Hi, everyone.
Welcome back to integralcalc.com.
Today we’re going to be talking about consumer and producer surplus.
And in this particular problem we’ve been asked to find consumer and producer surplus.
But before we do that, and this would always be the case, you need to find the equilibrium
quantity and the equilibrium price.
So first we go ahead and find equilibrium quantity and price and the way that we do
that is by setting the demand and supply curves equal to one another.
So we get -0.4q + 23 and we’ll set that equal to the supply curve so 0.03q^2 + 3.
And the way we usually end up solving for q in this case is by moving everything to
one side so that this equation is set equal to zero and then factoring.
So we’ll move everything from the left side over to the right side so that our q squared
term stays positive.
So we’ll get 0.03 q^2 and that means that we’ve added 0.4q and then we’ll subtract
23 from both sides and get a minus 23 + 3, which gives us a minus 20.
Now I’m going to go ahead and make this a little bit easier for ourselves and multiply
through by 100 so that we eliminate the decimals.
So (0.03)(100) = 3q^2.
(0.4)(100) = 40q and then (-20)(100) = -2000.
And now if we factor this, we’ll end up with 3q and I’m just going to go ahead and
put my factors in.
And then 2000, we can factor it as 20 and 100.
So we’d end up with minus 20 and plus 100.
So that would give us 3q^2 - 60q + 100q – 2000 = 3q^2 + 40q – 2000.
So now we can solve each of these factors separately for q.
If we solve the first one, 3q + 100 = 0, we would subtract the 100 from both sides and
then divide by 3 so we q = -100/3.
And if we solve this factor for q, obviously we’ll get q = 20.
Because quantity can’t be negative, this solution is not feasible.
So our only solution is q = 20.
So q = 20 is the equilibrium quantity.
Now we can use this, plug it back into either the demand or the supply equation and get
the equilibrium price.
So let’s go ahead and plug it back into the demand equation so we’ll say the demand
equation when we plug in 20 for q is going to give us -0.4(20) + 23.
So -0.4(20) is going to be -8 + 23 gives us 15.
So equilibrium price is going to be equal to 15.
So those are our equilibrium quantity and equilibrium price.
And now that we have those, we can go ahead and solve for consumer surplus and producer
surplus.
So let’s go ahead and start with consumer surplus.
I’ve got the formulas up here for consumer surplus and producer surplus.
And if we go ahead and find consumer surplus, we’re just plugging in equilibrium quantity
and price and the demand function into the formula.
So you can see that we’re going to integrate from zero to the equilibrium quantity which
we already know is 20.
And then we’re going to plug in the demand equation so -0.4q + 23 dq.
So that’s the end of our integral but then we have to subtract equilibrium quantity times
equilibrium price.
20(15) = 300 so we’ll end up subtracting 300.
And when we integrate, we’ll add one to the exponent here and then divide by the new
exponent which is going to be 2 so we end up with a -0.2q squared plus 23q.
And we’re going to evaluate that on the range 0 to 20 and then we’re going to subtract
300 from that.
So remember with a definite integral, we plug in the top number first.
So we’ll get 0.2 (20)^2 which is going to be 400 + 23(20).
And then we would subtract from that what we get when we plug in 0.
But if we plug in 0, obviously this term will be zero so we really just need to subtract
300.
And when we end up doing this on or calculator, we’ll end up with consumer surplus equal
to 80.
So that’s our consumer surplus.
And now we just need to find producer surplus.
So to do
find producer surplus, we will multiply quantity by equilibrium price which we already know
to be 300.
And then we’ll subtract from that the integral from 0 to 20 and then we plug in the supply
curve.
So the supply curve is 0.03q squared + 3 dq.
So now, we can integrate.
300 minus and we’ll go ahead and draw a bracket here because we only are going to
integrate from 0 to 20.
So for the integral, we’ll end up with 0.01q^3 + 3q on the range 0 to 20.
So we’ll end up with 300 minus 0.01(20)^3 + 3(20).
And normally we would subtract what we get when we lug in zero but again we’ll get
zero so we don’t really need to say minus zero.
When we do this out on our calculator, we will find that the producer surplus is going
to be equal to 160.
So that’s it.
That’s how you find the equilibrium quantity and price and then use those to find consumer
surplus and producer surplus.
So I hope that video helped you guys and I will see you in the next one.
Bye!