<h2>Economic profit</h2>
<p>I think there are some problems with the formulation in your question, first you heavily focus on opportunity cost but note accounting profit does not even properly capture all revenue firm gets.</p>
<p>I will first focus on economic profit more broadly and at the end go back to opportunity cost.</p>
<p>Following Varian Microeconomic Analysis pp 24 (emphasis mine),</p>
<blockquote>
<p><strong>Economic profit is defined to be the difference between the revenue a firm receives and the costs that it incurs. It is important to understand that all costs must be included in the calculation of profit.</strong> …</p>
<p>Both revenues and costs of a firm depend on the actions taken by the firm. These actions may take many forms: actual production activities, purchases of factors, and purchases of advertising are all examples of actions undertaken by a firm. At a rather abstract level, we can imagine that a firm can engage in a large variety of actions such as these. <strong>We can write revenue as a function of the level of operations of some <span class="math-container">$n$</span> actions, <span class="math-container">$R(a_1,...,a_n)$</span>, and costs as a function of these same <span class="math-container">$n$</span> activity levels, <span class="math-container">$C(a_1,...,a_n)$</span>.</strong></p>
</blockquote>
<p>The above is already rigorous definition for <em>economic</em> profit, <em>economic</em> revenue and <em>economic</em> cost.</p>
<p>Note accounting operates under its own sets of rules. <strong>Accounting, is not econometrics. Point of accounting is not to accurately estimate or measure economic relationships</strong>. So what you are doing above, defining economic profit at simply accounting profit minus opportunity cost is somewhat misguided (it would only hold in an unrealistic special case where we assume accounting only missed opportunity cost but accounting misses so much more).</p>
<p>First, following Varian we can start with a set of occurring actions <span class="math-container">$a$</span> that create revenue and costs. However, when it comes to accounting, not all of these will be reported. For example l, consider selling goods without issuing receipt to avoid paying taxes, which will not get reported as accounting profit. The same holds for many costs other than opportunity cost. For example, consider how depreciation costs are treated by accountants. From purely economic perspective most assets definitely do not depreciate in linear fashion yet accounting allows firms to use linear depreciation for most assets regarding what the true depreciation cost each period is. So the accounting depreciation costs will too be different from economic depreciation cost. Or consider the fact that many countries allow small business to just in their accounting declare portion of their profit as cost without actually even proving they actually incurred these costs (this is done in some countries to ease administrative burden for sole traders and similar one person businesses).</p>
<p>Consequently, it makes more sense to rather define accounting profit in terms of economic profit and then work backwards by inverting those functions if you would want to get back to economic profit from accounting one.</p>
<p>Call <span class="math-container">$A=(a_1,...,a_n)$</span>.  set of all economic actions that actually occur, and <span class="math-container">$R(A)$</span> would be <em>economic</em> revenue from those activities. Now apply some function <span class="math-container">$\rho$</span> which tells you based on <span class="math-container">$R(A)$</span> what actually accounting revenue is.</p>
<p>So accounting revenue <span class="math-container">$R_{acc}$</span> would be:</p>
<p><span class="math-container">$$R_{acc}= \rho(R(A))$$</span></p>
<p>Similarly accounting cost <span class="math-container">$C_{acc}$</span>, we will have actual economic cost firm incurs <span class="math-container">$C$</span> based on set of actions it takes <span class="math-container">$A$</span> and then we will have some function gamma that tells us what enters into accounting and what not so we would have that</p>
<p><span class="math-container">$$C_{acc} = \gamma (C(A))$$</span></p>
<p>Finally the <strong>accounting profit</strong> would be difference between the above so:</p>
<p><span class="math-container">$$\Pi_{acc} = \rho(R(A)) -\gamma (C(A))$$</span></p>
<p>Note what <span class="math-container">$\rho(.)$</span> and <span class="math-container">$\gamma(.)$</span> are will depend on local accounting regulations and laws. in some country government could declare that any firm has to put  always just 1000€ as their cost (yes it is unrealistic example but I am just showing it to prove a point) and in such case <span class="math-container">$\gamma(C(A))=1000$</span> regardless of anything that happens.</p>
<p>Whereas the true economic profit would just difference between economic revenues and economic costs:</p>
<p><span class="math-container">$$\pi=R(A)-C(A)$$</span>.</p>
<p>If you would want work your way back from accounting definitions then you could use inverse functions and economic profit calculated from accounting one would be:</p>
<p><span class="math-container">$$\pi= \rho^{-1}-\gamma^{-1}$$</span>.</p>
<p>Note the above does not really require definition of opportunity cost since here simply the function <span class="math-container">$C(A)$</span> is just some function where we assume that <span class="math-container">$C(A)$</span> assigns true economic cost to every action <span class="math-container">$a_1,…,a_n$</span>, for standard profit optimization it’s completely okay to treat <span class="math-container">$C(.)$</span> as a black box and thus you will not really require separate definition of opportunity cost. As a matter of fact note in situation where there are no alternatives like having mana (from the biblical story where it could only be immediately consumed not stored or used for anything else) opportunity cost would not even be part of <span class="math-container">$C$</span> but still there would be distinction and disconnect between accounting and economic profit.</p>
<p>For example if we have <span class="math-container">$C(A)=c_1 a_1+c_2 a_2… c_n a_n$</span> and if we define <span class="math-container">$C$</span> as a function that always assigns true economic cost then you definition <span class="math-container">$c_1$</span> includes all economic costs including opportunity cost, but the economic profit, economic revenue and economic costs are still rigorously defined (even if the constituents of costs such as opportunity costs aren’t).</p>
<h2>opportunity cost</h2>
<p>Next tackling the opportunity cost separately (which can be later used to build function <span class="math-container">$C$</span>), good places to start are <a href="https://link.springer.com/chapter/10.1007/978-1-349-21315-3_69" rel="nofollow noreferrer">Buchanan (1987)</a> or <a href="https://www.encyclopedia.com/social-sciences-and-law/sociology-and-social-reform/sociology-general-terms-and-concepts/cost" rel="nofollow noreferrer">Alchian (1968)</a>. The authors do not necessarily derive rigorous opportunity cost functions but give rigorous verbal definitions on which we can build.</p>
<p>For example, according by to <a href="https://link.springer.com/chapter/10.1007/978-1-349-21315-3_69" rel="nofollow noreferrer">Buchanan (1987) </a>:</p>
<blockquote>
<p>Opportunity cost is the anticipated value of 'that which might be' if choice were made differently. Note that it is not the value of 'that which might have been' without the qualifying reference to choice. In the absence of choice, it may be sometimes meaningful to discuss values of events that might have occurred but did not. It is not meaningful to define these values as opportunity costs, since the alternative scenario does not represent a lost or sacrificed opportunity. Once this basic relationship between choice and opportunity cost is acknowledged, several implications follow.</p>
</blockquote>
<blockquote>
<p>First, if choice is made among separately valued options, someone must do the choosing. That is to say, a chooser is required, a person who decides. From this the second implication emerges. The value placed on the option that is not chosen, the opportunity cost, must be that value that exists in the mind of the individual who chooses. It can find no other location. Hence, cost must be borne exclusively by the chooser; it can be shifted to no one else. A third necessary consequence is that opportunity cost must be subjective. It is within the mind of the chooser, and it cannot be objectified or measured by anyone external to the chooser. It cannot be readily translated into a resource, commodity, or money dimension. Fourth, opportunity cost exists only at the moment of decision when choice is made. It vanishes immediately thereafter. From this it follows that cost can never be realized; that which is rejected can never be enjoyed.</p>
</blockquote>
<p>since the opportunity cost is the value of the next best alternative you can rigorously define as follows.</p>
<p>Suppose that for each action that actually was taken in the set <span class="math-container">$A=a_1,…,a_n$</span>, there are alternative sets of alternative actions (e.g. each <span class="math-container">$a_i$</span>, has some alternative <span class="math-container">$b_i$</span>, <span class="math-container">$z_i$</span> and so on, and let’s assume that all the other alternatives are mutually exclusive with <span class="math-container">$a$</span>. Well then opportunity cost of each action <span class="math-container">$a$</span> is just value of the next best alternative so if we have that:</p>
<p><span class="math-container">$a_i\succ b_i \succ z_i … $</span></p>
<p>Then opportunity cost will simply be <span class="math-container">$V(b_1)$</span> where <span class="math-container">$V$</span> is simply just some function that assigns monetary value (or in consumer problem utility to <span class="math-container">$b_i$</span>).</p>
<p>If you want to then put this value into profit function explicitly then simply the cost function will be some composite function given by <span class="math-container">$C(a_i, V(b_i))$</span> (note this is ultimately just function of <span class="math-container">$a_i$</span> because the next best alternative necessarily depends on what you choose as <span class="math-container">$a_i$</span>, so in the end it’s just more complex way of writing <span class="math-container">$C(a_1))$</span></p>
<p>I think the above is rigorous even though it’s very simple (of course perhaps the notation I use could be made more precise).</p>

