<p>Greg Mankiw's answer:<a href="https://i.sstatic.net/Wss9i.jpg" rel="noreferrer"><img src="https://i.sstatic.net/Wss9i.jpg" alt="enter image description here"></a></p>

<p>plus extra characters </p>

CC BY-SA 3.0
2017-05-19T06:55:14.277Z
But why is this an explanation of **why we do not model profits**? Are implicit costs modeled? Can you elaborate? At the moment, it's not a helpful answer.
luchonacho
CC BY-SA 3.0
2017-05-19T09:43:39.017Z
one word for two concepts @luchonacho, economic models of perfect competition have profit, it's just that they are equal to zero in equilibrium. Zero economic profit corresponds to positive accounting profit. You should check Mankiw's textbook if you need more explanations, it has several chapters on perfect competition with and without free entry. Profits are not just modeled, they are central to the model, since they are the object being maximized. Zero economic profit is an optimum: without optimization economic profit could be negative (while accounting profit could remain positive).
PatrickT
CC BY-SA 3.0
2017-05-19T10:01:56.843Z
Maybe I'm not being clear. I understand the issue of economic profits, competition, optimisation, etc. I just do not understand why we should not be interested in modeling accounting profits, or ownership of firms, explicitly.
luchonacho
CC BY-SA 3.0
2017-05-19T10:11:54.777Z
@luchonacho, economists are interested in modeling ownership of firms explicitly of course, and you can find studies on that in the __Journal of Finance__ and other top journals, but in the basic, textbook model of perfect competition, ownership is irrelevant: if ownership were relevant, competition would not be perfect. You may also be interested in looking into Modigliani-Miller theorem for related irrelevance results in the idealized world of perfect information, perfect competition, perfect rationality, no distortions, no taxes, etc..
PatrickT
CC BY-SA 3.0
2017-05-25T10:46:56.373Z
See my answer, which is the type of explicit analysis I was looking for.
luchonacho
CC BY-SA 3.0
2017-03-25T11:12:21.027Z
2017-03-25T11:12:21.027Z
PatrickT
<blockquote>
  <p>It seems economic models do not explicitly include ownership of the firm (not of the physical capital, an input used to produce).</p>
</blockquote>

<p>Many economic model <em>do</em> have profits. We're just more careful about what we call it. See the definition of "<a href="http://en.wikipedia.org/wiki/Profit_%28economics%29" rel="noreferrer">economic profit</a>." When you think of profits, you need to ask yourself if the money you are making is simply a fair exchange for the costs you incur---explicit and implicit costs---or whether some portion of that profit represent money gained above and beyond the (true) costs of production (see "<a href="http://en.wikipedia.org/wiki/Economic_rent" rel="noreferrer">economic rent</a>"). Indeed, this is the only sensible way to think about the world.</p>

<blockquote>
  <p>It seems economic models do not explicitly include ownership of the firm</p>
</blockquote>

<p>Many economic models <em>do</em> in fact include ownership of the firm. However, most economic models are built for demonstrating some particular idea in relative isolation, as opposed to trying to model every single detail. So if someone wants to demonstrate some effect in a world without economic rents, then a lot of the time modeling the ownership of the <em>firm</em> is less important than modeling the ownership of the <em>capital</em>---and thus it might be excluded. </p>

<p>If you want to see models that dig deep into the issues of economic rents and the ownership of a firm, I would recommend taking a look at the subfields called "<a href="http://en.wikipedia.org/wiki/Industrial_organization" rel="noreferrer">industrial organization</a>" and "<a href="http://en.wikipedia.org/wiki/Corporate_finance" rel="noreferrer">corporate finance</a>." (For example, take a look at <a href="http://rads.stackoverflow.com/amzn/click/0691125562" rel="noreferrer">Jean Tirole's book on corporate finance</a>.</p>

<blockquote>
  <p>Meanwhile, in real life, even in competitive markets, firms do make profits on the long run.</p>
</blockquote>

<p>This is a bold claim. If you want to be scientific, you're going to have to be precise about how you (1) define profits and (2) define competitive markets, and (3) show (with data and sound analysis) that firms very commonly do make profits as you've defined them. This is how an economist would operate.</p>

<blockquote>
  <p>Firms are just an ethereal "thing" that take inputs and produce output with a technology freely available.</p>
</blockquote>

