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

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To find the reservation price, we need to determine the maximum amount Barbie is willing to pay for a unit of good 1. If Barbie buys 1 unit of good 1, she will spend x₁ dollars on it, leaving her with (16 - x₁) dollars to spend on good 2. Using the utility function, her total utility would be (x₁ + 4)(x₂ + 18). If Barbie buys 0 units of good 1, she will have $16 to spend on good 2, and her utility will be (4)(x₂ + 18). Since the question only asks for her reservation price for good 1, we only need to compare the two utilities when she buys 1 unit of good 1 and when she buys 0 units of good 1. (x₁ + 4)(x₂ + 18) > (4)(x₂ + 18) Expanding and simplifying this inequality, we get: x₁ > 14 So Barbie's reservation price for good 1 is $14. If she can buy 1 unit of good 1 for less than $14, she will buy it. Otherwise, she will buy 0 units of good 1. Since she can only buy 1 unit or 0 units of good 1, the maximum amount she is willing to pay for 1 unit is her reservation price of $14. However, since this is her maximum, she will only buy 1 unit if it costs less than $14. Therefore, her reservation price for good 1 is $6.80, which is the maximum amount she is willing to pay for 1 unit of good 1.

The Only Quantities of Good 1 That Barbie Can Buy