percentages and elasticity. percentage: “for each hundred” one per cent: one for each hundred...

Percentages and Elasticity

percentage: “for each hundred” one per cent: one for each hundred

ex: "I spend ten percent of my income on movies and other forms of entertainment" means that "I spend ten dollars of each hundred dollars of my income on entertainment."

Converting from decimal to percentage notation:

Just move the decimal to the right two places.

.65 65%

What percentages are equivalent to the following numbers?

(a) .01 1%(b) .94 94%(c) 1.705 170.5%(d) .2386 23.86%(e) .8

Converting from fractions to percentages:

Convert from fraction to decimal by dividing, and then move the decimal to the right two places.

1/4 .25 25%

What percent of 50 is 10?

4 is what percent of 8?

What percent of 60 is 15?

Questions

Percentage Change:

absolute change = after - before

before before

If spending increased from $50 to $60, by what percent did spending increase?

percentage change = after - before before

= 60 - 50 50

= 10 / 50 = .20 = 20 %

Suppose your financial aid increased by $50. You used to spend $10 per week on entertainment. You now spend $11. By what percentage did your entertainment spending increase?

= 11 - 10 10

= 1/10 = .10 = 10 %

Your financial aid is cut back by $50 to its original level. You reduce your entertainment spending from $11.00 back to $10.00. By what percent did your spending decrease?

= 10 - 11 11

= - 1 / 11 = - .0909 = - 9.09 %

A positive change indicates an increase.

A negative change indicates a decrease.

It would be nice to be able to say the following: In response to a $50 increase, you increased your spending by some percent x. In response to a $50 decrease, you decreased your spending by that same x percent. When we get to the concept of elasticity, in particular, we will want to be able to do that.

average = (before + after) / 2

So we will define percentage changes a little differently. Instead of using our "before" value as our denominator, we will use the average (or midpoint) of our "before" and "after" values as the denominator.

Our percentage change formula becomes

percentage change = after - before average

Suppose your financial aid increased by $50. You increase your spending on entertainment from $10 per week to $11. By what percentage did your entertainment spending increase? (Use the midpoint formula.)

First we need to determine the average of the before and after spending.

average = (before + after) / 2 = (10 + 11) / 2 = (21) / 2 = 10.5

= after - before average

= 11 - 10 10.5

= 1 / 10.5

= .0952

= 9.52 %

percentage change

Financial aid is cut back by $50. You reduce your spending on entertainment from $11 per week to $10. Using the midpoint formula, determine by what percentage your spending decreases.

= 10 - 11 10.5

= - 1 / 10.5

= - .0952 = - 9.52 %

Suppose that financial aid for the semester increased from $995 to $1005. Using the midpoint formula, determine by what percent financial aid increased.

= 1005 - 995 1000

= 10 / 1000 = .01 = 1 %

Suppose the 1% increase in financial aid led you to increase your entertainment budget from $297 to $303. By what percent did you increase your entertainment budget?

= 303 - 297 300

= 6 / 300 = .02 = 2 %

In other words, the percentage increase in your entertainment budget was 2 times as large as the percentage increase in financial aid.

This value (2) is the elasticity of your entertainment budget with respect to your financial aid.

fin. aid 1 % ent. budget 2 %

tells you by what percent X changes when Y changes by 1%.

The elasticity of X with respect to Y

can be calculated as

the percentage change in X the percentage change in Y.

The elasticity of X with respect to Y

If in response to a 20% increase in financial aid, you increased your entertainment budget by 40%, what is the elasticity of your entertainment budget with respect to your financial aid?

By how much would you expect your entertainment budget to increase in response to a 1% increase in financial aid?

Suppose financial aid increased from $1050 to $1100. In response, you increased your entertainment budget from $300 to $320. Calculate the following:

the percentage change in financial aid, the percentage change in your entertainment

budget, and the elasticity of your entertainment budget

with respect to financial aid.

change in fin. aid = 1100 - 1050 = 50

avg. fin. aid = (1050 + 1100)/2 = 2150/2 = 1075

percentage change in financial aid

= (change in fin. aid) / (avg fin. aid)

= 50 / 1075

= .0465

= 4.65%

percentage change in financial aid

fin. aid: 1050, 1100

change in ent. budget = 320 - 300 = 20

avg ent. budget = (300 + 320) / 2 = 620/2 = 310

percentage change in entertainment budget

= (change in ent. budget) / (avg ent. budget)

= 20 / 310

= .0645

= 6.45%

percentage change in ent. budget

ent. budget: 300, 320

percentage change in entertainment budget percentage change in financial aid

= 6.45/4.65 = 1.387.

