<p>Consider a one commodity, two consumer world with</p>
<ul>
<li>Preferences: <span class="math-container">$u_1(x_1, x_2) = x_1$</span> and <span class="math-container">$u_2(x_1, x_2) = x_1 + x_2$</span>, where <span class="math-container">$x_i$</span> is <span class="math-container">$i$</span>'s consumption of the good.</li>
<li>Endowment: Both consumers have 10 units of the good. So the total endowment is 20 units.</li>
</ul>
<p>Clearly, competitive equilibrium is any price <span class="math-container">$p&gt;0$</span> and the only equilibrium allocation is the endowment <span class="math-container">$(x_1^c,x_2^c) =(10, 10)$</span>.
And there is only one Pareto efficient allocation which is <span class="math-container">$(x_1^p, x_2^p) = (20,0)$</span>.</p>

CC BY-SA 4.0
2022-05-05T15:10:24.843Z
2022-05-05T15:10:24.843Z
Amit
<p>This is an old question. The first example of inefficiency caused by altruism that I know of is due to <em>Winter, S. (1969). A Simple Remark on the Second Optimality Theorem of Welfare Economics. Journal of Economic Theory, 1, 99–103</em>, but I am sure others were around before that.</p>
<p>Now, the example in Winter(1969) does not answer your question because it features altruism for <em>all</em> agents in the economy. As mentioned by The Almighty Bob you can find another example in Heidhues and Riedel (2007). Another good reference is <em>Dufwenberg, M., Heidhues, P., Kirchsteiger, G., Riedel, F., &amp; Sobel, J. (2011). Other-Regarding Preferences in General Equilibrium. The Review of Economic Studies, 78(2), 613–639</em>, which is a summary paper of Heidhues and Riedel (2007) and contributions on the topics by the other authors.</p>
<p>Without more constraints on the preferences, it is not hard to find an example satisfying the conditions of your question. I am sure you could find it yourself, if you haven't already. But in order to make the answer complete here is one:</p>
<blockquote>
<p><span class="math-container">$\Omega \equiv \{(1,1),(1,1)\}$</span> (individual endowements)</p>
<p><span class="math-container">$U_1 \equiv \min \{x_1,y_1\}$</span> (perfect complement, no altruism)</p>
<p><span class="math-container">$U_2 \equiv \min \{x_2, y_2\} + 2*U_1$</span> (perfect complement, with (strong) altruism toward <span class="math-container">$1$</span>)</p>
<p>Walrasian equilibrium = {(1,1),(1,1)}, together with whatever (finite) relative price you like.</p>
</blockquote>
<p>I guess it is clear that the Walrasian equilibrium is not Pareto efficient. Notice that this &quot;result&quot; highly depends on the definition of the consumption space. If you add</p>
<ul>
<li>Donation of good <span class="math-container">$x$</span> from <span class="math-container">$2$</span> to <span class="math-container">$1$</span>, and</li>
<li>Donation of good <span class="math-container">$y$</span> from <span class="math-container">$2$</span> to <span class="math-container">$1$</span>,</li>
</ul>
<p>to the consumption space, then you would recover efficiency (although it might take a little care to define a meaningful notion of Walrasian equilibrium in this case).</p>

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2015-04-29T17:23:07.493Z
2016-04-15T13:22:01.613Z
2020-06-17T09:40:54.643Z
Martin Van der Linden
<p>I'm looking for a theoretic example of an economy where one agent is altruistic, while the others are not, that would make a walrasian equilibrium not efficient.  </p>

CC BY-SA 3.0
2015-03-27T15:30:14.137Z
As far as I remember there is an answer somewhere in http://papers.ssrn.com/soL3/papers.cfm?abstract_id=1015228, however I was to lazy to look for it, so I won't post this as an answer ;-)
The Almighty Bob
CC BY-SA 3.0
2015-03-27T21:39:24.480Z
@TheAlmightyBob Thanks for the comment. However, I don't have acess to the paper... Could you make it available? or  write an answer here, please?
An old man in the sea.
CC BY-SA 3.0
2015-04-02T18:52:37.287Z
Sequential entry in spatial pricing has one such result: depending on the government's move order, the introduction of a government owned, welfare maximizing firm, can sometimes reduce the overall welfare.
RegressForward
CC BY-SA 3.0
2015-04-23T06:29:20.500Z
An old man:
You can usually find free to access version papers on the author's website by googling the exact name of the paper. Here is a link to the paper mentioned by The Almighty Bob:
http://pub.uni-bielefeld.de/luur/download?func=downloadFile&recordOId=2315688&fileOId=2319836
Giskard
CC BY-SA 3.0
2015-04-30T08:39:57.833Z
There is a paper of Alain Venditti (probably with some co-author) on Journal of Economic Theory with some altruistic and selfish agents. It may be useful maybe.
optimal control
<microeconomics>
Example of an economy where Equilibria may not be efficient, where one agent is altruistic
An old man in the sea.
# Example of an economy where Equilibria may not be efficient, where one agent is altruistic