<p>Hidden in this comment is an very interesting object of study. What is a firm, <em>really</em>? 
This was an important question (first?) and most famously raised by <a href="http://en.wikipedia.org/wiki/Ronald_Coase" rel="noreferrer">Ronald Coase</a>. How do we define a firm? Why do they exists? etc. See the "<a href="http://en.wikipedia.org/wiki/Theory_of_the_firm" rel="noreferrer">theory of the firm</a>."</p>

CC BY-SA 3.0
2015-02-14T23:16:50.563Z
If you change to *some firms to make profits on the long run*, I'm pretty certain that you can show that (long run being over the span of data available). I don't think he meant *all* firms, as obviously many fail and exit.
FooBar
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2015-02-14T23:17:33.783Z
Certainly. But I'd also like to point out that it's not something that should be taken for granted.
jmbejara
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2015-02-15T19:22:45.543Z
I get the point of profits and rents over the opportunity cost but that is exactly what I mean. It is very likely that the opportunity cost is **endogenously** determined and so it is interesting to model (e.g. think of Marx's rate of profit). Why does it need to be obscured/hidden behind "capital", which usually refers to physical capital, no about ownership of equity. In fact, people usually don't own the assembly lines, the aircrafts, the labs, etc, but the shares of companies. I have seen models of ownership, specially in macro, my area. I'm just puzzled why is is not a mainstream thing.
luchonacho
CC BY-SA 3.0
2015-02-14T23:07:44.987Z
2017-03-25T09:45:44.533Z
2017-03-25T09:45:44.533Z
jmbejara
<p>I assume that this is about Macroeconomics in particular (you might want to add that tag).</p>

<h3>Often we don't need profits</h3>

<p>The interaction of firm profits is often irrelevant. It is well summarized under capital rents. Having firm profits does not add any insights to, say, Solow's growth model or the standard Neoclassical model.</p>

<h3>Operative profits vs Life time profits</h3>

<p>In some models, however, we do need profitability of firms. That typically comes from additional costs of running the firm. For example </p>

<ul>
<li>Vacancy costs (search and matching model)</li>
<li>Entry costs (patents in Romer's growth model)</li>
</ul>

<p>Firms run positive operative profits, which exactly offset the entrance costs. That is, life time profits are zero. </p>

<p>Why? Because we start in many models from the first-best and gradually remove particular assumptions. Romer could have talked in his model about a world in which patents have limited enforceability, or competitors particularly catch up, but the standard models we research ( and teach) typically do not incorporate this.</p>

<p>Later on, Kortum (1997) and Aghion, Howard, Howit create research based growth models that allow for catching up of competitors. </p>

<h3>Profits are messy to model</h3>

<p>Profits mess up a lot of things. Think about the neo keynesian model, a standard example of where firms make profits (positive or negative) in every period, which do not necessarily cancel out over time. Allow for heterogeneity in households assets: some are poor, some are rich.</p>

<p>Perfect representation of the owner in the decision making process of the firm requires that the stochastic discount factor (SDF) of the firm represents the SDF of the owner. But, in this model, who is the owner? Rich and poor agents share the firm, with different degree of ownership. To do this properly, we would need to add a stock market for the firm and allow agents to trade it; track the ownership of the firm and adjust the SDF. </p>

<p>But no, you can't even look at the share of ownership: Say, $w$ denotes the wealth level, $m(w)$ denotes the measure of firm assets hold by agents with wealth $w$, and $\rho(w)$ denotes the discount rate of agents with wealth level $w$. Can you then just compute the firms SDF $\rho_F$ with</p>

<p>$$\rho_F = \int \rho(w)m(w) dw$$?</p>

<p>In reality, firm decisions are often voted over by simple majority. You'd need to model coalitions that get votes of $50%$ of asset holdings and see what SDF that gives you. </p>

<p>Of course, this is much too complicated to do correctly. Even the intermediate step with an asset market is not trivial, people very often fall back to assuming the SDF of the richest agents, predicting their model's outcome. But it's not nice: We like to have internality consistent models, and profits make this difficult.</p>