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2021-05-27T14:24:47.070Z
Economic profit: But I didn't make this up myself, I just follow the simplified usage of this term in econ textbooks (see link in edited version).
VARulle
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2021-05-27T14:27:53.353Z
Opportunity costs: It seems that your setting is just an extended - but still finite - version of my 2-alternatives setting. But what about the (continuous) monopoly example?
VARulle
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2021-05-27T15:42:21.870Z
@VARulle but you are citing **undergrad** textbook and not even one from really reputable author (for example Mankiw actually includes more nuance discussing, albeit very briefly in his textbook - although Mankiw undergrad textbook also does not include enough nuance). In addition also the open textbook elsewhere states "accounting profit is a cash concept" <- this is actually more nuanced because as written in my first half of the  answer it directly means that only cash/easily quantifiable transitions are included, so it excludes any hard to quantify transaction be it on rev or exp side
1muflon1
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2021-05-27T15:45:04.037Z
@VARulle you simply cannot expect undergraduate textbooks to provide rigorous treatment of profit in the same way as for example MWG or Varian does. Undergraduate physics textbooks also lack nuance and often do not explain terms rigorously. In addition, textbooks vary in quality the OpenStack textbooks are good enough given that they are completely free but they are not that far from just economics for dummies lets say
1muflon1
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2021-05-27T15:49:21.980Z
@VARulle regarding the opportunity cost the problem with your monopoly example is that you define the opportunity cost in terms of second best accounting profit that is definitely not correct. Next, in order for something to create opportunity cost it must be mutually exclusive with action you are taking. If you action a_1 is to produce Q=100 then action b_1 Q=50 is not mutually exclusive action since producing 50 units of the same product would be just subset of producing 100 units of the same product. Rather you should calculate opportunity cost from mutually exclusive activity
1muflon1
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2021-05-27T19:06:35.247Z
That's my problem: These concepts seem only to be discussed and (vaguely) defined in undergrad textbooks. I was wondering whether there actually is some more formal treatment. But in MWG the word opportunity costs doesn't even appear. Moreover, I'd expect that even undergrad textbooks present some *consistent* version of these concepts.
VARulle
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2021-05-27T19:11:22.780Z
"accounting profit is a cash concept": Yes, that's how I also use it here, revenue minus explicit (i.e., out-of-pocket) costs. This is talking about the producer theory model world of course, so I don't know what a "hard to quantify transaction" even means there.
VARulle
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2021-05-27T19:13:40.733Z
@VARulle but the graduate textbooks have rigorous definitions of profits which was still part of your question. Also, I agree that undergraduate textbooks should be more consistent but what you gonna do on that front :/, I think you see more consistency when you compare top textbooks like Mankiw vs Knight or Krugman, and you will see more deviations if you take textbooks from some unknown authors. Regarding opportunity cost the problem is that for 99.9% of applications it’s fine to treat $C(.)$ as a black box with just some regularity conditions. I don’t think I ever come across
1muflon1
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2021-05-27T19:14:25.037Z
Application where it would be necessary to explicitly model opportunity cost. As a result this is something that will rarely be done. I think way I introduced above would be valid way that then can be extended to various other situations
1muflon1
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2021-05-27T19:16:14.433Z
@VARulle but there are a lot of things that are not easily to quantify. For example, from labor economics you don’t get wage just in (explicit) terms of money but also in terms of implicit benefits eg nice and safe working environment. The same holds for firm itself. Cash transactions here would mean transactions that explicitly have some price tag attached to them
1muflon1
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2021-05-28T09:39:56.573Z
I also can't wrap my head around this "mutually exclusive actions" thing... My view is: You can choose between black boxes A and B. A gives you 10 dollars, B 15 dollars. If you choose A, you give up the opportunity of choosing B, so opportunity cost of choosing A is 15. You are telling me that, no, it raher depends on what happens inside the black boxes?
VARulle
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2021-05-28T14:32:21.433Z
@VARulle I am just saying that just because you list arbitrary two choices that does not mean they are mutually exclusive. Producing 3 liters of water is not mutually exclusive with producing 2liters of water if by producing 3 liters of water you produced two liters of water. If A and B can happen at the same time by definition they are not mutually exclusive. Saying I assume A and B are mutually exclusive events but then picking events that are not mutually exclusive will be inconsistent. Its like asking so  if a > b does the inequality depend of what a and b are well yes
1muflon1
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2021-05-28T14:34:26.477Z
you can't pick A= 5 and B= 10 and have the inequality hold
1muflon1
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2021-05-28T14:50:09.187Z
I just can't imagine why mutual exclusivity should be important for the calculation of opportunity costs. You seem to suggest that even if I know all the revenues and explicit costs of A and B I cannot calculate the opportunity costs of choosing A as long as I don't know whether the production processes hidden in A and B have some overlap.
VARulle
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2021-05-28T14:53:50.380Z
@VARulle but the mutual exclusivity is definitional condition of opportunity cost. If you can at the same time do both A and B you are not giving up anything. The opportunity cost is created (as the articles I referenced explain in greater detail) when there is some alternative thing you could be doing. This is why using mana (in biblical sense) has no opportunity cost - you can only eat it not save it or use it for some alternative use. And yes you can’t calculate opportunity cost if you don’t know if actions are mutually exclusive, like you can’t calculate x if x=10+y and you don’t know y
1muflon1
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2021-05-28T15:12:40.617Z
@VARulle I would really recommend reading the two articles cited in the second section they are not even long
1muflon1
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2021-05-29T08:12:33.783Z
Let A and B be mutually exclusive actions you value at 8 and 10, respectively, e.g. A is "drink 1 free beer" and B is "eat 1 free soup". Assume you can only choose one of these options at your favorite restaurant. If you choose A, you incur opportunity costs of 10, so profit is -2. If you choose B, you incur opportunity costs of 8, so profit is 2. You better choose B. That's standard, right? (contd. ...)
VARulle
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2021-05-29T08:15:42.583Z
(... contd.) Now drop the mutual exclusivity assumption, e.g. A is "drink 1 free beer" and B is "drink 2 free beers". What changes in the analysis? Nothing! Of course drinking 2 beers you don't forego drinking 1 beer - but you do forego the opportunity of drinking *only* 1 beer and then stopping consumption. Mutual exclusivity plays no role in this.
VARulle
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2021-05-29T08:16:49.920Z
I have also read both the Buchanan and the Alchian articles, but they do not provide the *formal* definition I asked for and don't help a lot.
VARulle
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2021-05-29T09:30:35.987Z
@VARulle I told you in advance they don’t have formal derivation but mention mutual exclusiveness and have more explanation. Also, the point about mutual exclusivity here is that the action should not be subset of another action, action of drinking 1 beer is just subset of action where you drink 1 bear and then drink second beer. As those sources explain an example of a situation where there is no opportunity cost is a situation where there is only choice of one good available without any alternative there is no opportunity cost regardless of how much you consume of that good.
1muflon1
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2021-05-29T09:34:16.403Z
@VARulle I think someone else in comments mentioned that in some work opportunity cost is equated to marginal rate of substitution, that is a way of measuring opportunity cost *on the margin* when deciding between two goods x and y as it tells you what sacrifice you have to make in terms of not consuming y to consume 1 more x, but it makes no sense to try to construct marginal rate of substitution if in model there is only one good let’s say U=x, then there is no opportunity cost as consumer can only consume x no other option available
1muflon1
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2021-05-31T06:58:27.003Z
Funnily enough, Buchanan mentions mutual exclusivity only in an example that actually contradicts your rule: "*There are two mutually exclusive thermostat settings for a building, High and Low.*" I guess in your interpretation "heating up to 18°C" is a subset of "heating up to 24°C", so there should be no opportunity costs of either choice. That's just the opposite of Buchanan's conclusions.
VARulle
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2021-05-31T07:55:27.850Z
@VARulle but you are getting too hang up on the example. The point is that they have to be mutually exclusive. Buchanan simply by construction claims that $High_T ∩ Low_T =∅$, we can now discuss whether that is a good representation of real life High vs Low temperature but in his case mutual exclusivity will hold by construction, so all you debate in previous comment is whether that is IRL mutually exclusive or not.  Going back to our previous example if we denote two action sets $A=\{x_1\}$ where $x_1$ is manufacturing first discrete Q of output, and then there is second set $B=\{x_1, x_2\}$
1muflon1
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2021-05-31T08:02:57.207Z
where $x_1$ is manufacturing of first unit and $x_2$ manufacturing of second. Here $A ∩ B \neq ∅$ and in fact $A⊆B$ so if you choose B, A is not a valid mutually exclusive action for opportunity cost calculation. If you would choose A then B would also not be valid mutually exclusive action but subset of B call it $B_1 = \{x_2\}$ would still be. This is how I think about it terms of set theory. Anyway, if you are willing to accept mutual exclusivity, then we are just arguing about whether temperature example was good example or bad example of mutually exclusive event not about OC definition
1muflon1
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2021-05-31T09:52:10.607Z
But I'm not willing to accept mutual exclusivity as necessary for opportunity cost calculation, because I have not yet seen a single example where violating it would result in incorrect opportunity cost calculation (and therefore wrong choice).
VARulle
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2021-05-31T10:03:02.820Z
@VARulle but that is by definition part of opportunity cost. That is like saying you are not willing to accept that by definition in economics cash is not capital. Well then we are just dealing with semantics. Regarding your requirement how can opportunity cost ever change a  decision. By definition, opportunity cost is the value of second best forgone alternative, if you have option of being gardener and accountant so we take acc wage as oc for being gardener, if that makes you to decide being acc then acc can no longer be oc as now it is your first choice and so oc will be different
1muflon1
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2021-05-31T10:05:16.530Z
the mutual exclusivity changes the profit calculation as without mutual exclusivity there is always a second best option just to replicate the first option -  in such case economic profit would not be possible, in fact if you are not willing to accept mutual exclusivity concept of opportunity cost is meaningless as then it is just value of your best option since you can always make arbitrarily many non-mutually exclusive options like option A: on fork go left vs B:on a fork go left but it won't affect the choice since what opportunity cost is, is conditional on choice you make *by definition*.
1muflon1
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2021-05-31T10:15:23.487Z
so first picking some choice is antecedent to having opportunity cost
1muflon1
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2021-05-31T12:08:54.267Z
How can you say "by definition" if there is no formal definition anywhere in the literature? "Mutually exclusive" for me just means that if the set of feasible actions is {a, b} then you can choose either a or b, but not both a and b. In your understanding you need to know something about what kind of actions a and b are. 1 beer or 1 wine - good, 1 beer or 2 beers - bad? I'd say calling the first case exclusive and the second case not exclusive is pure semantics. And no, you can't just replicate an action, because if a = "go left" and b = "go left", then {a, b} = {a} and there is no choice.
VARulle
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2021-05-31T12:26:58.383Z
@VARulle but there are definitions in literature, just because a definition is not mathematically formalized that does not make it not a definition. In fact, cash not being capital is not mathematically defined either. In fact the Buchanan definition is even listed as an entry in the official Palgrave dictionary of economics making it at least somewhat authoritative. In addition, the mutual exclusivity is implicit even in the textbook you listed and also Mankiw etc, because you will never find an example where someone would consider lower consumption of same good for opportunity cost
1muflon1
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2021-05-31T12:33:09.797Z
@VARulle  Next, you can then just additionally mathematically formalize it if you want. Note if you want to define opportunity cost in different way that would be possible but defining it in way that excludes mutual exclusiveness would go contrary to how OC is currently used. That is like asking why we should define capital in a way that excludes cash? You can have consistent definitions of capital that do include cash (e.g. finance). Same way you could have definition of OC that would make all economic profits zero (but I never seen such definition ever used in economics and it is not useful)
1muflon1
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2021-05-31T12:34:30.350Z
Lastly, again just because some definition is informal, that does not mean its not definition, you can just formalize the informally given definition that will follow the conditions set in the informal definition. As long as those conditions do not create contradictions there is no issue with that. Concept of utility existed long before it was formalized and made more precise, lack of formalized definition did not meant utility can be anything whatsoever
1muflon1
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2021-05-31T12:44:29.557Z
A definition has to be exact, otherwise it is open to discussion, as the current one shows. A formal definition would be exact, the Buchanan "definition" is not. He uses only discrete choice examples where there is no problem anyway, and the thermostat example even seems to run counter to your requirement. If a "definition" is so vague that it is left unclear whether "consuming" 18°C is a subset of "consuming" 24°C, then it is not a definition.
VARulle
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2021-05-31T12:54:19.777Z
@VARulle  You are right that if there some ambiguity in definition there is space for discussion, but that is not equivalent of saying well anything goes. In biology some animal categories are not 100% exactly defined so there might be discussion whether some sea critter should be classified as cephalopods or crustaceans because the animal might not neatly fall into either category, but that does not mean that the sea critter can be arbitrarily labeled mammal. The mutual  exclusiveness is not only feature of the Buchanan definition but also others whether explicitly or implicitly
1muflon1
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2021-05-31T12:56:57.220Z
@VARulle hence if you want to create a formal mathematical definition it should respect that. Or alternatively, if you want new definition it should at least be useful, defining OC in a way that would mean all economic texts would have to be rewritten and we would have to redefine our understanding of economic profit would not be useful at all. It would be like trying to now redefine capital in economics to include cash,  just on account that there likely is no formal definition of capital as set that does not include cash but only described verbally
1muflon1
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2021-05-31T13:02:56.933Z
@VARulle by the way lets go back to also what you said ", you can't just replicate an action, because if a = "go left" and b = "go left", then {a, b} = {a} and there is no choice." <- this is exactly my point. Consider following actions: first a composite action A: go left, go right at next turn; next simple action B: go left. You would claim that these are distinct action but that is just because actions A is a composite action. In fact action A is just $A= A_1 ∪ A_2$ where $A_1$ is go left an $A_2$ is go right at the next turn. Then exactly $\{A_1,B\} = \{A_1\}$ <- there is no choice at all
1muflon1
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2021-05-31T13:03:13.840Z
