A consumer has quasilinear utility (, 1, 2, ... , ) = + (1, 2, ... , ), where:

is the quantity of good ,

is money (left over after paying for goods 1, ... , ), and

is a differentiable strictly concave function (this just means that if the optimum is interior, you can find it by taking first-order conditions). She has income , and faces prices ₁, ₂, ... , .

a) Find the first-order conditions that determine how much of each good is consumed, assuming that the solution is interior.

b) What condition on is needed for an interior solution? [Hint: cannot be negative at the consumer's optimum.] Express your answer in terms of , ₁, ₂, ... , and the quantities from the optimum determined in part a, which you may denote ₁ , ₂ , ... , .

For the remainder of this question, assume that the solution is always interior, so that the answer from part a is valid.

c) Find / . [Hint: Do the first-order conditions depend on ?] Are goods 1, ... , normal, inferior, or neither?

Question

A consumer has quasilinear utility (, 1, 2, ... , ) = + (1, 2, ... , ), where:

is the quantity of good ,

is money (left over after paying for goods 1, ... , ), and

is a differentiable strictly concave function (this just means that if the optimum is interior, you can find it by taking first-order conditions). She has income , and faces prices ₁, ₂, ... , .

a) Find the first-order conditions that determine how much of each good is consumed, assuming that the solution is interior.

b) What condition on is needed for an interior solution? [Hint: cannot be negative at the consumer's optimum.] Express your answer in terms of , ₁, ₂, ... , and the quantities from the optimum determined in part a, which you may denote ₁ , ₂ , ... , .

For the remainder of this question, assume that the solution is always interior, so that the answer from part a is valid.

c) Find / . [Hint: Do the first-order conditions depend on ?] Are goods 1, ... , normal, inferior, or neither?