Computing competitive equilibrium( plus consumption rivalry)

[Dostre]

Homework Statement

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?

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.

 

[nate808]

Thank you for your post. Let me try to help you with your questions.

1. To compute each other's demands, you can follow the standard procedure of solving the optimization problem by forming a Lagrangian and taking the first-order conditions. However, as you mentioned, the presence of the other agent's consumption in the utility function complicates things a bit. In this case, you can use a technique called "substitution," where you substitute the budget constraint into the utility function and then solve the resulting unconstrained optimization problem. This should give you the demand functions for both agents.

2. To find the competitive equilibrium price and allocations, you need to solve for the price that equates the demand of one agent with the supply of the other agent. In this case, since there are only two goods and two agents, the price will be determined by the ratio of the two agents' marginal utilities of good 2. Once you have the equilibrium price, you can plug it into the demand functions to get the equilibrium allocations.

3. Finally, to analyze the effect of the parameter b on the equilibrium price and consumption allocations, you can start by considering the impact on each agent's demand function. How does changing b affect their demand for good 2? This, in turn, will affect the equilibrium price and allocations. For example, if b increases, the demand for good 2 will decrease, which will decrease the equilibrium price and allocations of good 2. On the other hand, if b decreases, the opposite will happen.

I hope this helps. Good luck with your analysis!