Calculate the demand functions of x1 and x2 for the following utility function  U1(x1, x2) = log x1 + 2 log x2, given
income I and prices p1 and p2. Writing a demand function is tantamount to express demand
for good x as a function of px, I, and perhaps py in the case of a rational consumer. Without
further information, you must:
1. Write the budget constraint I as a function of quantities and prices.
2. Write the Lagrangian equation.
3. Compute the first derivative of L with respect to x1, x2, and λ.
4. Express xi as a function of income I and, if relevant, xj , where j ̸= i and i, j ∈ {1, 2}.


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.

---

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.

Calculate the demand functions of x1 and x2 for the following utility function U1(x1, x2) = log x1 + 2 log x2, given income I and prices p1 and p2. Writing a demand function is tantamount to express demand for good x as a function of px, I, and perhaps py in the case of a rational consumer. Without further information, you must: 1. Write the budget constraint I as a function of quantities and prices. 2. Write the Lagrangian equation. 3. Compute the first derivative of L with respect to x1, x2, and λ. 4. Express xi as a function of income I and, if relevant, xj , where j ̸= i and i, j ∈ {1, 2}.

Learn how to calculate the demand functions for x1 and x2 from the utility function U(x1, x2) = log(x1) + 2 log(x2) given prices and income. This problem involves concepts like Lagrangian multipliers, utility maximization, and the derivation of Marshallian demand functions. Suitable for undergraduate economics students.

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