<p>Let us name three consumers A, B and C, and two goods X and Y. Equilibrium price vector $(p_x, p_y=1)$ and allocation $((x_A, y_A), (x_B, y_B), (x_C, y_C))$ satisfy the following:</p>

<p>Optimality Conditions (Allocation must solve the utility maximization problem of the three consumers, i.e. it must lie on the demand functions)</p>

<ul>
<li>$(x_A, y_A) = \begin{cases} \left(\frac{10p_x+10}{p_x}, 0\right)  &amp; \text{if } p_x \leq \frac{1}{10}  \\ \left(10,10\right) &amp; \text{if } \frac{1}{10} \leq p_x \leq  10 \\ \left(0, 10p_x+10\right) &amp; \text{if } p_x \geq 10\end{cases} $</li>
<li>$(x_B, y_B) = \left(\frac{2(4p_x + 6)}{5p_x}, \frac{3(4p_x + 6)}{5}\right)$ </li>
<li>$(x_C, y_C) = \left(\frac{3(4p_x + 4)}{5p_x}, \frac{2(4p_x + 4)}{5}\right)$</li>
</ul>

<p>Feasibility Conditions</p>

<ul>
<li>$x_A + x_B + x_C = 18$</li>
<li>$y_A + y_B + y_C = 20$</li>
</ul>

<p>Solving the above gives price vector $(p_x, p_y) = (1.2, 1)$ that supports the allocation $((x_A, y_A), (x_B, y_B), (x_C, y_C)) = ((10, 10), (3.6, 6.48), (4.4,3.52))$ in equilibrium.</p>

CC BY-SA 3.0
2017-03-04T14:40:32.800Z
so does it mean that, for example, if the equilibrium price solved for consumer 2 and 3 is $p_{1}>10$ or $p_{1}<\frac{1}{10}$ with $p_{2}=1$, there will be no equilibrium at all?
K. Rob
CC BY-SA 3.0
2017-03-04T03:12:07.313Z
2017-03-04T14:02:35.950Z
2017-03-04T14:02:35.950Z
Amit
<p>Please kindly instruct me on solving the following in a general equilibrium framework with standard budget constraint,
$$ u^{1}\left ( x  \right )= max\left [ \frac{x_{1}}{10}, \frac{x_{2}}{10}\right ]+ min\left [ x_{1},x_{2} \right ] $$
, with $e^{1} = (10,10)$</p>

<ol>
<li>What would this consumer demand at prices $p = (\frac{3}{4}; \frac{1}{4})$?</li>
</ol>

<p>Given the other two consumer utilities</p>

<p>$$u^{2}\left ( x  \right )= x_{1}^{\frac{2}{5}}x_{2}^{\frac{3}{5}} $$</p>

<p>$$u^{3}\left ( x  \right )= x_{1}^{\frac{3}{5}}x_{2}^{\frac{2}{5}}$$
, with $e^{2} = (4,6)$ , $e^{3} = (4,4)$ </p>

<ol start="2">
<li>Find one Walrasian equilibrium (and determine both equilibrium prices and the equilibrium allocation).</li>
</ol>

<p>I have completely no idea how to start with it...</p>

<p>Thank you!</p>

CC BY-SA 3.0
2017-03-03T20:09:51.937Z
For (2) I suggest you view this playlist on youtube: https://www.youtube.com/playlist?list=PLUJGfL_499TKsujAH6aeObLCw5VvSjzAx I am sure it will be helpful.
Amit
CC BY-SA 3.0
2017-03-03T20:13:53.157Z
For (1) I can give you the hint: $u(x_1, x_2) = \frac{x_1}{10} + x_2$ when $x_1 > x_2$ and $u(x_1, x_2) = \frac{x_2}{10} + x_1$ otherwise. Now plot the indifference curves, and the budget line.
Amit
CC BY-SA 3.0
2017-03-04T14:18:02.307Z
so is this the correct IC?
$$\left\{\begin{matrix}
x_{2}=-\frac{1}{10}x_{1}+\bar{U},    x_{1}> x_{2}
\\ 
x_{2}=-10x_{1}+10\bar{U},     x_{2}> x_{1}
\end{matrix}\right.$$
K. Rob
<utility><general-equilibrium>
General equilibrium regarding on U= max(ax,ay) + min(x,y)
K. Rob
# General equilibrium regarding on U= max(ax,ay) + min(x,y)

