We are tasked with calculating the demand functions for $x_1$ and $x_2$ for the utility function:

$U(x_1, x_2) = \log(x_1) + 2\log(x_2)$

given income $I$ and prices $p_1$ and $p_2$. We will proceed through the steps outlined to compute the demand functions.

Step 1: Write the Budget Constraint

The budget constraint is the equation that reflects the consumer's total expenditure, given by:

$p_1 x_1 + p_2 x_2 = I$

where:

• $p_1$ is the price of good 1,
• $p_2$ is the price of good 2,
• $I$ is the total income,
• $x_1$ and $x_2$ are the quantities of goods 1 and 2, respectively.

Step 2: Write the Lagrangian Equation

To solve the optimization problem, we use the method of Lagrange multipliers. The Lagrangian is:

$\mathcal{L}(x_1, x_2, \lambda) = \log(x_1) + 2\log(x_2) + \lambda (I - p_1 x_1 - p_2 x_2)$

where $\lambda$ is the Lagrange multiplier.

Step 3: Compute the First Derivatives

We now compute the first-order conditions by taking the partial derivatives of $\mathcal{L}$ with respect to $x_1$, $x_2$, and $\lambda$, and then setting them equal to zero.

Derivative with respect to $x_1$:

$\frac{\partial \mathcal{L}}{\partial x_1} = \frac{1}{x_1} - \lambda p_1 = 0$ $\Rightarrow \lambda = \frac{1}{p_1 x_1}$

Derivative with respect to $x_2$:

$\frac{\partial \mathcal{L}}{\partial x_2} = \frac{2}{x_2} - \lambda p_2 = 0$ $\Rightarrow \lambda = \frac{2}{p_2 x_2}$

Derivative with respect to $\lambda$ (enforces the budget constraint):

$\frac{\partial \mathcal{L}}{\partial \lambda} = I - p_1 x_1 - p_2 x_2 = 0$ $\Rightarrow p_1 x_1 + p_2 x_2 = I$

Step 4: Express $x_1$ and $x_2$ as Functions of Income and Prices

From the first two derivatives, we now have two expressions for $\lambda$:

1. $\lambda = \frac{1}{p_1 x_1}$
2. $\lambda = \frac{2}{p_2 x_2}$

Equating these two expressions:

$\frac{1}{p_1 x_1} = \frac{2}{p_2 x_2}$

Solving for $x_1$ in terms of $x_2$:

$x_2 = \frac{2 p_1}{p_2} x_1$

Now, substitute this into the budget constraint $p_1 x_1 + p_2 x_2 = I$:

$p_1 x_1 + p_2 \left( \frac{2 p_1}{p_2} x_1 \right) = I$ $p_1 x_1 + 2 p_1 x_1 = I$ $3 p_1 x_1 = I$ $x_1 = \frac{I}{3 p_1}$

Now, substitute $x_1$ into the expression for $x_2$:

$x_2 = \frac{2 p_1}{p_2} \cdot \frac{I}{3 p_1} = \frac{2 I}{3 p_2}$

Demand Functions

Thus, the demand functions for goods $x_1$ and $x_2$ are:

$x_1 = \frac{I}{3 p_1}$ $x_2 = \frac{2 I}{3 p_2}$

These are the Marshallian demand functions for goods 1 and 2.

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Would you like further details on any part of this solution? Here are five related questions to expand on this topic:

1. How do changes in prices $p_1$ and $p_2$ affect the demand for $x_1$ and $x_2$?
2. What are the elasticities of demand for $x_1$ and $x_2$?
3. Can we derive the indirect utility function from these demand functions?
4. How would the demand functions change if the utility function had a different form (e.g., CES utility)?
5. How do we interpret the Lagrange multiplier $\lambda$ in this context?

Tip: When solving utility maximization problems, always check that the first-order conditions satisfy the budget constraint, as this ensures optimal consumption.