(Asked by: username_0 on 2015-03-27)

I'm looking for a theoretic example of an economy where one agent is altruistic, while the others are not, that would make a walrasian equilibrium not efficient.

**Tags:** microeconomics

> **Comments:**
> **username_1**: As far as I remember there is an answer somewhere in http://papers.ssrn.com/soL3/papers.cfm?abstract_id=1015228, however I was to lazy to look for it, so I won't post this as an answer ;-)  
> **username_0**: @TheAlmightyBob Thanks for the comment. However, I don't have acess to the paper... Could you make it available? or  write an answer here, please?  
> **username_2**: Sequential entry in spatial pricing has one such result: depending on the government's move order, the introduction of a government owned, welfare maximizing firm, can sometimes reduce the overall welfare.  
> **username_3**: An old man: You can usually find free to access version papers on the author's website by googling the exact name of the paper. Here is a link to the paper mentioned by username_1: http://pub.uni-bielefeld.de/luur/download?func=downloadFile&recordOId=2315688&fileOId=2319836  
> **username_4**: There is a paper of Alain Venditti (probably with some co-author) on Journal of Economic Theory with some altruistic and selfish agents. It may be useful maybe.  

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## 2 answers

### Answer by username_6 on 2015-04-29. Score: 4 | Accepted answer

This is an old question. The first example of inefficiency caused by altruism that I know of is due to *Winter, S. (1969). A Simple Remark on the Second Optimality Theorem of Welfare Economics. Journal of Economic Theory, 1, 99–103*, but I am sure others were around before that.

Now, the example in Winter(1969) does not answer your question because it features altruism for *all* agents in the economy. As mentioned by username_1 you can find another example in Heidhues and Riedel (2007). Another good reference is *Dufwenberg, M., Heidhues, P., Kirchsteiger, G., Riedel, F., & Sobel, J. (2011). Other-Regarding Preferences in General Equilibrium. The Review of Economic Studies, 78(2), 613–639*, which is a summary paper of Heidhues and Riedel (2007) and contributions on the topics by the other authors.

Without more constraints on the preferences, it is not hard to find an example satisfying the conditions of your question. I am sure you could find it yourself, if you haven't already. But in order to make the answer complete here is one:

> $\Omega \equiv \{(1,1),(1,1)\}$ (individual endowements)
>
> $U\_1 \equiv \min \{x\_1,y\_1\}$ (perfect complement, no altruism)
>
> $U\_2 \equiv \min \{x\_2, y\_2\} + 2\*U\_1$ (perfect complement, with (strong) altruism toward $1$)
>
> Walrasian equilibrium = {(1,1),(1,1)}, together with whatever (finite) relative price you like.

I guess it is clear that the Walrasian equilibrium is not Pareto efficient. Notice that this "result" highly depends on the definition of the consumption space. If you add

* Donation of good $x$ from $2$ to $1$, and
* Donation of good $y$ from $2$ to $1$,

to the consumption space, then you would recover efficiency (although it might take a little care to define a meaningful notion of Walrasian equilibrium in this case).


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### Answer by username_5 on 2022-05-05. Score: 1

Consider a one commodity, two consumer world with

* Preferences: $u\_1(x\_1, x\_2) = x\_1$ and $u\_2(x\_1, x\_2) = x\_1 + x\_2$, where $x\_i$ is $i$'s consumption of the good.
* Endowment: Both consumers have 10 units of the good. So the total endowment is 20 units.

Clearly, competitive equilibrium is any price $p>0$ and the only equilibrium allocation is the endowment $(x\_1^c,x\_2^c) =(10, 10)$.
And there is only one Pareto efficient allocation which is $(x\_1^p, x\_2^p) = (20,0)$.


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