CC BY-SA 3.0
2015-02-15T19:09:43.593Z
I don't see how profits are well summarized in capital rents. Theoretically, capital depreciates, advances technologically (more productive), reaches a steady state. Empirically, capital is always measured as stock of machines and buildings. This is very different from equity (anyone can turn savings into shares, increasing their price if more value not created). It might be difficult to model profits per se (beyond the models you mention) but perhaps we are not trying hard enough. It might be better to have profits in with a very trivial ownership mechanism that having no profits at all.
luchonacho
CC BY-SA 3.0
2015-02-15T19:14:51.663Z
@luchonacho But you certainly agree that under the stochastic discount factor of the representative agent, expected utility of rents to capital and expected utility from profits are the same, right? And ex-ante expectations (as opposed to ex-post realizations) are the relevant metric (since they appear in all the objective functions).
FooBar
CC BY-SA 3.0
2015-02-15T19:29:53.527Z
sorry, I can't see this. Are you assuming that return on equity (RoE) is equal to return on investment (RoI)? (which are not).
luchonacho
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2015-02-15T20:00:55.807Z
Under representative SDF. Or, if you are familiar with mathematical finance, under the risk neutral measure.
FooBar
CC BY-SA 3.0
2015-02-15T20:49:05.727Z
I'm not into that literature but judging for that assumption's implication (RoE=RoI), seems a pretty bad assumption to make, specially in finance! I went to a seminar the other day where a guy had a DSGE model trying to reproduce equity risk premium data. I raised the same question. Why would he expect to replicate data coming from share returns when he is using a standard macro model with capital (as a factor in production) and thus actually modeling return on investment? Of course, his model had no explicit profits. Its funny how we try to have internally but not externally consistent models
luchonacho
CC BY-SA 3.0
2015-02-15T20:49:28.943Z
Thats not the assumption's implication. If you have questions regarding risk neutral measures or implications of no-arbitrage, I suggest you open another question. Comments should not be used for extended questions / discussions.
FooBar
CC BY-SA 3.0
2015-02-15T18:13:17.230Z
2015-02-15T18:13:17.230Z
FooBar
<p>I want to take Alecos' answer, which seems to be more explicit about the issue at hand, and provide a formalisation, which to me provides a better picture of why neoclassical models do not usually allow for accounting profits.</p>

<p>Imagine an economy, where a government had issue a bond which pays interest rate of $r$ <strong>per period</strong>. There is no inflation, so all is in real terms. There is a capitalist with a pool of capital of \$100. There is also a technology $F(\cdot)$ which is capable to transform labour power and capital services into final goods. Say that workers reservation wage is $w$ <strong>per period</strong>, exogenous. </p>

<p>Say the capitalist creates a for-profit firm with his capital. This capital (equity, E) is used to buy an equivalent worth of capital machines (K), which for simplicity are assumed to already exist (no need to be produced). In other words, the firm now <strong>owns</strong> the machines. The balance sheet of the firm is:</p>

<pre><code>+----------+-------------+
|  Assets  | Liabilities |
+----------+-------------+
| K: 100   | E: 100      |
+----------+-------------+
</code></pre>

<p>Now, the firm hires workers, to which it pays $w$, and uses its technology $F(\cdot)$ to produce $Y$ goods <strong>within one period</strong>. These are sold at a unit price $p$. Therefore, <strong>from an accounting point of view</strong>, the firm's profits <strong>at the end of the period</strong> are:</p>

<p>$$ \pi_c = pY - wL $$  </p>

<p>Then, it follows that the rate of return on equity (RoE, or $R$) is:</p>

<p>$$ R \equiv \frac{\pi_c}{K} $$</p>

<p>Additionally, since the opportunity cost <strong>during the period of production</strong> of the capitalist was $rE=rK$, <strong>economic profits</strong> of the firm are:</p>

<p>$$ \pi_e = pY - wL - rK = \pi_c - rK $$</p>

<p>Replacing the RoE in the above yields:</p>

<p>$$ \pi_e = RK - rK $$</p>

<p>Therefore, the firms make positive <strong>economic</strong> profits whenever the rate of return on equity is higher than the opportunity cost rate (the govt. bond yield).</p>

<p>It is expected that in competitive markets/long run, the price of the good will be adjusted so that the RoE equals $r$, where there are no <strong>economic</strong> profits ($\pi_e=0$). With this conditions, any capitalist is therefore indifferent between buying the government bond and setting up a firm.</p>

