<p><span class="math-container">$EO_2BO_1$</span> is 2's offer curve and <span class="math-container">$EO_1A$</span> is 1's offer curve. Set of competitive equilibrium allocations is given by the intersection of the two offer curves that is the set of allocations on the line segment <span class="math-container">$O_1B$</span>.
<a href="https://i.sstatic.net/Cc8B1.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Cc8B1.png" alt="Offer Curves" /></a></p>

CC BY-SA 4.0
2021-01-18T15:28:19.713Z
Hi Amit, can you please provide a solution to this problem https://economics.stackexchange.com/questions/42192/find-pareto-optimal-allocations-and-the-core-for-the-following-economies many thanks!
studentp
CC BY-SA 4.0
2021-01-12T08:12:02.293Z
2021-01-12T08:12:02.293Z
Amit
<p>QUESTION:</p>
<p>Consider simple two-person, two-good economy in which agents’ utility functions are given by</p>
<p><span class="math-container">$U_1(x_{11}, x_{21}) = min\{x_{11}, x_{21}\} $</span>, and  <span class="math-container">$U_2(x_{12}, x_{22}) = min\{4x_{12}, x_{22}\} $</span>.</p>
<p>Endowments are w1 =(30,0) and w2 =(0,20).</p>
<p>If neither agents can have negative consumption of either good, what is Walrasian equilibrium?</p>
<p>SOLUTION:</p>
<p>With Leontiev preferences, the indifference curves of both agents are right-angles. For agent 1 their vertices lie on the line <span class="math-container">$x_{11} = x_{21} $</span>, whereas, for agent 2, they lie on the line <span class="math-container">$x_{22} = 4x_{12}$</span>. These lines are respective offer curves of both agents when both prices are strictly positive. For the case when one of the prices is zero, we have following offer curves:</p>
<p>————</p>
<p><span class="math-container">$OC_1(P_1, P_2)=$</span></p>
<p>if <span class="math-container">$(P_1, P_2) \in $</span> {0} <span class="math-container">$\times R_+ $</span> then <span class="math-container">$OC_1(P_1, P_2)= \{(x_{11}, w_{21}): x_{11} &gt; w_{21} \}$</span>,</p>
<p>if <span class="math-container">$(P_1, P_2) \in $</span>  <span class="math-container">$R_+ \times $</span> {0} then <span class="math-container">$OC_1(P_1, P_2)= \{(w_{11}, x_{21}) : x_{21} &gt; w_{11} \}$</span>,</p>
<p><span class="math-container">$OC_2(P_1, P_2)=$</span></p>
<p>if <span class="math-container">$(P_1, P_2) \in $</span> {0} <span class="math-container">$\times R_+ $</span> then <span class="math-container">$OC_2(P_1, P_2)= \{(x_{11}, w_{21}) : x_{11} &gt; (1/4) w_{21} \} $</span>,</p>
<p>if <span class="math-container">$(P_1, P_2) \in $</span>  <span class="math-container">$R_+ \times $</span> {0} then <span class="math-container">$OC_2(P_1, P_2)= \{(w_{11}, x_{21}) : x_{21} &gt; 4 w_{11} \} $</span>,</p>
<p>my question is how to write this offer curve? I don't ask to solve this question. I dont understand the point where the offer curves are written. Especially I find to be difficult to write the OC2 for the second consumer. I mean writing OC_2 is especially difficult for me. If you show how to write offer curves, I will really glad. Thank you.</p>

<microeconomics><consumer-theory><self-study><general-equilibrium>
Question on General Equilibrium: how to write offer curves?
studentp
# Question on General Equilibrium: how to write offer curves?

(Asked by: username_0 on 2020-12-19)

QUESTION:

Consider simple two-person, two-good economy in which agents’ utility functions are given by

$U\_1(x\_{11}, x\_{21}) = min\{x\_{11}, x\_{21}\} $, and $U\_2(x\_{12}, x\_{22}) = min\{4x\_{12}, x\_{22}\} $.

Endowments are w1 =(30,0) and w2 =(0,20).

If neither agents can have negative consumption of either good, what is Walrasian equilibrium?

SOLUTION:

With Leontiev preferences, the indifference curves of both agents are right-angles. For agent 1 their vertices lie on the line $x\_{11} = x\_{21} $, whereas, for agent 2, they lie on the line $x\_{22} = 4x\_{12}$. These lines are respective offer curves of both agents when both prices are strictly positive. For the case when one of the prices is zero, we have following offer curves:

————

$OC\_1(P\_1, P\_2)=$

if $(P\_1, P\_2) \in $ {0} $\times R\_+ $ then $OC\_1(P\_1, P\_2)= \{(x\_{11}, w\_{21}): x\_{11} > w\_{21} \}$,

if $(P\_1, P\_2) \in $ $R\_+ \times $ {0} then $OC\_1(P\_1, P\_2)= \{(w\_{11}, x\_{21}) : x\_{21} > w\_{11} \}$,

$OC\_2(P\_1, P\_2)=$

if $(P\_1, P\_2) \in $ {0} $\times R\_+ $ then $OC\_2(P\_1, P\_2)= \{(x\_{11}, w\_{21}) : x\_{11} > (1/4) w\_{21} \} $,

if $(P\_1, P\_2) \in $ $R\_+ \times $ {0} then $OC\_2(P\_1, P\_2)= \{(w\_{11}, x\_{21}) : x\_{21} > 4 w\_{11} \} $,

my question is how to write this offer curve? I don't ask to solve this question. I dont understand the point where the offer curves are written. Especially I find to be difficult to write the OC2 for the second consumer. I mean writing OC\_2 is especially difficult for me. If you show how to write offer curves, I will really glad. Thank you.

**Tags:** microeconomics, consumer-theory, self-study, general-equilibrium

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## 1 answers

### Answer by username_1 on 2021-01-12. Score: 1 | Accepted answer

$EO\_2BO\_1$ is 2's offer curve and $EO\_1A$ is 1's offer curve. Set of competitive equilibrium allocations is given by the intersection of the two offer curves that is the set of allocations on the line segment $O\_1B$.
[![Offer Curves](https://i.sstatic.net/Cc8B1.png)](https://i.sstatic.net/Cc8B1.png)

> **Comments:**
> **username_0**: Hi Amit, can you please provide a solution to this problem https://economics.stackexchange.com/questions/42192/find-pareto-optimal-allocations-and-the-core-for-the-following-economies many thanks!  

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