and for $A_2$ you simply do not have any alternative action. If there would be some alternative action for $A_2$ like second $B_2$ action where on a second turn you go left you could talk about opportunity cost between $A_2$ an $B_2$ but there cannot be an opportunity cost created when choosing between $A_1$ and $B_1$ as there is no choice
1muflon1
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2021-05-31T23:03:43.323Z
But if the action set is defined as $\{A, B\}$ then there is no choosing between $A_1$ and $B$, simply because $A_1$ is not in the action set.
VARulle
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2021-05-31T23:13:01.973Z
See, in the binary choice case my formal definition is easily used to compute opportunity costs: If action set = {A,B} then opp.costs(A) = u(B) and opp.costs(B) = u(A). Then profit(A) = u(A)-opp.costs(A) = u(A)-u(B). Similarly, profit(B) = u(B)-opp.costs(A). So a utility-maximizing agent chooses the option with the higher profit and vice versa. With your requirement you can't even calculate opp.costs, because you don't know whether or not A and B are "mutually exclusive". This makes no sense to me.
VARulle
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2021-05-31T23:14:34.400Z
And let me repeat: The Buchanan article doesn't require "mutual exclusivity" anywhere in the text, as far as I remember. If I am wrong, please give me a reference.
VARulle
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2021-05-31T23:46:53.370Z
@VARulle 1. Buchanan mentions mutual exclusivity when he constructs the example 524. 2. "But if the action set is defined as {A,B} then there is no choosing between A1 and B, simply because A1 is not in the action set" <- but A_1 is simply a subset of set A so it is in the action set.  "With your requirement you can't even calculate opp.costs, because you don't know whether or not A and B are "mutually exclusive"" <- sorry but this is strawman, I gave you clear condition for mutual exclusivity $A∩B= ∅$  so you can know at all times whether two actions are mutually exclusive
1muflon1
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2021-05-31T23:48:49.653Z
3. "Then profit(A) = u(A)-opp.costs(A) = u(A)-u(B). Similarly, profit(B) = u(B)-opp.costs(A). " <- I think I misunderstood what you were saying before, you can use the opportunity cost where B and A are mutually exclusive in this way
1muflon1
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2021-06-01T00:02:57.257Z
@VARulle 4. Note in addition, every textbook and also Buchaman and the other article I cited and also specifically states that opportunity cost is cost of forgone alternative. I mean this directly implies mutual exclusivity You can't claim that when you are choosing between consumption of 2 or 3 apples, by consuming 3 apples you are forgoing consumption of two apples, when consumption of two apples is necessarily pre-requisite for consumption of 3 apples. What even more the definitions mention scarcity as well. Scarcity has far better and rigorous mathematical definition
1muflon1
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2021-06-01T00:06:39.740Z
which requires some demands not being met in optimum (see Giskards answer here for rigorous definition https://economics.stackexchange.com/a/32689/15517). Here if the choice is only between 3 and 2 apples then there is no scarcity in the problem in the first place and thus there cannot be any opportunity cost by definition either
1muflon1
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2021-06-01T07:43:06.343Z
Sorry, but I don't know what "example 524" is. I'm referring to the Palgrave reprint of Buchanan's article at https://link.springer.com/referenceworkentry/10.1057%2F978-1-349-95121-5_1433-2, that does not mention mutual exclusivity except for the thermostat example.
VARulle
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2021-06-01T07:52:34.643Z
"*but A_1 is simply a subset of set A so it is in the action set*": In my toy model, A is an action, not a set, so there is no "subset of set A". If you insist on modeling actions as sets, and require that subsets of actions are actions themselves, e.g. A ={A1, A2}, where A1 happens to be identical to B, then the action set in my model is simply {A, A1, A2}. Your understanding of mutual excl. then says that A has no opp.costs, but with my definition there's no reason why {A, A1, A2} should be treated differently from {A, X, Y}.
VARulle
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2021-06-01T07:57:58.923Z
The reason is that if a child chooses to eat 5 pieces of cake and then feels sick, it's mother might well ask "Why didn't you stop after the second piece?", thereby pointing to the *foregone opportunity* of choosing N = 2 instead of N = 5, without any problems due to alleged "non-mutual-exclusivity" of these two alternatives.
VARulle
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2021-06-01T08:04:03.363Z
@VARulle example on page 534. Thats what I meant, he then proceeds to build an example of opportunity cost base on that that
1muflon1
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2021-06-01T08:09:18.727Z
"In my toy model, A is an action, not a set, so there is no "subset of set A"." But this is like saying in your model you insist on splitting costs total costs into variable costs and fixed costs. You just aggregate actions that I prefer to keep separate because it is more useful.
1muflon1
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2021-06-01T08:20:23.557Z
@VARulle but the problem is that in your cake eating example there are no alternatives. Again, just break it down into the discrete actions. If you are eating 2 pieces of cake what you are doing is to commit yourself to action plan $A_1=\{ e_1, e_2\}$, if you want to eat 5 pieces of cake $A_2= \{e_1, e_2, e_3, e_4, e_5 \}$, it is simply impossible to eat 5 pieces of cake without first going through process of eating 2. The point is that either under $A_1$ and $A_2$ action $e_1$ and $e_2$ are present so $e_1$ and $e_2$ are not forgone, you still had to took those actions
1muflon1
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2021-06-01T09:55:07.930Z
@VARulle in addition your concept of opportunity cost simply does not work with rest of the economics. For example, in Mankiws principles on pp 25 Mankiw claims (with reference to PPF on pp24) that when we move from point A at PPF A= (2200PC, 600Cars) to point B=(2000PC, 700Cars) the OC of 100 extra cars will be 200 PC, but if I would follow your claims it should 200 PC + 99 Cars because you would claim that we are giving up the option to produce just 99 Cars as well <- that does not mesh with OC definitions and usage in current economics (I can give you analogous ex from almost any txbook)
1muflon1
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2021-06-01T11:59:56.690Z
You are misrepresenting my claims. In my model of the PPF example, if you can choose between A = (2200PC, 600Cars) and B = (2000PC, 700Cars) then there simply is no possibility to "produce just 99" extra cars. If you want to include this choice possibility, then you have to add it to the choice set, so there is an additional option C = (2002PC, 699Cars). There's nothing here to worry about.
VARulle
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2021-06-01T12:01:45.273Z
Moreover, the Mankiw example is just another example for an informal definition that doesn't fit with the standard (but also informal) definition: What is the "value of the best foregone alternative" if you choose B? This depends on what you mean by "value" and by "best". The Mankiw example is just a technical tradeoff and can't tell you whether A or B is the "better" choice.
VARulle
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2021-06-01T12:06:39.613Z
@VARulle what you mean misrepresentation, you can by definition choose any combination of cars within PPF point C not shown in Mankiw example but we could simply draw it in with C=(2002PC, 699Cars) is by definition in the choice set otherwise what is in the picture on pp 24 would not be PPF. So not only C is in a choice set but all other points within PPF. And yes Mankiw does not offer formal definition of opportunity cost, but the point is that your definition conflicts will all the informal definitions out there while mine is completely consistent.
1muflon1
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2021-06-01T12:08:42.713Z
The thing about definitions is that they are ultimately semantics any formal definition is in itself valid definition as long as it is internally consistent. For example, re-defining dogs as rocks, will be valid definition in a system where you simply define rocks to be also dogs, but introducing such new formal definition into biology would not make sense even if maybe you can make it more precise and rigorous than some less formal definition of a dog biology has. My point is if you want to construct a formal definition of OC it should respect the usage of OC in economics.
1muflon1
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2021-06-01T12:18:34.727Z
Or let me give you another analogy. For example, rationality in economics is nowadays rigorously defined (you can have look at MWG ch 1). But that was not always so, in times of Smith and Ricardo, rationality was only vaguely defined and even the first more narrowed (but still mathematically informal) definition by Mill was bit vague (equivalently to notion of OC in Mankiw), but then when 19-20 cent. economists formalized OC they did not went and just said screw all economics before today and make up arbitrary definition of rationality, they tried to formalize the notion of rationality
1muflon1
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2021-06-01T12:22:39.847Z
as it was used in economics. What you are doing basically is on fundamental level, you are dismissing the way how opportunity cost currently works in economics, just because textbooks do not present rigorous mathematical model, and then you construct rigorous formal definition of opportunity cost that has clear contradiction with usage of OC in economics. Well you can do but what is a point of formalizing definition if you are not respecting its usage in literature but make your own? You are not really formalizing the informal definition but making a new one.
1muflon1
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2021-06-01T12:31:49.470Z
Let us [continue this discussion in chat](https://chat.stackexchange.com/rooms/124945/discussion-between-1muflon1-and-varulle). We already generated too many comments and the comments are getting flagged by automatic system
1muflon1
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2021-06-01T13:59:36.733Z
Yes, let's move to chat.
VARulle
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2022-10-11T10:30:21.263Z
This:https://pressbooks.bccampus.ca/uvicecon103/chapter/1-2-opportunity-costs-sunk-costs/ seems to contradict what you said. See table 1.2b.
Vivaan Daga
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2022-10-11T11:56:59.167Z
@Shinrin-Yoku it does not contradict anything I say here in fact that site says pretty much exactly I do here just in more length and convoluted way
1muflon1
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2022-10-13T11:10:32.420Z
Thanks for the response! I think that site(table 1.2b) contradicts  the Alchian paper you linked, since it seems to add the explicit cost of A to the opportunity cost of A(as defined by Alchian) to get the opportunity cost of A(per the site).
Vivaan Daga
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2022-10-13T16:48:48.663Z
So is there a contradiction?
Vivaan Daga
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2022-10-13T17:45:13.240Z
@Shinrin-Yoku they add explicit costs there because the money you use on A could be saved or to buy something
1muflon1
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2022-10-14T01:45:32.567Z
But that contradicts answers here: https://economics.stackexchange.com/questions/20908/what-is-opportunity-cost/20953?noredirect=1#comment86581_20953 are these just alternative definitions.
Vivaan Daga
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2022-10-14T10:12:00.153Z
Are you using a different def?
Vivaan Daga
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2022-10-14T15:01:39.757Z
This is really killing me so an answer would be much appreciated
Vivaan Daga
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2021-05-26T16:21:35.230Z
2021-05-26T18:30:58.670Z
2021-05-26T18:30:58.670Z
1muflon1
<p>[<em>Edited to remove a mistake in a definition and some typos.</em>]</p>
<p>According to various econ textbooks (e.g. <a href="https://opentextbc.ca/principlesofeconomics/chapter/7-1-explicit-and-implicit-costs-and-accounting-and-economic-profit/" rel="nofollow noreferrer">this one</a>), there is a fundamental distinction between <em>economic profit</em> and <em>accounting profit</em>:</p>
<p><strong>(i)</strong> While <strong>accounting profit</strong> subtracts only <strong>explicit costs</strong> (out-of-pocket costs) from revenue,</p>
<p><strong>(ii)</strong> <strong>economic profit</strong> subtracts <strong>opportunity costs</strong>, also known as <strong>economic costs</strong>, which consist of explicit and <strong>implicit costs</strong>. Here,</p>
<p><strong>(iii)</strong> opportunity costs are generally defined as the &quot;value of the best foregone alternative&quot;. In the context of firms, &quot;value&quot; of course means profits.</p>
<p><strong>(iv)</strong> Furthermore, when economists talk about &quot;costs&quot; and &quot;profits&quot;, they always mean <em>economic costs</em> and <em>economic profits</em>. For example, firms are assumed to maximize economic profits.</p>
<p>All the definitions I found were verbal and informal as above. Is there a <em>formal</em> definition of these concepts? Do they make sense at all? Since I didn't find anything in the literature (by a quick search, admittedly), I tried to formalize it myself for the firm context:</p>
<p>Let <span class="math-container">$X$</span> be a set of alternatives the firm can choose from, and let <span class="math-container">$z\in X$</span>. Let's assume that alternative <span class="math-container">$z$</span> yields revenue <span class="math-container">$r(z)$</span> and generates explicit costs <span class="math-container">$c_e(z)$</span> for the firm. Accounting profit <span class="math-container">$\pi_a(z)$</span> is defined as revenue minus explicit costs, so <span class="math-container">$\pi_a(z)=r(z)-c_e(z)$</span>. Economic profit <span class="math-container">$\pi_e(z)$</span> is then defined as accounting profit minus implicit costs <span class="math-container">$c_i(z)$</span>, thus <span class="math-container">$\pi_e(z)=\pi_a(z)-c_i(z)=r(z)-(c_e(z)+c_i(z))=r(z)-c_o(z)$</span>.</p>
<p>Opportunity costs of choosing <span class="math-container">$z$</span> are the value of the best foregone alternative, so <span class="math-container">$c_o(z)=\max_{k\in X,\,k\ne z}\pi(k)$</span>.</p>
<p>Now, what is the &quot;profit&quot; <span class="math-container">$\pi(k)$</span> here? Economic profit or accounting profit? This might not be immediately obvious. Let's have a closer look.</p>
<p><strong>Finite sets of alternatives</strong></p>
<p>Assume for simplicity that there are only two alternatives, <span class="math-container">$X=\{x,y\}$</span>, with <span class="math-container">$r(x)&gt;r(y)$</span>.</p>
<p><strong>(i) Let's try economic profits, so <span class="math-container">$\pi(k):=\pi_e(k)$</span>:</strong></p>
<p>Then <span class="math-container">$c_o(x)=\pi_e(y)$</span> and <span class="math-container">$c_o(y)=\pi_e(x)$</span>.</p>
<p>Therefore <span class="math-container">$\pi_e(x)=r(x)-c_o(x)=r(x)-\pi_e(y)$</span>. Now, analogously, <span class="math-container">$\pi_e(y)=r(y)-c_o(y)=r(y)-\pi_e(x)$</span>. Substituting, <span class="math-container">$\pi_e(x)=r(x)-r(y)+\pi_e(x)$</span>, implying <span class="math-container">$r(x)=r(y)$</span>. This is a contradiction, so this approach makes no sense.</p>
<p><strong>(ii) Let's interpret &quot;profit&quot; as accounting profit, i.e. <span class="math-container">$\pi(k):=\pi_a(k)$</span>.</strong></p>
<p>Then <span class="math-container">$c_o(x)=\pi_a(y)$</span> and <span class="math-container">$c_o(y)=\pi_a(x)$</span>, so <span class="math-container">$\pi_e(x)=r(x)-c_o(x)=r(x)-\pi_a(y)=r(x)-r(y)+c_e(y)$</span>. Similarily, <span class="math-container">$\pi_e(y)=r(y)-c_o(y)=r(y)-\pi_a(x)=r(y)-r(x)+c_e(x)$</span>. This is at least not contradictory.</p>
<p><strong>Infinite sets of alternatives</strong></p>
<p>But there are rarely only two alternatives. Consider instead a standard quantity-setting monopolist. The set of alternatives is the set of quantity choices, so <span class="math-container">$X=\mathbb R^+$</span>. Let's denote by <span class="math-container">$q^*$</span> the firm's optimal choice. Then the firm accrues accounting profits denoted by <span class="math-container">$\pi_a(q^*)=r(q^*)-c_e(q^*)$</span>, where both <span class="math-container">$r$</span> and <span class="math-container">$c_e$</span> (and therefore <span class="math-container">$\pi_a$</span>) are assumed to be continuous. What is the firm's economic profit <span class="math-container">$\pi_e(q^*)$</span> then?</p>
<p>Well, the firm's &quot;(accounting) profit from it's best foregone alternative&quot; is the accounting profit resulting from the smallest strictly positive deviation from <span class="math-container">$q^*$</span>, which, unfortunately, doesn't exist. To the rescue, let's interpret the &quot;(accounting) profit from the best foregone alternative&quot; as the supremum (instead of the maximum) of the foregone accounting profits: <span class="math-container">$c_o(q^*)=\sup_{k\in X,\,k\ne q^*}\pi_a(k)$</span>. This, however, is just <span class="math-container">$\pi_a(q^*)$</span>. Then <span class="math-container">$\pi_e(q^*)=r(q^*)-\pi_a(q^*)=c_e(q^*)$</span>.</p>
<p>Economic profit of a profit-maximizing monopolist being equal to its out-of-pocket costs? This seems not to make sense.</p>
<p>So do the concepts of economic profit and opportunity costs as defined in (i)-(iv) make sense at all? Or which of (i)-(iv) above is wrong? Or is the &quot;value&quot; of the best foregone alternative in the context of firm decisions neither economic nor accounting profit of the best foregone alternative? What else?</p>

