Consider a consumer who consumes only two goods, x and y. His utility over these two goods is given by U (x, y) = xy. The budget constraint of the consumer is given by 3x + 9y = 216, where 3 is the price of good x, 9 is the price of good y, and 216 is the total income of the consumer.

(a) Find the optimal quantities of good x and y that the consumer is going to consume. Show the solution in a graph. What level of utility is the consumer going to achieve with this bundle?

(b) Now assume that the price of good x increases to 6. Find the new optimal consumption bundle and show it in a  graph.

(c) Find the income and substitution effects associated with the increase in the price of x. Show your result in a graph.

(d) Given your result in (c), what type of goods are x and y (normal, inferior, or Giffen)?