# Step-by-Step Solution ## Step 1: Find Conditional Factor Demands and Functions ###1.1 **Assumptions and Functions** - goods: Good1 and Good 2 - Two factors: Skilled labor (H, wage = \( v \)), Unskilled labor (L, wage = \( w \)) - Production functions (Cobb-Douglas, constant returns): - For good \( i \): \( Q_i = H_i^{\beta_i} L_i^{1-\beta_i} \) - \( \beta_1 = \frac{2}{3}, \quad \beta_2 = \frac{1}{3} \) ### 1.2. **Conditional Factor Demands** Given a target output \( Q_i \), cost minimization yields (via Lagrangian or Cobb-Douglas rules): - **Optimal factor ratios:** - \( \frac{H_i}{L_i} = \frac{\beta_i}{1-\beta_i} \cdot \frac{w}{v} \) - **Conditional demands (solving for \( H_i \) and \( L_i \) as functions of \( Q_i, v, w \)):** - \( H_i = \beta_i \left( \frac{C_i}{v^{\beta_i} w^{1-\beta_i}} \right) Q_i \) - \( L_i = (1 - \beta_i) \left( \frac{C_i}{v^{\beta_i} w^{1-\beta_i}} \right) Q_i \) - Where \( C_i \) is a proportionality constant, but for Cobb-Douglas with unit productivity, we can write: - \( H_i = \beta_i \cdot \frac{Q_i}{\left( \frac{v}{\beta_i} \right)^{\beta_i} \left( \frac{w}{1-\beta_i} \right)^{1-\beta_i}} \) - \( L_i = (1-\beta_i) \cdot \frac{Q_i}{\left( \frac{v}{\beta_i} \right)^{\beta_i} \left( \frac{w}{1-\beta_i} \right)^{1-\beta_i}} \) **Explanation:** This step gives the formulas for how much skilled and unskilled labor is needed to produce any output \( Q_i \), given wages \( v, w \) and technology parameters \( \beta_i \). --- ### 1.3. **Price as a Function of Wages** Under perfect competition, price equals minimum cost per unit: \( p_i = \min_{H_i, L_i} [v H_i + w L_i] \) such that \( Q_i = H_i^{\beta_i} L_i^{1-\beta_i} \) By Cobb-Douglas cost function properties: \( p_i = \frac{1}{A_i} \cdot v^{\beta_i} w^{1-\beta_i} \) (Assuming productivity parameter \( A_i = 1 \) for simplicity.) **For Good 1:** \( p_1 = v^{2/3} w^{1/3} \) **For Good 2:** \( p_2 = v^{1/3} w^{2/3} \) **Explanation:** This step describes how, in a competitive market, the price of each good equals its unit cost, which for Cobb-Douglas is a weighted geometric mean of the factor prices, with exponents equal to each factor’s output share. --- ## Step 2: Equilibrium Quantities and Factor Content ### 2.1. **Equilibrium Production and Consumption** - **Utility function:** \( U = C_1^{\alpha} C_2^{1-\alpha} \), \( \alpha = 5/7 \) - **Consumer optimization** (spend total income \( Y \)): - \( C_1 = \alpha \frac{Y}{p_1} \) - \( C_2 = (1-\alpha) \frac{Y}{p_2} \) **Market clearing:** \( Q_1 = C_1 \), \( Q_2 = C_2 \) - **Total income:** \( Y = v H^T + w L^T \) Plug in the prices from Step 1.3 and income: - \( Q_1 = \frac{5}{7} \cdot \frac{v H^T + w L^T}{v^{2/3} w^{1/3}} \) - \( Q_2 = \frac{2}{7} \cdot \frac{v H^T + w L^T}{v^{1/3} w^{2/3}} \) ### 2.2. **Factor Content Per Unit Output** From Step 1.2, per unit output: - For good \( i \): - Skilled labor per unit: \( h_i = \beta_i \) - Unskilled labor per unit: \( l_i = 1 - \beta_i \) So: - Good 1: \( h_1 = 2/3, \quad l_1 = 1/3 \) - Good 2: \( h_2 = 1/3, \quad l_2 = 2/3 \) **Explanation:** This step derives equilibrium production and consumption using the Cobb-Douglas utility and production functions, and identifies the skilled/unskilled labor proportions per unit output for each good. --- ## Final Answers ### **1. Prices as a function of wages:** - \( p_1 = v^{2/3} w^{1/3} \) - \( p_2 = v^{1/3} w^{2/3} \) ### **2. Equilibrium quantities:** - \( Q_1 = \frac{5}{7} \cdot \frac{v H^T + w L^T}{v^{2/3} w^{1/3}} \) - \( Q_2 = \frac{2}{7} \cdot \frac{v H^T + w L^T}{v^{1/3} w^{2/3}} \) ### **3. Factor content per unit production:** - Good 1: \( 2/3 \) skilled, \( 1/3 \) unskilled per unit - Good 2: \( 1/3 \) skilled, \( 2/3 \) unskilled per unit --- **Summary:** - Step 1 solved for factor demands and prices as functions of wages using Cobb-Douglas properties. - Step 2 used these to find equilibrium outputs and factor input requirements, linking utility maximization and production. - The framework sets up for calibration and further trade analysis as requested in your subsequent parts.