The image you uploaded contains a set of economic and mathematical questions. Here's a breakdown of the problem from the image and how to approach solving it:

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.