<p>So after solving a), you have the demand functions
<span class="math-container">$x_{11}(p)$</span>,
<span class="math-container">$x_{21}(p)$</span>,
<span class="math-container">$x_{12}(p)$</span>,
<span class="math-container">$x_{22}(p)$</span>,
right?</p>
<p>An equilibrium exists if there is such a <span class="math-container">$p$</span> for which there is no excess demand or excess supply*. In this scenario, the supply is constant, given by the endowment thus you are looking for a price <span class="math-container">$p$</span> for which
<span class="math-container">$$
x_{11}(p) + x_{21}(p) = \omega_{11} + \omega_{21} 
$$</span>
and
<span class="math-container">$$
x_{12}(p) + x_{22}(p) = \omega_{12} + \omega_{22}. 
$$</span>
If you have calculated the demand functions correctly it is enough to use one of these market clearing equations to determine for which parameters <span class="math-container">$a$</span>,<span class="math-container">$b$</span> the equation can hold, as the other equation follows from this one and the budget constraints.</p>
<hr />
<p><br> *A special case is if <span class="math-container">$p = 0$</span>, in which case excess supply is possible in equilibrum.</p>

CC BY-SA 4.0
2022-04-18T13:42:35.783Z
2022-04-18T13:42:35.783Z
Giskard
<p><a href="https://i.sstatic.net/n4mcy.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/n4mcy.png" alt="Question below" /></a></p>
<p>I was wondering how to tackle part b of this question. I've already completed part a) using the usual method of equating the MRS (kink point for the second utility function) then looking at feasibility but part b is confusing me as to where to start.</p>
<p>Do i look at the assumptions of the existence of a Walrasian equilibrium such as the utility functions being strictly increasing, concave and continuous and if so what do I do with the information? If anyone could give me a starting point i'd appreciate that.</p>
<p>Thanks</p>

<microeconomics><general-equilibrium>
Existence of a Walrasian Equilibrium
dreadpirateroberts
# Existence of a Walrasian Equilibrium

(Asked by: username_0 on 2022-04-18)

[![Question below](https://i.sstatic.net/n4mcy.png)](https://i.sstatic.net/n4mcy.png)

I was wondering how to tackle part b of this question. I've already completed part a) using the usual method of equating the MRS (kink point for the second utility function) then looking at feasibility but part b is confusing me as to where to start.

Do i look at the assumptions of the existence of a Walrasian equilibrium such as the utility functions being strictly increasing, concave and continuous and if so what do I do with the information? If anyone could give me a starting point i'd appreciate that.

Thanks

**Tags:** microeconomics, general-equilibrium


---

## 1 answers

### Answer by username_1 on 2022-04-18. Score: 1

So after solving a), you have the demand functions
$x\_{11}(p)$,
$x\_{21}(p)$,
$x\_{12}(p)$,
$x\_{22}(p)$,
right?

An equilibrium exists if there is such a $p$ for which there is no excess demand or excess supply\*. In this scenario, the supply is constant, given by the endowment thus you are looking for a price $p$ for which
$$
x\_{11}(p) + x\_{21}(p) = \omega\_{11} + \omega\_{21}
$$
and
$$
x\_{12}(p) + x\_{22}(p) = \omega\_{12} + \omega\_{22}.
$$
If you have calculated the demand functions correctly it is enough to use one of these market clearing equations to determine for which parameters $a$,$b$ the equation can hold, as the other equation follows from this one and the budget constraints.

---

\*A special case is if $p = 0$, in which case excess supply is possible in equilibrum.


---

