<p>Solving the profit maximisation problem of firm producing <span class="math-container">$x$</span>,</p>
<p><span class="math-container">$\max_{x\geq 0, l_X\geq 0} \ p_Xx - l_X$</span> subject to <span class="math-container">$x = 2\sqrt{l_X}$</span></p>
<p>gives the demand for labor by firm producing <span class="math-container">$x$</span> and supply of <span class="math-container">$x$</span> as:
<span class="math-container">$l_X(p_X)=p_X^2$</span>, <span class="math-container">$x^s(p_X)=2p_X$</span>.
Profits earned by this firm is <span class="math-container">$\pi_X(p_X)=p_X^2$</span>.</p>
<p>Solving the profit maximisation problem of firm producing <span class="math-container">$y$</span>,</p>
<p><span class="math-container">$\max_{x\geq 0, y\geq 0} \ p_Yy - p_Xx$</span> subject to <span class="math-container">$y = 2x$</span></p>
<p>yields the condition for positive production of <span class="math-container">$y$</span> as:
<span class="math-container">$p_Y = \dfrac{p_X}{2}$</span>.
Profits earned by this firm is <span class="math-container">$\pi_Y(p_X, p_Y)=0$</span>.</p>
<p>Now we can solve the utility maximisation problem of consumer <span class="math-container">$i$</span>
<span class="math-container">\begin{eqnarray*} \max_{l_i\geq 0,x_i\geq 0,y_i\geq 0} &amp; l_ix_i^2y_i^2 \\ \text{s.t. } &amp; p_Xx_i+p_Yy_i+l_i = m_i \end{eqnarray*}</span>
Here <span class="math-container">$m_1=6+\pi_X = 6+p_X^2$</span> and <span class="math-container">$m_2 = 6+\pi_Y = 6$</span>.</p>
<p>Solving the utility maximisation problem, we get the demand for leisure as
<span class="math-container">$l_1 = \dfrac{6+p_X^2}{5}$</span>, <span class="math-container">$l_2 = \dfrac{6}{5}$</span>. Using demand = supply condition in the labor-market, we get</p>
<p><span class="math-container">$l_1+l_2+l_X=12$</span> i.e.
<span class="math-container">$\dfrac{6+p_X^2}{5}+\dfrac{6}{5}+p_X^2 = 12$</span></p>
<p>which gives <span class="math-container">$p_X=2\sqrt{2}$</span>. Using <span class="math-container">$p_Y = \dfrac{p_X}{2}$</span>, we get <span class="math-container">$p_Y = \sqrt{2}$</span>. Corresponding demand for labor by firm producing <span class="math-container">$x$</span> equals <span class="math-container">$l_X=8$</span>, demands for leisures by the two individuals are respectively <span class="math-container">$l_1=2.8$</span> and <span class="math-container">$l_2=1.2$</span>.</p>

CC BY-SA 4.0
2023-10-30T02:57:15.223Z
2023-10-30T02:57:15.223Z
Amit
<p>Consider a production economy with 2 consumers and 2 firms. There are 3 goods: leisure <span class="math-container">$l$</span> and two goods <span class="math-container">$x,y$</span>. The utility function of both consumers are <span class="math-container">$u_i(l_i,x_i,y_i)=l_i(x_i)^2(y_i)^2$</span> and the production functions of the 2 firms are <span class="math-container">$x=2\sqrt{l}$</span> and <span class="math-container">$y=2x$</span>. Each consumer is only endowed with 6 units of leisure. Consumer 1 is the sole owner of the first firm (the one that produces <span class="math-container">$x$</span>) and consumer 2 is the sole owner of the second firm (the one that makes <span class="math-container">$y$</span>). Suppose the wage rate is 1.</p>
<p>How do I find a Walrasian equilibrium for this economy?</p>

CC BY-SA 4.0
2023-10-29T22:07:07.197Z
You solve for the demand of the consumers, the profit-maximizing plan of from 1, and use the fact that if firm 2 is active (necessary, since equilibria are Pareto efficient), constant returns to scale pins down the ratio of the prices of the two consumption goods. So  you just have to find one price ratio under which supply is equal to demand.
Michael Greinecker
<walrasian>
How to find a Walrasian equilibrium for this production economy?
Ludwig Gershwin
# How to find a Walrasian equilibrium for this production economy?

(Asked by: username_0 on 2023-10-29)

Consider a production economy with 2 consumers and 2 firms. There are 3 goods: leisure $l$ and two goods $x,y$. The utility function of both consumers are $u\_i(l\_i,x\_i,y\_i)=l\_i(x\_i)^2(y\_i)^2$ and the production functions of the 2 firms are $x=2\sqrt{l}$ and $y=2x$. Each consumer is only endowed with 6 units of leisure. Consumer 1 is the sole owner of the first firm (the one that produces $x$) and consumer 2 is the sole owner of the second firm (the one that makes $y$). Suppose the wage rate is 1.

How do I find a Walrasian equilibrium for this economy?

**Tags:** walrasian
> **Comments:**
> **username_1**: You solve for the demand of the consumers, the profit-maximizing plan of from 1, and use the fact that if firm 2 is active (necessary, since equilibria are Pareto efficient), constant returns to scale pins down the ratio of the prices of the two consumption goods. So  you just have to find one price ratio under which supply is equal to demand.  

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## 1 answers

### Answer by username_2 on 2023-10-30. Score: 4 | Accepted answer

Solving the profit maximisation problem of firm producing $x$,

$\max\_{x\geq 0, l\_X\geq 0} \ p\_Xx - l\_X$ subject to $x = 2\sqrt{l\_X}$

gives the demand for labor by firm producing $x$ and supply of $x$ as:
$l\_X(p\_X)=p\_X^2$, $x^s(p\_X)=2p\_X$.
Profits earned by this firm is $\pi\_X(p\_X)=p\_X^2$.

Solving the profit maximisation problem of firm producing $y$,

$\max\_{x\geq 0, y\geq 0} \ p\_Yy - p\_Xx$ subject to $y = 2x$

yields the condition for positive production of $y$ as:
$p\_Y = \dfrac{p\_X}{2}$.
Profits earned by this firm is $\pi\_Y(p\_X, p\_Y)=0$.

Now we can solve the utility maximisation problem of consumer $i$
\begin{eqnarray\*} \max\_{l\_i\geq 0,x\_i\geq 0,y\_i\geq 0} & l\_ix\_i^2y\_i^2 \\ \text{s.t. } & p\_Xx\_i+p\_Yy\_i+l\_i = m\_i \end{eqnarray\*}
Here $m\_1=6+\pi\_X = 6+p\_X^2$ and $m\_2 = 6+\pi\_Y = 6$.

Solving the utility maximisation problem, we get the demand for leisure as
$l\_1 = \dfrac{6+p\_X^2}{5}$, $l\_2 = \dfrac{6}{5}$. Using demand = supply condition in the labor-market, we get

$l\_1+l\_2+l\_X=12$ i.e.
$\dfrac{6+p\_X^2}{5}+\dfrac{6}{5}+p\_X^2 = 12$

which gives $p\_X=2\sqrt{2}$. Using $p\_Y = \dfrac{p\_X}{2}$, we get $p\_Y = \sqrt{2}$. Corresponding demand for labor by firm producing $x$ equals $l\_X=8$, demands for leisures by the two individuals are respectively $l\_1=2.8$ and $l\_2=1.2$.


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