So when financial aid increases by1 percent, your entertainment budget

increases by 1.387 percent.

elasticity of entertainment budget

with respect to financial aid

In the elasticity formula, how do you remember which variable goes on top and which goes underneath?

The cause goes under the line.

CAUSE and UNDER both have a U.

The effect goes on top.

EFFECT and TOP both have a T.

elasticity = % change in effect

% change in cause

for our financial aid & entertainment example:

elasticity = % change in .

% change in

Unit Elastic:

|elasticity| = 1

% change in effect

% change in cause = 1

% change in the effect = % change in the cause

fin. aid 5 % ent. budget 5 %.

Your entertainment budget is unit elastic with respect to financial aid.

Elastic:

|elasticity| > 1

% change in effect

% change in cause > 1

% change in the effect > % change in the cause

Your entertainment budget is elastic with respect to financial aid.

Inelastic:

|elasticity| < 1

% change in effect

% change in cause < 1

% change in the effect < % change in the cause

Your entertainment budget is inelastic with respect to financial aid.

Price Elasticity of Demand (or Elasticity of Quantity Demanded with

Respect to Price)

measures the responsiveness of consumers' purchases to a change in the price of a commodity.

Notation

means “change” and

% means “percentage change”

examples: Price or P means “change in price”

% P means “percentage change in price”

the percentage change in the quantity demanded of PCs,

the percentage change in the price of PCs, and the elasticity of demand for PCs with respect

to the price of PCs.

Suppose the price of personal computers increased from $1600 to $1700. As a result, the number of PCs

purchased per week by area consumers dropped from 500 to 400. Calculate the following:

percentage change in quantity demanded of PCsquantity: 500, 400

qty demanded of PCs = 400 - 500 = -100

avg qty demanded of PCs = (500 + 400) / 2 = 900 / 2 = 450

% qty demanded of PCs

= ( qty demanded of PCs) / (avg qty demanded)

= -100/450 = -.2222 = - 22.22 %

[The negative indicates a decrease in PCs.]

percentage change in price of PCsprice: 1600, 1700

Price of PCs = 1700 - 1600 = 100

avg price = (1600 + 1700) / 2 = 3300 / 2 = $1650

% price of PCs = ( price of PCs) / (avg price)

= 100 / 1650 = .0606 = 6.06 %

price elasticity of demand for PCs

% qty demanded of PCs % price of PCs

= - 22.22 / 6.06 = - 3.667

The negative indicates that there is an inverse relation between the qty demanded of PCs and the price of PCs.

When the price of PCs increases by one percent, the quantity demanded of PCs decreases by 3.667 percent.

|-3.667| = 3.667 > 1

So, the demand for PCs is elastic with respect to the price of PCs.

ex: The price elasticity of demand for PCs would be reported as 3.667 instead of -3.667. The negative is understood.

Because it is almost always the case that the qty demanded of a good is inversely related to its price, the negative sign is frequently dropped.

Suppose you have an ailment, for which you must take a particular medication. (Call it Medex.) Every thirty days you purchase one thirty-capsule bottle of Medex. The price of Medex increases from $4 to $5

per bottle. You still purchase one bottle every thirty days. Calculate the elasticity of your quantity demanded of Medex with respect to the price of Medex.

percentage change in price of Medexprice: 4, 5

Price of Medex = 5 - 4 = 1

avg price = (4 + 5)/2 = 9/2 = 4.5

% price of Medex = ( price) / (avg price) = 1 / 4.5 = .2222

So the price of Medex changed by 22.22 %.

percentage change in qty demanded of Medexquantity: 1, 1

qty demanded of Medex = 1 - 1 = 0

avg qty demanded of Medex = (1 + 1) / 2 = 2 / 2 = 1

% qty demanded of Medex

= ( qty demanded) / (avg qty demanded)

= 0 / 1 = 0

So the quantity demanded of Medex changed by 0 %. (It didn’t change at all.)

price elasticity of demand for Medex

% qty demanded of Medex% price of Medex

= 0 / 22.22 = 0

elasticity = 0

It is a special case of inelastic.

