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Elasticity

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Elasticity

On the demand side of a market we said

1) The demand curve is downward sloping and this is a reflection of idea that the price and the quantity demanded are inversely related, or move in opposite directions, and

2) The demand curve could shift if a determinant of demand should change. Why donâ€™t you write down here all the determinants of demand.

Example 1

PQTR

110 0

101 10

92 18

83 24

74 28

65 30

56 30

47 28

38 24

29 18

110 10

011 0

P

Q

TR

Q

On the previous slide letâ€™s say the demand for a product is such that when consumers come to the market as a group the demand curve is the P and Q columns and graphs out as the upper graph. For example, when the price is 6 the quantity demanded is 5.

What this means is that as a group if the price is 6, 5 units will be purchased by the consumers. The consumer expenditure would be P times Q (or 30 in the example at a price of 6). Note also that here the expenditure of the consumers is the total revenue (TR) of the producers. In economics we tend to focus on the TR of producers in this context.

One way to look at the demand curve and associated TR is to start at the highest price and move your way down to lower prices.

Note as price falls

1) The quantity demanded always rises (this is again the law of demand), and

2) The TR rises, levels off and then falls.

So, as price falls, TR does not always rise! What is the deal here?

Since TR = P times Q on a demand curve, and since when P falls Q rises, the change in TR takes into consideration a lower price and a higher quantity.

Technical detail (hey you, stay here and read on!)

If the percentage change in the quantity demanded is larger than the percentage change in price then the quantity going up should outweigh the price going down on the TR change and thus TR will go up.

But, if the % change in Qd is smaller than the % change in P then the % change in P should be the heavier factor and TR should fall.

If the % changes in P and Q are same the TR should not change.

Do you remember fractions from your prior school daze? I bet you do. Some of you may have seen N/D as a general fraction, where N stands for numerator or top of the fraction and D stands for denominator, or bottom of the fraction.

Some dude (or is it dud?) named Marshall devised a fraction that he called the price elasticity of demand. It looks at the % change in the quantity demanded as the numerator and the % change in the price as the denominator. So this fraction looks at exactly what we saw as the determining factors of what would make TR rise as the price falls.

Say you put $1 in the bank and at the end of the year you have $2 (this is a special bank!). The percentage change in the amount of money in your account is found by the basic formula

(end value â€“ start value) divided by start value and we have

(2 â€“ 1)/1 = 1 (and you could multiply by 100 to say 100 percentage increase).

As another example say you buy a share of stock for $10 and at the end of 1 year the stock has a price of $5. The percentage change in the price of the stock (5 â€“ 10)/10 = -0.5. The minus sign here is important because it is an indication that the stock price went down (and here the fall is 50%).

The price elasticity of demand =

% change in the quantity demanded divided by the

% change in the price.

What will be the numerical sign of the price elasticity of demand? It will always be negative because P and Q move in opposite directions (think of the examples on the previous page). BUT, most of the time we ignore the minus sign. Not always, but often because we know it will be negative.

If the price elasticity of demand (after ignoring the minus sign) is

1) Greater than 1 then TR will rise as the price falls because the % change in quantity demand has the larger impact,

2) Less than 1 the TR will fall as the price falls because the % change in price has the larger impact, and

3) Equal to 1 the TR will not change as the price falls because the two percentage changes are the same.

Letâ€™s look at these three cases in some graphs, shall we?

Own price elasticity and total revenue changes

Total revenue (TR) is price times quantity. Along the demand curve P and Q move in opposite directions. Knowledge of Ed assists in knowing how TR will change.

When we look at the collection of consumers in the market, at this time in our study we assume each consumer pays the same price per unit for the product.

Also at this time in our study the total expenditure of the consumers in the market would equal the total revenue (TR) to the sellers.

So, here we look at the whole demand side of the market in general.

P

TR in the market is equal to the price in the market multiplied by the quantity traded in the market. In this diagram TR equals the area of the rectangle made by P1, Q1 and

P1

Q

Q1

the horizontal and vertical axes. We know from math that the area of a rectangle is base times height and thus here that means P times Q. The area is TR!

Elasticity and total revenue relationship

P

Now in this graph when the price is P1 the TR = a + b(adding areas) and if the price is P2 the TR = b + c.

The change in TR if the price should fall

P1

P2

a

b c

Q

Q1 Q2

from P1 to P2 is (b + c) - (a + b) = c - a.

Similarly, if the price should rise from P2 to P1 the change in TR is a - c. I will focus on price declines next.

Elasticity and total revenue relationship

P

Since the change in TR is c - a, the value of the change will depend on whether c is bigger or smaller, or even equal to, a. In this diagram we see c > a and thus the change in TR > 0.

P1

P2

a

b c

Q

Q1 Q2

This means that as the price falls, TR rises. In the upper left of the demand the demand is price elastic because the percentage change in quantity demanded is the larger part. Thus if the price falls in the elastic range of demand TR rises.

You will note on the previous screen that I had c - a. In the graph c is indicating the change in TR because we are selling more units. The area a is indicating the change in TR when there is a price change. We have to bring the two together to get the change in TR.

Thus a lower price has a good and a bad.

Good - sell more units.

Bad - sell at lower price.

Elasticity and total revenue relationship

P

Now in this graph when the price is P1 the TR = a + b(adding areas) and if the price is P2 the TR = b + c.

In this diagram we see c < a and thus the change in TR < 0.

P1

P2

a

b

c

Q

Q1 Q2

In the lower right of the demand the demand is price inelastic. Thus if the price falls in the inelastic range of demand TR falls.

Elasticity and total revenue relationship

P

Now in this graph when the price is P1 the TR = a + b(adding areas) and if the price is P2 the TR = b + c.

