Currently the UK is looking to implement a windfall profit tax; i.e. taxing excess profits being earned by oil companies. Similar ideas seem to be floating around elsewhere.
I was reminded of an inflation penalty that I had read about. This was part of a small part of a system for handling inflation called “market anti-inflation plan” which I will not go into today. The authors did mention something akin to a profit tax/penalty, but declined to use it for reasons that I do not recall (I believe it was because they did not want to penalize changes in firm cost reduction efficiencies). Instead they advocate for another penalty, which I will call a “price increase penalty”. (It’s highly possible that it has another name in the literature, given it’s straightforwardness, however the authors just present it in math notation).
The penalty ends up modifying the firm’s profit equation:
Where TR is the total revenue, TC is the total cost, and the penalty is a new term added to the profit equation. The form of the penalty is up to you, so you could place a profit tax here, but I will be looking at a price increase tax which looks like this:
Here I’ve expanded the total revenue into price*quantity as usual. I show the total cost in the equation, but it’s not a focus of this post. I will be setting it to 0 as in Part I.
The elements of the penalty are as follows:
- k: this is just a number (usually between 0 and 1) which effectively dictates the relative power of the penalty, which I will go into later. For this post I will consider this value as fixed, but you should know that there are methods where k can vary freely, such as by a market mechanism.
- Q*P(Q): This is just the same as total revenue
- Q*P-bar: P-bar (the P with the bar above it) is the previous price which you want to penalize the company from straying too far from.
Effectively the penalty taxes the firm by some factor of the amount of total revenue it earned from increasing the price of oil.
You can rearrange terms and combine the penalty terms with the original total revenue. This will help with graphing the equations:
You now effectively have a “new” total revenue term, since it consists a of “new price” multiplied by Q, similar to how total revenue is usually defined:
So now we can graph these equations to see how the penalty will behave.
Say we use these curves for the firm’s profit equation (same as in Part I):
As seen in Part I, the curves result in an optimal price of $2.5 and a proportion of total customers of 37%.
Say we had previously seen oil sell for $1.92 with oil sold to 47% of customers (as seen in Part I with the competitor), and we want to penalize the firm for increasing the price above that value: we set P-bar to $1.92. Here is what the graph will look like for different k values once we set P-bar to $1.92.
I will be using red for the demand curve with the penalty active (Pnew), and yellow for the marginal revenue curve with the penalty active.
At k = 0:
When k=0 the new demand and marginal revenue curves are exactly the same as the un-penalized curves. This makes sense since when k=0, the penalty term vanishes and the profit equation goes back to being un-penalized.
At k = 0.23:
I picked this k value because in this case it leads to an optimal price of $1.92 with a 47% of customers served. This close to the same optimum that we obtained with the competitor/price control in Part I. We can also see what the penalty does to the curves: it skews and rotates them. By doing this, we can see there’s sometimes optimal settings which will both lower the profits earned by the oil company while also increasing the amount of oil supplied.
You may have noticed something peculiar about these curves which did not occur with a price control. The next k setting will highlight it.
At k = 1:
It may be hard to see in this case, but the marginal revenue curve lies underneath the red demand curve. This is just due to the definition of the marginal revenue and it will always behave like this when the demand curve is horizontal/flat.
The reason the curve is flat is because at k=1, the two Q*P(Q) terms (one from the original total revenue, one from the penalty) will cancel each other out and you are just left with Q*P-bar. P-bar is a fixed value, so you just end up with a flat curve.
Notice how at a certain point the red demand curve is actually higher than the un-penalized blue demand curve. This never happened the price control. It is important to go over what each of these curves mean in order to understand what’s happening.
The blue demand curve corresponds to the price per unit of oil that the customers will pay. With the penalty active, the customers will still pay the price as dictated by the blue curve. The red demand curve corresponds to the price per unit of oil sold that the company will take in as revenue. The difference between the red and the blue is due to the penalty, it is a tax that takes revenue away from the oil company. When the blue curve is above the red curve, customers are being charged high prices, and the company is being penalized, so they will only take in revenue per unit of oil as dictated by the red demand curve.
However, the penalty can actually go negative. For example, P-bar was set to $1.92. What happens if the oil company starts charging customers less than $1.92 per unit of oil? “Q*P(Q) — Q*P-bar” becomes negative, and thus money is not taken away from the firm, instead money is given to the firm. You can think of it now as a subsidy to the oil company in order to keep prices low.
Here are the resulting optimal values when k=1 (referring back to the k=1 graph above):
- Oil is sold to 100% of customers.
- The price that customers pay per unit of oil is $0.
- The amount that the oil company receives per unit of oil is $1.92. This is effectively all paid for by the government through the “penalty”.
Note that on it’s face it may seem that the penalty becoming negative is a bad outcome, this is not necessarily true. It’s true that the total revenue/profit of the oil firm is higher, but the revenue per unit of oil is lower than without the penalty. As stated above, the price would be $2.5 without the penalty.
So it’s up to you to decide whether this is desirable outcome. It’s not necessarily bad in a vacuum.
Lastly, you may be wondering what it would look like if we remove the possibility that the penalty can go negative. As an extra, I will show you what that looks like. In the following cases, the penalty term can never drop below zero.
At k = 1:
Here the red demand curve is flat up until the blue demand curve reaches $1.92 (the P-bar) value. After that the penalty is no longer active (it is zero, and cannot go lower), so the red and blue demand curves are the same. In this way the penalty with k=1 behaves just like a price ceiling.
At k = 0.23:
Similar intuition explains this graph. Once the price hits P-bar, the penalty is deactivated so the curves match the unpenalized curves.
Neat huh?
Along with price controls, this price increase penalty gives us another method of manipulating the firm’s profit equation to achieve desired outcomes. It shows that it is possible (in certain cases) to reduce profits and price while increasing supply.
