general equilibrium-HELP NEEDED

[cincinnatus]

 There are three commodities and two agents with endowments
ω1 = (0,3,0), ω2 = (1,0,1)
and utility functions
u1(x11,x12,x13) = x11 + 2x13, u2(x21,x22,x23) = x21 + x22 + x23
 Determine the Walrasian equilibria. Also determine the set of Pareto efficient allocations.

[knowpraveen]

In the above case, we employ the walras' Law which states that in a k-good market, if k-1 markets clear the kth market should clear as well.

So, it's clear from the above statement that the question boils down to clearing the markets for the first and the third good, which gives us two perfect substitutes as per the utility function. One with the ordinary perfect substitute utility function and the other a Marshallian one.

Hence the Pareto efficient allocations are the y axis for agent 1 and the x axis for agent 2.

[cincinnatus]

thanks a lot for replying @knowpraveen,
now in order to find the walrasian equilibrium and price ratios, do we consider the two utility functions to be
u1=x11+2x13 and u2=x21+x23 with endowments (0,0) and (1,1) while equilibrium allocations of good 2 remains x12=0 and x22=3...?

[knowpraveen]

I was hoping someone would have already answered the walrasian equilibria part. In fact, that's why I visited the forum in the first place.

Let's see if someone helps out on this. Share resources which you think might be of help to take me through the walrasian equilibria for non-unique equilibrium points as in two perfect complements, two perfect substitutes etc. Thanks in advance.