In this diagram we see c = a and thus the change in TR = 0.

P1

P2

a

b c

Q

Q1 Q2

In the middle of the demand the demand is unit elastic. Thus if the price falls in the unit elastic range of demand TR does not change.

If price TRif demand is

Fallsriseselastic

Fallsdoes not changeunit elastic

Fallsfallsinelastic

Risesfallselastic

Risesdoes not changeunit elastic

Risesrisesinelastic

As you look at the summary, here is a little memory device to help you put this all together.

If demand is elastic, price and TR move in opposite directions.

If demand is inelastic, price and TR move in the same direction.

If demand is unit elastic, TR does not change as the price changes.

Check this out

If you have a dollar and it grows to $2, then we say the gain was 100%.

But, if you start with $2 and it falls to $1, then we saw you lost 50%.

So, the movement between $1 and $2 is either 100% or 50%, depending on where you start.

This freaks some people out. We need to make this kind of calculation (twice actually, once for price and once for quantity) when we calculate an elasticity. So, we use a slightly different calculation method so people do not get freaked out. (Freaked is a highly technical term you usually only see in the journals.)

When you go back and consider our numerical example letâ€™s look at the two points (Q, P) of 1,10 and 2, 9

P

The most direct way to calculate a percentage change is take the later value minus the earlier and divide by the earlier.

For example, if the quantity goes from 1 to 2 we have as a percentage change in quantity (2 â€“ 1) / 1 = 1/1 and the associated percentage change in price is (9 â€“ 10)/10 = -1/10, so the elasticity is (1/1)/(-1/10) = -10. Again, some freak out though if we have the Q start at 2 and go to 1 because then the elasticity would be [(1 â€“ 2)/ 2]/[(10 â€“ 9)/9] = (-1/2)/(1/9) = -4.5

10

9

Q

1 2

Midpoint Formula

Here we show the midpoint formula for calculating elasticity of demand

Because of the way we have calculated the elasticity up to this point we see that if we look at two points on the demand curve we get different elasticity values depending upon which point from which we start. This bugs people, so a midpoint formula was created.

To calculate each percentage change, do this

change in value divided by the average value.

Say the price moves from 10 to 9

P

10

9

Q

1 2

From the example on the previous screen see the calculation as:

% change in Q(2 â€“ 1)/[(2 + 1)/2]

= (2 â€“ 1)/1.5

= 1/1.5, (Fran rule: ignore above if you do not need it)

% change in P(9 â€“ 10)/[(9 + 10)/2]

= (9 - 10)/9.5

= -1/9.5 (Fran rule still applies)

The elasticity of demand is thus

[1/1.5]/[-1/9.5]

=[1/1.5][-9.5] (times reciprocal)

= -6.33

So, using the midpoint formula we get the same value for the elasticity between two points. So, here we have

% change in quantity demanded divided by % change in price equaling -6.33. This means for every 1 percent change in price the quantity demanded changes by 6.33% in the opposite direction.

Thus, we know demand is elastic and if the price is lowered the total revenue will rise.

Determinants of elasticity of demand

Different products have different elasticity values. It is thought there are factors that lead to certain elasticity values.

P

We see the same price change along both curves, but the flatter curve has a greater quantity response. If you worked out the elasticity, you would see the flatter curve is more elastic than the steep curve in any price range.

Q

- Some determinants are
- The number of substitutes a product has,
- The % of the consumer budget the item costs,
- Time.

The more substitutes an item has the more likely the quantity demanded will respond to a price change.

Thus, the more subs. there are the greater the elasticity.

The greater the % of your budget the greater is the elasticity.

Letâ€™s use milk as an example. Milk purchases by the typical consumer likely takes a relatively small percentage of the budget for a consumer. Changes in the price elicit almost no change in the quantity demanded.

A few years back the government taxed yachts and this raised the price of yachts. The Gov thought rich people would go on buying. Many buyers though said Iâ€™ll keep my current yacht longer because this increase is too much of my budget. The price change had a significant impact on the quantity demanded.

- The longer the time period since the price change, the more elastic demand tends to be. To a certain extent this is because the longer time since the price change the more time we have to find substitutes. With home heating fuels if the price goes up in short term we buy but in long term we buy less and wear a sweater or search for other methods.

Other demand elasticities

There are other elasticities besides the own price elasticity of demand. Letâ€™s see a few here.

We saw that things like taste and preference, price of other goods, income and the number of buyers shift the demand curve if they change. How much do they shift the demand curve?

We use other elasticity concepts as an indication of how much the curve will shift given a change in one of these factors.

Edxy = % change in Qdx / % change in Py.

The bigger the value the more the demand shifts.

If the value is negative we have complements and if positive we have substitutes.

If the absolute value is between 0 and 1 the cross elasticity is inelastic, if = 1 unit elastic and if greater than 1 elastic.

Edxm = % change in Qdx / % change in M.

The bigger the value the more the demand shifts.

If the value is negative we have an inferior good and if positive we have a normal good.

If the absolute value is between 0 and 1 the income elasticity is inelastic, if = 1, unit elastic, and if greater than 1, elastic.

Say the cross price elasticity for Coke with respect to the price of Pepsi is 1.7. This means for every 1% Pepsi raises price the demand for Coke will go up 1.7%.

Say Coke execâ€™s hear Pepsi is going to raise price by 2.5%. How much will Coke execâ€™s expect demand to rise?

1.7 = x/2.5 or x = 1.7(2.5) or 4.25%

The elasticity of supply is used to indicate the percentage change in the quantity supplied given a percentage change in price.

The elasticity of supply is calculated in a manner similar to the other elasticities we have seen and has a similar interpretation in terms of the range of values the elasticity might take, i.e. elastic, inelastic and unit elastic.