CC BY-SA 3.0
2017-05-25T09:22:30.413Z
2017-05-25T10:45:33.260Z
2017-05-25T10:45:33.260Z
luchonacho
<p>Accounting (after tax) profits are net of depreciation and of interest paid on loans, or of any capital/equipment actually rented by the firm. So conceptually, they map to "net returns on <em>own</em> capital".  </p>

<p>Not necessarily "equilibrium" or "competitive", because such characterizations are some steps further down the modelling road, in making assumptions on the market structure and on the goals of economic agents. </p>

<p>It has been a historical habit in economics to use the word "profit" to describe "the existence of returns to capital above the competitive-level" (hence for example "monopolist makes a profit"). </p>

<p>In other words, Economics uses the word "profit" with a different meaning than the one used in Accounting and in Business, that's all.   </p>

CC BY-SA 3.0
2017-03-25T02:13:32.743Z
2017-03-25T02:13:32.743Z
Alecos Papadopoulos
<p>Consider a simple neoclassical model of an economy populated by firms with a constant returns to scale production function, using homogeneous labour and capital. Firms are equal, and are price takers in all markets. Thus, firms pay factors their value marginal product. In the long run, any <em>economic</em> profit is wiped out by changes in the price of the good, due to the menace of entry of new firms.</p>

<p>In this setting, no income remains after paying out competitive wages to workers (L) and competitive return to the capital hired for production (K). This is, for firm $i$, </p>

<p>$$pY_i = L_iw + K_ir $$</p>

<p>Now, in real markets, even in competitive ones, firms do make profits, even if these profits are not <strong>abnormal</strong>. Part of these profits might be distributed as dividends.</p>

<p>In the trivial economic model however, the ownership of the firm is not explicitly modeled. Firms do not pay dividends. </p>

<p>Why is this? Why is ownership not relevant? Are profits uninteresting? That is what the capitalistic firms cares about! I'm puzzled.</p>

CC BY-SA 3.0
2015-02-14T21:57:00.290Z
I think it's because profit in economics and profit in accounting are quite different. Profit in economics is revenue minus oppotunity cost. It is a windfall.
Epicurus
CC BY-SA 3.0
2015-02-14T22:04:07.667Z
Fair enough. However, ownership of shares is a huge source of income in current economies and thus it is bound to have an important impact on so many phenomenon. In the vanilla model there is no ownership. Firms are just an ethereal "thing" that take inputs and produce output with a technology freely available. It seems fragile to base a whole science in such a ridiculous model.
luchonacho
CC BY-SA 3.0
2015-02-15T18:16:19.250Z
@luchonacho Also, in the standard model, there is no involuntary unemployment - not much better. Point is, it is a basic (first best) model. You start with the most basic model that gives you insights, and then see whether insights are robust to (relevant) extensions.
FooBar
CC BY-SA 3.0
2015-02-17T17:38:17.393Z
Ownership of shares is ownership of capital, so the "huge source of income" referred to in your comment can be accounted for by the "competitive return to capital" referred to in your original question.  Why do you think there needs to be any more to the story?
Steven Landsburg
CC BY-SA 3.0
2015-02-18T11:06:53.510Z
Because return on capital is the rental rate of capital (R=MPk) whereas return on equity should be related to the opportunity cost, which should be related to the real interest rate. In a standard neoclassical macro model MPk is determined by technology whereas real interest rate is determined by demand composition. Two different things indeed. I see the gain in "tractability" by cofounding them but I'm arguing that this is an incorrect let alone misleading procedure.
luchonacho
<economic-terms><profits>
Why (neoclassical) economic models have no profits, unlike the real world?
luchonacho
# Why (neoclassical) economic models have no profits, unlike the real world?

(Asked by: username_0 on 2015-02-14)

Consider a simple neoclassical model of an economy populated by firms with a constant returns to scale production function, using homogeneous labour and capital. Firms are equal, and are price takers in all markets. Thus, firms pay factors their value marginal product. In the long run, any *economic* profit is wiped out by changes in the price of the good, due to the menace of entry of new firms.

In this setting, no income remains after paying out competitive wages to workers (L) and competitive return to the capital hired for production (K). This is, for firm $i$,

$$pY\_i = L\_iw + K\_ir $$

Now, in real markets, even in competitive ones, firms do make profits, even if these profits are not **abnormal**. Part of these profits might be distributed as dividends.

