1.Suppose that Sam has a utility function u(x, y)= xy² where x is the amount of good 1 and y is the amount of good 2. The price of good 1 is $10 and price of good 2 is $20, and the income is $ 90. The price of good 1 is denoted by p_x and the income is donated by m. Derive the equations for income-offer curve, Engel curve for good 1, demand curve for good 1 and solve for the optimal consumption of (x, y).

2.(9pts) We are in the exact same situation as we were in the above question #7, except for the $5 price decrease of good 1. Then what is the income necessary to make the old consumption bundle (the optimal bundle from the previous problem) just affordable? What is the substitution effect on the demand for good 1 (the change in demand for good 1 due only to the change in relative price, with the adjusted income, you should show the sign as well) and income effect on the demand for good 1?

3.(8pts) Suppose that Sam has a utility function u(x, y)= x+y where x is the amount of good 1 and y is the amount of good 2. The price of good 2 is $20, and the income is $ 90.

a.With the $20 price decrease of good 1 from $30 to $10, then what is the substitution effect and income effect on the demand for good 1?

b.With the $5 price decrease of good 1 from $10 to $5, then what is the substitution effect and income effect on the demand for good 1?

4.(16pts) In the above question, you can find the shape of Engel curve for good 1 for the Cobb-Douglas Preferences. Consider the general from, u(x, y)= x^ay^b

a.What is the definition of homothetic preference? (you can write one of the two definitions in the lecture slide)

b.This utility function represents homothetic preference? Explain your answer.

c.Are quasi linear preferences homothetic? Explain your answer.

d.Are Engel Curves straight lines if the consumer's preferences are homothetic? (If yes, prove it. Otherwise, show the counter example)

5.(16pts) Assume that the three properties of weak preference relation that you learned in the class.

a.Prove the transitivity of strict preference relation.

b.Prove that indifference curves cannot intersect.