Final Answer:
If a well-behaved differentiable homothetic utility function is considered, and good X is normal, then good Y can be characterized as A. Normal.
Step-by-step explanation:
In economics, a homothetic utility function indicates that the consumer's preferences are constant along rays from the origin in the space of goods. Denoted as U(X, Y), where X and Y represent quantities of goods, a homothetic utility function satisfies the property U(tX, tY) = U(X, Y) for all positive values of t, emphasizing that utility depends on the ratio of X and Y rather than their absolute quantities. In this scenario, if good X is considered normal, meaning its demand increases as income rises, and the utility function is homothetic, then good Y would also be characterized as normal. This relationship implies that both goods X and Y are normal goods.
Specifically, the term "well-behaved differentiable" indicates that the utility function has smooth and continuous properties, allowing for differentiation with respect to the quantities of goods. The condition of homotheticity ensures that the consumer's preferences remain consistent across different scales of consumption. If good X is normal and the utility function is homothetic, then good Y, being the other good in the utility function, would also exhibit the same characteristics in terms of income elasticity, hence leading to its characterization as a normal good as well.
Therefore, considering the properties of a well-behaved differentiable homothetic utility function and the normality of good X, the logical inference within this framework is that good Y would also be classified as a normal good, reflecting a positive relationship between income and the demand for both goods X and Y.