In this post we go over the economics of monopoly
pricing. We start with a demand function
and a total cost function, and are able to figure out the necessary
calculations to get to equilibrium quantity and price.
Summary:
1) We need to equate
marginal revenue (MR) to marginal cost (MC) and in order to do this we need to
figure out what the MR and MC functions are.
If these are known already, skip to step 4.
2) To get the MR function, we need to double the slope of the
inverse demand curve (make it twice as steep).
Because of the mathematical relationship between demand and revenue,
this is appropriate.
3) To get the MC function, we need to take the derivative of
the total cost function with respect to quantity. Another way to get marginal cost is to find
the slope of the total cost curve (if the TC curve is linear, the MC curve will
be horizontal).
4) Now set MC=MR and solve for Q, this will give us the
equilibrium quantity associated with the monopoly.
5) First, find the average total cost by calculating total
cost and dividing by quantity. Use this
equilibrium quantity with the demand function to figure out what the price paid
by the consumer is.
6) To find the monopolist’s profit you need to multiply the
equilibrium quantity by the difference between the monopolist’s cost (what we
found by plugging Q into MC or MR) and the price charged to the consumers
(found by plugging Q into the demand function).
7) Optionally, the deadweight loss can be found by multiplying
½ by the difference between cost and price, times the difference in quantity between
the monopoly Q and the perfectly competitive Q (which is found by equating the
MC function with the demand function).
Wow, huge summary, here is an example:
The demand function the monopoly faces is D(p) = 10 – 3p,
and the cost function is C(q) = 2q.
In order to get our marginal revenue function, we need to
double the slope of the inverse demand curve, so first we need an inverse
demand curve. We can get this by solving
our demand curve for p.
Qd (quantity demanded) = 10 -3p and we add 3p to both sides,
subtract Qd from both sides, then divide both sides by 3 to get:
P = 10/3 – Qd/3 which is our inverse demand function
(because Price is now a function of quantity)
If we double the slope of this curve, we will get marginal
revenue, so our marginal revenue curve will be:
MR = 10/3 -2Qd/3
We now need to find our marginal cost equation which is equal
to the derivative or slope of the total cost curve (with respect to q). Since our total cost function is 2q, our
marginal cost is going to be 2.
We set our MR = MC and we get:
10/3 – 2Qd/3 = 2 and we can subtract 2 from both sides, and
add 2Qd/3 to both sides to get
4/3 = 2Qd/3 and we can then multiply both sides by 3, and
divide both sides by 2 to get:
Qd = 2 so our equilibrium quantity in this scenario is going
to be 2.
Since our marginal cost is flat at 2, we know that the
average cost will be 2 as well, but we can confirm this by using the equation o
find average total cost. Since our total
cost equation is 2q, and we are producing 2 goods, our total cost is 4, and
since (again) we are producing 2 goods we divide by 2 to get an average total
cost of 2.
We plug in our equilibrium quantity of 2 into our demand
function to get:
2 = 10 – 3p and we add 3p to both sides and subtract 2 from
both sides to get
3p = 8 and we divide both sides by 3 to get
P = 8/3 which is equal to 2 2/3 which is higher than our
cost to the monopolist which was 2.
So the equilibrium price and quantity is q = 2, and p = 2
2/3 (for the consumer).
The (economic) profit for the monopoly is the difference
between price charged and average total cost (2 2/3 – 2 = 2/3) multiplied by
quantity (2) which ends up being 4/3.
The deadweight loss to the economy because of the existence of a monopoly
is ½*(2/3*2) which is because we are trying to find the area of the triangle
where the height is the difference between MC and MB, and the base is the
difference between quantity supplied in the monopoly market vs the perfectly
competitive one. This gives us an answer
of 2/3.
