<p>Your analysis seems to be spot on, except for some minor typos:
$$
x_A^* = 100 \frac{p_x + p_y}{p_x + p_y} = 100,
$$
because the value of A's endowment is $100 \cdot (p_x + p_y)$. And you could include the case of
$$
x_B^* \in [0,50]  \mbox{ if } p_x = p_y.
$$</p>

<p>As you state at the end the equilibrium price vector is not unique, there is equilibrium for all $\frac{p_x}{p_y} &lt; 1$, because with this price vector we have
$$
(x_A^*,y_A^*) = (100,100), \hskip 20pt (x_B^*,y_B^*) = (50,0) 
$$
and the market is then indeed in equilibrium as
$$
x_A^* + x_B^* = x_A + x_B, \hskip 20pt y_A^* + y_B^* = y_A + y_B.
$$
The price ratio $\frac{p_x}{p_y} = 1$ also results in equilibrium, because one of the optimal consumptions of B is still $(x_B^*,y_B^*) = (50,0)$.</p>

CC BY-SA 3.0
2015-06-20T17:36:24.590Z
I denote the optimal demand of good $x$ by consumer $i$ with $x_i^*$, and I denoted her initial endowment with $x_i$. Alternately you could denote the initial endowment with $\omega_i^x$ to make a more striking difference. This frees up $x_i$ for other uses, e.g. in the utility function.
Giskard
CC BY-SA 3.0
2015-06-20T17:32:57.270Z
2015-06-20T17:32:57.270Z
Giskard
<p>Equilibrium price vector $(p_x, p_y=1)$ and allocation $((x_A, y_A), (x_B, y_B))$ satisfy the following:</p>

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

<ul>
<li>$(x_A, y_A) = \left(\frac{100p_x + 100}{p_x + 1}, \frac{100p_x +
   100}{p_x+1}\right) = (100, 100)$ </li>
<li>$(x_B, y_B) = \begin{cases} \left(\frac{50p_x}{p_x}, 0\right)  = (50,0) &amp; \text{if } p_x \leq 1  \\ \left(0, 50p_x\right) &amp; \text{if } p_x &gt; 1 \end{cases} $</li>
</ul>

<p>Feasibility Conditions</p>

<ul>
<li>$x_A + x_B = 150$</li>
<li>$y_A + y_B = 100$</li>
</ul>

<p>Clearly, any price vector $(p_x, p_y)$ satisfying $p_x \leq 1$ and $p_y = 1$, and the allocation $((x_A, y_A), (x_B, y_B)) = ((100, 100), (50, 0))$ is the equilibrium.</p>

CC BY-SA 3.0
2017-03-01T08:35:01.203Z
How is this different from my answer?
Giskard
CC BY-SA 3.0
2017-03-01T08:49:32.623Z
I guess its the same. I didn't read your answer when I posted it. I will be careful in future to avoid duplication.
Amit
CC BY-SA 3.0
2017-02-28T14:59:27.067Z
2017-02-28T14:59:27.067Z
Amit
<p>The problem is simple. However, I'm having trouble deriving the price vector for the system. What exactly am I doing wrong/missing out? </p>

<p>Two consumers. </p>

<p>$$
U_A = \min(x,y), \hskip 20pt U_B = x+y.
$$    </p>

<p>Initial endowments
$$
(x_A,y_A)=(100,100), \hskip 20pt (x_B,y_B)=(50,0).
$$ </p>

<p>Putting A on the bottom left, the contract curve will be a line with slope 1 from A, from point (0,0) to (100,100), A's endowment point. [Seen from A's perspective.] All points on this contract curve are points of 'tangency' between A's kinked indifference curves, and B's linear indifference map.</p>

<p>$x_A= 100(p_x+p_y)/p_x $</p>

<p>$x_B = 0$  (if  $p_x&gt;p_y$)</p>

<p>$x_B = 50$  (if  $p_x&lt;p_y$)</p>

<p>A cannot benefit from more than (100,100), while B will benefit from as much of either good as she can get: i.e. endowment point lies on contract. Thus, the endowment point can be the competitive equilibrium allocation. </p>

<p>The question is, what will be the equilibrium price vector. </p>

<p>If we keep the price vector as (1,1), X can sell and buy back endowment, and so can B. So, (1,1) makes the endowment point affordable. However, so does any price vector where $p_x&lt;p_y$ (ensuring 0 demand for Y from B.) </p>

<p>Does a unique price vector (ratio) exist for this system? How do we derive it? And what am I doing wrong?</p>

<p>Any help (resources on CE problems of this kind) will be wonderful. Thanks. </p>

CC BY-SA 3.0
2017-02-28T09:20:16.847Z
You can refer to this course on youtube: https://www.youtube.com/playlist?list=PLUJGfL_499TKsujAH6aeObLCw5VvSjzAx It has lots of problems of the kind you posted here.
Amit
<general-equilibrium>
Unable to determine price ratio for a pure exchange economy
user795028
# Unable to determine price ratio for a pure exchange economy

(Asked by: username_0 on 2015-06-20)

The problem is simple. However, I'm having trouble deriving the price vector for the system. What exactly am I doing wrong/missing out?