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2021-05-26T16:59:57.487Z
I've seen opportunity costs interpreted as marginal rates of substitution, which do make mathematical sense.
Michael Greinecker
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2021-05-26T19:20:57.323Z
My first question would be what are the units of economic profit, economic cost, and opportunity cost? Accounting customs record transactions at the historic price in money units {mu} or assign a valuation of items in money units. Utils have no dimension, like radian angle, so economic value is dimensionless. So one must  specify units of measure and the measurement procedure used to assign a number and a unit to each quantity of interest. Opportunity cost is value difference based on counter-factual reasoning. The dog could have caught the rabbit if he did not stop to chase his tail instead.
SystemTheory
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2021-05-27T01:32:05.750Z
The definition of accounting profit quoted by the OP is too idealistic. In reality, "accounting profit" is an arbitrary number constructed in order to minimize liabilities such as tax payments. (Example: Amazon's accounts for 2020 show UK sales of $17.5bn, and UK tax payments of only £300m, i,e, a tax rate of about 2% that was achieved by exploiting loopholes in tax legislation)  Even at the level of small business accounting, one of the first questions a good accountant asks before drawing up the official accounts is "How much profit do you want to declare for last year?"
alephzero
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2021-05-27T08:08:49.250Z
@MichaelGreinecker, I'm not sure if I understand this interpretation, but anyway it seems to be applicable only in a narrow consumer theory context.
VARulle
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2021-05-27T08:10:04.103Z
@SystemTheory, I'd be happy with monetary units...
VARulle
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2021-05-27T08:12:46.570Z
@alephzero, I'm not talking about real-world accounting profit but about the simplified concept used to distinguish between explicit and implicit costs. And it's not my invention, it's all over the econ textbooks.
VARulle
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2021-05-27T08:19:36.873Z
@VARulle Marginal rates of substitution give you under the right regularity condition the value of increasing one variable in terms of the cost of decreasinganother variable. That works not just for consumer theory but for any setting where the function to be optimized and the constraint is sufficiently smooth.
Michael Greinecker
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2021-05-27T09:58:48.527Z
@MichaelGreinecker, o.k., but then it doesn't work for a discrete choice set, right?
VARulle
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2021-05-27T19:09:49.637Z
I had a [similar frustration](https://economics.stackexchange.com/questions/20908/what-is-opportunity-cost#comment59406_20926) with the concept of opportunity cost, and have since accepted that this concept, though intuitive, is open to individual interpretations/definitions, as this [well-known example](https://onlinelibrary.wiley.com/doi/abs/10.4284/0038-4038-2011.218) shows.
Herr K.
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2021-05-27T21:40:57.810Z
@VARulle Right.
Michael Greinecker
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2021-05-27T22:54:59.473Z
@HerrK., thanks for the reference, this is very interesting!
VARulle
<definition><cost><profits><opportunity-cost>
Is there a formal definition of opportunity costs, economic costs, and/or economic profits?
VARulle
# Is there a formal definition of opportunity costs, economic costs, and/or economic profits?

