# General Equilibrium allocation holding fixed a consumer's utility

I'm having some issues with solving this general equilibrium exercise.

The way I started off is by assuming that since consumer 2's utility is fixed, he will have a fixed utility function. Then consumer 1 would shift his indifference curve to the point where they are tangent to maximise his own utility. Thus deriving the contract curve, by $$MRS^1=MRS^2$$ $$64x_1=117x_2-5x_1x_2$$ Then using the utility function: $$6=x{_{1}}^{1/2}x{_{2}}^{1/3}$$ I should be able to solve this because it's only 2 unknowns with 2 equations, but I cannot get anywhere without using an online calculator. I think the solution is allocation for consumer 1 is (4,8) and consumer 2: (9,8), but I am not sure how to get to this.

Can anyone find a similar exercise, where one consumer's utility was held fixed? Any advice on how to solve this?

My workings so far

• $MRS^1=frac{2x_2}{3x_1}}$ and $MRS^2=frac{3x_2}{2x_1}}$, then made them equal as per the contract curve definition. Then used the "Hint" to replace 2 of the 4 unknowns, so it turned into $$2(16-x_2)/3(13-x_1)=3/2x_2/x_1$$ – Tommy Dec 5 '17 at 22:17
• Oh, okay. You jumped several steps ahead without mentioning it. Somewhat confusing. – Giskard Dec 5 '17 at 22:25
• Right, I can take a picture of my proposed solution that might clear the confusion. However, I don't know if I'm doing this right and I was not able to solve the equation without using an online calculator anyways, so there has to be a better solution – Tommy Dec 5 '17 at 22:28
• Anyway I try to do it I need to solve a high degree polinomial. It is possible that there is no solution method. It is fairly easy to work backwards from such a problem, that is figure out the nice numbers you want as a solution and then set the parameters in such a way that everything works out. This does not mean that an easy analytical solution exists. Perhaps your prof. made the mistake of not checking the analytical solution. – Giskard Dec 5 '17 at 22:38
• Thank you for checking. I arrive at the same issue with this method then. I am wondering if there is maybe another way of arriving at a solution, like minimising expenditure for consumer 2, would that mean that consumer 1 maximises utility at that point? Or maybe some way with calculating a budget constraint and find where that is tangent to the indifference curve? – Tommy Dec 5 '17 at 22:42