Consumer and Producer Surplus


Uploaded by TheIntegralCALC on 25.07.2011

Transcript:
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!