# Step-by-Step Solution ## 1. **Understanding the Concepts** ### Concave Utility Function - A **concave** function \( u(x) \) satisfies: \( u(\lambda x + (1-\lambda) y) \geq \lambda u(x) + (1-\lambda)u(y) \) for all \( x, y \) and \( \lambda \in [,1] \). - Implies **risk aversion** and **diminishing marginal utility**. ### Quasi-Concave Utility Function - A **quasi-concave** function \( u(x) \) satisfies: \( u(\lambda x + (1-\lambda) y) \geq \min\{u(x), u(y)\} \). - Implies **convex upper contour sets**. ### Why This Distinction Matters - **Quasi-concavity** of utility ensures **convex preferences**, meaning consumers prefer averages/mixes to extremes. - **Concavity** is stronger and not always required for convex preferences. --- ## 2. **Analyzing the Options** ### A) Convex preferences - **Quasi-concavity** of utility function **implies convex preferences**. - **Concavity** is not necessary; only quasi-concavity is needed. - This is correct. ### B) Linear Engel curves - Linear Engel curves are related to **homothetic preferences**, not directly to quasi-concavity. ### C) Zero income effects - Zero income effect is a property of **quasi-linear preferences**, not quasi-concavity. ### D) Cardinal utility - Quasi-concavity is about **ordinal properties** (shape of indifference curves), not cardinal measurement. --- ## 3. **Conclusion** ### **Final Answer** **A) Convex preferences** --- ## **Summary Table** | Option | Explanation | Correct? | |--------|--------------------------|----------| | A | Convex preferences | Yes | | B | Linear Engel curves | No | | C | Zero income effects | No | | D | Cardinal utility | No | --- **Quasi-concave utility implies convex preferences, which is the correct answer.**