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Consider two agents (Pascal and Friedman) in a pure exchange economy with two goods and no free disposal. Pascal has a preference relation give by the utility function

$$u^P(x_1^P,x_2^P)=a\ln (x_1^P)+(1-a)\ln(x_2^P-bx_2^F)\\\text{subject to the constraint}\;x_1^P+px_2^P\leq w_1+pw_2$$

while Friedman's preferences are

$$u^F(x_1^F,x_2^F)=a\ln (x_1^F)+(1-a)\ln(x_2^F-bx_2^P)\\\text{subject to the constraint}\;x_1^F+px_2^F\leq y_1+py_2$$

Here $0 and $0. Additionally the consumption of good 2 of one agent enters in the utility of the other agent.

Pascal's endownment is $\vec{w} ^P=(w_1,w_2)\geq 0$, while Friedman's is $\vec{w} ^F=(y_1,y_2)\geq 0$. Let $p$ be the price of good two in terms of good one.

  • Compute each other's demands of these goods.
    (find $x_1^P(w_1^P,w_2^P,p,x_2^P)$ and $x_2^P(w_1^P,w_2^P,p,x_2^P)$ and same for $x_1^F$ and $x_2^F$)
  • Find the competitive equilibrium price and allocations.
  • How are the equilibrium price and consumption allocations affected by he parameter b?

Attempt: I need to solve those optimization problems separately by the method of Lagrange. But, since each utility function has the consumption of good two of the other agent I do not know how to solve optimization problems like that. Any hints please.

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    Well, assuming "no free disposal" means that the total amount of goods is constant, couldn't you express $x_2$ in terms of $x_1$? I don't see this immediately, but I think this may be how to do it...2012-05-29
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    This sentence also might be a key to this, but I am afraid I am not sure how to interpret it... "Additionally the consumption of good 2 of one agent enters in the utility of the other agent."2012-05-29
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    If you look at the utility functions each of them has the consumption of good two of the other agent in it. Pascal's utility func has $x_2^F$ and Friedman's utility has $x_2^P$. So, it looks like each utility function depends now on three variables: Pascal $u^P(x_1^P,x_2^P,x_2^F)$ and Friedman $u^F(x_1^F,x_2^F,x_2^P)$.2012-05-29
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    Downvote and flag for moderator attention with the comment: "This question was deleted: http://math.stackexchange.com/questions/149693/calculating-a-competitive-equilibrium-consumption-rivarly and then recreated here, presumably to evade the negative impression of my downvote, close vote and comments, which I consider an abuse of the system."2012-05-29
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    Pascal takes Friedmans consumption as fixed when solving his consumption problem and vice versa. So the problem can be solved in the usual way.2012-05-29
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    @Dostre: Four days ago, you had posted this exact question [here](http://math.stackexchange.com/questions/149693/calculating-a-competitive-equilibrium-consumption-rivarly), and then deleted it when it was voted down. Do not try to evade downvotes in this way, it is considered abuse of the system, and is not acceptable. I say this as a warning. In the future just add the bounty to the original question so that it can get more attention.2012-05-29
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    @EricNaslund Guys I corrected the question and added additional info so it became a simple optimization problem. Now, nobody needs the economic theory to solve it. My intent was not to abuse the system and I deleted the other question not because it was downvoted.2012-05-29
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    @joriki stop hating on me. I corrected the question. Now, it is an optimization problem. Just math, no economic theory. And even though I deleted the other question i did not get my rep back, so it does not make any difference whether I put the bounty on the old question or delete it and create a new one.2012-05-29
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    This is not about hate; it's about preventing people from abusing the system. Clearly you yourself don't believe that it doesn't make a difference whether you keep the old question or delete it and create a new one; if you did you wouldn't have gone to the unnecessary effort of deleting it and creating a new one. By the way, the question still contains undefined extra-mathematical terms such as "competitive equilibrium price and allocations".2012-05-29
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    @Dostre: I believe you could have edited the original question and added a bounty. For now, I think you can simply apologize for your mistake and move on with Generic Human's advice. After all, there is hw to be done, is there not? :)2012-05-29
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    Dear Dostre, nothing is gained from assuming that other users act based on *hate* or other such grounds. The usual way to fix a question is to edit it into a better question, as opposed to deleting it and asking a new one. @joriki, it is probably best not to include such speculations in comments here.2012-05-29

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Pascal does not (and cannot) optimize for $x_2^F$, he only optimizes his allocation as a function of $x_2^F$. However he jointly optimizes for both $x_1^P$ and $x_2^P$, so I'm not sure why $x_1^P$ would depend on $x_2^P$: I can only assume this is a typo and what you're really looking for is $x_1^P(w_1^P,w_2^P,p,x_2^F)$ and $x_2^P(w_1^P,w_2^P,p,x_2^F)$. $x_2^F$ will only be determined later, when putting together Pascal and Friedman's optimal consumption allocations to solve for a global equilibrium.

Once you see that, the actual optimization is not difficult, and you don't even need the Lagrange method if you realize that since $u^P$ is increasing in $x_1^P$ the upper bound on $x_1^P$ is an equality when $u^P$ is maximal, so you can express $x_1^P$ as a function of $x_2^P$ and obtain a very simple univariate problem. Since this is homework I'm not going to post the solution, but the solution is linear in $x_2^F$ which makes it easy to solve for the equilibrium allocation.

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    I will try to solve taking into consideration what you said. If it works out the bounty is yours.2012-05-29