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### Question 1:
#### Demand functions for two goods (M and N):
- **\( Q_M = 30 - 6P_M + 12P_N \)**
- **\( Q_N = 36 + 6P_M - 7P_N \)**

Where \( P_M \) and \( P_N \) are the prices of goods \( M \) and \( N \), respectively.

You are asked to determine:
1. **Price elasticity of demand** for both goods \( M \) and \( N \), and the **elasticity nature** (elastic, inelastic, or unit elastic).
2. **Relationship** between these two goods (whether they are substitutes or complements).

To calculate the **elasticity**:
- The **price elasticity of demand** is given by the formula:
\[
E_p = \frac{\partial Q}{\partial P} \times \frac{P}{Q}
\]
Where:
- \( \frac{\partial Q}{\partial P} \) is the derivative of the demand function with respect to price.
- \( P \) is the price.
- \( Q \) is the quantity demanded.

For **substitutes/complements**:
- If the cross-price elasticity (the derivative of \( Q_M \) with respect to \( P_N \), and \( Q_N \) with respect to \( P_M \)) is positive, the goods are substitutes.
- If the cross-price elasticity is negative, the goods are complements.

### Question 2:
#### Lagrange method for utility maximization:
- **Total Utility (TU)** function: \( TU = X^{0.6} Y^{0.4} \)
- The budget constraint is given as:
\[
3X + Y = 165
\]
Where:
- The prices are \( P_X = 3 \) and \( P_Y = 1 \).
  
You are asked to:
1. Maximize utility by finding the optimal values of \( X \) and \( Y \) using the **Lagrange multiplier method**.
2. Calculate the **maximum utility**.

To solve this using the **Lagrange method**:
- Define the Lagrange function:
\[
\mathcal{L} = X^{0.6} Y^{0.4} + \lambda (165 - 3X - Y)
\]
- Differentiate the Lagrange function with respect to \( X \), \( Y \), and \( \lambda \) and solve the system of equations to find the values of \( X \), \( Y \), and \( \lambda \).

Would you like a step-by-step solution for each part? Also, here are some related questions that could help deepen your understanding:

1. How does cross-price elasticity determine whether goods are substitutes or complements?
2. Can you explain how the Lagrange multiplier helps in constrained optimization problems?
3. How does price elasticity impact business decisions regarding pricing strategies?
4. What are the economic implications if the goods are found to be substitutes vs. complements?
5. How does a change in budget affect the optimal consumption of goods \( X \) and \( Y \)?

**Tip:** Elasticity values greater than 1 indicate elastic demand, less than 1 indicate inelastic demand, and equal to 1 means unitary elastic demand.

Suatu perusahaan memproduksi barang M dan N. Fungsi permintaan barang M adalah QM=30-6PM+12PN dan barang N adalah QN=36+6PM-7PN. Jika harga per unit barang M sebesar Rp.3,- dan N sebesar Rp.5,-, maka: a. Berapa elastisitas permintaan untuk masing-masing barang M dan barang N serta bagaimana sifat elastisitasnya? b. Bagaimana hubungan kedua macam barang tersebut. 2. Selesaikan dengan metode Lagrange. Kepuasan konsumen dari mengkonsumsi barang X dan Y ditunjukkan oleh TU=X^0.6Y^0.4. Dana yang dianggarkan untuk membeli barang tersebut hanya Rp.165,- dan harga per unit masing-masing barang X dan Y sebesar Rp.3,- dan Rp.1,-. a. Hitunglah berapa barang X dan Y harus dikonsumsi agar kepuasan maksimum. b. Berapa besarnya kepuasan maksimumnya.

This problem covers demand elasticity for goods M and N, as well as utility maximization for goods X and Y using the Lagrange method. It includes detailed formulas for price elasticity and Lagrange multipliers, making it suitable for advanced high school or undergraduate economics students.

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