<p>Given the economy with two consumers: <span class="math-container">$u_1=\min(2m_1,h_1)$</span> and <span class="math-container">$u_2=m_2+h_2$</span> with endowment allocation <span class="math-container">$E=((60,40),(40,60))$</span>, we observe the case where the price ratio is <span class="math-container">$\frac{p_M}{p_H}\in (0,1)$</span>. In such a case, as depicted in the figure, point <span class="math-container">$B$</span> is the optimal choice of consumer <span class="math-container">$1$</span>, and point <span class="math-container">$C$</span> is the optimal choice of consumer <span class="math-container">$2$</span> and it will offer higher satisfaction to consumer <span class="math-container">$2$</span> in comparison to point <span class="math-container">$B$</span> and is also affordable. Therefore, such a price ratio cannot be competitive equilibrium as there is excess demand for commodity <span class="math-container">$M$</span>.</p>
<p><a href="https://i.sstatic.net/tQMYe.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/tQMYe.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2024-04-10T21:16:26.053Z
2024-04-10T21:16:26.053Z
Amit
<p>I will denote the price ratio <span class="math-container">$p_M/p_H$</span> by <span class="math-container">$p$</span>.</p>
<p>The <span class="math-container">$m$</span> demand of <span class="math-container">$A$</span> is:
<span class="math-container">$$
m_A = \frac{60p + 40}{p + 2}
$$</span>
The <span class="math-container">$m$</span> demand of <span class="math-container">$B$</span> is:
<span class="math-container">\begin{align*}
m_B = \left\{
\begin{array}{ll}
0 &amp; \text{ if } p &gt; 1 \\
[0,100] &amp; \text{ if } p = 1 \\
\frac{40p+60}{p} &amp; \text{ if } p &lt; 1  
\end{array}
\right.
\end{align*}</span>
Middle part represents that any <span class="math-container">$m_B$</span> consumption in the <span class="math-container">$[0,100]$</span> interval can be optimal. Investigating the three cases (<span class="math-container">$p&gt;1$</span>, <span class="math-container">$p=1$</span>, <span class="math-container">$p&lt;1$</span>) separately, you will find there is only one price where demand equals supply
<span class="math-container">$$
m_A + m_B = 60 + 40.
$$</span></p>

CC BY-SA 4.0
2024-04-07T10:31:06.863Z
Apparently I did not do such a good job formulating my question. I understand the notion of agents maximising utility according to their utility functions. The question I really wanted to ask was, under the assumption of full information, which is one of the assumption of competitive markets, why doesn't player B account for player A's preferences and decide to meet him at point B, knowing that this will bring him higher utility than in the inital endowment. Does the assumption on perfect information not apply to general equilibriums?
Jovan Jezdic
CC BY-SA 4.0
2024-04-07T20:05:21.783Z
There aren't any strategic decisions in competitive equilibrium, just demands given the price, So I am not sure how B would meet A in that point.
Giskard
CC BY-SA 4.0
2024-04-06T10:10:13.840Z
2024-04-06T10:10:13.840Z
Giskard
<p>Recently came across this question on a microeconomics test and there was something that did not sit quite right with me.</p>
<p>In an economy with two agents, A and B, and two goods, milk and honey, the preferences of each consumer are the following:</p>
<p><span class="math-container">$$U_A=\min(2m_A,h_A)$$</span>
<span class="math-container">$$U_B=m_B+h_B$$</span>
There is 100 of each good, no production in the economy, and the initial endowment is:</p>
<p><span class="math-container">$$E_A(m_A,h_A)=(60,40)$$</span>
<span class="math-container">$$E_B(m_B,h_B)=(40,60)$$</span></p>
<p>What is the competitive equilibrium allocation with this endowment?</p>
<p><a href="https://i.sstatic.net/OxBsi.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/OxBsi.jpg" alt="enter image description here" /></a></p>
<p>After drawing the Edgeworth box, it is easy to see that the contract curve is defined by consumer A because of his preferences. Following that, we can see one trivial solution in point A, where the price ratio is 1. My problem with this is that I cannot comprehend that this would be the only solution in such an economy. For any other price ratio <span class="math-container">${{p_M}\over{p_H}}\in (0,1)$</span>, couldn't the equilibrium be in point B drawn on the graph? If not, why not? Coming from an assumption of rationality and utility maximisation of the two players, although B cannot reach its highest indifference curve, <span class="math-container">$I_3$</span>, he can reach <span class="math-container">$I_2$</span>, which clearly provides more utility to him than <span class="math-container">$I_1$</span>. Please help me understand whether I have overlooked something, or if my thought process is wrong.</p>