In the trivial economic model however, the ownership of the firm is not explicitly modeled. Firms do not pay dividends.

Why is this? Why is ownership not relevant? Are profits uninteresting? That is what the capitalistic firms cares about! I'm puzzled.

**Tags:** economic-terms, profits

> **Comments:**
> **username_1**: I think it's because profit in economics and profit in accounting are quite different. Profit in economics is revenue minus oppotunity cost. It is a windfall.  
> **username_0**: Fair enough. However, ownership of shares is a huge source of income in current economies and thus it is bound to have an important impact on so many phenomenon. In the vanilla model there is no ownership. Firms are just an ethereal "thing" that take inputs and produce output with a technology freely available. It seems fragile to base a whole science in such a ridiculous model.  
> **username_2**: @username_0 Also, in the standard model, there is no involuntary unemployment - not much better. Point is, it is a basic (first best) model. You start with the most basic model that gives you insights, and then see whether insights are robust to (relevant) extensions.  
> **username_3**: Ownership of shares is ownership of capital, so the "huge source of income" referred to in your comment can be accounted for by the "competitive return to capital" referred to in your original question.  Why do you think there needs to be any more to the story?  
> **username_0**: Because return on capital is the rental rate of capital (R=MPk) whereas return on equity should be related to the opportunity cost, which should be related to the real interest rate. In a standard neoclassical macro model MPk is determined by technology whereas real interest rate is determined by demand composition. Two different things indeed. I see the gain in "tractability" by cofounding them but I'm arguing that this is an incorrect let alone misleading procedure.  

---

## 5 answers

### Answer by username_6 on 2017-03-25. Score: 4 | Accepted answer

Accounting (after tax) profits are net of depreciation and of interest paid on loans, or of any capital/equipment actually rented by the firm. So conceptually, they map to "net returns on *own* capital".

Not necessarily "equilibrium" or "competitive", because such characterizations are some steps further down the modelling road, in making assumptions on the market structure and on the goals of economic agents.

It has been a historical habit in economics to use the word "profit" to describe "the existence of returns to capital above the competitive-level" (hence for example "monopolist makes a profit").

In other words, Economics uses the word "profit" with a different meaning than the one used in Accounting and in Business, that's all.


---

### Answer by username_5 on 2015-02-14. Score: 12

> It seems economic models do not explicitly include ownership of the firm (not of the physical capital, an input used to produce).

Many economic model *do* have profits. We're just more careful about what we call it. See the definition of "[economic profit](http://en.wikipedia.org/wiki/Profit_%28economics%29)." When you think of profits, you need to ask yourself if the money you are making is simply a fair exchange for the costs you incur---explicit and implicit costs---or whether some portion of that profit represent money gained above and beyond the (true) costs of production (see "[economic rent](http://en.wikipedia.org/wiki/Economic_rent)"). Indeed, this is the only sensible way to think about the world.

> It seems economic models do not explicitly include ownership of the firm

Many economic models *do* in fact include ownership of the firm. However, most economic models are built for demonstrating some particular idea in relative isolation, as opposed to trying to model every single detail. So if someone wants to demonstrate some effect in a world without economic rents, then a lot of the time modeling the ownership of the *firm* is less important than modeling the ownership of the *capital*---and thus it might be excluded.

If you want to see models that dig deep into the issues of economic rents and the ownership of a firm, I would recommend taking a look at the subfields called "[industrial organization](http://en.wikipedia.org/wiki/Industrial_organization)" and "[corporate finance](http://en.wikipedia.org/wiki/Corporate_finance)." (For example, take a look at [Jean Tirole's book on corporate finance](http://rads.stackoverflow.com/amzn/click/0691125562).

> Meanwhile, in real life, even in competitive markets, firms do make profits on the long run.

This is a bold claim. If you want to be scientific, you're going to have to be precise about how you (1) define profits and (2) define competitive markets, and (3) show (with data and sound analysis) that firms very commonly do make profits as you've defined them. This is how an economist would operate.

> Firms are just an ethereal "thing" that take inputs and produce output with a technology freely available.