Two consumers.

$$
U\_A = \min(x,y), \hskip 20pt U\_B = x+y.
$$

Initial endowments
$$
(x\_A,y\_A)=(100,100), \hskip 20pt (x\_B,y\_B)=(50,0).
$$

Putting A on the bottom left, the contract curve will be a line with slope 1 from A, from point (0,0) to (100,100), A's endowment point. [Seen from A's perspective.] All points on this contract curve are points of 'tangency' between A's kinked indifference curves, and B's linear indifference map.

$x\_A= 100(p\_x+p\_y)/p\_x $

$x\_B = 0$ (if $p\_x>p\_y$)

$x\_B = 50$ (if $p\_x<p\_y$)

A cannot benefit from more than (100,100), while B will benefit from as much of either good as she can get: i.e. endowment point lies on contract. Thus, the endowment point can be the competitive equilibrium allocation.

The question is, what will be the equilibrium price vector.

If we keep the price vector as (1,1), X can sell and buy back endowment, and so can B. So, (1,1) makes the endowment point affordable. However, so does any price vector where $p\_x<p\_y$ (ensuring 0 demand for Y from B.)

Does a unique price vector (ratio) exist for this system? How do we derive it? And what am I doing wrong?

Any help (resources on CE problems of this kind) will be wonderful. Thanks.

**Tags:** general-equilibrium

> **Comments:**
> **username_1**: You can refer to this course on youtube: https://www.youtube.com/playlist?list=PLUJGfL_499TKsujAH6aeObLCw5VvSjzAx It has lots of problems of the kind you posted here.  

---

## 2 answers

### Answer by username_2 on 2015-06-20. Score: 2

Your analysis seems to be spot on, except for some minor typos:
$$
x\_A^\* = 100 \frac{p\_x + p\_y}{p\_x + p\_y} = 100,
$$
because the value of A's endowment is $100 \cdot (p\_x + p\_y)$. And you could include the case of
$$
x\_B^\* \in [0,50] \mbox{ if } p\_x = p\_y.
$$

As you state at the end the equilibrium price vector is not unique, there is equilibrium for all $\frac{p\_x}{p\_y} < 1$, because with this price vector we have
$$
(x\_A^\*,y\_A^\*) = (100,100), \hskip 20pt (x\_B^\*,y\_B^\*) = (50,0)
$$
and the market is then indeed in equilibrium as
$$
x\_A^\* + x\_B^\* = x\_A + x\_B, \hskip 20pt y\_A^\* + y\_B^\* = y\_A + y\_B.
$$
The price ratio $\frac{p\_x}{p\_y} = 1$ also results in equilibrium, because one of the optimal consumptions of B is still $(x\_B^\*,y\_B^\*) = (50,0)$.

> **Comments:**
> **username_2**: I denote the optimal demand of good $x$ by consumer $i$ with $x_i^*$, and I denoted her initial endowment with $x_i$. Alternately you could denote the initial endowment with $\omega_i^x$ to make a more striking difference. This frees up $x_i$ for other uses, e.g. in the utility function.  

---

### Answer by username_1 on 2017-02-28. Score: 1

Equilibrium price vector $(p\_x, p\_y=1)$ and allocation $((x\_A, y\_A), (x\_B, y\_B))$ satisfy the following:

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

* $(x\_A, y\_A) = \left(\frac{100p\_x + 100}{p\_x + 1}, \frac{100p\_x +
  100}{p\_x+1}\right) = (100, 100)$
* $(x\_B, y\_B) = \begin{cases} \left(\frac{50p\_x}{p\_x}, 0\right) = (50,0) & \text{if } p\_x \leq 1 \\ \left(0, 50p\_x\right) & \text{if } p\_x > 1 \end{cases} $

Feasibility Conditions

* $x\_A + x\_B = 150$
* $y\_A + y\_B = 100$

Clearly, any price vector $(p\_x, p\_y)$ satisfying $p\_x \leq 1$ and $p\_y = 1$, and the allocation $((x\_A, y\_A), (x\_B, y\_B)) = ((100, 100), (50, 0))$ is the equilibrium.

> **Comments:**
> **username_2**: How is this different from my answer?  
> **username_1**: I guess its the same. I didn't read your answer when I posted it. I will be careful in future to avoid duplication.  

---