CC BY-SA 4.0
2024-04-05T03:40:02.850Z
Have you considered solving this competitive equilibrium exercise with demand functions?
Giskard
CC BY-SA 4.0
2024-04-05T10:16:10.607Z
Yes, but I am not sure how would I derive these demand functions considering the nature of the two goods.
Jovan Jezdic
CC BY-SA 4.0
2024-04-05T11:07:42.977Z
*Hint:* For any price ratio, consumer B would always demand 100 of the cheaper good and 0 of the more expensive one. So it seems like the market could never clear. But there is a single exception...
VARulle
CC BY-SA 4.0
2024-04-05T12:46:36.240Z
If you’re talking about the exception being when the prices are the same I already mentioned that as one solution. If you’re not getting at that, I’m not sure what you’re getting at. I understand that with any other price ratio other than 1, B would demand only one good, but does that mean that in this model, any price ratio other than one produces no equilibria? As I’ve mentioned, this to me, does not seem as rational behaviour from the consumers.
Jovan Jezdic
<general-equilibrium><competitive-equilibrium><perfect-complements><perfect-substitutes>
Competitive equilibrium in a two-person economy with substitutes and complements
Jovan Jezdic
# Competitive equilibrium in a two-person economy with substitutes and complements

(Asked by: username_0 on 2024-04-04)

Recently came across this question on a microeconomics test and there was something that did not sit quite right with me.

In an economy with two agents, A and B, and two goods, milk and honey, the preferences of each consumer are the following:

$$U\_A=\min(2m\_A,h\_A)$$
$$U\_B=m\_B+h\_B$$
There is 100 of each good, no production in the economy, and the initial endowment is:

$$E\_A(m\_A,h\_A)=(60,40)$$
$$E\_B(m\_B,h\_B)=(40,60)$$

What is the competitive equilibrium allocation with this endowment?

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

After drawing the Edgeworth box, it is easy to see that the contract curve is defined by consumer A because of his preferences. Following that, we can see one trivial solution in point A, where the price ratio is 1. My problem with this is that I cannot comprehend that this would be the only solution in such an economy. For any other price ratio ${{p\_M}\over{p\_H}}\in (0,1)$, couldn't the equilibrium be in point B drawn on the graph? If not, why not? Coming from an assumption of rationality and utility maximisation of the two players, although B cannot reach its highest indifference curve, $I\_3$, he can reach $I\_2$, which clearly provides more utility to him than $I\_1$. Please help me understand whether I have overlooked something, or if my thought process is wrong.

**Tags:** general-equilibrium, competitive-equilibrium, perfect-complements, perfect-substitutes

> **Comments:**
> **username_1**: Have you considered solving this competitive equilibrium exercise with demand functions?  
> **username_0**: Yes, but I am not sure how would I derive these demand functions considering the nature of the two goods.  
> **username_2**: *Hint:* For any price ratio, consumer B would always demand 100 of the cheaper good and 0 of the more expensive one. So it seems like the market could never clear. But there is a single exception...  
> **username_0**: If you’re talking about the exception being when the prices are the same I already mentioned that as one solution. If you’re not getting at that, I’m not sure what you’re getting at. I understand that with any other price ratio other than 1, B would demand only one good, but does that mean that in this model, any price ratio other than one produces no equilibria? As I’ve mentioned, this to me, does not seem as rational behaviour from the consumers.  

---

## 2 answers

### Answer by username_3 on 2024-04-10. Score: 1 | Accepted answer

Given the economy with two consumers: $u\_1=\min(2m\_1,h\_1)$ and $u\_2=m\_2+h\_2$ with endowment allocation $E=((60,40),(40,60))$, we observe the case where the price ratio is $\frac{p\_M}{p\_H}\in (0,1)$. In such a case, as depicted in the figure, point $B$ is the optimal choice of consumer $1$, and point $C$ is the optimal choice of consumer $2$ and it will offer higher satisfaction to consumer $2$ in comparison to point $B$ and is also affordable. Therefore, such a price ratio cannot be competitive equilibrium as there is excess demand for commodity $M$.

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


---

### Answer by username_1 on 2024-04-06. Score: 0

I will denote the price ratio $p\_M/p\_H$ by $p$.

The $m$ demand of $A$ is:
$$
m\_A = \frac{60p + 40}{p + 2}
$$
The $m$ demand of $B$ is:
\begin{align\*}
m\_B = \left\{
\begin{array}{ll}
0 & \text{ if } p > 1 \\
[0,100] & \text{ if } p = 1 \\
\frac{40p+60}{p} & \text{ if } p < 1
\end{array}
\right.
\end{align\*}
Middle part represents that any $m\_B$ consumption in the $[0,100]$ interval can be optimal. Investigating the three cases ($p>1$, $p=1$, $p<1$) separately, you will find there is only one price where demand equals supply
$$
m\_A + m\_B = 60 + 40.
$$

> **Comments:**
> **username_0**: Apparently I did not do such a good job formulating my question. I understand the notion of agents maximising utility according to their utility functions. The question I really wanted to ask was, under the assumption of full information, which is one of the assumption of competitive markets, why doesn't player B account for player A's preferences and decide to meet him at point B, knowing that this will bring him higher utility than in the inital endowment. Does the assumption on perfect information not apply to general equilibriums?  
> **username_1**: There aren't any strategic decisions in competitive equilibrium, just demands given the price, So I am not sure how B would meet A in that point.  

---

