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

According to various econ textbooks (e.g. this one), 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?

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    $\begingroup$ I've seen opportunity costs interpreted as marginal rates of substitution, which do make mathematical sense. $\endgroup$ May 26 at 16:59
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    $\begingroup$ 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. $\endgroup$ May 26 at 19:20
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    $\begingroup$ 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?" $\endgroup$
    – alephzero
    May 27 at 1:32
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    $\begingroup$ @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. $\endgroup$ May 27 at 8:19
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    $\begingroup$ I had a similar frustration 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 shows. $\endgroup$
    – Herr K.
    May 27 at 19:09
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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) or Alchian (1968). 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) :

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).

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  • $\begingroup$ 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). $\endgroup$
    – VARulle
    May 27 at 14:24
  • $\begingroup$ 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? $\endgroup$
    – VARulle
    May 27 at 14:27
  • $\begingroup$ @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 $\endgroup$
    – 1muflon1
    May 27 at 15:42
  • $\begingroup$ @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 $\endgroup$
    – 1muflon1
    May 27 at 15:45
  • $\begingroup$ @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 $\endgroup$
    – 1muflon1
    May 27 at 15:49

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