An economics application elasticity of demand. Our goals are to find
the elasticity of demand function, find the maximum of a total revenue function,
and characterize demand in terms of elasticity. Retailers and manufacturers often need to know
how a small change in price will affect the demand for a product. For example, if a small
increase in price produces no change in demand, a price increase will increase revenue. If a small
increase in price creates a drop in demand, the increase will probably decrease revenue.
In order to measure the sensitivity of demand to an increase in price,
we use what's called elasticity of demand. The elasticity of demand e is given as a function
of price x by elasticity of x is equal to x times the derivative of the demand function
divided by the demand function. This formula can be derived from the ratio of the percent
change in quantity to percent change in price, which I will show on the next slide.
Now these next three components are the most important ideas you need to be familiar with
when we talk about elasticity. The total revenue is increasing at those x values for
which the elasticity is less than one. Demand is inelastic. An increase in price will bring
an increase in revenue. If demand is inelastic, then revenue is increasing. Total revenue is
decreasing at those x values for which the elasticity is greater than one. Demand is
elastic. An increase in price will bring a decrease in revenue. If demand is elastic,
then revenue is decreasing. Total revenue is maximized at the values for which e of x is
equal to one. Demand is unit elastic. The demand has unit elasticity when revenue is at a maximum.
Here's kind of the derivation of the elasticity formula. I'm not going to
go through all this. You may want to pause the video and make sure this makes sense to you,
but this is how you can derive the elasticity formula starting
with the ratio of the percent change in quantity to the percent change in price.
Let's take a look at some of the price elasticities in the US economy. On top,
we have items listed that are elastic demands, and below we have inelastic demands. Remember
when e is greater than one, an increase in price will bring a decrease in revenue. So for example,
if we take a look at furniture listed at one point twenty-six, we're saying if there's an
increase in price, revenue will decrease. Probably because if the price of furniture goes too high,
we will stop buying it or we'll buy less of it.
Down below, when we have an elastic demand, for example, clothing and housing,
those are items where if the price goes up, we're still going to have to purchase them. We
have to have clothes. We have to have a place to live. That's why if the price increases,
you can expect an increase in revenue. Remember that revenue is maximized when the elasticity is
equal to one, so you can see the item here that are very close to the value of one.
Okay, let's go ahead and take a look at a couple of examples. A company found the demand for its
product to be q equals d of x which is equal to four hundred minus x. Find the elasticity
function. Okay, so let's go ahead and write our elasticity function again so we have it here. So
it's x times the derivative of the demand function divided by the demand function.
Okay, so for this problem, it's pretty straightforward. One,
we have x times the derivative of the demand function which we can see here
would be negative one divided by the demand function four hundred minus x.
Simplifying notice we have x in the numerator and four hundred minus x in the denominator.
Find the elasticity at the price x equals one hundred twenty-four. Determine if demand
is elastic or inelastic and interpret. So we need to find the elasticity when x is
one hundred twenty-four. Pretty straightforward. I've already calculated this value to save
time. It's zero point four four nine. So the conclusion here is that demand is inelastic,
which means if we increase price, I'm going to use some abbreviations here.
If we increase price, we can expect to have an increase in revenue.
We could also say that revenue is increasing. Find the price or value
of x for which revenue is a maximum. Remember that revenue is maximized
when elasticity is equal to one. So what we can do is set our elasticity function equal
to one and solve. There's our elasticity function. We set it equal to one and solve.
There's several ways of doing this. I'm going to go ahead and put one over one,
and then I'll do cross product. So I have x must equal four hundred minus x times
one, four hundred minus x. Solving this equation, I'd add x to both sides. Two x
is equal to four hundred. Dividing by two, we can see that x will equal two hundred.
So the price should be set at two hundred dollars to maximize revenue. As you can see from part b,
when it was one hundred twenty-four, it was inelastic meaning we should increase price,
and in fact, we should increase it to two hundred dollars.