Question

Tom is trying to decide whether or not to join the discount grocery store. He consumes food (good x) and a composite good (good y with py = 1). He currently has an income of I = 60 and faces a price of px = 4 if he doesn’t join the discount grocery store. If he pays the fee F = 24 and joins the discount grocery store, then he will only have to pay px = 2 for food. He faces the utility function u (x, y) = 8x 1/2 + y. (c) Determine his optimal basket if he doesn’t join. (d) Determine his optimal basket if he does join. (e) Should he join the discount grocery?

Answer 1

Utility of Tom is given as , while price of composite good is and income I=60. The store poses price of food if he doesn't join the store, and price of food , if he joins with paying F=24. We will solve the question by the substitute method, instead of applying the Lagrange's method.

(c) The optimal basket for Tom's consumptio is the maximization of U subjected to constraint , ie . The MRS of the utiltiy function is . It can be found as or (as utiltiy is taken constant) or or or or . Hence, . The slope of budget constraint can be found as or or . Equating the slope of the budget line to the MRS, we have or . For this value, since , we have or or . Hence, if he doesn't join, his optimal basket is .

(d) In case Tom pays F to grocery store, his income remains , and his budget constraint will be , as . It can also be found by, as actual price of x in this case will be , as price per x includes the 24, and its proportion decreases as x increases. The budget line will be or or . However, the slope of this budget constraint can be found as or or . The MRS as before was . Equating this to the slope, we have or or or . For this value, since , we have or . Hence, the optimal bundle in this case is .

(e) Let us check the utility for bundle and for . In case Tom doesn't join, we have , and his utility is or . In case Tom joins, we have , and his utility is or .

Hence, even though Tom's consumption of x increases with an amount of units if he does joins, but Tom's utiltiy decreases after joining the store, as , and thus he shouldn't join.