Calculating the deadweight loss from a subsidy

This post goes over the economics of a deadweight loss causes by a subsidy.  For information on deadweight loss look here.  This part of economics is fairly algebra intensive and the trick to solving these problems is knowing how to manipulate the demand and supply functions to get what you want.  After that trick, it is a simple exercise in algebra to find equilibrium price and quantity.  Here is the question in the context of bio-fuels:
Suppose demand for bio-fuels is given as Qd=420-30p and supply is Qs=-44+24p.  What is dead weight loss created by a subsidy of $3.87 per unit paid to supplier?

(The subsidy inclusive price received by suppliers is $3.87 higher than the paid price paid by consumers)
To solve this problem we need to follow these steps:
  1. Calculate equilibrium price and quantity without the subsidy.
  2. Calculate equilibrium price and quantity with the subsidy.
  3. Figure out the base and height of the resulting triangle that represents deadweight loss.
Before I go through the associated math, let’s first look at a graph representing the problem.  We know the appropriate demand and supply functions, and we know that without the subsidy, we will be in long run equilibrium.  The addition of the subsidy will result in a higher price received by the suppliers, a lower price paid by consumers, and a higher quantity being supplied/demanded than the original market equilibrium.  Remember that a subsidy is like a reverse tax, so it INCREASES supply because essentially the cost of supplying the goods has declined.
 First, we need to find the original market equilibrium.  One way to do this is to set Qd=Qs and solve for price.  Then we can substitute that price back into our demand and supply functions to find what Qs and Qd are (they should be equal).  For more info on this process, see the article showing how to find equilibrium price and quantity.
So setting our original Qd and Qs equations equal gives us:
420-30p = 44+24p
We subtract 44 and add 30p to both sides to get:
376=54p or p =6.96 (rounded)
We then plug our p into our Qd or Qs equations and we will get about 211.1 (depending on rounding):
Qd = 420-30(6.96) = 420-208.8 = 211.2
Qs = 44+24(6.96) = 44 + 167.04 = 211.04  (Close enough given rounding)
So equilibrium quantity is 211.1, now we need to find equilibrium price and quantity given the subsidy.  This is where it gets tricky.  Since the subsidy only affects the price suppliers receive, we need to add in the subsidy to the supply equation, and keep the demand equation the way it is.  This means we have the following NEW supply equation:
Qs(subsidy) = 44+24(p+3.87)
We now set Qd equal to Qs(subsidy) and solve for price (which gives us the price paid by the consumers).
420-30p = 44 + 24(p+3.87)  => 420-30p = 44 + 24p + 92.88
Now we subtract 136.88 (44+92.88) and add 30p to both sides to get:
283.12=54p or p = 5.24 (rounded)
So this p is our price paid by consumers given the subsidy on suppliers.  The price received by sellers is this p plus the subsidy or $9.11 (5.24+3.87).  Now we plug the demand price into the demand equation to solve for Qd:
Qd = 420-30(5.24) = 420-157.2 = 262.8
And we can plug the suppliers price into the supply function to get:
Qs = 44 + 24(9.11) = 44 + 218.64 = 262.6
These are pretty close, so we can say that equilibrium quantity given the subsidy is 262.7 (because of rounding to the nearest penny before).
Now to get the deadweight loss we have to find the area of the triangle.  We know that the height of the triangle is the subsidy (3.87) and the base of the triangle is the difference between the two equilibrium quantities, meaning the one before and after the subsidy.  Since our original equilibrium quantity was 211.1 and our equilibrium with the subsidy is 262.7 we can find the difference between these two to get the base of the triangle.
262.7 – 211.1 = 51.6
Now we use the equation for finding the area of a triangle to calculate this deadweight loss.
Area of a triangle = ½ (base * height)
Deadweight loss = ½ (51.6 * 3.87) = 99.85 or about 100.
So the deadweight loss from this policy (the enacting of the subsidy) results in a deadweight loss of about $100 or whatever units the quantity happens to be in.

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