Monopoly Pricing

Our next application is monopoly pricing and since we've seen that in the elasticity spreadsheets we're going do that one a little bit quickly. But I'd like to work through the algebra that I gave short shrift to the last time. So let's look at algebra over here. We're looking at a linear demand curve. The price intercept is "A". The slope of the demand is minus "B". So the equation of the demand curve is P equals A minus B times Q. "Q" is the output. Total revenue, which is price times quantity, is therefore A times Q minus B Q squared. That's price times quantity. Marginal revenue is the derivative of total revenue with respect to output. So that if I differentiate this with respect Q I get A minus 2 B Q, cause this is a quadratic term here so I get the coefficient -2 in front here. Marginal cost is a linear function in output and it's..... we're going to assume it's an upward sloping linear function. So now let's take a look at the graph. What the graph does is here is the marginal cost curve. This is the marginal revenue curve. It's twice as steep as the demand curve. The monopoly output is where marginal cost crosses marginal revenue. And when you have that output you go back up to the demand curve to determine the monopoly price. So the monopoly price is 65.89 at these values and the the output is 31.08. I also have indicated in the diagram where marginal cost crosses demand. That is at 50, 50 just like it was in the previous examples. So that explains monopoly pricing. Then you can look at the impact of changing the demand elasticity at this point. Let's make it more elastic at this point. The Lerner index falls. The Monopoly price falls. And you can make marginal cost more elastic which actually is raising marginal cost at all output less than 50. And because marginal cost is going up the price is actually rising. But the Lerner index is falling. So marginal cost is going up if you will faster than the price. So you can get a sense of how the monopoly solution changes as you change the demand elasticity or the supply elasticity around the point where price equals marginal cost.