Generalise the model
I denote \(c_1, l_1, L_1, y_1\) as the consumption, labour supply and demand, and outputs of relative high-quality goods. Also, denote \(c_2, l_2, L_2, y_2\) as those of relative low-quality goods. Note that the high-low quality stated in this working blog only refers to relative quality.
Consumers maximise their utility function subject to the budget constraint. For a representative consumer, the utility function is,
$$ \max_{c_1, c_2, l_1, l_2} u(c_1,c_2,1-l_1,1-l_2) $$
$$ s.t. \quad (l_1\cdot w_1)^i (l_2 \cdot w_2)^{1-i}\geq P_1 c_1 +P_2 c_2 $$
$$ i\in \{0,1\} $$
The wealth of consumers is in Bernoulli form because we assume each consumer can only provide a unique kind of labour in productions. \(i\) denotes the individual’s decision of providing labour for high-quality products or low-quality one.
Consumers provide labours \(l_1\) or \(l_2\), and consume goods \(c_1\) or \(c_2\).
Firms maximise profits. I simplify the model by considering only labour inputs as the factors input. The model could be further expanded by including capital term and letting the technology term be depending on other factor inputs. E.G. \(F( L, K )\).
$$ \max_{L_1, L_2} \pi = \max_{L_1, L_2} P_1 F(L_1)+P_2 F(L_2) – w_1 L_1 -w_2L_2 $$
Solve the model
Consumer:
$$ \frac{{\partial} \mathcal{L}}{\partial l_1}: u’_3=i\cdot \lambda (w_1 l_1)^{i-1} (w_2 l_2)^{1-i} \quad (1)$$
$$ \frac{{\partial} \mathcal{L}}{\partial l_2}: u’_4=(1-i)\cdot \lambda (w_1 l_1)^{i} (w_2 l_2)^{-i} \quad (2)$$
$$ \frac{\partial \mathcal{L}}{\partial c_1}: u’_1=\lambda P_1 \quad (3)$$
$$ \frac{\partial \mathcal{L}}{\partial c_2}: u’_2=\lambda P_2 \quad (4)$$
And I can get,
$$ \frac{u’_4}{u’_3}=\frac{1-i}{i}\frac{w_1 l_1}{w_2 l_2} \quad (5)$$
$$ \frac{P_1}{P_2}=\frac{u’_1(c_1)}{u’_2(c_2)} \quad (6)$$
Firms:
$$ \frac{\partial \pi}{\partial l_1}: P_1 F’_{l_1}=w_1 \quad(7)$$
$$ \frac{\partial \pi}{\partial l_1}: P_2 F’_{l_2}=w_2 \quad(8)$$
And get,
$$ \frac{w_1}{w_2}=\frac{P_1 F’_{l_1}}{P_2 F’_{l_2}} \quad(9)$$
Combine \((5)\), \((6)\) and \((9)\),
$$ \frac{u’_4}{u’_3}\frac{i}{1-i}\frac{l_2}{l_1}=\frac{u’_1}{u’_2}\cdot \frac{F’_1}{F’_2} $$
Apply the markets clearing condition,
$$\int l_1 =L_1$$
$$\int l_2 = L_2$$
$$ y_1=c_1 $$
$$ y_2=c_2 $$
The equilibrium condition could be rewritten as,
$$ \frac{u’_4}{u’_3}\cdot \frac{u’_2}{u’_1}=\frac{F’_1}{F’_2} $$
By aggregating individuals \(l_1\) and \(l_2\), \(i\) could then represent the proportion of people who provide labours for high-quality products or low-quality products. Thus, \(\frac{i}{1-i}=\frac{L_1}{L_2}\).
The Functional Form
Assume \(y_1,y_2=F(l_1,l_2)=A[l_1^{\alpha}+l_2^{\alpha}]\). Production depends purely on labour inputs and
\left\{
\begin{aligned}
F’_1=\alpha A l_1^{\alpha-1} \\
F’_2=\alpha A l_2^{\alpha-1}
\end{aligned}
\right.
\quad \Rightarrow \quad \frac{F’_1}{F’_2}=(\frac{l_1}{l_2})^{\alpha-1}
To be continued.
