Handout: Some Math Behind Elasticities
Price Elasticity of Demand (Own and Cross)
Own Price
Own price elasticity of demand tells us how much the quantity demanded of a good is
going to change when the price of that good changes, with all other factors held constant 1. For you mathematically minded folks, that’s . . .
e d =
Where e d is the price elasticity of demand 2. This makes a lot more n if we have a graphical example.日工
Say the graph here reprents the demand for CDs. Consider moving from point A to B. That’s a chang
e in price of $5, and a change in quantity demanded of 10. That means the price elasticity of demand is
=4.33 i.e. a 1% change in price will bring about a 4.33% change in quantity demanded 3. Now consider moving from point C to point D. That’s the
same price and quantity demanded increment change as before. But now . . . 1 Becau of the Law of Demand that we learned before, for all normal goods the change in quantity
demanded will move in the opposite direction of change in price. So elasticity of demand for normal goods will be a negative number. But the negative is often dropped for the sake of simplicity.
有哪些动物冬眠2 For tho of you unfamiliar with Greek letters, !is called “delta”, and it means “change in”. So
"Q d Q d is the equivalent of saying “percentage change in”, as it’s the change in quantity demanded divided by the total quantity demanded. Similar logic follows for price. 3 Note that I’ve ud our textbook’s convention of using the “midpoint formula” to calculate elasticity – we evaluate the elasticity at the average of the prior and post values for Q and P. Price ($) 10 20 30 40 50 60 70 80 5
10 15
20 25 30 35 40
A B C D Quantity
Demanded (in millions)
元宵节的古诗有哪些=0.23
< a 1% change in price will bring about a 0.23% change in quantity demanded. That’s quite a different result from before. What’s the deal? Let’s break the elasticity equation into two parts.
d
d d Q P P Q
e *!!=
The first part of the equation, "Q d "P , is the same in both cas (10/5), since we have a curve with a constant slope. But the cond part, d
Q P , is different as we move along the curve, becau at each point were dealing with different prices and quantities demanded. In the first ca, it was 32.5/15, and in the cond ca, it was 5/65.
What’s the moral of the story? Depending on the shape of the demand curve and草莓巧克力
where we are on the demand curve, price elasticity for the same good can take on different values.
There are demand curves that can be constructed that will have a constant elasticity. However, with at constant slope demand curve (where slope is not equal to 0), the elasticity will vary along the curve.
Cross Price
描写母爱的古诗Cross price elasticity of demand is similar to own price elasticity, but now, we’re looking at two goods at once. Specifically, we’re asking what percentage change in quantity demanded we’ll e for good A when the price of good B changes. Let’s go back to the math?
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e A ,B =
Here, I’ve chon the A,B subscript on the e as a way of indicating that we’re talking about the demand of good A and the price of good B. Note that we’re holding the price of good A constant.
The sign of the cross price elasticity of demand tells us about the relationship between the two good
s. For example, if e A,B is negative, that means as the price of good B goes up, the demand for good A goes down, and the goods are compliments. However, if the function is positive, as the price of good B increas, the demand for good A increas as well, and the two goods are substitutes.
“Size” of Price Elasticity
If e d is greater than 1, a 1% change in price will have a greater than 1% change in quantity demanded, and the good is called elastic. If e d is equal to one, the good is unit elastic, and if it is less than 1, the good is inelastic. In our earlier constant slope demand curve example, in the first ca we had elastic demand, but in the cond, it was inelastic. If e d is infinite, then quantity demanded is so amazingly nsitive to price changes that the tiniest change in price results in huge changes in quantity demanded, and the good is called perfectly elastic, which corresponds to a totally horizontal demand curve. On the other side of things, if e d is equal to 0, then no matter what happens to price, quantity demanded won’t change at all, and the good is called perfectly inelastic, which corresponds to a perfectly vertical demand curve4.
Own Price Elasticity and Revenue
How nsitive the quantity demanded for a good is to price changes also tells us about how revenue
for llers will change if price changes. If the price elasticity is greater than 1, the good is elastic, and quantity demanded is nsitive to price changes. The revenue gained by increasing the price is outweighed by the revenue lost by the decrea in quantity sold. So raising prices when e d > 1 results in an overall decrea in revenue. On the other side of the coin, if price elasticity is less than 1 (e d < 1), the revenue gained by increasing the price is greater than the revenue lost by the decrea in quantity sold, and overall revenue increas.
Who cares? For one, llers do. Take our above example again. If I’m in the music industry, and I know that we’re at point B on the demand curve, I know that price elasticity is greater than 1, so an increa in the price of CDs is going to result in a loss of revenue. But if I know I’m at point D a higher price means good news for me, becau I’ll make more revenue in the end.
4 It’s kind of hard to imagine that the demand for anything is perfectly elastic or inelastic. Is there really some good out there that, even if the price was incread twenty-fold, quantity demanded wouldn’t change? Probably not. They are largely theoretical concepts. But there are goods that are extreme cas in both directions.
Income Elasticity of Demand
As a concept, it’s almost identical to price elasticity, but now, we’re talking about . . . you guesd it, income. The income elasticity of demand measures how much the quantity demand of a good will change (at any given price) when income changes.
Note the part in parenthesis . . . at any given price. In the first ca, we held income constant, and asked what happened with quantity demanded when price changed
(movements along the demand curve). Now, we’re holding prices constant, and asking what happens to quantity demand when income changes (shifts of the demand curve).
The math looks almost the same, with a little change of what letters we throw in. Let I reprent income. Then our equation is for income elasticity of demand (e d,w ) is
e w d =,
Now, however, we have to watch out for negatives. Unlike price elasticity of demand, it isn’t assumed
that this is going to be a positive value. In fact, the sign of e d,w tells you a lot about the good we’re examining. If we get a positive value, then the good is a normal good, i.e. consumption of it increas as income increas. If we get a negative value, the good is an inferior good, meaning as our income increas, we consume less 5.
Price Elasticity of Supply
仪态万千Imagine you had a price elasticity of demand equation. Now imagine replacing Q d with Q s . If you were imagining right, you’d have:
e s =
Nothing new here, people. Just the same basic idea, but now we’re asking by what
percentage quantity supplied changes when price changes by a certain percentage. Other than that, it’s just about the same deal as price elasticity of demand, and the terminology is similar (inelastic supply, elastic supply, unit elastic supply, etc. etc. etc.).
5 An example of an inferior good we've discusd is Ramen noodles. Sure, they’re cheap and you can live off them for a little while. But if your income suddenly tripled, chances are you aren’t going to increa your Ramen consumption. Instead, you’ll probably start eating more “real” food, and less Ramen.
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