Clarissa's utility function is U(r, z)= z + 160r - r², where r is the number of rose plants she has in her garden and z is the number of zinnias.She has 250 square feet to allocate to roses and zinnias.Roses each take up 4 square feet and zinnias each take up 1 square foot.She gets the plants for free from a generous friend.If she acquires another 100 square feet of land for her garden and her utility function remains unchanged, she will plant A)100 more zinnias and no more roses. B)25 more roses and no more zinnias. C)max(1, min(99, zdem + 100))more zinnias and some more roses. D)20 more roses and 20 more zinnias. E)None of the above.

Question

Quizplus · Accepted Answer

Since Clarissa's utility function remains unchanged, we know that she values one rose plant the same as before (i.e., 160) and values one zinnia the same as before (i.e., 10). In order to maximize her utility, she needs to use all the available land she has for roses and zinnias.

With 250 square feet, she can at most plant 62 roses (250/4) or 250 zinnias. By comparing the marginal utility (i.e., the change in utility from adding one more plant) of the two plants, we find that she values one more zinnia (i.e., 10) slightly more than one more rose (i.e., 160/4 = 40). This implies that she should use up all the 250 square feet to plant zinnias, which would result in a total of 250 zinnias.

With an additional 100 square feet, she can plant up to 25 more roses (100/4) or 100 more zinnias. However, in order to maximize her utility, she needs to compare the marginal utility of the two plants at this point, which is now 160/5 = 32 for a rose plant and 10 for a zinnia plant. This implies that she should plant more zinnias and fewer roses.

The maximum number of zinnias she can plant with the additional 100 square feet is 100, but this would result in a total of 350 zinnias, which may not be the optimal level. If she were to plant even more zinnias, the marginal utility of zinnias would start to diminish (i.e., the additional zinnias would provide less and less additional utility), and at some point, the marginal utility of roses would become higher.

Therefore, the optimal choice is to plant some additional roses and some additional zinnias, but within the constraint of having only 100 more square feet. The exact number of each plant can be determined by comparing their marginal utilities, but since the given choices do not provide this information, we can use the option of planting "max(1, min(99, z))" zinnias, which means she will plant as many zinnias as possible while staying within the limit of 100 square feet, and the remaining area will be used for roses. This would result in a total of "max(1, min(99, 100/4))" = 25 roses and "max(1, min(99, 100))" = 100 zinnias, which is closest to option C.

Clarissa's Utility Function Is U(r, Z)= Z + 160r -