(Asked by: username_0 on 2021-05-26)

[*Edited to remove a mistake in a definition and some typos.*]

According to various econ textbooks (e.g. [this one](https://opentextbc.ca/principlesofeconomics/chapter/7-1-explicit-and-implicit-costs-and-accounting-and-economic-profit/)), there is a fundamental distinction between *economic profit* and *accounting profit*:

**(i)** While **accounting profit** subtracts only **explicit costs** (out-of-pocket costs) from revenue,

**(ii)** **economic profit** subtracts **opportunity costs**, also known as **economic costs**, which consist of explicit and **implicit costs**. Here,

**(iii)** opportunity costs are generally defined as the "value of the best foregone alternative". In the context of firms, "value" of course means profits.

**(iv)** Furthermore, when economists talk about "costs" and "profits", they always mean *economic costs* and *economic profits*. For example, firms are assumed to maximize economic profits.

All the definitions I found were verbal and informal as above. Is there a *formal* definition of these concepts? Do they make sense at all? Since I didn't find anything in the literature (by a quick search, admittedly), I tried to formalize it myself for the firm context:

Let $X$ be a set of alternatives the firm can choose from, and let $z\in X$. Let's assume that alternative $z$ yields revenue $r(z)$ and generates explicit costs $c\_e(z)$ for the firm. Accounting profit $\pi\_a(z)$ is defined as revenue minus explicit costs, so $\pi\_a(z)=r(z)-c\_e(z)$. Economic profit $\pi\_e(z)$ is then defined as accounting profit minus implicit costs $c\_i(z)$, thus $\pi\_e(z)=\pi\_a(z)-c\_i(z)=r(z)-(c\_e(z)+c\_i(z))=r(z)-c\_o(z)$.

Opportunity costs of choosing $z$ are the value of the best foregone alternative, so $c\_o(z)=\max\_{k\in X,\,k\ne z}\pi(k)$.

Now, what is the "profit" $\pi(k)$ here? Economic profit or accounting profit? This might not be immediately obvious. Let's have a closer look.

**Finite sets of alternatives**

Assume for simplicity that there are only two alternatives, $X=\{x,y\}$, with $r(x)>r(y)$.

**(i) Let's try economic profits, so $\pi(k):=\pi\_e(k)$:**

Then $c\_o(x)=\pi\_e(y)$ and $c\_o(y)=\pi\_e(x)$.

Therefore $\pi\_e(x)=r(x)-c\_o(x)=r(x)-\pi\_e(y)$. Now, analogously, $\pi\_e(y)=r(y)-c\_o(y)=r(y)-\pi\_e(x)$. Substituting, $\pi\_e(x)=r(x)-r(y)+\pi\_e(x)$, implying $r(x)=r(y)$. This is a contradiction, so this approach makes no sense.

**(ii) Let's interpret "profit" as accounting profit, i.e. $\pi(k):=\pi\_a(k)$.**

Then $c\_o(x)=\pi\_a(y)$ and $c\_o(y)=\pi\_a(x)$, so $\pi\_e(x)=r(x)-c\_o(x)=r(x)-\pi\_a(y)=r(x)-r(y)+c\_e(y)$. Similarily, $\pi\_e(y)=r(y)-c\_o(y)=r(y)-\pi\_a(x)=r(y)-r(x)+c\_e(x)$. This is at least not contradictory.

**Infinite sets of alternatives**

But there are rarely only two alternatives. Consider instead a standard quantity-setting monopolist. The set of alternatives is the set of quantity choices, so $X=\mathbb R^+$. Let's denote by $q^\*$ the firm's optimal choice. Then the firm accrues accounting profits denoted by $\pi\_a(q^\*)=r(q^\*)-c\_e(q^\*)$, where both $r$ and $c\_e$ (and therefore $\pi\_a$) are assumed to be continuous. What is the firm's economic profit $\pi\_e(q^\*)$ then?

Well, the firm's "(accounting) profit from it's best foregone alternative" is the accounting profit resulting from the smallest strictly positive deviation from $q^\*$, which, unfortunately, doesn't exist. To the rescue, let's interpret the "(accounting) profit from the best foregone alternative" as the supremum (instead of the maximum) of the foregone accounting profits: $c\_o(q^\*)=\sup\_{k\in X,\,k\ne q^\*}\pi\_a(k)$. This, however, is just $\pi\_a(q^\*)$. Then $\pi\_e(q^\*)=r(q^\*)-\pi\_a(q^\*)=c\_e(q^\*)$.

Economic profit of a profit-maximizing monopolist being equal to its out-of-pocket costs? This seems not to make sense.

So do the concepts of economic profit and opportunity costs as defined in (i)-(iv) make sense at all? Or which of (i)-(iv) above is wrong? Or is the "value" of the best foregone alternative in the context of firm decisions neither economic nor accounting profit of the best foregone alternative? What else?

**Tags:** definition, cost, profits, opportunity-cost
> **Comments:**
> **username_1**: I've seen opportunity costs interpreted as marginal rates of substitution, which do make mathematical sense.  
> **username_2**: My first question would be what are the units of economic profit, economic cost, and opportunity cost? Accounting customs record transactions at the historic price in money units {mu} or assign a valuation of items in money units. Utils have no dimension, like radian angle, so economic value is dimensionless. So one must  specify units of measure and the measurement procedure used to assign a number and a unit to each quantity of interest. Opportunity cost is value difference based on counter-factual reasoning. The dog could have caught the rabbit if he did not stop to chase his tail instead.  
> **username_3**: The definition of accounting profit quoted by the OP is too idealistic. In reality, "accounting profit" is an arbitrary number constructed in order to minimize liabilities such as tax payments. (Example: Amazon's accounts for 2020 show UK sales of $17.5bn, and UK tax payments of only £300m, i,e, a tax rate of about 2% that was achieved by exploiting loopholes in tax legislation)  Even at the level of small business accounting, one of the first questions a good accountant asks before drawing up the official accounts is "How much profit do you want to declare for last year?"  
> **username_0**: @MichaelGreinecker, I'm not sure if I understand this interpretation, but anyway it seems to be applicable only in a narrow consumer theory context.  
> **username_0**: @SystemTheory, I'd be happy with monetary units...  
> **username_0**: @alephzero, I'm not talking about real-world accounting profit but about the simplified concept used to distinguish between explicit and implicit costs. And it's not my invention, it's all over the econ textbooks.  
> **username_1**: @VARulle Marginal rates of substitution give you under the right regularity condition the value of increasing one variable in terms of the cost of decreasinganother variable. That works not just for consumer theory but for any setting where the function to be optimized and the constraint is sufficiently smooth.  
> **username_0**: @MichaelGreinecker, o.k., but then it doesn't work for a discrete choice set, right?  
> **username_4**: I had a [similar frustration](https://economics.stackexchange.com/questions/20908/what-is-opportunity-cost#comment59406_20926) with the concept of opportunity cost, and have since accepted that this concept, though intuitive, is open to individual interpretations/definitions, as this [well-known example](https://onlinelibrary.wiley.com/doi/abs/10.4284/0038-4038-2011.218) shows.  
> **username_1**: @VARulle Right.  
> **username_0**: @HerrK., thanks for the reference, this is very interesting!  

