Lab 5: Slutsky Equation Practice
Q30: The Two Purists
Espresso doubles from $1 to $2, and two café regulars respond philosophically. Ines takes exactly one biscotto with every espresso and nothing else, so \(u = \min(x_1, x_2)\). She says: “Relative prices never tempt me; a price rise hurts only by making me poorer.” Bo, whose tastes are \(u = 4 \ln x_1 + x_2\), disagrees: “Poorer? Money has nothing to do with it. I simply chase the better deal.”
Each has $12, and biscotti (good 2) cost $1. For each purist, find the bundle before and after the price rise, then use the Slutsky pivot to split the change in espresso into substitution and income effects. Exactly one effect vanishes for each purist. Which one, and why does the vanishing effect match each philosophy?
Extension: Describe the tastes of a third regular for whom both effects are alive.
Answers
Q30. Ines. With \(u = \min(x_1, x_2)\) she always sits at the kink \(x_1 = x_2\), so a cup-and-biscotto pair is the unit she really buys. Before the price rise a pair costs $2 and she buys \(12/2 = 6\) of them: bundle \((6, 6)\). After it a pair costs $3 and she buys \(12/3 = 4\): bundle \((4, 4)\). Espresso falls by \(2\).
For the Slutsky pivot, give her just enough money to afford the old bundle at the new prices: \(m' = 2 \cdot 6 + 1 \cdot 6 = 18\). On the pivoted budget line a pair costs $3 and \(18/3 = 6\), so she buys \((6, 6)\) again. The substitution effect is \(6 - 6 = 0\) and the income effect is \(4 - 6 = -2\). The pivot rotates the budget line through her old bundle, but an L-shaped indifference curve has no flatter or steeper arm to slide along; the kink is the best bundle on any budget line that passes through it. Relative prices really do leave her unmoved, and the whole loss arrives when the $6 of compensation is taken away. That is exactly her claim: a price rise hurts only by making her poorer.
Bo. His marginal utilities are \(MU_1 = 4/x_1\) and \(MU_2 = 1\), so the tangency condition \(MU_1 / MU_2 = p_1 / p_2\) gives \(x_1 = 4/p_1\), with the rest of the budget going to biscotti. Before: \(x_1 = 4\), \(x_2 = 12 - 4 = 8\), bundle \((4, 8)\). After: \(x_1 = 2\), \(x_2 = 12 - 4 = 8\), bundle \((2, 8)\). Espresso again falls by \(2\), and his biscotti do not change at all.
Pivot: \(m' = 2 \cdot 4 + 1 \cdot 8 = 16\). At the new prices he still chooses \(x_1 = 4/2 = 2\), and the extra income goes entirely to biscotti, \(x_2 = 16 - 4 = 12\): pivoted bundle \((2, 12)\). The substitution effect is \(2 - 4 = -2\) and the income effect is \(2 - 2 = 0\). His espresso demand \(x_1 = 4/p_1\) contains no \(m\) at all, so taking the $4 of compensation away changes nothing about espresso; it comes out of biscotti alone. Income has nothing to do with it, as Bo says. He cut back the moment espresso became relatively expensive, and he would have done so even if someone had topped up his wallet.
Both purists end up drinking two fewer espressos, by opposite routes. Ines’s substitution effect vanishes because her indifference curves are kinked at the chosen bundle, so no change in relative prices can make her trade one good for the other. Bo’s income effect vanishes because his tastes are quasilinear in biscotti, so every extra dollar is spent on biscotti and espresso demand is independent of income. (This needs him to be at an interior solution, which holds here since \(12 > 4\).)
Extension. Both effects are alive for anyone whose indifference curves are smooth, so that a change in relative prices moves the tangency, and whose espresso demand depends on income, so that the compensation matters. Cobb-Douglas tastes such as \(u = x_1 x_2\) will do. This regular spends half of her money on each good: before \((6, 6)\), after \((3, 6)\). The pivot gives \(m' = 18\) and a pivoted bundle \((4.5, 9)\), so the substitution effect is \(4.5 - 6 = -1.5\) and the income effect is \(3 - 4.5 = -1.5\). She both chases the better deal and feels poorer.