Thinking about opportunity cost mathematically
My last post was on contextualizing opportunity costs. After writing that piece, out of curiosity I started to think about how can we mathematically think about opportunity cost. In other words, I wanted to know if there was a formula or a mathematically approximate way to calculate opportunity costs, so that we can make better decisions. So I got to thinking and came up with a rough mathematical framework in which we may be able to think about opportunity costs. Of course, the natural question is why didn’t I look this up on the internet. But the embarrassing answer to this is that I came up with this formulation while I was taking a shower.1 And then just decided to have fun with this idea.
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Calculating Opportunity Costs ground up
I must begin by saying that I am not an economics student. And thus whatever I am about to say is most likely either false, or a misunderstanding of what the actual economics is. My only claim to be able to think mathematically is that I hold a minor in Mathematics, and that I like Mathematics.
So here’s a lazy and novice attempt at a proof:
Let us assume that we’re acting under only two constraints i.e. time and energy. Energy being both mental and physical. Let the available time be T(I) and energy be E(I). What that means is that we only have this much quantity of time and energy to spend on anything that we choose to do. That is our constraint.
The problem statement that we aim to resolve is - “Given the constrained environment, how do we decide what is the best use of our scarce resources, which in this case are our time and energy”. To be able to answer that question, at least well enough to actually make an educated guess - we’ll have to think in terms of opportunity costs of various tasks relative to other tasks that can be done using the same time and energy i.e. under the same constraints. What is really important to understand is that we are - firstly, defining opportunity cost not as an absolute quantity2 but as a relative quantity, in relation to other activities that can be done. Secondly, that we are only defining opportunity costs relatively to the tasks that can be performed under the same constraints, and not any other configuration. What that simply means is that our calculation of opportunity cost is a function of our constraints. Our constraints define what tasks we can do.
Now, let us assume that we have certain tasks that use the same time and energy, in other words are bound by the same constraints, as ranging from,
T1, T2, T3 …………….. Tn.
To qualitatively rank these terms within themselves, we cannot only look at either the costs or the benefits of each tasks. Because every task has both - a cost and a benefit. Thus the mathematically appropriate measurement of each task that will help us rank these tasks is the ratio between benefit and cost (B/C). Because every task has different benefits and costs attached to it, the ratio of benefit to cost will be different for every task. The corollary to this is the fact that it makes no difference to choose from two tasks that have the same ratio. If that is the whole range of options available to you - you incur no opportunity cost by choosing one over the other. But of course, as economists will tell you, such a case is impossible. Firstly because there are far too many constraints than just two as in our example, and secondly because at the core of the subject is the idea that there is always a better thing that you can do under the same constraints. Always.
So, let us define measurement of tasks (T) as the ratio of attached benefit over cost(B/C). So T = B1/C1, T2 = B2/C2, T3 = B3/C3 and so on. Let us also assume that T(n+1) > T(n) > T(n-1) and so on. Therefore we get the ranking system in which, T1 < T2 <T3 < T4 < T5 and so on. What that really means is B1/C1 < B2/C2 <B3/C3 < B4/C4 < B5/C5 and so on.3
So, we should define the opportunity cost not as the cost incurred for not being able to do the ‘next best thing’ under the same constraints, but as the cost incurred for not being able to do ‘the best thing’ under the same constraints. Now here I am honestly unclear on what is meant by the term ‘next best thing’. Working in the same definition of there being multiple ranked tasks (T1, T2, T3 and so on) - does the next best thing definition mean that if we were to calculate the opportunity cost of T3, we would define as the cost incurred for not being able to do the ‘next best thing’ i.e. T4 or the does the next best thing actually refer to the best thing that you could do under the same constraints (in this case Tn). The reason why that is important to resolve is because if the understanding is the ‘next best thing’ (T3 & T4) instead of ‘the best thing’ (T3 and Tn) - we are calculating opportunity cost wrongly.
The reason why I say this is because under the next best thing definition, the opportunity cost of T3 will be
OC3 = C3 - C4 (C3 > C4, because the higher the n value of a task, or simply rank of a task, the larger the ration B/C is. This implies that the as we go higher in n, we see increasing values of B and decreasing values of C. B is directly proportionally to n and n is inversely proportional to C).
So such an articulation is mathematically imprecise because of two things:
Because it is imprecise to only think in terms of costs and not benefits. Going back to our discussion on the term B/C - the better mathematical quantity to use in order to find out opportunity cost should be the ration of benefit over cost and not just cost.
The next best thing definition does not actually tell us the maximum relative opportunity cost one incurs on choosing to do that task, and only tells us the cost relative to the next best task on a ranked system of tasks.
Thus, in my opinion the corrected formula for opportunity cost should be - assuming you’re doing task Tp and Tb is the best thing that can be done under the same constraints (i.e. is the best use of our time and energy - T(I) and E(E))
O.C(b) = B(b)/C(b) - B(p)/C(p) = 𝛿B/𝛿C
I thank Niha Satyaprakash and Ayushi Baloni for their valuable comments to this piece. The insight I was able to gain from them was that standard economics actually does consider both benefits and costs and not merely costs when evaluating opportunity costs. And also that perhaps a better term to use is C/B instead of B/C because of the cost dependence of calculations.
Notice how this is nothing other than signaling status. By revealing that I came up with this framework while I was showering, I am signaling to all my readers that I am ‘smart’ and that I think smart things even while I do mundane things. I usually prefer to hold myself to the standard - ‘do not signal status until it is a pre requisite’. In that sense the only reason I revealed the information is to be able to explain why I did not rely on the internet. Status, if signaled in any context that does not require it is nothing but zero sum game.
Simply because that is impossible and also because it doesn’t make any sense. I only want to know what is the best use of my time given that I have these many things I can do, and not in some absolute terms. That doesn’t help me chose what to do.
Now here I must confess that we have taken a huge leap of faith (in real life). The leap is assuming that quantities such as benefit and cost, even if relative, are actually calculable. The reason why I say that’s an assumption is because in real life there are so many constraints and background conditions that virtually makes it impossible to be able to make even an educated quantification of these terms. Again, I am no economist. That is my educated mathematical intuition. We may, in fact, have a system of calculating these quantities. If you know of any such formula, let me know!