---

## 1 answers

### Answer by username_5 on 2021-05-26. Score: 2

## Economic profit

I think there are some problems with the formulation in your question, first you heavily focus on opportunity cost but note accounting profit does not even properly capture all revenue firm gets.

I will first focus on economic profit more broadly and at the end go back to opportunity cost.

Following Varian Microeconomic Analysis pp 24 (emphasis mine),

> **Economic profit is defined to be the difference between the revenue a firm receives and the costs that it incurs. It is important to understand that all costs must be included in the calculation of profit.** …
>
> Both revenues and costs of a firm depend on the actions taken by the firm. These actions may take many forms: actual production activities, purchases of factors, and purchases of advertising are all examples of actions undertaken by a firm. At a rather abstract level, we can imagine that a firm can engage in a large variety of actions such as these. **We can write revenue as a function of the level of operations of some $n$ actions, $R(a\_1,...,a\_n)$, and costs as a function of these same $n$ activity levels, $C(a\_1,...,a\_n)$.**

The above is already rigorous definition for *economic* profit, *economic* revenue and *economic* cost.

Note accounting operates under its own sets of rules. **Accounting, is not econometrics. Point of accounting is not to accurately estimate or measure economic relationships**. So what you are doing above, defining economic profit at simply accounting profit minus opportunity cost is somewhat misguided (it would only hold in an unrealistic special case where we assume accounting only missed opportunity cost but accounting misses so much more).

First, following Varian we can start with a set of occurring actions $a$ that create revenue and costs. However, when it comes to accounting, not all of these will be reported. For example l, consider selling goods without issuing receipt to avoid paying taxes, which will not get reported as accounting profit. The same holds for many costs other than opportunity cost. For example, consider how depreciation costs are treated by accountants. From purely economic perspective most assets definitely do not depreciate in linear fashion yet accounting allows firms to use linear depreciation for most assets regarding what the true depreciation cost each period is. So the accounting depreciation costs will too be different from economic depreciation cost. Or consider the fact that many countries allow small business to just in their accounting declare portion of their profit as cost without actually even proving they actually incurred these costs (this is done in some countries to ease administrative burden for sole traders and similar one person businesses).

Consequently, it makes more sense to rather define accounting profit in terms of economic profit and then work backwards by inverting those functions if you would want to get back to economic profit from accounting one.

Call $A=(a\_1,...,a\_n)$. set of all economic actions that actually occur, and $R(A)$ would be *economic* revenue from those activities. Now apply some function $\rho$ which tells you based on $R(A)$ what actually accounting revenue is.

So accounting revenue $R\_{acc}$ would be:

$$R\_{acc}= \rho(R(A))$$

Similarly accounting cost $C\_{acc}$, we will have actual economic cost firm incurs $C$ based on set of actions it takes $A$ and then we will have some function gamma that tells us what enters into accounting and what not so we would have that

$$C\_{acc} = \gamma (C(A))$$

Finally the **accounting profit** would be difference between the above so:

$$\Pi\_{acc} = \rho(R(A)) -\gamma (C(A))$$

Note what $\rho(.)$ and $\gamma(.)$ are will depend on local accounting regulations and laws. in some country government could declare that any firm has to put always just 1000€ as their cost (yes it is unrealistic example but I am just showing it to prove a point) and in such case $\gamma(C(A))=1000$ regardless of anything that happens.

Whereas the true economic profit would just difference between economic revenues and economic costs:

$$\pi=R(A)-C(A)$$.

If you would want work your way back from accounting definitions then you could use inverse functions and economic profit calculated from accounting one would be:

$$\pi= \rho^{-1}-\gamma^{-1}$$.

Note the above does not really require definition of opportunity cost since here simply the function $C(A)$ is just some function where we assume that $C(A)$ assigns true economic cost to every action $a\_1,…,a\_n$, for standard profit optimization it’s completely okay to treat $C(.)$ as a black box and thus you will not really require separate definition of opportunity cost. As a matter of fact note in situation where there are no alternatives like having mana (from the biblical story where it could only be immediately consumed not stored or used for anything else) opportunity cost would not even be part of $C$ but still there would be distinction and disconnect between accounting and economic profit.

For example if we have $C(A)=c\_1 a\_1+c\_2 a\_2… c\_n a\_n$ and if we define $C$ as a function that always assigns true economic cost then you definition $c\_1$ includes all economic costs including opportunity cost, but the economic profit, economic revenue and economic costs are still rigorously defined (even if the constituents of costs such as opportunity costs aren’t).

## opportunity cost

Next tackling the opportunity cost separately (which can be later used to build function $C$), good places to start are [Buchanan (1987)](https://link.springer.com/chapter/10.1007/978-1-349-21315-3_69) or [Alchian (1968)](https://www.encyclopedia.com/social-sciences-and-law/sociology-and-social-reform/sociology-general-terms-and-concepts/cost). The authors do not necessarily derive rigorous opportunity cost functions but give rigorous verbal definitions on which we can build.

For example, according by to [Buchanan (1987)](https://link.springer.com/chapter/10.1007/978-1-349-21315-3_69) :

> Opportunity cost is the anticipated value of 'that which might be' if choice were made differently. Note that it is not the value of 'that which might have been' without the qualifying reference to choice. In the absence of choice, it may be sometimes meaningful to discuss values of events that might have occurred but did not. It is not meaningful to define these values as opportunity costs, since the alternative scenario does not represent a lost or sacrificed opportunity. Once this basic relationship between choice and opportunity cost is acknowledged, several implications follow.

> First, if choice is made among separately valued options, someone must do the choosing. That is to say, a chooser is required, a person who decides. From this the second implication emerges. The value placed on the option that is not chosen, the opportunity cost, must be that value that exists in the mind of the individual who chooses. It can find no other location. Hence, cost must be borne exclusively by the chooser; it can be shifted to no one else. A third necessary consequence is that opportunity cost must be subjective. It is within the mind of the chooser, and it cannot be objectified or measured by anyone external to the chooser. It cannot be readily translated into a resource, commodity, or money dimension. Fourth, opportunity cost exists only at the moment of decision when choice is made. It vanishes immediately thereafter. From this it follows that cost can never be realized; that which is rejected can never be enjoyed.

since the opportunity cost is the value of the next best alternative you can rigorously define as follows.

Suppose that for each action that actually was taken in the set $A=a\_1,…,a\_n$, there are alternative sets of alternative actions (e.g. each $a\_i$, has some alternative $b\_i$, $z\_i$ and so on, and let’s assume that all the other alternatives are mutually exclusive with $a$. Well then opportunity cost of each action $a$ is just value of the next best alternative so if we have that:

$a\_i\succ b\_i \succ z\_i … $

Then opportunity cost will simply be $V(b\_1)$ where $V$ is simply just some function that assigns monetary value (or in consumer problem utility to $b\_i$).

If you want to then put this value into profit function explicitly then simply the cost function will be some composite function given by $C(a\_i, V(b\_i))$ (note this is ultimately just function of $a\_i$ because the next best alternative necessarily depends on what you choose as $a\_i$, so in the end it’s just more complex way of writing $C(a\_1))$

I think the above is rigorous even though it’s very simple (of course perhaps the notation I use could be made more precise).

