Competitive equilibrium for an economy with a consumer and a producer

Question

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.

Answer 1

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 $24a$.