(Asked by: username_0 on 2017-03-03)

Please kindly instruct me on solving the following in a general equilibrium framework with standard budget constraint,
$$ u^{1}\left ( x \right )= max\left [ \frac{x\_{1}}{10}, \frac{x\_{2}}{10}\right ]+ min\left [ x\_{1},x\_{2} \right ] $$
, with $e^{1} = (10,10)$

1. What would this consumer demand at prices $p = (\frac{3}{4}; \frac{1}{4})$?

Given the other two consumer utilities

$$u^{2}\left ( x \right )= x\_{1}^{\frac{2}{5}}x\_{2}^{\frac{3}{5}} $$

$$u^{3}\left ( x \right )= x\_{1}^{\frac{3}{5}}x\_{2}^{\frac{2}{5}}$$
, with $e^{2} = (4,6)$ , $e^{3} = (4,4)$

2. Find one Walrasian equilibrium (and determine both equilibrium prices and the equilibrium allocation).

I have completely no idea how to start with it...

Thank you!

**Tags:** utility, general-equilibrium

> **Comments:**
> **username_1**: For (2) I suggest you view this playlist on youtube: https://www.youtube.com/playlist?list=PLUJGfL_499TKsujAH6aeObLCw5VvSjzAx I am sure it will be helpful.  
> **username_1**: For (1) I can give you the hint: $u(x_1, x_2) = \frac{x_1}{10} + x_2$ when $x_1 > x_2$ and $u(x_1, x_2) = \frac{x_2}{10} + x_1$ otherwise. Now plot the indifference curves, and the budget line.  
> **username_0**: so is this the correct IC? $$\left\{\begin{matrix} x_{2}=-\frac{1}{10}x_{1}+\bar{U},    x_{1}> x_{2} \\  x_{2}=-10x_{1}+10\bar{U},     x_{2}> x_{1} \end{matrix}\right.$$  

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## 1 answers

### Answer by username_1 on 2017-03-04. Score: 2 | Accepted answer

Let us name three consumers A, B and C, and two goods X and Y. Equilibrium price vector $(p\_x, p\_y=1)$ and allocation $((x\_A, y\_A), (x\_B, y\_B), (x\_C, y\_C))$ satisfy the following:

Optimality Conditions (Allocation must solve the utility maximization problem of the three consumers, i.e. it must lie on the demand functions)

* $(x\_A, y\_A) = \begin{cases} \left(\frac{10p\_x+10}{p\_x}, 0\right) & \text{if } p\_x \leq \frac{1}{10} \\ \left(10,10\right) & \text{if } \frac{1}{10} \leq p\_x \leq 10 \\ \left(0, 10p\_x+10\right) & \text{if } p\_x \geq 10\end{cases} $
* $(x\_B, y\_B) = \left(\frac{2(4p\_x + 6)}{5p\_x}, \frac{3(4p\_x + 6)}{5}\right)$
* $(x\_C, y\_C) = \left(\frac{3(4p\_x + 4)}{5p\_x}, \frac{2(4p\_x + 4)}{5}\right)$

Feasibility Conditions

* $x\_A + x\_B + x\_C = 18$
* $y\_A + y\_B + y\_C = 20$

Solving the above gives price vector $(p\_x, p\_y) = (1.2, 1)$ that supports the allocation $((x\_A, y\_A), (x\_B, y\_B), (x\_C, y\_C)) = ((10, 10), (3.6, 6.48), (4.4,3.52))$ in equilibrium.

> **Comments:**
> **username_0**: so does it mean that, for example, if the equilibrium price solved for consumer 2 and 3 is $p_{1}>10$ or $p_{1}<\frac{1}{10}$ with $p_{2}=1$, there will be no equilibrium at all?  

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