# Competitive equilibrium for an economy with a consumer and a producer

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.

• Why do you think your final two equations are wrong? Apr 2, 2022 at 6:32
• @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? Apr 2, 2022 at 6:35

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 $$24a$$.