Given the supply and demand functions, we've now found the equilibrium point as well as
the consumer surplus. We want to finish by determining the producer surplus given
the same demand and supply functions and equilibrium point. The formula
for producer surplus is given here, but before we calculate this, let's review this concept.
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. Looking at this
red supply function, if a producer was willing to supply a certain amount of items at a lower price,
but they actually sold the items at a higher price, it makes sense there’s going to be a
benefit to the producer. We can measure this benefit by determining the producer's surplus.
So if this is our point of equilibrium where the x-coordinate represents the quantity sold and the
y-coordinate represents the price, the area of this rectangle here would represent the total
revenue received from selling all of those items, and the area of this rectangle would be q times p.
But then if we were to subtract out the area under the supply function on the same interval,
or this area here, this remaining area in blue would represent the producer surplus,
which would be the extra revenue received for not having to sell these items at a lower price.
And this revenue would contribute to profit. On a more personal level,
to get a better understanding of what this means, let's say you're trying to
sell an old computer and you decide you're not willing to sell it for less than one
hundred dollars. If someone comes along and offers you one hundred fifty dollars, the
extra fifty dollars that you receive could be considered producer surplus.
Now let's go back and take a look at our example. So the producer surplus is going to be equal to
q times p, or one thousand six hundred times eighty, minus the integral from zero to q,
or from zero to one thousand six hundred. Our integrand is going to be the supply function,
so I write this as two times x raised to the power of one twelfth dx.
So we'll have one thousand six hundred times eighty minus this is going to be
two times x raised to the power of twelve plus one. That's three,
so it's x raised to the power of three, divided by three, but we'll multiply by
two-thirds instead. We'll evaluate this at one thousand six hundred and zero.
One thousand six hundred times eighty is one hundred twenty-eight thousand, minus this is
going to be four-thirds, and then we'll replace x with one thousand six hundred, so we'll have one
thousand six hundred raised to the power of three, minus this will be zero to the power of three.
So we'll have one hundred twenty-eight thousand minus four-thirds times one
thousand six hundred raised to the power of three halves is sixty-four thousand. So we have
one hundred twenty-eight thousand minus, let's round this to the nearest penny,
so it's going to be eighty-five thousand three hundred thirty-three point thirty-three.
So our producer surplus will be this difference, which will be forty-two
thousand six hundred sixty-six point sixty-seven. And again, this represents the extra revenue
received for selling more items at a higher price and will contribute to their profit.