Jordan's preferences for toothpaste (goodx) and all other goods (goody) are represented by the utility functionu(x, y) = 9 ln(x) +y.

(a) LetMdenote his income, andp_xandp_ythe prices of both goods. Find Jordan's demand functions for goodsxandy, using the Lagrange function method. What is his marginal utility of income? Note: His incomeMis in the millions of dollars, and it will not be important for this question to know its exact value.

(b) Supposep_y= 1 so that we can interpretyas the income spent on all goods other than toothpaste. At the store where Jordan buys his toothpaste, there is a sale so that the original pricep_x= 3 has been lowered top_x= 1. Evaluate the change in utility that Jordan experiences as a result.

(c) Can you evaluate Jordan's welfare change after the sale, but using the notion of consumer's surplus? Compare your answer to part (b).