# Computing competitive equilibrium( plus consumption rivalry)

1. May 25, 2012

### Dostre

1. The problem statement, all variables and given/known data
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

$$u^P(x_1^P,x_2^P)=a\ln (x_1^P)+(1-a)\ln(x_2^P-bx_2^F)$$

while Friedman's preferences are

$$u^F(x_1^F,x_2^F)=a\ln (x_1^F)+(1-a)\ln(x_2^F-bx_2^p)$$

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. Both utility functions are subject to the constraint $$x_1^i+px_2^i\leq w_1^i+pw_2^i$$

1. Compute each other's demands of these goods.
2. Find the competitive equilibrium price and allocations.
3. How are the equilibrium price and consumption allocations affected by he parameter b?

3. The attempt at a solution
I think I should treat those utility functions separately. When solving the optimization problem of maximizing their utility I should form a Lagrangian. But this problem is weird because I have the consumption of good 2 ($x_{2}^i$) of one agent int he utility of the other agent. This confuses me and I do not know if a Lagrangian would somewhat different due to this stipulation. Any hints on solving all parts of this problem please.