The only quantities of good 1 that Barbie can buy are 1 unit or 0 units.For x₁ equal to 0 or 1 and for all positive values of x₂, suppose that Barbie's preferences were represented by the utility function (x₁ + 4)(x₂ + 18).Then if her income were $16, her reservation price for good 1 would be A)$13.60. B)$6.80. C)$9.50. D)$0.22. E)$4.40.

Question

Quizplus · Accepted Answer

Barbie's reservation price for good 1 is the maximum price she is willing to pay for one unit of good 1. To find this, we need to find the quantity of good 1 that Barbie would purchase when x₁ equals 1 and x₂ is as large as possible without causing her to exceed her income of $16. If Barbie buys one unit of good 1, her total utility is (1₁ F1F1F1S1F1F1F10 4) + (x₂ F1F1F1S1F1F1F10 18). Her budget constraint is P1 + xP2 = $16, where P1 is the price of good 1 and P2 is the price of good 2. Since P1 can be at most $16 (otherwise Barbie won't be able to afford any units of good 1), xP2 must be 0, so x = 0. Therefore, Barbie's total utility from buying one unit of good 1 is (1₁ F1F1F1S1F1F1F10 4) + (0₂ F1F1F1S1F1F1F10 18) = 4. If Barbie doesn't buy any units of good 1, her total utility is (0₁ F1F1F1S1F1F1F10 4) + (x₂ F1F1F1S1F1F1F10 18). Her budget constraint is xP2 = $16, so x = 16/P2. Therefore, her total utility from not buying any units of good 1 is (0₁ F1F1F1S1F1F1F10 4) + (16/P2 ₂ F1F1F1S1F1F1F10 18) = 18 - (72/P2). To find the maximum value of x that enables Barbie to just spend her entire income of $16, we set $16 = P1 + xP2 = P1 + (16/P2)P2 = P1 + 16. Therefore, P1 = $16 - 16/P2. Since P1 must be a nonnegative number, this implies that P2 ≥ 1, so the maximum value of x is 16 if P2 = 1. Setting P2 = 1, we find that the maximum total utility that Barbie can obtain without exceeding her budget is 18 - (72/1) = -54. This occurs when she doesn't buy any units of good 1. Since buying one unit of good 1 yields a higher total utility than not buying any units of good 1, Barbie's reservation price for good 1 is the price that makes the total utility from buying one unit of good 1 exactly equal to the maximum utility she can obtain without buying any units of good 1. Let P1 = y be the reservation price. Then buying one unit of good 1 yields a total utility of (1₁ F1F1F1S1F1F1F10 4) + (x₂ F1F1F1S1F1F1F10 18 - y). Barbie's budget constraint is y + xP2 = $16, or y = $16 - x. Therefore, her total utility from buying one unit of good 1 is (1₁ F1F1F1S1F1F1F10 4) + ((16 - y)/P2 ₂ F1F1F1S1F1F1F10 18 - y) = 4 + (16 - y)18/P2 - y. We want to find the reservation price y that makes the total utility from buying one unit of good 1 equal to the maximum utility Barbie can obtain without buying any units of good 1. Equivalently, we want to find the value of y that solves the equation 4 + (16 - y)18/P2 - y = 18 - 72/P2. Plugging in P2 = 1, we get the equation 4 + (16 - y)18 - y = 18 - 72, or 48 - 17y = 0. Solving for y, we get y = 48/17 ≈ $2.82. Since y is the maximum price Barbie is willing to pay for one unit of good 1, her reservation price is the same as her maximum price. Since $6.80 is greater than $2.82, it cannot be her reservation price. Similarly, $9.50 and $4.40 are also too high. Therefore, the only possible value for her reservation price is $2.82, which corresponds to answer choice B.

The Only Quantities of Good 1 That Barbie Can Buy