<p>Parallel to Arrow and Debreu, there is the approach of Lionel McKenzie, in which no ownership is specified and all technology has constant returns to scale. In such a model, firms can make no profit.</p>
<p>If all firms in an Arrow-Debreu economy have constant returns to scale, equilibrium profits must necessarily be zero. A firm then always has the option to produce nothing without any input, so profits cannot be negative. If a firm were to make positive profits, it could double the profit by doubling the production plan, which would contradict profits being maximal. So the Arrow-Debreu approach is seemingly more general than the McKenzie approach.</p>
<p>However, there is a way to represent the Arrow-Debreu approach within McKenzie's approach. Let the commodity space be <span class="math-container">$\mathbb{R}^l$</span>. If <span class="math-container">$Y\subseteq\mathbb{R}^l$</span> is a convex production set, then there exists a convex production set with constant returns to scale and an additional factor, <span class="math-container">$Y'\subseteq\mathbb{R}^{l+1}$</span> such that
<span class="math-container">$$Y=\big\{y\in\mathbb{R}^l:(y_1,\ldots,y_l,-1)\in Y'\big\}.$$</span>
Indeed, the set <span class="math-container">$Y'$</span> must be given by <span class="math-container">$$Y'=\big\{\alpha(y_1,\ldots,y_l,-1):y\in Y, \alpha\geq 0\big\}.$$</span>
So one can always think of decreasing returns to scale as constant returns to scale with an unmodeled factor whose net supply is <span class="math-container">$1$</span>. Now, in terms of profits, let <span class="math-container">$p=(p_1,\ldots,p_{l+1})$</span> be a price system for the extended commodity space. The firm with the production set <span class="math-container">$Y'$</span> must make zero profits in equilibrium. So if <span class="math-container">$(y_1,y_2,\ldots,y_l,-1)$</span> is a profit-maximizing production plan we must have, by the mentioned zero profit condition, that
<span class="math-container">$$p_1 y_1+p_2 y_2+\cdots p_l y_l + p_{l+1}(-1)=0$$</span>
<span class="math-container">$$p_1 y_1+p_2 y_2+\cdots p_l y_l=p_{l+1}.$$</span>
So profits for the firm with the original production set <span class="math-container">$Y$</span> can be interpreted as returns to the unmodeled production factor.</p>
<p>The economic relevance of these results is that one can always view profits as returns to unspecified factors in an economy that actually satisfies a zero-profit condition.</p>

CC BY-SA 4.0
2022-01-03T10:12:18.427Z
Thx for the answer. Is there perhaps some reference you would recommend? Preferable not book length.
Jesper Hybel
CC BY-SA 4.0
2022-01-03T10:26:26.353Z
Perhaps this one:  McKenzie, Lionel. "On the Existence of General Equilibrium for a Competitive Market," Econometrica, Jan. 1959, 27(1), pp. 5471 ??
Jesper Hybel
CC BY-SA 4.0
2022-01-03T11:12:49.007Z
The original model of McKenzie was from 1954 ("On Equilibrium in Graham’s Model of World Trade and Other Competitive Systems"), but Section 7 of the paper you mention gives essentially the argument in my post. I'm not sure there is much more to say; I learned this from a few line of small print and an exercise in MWG.
Michael Greinecker
CC BY-SA 4.0
2022-01-03T11:39:02.433Z
I personally think this article is more interesting than the McKenzie approach:  General Equilibrium with Free Entry: A Synthetic Approach to the Theory of Perfect Competition*  By WILLIAM NOVSHEK and HUGO SONNENSCHEIN.
Jesper Hybel
CC BY-SA 4.0
2022-01-03T11:51:21.207Z
In the words of Novshek and Sonnenschein: 'We believe the traditional Arrow-Debreu-McKenzie theory dismays many Marshallians because it contains no role for marginal
firm', the point being that positive profit is driven to zero by firm entry.
Jesper Hybel
CC BY-SA 4.0
2022-01-03T09:31:37.840Z
2022-01-03T09:31:37.840Z
Michael Greinecker
<p>I'm reading a book where the definition of an equilibrium for a competitive economy is given as in</p>
<p>Kenneth J. Arrow; Gerard Debreu (1954) Existence of an Equilibrium for a Competitive Economy <em>Econometrica</em>, Vol. 22, No. 3. , pp. 265-290.</p>
<p>where the authors define an equilibrium roughly as a price vector <span class="math-container">$p$</span> where households maximize utility, firms maximize profit, and consumption of households satisfies the income restraint.</p>
<p>However, it seems to me that <strong>the zero profit rule</strong> which is often associated with equilibrium in a competitive economy is not assumed (nor implied).</p>
<p>So my question is how does the zero profit rule enter into general equilibrium theory?</p>
<p>Is it imposed as a further assumption and if so are there any seminal papers discussing the role of the zero-profit rule and perhaps like the above article giving conditions for the existence of competitive equilibrium?</p>
<p>Handbook chapters could also be interesting but I am not sure which handbooks one should consult when looking for microeconomic theory and general equilibrium theory.</p>