> **Comments:**
> **username_0**: Economic profit: But I didn't make this up myself, I just follow the simplified usage of this term in econ textbooks (see link in edited version).  
> **username_0**: Opportunity costs: It seems that your setting is just an extended - but still finite - version of my 2-alternatives setting. But what about the (continuous) monopoly example?  
> **username_5**: @VARulle but you are citing **undergrad** textbook and not even one from really reputable author (for example Mankiw actually includes more nuance discussing, albeit very briefly in his textbook - although Mankiw undergrad textbook also does not include enough nuance). In addition also the open textbook elsewhere states "accounting profit is a cash concept" <- this is actually more nuanced because as written in my first half of the  answer it directly means that only cash/easily quantifiable transitions are included, so it excludes any hard to quantify transaction be it on rev or exp side  
> **username_5**: @VARulle you simply cannot expect undergraduate textbooks to provide rigorous treatment of profit in the same way as for example MWG or Varian does. Undergraduate physics textbooks also lack nuance and often do not explain terms rigorously. In addition, textbooks vary in quality the OpenStack textbooks are good enough given that they are completely free but they are not that far from just economics for dummies lets say  
> **username_5**: @VARulle regarding the opportunity cost the problem with your monopoly example is that you define the opportunity cost in terms of second best accounting profit that is definitely not correct. Next, in order for something to create opportunity cost it must be mutually exclusive with action you are taking. If you action a_1 is to produce Q=100 then action b_1 Q=50 is not mutually exclusive action since producing 50 units of the same product would be just subset of producing 100 units of the same product. Rather you should calculate opportunity cost from mutually exclusive activity  
> **username_0**: That's my problem: These concepts seem only to be discussed and (vaguely) defined in undergrad textbooks. I was wondering whether there actually is some more formal treatment. But in MWG the word opportunity costs doesn't even appear. Moreover, I'd expect that even undergrad textbooks present some *consistent* version of these concepts.  
> **username_0**: "accounting profit is a cash concept": Yes, that's how I also use it here, revenue minus explicit (i.e., out-of-pocket) costs. This is talking about the producer theory model world of course, so I don't know what a "hard to quantify transaction" even means there.  
> **username_5**: @VARulle but the graduate textbooks have rigorous definitions of profits which was still part of your question. Also, I agree that undergraduate textbooks should be more consistent but what you gonna do on that front :/, I think you see more consistency when you compare top textbooks like Mankiw vs Knight or Krugman, and you will see more deviations if you take textbooks from some unknown authors. Regarding opportunity cost the problem is that for 99.9% of applications it’s fine to treat $C(.)$ as a black box with just some regularity conditions. I don’t think I ever come across  
> **username_5**: Application where it would be necessary to explicitly model opportunity cost. As a result this is something that will rarely be done. I think way I introduced above would be valid way that then can be extended to various other situations  
> **username_5**: @VARulle but there are a lot of things that are not easily to quantify. For example, from labor economics you don’t get wage just in (explicit) terms of money but also in terms of implicit benefits eg nice and safe working environment. The same holds for firm itself. Cash transactions here would mean transactions that explicitly have some price tag attached to them  
> **username_0**: I also can't wrap my head around this "mutually exclusive actions" thing... My view is: You can choose between black boxes A and B. A gives you 10 dollars, B 15 dollars. If you choose A, you give up the opportunity of choosing B, so opportunity cost of choosing A is 15. You are telling me that, no, it raher depends on what happens inside the black boxes?  
> **username_5**: @VARulle I am just saying that just because you list arbitrary two choices that does not mean they are mutually exclusive. Producing 3 liters of water is not mutually exclusive with producing 2liters of water if by producing 3 liters of water you produced two liters of water. If A and B can happen at the same time by definition they are not mutually exclusive. Saying I assume A and B are mutually exclusive events but then picking events that are not mutually exclusive will be inconsistent. Its like asking so  if a > b does the inequality depend of what a and b are well yes  
> **username_5**: you can't pick A= 5 and B= 10 and have the inequality hold  
> **username_0**: I just can't imagine why mutual exclusivity should be important for the calculation of opportunity costs. You seem to suggest that even if I know all the revenues and explicit costs of A and B I cannot calculate the opportunity costs of choosing A as long as I don't know whether the production processes hidden in A and B have some overlap.  
> **username_5**: @VARulle but the mutual exclusivity is definitional condition of opportunity cost. If you can at the same time do both A and B you are not giving up anything. The opportunity cost is created (as the articles I referenced explain in greater detail) when there is some alternative thing you could be doing. This is why using mana (in biblical sense) has no opportunity cost - you can only eat it not save it or use it for some alternative use. And yes you can’t calculate opportunity cost if you don’t know if actions are mutually exclusive, like you can’t calculate x if x=10+y and you don’t know y  
> **username_5**: @VARulle I would really recommend reading the two articles cited in the second section they are not even long  
> **username_0**: Let A and B be mutually exclusive actions you value at 8 and 10, respectively, e.g. A is "drink 1 free beer" and B is "eat 1 free soup". Assume you can only choose one of these options at your favorite restaurant. If you choose A, you incur opportunity costs of 10, so profit is -2. If you choose B, you incur opportunity costs of 8, so profit is 2. You better choose B. That's standard, right? (contd. ...)  
> **username_0**: (... contd.) Now drop the mutual exclusivity assumption, e.g. A is "drink 1 free beer" and B is "drink 2 free beers". What changes in the analysis? Nothing! Of course drinking 2 beers you don't forego drinking 1 beer - but you do forego the opportunity of drinking *only* 1 beer and then stopping consumption. Mutual exclusivity plays no role in this.  
> **username_0**: I have also read both the Buchanan and the Alchian articles, but they do not provide the *formal* definition I asked for and don't help a lot.  
> **username_5**: @VARulle I told you in advance they don’t have formal derivation but mention mutual exclusiveness and have more explanation. Also, the point about mutual exclusivity here is that the action should not be subset of another action, action of drinking 1 beer is just subset of action where you drink 1 bear and then drink second beer. As those sources explain an example of a situation where there is no opportunity cost is a situation where there is only choice of one good available without any alternative there is no opportunity cost regardless of how much you consume of that good.  
> **username_5**: @VARulle I think someone else in comments mentioned that in some work opportunity cost is equated to marginal rate of substitution, that is a way of measuring opportunity cost *on the margin* when deciding between two goods x and y as it tells you what sacrifice you have to make in terms of not consuming y to consume 1 more x, but it makes no sense to try to construct marginal rate of substitution if in model there is only one good let’s say U=x, then there is no opportunity cost as consumer can only consume x no other option available  
> **username_0**: Funnily enough, Buchanan mentions mutual exclusivity only in an example that actually contradicts your rule: "*There are two mutually exclusive thermostat settings for a building, High and Low.*" I guess in your interpretation "heating up to 18°C" is a subset of "heating up to 24°C", so there should be no opportunity costs of either choice. That's just the opposite of Buchanan's conclusions.  
> **username_5**: @VARulle but you are getting too hang up on the example. The point is that they have to be mutually exclusive. Buchanan simply by construction claims that $High_T ∩ Low_T =∅$, we can now discuss whether that is a good representation of real life High vs Low temperature but in his case mutual exclusivity will hold by construction, so all you debate in previous comment is whether that is IRL mutually exclusive or not.  Going back to our previous example if we denote two action sets $A=\{x_1\}$ where $x_1$ is manufacturing first discrete Q of output, and then there is second set $B=\{x_1, x_2\}$  
> **username_5**: where $x_1$ is manufacturing of first unit and $x_2$ manufacturing of second. Here $A ∩ B \neq ∅$ and in fact $A⊆B$ so if you choose B, A is not a valid mutually exclusive action for opportunity cost calculation. If you would choose A then B would also not be valid mutually exclusive action but subset of B call it $B_1 = \{x_2\}$ would still be. This is how I think about it terms of set theory. Anyway, if you are willing to accept mutual exclusivity, then we are just arguing about whether temperature example was good example or bad example of mutually exclusive event not about OC definition  
> **username_0**: But I'm not willing to accept mutual exclusivity as necessary for opportunity cost calculation, because I have not yet seen a single example where violating it would result in incorrect opportunity cost calculation (and therefore wrong choice).  
> **username_5**: @VARulle but that is by definition part of opportunity cost. That is like saying you are not willing to accept that by definition in economics cash is not capital. Well then we are just dealing with semantics. Regarding your requirement how can opportunity cost ever change a  decision. By definition, opportunity cost is the value of second best forgone alternative, if you have option of being gardener and accountant so we take acc wage as oc for being gardener, if that makes you to decide being acc then acc can no longer be oc as now it is your first choice and so oc will be different  
> **username_5**: the mutual exclusivity changes the profit calculation as without mutual exclusivity there is always a second best option just to replicate the first option -  in such case economic profit would not be possible, in fact if you are not willing to accept mutual exclusivity concept of opportunity cost is meaningless as then it is just value of your best option since you can always make arbitrarily many non-mutually exclusive options like option A: on fork go left vs B:on a fork go left but it won't affect the choice since what opportunity cost is, is conditional on choice you make *by definition*.  
> **username_5**: so first picking some choice is antecedent to having opportunity cost  
> **username_0**: How can you say "by definition" if there is no formal definition anywhere in the literature? "Mutually exclusive" for me just means that if the set of feasible actions is {a, b} then you can choose either a or b, but not both a and b. In your understanding you need to know something about what kind of actions a and b are. 1 beer or 1 wine - good, 1 beer or 2 beers - bad? I'd say calling the first case exclusive and the second case not exclusive is pure semantics. And no, you can't just replicate an action, because if a = "go left" and b = "go left", then {a, b} = {a} and there is no choice.  
> **username_5**: @VARulle but there are definitions in literature, just because a definition is not mathematically formalized that does not make it not a definition. In fact, cash not being capital is not mathematically defined either. In fact the Buchanan definition is even listed as an entry in the official Palgrave dictionary of economics making it at least somewhat authoritative. In addition, the mutual exclusivity is implicit even in the textbook you listed and also Mankiw etc, because you will never find an example where someone would consider lower consumption of same good for opportunity cost  
> **username_5**: @VARulle  Next, you can then just additionally mathematically formalize it if you want. Note if you want to define opportunity cost in different way that would be possible but defining it in way that excludes mutual exclusiveness would go contrary to how OC is currently used. That is like asking why we should define capital in a way that excludes cash? You can have consistent definitions of capital that do include cash (e.g. finance). Same way you could have definition of OC that would make all economic profits zero (but I never seen such definition ever used in economics and it is not useful)  