Hidden in this comment is an very interesting object of study. What is a firm, *really*?
This was an important question (first?) and most famously raised by [Ronald Coase](http://en.wikipedia.org/wiki/Ronald_Coase). How do we define a firm? Why do they exists? etc. See the "[theory of the firm](http://en.wikipedia.org/wiki/Theory_of_the_firm)."

> **Comments:**
> **username_2**: If you change to *some firms to make profits on the long run*, I'm pretty certain that you can show that (long run being over the span of data available). I don't think he meant *all* firms, as obviously many fail and exit.  
> **username_5**: Certainly. But I'd also like to point out that it's not something that should be taken for granted.  
> **username_0**: I get the point of profits and rents over the opportunity cost but that is exactly what I mean. It is very likely that the opportunity cost is **endogenously** determined and so it is interesting to model (e.g. think of Marx's rate of profit). Why does it need to be obscured/hidden behind "capital", which usually refers to physical capital, no about ownership of equity. In fact, people usually don't own the assembly lines, the aircrafts, the labs, etc, but the shares of companies. I have seen models of ownership, specially in macro, my area. I'm just puzzled why is is not a mainstream thing.  

---

### Answer by username_4 on 2017-03-25. Score: 9

Greg Mankiw's answer:[![enter image description here](https://i.sstatic.net/Wss9i.jpg)](https://i.sstatic.net/Wss9i.jpg)

plus extra characters

> **Comments:**
> **username_0**: But why is this an explanation of **why we do not model profits**? Are implicit costs modeled? Can you elaborate? At the moment, it's not a helpful answer.  
> **username_4**: one word for two concepts @username_0, economic models of perfect competition have profit, it's just that they are equal to zero in equilibrium. Zero economic profit corresponds to positive accounting profit. You should check Mankiw's textbook if you need more explanations, it has several chapters on perfect competition with and without free entry. Profits are not just modeled, they are central to the model, since they are the object being maximized. Zero economic profit is an optimum: without optimization economic profit could be negative (while accounting profit could remain positive).  
> **username_0**: Maybe I'm not being clear. I understand the issue of economic profits, competition, optimisation, etc. I just do not understand why we should not be interested in modeling accounting profits, or ownership of firms, explicitly.  
> **username_4**: @username_0, economists are interested in modeling ownership of firms explicitly of course, and you can find studies on that in the __Journal of Finance__ and other top journals, but in the basic, textbook model of perfect competition, ownership is irrelevant: if ownership were relevant, competition would not be perfect. You may also be interested in looking into Modigliani-Miller theorem for related irrelevance results in the idealized world of perfect information, perfect competition, perfect rationality, no distortions, no taxes, etc..  
> **username_0**: See my answer, which is the type of explicit analysis I was looking for.  

---

### Answer by username_2 on 2015-02-15. Score: 1

I assume that this is about Macroeconomics in particular (you might want to add that tag).

### Often we don't need profits

The interaction of firm profits is often irrelevant. It is well summarized under capital rents. Having firm profits does not add any insights to, say, Solow's growth model or the standard Neoclassical model.

### Operative profits vs Life time profits

In some models, however, we do need profitability of firms. That typically comes from additional costs of running the firm. For example

* Vacancy costs (search and matching model)
* Entry costs (patents in Romer's growth model)

Firms run positive operative profits, which exactly offset the entrance costs. That is, life time profits are zero.

Why? Because we start in many models from the first-best and gradually remove particular assumptions. Romer could have talked in his model about a world in which patents have limited enforceability, or competitors particularly catch up, but the standard models we research ( and teach) typically do not incorporate this.

Later on, Kortum (1997) and Aghion, Howard, Howit create research based growth models that allow for catching up of competitors.

### Profits are messy to model

Profits mess up a lot of things. Think about the neo keynesian model, a standard example of where firms make profits (positive or negative) in every period, which do not necessarily cancel out over time. Allow for heterogeneity in households assets: some are poor, some are rich.

Perfect representation of the owner in the decision making process of the firm requires that the stochastic discount factor (SDF) of the firm represents the SDF of the owner. But, in this model, who is the owner? Rich and poor agents share the firm, with different degree of ownership. To do this properly, we would need to add a stock market for the firm and allow agents to trade it; track the ownership of the firm and adjust the SDF.

