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How to Determine the Profit-Maximizing Price of a Sandwich

My nephew, Paul, works for a sandwich shop chain. Recently, he was promoted to the chain’s management team. Paul and I celebrated his promotion over a couple of beers. While chatting about the chain’s business model, Paul asked: “Any idea how to optimally set the price of a sandwich?” An Economics 101 question, so it was not hard for me to give the answer: “Simply equate the marginal cost to marginal revenue to find the profit-maximizing price.” Paul looked at me as if I had tried to explain the fourth-order Runge-Kutta method to approximate solutions of differential equations. The following conversation unfolded.

Paul: “Marginal revenue? Marginal cost? Equate? Maybe I should have some of what you’re drinking!”

Me: “Let me try and explain this econ lingo in plain words. The marginal cost of a sandwich is what it costs for the sandwich shop to produce one extra sandwich.”

Paul: “Ah. So, when determining the marginal cost, I guess I should ignore fixed costs like the rent the shop pays to the landlord, equipment costs, and personnel costs.”

Me: “Absolutely.”

Paul: “Well, in that case, the marginal cost of a typical sandwich is around 50 cents, which is mainly the cost of the bread and the spreads.”

Me: “What about the marginal revenue?”

Paul: “The price of most sandwiches are €2. Is that what you’re looking for?”

Me: “Nope. That would be the average revenue, not the marginal revenue. To determine marginal revenue, imagine that you have decreased the price in such a way that you sell one more sandwich. The additional revenue that you gain over all sandwiches that you sell is the marginal revenue.”

Paul: “I see. Let’s say that if I decrease the price by one cent, the shop sells 100 sandwiches an hour instead of 99 sandwiches at the current price of €2. Revenue increases from 99x€2=€198 to 100x€1.99=€199. So, the marginal revenue is €1.”

Me: “Yes, indeed. In fact, it is higher than the marginal cost of 50 cents, which suggests that it may be profitable to decrease the price. By selling more sandwiches the shop will increase its profits because the gains on the additional sandwiches sold (marginal revenue) are greater than the costs.”

On the back of a beer mat, I drew the following figure:

Me: “You see. Just find the quantity (Q*) where marginal costs and marginal revenue are equal and then find the corresponding price (P*).”

Paul: “Wait a minute. You seem to forget that Broodje Bart and Subway, our main competitors, will decrease their price as a response to our price decrease.”

Me: “Not quite. The demand curve that I sketched does not represent the total demand for sandwiches in the neighborhood but the demand for your firm’s sandwiches… taking competitors’ responses into account.”

Paul: “All very well. But is there some way I can find out whether the marginal revenue is indeed €1? It seems to be much harder to determine than the marginal cost.”

Paul had a point, making my advice based on Econ 101 insights pretty worthless. And I’m afraid that follow-up courses in the economics programs are not very helpful either. Luckily, Paul made an illuminating remark: “I guess we can learn a bit from price variations in the past. A couple of months ago, we raised the price by 10 cents. And last week we sold sandwiches for €1 as part of a marketing campaign to attract consumers’ attention to a recently opened shop. The shop almost exploded!”

Indeed, firms can learn a lot from price variations about the structure of demand and so about marginal revenue and optimal pricing. In fact, many large retailers, like supermarket chains, record all individual transactions. The resulting huge data sets contain loads of information about the demand for a large range of products. At the same time, ‘big data’ may be of limited value if it is not obtained using randomized experiments. A location manager might decide to give temporary discounts on some products just to get rid of excess inventory. If this excess inventory is caused by unforeseen random events, the revenue effect of the price decrease is not very informative. Suppose the supermarket sells little ice cream in a particular month because of unforeseen bad weather. A price decrease might then boost demand in the next sunny month by more than it would have compared to a situation where the location manager kept the same price.

Randomized experiments may be much more informative about consumers’ price responses than big data. Barron et al. (2008) conducted such an experiment varying retail prices at 54 gasoline stations of a major retailer in California. Randomly selected petrol stations increased or decreased the price by two cents. The beauty of their experiment is that the price changes were effective for an entire week so that nearby competitors could respond to the price changes. At the end of the week, the researchers obtained a convincing measure of the effect of the price changes on the petrol stations’ demand.

Sometimes price experiments produce surprising results. Gneezy et al. (2014) varied prices of bottles of Cabernet Sauvignon at a small Californian winery on different days over the course of several weeks. Customers typically visit the winery as part of a wine trip through California. As a result, the vast majority of winery customers were one-time visitors who were unaware that they were taking part in an experiment. This is important because otherwise customers may be inclined to postpone their purchase in the hope of getting a better deal later. In any case, the surprising observation was that demand increased when the price was increased from a low price to an intermediate price. An upward sloping demand curve! Rare like a unicorn in an Econ 101 course. And very useful information for the winery.

My face broadened into a smile. I felt I had convinced Paul that two steps are essential in determining the optimal price: (1) ignoring fixed costs, and (2) experimentation. And when experimenting, to make sure that (1) consumers are unaware of this so that they behave as they usually would, and (2) the experiment takes place over a sufficiently long time span to allow competitors to react to the price changes.

We ordered another imperial stout and started to chat about football.



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