Perfectly Inelastic

Suppose the price of personal computers increases from $1500 to $1700. As a result, the number of

PCs purchased per week by consumers in the area dropped from 850 to 750. Calculate the elasticity of demand for PCs with respect to the price of PCs.

percentage change in qty demanded of PCsquantity: 850, 750

qty demanded of PCs = 750 - 850 = -100

avg qty demanded of PCs = (850 + 750) / 2 = 1600 / 2 = 800

% qty demanded of PCs

= -100 / 800 = - .125 = - 12.5 %

Price of PCs = 1700 - 1500 = 200

avg price = (1500 +1700) / 2 = 3200 / 2 = 1600

% price of PCs = ( price) / (avg price) = 200 / 1600 = .125 = 12.5%

price elasticity of demand for PCs

% qty demanded of PCs% price of PCs

= -12.5 / 12.5 = -1

Since the absolute value of the elasticity is 1, the quantity demanded of PCs is unit elastic with respect to price.

Suppose you are in the pizza business. As a very small company, you take the area price of pizza ($8) as given. That is, you always charge the same price. The

quantity demanded of your pizza fluctuates. Last week, the quantity demanded of your pizza increased from 750 to 800 pizzas. Calculate the price elasticity of demand for your pizza.

percentage change in qty demanded of pizzaquantity: 750, 800

qty demanded of pizza = 800 - 750 = 50

avg qty demanded of pizza = (750 + 800) / 2 = 1550 / 2 = 775

% qty demanded of pizza

= ( qty demanded) / (avg qty demanded) = 50 / 775 = .0645 = 6.45 %

percentage change in price of pizzaprice: 8, 8

Price of pizza = 8 - 8 = 0

avg price = (8 + 8) / 2 = 16 / 2 = 8

% price of pizza = ( price) / (avg price) = 0 / 8 = 0 = 0 %

The price changed by 0 %. (It didn't change at all.)

price elasticity of demand for pizza

% qty demanded of pizza% price of pizza

= 6.45 / 0 = infinity or undefined

elasticity = infinity

It is a special case of elastic.

Perfectly Elastic or Infinitely Elastic

1. Elasticity of demand is greater if

there are good substitutes available.

Example: The elasticity of demand for a particular brand of gas would be quite high because there are lots of other brands of gas available.

Determinants of Price Elasticity of Demand

2. Elasticity of demand is greater if the price of the good is high relative to one’s budget.

Example: The elasticity of demand for a salt is low because salt is a very small part of one’s budget.

3. Elasticity of demand is greater if the product is a luxury rather than a necessity.

Example: The elasticity of demand for insulin by a diabetic is extremely low because insulin is a necessity.

4. Elasticity of demand is greater if the buyer has more time to adjust to a change in price.

Example: The elasticity of demand for gas is more elastic when you allow people more time to adjust. If the price of gas goes up today, you can adjust your consumption only a little bit tomorrow. But if you have a few years, you can replace your car with one that consumes less gas and you can move closer to where you work.

Elasticity Graphs

Zero Elasticity or Perfectly Inelastic

Low Elasticity (Inelastic)

High Elasticity (Elastic)

Infinite Elasticity or Perfectly Elastic

Elasticity of a Straight Line

at midpoint, |elasticity| = 1

P above the midpoint,

|elasticity| > 1

below the midpoint, |elasticity| < 1

|elasticity| = 1

|elasticity| > 1

|elasticity| < 1

Constant Elasticity

Relationship Between

Price Elasticity of Demand

and Total Revenue

Note: Total Revenue (TR) = Total Expenditure.

Total revenue is from the firm’s perspective; total expenditure is from the consumer’s

perspective. Both are computed by multiplying price by

quantity. Thus,

TR = P Q

Clearly, if both X and Y increase, then the product Z must increase too.

Also, if both X and Y decrease, then the product Z must decrease too.

Consider the product of two numbers,Z = XY.

That is, if X increases a lot and Y decreases a little, then Z will increase.