CC BY-SA 4.0
2022-01-02T16:14:57.897Z
"*my question is how does the zero profit rule enter into general equilibrium theory*" **Does** the zero profit rule enter **general** equilibrium theory? Not all models with perfectly competitive markets are general equilibrium models. If I remember correctly (?), Arrow and Debreu do not make the zero profit assumption, in fact there is an ownership matrix of the economy which tells how firms divide their profits among the consumer/owners. So can you please support the implied claim in your qouted sentence with a reference?
Giskard
CC BY-SA 4.0
2022-01-02T16:18:26.870Z
A reference for the [ownership matrix](https://books.google.hu/books?id=PCRzl7ZcqYwC&pg=PA13&lpg=PA13&dq=arrow+economy+ownership+matrix&source=bl&ots=3G5t0ihapB&sig=ACfU3U32kDZurp2MkjuKCd4zALNWsyCCzQ&hl=en&sa=X&redir_esc=y#v=onepage&q=arrow%20economy%20ownership%20matrix&f=false).
Giskard
CC BY-SA 4.0
2022-01-02T18:13:29.093Z
@Giskard 'Arrow and Debreu do not make the zero profit assumption' I agree with that reading of the cited article, which is exactly what motivates my question of how is this condition integrated into a GE setting? Perhaps the answer is simply that it is not in the sense that no one has considered this but I somehow find this hard to believe.
Jesper Hybel
CC BY-SA 4.0
2022-01-02T18:16:08.607Z
For example: Walras (1874–7,ch. 18), whilst differentiating clearly between the
rate of net income which is the return on capital, and the equilibrium condition of zero excess profits, nonetheless regarded excess profits as the reward to entrepreneurship and hence argued that in equilibrium, the return to entrepreneurship would be zero. Walras, L. 1874–7. Elements of pure economics. Trans. ed. W. Jaffe. Homewood: Irwin, 1954.
Jesper Hybel
CC BY-SA 4.0
2022-01-02T18:16:11.710Z
cited by Eatwell, John (2016), "Zero-Profit Condition", The New Palgrave Dictionary of Economics, London: Palgrave Macmillan UK, pp. 1–2, doi:10.1057/978-1-349-95121-5_1302-1, ISBN 978-1-349-95121-5, retrieved 2021-11-21
Jesper Hybel
CC BY-SA 4.0
2022-01-02T19:01:39.427Z
The zero-profit condition is essentially used to pin down the level of output produced in general equilibrium, conditional on wages and rental rates. This is a consequence of Euler's theorem and CRS technologies. If/when you deviate from CRS technologies, super normal profits exist, and you have to determine an optimal supply curve (i.e. a level of output such that profits are maximized).
ChinG
<microeconomics><general-equilibrium><competitive-equilibrium><profits>
Perfect Competition, Zero profit rule and General Equilibrium
Jesper Hybel
# Perfect Competition, Zero profit rule and General Equilibrium

(Asked by: username_0 on 2022-01-02)

I'm reading a book where the definition of an equilibrium for a competitive economy is given as in

Kenneth J. Arrow; Gerard Debreu (1954) Existence of an Equilibrium for a Competitive Economy *Econometrica*, Vol. 22, No. 3. , pp. 265-290.

where the authors define an equilibrium roughly as a price vector $p$ where households maximize utility, firms maximize profit, and consumption of households satisfies the income restraint.

However, it seems to me that **the zero profit rule** which is often associated with equilibrium in a competitive economy is not assumed (nor implied).

So my question is how does the zero profit rule enter into general equilibrium theory?

Is it imposed as a further assumption and if so are there any seminal papers discussing the role of the zero-profit rule and perhaps like the above article giving conditions for the existence of competitive equilibrium?

Handbook chapters could also be interesting but I am not sure which handbooks one should consult when looking for microeconomic theory and general equilibrium theory.

