4/12/12

Monopoly math problem with a tax



This post goes over the algebraic methods necessary to solve common economics monopoly problems.  We assume that you are given a basic demand function and marginal cost function, and are asked to derive marginal revenue function and find out what the monopoly price and quantity will end up being.

First we are probably given either a demand function (solved for Q) or an inverse demand function (solved for P).  We need the inverse demand function because this gives us the slope of the demand curve (since P is on the Y axis).  Once we have the inverse demand function we can solve for the marginal revenue function by doubling the slope (making it steeper).  A past post goes over the math behind calculating monopoly equilibrium price and quantity, so I will go over another example really quickly then introduce the idea of combining demand curves and adding a tax into the mix.


First, we begin with an inverse demand function:

P = 100,000 – 20q

And a constant marginal cost function of 20,000

The first thing we need to do is find our marginal revenue function, and this will be the same as our inverse demand function except it will have double the slope, so if we double the slope of our inverse demand function we will get:

MR = 100,000 – 40q

Example of a monopoly graph
We now set our marginal revenue function equal to our marginal cost function and solve for quantity to get our equilibrium quantity (remember in economics MB (or in this case MR) always equals MC).

MR = MC = 100,000 – 40q = 20,000 so we can add 40q to both sides and subtract 20,000 from both sides to get:

80,000 = 40q now divide both sides by 40 to get:

q = 2,000 which is our equilibrium quantity under the monopoly scenario.  We can plug this quantity into our original inverse demand function to get equilibrium price:

P = 100,000 – 20(2,000) = 100,000 – 40,0000 = 60,000

So our equilibrium price is going to be 60,000.

Now what happens if the amount of consumers in the economy doubles?  We will have to aggregate the two separate demand curves.  I have found that the easiest way to do this is to solve for our original demand equation, and then divide the amount demanded by 2 (to show that it represents only half of the demand).  Then we multiply both sides by 2 to get rid of the fraction and we are left with our aggregated demand function.

For example, using the original inverse demand function above and solving for q gives us:

q = 5,000 – P/20 and we divide our q by 2 to get:
q/2 = 5,000 – P/20 now we multiply both sides by 2 to get:
q = 10,000 – P/10 which is our new aggregate demand function.

We can take this function and solve for P to get our inverse demand function back:

P = 100,000 – 10q

We then double the slope to get the marginal revenue function:

MR = 100,000 – 20q and set this equal to our marginal cost to get:

100,000 – 20q = 20,000 and solve for q to get:

q = 4,000 (note that you can get equilibrium price by plugging this amount into the new inverse demand function following the same steps as before).

Which is our new equilibrium quantity given the additional consumers.  This result is pretty cool because we simply doubled the amount of consumers and we were able to double our equilibrium quantity.  Note that this is due to the fact that we have a horizontal marginal cost curve; if supply/marginal cost was upward sloping we would NOT see equilibrium quantity double.

Finally, was if there was a tax of $2,000 per input placed on the good?  We can apply this to either the consumer or the producer and the result will be the same, check this out:

Remember that we need to set MR = MC, so what if we add the tax to the demand curve which will shift the MR curve down, we will get:

98,000 – 20q = 20,0000

Which means that we end up with an equilibrium q of 3,900.  If we place the tax on the producer (which will shift the supply curve up – decreasing it), we will get:

100,000 – 20q = 22,000

And solving for q will again give us 3,9000.  We can find equilibrium price by plugging this value into our inverse demand function above.

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