Alice has homothetic preferences over n goods, q_i, i=1,...,n. Her Marshallian demands are f_i(y,p₁,...,p_n), i=1,...,n, where y is total budget and p_i, i=1,...,n, is the price of the ith good.

Which of the following is necessarily true?

Agatha consumes two goods, q₁ and q₂. Her marginal rate of substitution is -q₂ exp(-q₁).  Income expansion paths all have which of the following forms (each for some different value of A)?

Suppose total budget elasticities are independent of total budget for all goods.  What must be true of preferences?

If preferences are strictly convex then there can be multiple tangencies between indifference curves and the budget constraint only if the budget set is not convex.

Assuming no corner solutions, an income expansion path is a locus of points of equal MRS.

Vlad consumes goods q. His preferences are homothetic.  What must be true of any utility function u(q) representing his preferences?

Maya consumes two goods, lipstick q₁ and mascara q₂. The figure below shows her indifference map. 

Which of the following accurately describes her preferences? 

Sofia consumes two goods, salt q₁ and vinegar q₂. She currently consumes a bundle (1,1) and is indifferent between this and any other bundle such that q₁+√q₂=2.  The rest of her indifference map consists of indifference curves defined by  q₁/υ+√(q₂/υ)=2 for different values of υ.  

What class of preferences does she belong to?

John consumes two goods, q₁ and q₂. His preferences are captured by utility function u(q₁,q₂)=q₁q₂+q₁².

John's preferences are