Consumption rivalry

Dostre

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.

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?

Attempt: First of all this should hold (market clearing): $$x_1^P+px_2^P=w_1^P+pw_2^P$$
I think I should start with forming the Lagrangian. How will it look? Then, once the Langrangian is formed what am I solving for? Any hints please.

Evo

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