Bob's preferences can be represented by the utility function U = (XY) where X and Y are the quantities of the two different commodities which Bob can choose to consume each week. He faces prices Px, Py and income I. a. Using the Lagrangian method, find Bob's demand functions for good X and good Y. Show all your work. b. If Bob faces prices $20 for good X and $10 for good Y and he earns $240 each week, how much of each commodity will he buy each week? Show Bob's choice on a IC-BL diagram. c. If the government puts a tax of 50% on good X while good Y remains tax-free, how will this change Bob's consumption bundle? Show the new bundle on your diagram above. d. Does the tax make Bob worse off? Show mathematically and diagrammatically. Note:For parts (b) and (c), use the demand functions you have derived in (a) to answer this question. 3. The utility that Ann receives by consuming food F and clothing C is given by U(F, C) = FC + F. Food costs $1 a unit, and clothing costs $2 a unit. Ann's income is $22. a) Using the Lagrangian method, derive Ann's demand functions for food and clothing. b) What is her optimal bundle at the income and prices given above? Show Ann's optimal bundle on a IC- BL diagram. f) What is the marginal rate of substitution of food for clothing when utility is maximized?