But no, you can't even look at the share of ownership: Say, $w$ denotes the wealth level, $m(w)$ denotes the measure of firm assets hold by agents with wealth $w$, and $\rho(w)$ denotes the discount rate of agents with wealth level $w$. Can you then just compute the firms SDF $\rho\_F$ with

$$\rho\_F = \int \rho(w)m(w) dw$$?

In reality, firm decisions are often voted over by simple majority. You'd need to model coalitions that get votes of $50%$ of asset holdings and see what SDF that gives you.

Of course, this is much too complicated to do correctly. Even the intermediate step with an asset market is not trivial, people very often fall back to assuming the SDF of the richest agents, predicting their model's outcome. But it's not nice: We like to have internality consistent models, and profits make this difficult.

> **Comments:**
> **username_0**: I don't see how profits are well summarized in capital rents. Theoretically, capital depreciates, advances technologically (more productive), reaches a steady state. Empirically, capital is always measured as stock of machines and buildings. This is very different from equity (anyone can turn savings into shares, increasing their price if more value not created). It might be difficult to model profits per se (beyond the models you mention) but perhaps we are not trying hard enough. It might be better to have profits in with a very trivial ownership mechanism that having no profits at all.  
> **username_2**: @username_0 But you certainly agree that under the stochastic discount factor of the representative agent, expected utility of rents to capital and expected utility from profits are the same, right? And ex-ante expectations (as opposed to ex-post realizations) are the relevant metric (since they appear in all the objective functions).  
> **username_0**: sorry, I can't see this. Are you assuming that return on equity (RoE) is equal to return on investment (RoI)? (which are not).  
> **username_2**: Under representative SDF. Or, if you are familiar with mathematical finance, under the risk neutral measure.  
> **username_0**: I'm not into that literature but judging for that assumption's implication (RoE=RoI), seems a pretty bad assumption to make, specially in finance! I went to a seminar the other day where a guy had a DSGE model trying to reproduce equity risk premium data. I raised the same question. Why would he expect to replicate data coming from share returns when he is using a standard macro model with capital (as a factor in production) and thus actually modeling return on investment? Of course, his model had no explicit profits. Its funny how we try to have internally but not externally consistent models  
> **username_2**: Thats not the assumption's implication. If you have questions regarding risk neutral measures or implications of no-arbitrage, I suggest you open another question. Comments should not be used for extended questions / discussions.  

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### Answer by username_0 on 2017-05-25. Score: 1

I want to take Alecos' answer, which seems to be more explicit about the issue at hand, and provide a formalisation, which to me provides a better picture of why neoclassical models do not usually allow for accounting profits.
Imagine an economy, where a government had issue a bond which pays interest rate of $r$ per period. There is no inflation, so all is in real terms. There is a capitalist with a pool of capital of \$100. There is also a technology $F(\cdot)$ which is capable to transform labour power and capital services into final goods. Say that workers reservation wage is $w$ per period, exogenous. 
Say the capitalist creates a for-profit firm with his capital. This capital (equity, E) is used to buy an equivalent worth of capital machines (K), which for simplicity are assumed to already exist (no need to be produced). In other words, the firm now owns the machines. The balance sheet of the firm is:
+----------+-------------+
|  Assets  | Liabilities |
+----------+-------------+
| K: 100   | E: 100      |
+----------+-------------+

Now, the firm hires workers, to which it pays $w$, and uses its technology $F(\cdot)$ to produce $Y$ goods within one period. These are sold at a unit price $p$. Therefore, from an accounting point of view, the firm's profits at the end of the period are:
$$ \pi_c = pY - wL $$  
Then, it follows that the rate of return on equity (RoE, or $R$) is:
$$ R \equiv \frac{\pi_c}{K} $$
Additionally, since the opportunity cost during the period of production of the capitalist was $rE=rK$, economic profits of the firm are:
$$ \pi_e = pY - wL - rK = \pi_c - rK $$
Replacing the RoE in the above yields:
$$ \pi_e = RK - rK $$
Therefore, the firms make positive economic profits whenever the rate of return on equity is higher than the opportunity cost rate (the govt. bond yield).
It is expected that in competitive markets/long run, the price of the good will be adjusted so that the RoE equals $r$, where there are no economic profits ($\pi_e=0$). With this conditions, any capitalist is therefore indifferent between buying the government bond and setting up a firm.


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