Welcome to an example of producer surplus. We're given that x is the number of items. The demand
function is d of x, and the supply function is s of x. Number one, we want to determine the
equilibrium point, and then number two, determine the producer surplus at the equilibrium point.
Let's begin by reviewing producer surplus. The benefit a producer receives when supplying
more units at a price that is higher than a price at which fewer units would be supplied
is called producer surplus. One way to think of this was if you were selling a used car
and you said you wanted to sell the car for two thousand five hundred dollars, but you
listed the car at three thousand dollars, and it actually sold at three thousand dollars. The
extra five hundred dollars that you received unexpectedly could be considered producer surplus.
If we take a look at the supply function on the coordinate plane, as we see here where we have
quantity on the horizontal axis and price on the vertical axis, if the point q comma
p is the equilibrium point, this area here would represent the producer surplus. Notice how there
are producers that would have been willing to supply the product at a lower price, but it's
actually selling at a higher price. Therefore, the extra money received is producer surplus.
Producer surplus is equal to q times p minus the integral of s of x from zero to q. So,
we're finding the area of this rectangle here, which is q times p, and then subtracting the
area under the supply function from zero to q, leaving us with producer surplus.
So, going back to our example, the first step is to find the equilibrium point, which is the point
where the supply and demand are equal to each other. So, if we want to determine where d of x
equals s of x, that would occur when four hundred minus zero point five x equals zero point three x.
To solve for x, let's add zero point five x to both sides, giving us
four hundred equals zero point eight x. Dividing both sides by zero point eight, we have x equals
four hundred divided by zero point eight, or five hundred. So, five hundred would be the quantity
of the equilibrium point. But we still have to find the equilibrium price. To do this, we can use
either the supply function or the demand function. Let's go ahead and use the supply function.
s of five hundred would be equal to zero point three times five hundred, or one hundred fifty. So
the equilibrium point is five hundred comma one hundred fifty, where again,
five hundred is the equilibrium quantity and one hundred fifty is the equilibrium price.
Now, before we find the producer surplus, let's look at the graph of d of x and s
of x on the coordinate plane. The red line is the demand function d of x,
and the blue line is the supply function s of x. We just found this point here,
the point of equilibrium where the quantity is five hundred and the
price is one hundred fifty dollars. This shaded area here is the producer surplus.
So, we'll find the area of this rectangle and subtract the area under the supply function
over the interval from zero to five hundred. The producer surplus is going to be equal to
q times p, which is five hundred times one hundred fifty, minus the integral of s of x,
or zero point three x, from zero to five hundred.
Five hundred times one hundred fifty is seventy-five thousand, so we have
seventy-five thousand, and then minus the definite integral. To evaluate this,
we'll first find the antiderivative of zero point three x, which would be zero point three times x
squared divided by two. Well, zero point three times one-half would be zero point one five.
So now we'll evaluate this at five hundred and then zero and then find the difference.
So when x is five hundred, we'll have zero point one five times five hundred squared,
minus when x is zero, of course this is zero.
So this is going to be seventy-five thousand minus this comes out to thirty-seven thousand
five hundred, and this difference is thirty-seven thousand five hundred, which means the producer
surplus is thirty-seven thousand five hundred dollars. I hope you found this helpful.