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.
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.