3/16/12

How to find monopoly price and quantity



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.





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