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taxes and supply and demand

The supply curve for a certain industry is given by S(p) = pāˆ’2 and demand by D(p) = 30 āˆ’ 3p.

What is the total tax revenue and the producer and consumer surplus under a $2 tax? What is the deadweight loss (DWL)?

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Andy C. | Math/Physics TutorMath/Physics Tutor
4.9 4.9 (21 lesson ratings) (21)
Well the tax hike was $2.
So the tax in terms of price p is 2/p.
The revenue would then be the 2/p * R(x) where R is the profit function before taxes.
Now here is where I have trouble because I am used to plotting the supply and demand curves
with the price on the x - axis and the qty on the y-axis.  Price affects the supply and demand,
not the other way around, at least not entirely. Most of the examples I read graph it the opposite.
But from what I read, it is that area of the triangle thingy again. It goes something like this:
  Consumer surplus is the area of the triangle above the equilibrium price but below the demand line.
  The equilibrium price is when S(p)=D(p) ---> p - 2 = 30 - 3p
                                                                   4p = 32
                                                                      p = 8.
   So the equilibrium price is $8 and the ideal (equilibrium) inventory qty is 8 -2 = 30 - 3(8) = 6
   So keeping 6 of them on the shelf at all times will sell steady at $8 each.
 The price must start at $2 , so that the supply is not zero or negative.
  That makes the base of the triangle 8-2 or 6.
  The demand changes from D(2)= 30 - 3(2) = 30 - 6 = 24           to
                                         D(8) = 30 - 3(8) = 30 -24 = 6
    for a whopping change of 18 in customer demand over the $6 price hike.
 That would make the height of the rectangle 18.
  The consumer surplus is the the area of this rectangle which is 1/2*6*18 = 54.
  So 54 customers are out there looking to buy from the competition. s m h. sad, very sad.
 For the producer surplus, it's a similar thing except it is the supply line rather than the demand
line that is the height of the triangle. The base is the same as before, that is the change in price
which is $6.
S(2) = 0  while S(8) = 8-2 = 6.  So there are 6 more vendors willing to hike the price $6.
As a result, 1/2*6*6 = 18. So 18 vendors are reluctant to raise the price at all, or not as much.
I have already done the DWL for you in the last problem.