Deterministic Preferences Practice

Q1. Preference Relations

Which condition is NOT part of the standard preference axioms for deterministic utility representation on R^n_+?

Q2. Utility Existence

The Debreu representation theorem states that a continuous, strictly monotone preference on a convex, compact set admits what kind of utility function?

Q3. Cobb–Douglas Insight

For U(x_1, x_2) = x_1^{0.4} x_2^{0.6}, what is the marginal rate of substitution MRS_{12}?

Q4. Convexity Test

Which example exhibits strictly convex preferences?

Q5. Leontief Geometry

For Leontief preferences U(x_1, x_2) = min{x_1/2, x_2}, what describes the optimal consumption bundle on a given budget line?

Q6. First-Order Conditions

In a two-good utility maximization with smooth, strictly convex preferences, the interior solution satisfies which condition?

Q7. Comparative Statics

If preferences are homothetic, how does scaling income affect the optimal bundle?

Q8. Ordinal Transformations

Given U(x) = x_1 + x_2, which transformation produces an equivalent ordinal utility?

Q9. Corner Solutions

Under linear utility U(x_1, x_2) = 3 x_1 + 2 x_2, when do corner solutions arise?

Q10. Envelope Intuition

For a Lagrangian L = U(x_1, x_2) + λ (w - p_1 x_1 - p_2 x_2), what does λ represent at the optimum?