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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.

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  • $\begingroup$ As far as I remember there is an answer somewhere in 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 ;-) $\endgroup$ – The Almighty Bob Mar 27 '15 at 15:30
  • $\begingroup$ @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? $\endgroup$ – An old man in the sea. Mar 27 '15 at 21:39
  • $\begingroup$ 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. $\endgroup$ – RegressForward Apr 2 '15 at 18:52
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    $\begingroup$ 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: pub.uni-bielefeld.de/luur/… $\endgroup$ – Giskard Apr 23 '15 at 6:29
  • $\begingroup$ 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. $\endgroup$ – optimal control Apr 30 '15 at 8:39
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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 The Almighty Bob 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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