Cournot competitionunder a binding price ceiling: equilibrium structure and exact pass-through

In the world of economics, there is a classic scenario where a few companies compete by deciding how much of a product to make. This is known as Cournot competition. Imagine a group of fuel importers in a landlocked country. Each company has a physical limit on how much fuel it can bring across the border in a month, and each has its own costs for doing so. They all watch the market price, which drops as the total amount of fuel available increases. If they act purely in their own self-interest, they will each choose a production level that maximizes their profit, given what the others are doing. Usually, economists can predict exactly what will happen: there is one specific amount of fuel that will be sold, and one specific way that amount will be split among the companies.

However, governments sometimes step in to set a maximum price, or a price ceiling, to keep costs down for consumers. When this ceiling is set low enough to actually matter, it changes the game entirely. The price can no longer rise above that limit, no matter how much fuel is demanded. This creates a sharp, sudden break in the rules of the game. The usual mathematical tools that economists use to find the solution break down because the profit curve for each company becomes jagged at that exact price point. For a long time, it was unclear what would happen in this situation. Would the companies produce more or less? Would there be one clear outcome, or many? And if a government decided to release extra fuel from a national reserve, how much of that would actually reach the market?

A researcher at the University of Finance and Economics in Mongolia has now mapped out exactly what happens in this complex scenario. By studying a model where companies face both production limits and a strict price cap, the study reveals that the usual certainty of a single solution disappears. Instead of finding one specific way to split the market, the companies can settle on a wide range of different production plans, all of which are equally stable and profitable. The total amount of fuel sold is fixed and predictable, but the share each company gets is not. The study also shows that when the government adds extra fuel to the market, the result is an all-or-nothing situation: either the private companies pull back their own production to make room for the new supply, leaving the total amount unchanged, or they do not pull back at all, and the total supply increases by the full amount of the new fuel. There is no middle ground.

The researcher began by looking at the shape of the companies' profits. In a normal market, a company's profit curve is smooth, like a gentle hill, making it easy to find the peak where profit is highest. But when a price ceiling is active, the curve gets a sharp corner, or a kink, right where the equilibrium sits. The researcher proved that even with this jagged edge, the curve still holds a property that guarantees a solution exists. The profit function remains shaped in a way that prevents wild, unpredictable swings, ensuring that the market will always settle down into a stable state, even if that state is not unique.

The most striking discovery concerns the nature of that stable state when the price ceiling is binding. The study found that the market clears at the exact price limit, and the total quantity sold is determined by the point where the demand curve hits that price cap. However, the way this total quantity is divided among the companies is not fixed. As long as each company produces an amount between two specific boundaries—one based on what they would produce if they ignored their effect on the price, and another based on what they would produce if they acted strategically—they are all happy. This creates a whole family of possible outcomes. In a specific example calculated in the study involving four fuel importers, the total supply was fixed at roughly 154,170 tonnes per month. Yet, the split between the first two companies could vary by as much as 5.83 percent of that total. One company could produce significantly more while the other produced less, and the market would still be in balance. The study describes this collection of possible outcomes as a geometric shape called a polytope, a multi-sided figure where every point inside represents a valid, stable market arrangement.

This multiplicity of solutions has a direct impact on how we understand government intervention. The study tested what happens when a public authority releases extra fuel into the market. The result was a sharp dichotomy. If the private companies are already producing as much as they want to at the capped price, any new fuel released by the government is simply replaced by the private companies cutting back their own output. The total supply remains exactly the same, and the price stays at the ceiling. The pass-through rate is zero. On the other hand, if the private companies are constrained by their physical capacity and cannot produce more even if they wanted to, then any new fuel released adds directly to the total supply. The pass-through rate is one hundred percent. The study found that there is no scenario where the pass-through is somewhere in between, like fifty percent. The switch between these two states happens at a precise, calculable point.

The researcher also explored how the total supply changes as the government adjusts the price ceiling itself. The relationship is not a simple line. As the ceiling is raised from a very low level, total supply increases. It reaches a peak at a specific price point, which the study identifies as the optimal ceiling for maximizing the amount of fuel available. Beyond this point, raising the ceiling further causes the total supply to drop. This happens because once the price is high enough, the companies start to hold back production to keep the price from falling, effectively creating a shortage relative to what could be supplied. The study shows that the range of prices where this drop occurs is exactly the same range where the market split becomes uncertain. In the specific example used, the optimal ceiling was found to be around 1,222.87 dollars per tonne. At this price, the total supply was maximized at 177,295.50 tonnes. If the ceiling were set anywhere between roughly 1,222 and 1,338 dollars, the total supply would be lower, and the market would be stuck in a state of uncertainty regarding who supplies how much.

To verify these theoretical findings, the researcher ran a series of computer simulations using the specific numbers from the fuel import example. They used a standard method for finding market equilibria, which involves companies repeatedly adjusting their output to match what the others are doing. When the simulation started from different random positions, it converged to different final outcomes every time. In one set of 100 runs, the algorithm found 50 distinct ways to split the market, all of which were mathematically valid equilibria. The total amount of fuel was the same in every single case, matching the prediction to nine significant figures, but the individual shares varied widely. This confirmed that the uncertainty is not a flaw in the calculation method, but a real feature of the market structure. The study also noted that if the companies adjusted their output one by one, rather than all at once, the system would eventually cover the entire range of possible splits, further proving that the solution is a continuous band rather than a single point.

The research concludes that the presence of a binding price ceiling fundamentally alters the predictability of an oligopoly. While the total amount of goods in the market remains a fixed, calculable number, the distribution of that market among the competitors becomes fluid. This has profound implications for policymakers. If a regulator wants to maximize the supply of a critical good like fuel, they must set the price ceiling at a very specific point. Setting it too low suppresses the willingness of companies to supply, while setting it too high allows them to restrict output. Furthermore, the regulator cannot rely on the market to naturally select a single, fair distribution of production. The study suggests that in any real-world application where a unique outcome is assumed, that assumption may be hiding a wide range of equally valid possibilities. The findings offer a clear, rigorous map of a market that was previously thought to be too jagged to analyze, showing that while the total volume is certain, the path to get there is not.