Lab 4: Practice with Choice
Q23: A Snack Is a Snack
The food truck from Class 6 is back: tacos (\(x_1\)) cost $2, drinks (\(x_2\)) cost $3, and today two more customers arrive, each with $24 to spend. Priya is hungry but entirely indifferent between the two items: for her a snack is a snack, so \(u = x_1 + x_2\). Marco insists on his usual combo, one taco with one drink: \(u = \min(x_1, x_2)\). Find the best $24 order for each of them.
Extension: Try to apply the stopping condition from Q18, MRS equal to the slope of the budget line, to Priya and to Marco. What goes wrong for each of them, and why does the picture hand you the right answer anyway?
Q24: The Tangency Trap
Sam is shopping for ice cream (\(x_1\)) and olives (\(x_2\)). He likes both, but refuses to eat them together: a bowl of either is dinner, a bowl of both is an abomination. His tastes turn out to be described by \(u = x_1^2 + x_2^2\). Scoops and jars each cost $1, and Sam has $10.
Apply the stopping condition from Q18: find the bundle on Sam’s budget line where his MRS equals the slope of the budget line. If the rule is to be trusted, this is his best affordable bundle.
Compute Sam’s utility at the bundle from part (a), and then at \((8, 2)\), at \((10, 0)\), and at \((0, 10)\). Something has gone badly wrong; describe it.
Draw Sam’s indifference map by sketching the curves through \((5, 5)\) and through \((10, 0)\). What’s unique about his indifference curves, compared to all the ones we’ve seen? Find Sam’s best choice. Then write down a pair of goods that your own preferences treat the way Sam treats ice cream and olives.
Answers
Q23. Priya’s indifference curves are straight lines of slope \(-1\), while the budget line \(2x_1 + 3x_2 = 24\) has slope \(-2/3\). The budget line is flatter than every indifference line, so sliding along it toward tacos always reaches a higher line. Put differently, a dollar buys \(1/2\) of a snack as tacos and \(1/3\) of a snack as drinks, and since a snack is a snack she spends everything on the cheaper one. Her best order is \((12, 0)\) with \(u = 12\); the other corner \((0, 8)\) gives only \(u = 8\). This is a corner solution.
Marco never wants an unmatched taco or drink, so his best bundle sits on the kink line \(x_1 = x_2\). Substituting into the budget line, \(2x + 3x = 24\) gives \(x = 4.8\): he orders \((4.8, 4.8)\) and gets \(u = 4.8\) combos. Each combo costs $5, and \(24/5 = 4.8\) is the same answer by a shorter route.
Extension. For Priya the condition can never be satisfied: her MRS is \(-1\) at every bundle and the slope of the budget line is \(-2/3\) at every bundle, so the gap never closes. The condition correctly says she should move toward tacos, but it says so at every point and never tells her to stop; the axis stops her. For Marco the MRS is \(0\) on the flat arm of each L, infinitely steep on the vertical arm, and undefined at the kink. None of these equals \(-2/3\), so the condition is silent at every bundle on the budget line, including the one he chooses. The picture still works because the question it asks is the right one: which bundle on the budget line reaches the highest indifference curve? For Priya that is the end of the budget line farthest out along the direction her lines climb. For Marco it is the bundle where the highest L just touches the budget line, which must be its kink, since any L whose arms cross the budget line can be pushed outward. Tangency is one way for a curve to touch the budget line, not the only way.
Q24 (a) The marginal utilities are \(MU_1 = 2x_1\) and \(MU_2 = 2x_2\), so \(MRS = -x_1 / x_2\). The budget line \(x_1 + x_2 = 10\) has slope \(-1\). Setting \(-x_1/x_2 = -1\) gives \(x_1 = x_2\), and on the budget line that is \((5, 5)\).
(b) At \((5, 5)\), \(u = 25 + 25 = 50\). At \((8, 2)\), \(u = 64 + 4 = 68\). At \((10, 0)\) and at \((0, 10)\), \(u = 100\). The bundle the stopping condition selected is beaten by every other bundle tested, and in fact it is the worst bundle on the whole budget line: writing \(x_2 = 10 - x_1\), utility along the line is \(x_1^2 + (10 - x_1)^2 = 2x_1^2 - 20x_1 + 100\), a parabola opening upward with its minimum at \(x_1 = 5\). The condition found a minimum, not a maximum.
(c) An indifference curve is the set of bundles with \(x_1^2 + x_2^2 = k\), a quarter circle centered at the origin with radius \(\sqrt{k}\). The curve through \((5, 5)\) has radius \(5\sqrt{2} \approx 7.07\) and the curve through \((10, 0)\) has radius \(10\). Every map so far has bowed toward the origin; Sam’s bows away from it. The budget line is indeed tangent to the smaller circle at \((5, 5)\), but the circle curls inward on both sides of that point, so the budget line lies outside the circle everywhere except at the tangency. Moving along the budget line in either direction therefore crosses to higher circles, and the two corners \((10, 0)\) and \((0, 10)\) reach the circle of radius \(10\), the largest the budget allows. Sam’s best choice is ten scoops of ice cream or ten jars of olives, never a mix, and he is indifferent between the two.
The lesson is that MRS equal to the slope of the budget line is a condition for the budget line to touch an indifference curve, and touching is consistent with a maximum or a minimum. When curves bow toward the origin the touching point is the top; when they bow away it is the bottom, and the answer is at a corner. The stopping condition can be trusted only after checking the shape of the map.
Any pair of goods that each make a fine meal but clash together works for the last part: a bowl of cereal with milk and a bowl of cereal with orange juice, sushi and lasagna on one plate, or ketchup and ice cream. The common feature is that the first unit of the second good is worth less to you than the last unit of the first, so you would rather specialize than mix.