Robert's utility function for toothpaste (good x) and other stuff (good y) is u(x, y) = ln x + y. Assume that p^y = 1. Let M be her income. Suppose now that M = 10 and that p_x = 1/2.

(a) What is Robert's optimal choice?

(b) Find Robert's standard demand function and her compensated demand function for toothpaste, in this situation.

(c) Suppose the price of toothpaste goes down to 1/3. Calculate Roberts welfare change using the difference in her utility levels, and compare it to what you would get using the notion of consumer's surplus.