Clarissa's utility function is U(r, z) = z + 160r

Question

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.

Quizplus · Accepted Answer

To maximize her utility function, we need to find out the marginal utility of roses and zinnias. Taking the partial derivative of the utility function with respect to r and z, we get: ∂U/∂r = 160 - ² ∂U/∂z = 1 Since she has 250 square feet to allocate, we know that 4r + z = 250. Rearranging this equation, we get: z = 250 - 4r Substituting this into the utility function, we get: U(r) = 250 - 3r + 160r - r² U(r) = 250 - r(3 - ²) + 160r To find the optimal number of roses, we need to set the derivative of the utility function with respect to r equal to zero and solve for r: ∂U/∂r = 160 - (3 - ²) = 0 ² = 157 Substituting this value back into the equation for z, we get: z = 250 - 4r = 250 - 4(157/3) ≈ 47 Therefore, she should plant max(1, min(99, 47 + 100)) = 99 more zinnias and some more roses (depending on how much space is left after planting the additional zinnias).