# Finding competitve equilibrium(consumption rivalry)

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 $0<a<1$ and $0<b<1$. 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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Downvote and flag for moderator attention with the comment: "This question was deleted: math.stackexchange.com/questions/149693/… 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." – joriki May 29 '12 at 6:30
Pascal takes Friedmans consumption as fixed when solving his consumption problem and vice versa. So the problem can be solved in the usual way. – Michael Greinecker May 29 '12 at 7:38
@Dostre: Four days ago, you had posted this exact question here, 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. – Eric Naslund May 29 '12 at 8:47
@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. – Koba May 29 '12 at 18:34
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. – Mariano Suárez-Alvarez May 29 '12 at 19:59

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.