Microeconomics questions attached below

PROBLEM 3 - Slutsky Equation, Income Effect, Substitution Effect, and Total Effect (25 Points) There are two goods which quantities are to be denoted by x and y, while prices are denoted by Pa and Py, respectively. There is a consumer whose income is to be denoted by / and utility by u. His expendi I =I(Px, Py, u) = 3 1 (27 P3 Py 4 a. Find the consumer's uncompensated demand function for good x and good y (7 Points) b. Suppose the consumer already spend $90 on good x which cost $1 and good y which cost $1, and initially purchase 60 of good x and 30 of good y (3 Points) i. If the price of good x increase to $2, how many will he purchase each of the goods? ii. How much of the decrease in his demand for good x is due to the fact that they have become relatively more expensive? How much is due to the fact that his overall purchasing power has decreased? [Hint: find first the initial utility before the price change and income level needed to reach those initial utility with new prices. Slope of the indifference curve is given by MRS = -~] (10 Points) iii. Based on your answer, draw it in a graph indicate the direction of each effect and derive the compensated and uncompensated demand curve for good x. Are they different or same? Which one is steeper? Why? (5 Points) C. [BONUS] Using slutsky equation, decompose the effect of an infinitesimal increase in P on demand of good x (5 Points)