A consumer has quasilinear utility (,1,2,...,$(,1,2,...,$

)=+(1,2,...,$)=+(1,2,...,$

),$),$ where:

$$

is the quantity of good ,$,$

$$ is money ($($left over after paying for goods 1,...,),$1,...,),$ 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,...,,$1,2,...,,$ all measured in the same units as .$.$

a)$)$ Write the consumer$$s budget constraint.

b)$)$ Find the first−$-$order conditions that determine how much of each good is consumed,

assuming that the solution is interior.

[$[$Hint: Use the substitution method. If you$$re having trouble doing the problem for an

arbitrary function ,$,$ do the case where =1$=1$ or 2$2$ and (1,2,...,$(1,2,...,$

)=1$)=1$ or

1+2$1+2$ and see if you can generalize. Without a specific functional form, you can

simply write the partial derivative of (1,2,...,$(1,2,...,$

)$)$ with respect to $$ as

(1,...,)$(1,...,)$

$$

.]$.]$

c)$)$ 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,...,$,1,2,...,$ and the quantities

from the optimum determined in part a,$,$ which you may denote 1$1$

$$

,2$,2$

$$

,...,$,...,$

$$

.$.$

For the remainder of this question, assume that the solution is always interior, so that your

answer from part a is valid.

d)$)$ Find $$

$$

$$

.[$.[$Hint: Do your first−$-$order conditions depend on ?]$?]$ Are goods 1,...,$1,...,$ normal,

inferior, or neither?

Suppose the price of good 1$1$ increases from 1$1$ t