<p>I'm having some issues with solving this general equilibrium exercise.</p>

<p><a href="https://i.sstatic.net/9Jw1a.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/9Jw1a.png" alt="General Equilibrium"></a></p>

<p>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.</p>

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

<p>My workings so far<a href="https://i.sstatic.net/NcQQJ.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/NcQQJ.jpg" alt="enter image description here"></a></p>

CC BY-SA 3.0
2017-12-05T22:17:48.920Z
$MRS^1=frac{2x_2}{3x_1}}$ 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
CC BY-SA 3.0
2017-12-05T22:25:40.310Z
Oh, okay. You jumped several steps ahead without mentioning it. Somewhat confusing.
Giskard
CC BY-SA 3.0
2017-12-05T22:28:12.627Z
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
CC BY-SA 3.0
2017-12-05T22:38:07.263Z
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
CC BY-SA 3.0
2017-12-05T22:42:16.883Z
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
CC BY-SA 3.0
2017-12-05T22:42:30.007Z
E.g. I can easily see the roots of the polinomial $(x - 1.1)(x - 1.2)(x - 1.3)$ but if I transform this to its usual $x^3 + ax^2 + bx + c$ form you will have a very hard time figuring out the roots. So would I.
Giskard
CC BY-SA 3.0
2017-12-05T22:48:18.010Z
I appreciate the time you are spending on this. That is why I wonder if there is a different method to solve, perhaps with some kind of budget constraint, as this method yields the high degree polynomial.
Tommy
CC BY-SA 3.0
2017-12-05T22:54:14.673Z
I understand, but I find it unlikely, and a budget constraint would be meaningless given the text of the exercise. Your micro work is sound. It is perfectly acceptable to e-mail the prof at this point giving him your equations, telling him that you can verify that the solution (9,8)(4,8) but it seems to you it is not possible to analitically derive it.
Giskard
CC BY-SA 3.0
2017-12-05T22:57:55.840Z
Thank you for your help. I'll do that, and I'll get back here once I receive an answer. In the meantime, I'll leave this exercise up here, if someone comes along with any possible solutions. Thanks!
Tommy
CC BY-SA 3.0
2017-12-05T23:00:09.583Z
Until then, would you be able to find any similar exercises and solutions, where one consumer's utility is fixed while trying to maximise another consumer's utility? It might be my search terms, but I am struggling to find anything similar to this problem.
Tommy
<microeconomics><utility><general-equilibrium><competitive-equilibrium><walrasian>
General Equilibrium allocation holding fixed a consumer's utility
Tommy
# General Equilibrium allocation holding fixed a consumer's utility

(Asked by: username_0 on 2017-12-05)

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

[![General Equilibrium](https://i.sstatic.net/9Jw1a.png)](https://i.sstatic.net/9Jw1a.png)

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[![enter image description here](https://i.sstatic.net/NcQQJ.jpg)](https://i.sstatic.net/NcQQJ.jpg)

**Tags:** microeconomics, utility, general-equilibrium, competitive-equilibrium, walrasian

> **Comments:**
> **username_0**: $MRS^1=frac{2x_2}{3x_1}}$ 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 $$2(16-x_2)/3(13-x_1)=3/2x_2/x_1$$  
> **username_1**: Oh, okay. You jumped several steps ahead without mentioning it. Somewhat confusing.  
> **username_0**: 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  
> **username_1**: 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.  
> **username_0**: 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?  
> **username_1**: E.g. I can easily see the roots of the polinomial $(x - 1.1)(x - 1.2)(x - 1.3)$ but if I transform this to its usual $x^3 + ax^2 + bx + c$ form you will have a very hard time figuring out the roots. So would I.  
> **username_0**: I appreciate the time you are spending on this. That is why I wonder if there is a different method to solve, perhaps with some kind of budget constraint, as this method yields the high degree polynomial.  
> **username_1**: I understand, but I find it unlikely, and a budget constraint would be meaningless given the text of the exercise. Your micro work is sound. It is perfectly acceptable to e-mail the prof at this point giving him your equations, telling him that you can verify that the solution (9,8)(4,8) but it seems to you it is not possible to analitically derive it.  
> **username_0**: Thank you for your help. I'll do that, and I'll get back here once I receive an answer. In the meantime, I'll leave this exercise up here, if someone comes along with any possible solutions. Thanks!  
> **username_0**: Until then, would you be able to find any similar exercises and solutions, where one consumer's utility is fixed while trying to maximise another consumer's utility? It might be my search terms, but I am struggling to find anything similar to this problem.  

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## 0 answers

