To find a direct utility function $u(x_1, x_2)$ that rationalizes the consumer's demand behavior given the expenditure function $e(p_1, p_2, u) = \frac{up_1p_2}{p_1 + p_2}$, we can proceed as follows:

Step 1: Identify the indirect utility function

The expenditure function is typically related to the indirect utility function $v(p_1, p_2, m)$ (where $m$ is income) through the relationship: $v(p_1, p_2, m) = u \quad \text{where} \quad m = e(p_1, p_2, u)$ So, we have: $v(p_1, p_2, m) = \frac{m(p_1 + p_2)}{p_1p_2}$

Step 2: Derive the Marshallian demand functions

To derive the demand functions, we can use Roy's identity: $x_1(p_1, p_2, m) = -\frac{\partial v/\partial p_1}{\partial v/\partial m}, \quad x_2(p_1, p_2, m) = -\frac{\partial v/\partial p_2}{\partial v/\partial m}$ First, we compute the partial derivatives of the indirect utility function $v(p_1, p_2, m)$: $\frac{\partial v}{\partial p_1} = \frac{m \cdot p_2}{p_1^2 p_2}, \quad \frac{\partial v}{\partial p_2} = \frac{m \cdot p_1}{p_1 p_2^2}, \quad \frac{\partial v}{\partial m} = \frac{p_1 + p_2}{p_1 p_2}$ Thus, the demand functions are: $x_1(p_1, p_2, m) = -\frac{\frac{m \cdot p_2}{p_1^2 p_2}}{\frac{p_1 + p_2}{p_1 p_2}} = \frac{m \cdot p_2}{p_1(p_1 + p_2)}$ $x_2(p_1, p_2, m) = -\frac{\frac{m \cdot p_1}{p_1 p_2^2}}{\frac{p_1 + p_2}{p_1 p_2}} = \frac{m \cdot p_1}{p_2(p_1 + p_2)}$

Step 3: Find the utility function $u(x_1, x_2)$

The Marshallian demand functions $x_1$ and $x_2$ suggest that the utility function should take the form where $x_1$ and $x_2$ are homogeneous of degree one in $m$. Consider a Cobb-Douglas utility function of the form: $u(x_1, x_2) = x_1^{\alpha} x_2^{\beta}$ Given that the demands derived from $e(p_1, p_2, u)$ show a symmetry and depend on the ratio $\frac{p_2}{p_1}$, the parameters $\alpha$ and $\beta$ should be equal, implying: $u(x_1, x_2) = x_1^{\frac{1}{2}} x_2^{\frac{1}{2}} = \sqrt{x_1 x_2}$

Conclusion

The direct utility function $u(x_1, x_2)$ that rationalizes this consumer's demand behavior is: $u(x_1, x_2) = \sqrt{x_1 x_2}$ This utility function is consistent with the given expenditure function.

Would you like any further details or have any questions?

Here are some related questions:

1. How would you derive the demand functions using the utility function $u(x_1, x_2) = \sqrt{x_1 x_2}$?
2. Can you explain how the duality between the expenditure function and the utility function works?
3. How does the homogeneity of the utility function influence the demand functions?
4. What role does the Cobb-Douglas utility function play in consumer theory?
5. How does Roy's identity help in deriving demand functions?

Tip: When dealing with expenditure and utility functions, understanding the relationship between indirect and direct utility functions can greatly simplify the derivation of demand functions.