An economics application: consumer surplus and producer surplus. Our goal today is, given demand
and supply functions, to find the consumer surplus and producer surplus at the equilibrium point.
The consumer's demand curve, p equals d of x, gives the price per unit
that the consumer is willing to pay for x units.
It is usually a decreasing function. Utility is a function often considered in economics.
When a consumer receives x units of a product, a certain amount of pleasure or utility, u, is
derived. Consumer surplus is the utility received by consumers for which there is no payment.
Graphically, consumer surplus is the area of the orange region on the graph below. So what we have
here is a demand curve as a decreasing function. p is the equilibrium point, so this is the price
the market is charging, and the consumer surplus is this orange region here. You can kind of think
of consumer surplus like this: let's say you want to go buy a new cell phone or a new iPod,
and you're willing to pay one hundred dollars for it, but the market is only charging fifty dollars.
You think you're getting a smoking deal and you're getting about fifty dollars
worth of pleasure from purchasing it for fifty dollars rather than one hundred.
Therefore, what you're feeling is what you can consider consumer surplus.
If you consider the point q, p, the blue region represents the amount paid for x units,
and again the orange region is your consumer surplus. So in order to find the orange region,
what we have to do is take the integral of the demand function
from zero to q and then subtract out the area of this rectangle, which would be q times p.
Now, the producer surplus curve, p equals s of x,
gives the price per unit that the seller is willing to accept when supplying x units. It
is usually an increasing function. The benefit a producer receives when supplying more units
at a price that is higher than a price at which fewer units will be supplied is called producer
surplus. Graphically, the green area below represents the producer surplus.
So the supply curve here is increasing. As price
increases, more and more producers are willing to supply the product.
Let's suppose that a producer is willing to supply our product to market for forty dollars,
but the market price is actually fifty dollars. Then they are receiving ten dollars worth of
producer surplus. If you consider the point q, p, the yellow region is the total cost per person,
the green region is the producer surplus, or the surplus over cost. It is a contribution to profit.
The area of the yellow and green regions represents the total receipts.
So in order to find the area of this green region, we have to take the area of the entire rectangle,
q times p, and subtract out the area under the curve in yellow.
The equilibrium point, x, comma p, is the point at which the supply and demand curves intersect. It
is at that point at which sellers and buyers come together, and purchases and sales actually occur.
So this graph combines both the graphics of the consumer surplus and the producer surplus. Often
it's a good idea to graph both the demand curve and the supply curve to see their relationship.
To find the equilibrium point, we need to set the supply and demand
functions equal to each other and solve for x. To find the price p, we need to substitute
x into either the supply or demand functions. Since we are finding the point of intersection,
let's go ahead and take a look at an example. d of x is the price that consumers are willing to pay
for x units of an item, and s of x is the price that producers are willing to accept for x units.
Find the equilibrium point. To start this off, we know the equilibrium point
occurs when the supply function is equal to the demand function. So in order to find that
point of equilibrium, we have to set x minus four squared equal to x squared plus two x plus six.
So let's go ahead and expand this and solve for x.
It works out pretty well for us because the x squared terms
would simplify out, solving this for x. Now, maybe I would subtract two x on both sides,
and then I'm also going to subtract sixteen at the same time. So what we'd have here on the left:
negative ten x is equal to looks like negative ten. Therefore, x is equal to one.
Remember, the equilibrium point consists of two coordinates: the quantity x and then the price.
Now, in order to find the price, we need to substitute x equals one into one of these
functions. Here, I'll go ahead and do it mentally. One minus four would be negative three. Squared
would give us nine, so nine dollars. Remember that each of these functions we could set equal to p.
Okay, so our equilibrium point is the point one, nine.
What we want to do now is find the consumer surplus at the equilibrium point. So again, our
equilibrium point is quantity of one and a price of nine dollars. Now we don't know the units of x.
One item seems obviously very low. One could be in the hundreds or thousands or millions,
as far as we know, but we'll go ahead and just use the value of one that we found.
Find the consumer surplus at the equilibrium point. So here's our formula for consumer surplus.
Let's go ahead and fill in the information. We're going to integrate from zero to one of the demand
function, which was x minus four squared from the previous screen, and then we're going to
subtract out the area of that rectangle, q times p, which of course is going to be one times nine.
Okay, in order to integrate this, we are going to have to expand this.
Now we'll integrate,
and we need to evaluate this at zero and one and then subtract nine. To save a little bit of time,
I've already done this, and we're going to get three dollars and thirty-three cents.
Let's verify this on our graphing calculators. Okay, so remember, if you hit math
and then option nine, function integration, we can integrate x minus four squared
with respect to x from zero to one and then subtract nine,
which verifies our answer. So the consumer is getting three dollars
and thirty-three cents worth of utility by purchasing the item at nine dollars.
Now we'll take a look at it from the producer side. We want to find the producer surplus
at this equilibrium point. So we have a formula:
qp minus the integral of the supply function from zero to q. So let's go ahead and do that.
Our producer surplus is going to be equal to one times nine minus the integral, again from zero
to one of our supply function, which was given on the previous screen as x squared plus two x plus
six. Okay, let's show one more step, and then we'll verify it with the graphing calculator.
So, of course, one times nine is nine minus, now we subtract this
entire integral. So now we need to find the indefinite integral and then evaluate it at
one and zero. I'm not going to take the time to simplify this and do the substitution;
I did this ahead of time, and it came out to one point sixty-seven.
However, let's go ahead and verify that on the graphing calculator as well.
Okay, so we'd have nine minus, remember it was math, option nine,
and then we can go ahead and type in our integrand
with respect to x from zero to one, and again that verifies our solution.
Remember, this is the utility that's received by the producer;
that is a contribution to profit. I hope that helps explain a little bit about consumer surplus,
producer surplus, and what it all means. Thank you for tuning in, and have a good day.