The problem asks to derive the direct utility function $u(x_1, x_2)$ from the given indirect utility function:

$v(p_1, p_2, y) = \sqrt{\frac{y(p_1 + p_2)}{p_1 p_2}}$

Here's how we approach the problem step by step.

1. Recall the indirect utility function:

The indirect utility function gives the maximum utility that a consumer can achieve given prices $p_1$, $p_2$, and income $y$. The goal is to find the direct utility function $u(x_1, x_2)$, which expresses utility as a function of the quantities $x_1$ and $x_2$ of goods consumed.

2. Deriving Marshallian demand functions:

To find the direct utility function, we need to derive the demand functions for $x_1$ and $x_2$. The Marshallian demand for each good can be found by maximizing the utility subject to the budget constraint.

Given that:

• $v(p_1, p_2, y) = u(x_1, x_2)$,
• The budget constraint is $p_1 x_1 + p_2 x_2 = y$.

We start by obtaining the demand functions. Since the indirect utility function is homogenous of degree zero in prices and income, we can infer the structure of the demand functions. From the form of the indirect utility function, we assume a Cobb-Douglas utility function, because the given indirect utility suggests that the expenditure shares are constant.

The general form of the demand functions for a Cobb-Douglas utility function would be:

$x_1 = \frac{\alpha y}{p_1}, \quad x_2 = \frac{(1 - \alpha) y}{p_2}$

where $\alpha$ is a constant that represents the share of income spent on good 1.

3. Guessing the direct utility function form:

The structure of the indirect utility function $v(p_1, p_2, y)$ suggests that the direct utility function is likely of the form:

$u(x_1, x_2) = A x_1^\alpha x_2^\beta$

where $A$, $\alpha$, and $\beta$ are constants to be determined.

4. Using the indirect utility function to infer the direct utility:

Comparing this with the indirect utility function $v(p_1, p_2, y) = \sqrt{\frac{y(p_1 + p_2)}{p_1 p_2}}$, we can infer that the utility function is symmetric with respect to $p_1$ and $p_2$, which implies $\alpha = \beta = 1/2$. The direct utility function thus becomes:

$u(x_1, x_2) = \sqrt{x_1 x_2}$

This is a Cobb-Douglas utility function where utility depends on the geometric mean of the two goods consumed.

Final Answer:

The direct utility function is:

$u(x_1, x_2) = \sqrt{x_1 x_2}$

Would you like more details on any of these steps?

Here are 5 follow-up questions you could explore:

1. How do you derive Marshallian demand functions from a given utility function?
2. What role does homogeneity of the indirect utility function play in determining demand?
3. Why does the given indirect utility function suggest a Cobb-Douglas form for the direct utility function?
4. How can the relationship between prices and quantities consumed be represented using duality in economics?
5. Can this approach be extended to more than two goods, and how?

Tip:

When solving for direct and indirect utility functions, understanding how the budget constraint interacts with the utility maximization problem is key to deriving accurate demand functions.