Welcome to Unit Demand's first blog post! Today we're learning about how a firm's optimal profit can be found with **price elasticity of demand** by utilizing the Lerner index, named after Economist Abba Lerner. Price Elasticity of demand is a metric that Unit Demand currently tracks for free. You can find more details here.

In economics, it's taught that you are a monopolist if your **marginal cost** (MC) of producing a product is equal to your **marginal revenue** (MR). In plain english, this is equation implies that you are at an optimal level of production when **MC=MR**, and you are **maximizing profit**. You are extracting all profit from the market (contingent on your production function).

Make sure you remember this identity, as it is crucial for the rest of the equation, and profit optimization/economics in general.

The Lerner Index measures a firm's monopoly power. It ranges from 0 to 1, with 0 implying that a firm has no monopoly power (operating under perfect competiiton) and 1 implying that a firm has complete monopoly power.

It's simple to derive the Lerner Index, first we'll show the formal equation, then we'll dig into the intuition.

The numbers say that the Lerner Index (L) can be found by the difference in your offered price for a unit, and the cost it takes you to make it (P - MC) over the priced offered (P). Assuming that your company is rational and accounts for the long-run, your marginal cost will be bound from 0 to P.

Therefore, when your marginal cost to produce a product is very close to the price that you sell it at, the Lerner Index gets closer to 0, and you have very little market power. Conversely, if you have an extreme comparative advantage, and you can produce an additional unit at no cost (think industries like software) the Lerner Index rules in your favor -- congratulations, you are a monopolist. Enjoy your profits.

With an understanding of what defines a monopolist (MR=MC) and a quantitative index of how to measure a monopolist in the market (Lerner Index) -- it's time to introduce a special underlying relationship that can perfect your product pricing.

Hold on tight.

While this looks scary, it's simply an identity in economics -- something that always holds true. We see our old friend, the Lerner Index on the left side of the equation. In the context of this equation, it is called the Lerner Rule. Quite simply, in order to maximize profits, you must fill in all of the variables from the left side of the equation to match the right-side of the equation.

The right side of the equation is the inverse of price elasticity of demand. Price Elasticity of demand measures **how quantitiy changes in response to price changes**. This is a core metric that Unit Demand tracks, so a quick refresher: it ranges from 0 to infinity. If it is less than 1, it means that changing your product price by 1% will yield a change in quantitiy demanded of less than 1%. This is fantastic, as there is room for you to raise prices. The other side of the coin entails that a price elasticity of demand greater than 1 means that raising the product price by 1% will yield a change in quantity greater than 1%, which means that your customers are price sensitive, and the margin for profit is slim. Price Elasticity of demand is negative, so -1/Ed lets us get what is the absolute value.

The right side of the equation is the inverse of price elasticity of demand. Price Elasticity of demand measures **how quantitiy changes in response to price changes**. This is a core metric that Unit Demand tracks, so a quick refresher: it ranges from 0 to infinity. If it is less than 1, it means that changing your product price by 1% will yield a change in quantitiy demanded of less than 1%. This is fantastic, as there is room for you to raise prices. The other side of the coin entails that a price elasticity of demand greater than 1 means that raising the product price by 1% will yield a change in quantity greater than 1%, which means that your customers are price sensitive, and the margin for profit is slim. Price Elasticity of demand is negative, so -1/Ed lets us get what is the absolute value.

-1/Ed, when price elasticity of demand (Ed) is known, is just a constant, so the left side of the equation (the Lerner Index) becomes crucial, as those variables are core to your company. Luckly, with some algebra, we can rearrange those variables to maximize our profit.

Here it is, ladies and gentlemen, the fruit of your hard labor in reading this blog post. To find the optimal profit-maximizing price of a product for your firm, you should know two things: what it costs to make a unit of the product (marginal cost) and the price elasticity of demand for the product (Ed). **Plug those variables into the right side of the equation, and out comes profit**.

The above equation is golden since it is derived from rearranging our original monopolist identity of MR=MC. Imaging taking your firm's profit function: R(Q) - C(Q), where R is the revenue function that takes in quantity, and C is the cost function that takes in quantity. When you take the first derivative, you would find that optimal price would be equal to plugging in those two variables (marginal cost and elasticitiy of demand) into the optimal price equation described before. The beauty of the optimal price equation is that the variables that it requires may be more readily found through the use of econometrics software, reducing your headaches and increasing your profits.

Are you interested in finding the price elasticity of demand for your catalogue of products? Give Unit Demand a shot, check out what we offer **here**.