question 4(7) please

Question 4: Quasilinear preferences [30 points) A consumer has the following utility function: U(x, y) = y +x3. Let P. and py be the corresponding prices and m her income. 1. Write down the corresponding Lagrangian function and find the first order conditions (FOCs). [3 points] 2. Use these FOCs to find the expression for the marginal rate of substitution (MRS). Bearing in mind that the MRS is the slope of the indifference curve, what can you tell about the indifference curves of a quasilinear utility function? [3 points 3. Find the Marshallian demand for x and y. In other words, find x* (Pa, Py, m) and y* (P2, Py, m). [3 points) 2 4. Let Pc = Py = 1 and m = 1/9. What are the optimal consumption levels for x and y? (Hint: Consumption levels can never be negative! [4 points) 5. How does your answer to the previous part change if m= 1 and still pa = Py = 1? [4 points) 6. For Pc = Py = 1, draw the income offer curve and the Engel curve for good x. [4 points) 7. Is x a normal or an inferior good? In terms of share of income spent on x, what happens to it as m increases? (4 points) 8. Is y a normal or an inferior good? Justify your answer. Give an example of two goods for which an "average consumer is likely to have quasilinear preferences. (5 points]