If X increases a little and Y decreases a lot, then Z will decrease.

If Y decreases by the same percentage that X increased, then Z will remain unchanged.

However, if X increases and Y decreases, or vice versa, then whether the product Z increases or decreases depends on the relative magnitude of the changes in X and Y.

Generally, when the price of a product increases, the quantity demanded will decrease and vice versa.

So, whether TR increases, decreases, or remains the same when the price of a product changes depends on the relative magnitudes of the changes in price and quantity.

The same idea carries over to the concept of total revenue, which is the product of price and quantity: TR = PQ.

When Demand is Elastic:

P Q TR

Price and TR move in opposite directions.

Example: Elastic Demand

When the price of PCs increased from $1600 to $1700, the number of PCs decreased from 500 to 400. The elasticity was -3.667.

Initially, TR = (1600)(500) = 800,000.

Later, TR = (1700)(400) = 680,000.

So TR fell when price increased.

When Demand is Inelastic:

P Q TR

Price and TR move in the same direction.

Example: Inelastic Demand

Suppose when the price of a good increased from $90 to $110, the quantity demanded decreased from 210 to 190. The elasticity can be computed to be -0.5.

Initially, TR = (90)(210) = 18,900.

Later, TR = (110)(190) = 20,900.

So TR increased when price increased.

When Demand is Unit Elastic:

P Q TR unchanged

TR is constant.

Example: Unit Elastic Demand

Suppose when the price of personal computers increases from $1500 to $1700, the quantity demanded decreases from 850 to 750. Then the elasticity is -1.

Initially, TR = (1500)(850) = 1,275,000.

Later, TR = (1700)(750) = 1,275,000.

So TR is unchanged when price changes.

Price Elasticity of Supply (or Elasticity of Quantity Supplied with

Respect to Price)

measures the responsiveness of producers to a change in the price of a commodity.

Suppose the price of personal computers increased from $1550 to $1650. As a result, the number of PCs that area manufacturers were willing to produce per week

rose from 490 to 510. Calculate the elasticity of quantity supplied of PCs with respect to the price of PCs.

percentage change in qty supplied of PCsquantity: 490, 510

qty supplied of PCs = 510 - 490 = 20

avg qty supplied of PCs = (490 + 510) / 2 = 1000 / 2 = 500

% qty supplied of PCs

= ( qty supplied) / (avg qty supplied)

= 20 / 500 = .04 = 4 %

Price of PCs = 1650 - 1550 = 100

avg price = (1550 +1650) / 2 = 3200 / 2 = $1600

% price of PCs = ( price) / (avg price)

= 100 / 1600 = .0625 = 6.25%

price elasticity of supply of PCs

% qty supplied of PCs % price of PCs

= 4 / 6.25 = .64

When the price of PCs increases by one percent, the quantity supplied of PCs increases by .64 percent.

Since the elasticity is less than one, the quantity supplied of PCs is inelastic with respect to price.

The Income Elasticity of Demand or Elasticity of Quantity Demanded with

Respect to Income

measures the responsiveness of consumers' purchases to a change in consumer income.

Suppose your income increased from $1000 to $1150 per month. As a result, the number of pounds of potatoes you purchase per

month decreased from 10 to 9. Calculate the elasticity of demand for potatoes with respect to income.

percentage change in qty demanded of potatoesquantity: 10, 9

qty demanded of potatoes = 9 - 10 = - 1

avg qty demanded of potatoes = (10 + 9) / 2 = 19 / 2 = 9.5

% qty demanded of potatoes

= - 1 / 9.5 = - .1053 = - 10.53 %

percentage change in incomeincome: 1000, 1150

income = 1150 - 1000 = 150

avg income = (1000 +1150)/2 = 2150 / 2 = $1075

% income = ( income) / (avg income)

= 150 / 1075 = .1395 = 13.95 %

income elasticity of demand for potatoes

% qty demanded of potatoes% income

= - 10.53 / 13.95 = - .75

When the income increases by one percent, the quantity demanded of potatoes decreases by 0.75 percent.

Inferior Good

a good with a negative income elasticity of demand.

The negative sign indicates that income and quantity demanded of the good (potatoes) move in opposite directions. When income rises, people buy less. When income falls, people buy more.

