One and a Half Times as Good

When does it make sense to use ratios to compare prices?

         Lets consider two different anecdotes. In each anecdote the actors will use similarly sounding reasoning but in only one of the cases do I believe that this reasoning is valid.

            Imagine a group of friends organizing a weekend getaway to the beach, they have two options on where they can stay. The first option is nicer but also more expensive, it would cost $600 per person. The second option is not quite as nice but is cheaper, it would only cost $400 per person. The friends are trying to decide whether or not it would be worth it to spring for the nicer accommodations. One friend reasons “I have about $1200 in my vacation budget for this summer, if we go with the cheaper option I could afford to take three trips like this over the summer but if we go for the more expensive option I could only afford two trips over the summer. The nicer place is cool but staying there would only make my weekend a little bit better, not one and a half times as good so I would get more enjoyment from the three trips to the beach staying at the cheaper place than two trips staying at the more expensive place so I think staying at the cheaper place is the better option” The other friends agree that this is solid reasoning and they book the cheaper rooms.

            Now imagine a couple shopping at the grocery store. The couple wants to buy some pasta. The grocery store stocks two different brands of pasta, a cheaper basic brand that costs 80 cents per box and a more expensive fancier brand that costs $1.20 per box. Having tried both brands in the past the couple knows that they prefer the taste of the fancy brand just a little bit over the taste of the basic brand. The husband reasons “The fancy brand is only a little bit better but it costs one a half times as much, I know its not one and a half times as tasty so I think that we should stick with the basic brand.”

            But the wife counters “I don't agree, I think the fancy brand is worth it. Think about it this way, the fancy brand is only 40 cents more per box and the box has 8 servings. That's only 5 cents more per serving. Don't you think the better taste is worth more than 5 cents per serving”

            The husband agrees that 5 cents per serving seems worth it for the better tasting pasta so the couple puts a box of the fancy brand pasta in their cart and moves along.

            In both of these situations one of the characters tried to reason about which of two options was preferable by comparing the ratio of their benefits to the ratio of their costs. In the first scenario this seemed like really good reasoning but in the second scenario it seemed to lead them astray. So what gives? Is this valid reasoning or not? What is the difference between these two scenarios?

            I believe that one salient difference is that in the first scenario the difference in price would cause a difference in consumption, the friends would be taking three beach trips over the summer instead of only two but in the second scenario the difference in price would not cause a difference in consumption, the price of pasta is likely not the limiting factor in the couples consumption of pasta. There are other factors like how hungry the couple is and how frequently they could eat pasta before growing tired of it so the couple would likely eat about the same amount of pasta no matter which brand they ended up picking.

           Another, more precise, way to think about this is that in the pasta example the monetary cost of the pasta is not the entire cost. In order to gain the benefit of eating the pasta the couple has to pay not just the monetary cost but also the opportunity cost of having something other than pasta to eat for a meal. Pasta is very cheap so for all but the most destitute of people the opportunity cost will be the dominant part of the cost and the monetary cost will be mostly irrelevant.

            But doesn't this apply to the friends going to the beach as well? Aren't they also paying an opportunity cost? Yes they are. But if the friends really like going to the beach and would want to go as much as they can afford then the monetary cost could be the dominant factor on the cost side so comparing the ratio of monetary costs to the ratio of benefits could be a good first order approximation.

            Now what about the wife's reasoning where she looked at the price differential? Was this valid reasoning? Could the friends going to the beach have used similar reasoning? What are the advantages and disadvantages of this approach?

           I believe that the wife did have good reasoning. All she did was look at the difference between the two choices and netted out the difference in cost (5 cents per serving) and the difference in benefit (slightly better pasta for that serving). Once she did that it was clear the benefit was worth the cost because the cost was very small. This is a line of reasoning which can always be used but in situations where both the cost and the benefit are significant it may still be a nontrivial question whether or not it is worth it.

            If the friends at the beach had used this reasoning they would have ended up at the question “Is the benefit of having a nicer place to stay for the weekend worth $200?” This question is still not obvious. One strategy we can use to help us answer such a question is to try to convert $200 to something more similar to the vacation differential in order to make the question more intuitive. This is essentially what the one friend did when he looked at the multiple. He said “$200 is enough to pay for half of another trip to the beach staying at the cheap place, I know that I value half of another trip more than I value having nicer accommodations for this trip so $200 for nicer accommodations isn't worth it.”

            The takeaway here is that our intuition often has a much easier time comparing how much to value things when the things are similar (like 2 vacations staying at the nice place vs 3 vacations staying at the less nice place) and a much more difficult time comparing things that are dissimilar (like $200 versus the differential in enjoyment between a vacation with a nicer room and a vacation with a less nice room). Thus in some situations we can maximize the power of our intuition by looking at ratios of costs and benefits which is likely to involve a comparison based on things which are more similar to each other. However we need to make sure that we only apply this line of reasoning in scenarios where it makes sense. That is, only apply it in scenarios where the monetary cost (or whatever cost we are looking at ratios of) is the dominant factor and thus a change in this cost would actually cause us to adjust our consumption. In scenarios with other hidden costs this ratio is not meaningful. Checking to see how you would adjust your consumption is a good way to detect hidden costs. Opportunity costs can be non-obvious, the couple deciding which box of pasta to buy could easily miss the fact that in order to gain the benefit of the tastiness of the pasta they have to pay not only the monetary cost to buy the box of pasta but also the opportunity cost of not having something else to eat that night. However the couple will not miss the fact that their consumption does not adjust in line with the monetary cost. They know that they will not eat one and a half times as much pasta if they buy the cheap kind instead of the more expensive kind and this observations tells them that there are likely to be some hidden costs they have not yet considered. It's important to keep in mind though that although consumption adjustment is a good heuristic to find hidden costs it's not a hard and fast rule. If a good has a quickly decreasing marginal utility with extra consumption you might not adjust your consumption upwards much even if all costs associated with it dropped.