q$ ‍. What is the elasticity of supply as price rises from 3 to 4? What is the elasticity of supply as the price rises from 7 to 8? Would you expect these answers to be the same?
The equation for a supply curve is $P = 3 Q - 8$ ‍. What is the elasticity in moving from a price of 4 to a price of 7?

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Jm-punk
Posted 7 years ago. Direct link to Jm-punk's post “Transatlantic air travel ...”
Transatlantic air travel in business class has an estimated elasticity of demand of 0.40 less than transatlantic air travel in economy class, with an estimated price elasticity of 0.62. Why do you think this is the case? For me, I feel that it is because business flights has a higher degree of necessity as compared to economy flights meant for leisure. As such, a change in airfare prices for business flights wouldn't impact much of the quantity demanded due to its higher indispensability.
Button navigates to signup pageComment on Jm-punk's post “Transatlantic air travel ...”
(11 votes)
Answer
- Tejas
  Posted 7 years ago. Direct link to Tejas's post “I would say that it is si...”
  I would say that it is simply because business class travelers care less about the price, given that they are already not buying the cheapest option.
  Button navigates to signup page
  (35 votes)
Nitigya Golait
Posted 7 years ago. Direct link to Nitigya Golait's post “What is the answer to cri...”
What is the answer to critical question No.1
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(5 votes)
Answer
- Savaş Can Akçay
  Posted 7 years ago. Direct link to Savaş Can Akçay's post “The change in price has a...”
  The change in price has a less impact on preferences of the people with higher incomes, who prefer the business class. The price is not the most important criterion for these people, that is why any given percentage change in price will cause a smaller percentage change in quantity demanded.
  The influence of the change in price is bigger for the people with less incomes, who prefer the economy class. This statistic shows us that the price has a remarkable significance for these people, because their purchasing power is less than people in business class, that is why the price should be a more important criterion for them. Additionally, despite the wage gap between the consumers the demand curve is inelastic for both classes in Transatlantic Air Travel.
  Comment on Savaş Can Akçay's post “The change in price has a...”
  (17 votes)
Fayez Khan
Posted 7 years ago. Direct link to Fayez Khan's post “Hey for the first set of ...”
Hey for the first set of elasticity, from going to G to H it says the % change in price is 7.81 but when I divide 10 by 125 I get 0.08 which is 8%, so am I doing it wrong or is the answer wrong?
Button navigates to signup pageComment on Fayez Khan's post “Hey for the first set of ...”
(7 votes)
Answer
- Stefan van der Waal
  Posted 7 years ago. Direct link to Stefan van der Waal's post “You're right. It's suppos...”
  You're right. It's supposed to be 8%. The answer is wrong.
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  (5 votes)
muhammad.alenzy
Posted 8 years ago. Direct link to muhammad.alenzy's post “to find the price elastic...”
to find the price elasticity of the entire demand curve, is it correct to find the elasticity for all points and find the average of those values?
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(3 votes)
Answer
- Stefan van der Waal
  Posted 8 years ago. Direct link to Stefan van der Waal's post “No, there is no such thin...”
  No, there is no such thing as a price elasticity of the entire curve. The price elasticity is different for every point.
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  (9 votes)
Felix Kaloczy
Posted 7 years ago. Direct link to Felix Kaloczy's post “could someone please give...”
could someone please give the answers to the problems?
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(0 votes)
Answer
- Qi.Z
  Posted 7 years ago. Direct link to Qi.Z's post “Here's my answers to the ...”
  Here's my answers to the problems:
  (1) E = |(1/5.5)/(-3/31.5)| = 1.91 Thus the demand curve is elastic here.
  (2) E1 = |(0.1/0.45)/(-1/4.5)| = 1 E2 = | [(1/36)/(17/72)]/(-1/8.5) | = 1 The demand curve at the two points are both unit elastic. In fact, the elasticity of price at every point of the curve equals to 1. Since P always equals to 2/Q on this curve, ΔP = P(new) - P(old) = 2/Q(new) - 2/Q(old) = 2(Q(old) - Q(new))/Q(new)*Q(old) = 2ΔQ/[Q(new)*Q(old)]; similarly, P(average) = (Q(new) + Q(old)) / Q(new)*Q(old). Thus we get %ΔP = 2ΔQ/(Q(new) + Q(old)) and plug it into the formula E = %ΔQ/%ΔP, we get E = (ΔQ * (Q(new) + Q(old))) / (0.5 * (Q(new) + Q(old)) * 2ΔQ) = 1.
  (3) The method is the same as above. E1 = E2 = 1.
  (4) E = 0.41
  Hope it helps! :)
  Comment on Qi.Z's post “Here's my answers to the ...”
  (7 votes)
Nicolas Eleftheriou
Posted 7 years ago. Direct link to Nicolas Eleftheriou's post “Should i use the absolute...”
Should i use the absolute values for the change of quantity and price if i want to measure elasticity? Because if i don't most of the problems turn out to be inelastic since the Elasticity is smaller than 1. is that correct?
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(2 votes)
Answer
kimiyama_haruhi
Posted 7 years ago. Direct link to kimiyama_haruhi's post “Another question is, what...”
Another question is, what happen if both demand and supply increase, but there is a larger increase in demand than the supply? How would the graph looks like? Pls do help me
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(2 votes)
Answer
Hadorph
Posted 8 years ago. Direct link to Hadorph's post “In the example above (apa...”
In the example above (apartments), when calculating the price elasticity of supply, why do you also split the change in price by two?
According to the example the answer is 7.4 %, shouldn't it be 14.8 % ?
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(2 votes)
Answer
- Adena Daniel
  Posted 7 years ago. Direct link to Adena Daniel's post “Incorrect. He made a wron...”
  Incorrect. He made a wrong input in the y value but did the math correct. It was supposed to be 700-650 not 600.
  Comment on Adena Daniel's post “Incorrect. He made a wron...”
  (1 vote)
Bobby Hickson
Posted 7 years ago. Direct link to Bobby Hickson's post “Are there answers to the ...”
Are there answers to the problems?
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(1 vote)
Answer
- Stefan van der Waal
  Posted 7 years ago. Direct link to Stefan van der Waal's post “The answers to the self-c...”
  The answers to the self-check questions are below the questions. The answers to the other questions aren't given.
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  (2 votes)
hugofetiza
Posted 7 years ago. Direct link to hugofetiza's post “I feel as if the apartmen...”
I feel as if the apartment problem was not worded properly. In the first sentence, "rented" should be "supplied" since what is rented isn't always the full supply.

For reference, I put the problem below:

Assume that an apartment rents for $650 per month and, at that price, 10,000 units are rented—you can see these number represented graphically below. When the price increases to $700 per month, 13,000 units are supplied into the market.
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(1 vote)
Answer

Course: Microeconomics > Unit 3

Price elasticity of demand and price elasticity of supply

Key points

What is price elasticity?

Using the midpoint method to calculate elasticity

Using the point elasticity of demand to calculate elasticity

Calculating price elasticity of demand

Calculating the price elasticity of supply

Summary

Self-check questions

Review Questions

Critical-thinking questions

Problems

Attribution

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