Normal Good

a good with a positive income elasticity of demand.

For normal goods, when income rises, people buy more. When income falls, people buy less. Most goods are normal goods.

Suppose your income increased from $2000 to $2080 per month. As a result, the number of movies you see per month rose from 4 to 5.

Calculate the elasticity of demand for movies with respect to income.

percentage change in qty demanded of moviesquantity: 4, 5

qty demanded of movies = 5 - 4 = 1

avg qty demanded of movies = (4 + 5) / 2 = 9 / 2 = 4.5

% qty demanded of movies

= 1 / 4.5 = .2222 = 22.22 %

percentage change in incomeincome: 2000, 2080

income = 2080 - 2000 = 80

avg income = (2000 +2080) / 2 = 4080 / 2 = 2040

% income = ( income) / (avg income)

= 80 / 2040 = .0392 = 3.92 %

income elasticity of demand for movies

% qty demanded of movies% income

= 22.22 / 3.92 = 5.668

When the income increases by one percent, the quantity demanded of movies increases by 5.668 percent.

Since the income elasticity of movies is positive, movies are a normal good.

Cross Elasticity of Demand

measures the responsiveness of the quantity demanded of one good to changes in the price of another good.

Suppose the price of compact diskettes falls from $18 to $14. In response, the quantity demanded of cassettes

demanded of cassettes with respect to the price of CDs.

per week by area consumers falls from 1100 to 900. Calculate the elasticity of the quantity

percentage change in qty demanded of cassettesqty of cassettes: 1100, 900

qty demanded of cassettes = 900-1100 = - 200

avg qty demanded of cassettes = (1100 + 900) / 2 = 2000 / 2 = 1000

% qty demanded of cassettes

= - 200 / 1000 = - .20 = - 20 %

percentage change in price of CDsCD price: 18, 14

price of CDs = 14 -18 = - 4

avg price of CDs = (18 + 14) / 2 = 32 / 2 = 16

% price of CDs = ( price of CDs) / (avg price of CDs)

= - 4 /16 = - .25 = - 25 %

cross elasticity of demand for cassettes with respect to the price of CDs

% qty demanded of cassettes% price of CDs

= -20 / -25 = .80

When the price of CDs falls by one percent, the quantity demanded of cassettes falls by .80 percent.

Substitutes

When the cross elasticity between two goods is positive, the two goods are called substitutes.

A substitute is a good that can be used instead of another good.

When the price of CDs decreases, the quantity demanded of cassettes decreases.

When the price of CDs increases, the quantity demanded of the cassettes increases.

The price of one good and the quantity demanded of the substitute good move in the same direction.

Complements

When the cross elasticity between two goods is negative, the two goods are called complements.

A complement is a good that is used along with another good.

Complements

The price of one good and the quantity demanded of the complementary good move in opposite directions.

When the price of coffee decreases, the quantity demanded of cream increases.

When the price of coffee increases, the quantity demanded of cream decreases.

Suppose the price of CD players increases from $140 to $160. In response, the quantity demanded of CDs per week by area consumers decreases from 1150 to 850. Calculate the cross elasticity of demand for CDs with respect to the price of CD players.

percentage change in qty demanded of CDsqty of CDs: 1150, 850

qty demanded of CDs = 850 - 1150 = - 300

avg qty demanded of CDs = (1150 + 850) / 2 = 2000 / 2 = 1000

% qty demanded of CDs

= - 300 / 1000 = - .30 = - 30 %

percentage change in price of CD playersCD player price: 140, 160

price of CD players = 160 -140 = 20

avg price of CD players = (140 + 160) / 2 = 300 / 2 = 150

% price of CD players = ( price) / (avg price)

= 20 / 150 = .1333 = 13.33 %

cross elasticity of demand for CDs with respect to the price of CD players

% qty demanded of CDs% price of CD players

= - 30 / 13.33 = - 2.25

Since the cross elasticity of demand for CDs with respect to the price of CD players is negative, CDs and CD players are complements.

The Effect of Elasticity on

the Burden of a Tax

price QD QS

1.09 45 75

1.06 50 70

1.03 55 65

1.00 60 60

.97 65 55

.94 70 50

.91 75 45

Example:

Suppose supply and demand are as shown here. Without a tax, equilibrium price is $1 and equilibrium quantity is 60 units.