**Tags:** microeconomics, general-equilibrium, competitive-equilibrium, profits
> **Comments:**
> **username_1**: "*my question is how does the zero profit rule enter into general equilibrium theory*" **Does** the zero profit rule enter **general** equilibrium theory? Not all models with perfectly competitive markets are general equilibrium models. If I remember correctly (?), Arrow and Debreu do not make the zero profit assumption, in fact there is an ownership matrix of the economy which tells how firms divide their profits among the consumer/owners. So can you please support the implied claim in your qouted sentence with a reference?  
> **username_1**: A reference for the [ownership matrix](https://books.google.hu/books?id=PCRzl7ZcqYwC&pg=PA13&lpg=PA13&dq=arrow+economy+ownership+matrix&source=bl&ots=3G5t0ihapB&sig=ACfU3U32kDZurp2MkjuKCd4zALNWsyCCzQ&hl=en&sa=X&redir_esc=y#v=onepage&q=arrow%20economy%20ownership%20matrix&f=false).  
> **username_0**: @Giskard 'Arrow and Debreu do not make the zero profit assumption' I agree with that reading of the cited article, which is exactly what motivates my question of how is this condition integrated into a GE setting? Perhaps the answer is simply that it is not in the sense that no one has considered this but I somehow find this hard to believe.  
> **username_0**: For example: Walras (1874–7,ch. 18), whilst differentiating clearly between the rate of net income which is the return on capital, and the equilibrium condition of zero excess profits, nonetheless regarded excess profits as the reward to entrepreneurship and hence argued that in equilibrium, the return to entrepreneurship would be zero. Walras, L. 1874–7. Elements of pure economics. Trans. ed. W. Jaffe. Homewood: Irwin, 1954.  
> **username_0**: cited by Eatwell, John (2016), "Zero-Profit Condition", The New Palgrave Dictionary of Economics, London: Palgrave Macmillan UK, pp. 1–2, doi:10.1057/978-1-349-95121-5_1302-1, ISBN 978-1-349-95121-5, retrieved 2021-11-21  
> **username_2**: The zero-profit condition is essentially used to pin down the level of output produced in general equilibrium, conditional on wages and rental rates. This is a consequence of Euler's theorem and CRS technologies. If/when you deviate from CRS technologies, super normal profits exist, and you have to determine an optimal supply curve (i.e. a level of output such that profits are maximized).  

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## 1 answers

### Answer by username_3 on 2022-01-03. Score: 8 | Accepted answer

Parallel to Arrow and Debreu, there is the approach of Lionel McKenzie, in which no ownership is specified and all technology has constant returns to scale. In such a model, firms can make no profit.

If all firms in an Arrow-Debreu economy have constant returns to scale, equilibrium profits must necessarily be zero. A firm then always has the option to produce nothing without any input, so profits cannot be negative. If a firm were to make positive profits, it could double the profit by doubling the production plan, which would contradict profits being maximal. So the Arrow-Debreu approach is seemingly more general than the McKenzie approach.

However, there is a way to represent the Arrow-Debreu approach within McKenzie's approach. Let the commodity space be $\mathbb{R}^l$. If $Y\subseteq\mathbb{R}^l$ is a convex production set, then there exists a convex production set with constant returns to scale and an additional factor, $Y'\subseteq\mathbb{R}^{l+1}$ such that
$$Y=\big\{y\in\mathbb{R}^l:(y\_1,\ldots,y\_l,-1)\in Y'\big\}.$$
Indeed, the set $Y'$ must be given by $$Y'=\big\{\alpha(y\_1,\ldots,y\_l,-1):y\in Y, \alpha\geq 0\big\}.$$
So one can always think of decreasing returns to scale as constant returns to scale with an unmodeled factor whose net supply is $1$. Now, in terms of profits, let $p=(p\_1,\ldots,p\_{l+1})$ be a price system for the extended commodity space. The firm with the production set $Y'$ must make zero profits in equilibrium. So if $(y\_1,y\_2,\ldots,y\_l,-1)$ is a profit-maximizing production plan we must have, by the mentioned zero profit condition, that
$$p\_1 y\_1+p\_2 y\_2+\cdots p\_l y\_l + p\_{l+1}(-1)=0$$
$$p\_1 y\_1+p\_2 y\_2+\cdots p\_l y\_l=p\_{l+1}.$$
So profits for the firm with the original production set $Y$ can be interpreted as returns to the unmodeled production factor.

The economic relevance of these results is that one can always view profits as returns to unspecified factors in an economy that actually satisfies a zero-profit condition.

> **Comments:**
> **username_0**: Thx for the answer. Is there perhaps some reference you would recommend? Preferable not book length.  
> **username_0**: Perhaps this one:  McKenzie, Lionel. "On the Existence of General Equilibrium for a Competitive Market," Econometrica, Jan. 1959, 27(1), pp. 5471 ??  
> **username_3**: The original model of McKenzie was from 1954 ("On Equilibrium in Graham’s Model of World Trade and Other Competitive Systems"), but Section 7 of the paper you mention gives essentially the argument in my post. I'm not sure there is much more to say; I learned this from a few line of small print and an exercise in MWG.  
> **username_0**: I personally think this article is more interesting than the McKenzie approach:  General Equilibrium with Free Entry: A Synthetic Approach to the Theory of Perfect Competition*  By WILLIAM NOVSHEK and HUGO SONNENSCHEIN.  
> **username_0**: In the words of Novshek and Sonnenschein: 'We believe the traditional Arrow-Debreu-McKenzie theory dismays many Marshallians because it contains no role for marginal firm', the point being that positive profit is driven to zero by firm entry.  

---

