<p>This is the utility maximisation problem of the consumer:
<span class="math-container">\begin{eqnarray*}\max_{c_D, l_S} &amp; \ c_D^a(24-l_S)^{1-a} \\ \text{s.t.} &amp; \ pc_D = wl_S \\ \text{and} &amp; \ c_D\geq 0 , 0\leq l_S\leq 24\end{eqnarray*}</span>
Solving this problem, we get
<span class="math-container">$c_D = \dfrac{24aw}{p}$</span> and <span class="math-container">$l_S = 24a$</span>.</p>
<p>Now we can solve the firm's profit maximisation problem:
<span class="math-container">\begin{eqnarray*}\max_{c_S, l_D} &amp; \ pc_S - wl_D \\ \text{s.t.} &amp; \ c_S = Al_D \\ \text{and} &amp; \ c_S\geq 0 , l_D\geq 0\end{eqnarray*}</span></p>
<p>Solving this problem, we get
<span class="math-container">\begin{eqnarray*} c_S \in \begin{cases} \emptyset &amp; \text{if } p &gt; \dfrac{w}{A} \\ \mathbb{R}_+ &amp; \text{if } p = \dfrac{w}{A} \\ \{0\} &amp; \text{if } p &lt; \dfrac{w}{A}\end{cases} \end{eqnarray*}</span>
and the corresponding <span class="math-container">$l_D = \dfrac{c_S}{A}$</span>.</p>
<p>To find the competitive equilibrium, we can take one of the markets, say market for <span class="math-container">$c$</span>, and proceed by solving <span class="math-container">$c_S = c_D$</span> and we'll get the competitive equilibrium price ratio as:
<span class="math-container">$p =\dfrac{w}{A}$</span>, or equivalently, <span class="math-container">$\dfrac{w}{p} = A$</span>.
Corresponding value of <span class="math-container">$c$</span> consumed in equilibrium is <span class="math-container">$24Aa$</span> and labor employed equals <span class="math-container">$24a$</span>.</p>

CC BY-SA 4.0
2022-04-02T16:50:12.177Z
Thank you sir. Your Econschool is truly helpful.
Kushh Raghav
CC BY-SA 4.0
2022-04-02T16:15:35.957Z
2022-04-02T16:15:35.957Z
Amit
<p>A representative agent’s preference over consumption <span class="math-container">$(c)$</span> and labour supply <span class="math-container">$(l)$</span> is given by the utility function <span class="math-container">$$ u(c_D, l_S)= c_D^a .(24-l_S)^{1-a}$$</span>
Production of the consumption good <span class="math-container">$c$</span> is given by the production function <span class="math-container">$c = Al$</span>, where <span class="math-container">$A &gt; 0$</span> is the productivity of labour. Both the commodity market and labour market are perfectly competitive: the buyers and sellers take the price as given while takin demand and supply decisions. Let us denote the hourly wage rate by <span class="math-container">$w &gt; 0$</span> and price of the consumption good by <span class="math-container">$p &gt; 0$</span>
The question is asking about the competitive equilibrium, so I’ve found out the utility maximising <span class="math-container">$c_D =\frac {24aw}{p} ,_\ l_S= 24a$</span></p>
<p>I’m just confused about the profit maximising <span class="math-container">$l_D,c_S$</span>. It is coming out as <span class="math-container">$A= _\frac{w}{p}$</span> and upon further solving is coming out as <span class="math-container">$c_S = _\frac {w}{p} l_D$</span> which I think is wrong.</p>
<p>Can someone tell me what am I doing wrong? Thank you.</p>

CC BY-SA 4.0
2022-04-02T06:32:20.280Z
Why do you think your final two equations are wrong?
Giskard
CC BY-SA 4.0
2022-04-02T06:35:06.060Z
@Giskard I’m not sure, but I can’t seem to the find the answer to competitive equilibrium so I presumed my equations are wrong. Is it correct?
Kushh Raghav
<microeconomics><labor-economics><general-equilibrium><profit-maximization><competitive-equilibrium>
Competitive equilibrium for an economy with a consumer and a producer
Kushh Raghav
# Competitive equilibrium for an economy with a consumer and a producer

(Asked by: username_0 on 2022-04-02)

A representative agent’s preference over consumption $(c)$ and labour supply $(l)$ is given by the utility function $$ u(c\_D, l\_S)= c\_D^a .(24-l\_S)^{1-a}$$
Production of the consumption good $c$ is given by the production function $c = Al$, where $A > 0$ is the productivity of labour. Both the commodity market and labour market are perfectly competitive: the buyers and sellers take the price as given while takin demand and supply decisions. Let us denote the hourly wage rate by $w > 0$ and price of the consumption good by $p > 0$
The question is asking about the competitive equilibrium, so I’ve found out the utility maximising $c\_D =\frac {24aw}{p} ,\_\ l\_S= 24a$

I’m just confused about the profit maximising $l\_D,c\_S$. It is coming out as $A= \_\frac{w}{p}$ and upon further solving is coming out as $c\_S = \_\frac {w}{p} l\_D$ which I think is wrong.

Can someone tell me what am I doing wrong? Thank you.

**Tags:** microeconomics, labor-economics, general-equilibrium, profit-maximization, competitive-equilibrium

> **Comments:**
> **username_1**: Why do you think your final two equations are wrong?  
> **username_0**: @username_1 I’m not sure, but I can’t seem to the find the answer to competitive equilibrium so I presumed my equations are wrong. Is it correct?  

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## 1 answers

### Answer by username_2 on 2022-04-02. Score: 4 | Accepted answer

This is the utility maximisation problem of the consumer:
\begin{eqnarray\*}\max\_{c\_D, l\_S} & \ c\_D^a(24-l\_S)^{1-a} \\ \text{s.t.} & \ pc\_D = wl\_S \\ \text{and} & \ c\_D\geq 0 , 0\leq l\_S\leq 24\end{eqnarray\*}
Solving this problem, we get
$c\_D = \dfrac{24aw}{p}$ and $l\_S = 24a$.

Now we can solve the firm's profit maximisation problem:
\begin{eqnarray\*}\max\_{c\_S, l\_D} & \ pc\_S - wl\_D \\ \text{s.t.} & \ c\_S = Al\_D \\ \text{and} & \ c\_S\geq 0 , l\_D\geq 0\end{eqnarray\*}

Solving this problem, we get
\begin{eqnarray\*} c\_S \in \begin{cases} \emptyset & \text{if } p > \dfrac{w}{A} \\ \mathbb{R}\_+ & \text{if } p = \dfrac{w}{A} \\ \{0\} & \text{if } p < \dfrac{w}{A}\end{cases} \end{eqnarray\*}
and the corresponding $l\_D = \dfrac{c\_S}{A}$.

To find the competitive equilibrium, we can take one of the markets, say market for $c$, and proceed by solving $c\_S = c\_D$ and we'll get the competitive equilibrium price ratio as:
$p =\dfrac{w}{A}$, or equivalently, $\dfrac{w}{p} = A$.
Corresponding value of $c$ consumed in equilibrium is $24Aa$ and labor employed equals $24a$.

> **Comments:**
> **username_0**: Thank you sir. Your Econschool is truly helpful.  

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