12 Nov 2021
1. Consider models including effective labours.
2. For producers (firms). How consumers preference of goods quality could be conveyed to producers. 消费者对质量好与坏的产品的需求倾向如何向生产者传递。
Consider a simplified model.
Firms:
$$ Y_{Good}=A\cdot L_{Good}^{\alpha} $$
$$ Y_{Bad}=A\cdot L_{Bad}^{\alpha} $$
\( s.t. \quad L_{Bad}\leq L_{Good} \)
That means good quality products take more factor inputs to produce.
$$ \pi_{Good}=P_{Good}\cdot Y_{Good}-W\cdot L_{Good} $$
\pi_{Bad}=P_{Bad}\cdot Y_{Bad}-W\cdot L_{Bad}
Consumers:
$$ u(C_{Good},C_{Bad})-v(L) $$
\( s.t. \quad P_{Good}C_{Good}+P_{Bad}C_{Bad}<=WL \) Budget Constraint (Non-negative constraint is ignored just now)
CE
Consumers:
$$\mathcal{L}= v(L) +\lambda (WL-P_{Good}C_{Good}-P_{Bad}C_{Bad}) $$
F.O.C.
$$ u_{C_{Good}}’-\lambda \cdot P_{Good}=0 $$
u_{C_{Bad}}’-\lambda \cdot P_{Bad}=0
$$ -v'(L)+\lambda W=0$$
$$So \frac{u_{C_{Good}}’}{P_{Good}}=\frac{u_{C_{Bad}}’}{P_{Bad}}, and \frac{u_{C_{Good}}’}{u_{C_{Bad}}’}=\frac{P_{Good}}{P_{Bad}}$$
Firms:
F.O.C.
$$ \alpha P_{Good} A L^{\alpha-1}_{Good}-W=0$$
\alpha P_{Bad} A L^{\alpha-1}_{Bad}-W=0
$$ So \frac{P_{Good}}{P_{Bad}}=( \frac{L_{Good}}{L_{Bad}} )^{1-\alpha} $$
Combining consumers and firms euler condition, we get
\frac{u_{C_{Good}}’}{u_{C_{Bad}}’} =( \frac{L_{Good}}{L_{Bad}} )^{1-\alpha}
Problems exist, reconsider it. Intra-temporal model + and intertemporal model in RBC framework.
3. 对于生产者公司模型的研究
Separate firms production \(Y^S=Y^{i}+Y^{-i}\), firms maxi long term total profits
$$ \max_{Y^{i}_t, Cost^{i}_t, Y^{-i}_t, Cost^{-i}_t } \sum_{t=0}^{\infty} \pi_t=A(s_t, m_t)P^{i}_t (Y^{i}_t-Cost^{i}_t)+P^{-i}_t(P^{-i}_t-Cost^{-i}_t) $$
s.t. \(Cost^{i}>Cost^{-i} \forall t \)
, where \(A(\cdot,\cdot)\) is a ratio function of \(s_t\) the market share and \(m_t\) the cross market occupation ratio (跨市场经营指数). Meanwhile, \(m_t\) and \(s_t\) are all functions of \(Y^{i}_t, Cost^{i}_t, Y^{-i}_t, Cost^{-i}_t \).
P.S. consider what is the competitive equilibrium in the long run. At the steady-state condition. VIE application and impacts to cross-market operations such as Tencent Alibaba and Facebook.
4. 消费者传递链条， 生产者->平台->消费者。 考虑情况：上游利润低是因为平台榨干上游利润，最终消费者剩余多，但是生产者剩余极低。导致Aggregate Supply 减少。
Revisit 24 Nov 2021
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.
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:
$$ \frac{{\partial} \mathcal{L}}{\partial l_1}: u’_3=i\cdot \lambda (w_1 l_1)^{i-1} (w_2 l_2)^{1-i} $$
$$ \frac{{\partial} \mathcal{L}}{\partial l_2}: u’_4=(1-i)\cdot \lambda (w_1 l_1)^{i} (w_2 l_2)^{-i} $$
$$ \frac{\partial \mathcal{L}}{\partial c_1}: u’_1=\lambda P_1 $$
$$ \frac{\partial \mathcal{L}}{\partial c_2}: u’_2=\lambda P_2 $$
And I can get,
$$ \frac{u’_4}{u’_3}=\frac{1-i}{i}\frac{w_1 l_1}{w_2 l_2} $$
$$ \frac{P_1}{P_2}=\frac{u’_1(c_1)}{u’_2(c_2)} $$
Firms:
$$ \frac{\partial \pi}{\partial l_1}: P_1 F’_{l_1}=w_1 $$
$$ \frac{\partial \pi}{\partial l_1}: P_2 F’_{l_2}=w_2 $$
And get,
$$ \frac{w_1}{w_2}=\frac{P_1 F’_{l_1}}{P_2 F’_{l_2}} $$
