To answer this question, we need to understand the context provided by the quasilinear preference and how it affects the Hicksian demand functions and the expenditure function.

Quasilinear Preferences

A utility function is quasilinear in good 1 if it can be written in the form:

$u(x_1, x_2, \ldots, x_n) = x_1 + v(x_2, \ldots, x_n),$

where:

• $x_1$ is the quantity of the numeraire good (good 1).
• $x_2, \ldots, x_n$ are the quantities of the other goods.
• $v(x_2, \ldots, x_n)$ is some function representing the utility from the consumption of the other goods.

Quasilinear preferences imply that utility is linear in good 1 and nonlinear in the other goods.

Hicksian Demand Functions

Hicksian demand functions $h_i(p,u)$ are derived from the expenditure minimization problem, which is finding the minimum expenditure required to achieve a given level of utility $u$. For quasilinear preferences, the Hicksian demand for good 1 depends on the utility level $u$, while the Hicksian demands for the remaining goods $x_{-1} = (x_2, \ldots, x_n)$ depend only on the prices and not on the utility level $u$.

Since the utility function is quasilinear in good 1, the expenditure function can be easily characterized.

Expenditure Function for Quasilinear Preferences

The expenditure function $e(p, u)$ represents the minimum cost of achieving a utility level $u$ given prices $p = (p_1, p_2, \ldots, p_n)$.

Given quasilinear preferences, the expenditure function takes the form:

$e(p, u) = u + \tilde{e}(p_{-1}),$

where:

• $u$ is the utility level.
• $\tilde{e}(p_{-1})$ is the cost of achieving the baseline utility level (zero utility from good 1) given the prices of the other goods $p_{-1} = (p_2, \ldots, p_n)$.

Here’s why this form makes sense:

1. Utility Decomposition: In the quasilinear case, the utility derived from good 1 is separate and linear, $x_1$, and the rest is captured in the function $v(x_{-1})$.
2. Cost of Achieving Utility: To reach a utility level $u$, we need $u$ units of good 1 (since it’s linear in $x_1$) plus the minimum cost of obtaining utility level $v(x_{-1}) = 0$ from the other goods.
3. Independence of Hicksian Demands: The Hicksian demand for the remaining goods does not depend on $u$, just on prices $p_{-1}$, which further simplifies the expenditure function to a function of prices $p_{-1}$.

Conclusion

The expenditure function under quasilinear preferences with respect to good 1 is given by:

$e(p, u) = u + \tilde{e}(p_{-1}),$

where $\tilde{e}(p_{-1})$ is the cost of the optimal consumption bundle of the remaining goods, which is independent of $u$.

Would you like more details or have any questions? Here are some related questions that might help further your understanding:

1. What are the properties of the expenditure function in general?
2. How do quasilinear preferences affect consumer surplus?
3. What is the relationship between Hicksian demand and Marshallian demand under quasilinear preferences?
4. How does the quasilinear utility function impact the marginal rate of substitution between goods?
5. Can you provide an example of quasilinear preferences and compute the Hicksian demand functions?

Tip: Remember that with quasilinear preferences, the utility derived from one good (usually the numeraire) is linear, making calculations involving demand and expenditure functions more straightforward.