Existence of a Walrasian Equilibrium

Question

I was wondering how to tackle part b of this question. I've already completed part a) using the usual method of equating the MRS (kink point for the second utility function) then looking at feasibility but part b is confusing me as to where to start.

Do i look at the assumptions of the existence of a Walrasian equilibrium such as the utility functions being strictly increasing, concave and continuous and if so what do I do with the information? If anyone could give me a starting point i'd appreciate that.

Thanks

Answer 1

So after solving a), you have the demand functions $x_{11}(p)$, $x_{21}(p)$, $x_{12}(p)$, $x_{22}(p)$, right?

An equilibrium exists if there is such a $p$ for which there is no excess demand or excess supply*. In this scenario, the supply is constant, given by the endowment thus you are looking for a price $p$ for which $$ x_{11}(p) + x_{21}(p) = \omega_{11} + \omega_{21} $$ and $$ x_{12}(p) + x_{22}(p) = \omega_{12} + \omega_{22}. $$ If you have calculated the demand functions correctly it is enough to use one of these market clearing equations to determine for which parameters $a$,$b$ the equation can hold, as the other equation follows from this one and the budget constraints.

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*A special case is if $p = 0$, in which case excess supply is possible in equilibrum.