Consider Arturo, an individual with utility for a composite good \((X)\) and indoor air quality \((Q): U=X Q\). Indoor air quality depends on pollution levels outside \((P)\) and defensive expenditures \((D): D=P Q^{2}\). Let Arturo's income be \(Y\).

a. Write Arturo's problem of utility maximization subject to a budget constraint, mathematically. Draw Arturo's budget constraint and several indifference curves.

b. Suppose \(Y=10\) and \(P=1\). What are the optimal amounts of \(X\) and \(Q\) for Arturo?

c. What happens to the optimal \(X\) and \(Q\) (show your work) when 

(i) \(P\) increases to 2 

(ii) \(Y\) increases to 20 ?

d. It is observed that low income individuals are exposed to more pollution. Is your answer to part (c) consistent with this?