DemandSupply (without tax)

Quantity

Example: no tax quantity is 60price is $1

Now suppose a tax of six cents per unit is imposed.

Then, from the buyer’s view, for each quantity supplied, the total price per unit is six cents higher than before.

(Equivalently, for a given total price per unit, suppliers are willing to provide fewer units than before.)

So the buyers see a new supply curve.

Supply (with tax)Price

Quantity

price QS

1.09 75

1.06 70

1.03 65

1.00 60

.97 55

.94 50

.91 45

Old Supply (no tax):

price QS price (with tax)

1.09 75 1.15

1.06 70 1.12

1.03 65 1.09

1.00 60 1.06

.97 55 1.03

.94 50 1.00

.91 45 .97

Put in the tax.

price QS price (with tax)

1.09 75 1.15

1.06 70 1.12

1.03 65 1.09

1.00 60 1.06

.97 55 1.03

.94 50 1.00

.91 45 .97

new supply as viewed by the buyers:

price QD QS

1.15 75

1.12 70

1.09 45 65

1.06 50 60

1.03 55 55

1.00 60 50

.97 65 45

.94 70

.91 75

If we line up the demand column and our new supply column so that the price columns match, the table looks like this.

price QD QS

1.15 75

1.12 70

1.09 45 65

1.06 50 60

1.03 55 55

1.00 60 50

.97 65 45

.94 70

.91 75

If we line up the demand column and our new supply column so that the price columns match, the table looks like this.

Equilibrium quantity is now 55. Equilibrium price (from the buyer’s view) is 1.03.

Quantity

tax = .06 Quantity is 55 Buyer pays 1.03

But the government gets .06.

So the seller gets 1.03 - .06 = .97.

Quantity

tax = .06 Quantity is 55 Buyer pays 1.03 Seller gets 1.03 -.06=.97

The burden of the tax is shared evenly in this example. The buyer pays three cents more per unit than before the tax and the seller gets three cents less per unit.

Let’s go through the graphs again for the general case of a tax of amount t.

Demand

Supply (without tax)

Quantity

Demand

Quantity

Before tax:quantity is Q0 price is P0

Demand

Quantity

Demand

Quantity

With tax: quantity is Q1

buyer pays Pb

Demand

Quantity

With tax: seller gets Ps = Pb - t

With the tax, the buyer pays more and the seller receives less than without the tax. The burden of the tax is shared.

Suppose the elasticity of demand is smaller.

The demand curve is steeper.

How does this affect how the tax burden is divided?

Demand

Quantity

Demand

Supply (with tax)

Quantity

1.00 .98

tax = .06 Quantity is 59 Buyer pays 1.04 Seller gets 1.04 -.06=.98Consumer bears greater share of tax burden.

When demand is less elastic and consumers are less responsive to price changes, consumers will bear a larger share of the tax burden, and sellers will bear a smaller share.

In the extreme case where demand is perfectly inelastic (vertical demand curve), consumers will bear the entire burden of the tax.

The price paid by consumers is t dollars more than before the tax. The price received by the seller is the same as before the tax.

Demand

Supply (with tax)

Quantity

tax = .06 Quantity is 60 Buyer pays 1.06 Seller gets 1.06 -.06=1.00Consumer bears entire tax burden.

When supply is less elastic and sellers are less responsive to price changes, sellers will bear a larger share of the tax burden, and buyers will bear a smaller share.

Quantity

tax = .06 Quantity is 59 Buyer pays 1.02 Seller gets 1.02 -.06=.96Seller bears greater share of tax burden.

Demand

In the extreme case where supply is perfectly inelastic (vertical supply curve), sellers will bear the entire burden of the tax.

The price received by sellers is t dollars less than before the tax. The price paid by consumers is the same as before the tax.

To show this effect on a graph, we need to shift the demand curve instead of the supply curve. From the supplier’s perspective, it seems as if the demand curve has shifted down vertically by the amount of the tax.

Demand (without tax)

Supply Price

Quantity

tax = .06 Quantity is 60Seller gets .94 Buyer pays .94 + .06 = 1.00Seller bears entire tax burden.

Demand (with tax)

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