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!

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!