> **username_5**: Lastly, again just because some definition is informal, that does not mean its not definition, you can just formalize the informally given definition that will follow the conditions set in the informal definition. As long as those conditions do not create contradictions there is no issue with that. Concept of utility existed long before it was formalized and made more precise, lack of formalized definition did not meant utility can be anything whatsoever  
> **username_0**: A definition has to be exact, otherwise it is open to discussion, as the current one shows. A formal definition would be exact, the Buchanan "definition" is not. He uses only discrete choice examples where there is no problem anyway, and the thermostat example even seems to run counter to your requirement. If a "definition" is so vague that it is left unclear whether "consuming" 18°C is a subset of "consuming" 24°C, then it is not a definition.  
> **username_5**: @VARulle  You are right that if there some ambiguity in definition there is space for discussion, but that is not equivalent of saying well anything goes. In biology some animal categories are not 100% exactly defined so there might be discussion whether some sea critter should be classified as cephalopods or crustaceans because the animal might not neatly fall into either category, but that does not mean that the sea critter can be arbitrarily labeled mammal. The mutual  exclusiveness is not only feature of the Buchanan definition but also others whether explicitly or implicitly  
> **username_5**: @VARulle hence if you want to create a formal mathematical definition it should respect that. Or alternatively, if you want new definition it should at least be useful, defining OC in a way that would mean all economic texts would have to be rewritten and we would have to redefine our understanding of economic profit would not be useful at all. It would be like trying to now redefine capital in economics to include cash,  just on account that there likely is no formal definition of capital as set that does not include cash but only described verbally  
> **username_5**: @VARulle by the way lets go back to also what you said ", you can't just replicate an action, because if a = "go left" and b = "go left", then {a, b} = {a} and there is no choice." <- this is exactly my point. Consider following actions: first a composite action A: go left, go right at next turn; next simple action B: go left. You would claim that these are distinct action but that is just because actions A is a composite action. In fact action A is just $A= A_1 ∪ A_2$ where $A_1$ is go left an $A_2$ is go right at the next turn. Then exactly $\{A_1,B\} = \{A_1\}$ <- there is no choice at all  
> **username_5**: and for $A_2$ you simply do not have any alternative action. If there would be some alternative action for $A_2$ like second $B_2$ action where on a second turn you go left you could talk about opportunity cost between $A_2$ an $B_2$ but there cannot be an opportunity cost created when choosing between $A_1$ and $B_1$ as there is no choice  
> **username_0**: But if the action set is defined as $\{A, B\}$ then there is no choosing between $A_1$ and $B$, simply because $A_1$ is not in the action set.  
> **username_0**: See, in the binary choice case my formal definition is easily used to compute opportunity costs: If action set = {A,B} then opp.costs(A) = u(B) and opp.costs(B) = u(A). Then profit(A) = u(A)-opp.costs(A) = u(A)-u(B). Similarly, profit(B) = u(B)-opp.costs(A). So a utility-maximizing agent chooses the option with the higher profit and vice versa. With your requirement you can't even calculate opp.costs, because you don't know whether or not A and B are "mutually exclusive". This makes no sense to me.  
> **username_0**: And let me repeat: The Buchanan article doesn't require "mutual exclusivity" anywhere in the text, as far as I remember. If I am wrong, please give me a reference.  
> **username_5**: @VARulle 1. Buchanan mentions mutual exclusivity when he constructs the example 524. 2. "But if the action set is defined as {A,B} then there is no choosing between A1 and B, simply because A1 is not in the action set" <- but A_1 is simply a subset of set A so it is in the action set.  "With your requirement you can't even calculate opp.costs, because you don't know whether or not A and B are "mutually exclusive"" <- sorry but this is strawman, I gave you clear condition for mutual exclusivity $A∩B= ∅$  so you can know at all times whether two actions are mutually exclusive  
> **username_5**: 3. "Then profit(A) = u(A)-opp.costs(A) = u(A)-u(B). Similarly, profit(B) = u(B)-opp.costs(A). " <- I think I misunderstood what you were saying before, you can use the opportunity cost where B and A are mutually exclusive in this way  
> **username_5**: @VARulle 4. Note in addition, every textbook and also Buchaman and the other article I cited and also specifically states that opportunity cost is cost of forgone alternative. I mean this directly implies mutual exclusivity You can't claim that when you are choosing between consumption of 2 or 3 apples, by consuming 3 apples you are forgoing consumption of two apples, when consumption of two apples is necessarily pre-requisite for consumption of 3 apples. What even more the definitions mention scarcity as well. Scarcity has far better and rigorous mathematical definition  
> **username_5**: which requires some demands not being met in optimum (see Giskards answer here for rigorous definition https://economics.stackexchange.com/a/32689/15517). Here if the choice is only between 3 and 2 apples then there is no scarcity in the problem in the first place and thus there cannot be any opportunity cost by definition either  
> **username_0**: Sorry, but I don't know what "example 524" is. I'm referring to the Palgrave reprint of Buchanan's article at https://link.springer.com/referenceworkentry/10.1057%2F978-1-349-95121-5_1433-2, that does not mention mutual exclusivity except for the thermostat example.  
> **username_0**: "*but A_1 is simply a subset of set A so it is in the action set*": In my toy model, A is an action, not a set, so there is no "subset of set A". If you insist on modeling actions as sets, and require that subsets of actions are actions themselves, e.g. A ={A1, A2}, where A1 happens to be identical to B, then the action set in my model is simply {A, A1, A2}. Your understanding of mutual excl. then says that A has no opp.costs, but with my definition there's no reason why {A, A1, A2} should be treated differently from {A, X, Y}.  
> **username_0**: The reason is that if a child chooses to eat 5 pieces of cake and then feels sick, it's mother might well ask "Why didn't you stop after the second piece?", thereby pointing to the *foregone opportunity* of choosing N = 2 instead of N = 5, without any problems due to alleged "non-mutual-exclusivity" of these two alternatives.  
> **username_5**: @VARulle example on page 534. Thats what I meant, he then proceeds to build an example of opportunity cost base on that that  
> **username_5**: "In my toy model, A is an action, not a set, so there is no "subset of set A"." But this is like saying in your model you insist on splitting costs total costs into variable costs and fixed costs. You just aggregate actions that I prefer to keep separate because it is more useful.  
> **username_5**: @VARulle but the problem is that in your cake eating example there are no alternatives. Again, just break it down into the discrete actions. If you are eating 2 pieces of cake what you are doing is to commit yourself to action plan $A_1=\{ e_1, e_2\}$, if you want to eat 5 pieces of cake $A_2= \{e_1, e_2, e_3, e_4, e_5 \}$, it is simply impossible to eat 5 pieces of cake without first going through process of eating 2. The point is that either under $A_1$ and $A_2$ action $e_1$ and $e_2$ are present so $e_1$ and $e_2$ are not forgone, you still had to took those actions  
> **username_5**: @VARulle in addition your concept of opportunity cost simply does not work with rest of the economics. For example, in Mankiws principles on pp 25 Mankiw claims (with reference to PPF on pp24) that when we move from point A at PPF A= (2200PC, 600Cars) to point B=(2000PC, 700Cars) the OC of 100 extra cars will be 200 PC, but if I would follow your claims it should 200 PC + 99 Cars because you would claim that we are giving up the option to produce just 99 Cars as well <- that does not mesh with OC definitions and usage in current economics (I can give you analogous ex from almost any txbook)  
> **username_0**: You are misrepresenting my claims. In my model of the PPF example, if you can choose between A = (2200PC, 600Cars) and B = (2000PC, 700Cars) then there simply is no possibility to "produce just 99" extra cars. If you want to include this choice possibility, then you have to add it to the choice set, so there is an additional option C = (2002PC, 699Cars). There's nothing here to worry about.  
> **username_0**: Moreover, the Mankiw example is just another example for an informal definition that doesn't fit with the standard (but also informal) definition: What is the "value of the best foregone alternative" if you choose B? This depends on what you mean by "value" and by "best". The Mankiw example is just a technical tradeoff and can't tell you whether A or B is the "better" choice.  
> **username_5**: @VARulle what you mean misrepresentation, you can by definition choose any combination of cars within PPF point C not shown in Mankiw example but we could simply draw it in with C=(2002PC, 699Cars) is by definition in the choice set otherwise what is in the picture on pp 24 would not be PPF. So not only C is in a choice set but all other points within PPF. And yes Mankiw does not offer formal definition of opportunity cost, but the point is that your definition conflicts will all the informal definitions out there while mine is completely consistent.  
> **username_5**: The thing about definitions is that they are ultimately semantics any formal definition is in itself valid definition as long as it is internally consistent. For example, re-defining dogs as rocks, will be valid definition in a system where you simply define rocks to be also dogs, but introducing such new formal definition into biology would not make sense even if maybe you can make it more precise and rigorous than some less formal definition of a dog biology has. My point is if you want to construct a formal definition of OC it should respect the usage of OC in economics.  
> **username_5**: Or let me give you another analogy. For example, rationality in economics is nowadays rigorously defined (you can have look at MWG ch 1). But that was not always so, in times of Smith and Ricardo, rationality was only vaguely defined and even the first more narrowed (but still mathematically informal) definition by Mill was bit vague (equivalently to notion of OC in Mankiw), but then when 19-20 cent. economists formalized OC they did not went and just said screw all economics before today and make up arbitrary definition of rationality, they tried to formalize the notion of rationality  
> **username_5**: as it was used in economics. What you are doing basically is on fundamental level, you are dismissing the way how opportunity cost currently works in economics, just because textbooks do not present rigorous mathematical model, and then you construct rigorous formal definition of opportunity cost that has clear contradiction with usage of OC in economics. Well you can do but what is a point of formalizing definition if you are not respecting its usage in literature but make your own? You are not really formalizing the informal definition but making a new one.  
> **username_5**: Let us [continue this discussion in chat](https://chat.stackexchange.com/rooms/124945/discussion-between-1muflon1-and-varulle). We already generated too many comments and the comments are getting flagged by automatic system  
> **username_0**: Yes, let's move to chat.  
> **username_6**: This:https://pressbooks.bccampus.ca/uvicecon103/chapter/1-2-opportunity-costs-sunk-costs/ seems to contradict what you said. See table 1.2b.  
> **username_5**: @Shinrin-Yoku it does not contradict anything I say here in fact that site says pretty much exactly I do here just in more length and convoluted way  
> **username_6**: Thanks for the response! I think that site(table 1.2b) contradicts  the Alchian paper you linked, since it seems to add the explicit cost of A to the opportunity cost of A(as defined by Alchian) to get the opportunity cost of A(per the site).  
> **username_6**: So is there a contradiction?  
> **username_5**: @Shinrin-Yoku they add explicit costs there because the money you use on A could be saved or to buy something  
> **username_6**: But that contradicts answers here: https://economics.stackexchange.com/questions/20908/what-is-opportunity-cost/20953?noredirect=1#comment86581_20953 are these just alternative definitions.  
> **username_6**: Are you using a different def?  
> **username_6**: This is really killing me so an answer would be much appreciated  

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