So we’ve just started talking about monopoly in my microeconomics class, which is an extreme form of a market in which a single seller that makes a good with no close substitutes dominates a market. The prime example we talked about (and that my professor got heated about) was professional sports leagues, and the unfortunate truth that at some universities, coaches of winning college teams get paid three times (or more) what the president of that university makes.

What’s really interesting about monopolies in a mathy sense is deriving the marginal revenue from the demand curve using the half way rule.

Just as a side, *marginal revenue* is the amount of money a company makes from each extra unit it produces (i.e. apple makes $100 off of every extra iPhone 5 it makes after it’s original production quantity). This is different from *average product*, which is simply the average amount of revenue that each good makes the supplier (i.e. on average, apple makes $500 for every iPhone they sell).

The *demand curve* is simply a consumer’s desire to maximize their utility, shown in graph form. As the price of a good decreases, consumers want and can buy more of that good with the same income, which is why the line has a negative slope.

Okay, so back to the half way rule. The half way rule links marginal revenue with the elasticity of demand. *Elasticity *refers to how demand reacts to a change in price. For example, a market is relatively elastic if a small change in price creates a big change in demand.

∈ = ΔQ/ΔP × P/Q (in real words, elasticity is equal to the ratio of a change in quantity of price, times the ratio of price over quantity)

Next, we’ll make elasticity positive (|∈|), as we’re relating this equation back to a previous equation, MR = P – (ΔP/ΔQ)Q.

So now we have: |∈| = ΔQ/ΔP × P/Q. We’re now going to solve this equation for ΔP/ΔQ so that we can substitute in the previous MR equation.

This gives us: MR_{Q} = P (1 – 1/|∈|).

This equation is actually really cool, because it tells us that as demand becomes less affected by price (less elastic), the difference between price and marginal revenue gets bigger and bigger.

So when we have an infinite elasticity (which means that any minute change in price creates an astronomical change in quantity demanded), 1/|∈| becomes zero, and MR = P, which is what we learned is true in a perfectly competitive market!

This equation explains the graph for marginal revenue and demand of monopoly. At zero, both demand and marginal revenue will have the same y-intercept, as there is no price change to affect elasticity when nothing is being produced (x=0).

When you start graphing points through this equation given different quantities, you create a marginal revenue line th

at is much like the demand curve; a negatively sloped straight line. But, the marginal revenue has a slope that is twice that of the given demand curve.

Put another way, P = a – bQ, and MR = a – 2bQ.

So basically, we took two equations, put them together, and found the relationship between consumer behavior and how much a company can make off of a product.

Also, if you’re having a challenging hump day, I recommend you just watch this video. It’s what got me through my morning. Cheers!