Question:
A Constant Elasticity of Substitution (CES) utility function is given as:
\[
U_{{CES}}(z_1, z_2) = \frac{1}{\delta} \left( z_1^\delta + z_2^\delta \right)
\]
where \( z_1 \) and \( z_2 \) are two goods, and \( \delta \leq 1, \delta \neq 0 \). A Quasi-linear (QL) utility function is given as:
\[
U_{{QL}}(z_1, z_2) = 2z_1 + \log z_2
\]
where \( z_1 \) and \( z_2 \) are two goods. Which of the following statements is/are NOT CORRECT?
A Constant Elasticity of Substitution (CES) utility function is given as:
\[
U_{{CES}}(z_1, z_2) = \frac{1}{\delta} \left( z_1^\delta + z_2^\delta \right)
\]
where \( z_1 \) and \( z_2 \) are two goods, and \( \delta \leq 1, \delta \neq 0 \). A Quasi-linear (QL) utility function is given as:
\[
U_{{QL}}(z_1, z_2) = 2z_1 + \log z_2
\]
where \( z_1 \) and \( z_2 \) are two goods. Which of the following statements is/are NOT CORRECT?
Show Hint
In CES utility functions, the parameter \( \delta \) determines the elasticity of substitution. For \( \delta = 1 \), it simplifies to a linear utility function. Remember that the Marginal Rate of Substitution (MRS) is important when analyzing consumer choice and optimal consumption bundles.
Updated On: Apr 20, 2025
- The CES utility function is homothetic but the QL utility function is nonhomothetic.
- For \( \delta = 1 \), the CES utility function is not strictly convex.
- The Marginal Rate of Substitution (MRS\( z_1, z_2 \)) for the CES utility function and the QL utility function are dependent on both \( z_1 \) and \( z_2 \).
- If \( z_1 = z_2 \), the Marginal Rate of Substitution (MRS\( z_1, z_2 \)) is 2 for both the CES and the QL utility functions.
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The Correct Option is C, D
Solution and Explanation
Analyze the properties of the CES utility function.
(A) is correct: The CES utility function is homothetic, meaning that the MRS depends only on the ratio of the two goods, and the utility function retains its shape when scaled. The QL utility function, however, is nonhomothetic because the utility function is linear in one good and logarithmic in the other.
(B) is incorrect: For \( \delta = 1 \), the CES utility function becomes a linear function (i.e., \( U_{{CES}}(z_1, z_2) = z_1 + z_2 \)), which is strictly convex. Therefore, the statement is not correct.
(C) is correct: The MRS for both CES and QL utility functions depend on both \( z_1 \) and \( z_2 \). In the CES utility function, MRS depends on the ratio of goods, and for the QL utility function, it depends on both goods explicitly.
(D) is correct: If \( z_1 = z_2 \), the MRS for both the CES and QL utility functions is 2, which can be derived from the respective MRS formulas for both utility functions.
(A) is correct: The CES utility function is homothetic, meaning that the MRS depends only on the ratio of the two goods, and the utility function retains its shape when scaled. The QL utility function, however, is nonhomothetic because the utility function is linear in one good and logarithmic in the other.
(B) is incorrect: For \( \delta = 1 \), the CES utility function becomes a linear function (i.e., \( U_{{CES}}(z_1, z_2) = z_1 + z_2 \)), which is strictly convex. Therefore, the statement is not correct.
(C) is correct: The MRS for both CES and QL utility functions depend on both \( z_1 \) and \( z_2 \). In the CES utility function, MRS depends on the ratio of goods, and for the QL utility function, it depends on both goods explicitly.
(D) is correct: If \( z_1 = z_2 \), the MRS for both the CES and QL utility functions is 2, which can be derived from the respective MRS formulas for both utility functions.
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