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)

(The subsidy inclusive price received by suppliers is $3.87 higher than the paid price paid by consumers)

- Calculate equilibrium price and quantity without the subsidy.
- Calculate equilibrium price and quantity with the subsidy.
- 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.