Question: (a) Explain the meaning of homotheticity of preferences and its implication for the shape of income expansion paths.

(b) Theo consumes n types of chocolate in quantities \({{q}_{i}}\), \(i=1,...,n\). His preferences are captured by the utility function \(u\left( q \right)=\sum\limits_{i=1}^{n}{\frac{1}{i}\ln {{q}_{i}}}\), where \({{q}_{i}}\) is the quantity consumed of chocolate of type i. Chocolate of type i is of better quality and sold at a higher price than chocolate of type j if i > j. Are the following statements true or false? Briefly justify your answers.

(i) If Theo's total budget increases then he will spend a higher proportion of his budget on the highest quality of chocolate than he did before the increase.

(ii) Whatever Theo's budget, he will spend a higher share of his budget on the highest quality chocolate than on the lowest quality chocolate.