General Equilibrium question...

[BJE]

I'm just trying to guess at some of the questions on my midterm, and I can't find any example of this type of question, and I am having some trouble solving myself, can someone point me in the right direction? (I've just thought of this question, so I don't know if it has a solution)

 

It is a general equilibrium with question: with two consumers (1,2) , two producers (X,Y) *producer X only produces good x, producer y only produces good y. Consumer have asymmetric cobb-douglas preferences, and the production possibilities frontier is concave (the marginal rate of transformation depends on the quantities).. .. and consumers are only endowed with different amounts of labor.

 

Is it possible to solve for a equilibrium prices, quantities, and wages?

 

If someone could point me in the direction of a book with this type of question, that would be great. I have Jehle and Reny in front of me, but I don't see an example, pure exchange economies with asymmetric preferences, and interesting endowments seems to be as close as it gets :hmm:

[FreeMarkets]

Your production functions are presumably continuous, strictly increasing, strictly quasi-concave, weakly concave, and satisfy no free lunch. That implies your constraint set is compact. Your utility functions are continuous. That implies existence of a solution. Utility being strictly concave implies that the solution is a unique one. See Sundaram or any other optimization book.

[BJE]

the solution is a unique one. See Sundaram or any other optimization book.

Thanks, I'm having trouble with trying to see how there is a unique solution for BOTH consumers (the same solution), I really dont see how there is a unique solution for asymmetric utility functions because each utility function has a unique tangency point with the PPF, however they are not the same. Anyway, I'll look up the book. Thanks for the reply.

[FreeMarkets]

A unique solution doesn't mean that consumers have the same consumption bundles. It just means that given equilibrium prices, there is a unique allocation that solves the consumers' and firms' problems.

[treblekicker]

if you normalize the price of one consumption good, then you will be able to solve for unique equilibrium allocations and prices. this is a pretty standard problem whose method of solution should be found in most any advanced undergrad micro text.

[BJE]

Thanks for the replies. I guess the price you pay for the help is sounding a little stupid. .. I'm just looked up from my notepad, having solved it, and it seems so obvious now.